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\section{Introduction}
Suppose $E$ is a commutative $S$-algebra, in the sense of \cite{EKMM},
and $A$ is a commutative $E$-algebra. We want to capture the properties
and underlying structure of the homotopy groups $\pi_* A = A_*$ of $A$,
by studying operations associated to the cohomology theory that $E$
represents.
An important family of cohomology operations, called {\em power
operations}, is constructed via the extended powers. Specifically,
consider the {\em $m$'th extended power} functor
\[
{\mb P}_E^m (-) \coloneqq (-)^{\wedge_E m} / \Sigma_m \colon\thinspace {\rm Mod}_E \to {\rm Mod}_E
\]
on the category of $E$-modules, which sends an $E$-module to its
$m$-fold smash product over $E$ modulo the action by the symmetric group
on $m$ letters. The ${\mb P}_E^m (-)$'s assemble together to give the {\em
free commutative $E$-algebra} functor
\[
{\mb P}_E (-) \coloneqq \bigvee_{m \geq 0} {\mb P}_E^m (-) \colon\thinspace {\rm Mod}_E \to {\rm Alg}_E
\]
from the category of $E$-modules to the category of commutative
$E$-algebras. These functors descend to homotopy categories. In
particular, each $\alpha \in \pi_{d+i}~{\mb P}_E^m (\Sigma^d E)$ gives rise to a
power operation
\[
Q_\alpha \colon\thinspace A_d \to A_{d+i}
\]
(cf.~\cite[Sections I.2 and IX.1]{H_infty} and \cite[Section 3]{cong}).
Under the action of power operations, $A_*$ is an algebra over some
operad in $E_*$-modules involving the structure of $E_* B\Sigma_m$ for
all $m$. This operad is traditionally called a {\em Dyer-Lashof~ algebra}, or
more precisely, a Dyer-Lashof~ {\em theory} as the {\em algebraic theory} of
power operations acting on the homotopy groups of commutative
$E$-algebras (cf.~\cite[Chapters III, VIII, and IX]{H_infty} and
\cite[Section 9]{lpo}).
A specific case is when $E$ represents a Morava $E$-theory of height $n$
and $A$ is $K(n)$-local. Morava $E$-theory spectra play a crucial role
in modern stable homotopy theory, particularly in the work of Ando,
Hopkins, and Strickland on the topological approach to elliptic genera
(see \cite{cube}). As recalled in \cite[1.5]{cong}, the $K(n)$-local
$E$-Dyer-Lashof~ theory is largely understood based on work of those authors. In
\cite{cong}, Rezk maps out the foundations of this theory. He gives a
congruence criterion for an algebra over the Dyer-Lashof~ theory
(\cite[Theorem A]{cong}). This enables one to study the Dyer-Lashof~
{\em theory}, which models all the algebraic structure naturally
adhering to $A_*$, by working with a certain associative ring $\Gamma$ as
the Dyer-Lashof~ {\em algebra}. Moreover, Rezk provides a geometric description
of this congruence criterion, in terms of sheaves on the moduli problem
of deformations of formal groups and Frobenius isogenies (see
\cite[Theorem B]{cong}). This connects the structure of $\Gamma$ to the
geometry underlying $E$, moving one step forward from a workable object
$\Gamma$ to things that are computable. In a companion paper \cite{h2p2},
Rezk gives explicit calculations of the Dyer-Lashof~ theory for a specific Morava
$E$-theory of height $n = 2$ at the prime 2.
The purpose of this paper is to make available calculations analogous to
some of the results in \cite{h2p2}, at the prime 3, together with
calculations of the corresponding power operations on the
$K(1)$-localization of the Morava $E$-theory spectrum.
\subsection{Outline of the paper}
As in \cite{h2p2}, the computation of power operations in this paper
follows the approach of \cite{steenrod}: one first defines a total power
operation, and then uses the computation of the cohomology of the
classifying space $B\Sigma_m$ for the symmetric group to obtain
individual power operations. These two steps are carried out in
Sections \ref{sec:total} and \ref{sec:individual} respectively.
In Section \ref{sec:total}, by doing calculations with elliptic curves
associated to our Morava $E$-theory $E$, we give formulas for the total
power operation $\psi^3$ on $E_0$ and the ring $S_3$ which represents the
corresponding moduli problem.
In Section \ref{sec:individual}, based on calculations of
$E^* B\Sigma_m$ in \cite{Str98} as reflected in the formula for $S_3$,
we define individual power operations, and derive the relations they
satisfy. In view of the general structures studied in \cite{cong}, we
then get an explicit description of the Dyer-Lashof~ algebra $\Gamma$ for
$K(2)$-local commutative $E$-algebras.
In Section \ref{sec:K(1)}, we describe the relationship between the
total power operation $\psi^3$, at height 2, and the corresponding
$K(1)$-local power operations. We then derive formulas for the latter
from the calculations in Section \ref{sec:total}.
\begin{rmk}
\label{rmk:grading}
In Section \ref{sec:total}, we do calculations with a universal
elliptic curve over {\em all} of the moduli stack which is an affine
open subscheme of a weighted projective space (cf.~Proposition
\ref{prop:C}). At the prime 3, the supersingular locus consists of a
single closed point, and the corresponding Morava $E$-theory arises
{\em locally} in an affine coordinate chart of this weighted projective
space containing the supersingular locus. In this paper we choose a
particular affine coordinate chart for computing the homotopy groups of
the $E$-theory spectrum and the power operations; we hope that the
generality of the calculations in Section \ref{sec:total} makes it
easier to work with other coordinate charts as well.
\end{rmk}
\begin{rmk}
\label{rmk:parameter}
The ring $S_3$ turns out to be an algebra with one generator over the
base ring where our elliptic curve is defined (cf.~\isog{i} and
\eqref{S_3}). This generator appears as a parameter in the formulas
for the total power operation $\psi^3$, and is responsible for how the
individual power operations are defined and how their formulas look.
Different choices of this parameter result in different bases of the
Dyer-Lashof~ algebra $\Gamma$. The parameter in this paper comes from the relative
cotangent space of the elliptic curve at the identity (see \isog{iv},
Corollary \ref{cor:K'}, and Remark \ref{rmk:K'}). This choice is
convenient for deriving Adem relations in \q{iv}, and it fits into the
treatment of gradings in \cite[Section 2]{cong} (see \go{ii} and
Theorem \ref{thm:gamma}). We should point out that our choice is by no
means canonical. We do not know yet, as part of the structure of the
Dyer-Lashof~ algebra, if there is a canonical basis which is both geometrically
interesting and computationally convenient. Somewhat surprisingly,
although it appears to come from different considerations, our choice
has an analog at the prime 2 which coincides with the parameter used in
\cite{h2p2} (see Remarks \ref{rmk:K} and \ref{rmk:KK'}). The
calculations follow a recipe in hope of generalizing to other Morava
$E$-theories of height 2; we hope to address these matters and
recognize more of the general patterns based on further computational
evidence.
\end{rmk}
\subsection{Acknowledgements}
I thank Charles Rezk for his encouragement on this work, and for his
observation in a correspondence which led to Proposition
\ref{prop:frob^2} and Corollary \ref{cor:K'}.
I thank Kyle Ormsby for helpful discussions on Section \ref{sec:total},
and for directing me to places in the literature.
I thank Tyler Lawson for the sustained support from him I received as a
student.
\subsection{Conventions}
Let $p$ be a prime, $q$ a power of $p$, and $n$ a positive integer. We
use the symbols
\[
{\mb F}_q\text{,}~~{\mb Z}_q\text{,}~~{\rm and}~~{\mb Z}/n
\]
to denote a field with $q$ elements, the ring of $p$-typical Witt
vectors over ${\mb F}_q$, and the additive group of integers modulo $n$,
respectively.
If $R$ is a ring, then $R\llbracket x \rrbracket$ and $R (\!(x)\!)$
denote the rings of formal power series and formal Laurent series over
$R$ in the variable $x$ respectively. If $I \subset R$ is an ideal,
then $R_I^\wedge$ denotes the completion of $R$ with respect to $I$.
If $E$ is an elliptic curve and $m$ is an integer, then $[m]$ denotes
the multiplication-by-$m$ map on $E$, and $E[m]$ denotes the $m$-torsion
subgroup scheme of $E$.
All formal groups mentioned in this paper will be commutative and
one-dimensional.
The terminology for the structure of the Dyer-Lashof~ theory will follow
\cite{cong} and \cite{h2p2}; some of the notions there are taken in turn
from \cite{BW} and \cite{V}.
\section{Total power operations}
\label{sec:total}
\subsection{A universal elliptic curve and a Morava $E$-theory spectrum}
\label{subsec:ec}
A Morava $E$-theory of height 2 at the prime 3 has its formal group as
the universal deformation of a height-2 formal group over a perfect
field of characteristic 3. Given a supersingular elliptic curve over
such a field, its formal completion at the identity produces a formal
group of height 2. To study power operations for the corresponding
$E$-theory, we do calculations with the universal deformation of that
supersingular elliptic curve which is a family of elliptic curves with a
$\Gamma_1(N)$-structure (see \cite[Section 3.2]{KM}) where $N$ is prime to 3.
Here is a specific model (cf.~\cite[4(4.6a)]{husemoller}).
\begin{prop}
\label{prop:C}
Over ${\mb Z} [1/4]$, the moduli problem of nonsingular elliptic curves
with a choice of a point of exact order 4 and a nowhere-vanishing
invariant one-form is represented by
\begin{equation}
\label{Cxy}
C \colon\thinspace y^2 + a x y + a b y = x^3 + b x^2
\end{equation}
with chosen point $(0,0)$ and one-form
$dx / (2 y + a x + a b) = dy / (3 x^2 + 2 b x - a y)$ over the graded
ring
\[
S^\bullet \coloneqq {\mb Z} [1/4] [a, b, \Delta^{-1}]
\]
where $|a| = 1$, $|b| = 2$, and $\Delta = a^2 b^4 (a^2 - 16 b)$.
\end{prop}
\begin{proof}
Let $P$ be the chosen point of exact order 4. Since $2P$ is 2-torsion,
the tangent line of the elliptic curve at $P$ passes through $2P$, and
the tangent line at $2P$ passes through the identity at the infinity.
With this observation, the rest of the proof is analogous to that of
\cite[Proposition 3.2]{tmf3}.
\end{proof}
Over a finite field of characteristic 3, $C$ is supersingular precisely
when the quantity
\begin{equation}
\label{H}
H \coloneqq a^2 + b
\end{equation}
vanishes (cf.~\cite[V.4.1a]{AEC}). As $(3,H)$ is a homogeneous maximal
ideal of $S^\bullet$ corresponding to the closed subscheme ${\rm Spec\thinspace} {\mb F}_3$, the
supersingular locus consists of a single closed point, and $C$ restricts
to ${\mb F}_3$ as
\[
C_0 \colon\thinspace y^2 + x y - y = x^3 - x^2.
\]
From the above universal deformation $C$ of $C_0$, we next produce a
Morava $E$-theory spectrum which is 2-periodic. We follow the
convention that elements in algebraic degree $n$ lie in topological
degree $2n$, and work in an affine \'etale coordinate chart of the
weighted projective space ${\rm Proj\thinspace} {\mb Z} [1/4] [a, b]$ (see Remark
\ref{rmk:grading}). Define elements $u$ and $c$ such that
\[
a = u c \qquad {\rm and} \qquad b = u^2.
\]
Consider the graded ring
\[
S^\bullet [u^{-1}] \cong {\mb Z} [1/4] [a, \Delta^{-1}] [u^{\pm1}]
\]
where $|u| = 1$, and denote by $S$ its subring of elements in degree 0
so that
\begin{equation}
\label{S}
S \cong {\mb Z} [1/4] [c, \delta^{-1}]
\end{equation}
where $\delta = u^{-12} \Delta = c^2 (c^2 - 16)$. Write
\[
\widehat{S} = {\mb Z}_9 \llbracket h \rrbracket
\]
where
\begin{equation}
\label{h}
h \coloneqq u^{-2} H = c^2 + 1.
\end{equation}
Let $i$ be an element generating ${\mb Z}_9$ over ${\mb Z}_3$ with $i^2 = -1$.
We may choose
\[
c \equiv i ~~{\rm mod}~ (3,h)
\]
and we have
\[
\delta \equiv -1 ~~{\rm mod}~ (3,h)
\]
where $(3,h)$ is the maximal ideal of the complete local ring $\widehat{S}$.
Then by Hensel's lemma, both $c$ and $\delta$ lie in $\widehat{S}$, and both are
invertible. Thus
\[
\widehat{S} \cong S_{(3,h)}^\wedge.
\]
Now $C$ restricts to $S$ as
\begin{equation}
\label{Cc}
y^2 + c x y + c y = x^3 + x^2.
\end{equation}
Let $\widehat{C}$ be the formal completion of $C$ over $S$ at the identity. It
is a formal group over $\widehat{S}$, and its reduction to ${\mb F}_9 = \widehat{S} / (3,h)$
is a formal group ${\mb G}$ of height 2 in view of \eqref{h} and \eqref{H}.
By the Serre-Tate theorem (see \cite[2.9.1]{KM}), 3-adically the
deformation theory of an elliptic curve is equivalent to the deformation
theory of its 3-divisible group, and thus $\widehat{C}$ is the universal
deformation of ${\mb G}$ in view of Proposition \ref{prop:C}. Let $E$ be
the $E_\infty$-ring spectrum which represents the Morava $E$-theory
associated to ${\mb G}$ (see \cite[Corollary 7.6]{GH}). Then
\[
E_* \cong {\mb Z}_9 \llbracket h \rrbracket [u^{\pm 1}]
\]
where $u$ is in topological degree 2, and it corresponds to a local
uniformizer at the identity of $C$.
\subsection{Points of exact order 3}
To study $C$ in a formal neighborhood of the identity, it is convenient
to make a change of variables. Let
\[
u = \frac{x}{y} \quad {\rm and} \quad v = \frac{1}{y}, \qquad {\rm so} \qquad x = \frac{u}{v} \quad {\rm and} \quad y = \frac{1}{v}.
\]
The identity of $C$ is then $(u,v) = (0,0)$, with $u$ a local
uniformizer. The equation \eqref{Cxy} of $C$ becomes
\begin{equation}
\label{Cuv}
v + a u v + a b v^2 = u^3 + b u^2 v.
\end{equation}
\begin{prop}
\label{prop:tors}
On the elliptic curve $C$ over $S^\bullet$, the $uv$-coordinates $(d,e)$ of
any nonzero 3-torsion point satisfy the identities
\begin{equation}
\label{f}
f(d) = 0
\end{equation}
and
\begin{equation}
\label{g}
e = g(d)
\end{equation}
where $f, g \in S^\bullet [u]$ are given by
\begin{equation*}
\begin{split}
f(u) = & ~ b^4 u^8 + 3 a b^3 u^7 + 3 a^2 b^2 u^6 + (a^3 b + 7 a b^2) u^5 + (6 a^2 b - 6 b^2) u^4 + 9 a b u^3 \\
& + (-a^2 + 8 b) u^2 - 3 a u - 3, \\
g(u) = & -\frac{1}{a (a^2 - 16 b)} \big( a b^3 u^7 + (3 a^2 b^2 - 2 b^3) u^6 + (3 a^3 b -6 a b^2) u^5 + (a^4 + a^2 b \\
& + 2 b^2) u^4 + (4 a^3 - 15 a b) u^3 + 18 b u^2 - 12 a u - 18 \big).
\end{split}
\end{equation*}
\end{prop}
\begin{proof}
\footnote{See Appendix \ref{apx:tors} for explicit formulas for the
polynomials $\widetilde{f}$, $Q_1$, $R_1$, $Q_2$, $R_2$, $K$, $L$, $M$, and $N$
that appear in the proof. }
Given the elliptic curve $C$ with equation \eqref{Cxy}, a nonzero point
$Q$ is 3-torsion if and only if the polynomial
\[
\psi_3 (x) \coloneqq 3 x^4 + (a^2 + 4 b) x^3 + 3 a^2 b x^2 + 3 a^2 b^2 x + a^2 b^3
\]
vanishes at $Q$ (cf.~\cite[Exercise 3.7f]{AEC}). Substituting
$x = u / v$ and clearing the denominators, we get a polynomial
\[
\widetilde{\psi}_3(u,v) \coloneqq 3 u^4 + (a^2 + 4 b) u^3 v + 3 a^2 b u^2 v^2 + 3 a^2 b^2 u v^3 + a^2 b^3 v^4.
\]
As $Q = (d,e)$ in $uv$-coordinates, we then have
\begin{equation}
\label{Tp}
\widetilde{\psi}_3(d,e) = 0.
\end{equation}
To get the polynomial $f$, we take $v$ as variable and rewrite
\eqref{Cuv} as a quadratic equation
\begin{equation}
\label{quadratic}
a b v^2 + (-b u^2 + a u + 1) v - u^3 = 0,
\end{equation}
where the leading coefficient $a b$ is invertible in
$S^\bullet = {\mb Z} [1/4] [a, b, \Delta^{-1}]$ as $\Delta = a^2 b^4 (a^2 - 16 b)$.
Define
\begin{equation}
\label{Tfdef}
\widetilde{f}(u) \coloneqq \widetilde{\psi}_3(u,v) \widetilde{\psi}_3(u,\bar{v})
\end{equation}
where $v$ and $\bar{v}$ are formally the conjugate roots of
\eqref{quadratic} so that we compute $\widetilde{f}$ in terms of $u$ by
substituting
\[
v + \bar{v} = \frac{b u^2 - a u - 1}{a b} \qquad {\rm and} \qquad v \bar{v} = -\frac{u^3}{a b}.
\]
We then factor $\widetilde{f}$ over $S^\bullet$ as
\begin{equation}
\label{Tffactor}
\widetilde{f}(u) = -\frac{u^4 f(u)}{a^2 b}
\end{equation}
with $f$ the stated polynomial of order 8. We check that $f$ is
irreducible by applying Eisenstein's criterion to the homogeneous prime
ideal $(3,H)$ of $S^\bullet$.
We have $\widetilde{f}(d) = 0$ by \eqref{Tfdef} and \eqref{Tp}. To see
$f(d) = 0$, consider the closed subscheme $D \subset C[3]$ of points of
exact order 3. By \cite[2.3.1]{KM} it is finite locally free of rank 8
over $S^\bullet$. By the Cayley-Hamilton theorem, as a global section of $D$,
$u$ locally satisfies a homogeneous monic equation of order 8, and this
equation locally defines the rank-8 scheme $D$. Since $D$ is affine,
it is then globally defined by such an equation. In view of
$\widetilde{f}(d) = 0$ and \eqref{Tffactor}, we determine this equation, and (up
to a unit in $S^\bullet$) get the first stated identity \eqref{f}.
To get the polynomial $g$, we note that both the quartic polynomial
\[
A(v) \coloneqq \widetilde{\psi}_3(d,v)
\]
and the quadratic polynomial
\[
B(v) \coloneqq a b v^2 + (-b d^2 + a d + 1) v - d^3
\]
vanish at $e$, and thus so does their greatest common divisor (gcd).
Applying the Euclidean algorithm (see Appendix \ref{apx:tors} for
explicit expressions), we have
\begin{equation*}
\begin{split}
A(v) = & ~ Q_1(v) B(v) + R_1(v), \\
B(v) = & ~ Q_2(v) R_1(v) + R_2,
\end{split}
\end{equation*}
where
\[
R_1(v) = K(d) v + L(d)
\]
for some polynomials $K$ and $L$, and $R_2 = 0$ in view of \eqref{f}.
Thus $R_1(v)$ is the gcd of $A(v)$ and $B(v)$, and hence
\[
K(d) e + L(d) = R_1(e) = 0.
\]
To write $e$ in terms of $d$ from the above identity, we apply the
Euclidean algorithm to $f$ and $K$. Their gcd turns out to be 1, and
thus there are polynomials $M$ and $N$ with
\[
M(u) f(u) + N(u) K(u) = 1.
\]
By \eqref{f} we then have $N(d) K(d) = 1$, and thus
\[
e = -N(d) L(d) = g(d)
\]
where $g$ is as stated.
\end{proof}
\begin{rmk}
\label{rmk:dmod3}
The formula for $f$ in Proposition \ref{prop:tors} satisfies a
congruence
\[
f(u) \equiv u^2 (b^4 u^6 + a b H u^3 - H) ~~{\rm mod}~ 3.
\]
The two roots (counted with multiplicity) of $f(u)$ which reduce to
zero modulo 3 correspond to the two nonzero points in the unique
order-3 subgroup of $C$ in a formal neighborhood of the identity.
\end{rmk}
\subsection{A universal isogeny and a total power operation}
\begin{prop}
\label{prop:isog}
\mbox{}
\begin{enumerate}[(i)]
\item \label{isog(i)} The universal degree-3 isogeny $\psi$ with
source $C$ is defined over the graded ring
\[
S^\bullet_3 \coloneqq S^\bullet [\kappa] \big/ \big( W(\kappa) \big)
\]
where $|\kappa| = -2$ and
\begin{equation}
\label{W}
W(\kappa) = \kappa^4 - \frac{6}{b^2} ~ \kappa^2 + \frac{a^2 - 8 b}{b^4} ~ \kappa - \frac{3}{b^4},
\end{equation}
and has target the elliptic curve
\[
C' \colon\thinspace v + a' u v + a' b' v^2 = u^3 + b' u^2 v
\]
where
\begin{equation*}
\begin{split}
a' = & ~ \frac{1}{a} \big( (a^2 b^4 - 4 b^5) \kappa^3 + 4 b^4 \kappa^2 + (-6 a^2 b^2 + 20 b^3) \kappa + a^4 - 12 a^2 b + 12 b^2 \big), \\
b' = & ~ b^3.
\end{split}
\end{equation*}
\item \label{isog(ii)} The kernel of $\psi$ is generated by a point
$Q$ of exact order 3 with coordinates $(d,e)$ satisfying
\begin{equation}
\label{K}
\begin{split}
\kappa = & -\frac{1}{a^2 - 16 b} \big( a b^3 d^7 + (3 a^2 b^2 - 2 b^3) d^6 + (3 a^3 b - 6 a b^2) d^5 + (a^4 \\
& + a^2 b + 2 b^2) d^4 + (4 a^3 - 15 a b) d^3 + (a^2 + 2 b) d^2 - 12 a d - 18 \big) \\
= & ~ a e - d^2.
\end{split}
\end{equation}
\item \label{isog(iii)} The restriction of $\psi$ to the supersingular
locus at the prime 3 is the 3-power Frobenius endomorphism.
\item \label{isog(iv)} The induced map $\psi^*$ on the relative
cotangent space of $C'$ at the identity sends $du$ to $\kappa du$.
\end{enumerate}
\end{prop}
\begin{proof}
\footnote{See Appendix \ref{apx:isog} for the power series expansion of
$v$ and details of the calculations involving the group law on $C$ that
appear in the proof. }
Let $P = (u,v)$ be a point on $C$, and $Q = (d,e)$ be a nonzero
3-torsion point. Rewriting \eqref{Cuv} as
\[
v = u^3 + b u^2 v - a u v - a b v^2,
\]
we express $v$ as a power series in $u$ by substituting this equation
into itself recursively. For the purpose of our calculations, we take
this power series up to $u^{12}$ as an expression for $v$, and write
$e = g(d)$ as in \eqref{g}.
Define functions $u'$ and $v'$ by
\begin{equation}
\label{u'v'}
\begin{split}
u' \coloneqq & ~ u(P) \cdot u(P-Q) \cdot u(P+Q), \\
v' \coloneqq & ~ v(P) \cdot v(P-Q) \cdot v(P+Q),
\end{split}
\end{equation}
where $u(-)$ and $v(-)$ denote the $u$-coordinate and $v$-coordinate of
a point respectively. By computing the group law on $C$, we express
$u'$ and $v'$ as power series in $u$:
\begin{equation}
\label{KL}
\begin{split}
u' = & ~ \kappa u + (\text{higher-order terms}), \\
v' = & ~ \lambda u^3 + (\text{higher-order terms}),
\end{split}
\end{equation}
where the coefficients ($\kappa$, $\lambda$, etc.)~involve $a$, $b$, and
$d$. In particular, in view of \eqref{f}, we compute that $\kappa$
satisfies $W(\kappa) = 0$ with $|\kappa| = -2$ as stated in \eqref{isog(i)}.
Now define the isogeny $\psi \colon\thinspace C \to C'$ by
\begin{equation}
\label{psi}
u\big( \psi(P) \big) \coloneqq u' \qquad {\rm and} \qquad v\big( \psi(P) \big) \coloneqq \frac{\kappa^3}{\lambda} \cdot v',
\end{equation}
where we introduce the factor $\kappa^3 / \lambda$ so that the equation of
$C'$ will be in the Weierstrass form. Using \eqref{KL} (see Appendix
\ref{apx:isog} for explicit expressions), we then determine the
coefficients in a Weierstrass equation and get the stated equation of
$C'$.
We next check the statement of \eqref{isog(ii)}. In view of
\eqref{psi} and \eqref{u'v'}, the kernel of $\psi$ is the order-3
subgroup generated by $Q$. In \eqref{K}, the first identity is
computed in \eqref{KL}; we then compare it with the formula for $g$ in
Proposition \ref{prop:tors} and get the second identity.
For \eqref{isog(iii)}, recall from Section \ref{subsec:ec} that the
supersingular locus at the prime 3 is ${\rm Spec\thinspace} {\mb F}_3$. Over ${\mb F}_3$,
since $C[3] = 0$ by \cite[V.3.1a]{AEC}, $Q$ coincides with the identity,
and thus
\[
u\big( \psi(P) \big) = u(P) \cdot u(P-Q) \cdot u(P+Q) = \big( u(P) \big)^3.
\]
As the $u$-coordinate is a local uniformizer at the identity, $\psi$
then restricts to ${\mb F}_3$ as the 3-power Frobenius endomorphism.
The statement of \eqref{isog(iv)} follows by definition of $\kappa$ in
\eqref{KL}.
\end{proof}
\begin{rmk}
In view of \isog{iii}, the formal completion of $\psi \colon\thinspace C \to C'$ at
the identity of $C$ is a {\em deformation of Frobenius} in the sense of
\cite[11.3]{cong}. When it is clear from the context, we will simply
call $\psi$ itself a deformation of Frobenius.
\end{rmk}
\begin{rmk}
\label{rmk:K}
From \eqref{u'v'} and \eqref{KL} we have
\begin{equation}
\label{norm}
u(P-Q) \cdot u(P+Q) = \kappa + u \cdot (\text{higher-order terms}).
\end{equation}
In particular $u(-Q) \cdot u(Q) = \kappa$
(cf.~\cite[Proposition 7.5.2 and Section 7.7]{KM}). The analog of $\kappa$
at the prime 2 coincides with $d$ as studied in \cite[Section 3]{h2p2}.
\end{rmk}
Recall from Section \ref{subsec:ec} that
\[
E^0 \cong {\mb Z}_9 \llbracket h \rrbracket = \widehat{S} \cong S_{(3,h)}^\wedge
\]
in which $c$ and $i$ are elements with $c^2 + 1 = h$ and $i^2 = -1$.
Given the graded ring $S^\bullet_3$ in \isog{i}, define
\begin{equation}
\label{S_3}
S_3 \coloneqq S [\alpha] / \big( w(\alpha) \big)
\end{equation}
where
\begin{equation}
\label{w}
w(\alpha) = \alpha^4 - 6 \alpha^2 + (c^2 - 8) \alpha - 3
\end{equation}
(cf.~the definition of $S$ from $S^\bullet$ in \eqref{S}). By
\cite[Theorem 1.1]{Str98} we have
\[
E^0 B\Sigma_3 / I \cong \big( S_3 \big)_{(3,h)}^\wedge
\]
where
\begin{equation}
\label{transfer}
I \coloneqq \bigoplus_{0<i<3} {\rm image} \big( E^0 B(\Sigma_i \times \Sigma_{3-i}) \xrightarrow{\rm transfer} E^0 B\Sigma_3 \big)
\end{equation}
is the {\em transfer ideal}.
In view of this and the construction of {\em total power operations} for
Morava $E$-theories in \cite[3.23]{cong}, we have the following
corollary.
\begin{cor}
\label{cor:psi3}
The total power operation
\[
\psi^3 \colon\thinspace E^0 \to E^0 B\Sigma_3 / I \cong E^0 [\alpha] \big/ \big( w(\alpha) \big)
\]
is given by
\begin{equation*}
\begin{split}
\psi^3(h) = & ~ h^3 + (\alpha^3 - 6 \alpha - 27) h^2 + 3 (-6 \alpha^3 + \alpha^2 + 36 \alpha + 67) h \\
& + 57 \alpha^3 - 27 \alpha^2 - 334 \alpha - 342, \\
\psi^3(c) = & ~ c^3 + (\alpha^3 - 6 \alpha - 12) c - 4 (\alpha + 1)^2 (\alpha - 3) c^{-1}, \\
\psi^3(i) \thinspace = & -i,
\end{split}
\end{equation*}
where
\begin{equation}
\label{Amod3}
\alpha \equiv 0 ~~{\rm mod}~ 3.
\end{equation}
\end{cor}
\begin{proof}
By \isog{i}, in $xy$-coordinates, $C'$ restricts to $S_3$ as
\[
y^2 + c' x y + c' y = x^3 + x^2
\]
where
\[
c' = \frac{1}{c} \big( (c^2 - 4) \alpha^3 + 4 \alpha^2 + (-6 c^2 + 20) \alpha + c^4 - 12 c^2 + 12 \big).
\]
By \cite[Theorem B]{cong}, since the above equation is in the form of
\eqref{Cc}, there is a correspondence between the restriction to $S_3$
of the universal isogeny $\psi$, which is a deformation of Frobenius,
and the total power operation $\psi^3$. In particular $\psi^3(c)$ is given by
$c'$. As $\psi^3$ is a ring homomorphism, we then get the formula for
$\psi^3(h) = \psi^3(c^2 + 1)$. We also have
\[
\big( \psi^3(i) \big)^2 = \psi^3(-1) = -1,
\]
and thus $\psi^3(i) = i$ or $-i$. We exclude the former possibility in
view of the congruence
\[
\psi^3(i) \equiv i^3 ~~{\rm mod}~ 3
\]
by \cite[Propositions 3.25 and 10.5]{cong}.
The congruence \eqref{Amod3} follows from Remark \ref{rmk:dmod3} and
\eqref{K}.
\end{proof}
\section{Individual power operations}
\label{sec:individual}
\subsection{A composite of deformations of Frobenius}
Recall from Proposition \ref{prop:isog} that over $S^\bullet_3$ we have the
universal degree-3 isogeny $\psi \colon\thinspace C \to C' = C/G$ where $G$ is an
order-3 subgroup of $C$; in particular, $\psi$ is a deformation of the
3-power Frobenius endomorphism over the supersingular locus. We want to
construct a similar isogeny $\psi'$ with source $C'$ so that the
composite $\psi' \circ \psi$ will correspond to a composite of total
power operations via \cite[Theorem B]{cong}.
Let $G' \coloneqq C[3]/G$ which is an order-3 subgroup of $C'$. Recall
from Section \ref{subsec:ec} that $C$ is the universal deformation of a
supersingular elliptic curve $C_0$. Since the 3-divisible group of
$C_0$ is formal, $C_0[3]$ is connected. Thus over a formal neighborhood
of the supersingular locus, if $G$ is the unique connected order-3
subgroup of $C$, $G'$ is then the unique connected order-3 subgroup of
$C'$. As in the proof of Proposition \ref{prop:isog}, we define
$\psi' \colon\thinspace C' \to C'/G'$ using a nonzero point in $G'$ (see \eqref{u'v'}
and \eqref{psi}), and $\psi'$ is then a deformation of Frobenius. Over
the supersingular locus, the pair $(\psi, \psi')$ is {\em cyclic in
standard order} in the sense of \cite[6.7.7]{KM}. We describe it more
precisely as below.
\begin{prop}
\label{prop:frob^2}
The following diagram of elliptic curves over $S^\bullet_3$ commutes:
\begin{equation}
\label{frob^2}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (LT) at (0, 2) {$C$};
\node (MT) at (3.8, 2) {$C/G = $};
\node (RT) at (4.65, 2.04) {$C'$};
\node (LB) at (1.9, 0) {$C/C[3]$};
\node (MB) at (3.5, 0) {$\cong \frac{C/G}{C[3]/G} = $};
\node (RB) at (4.65, 0.025) {$\frac{C'}{G'}$};
\node at (4.95, -0.15) {.};
\draw [->] (LT) -- node [above] {$\scriptstyle \psi$} (MT);
\draw [->] (LT) -- node [left] {$\scriptstyle [-3]$} (LB);
\draw [->] (RT) -- node [right] {$\scriptstyle \psi'$} (RB);
\end{tikzpicture}
\end{equation}
\end{prop}
\begin{proof}
By \cite[2.4.2]{KM}, since ${\rm Proj\thinspace} S^\bullet_3$ is connected, we need only show
that the locus over which $\psi' \circ \psi = [-3]$ is not empty, where
by abuse of notation $[-3]$ denotes the map $[-3]$ on $C$ composed with
the canonical isomorphism from $C/C[3]$ to $C'/G'$.
Recall from Section \ref{subsec:ec} that $C$ restricts to the
supersingular locus ${\mb F}_3$ as
\[
C_0 \colon\thinspace y^2 + x y - y = x^3 - x^2.
\]
By \isog{iii} both $\psi$ and $\psi'$ restrict as the 3-power Frobenius
endomorphism $\psi_0$. By \cite[2.6.3]{KM}, in the endomorphism ring
of $C_0$, $\psi_0$ is a root of the polynomial
\begin{equation}
\label{charpoly}
X^2 - {\rm trace}(\psi_0) \cdot X + 3
\end{equation}
with ${\rm trace}(\psi_0)$ an integer satisfying
\[
\big( {\rm trace}(\psi_0) \big)^2 \leq 12.
\]
Moreover by \cite[Exercise 5.10a]{AEC}, since $C_0$ is supersingular,
we have
\[
{\rm trace}(\psi_0) \equiv 0 ~~{\rm mod}~ 3.
\]
Thus ${\rm trace}(\psi_0) = 0$, 3, or $-3$. We exclude the latter two
possibilities by checking the action of $\psi_0$ at the 2-torsion point
$(1,0)$. It then follows from \eqref{charpoly} that
$\psi_0 \circ \psi_0$ agrees with $[-3]$ on $C_0$ over ${\mb F}_3$.
\end{proof}
Analogous to \isog{iv}, let $\kappa'$ be the element in $S^\bullet_3$ such that
$(\psi')^*$ sends $du$ to $\kappa' du$. Note that $|\kappa'| = -6$.
\begin{cor}
\label{cor:K'}
The following relations hold in $S^\bullet_3$:
\[
b^4 \kappa \kappa' + 3 = 0
\]
and
\[
\kappa' = -\kappa^3 + \frac{6}{b^2} ~ \kappa - \frac{a^2 - 8 b}{b^4}.
\]
\end{cor}
\begin{proof}
The isogenies in \eqref{frob^2} induce maps on relative cotangent
spaces at the identity. By \isog{iv} we then have a commutative
diagram
\begin{center}
\begin{tikzpicture}
\node (LT) at (0, 2) {$\kappa \kappa' du$};
\node (RT) at (4.65, 2) {$\kappa' du$};
\node (LB) at (1.9, 0) {$du$};
\node (RB) at (4.65, 0) {$du$};
\node at (4.95, -0.125) {.};
\draw [|->] (RT) -- node [above] {$\scriptstyle \psi^*$} (LT);
\draw [|->] (LB) -- node [left] {$\scriptstyle [-3]^*$} (LT);
\draw [|->] (RB) -- node [right] {$\scriptstyle (\psi')^*$} (RT);
\draw [double distance=1.3pt] (LB) -- (RB);
\end{tikzpicture}
\end{center}
Thus for the first stated relation we need only show that $[3]^*$ sends
$du$ to $3 du / b^4$.
For $i = 1$, 2, 3, and 4, let $Q_i$ be a generator for each of the four
order-3 subgroups of $C$. Each $Q_i$ can be chosen as $Q$ in
\eqref{u'v'}, and we denote the corresponding quantity $\kappa$ in
\eqref{KL} by $\kappa_i$. Define an isogeny $\Psi$ with source $C$ by
\begin{equation*}
\begin{split}
u\big( \Psi(P) \big) \coloneqq & ~ u(P) \prod_{i=1}^4 \big( u(P-Q_i) \cdot u(P+Q_i) \big), \\
v\big( \Psi(P) \big) \coloneqq & ~ v(P) \prod_{i=1}^4 \big( v(P-Q_i) \cdot v(P+Q_i) \big).
\end{split}
\end{equation*}
In view of \eqref{norm}, since $[3]$ has the same kernel as $\Psi$, we
have
\begin{equation}
\label{s}
[3]^* (du) = s \cdot \kappa_1 \kappa_2 \kappa_3 \kappa_4 \cdot du
\end{equation}
where $s$ is a degree-0 unit in $S^\bullet$ coming from an automorphism of $C$
over $S^\bullet$. In view of \eqref{W} we have
\[
\kappa_1 \kappa_2 \kappa_3 \kappa_4 = -\frac{3}{b^4}.
\]
We then compute that $s = -1$ by comparing the restrictions of the two
sides of \eqref{s} to $S$ (see \eqref{S} for the definition of $S$):
$[3]^*$ becomes the multiplication-by-3 map, and $-3 / b^4$ becomes
$-3$ (cf.~the constant term in \eqref{w}). Thus $[3]^*$ sends $du$ to
$3 du / b^4$.
The second stated relation follows by a computation from the first
relation and the relation $W(\kappa) = 0$ as in \isog{i}.
\end{proof}
\begin{rmk}
\label{rmk:KK'}
As noted in Remark \ref{rmk:K}, the (local) analog of $\kappa$ at the prime
2 coincides with the parameter $d$ in \cite[Section 3]{h2p2}. In
particular, with the notations there and the equation in
\cite[Proposition 3.2]{tmf3}, $d$ and $d'$ satisfy an analogous
relation $A_3 d d' + 2 = 0$ which locally reduces to $d d' + 2 = 0$
(the analog of the factor $s$ in the proof of Corollary \ref{cor:K'}
equals 1; cf.~\cite[Theorem 2.5.7]{andoduke}). These arise as examples
of \cite[Lemma 3.21]{poonen}.
\end{rmk}
\begin{rmk}
\label{rmk:K'}
In view of \eqref{frob^2}, $-\psi'$ (composed with the canonical
isomorphism on the target) turns out to be the dual isogeny of $\psi$
(cf.~the proof of \cite[2.9.4]{KM}). If $G$ is the unique order-3
subgroup of $C$ in a formal neighborhood of the identity, then
\[
\kappa \equiv 0 ~~{\rm mod}~ 3
\]
by Remark \ref{rmk:dmod3} and \eqref{K}. Thus in view of Corollary
\ref{cor:K'} and \eqref{H} we have
\[
-\kappa' = \kappa^3 - \frac{6}{b^2} ~ \kappa + \frac{a^2 - 8 b}{b^4} \equiv \frac{H}{b^4} ~~{\rm mod}~ 3.
\]
This congruence agrees with the interpretation of $H$ as defined by the
tangent map of the Verschiebung isogeny over ${\mb F}_3$ (see
\cite[12.4.1]{KM}).
\end{rmk}
\subsection{Individual power operations}
Let $A$ be a $K(2)$-local commutative $E$-algebra. By \cite[3.23]{cong}
and Corollary \ref{cor:psi3}, we have a total power operation
\[
\psi^3 \colon\thinspace A_0 \to A_0 \otimes_{E_0} (E^0 B\Sigma_3 / I) \cong A_0 [\alpha] \big/ \big( w(\alpha) \big).
\]
We also have a composite of total power operations
\begin{equation}
\label{psi3^2}
\begin{split}
A_0 \stackrel{\psi^3}{\longrightarrow} A_0 \otimes_{E_0} (E^0 B\Sigma_3 / I) \stackrel{\psi^3}{\longrightarrow}
& ~ \big( A_0 \otimes_{E_0} (E^0 B\Sigma_3 / I) \big) \tensor[^\psi^3]{\otimes}{_{E_0 [\alpha]}} (E^0 B\Sigma_3 / I) \\
\cong \thinspace \thinspace & ~ \Big( A_0 [\alpha] \big/ \big( w(\alpha) \big) \Big) \tensor[^\psi^3]{\otimes}{_{E_0 [\alpha]}} \Big( E^0 [\alpha] \big/ \big( w(\alpha) \big) \Big),
\end{split}
\end{equation}
where the elements in the target $M \tensor[^\psi^3]{\otimes}{_R} N$ are
subject to the equivalence relation (as well as other ones in a usual
tensor product)
\[
m \otimes (r \cdot n) \sim \big( m \cdot \psi^3(r) \big) \otimes n
\]
for $m \in M$, $n \in N$, and $r \in R$ with
\[
\psi^3(\alpha) = -\alpha^3 + 6 \alpha - h + 9
\]
in view of Corollary \ref{cor:K'}.
\begin{defn}
Define the {\em individual power operations}
\[
Q_k \colon\thinspace A_0 \to A_0
\]
for $k = 0$, 1, 2, and 3 by
\begin{equation}
\label{Q_k}
\psi^3 (x) = Q_0(x) + Q_1(x) \alpha + Q_2(x) \alpha^2 + Q_3(x) \alpha^3.
\end{equation}
\end{defn}
\begin{prop}
\label{prop:Q}
The following relations hold among the individual power operations
$Q_0$, $Q_1$, $Q_2$, and $Q_3$:
\begin{enumerate}[(i)]
\item \label{Q(i)} $Q_0(1) = 1, \quad Q_1(1) = Q_2(1) = Q_3(1) = 0;$
\item \label{Q(ii)} $Q_k(x+y) = Q_k(x) + Q_k(y) \text{~for all~} k;$
\item \label{Q(iii)} {\em Commutation relations }
\begin{equation*}
\begin{split}
Q_0(h x) = & ~ (h^3 - 27 h^2 + 201 h - 342) Q_0(x) + (3 h^2 - 54 h + 171) Q_1(x) \qquad \qquad \\
& + (9 h - 81) Q_2(x) + 24 Q_3(x), \\
Q_1(h x) = & ~ (-6 h^2 + 108 h - 334) Q_0(x) + (-18 h + 171) Q_1(x) + (-72) Q_2(x) \\
& + (h - 9) Q_3(x), \\
Q_2(h x) = & ~ (3 h - 27) Q_0(x) + 8 Q_1(x) + 9 Q_2(x) + (-24) Q_3(x), \\
Q_3(h x) = & ~ (h^2 - 18 h + 57) Q_0(x) + (3 h - 27) Q_1(x) + 8 Q_2(x) + 9 Q_3(x), \\
Q_0(c x) = & ~ (c^3 - 12 c + 12 c^{-1}) Q_0(x) + (3 c - 12 c^{-1}) Q_1(x) + (12 c^{-1}) Q_2(x) \\
& + (-12 c^{-1}) Q_3(x), \\
Q_1(c x) = & ~ (-6 c + 20 c^{-1}) Q_0(x) + (-20 c^{-1}) Q_1(x) + (- c + 20 c^{-1}) Q_2(x) \\
& + (4 c - 20 c^{-1}) Q_3(x), \\
Q_2(c x) = & ~ (4 c^{-1}) Q_0(x) + (-4 c^{-1}) Q_1(x) + (4 c^{-1}) Q_2(x) + (- c - 4 c^{-1}) Q_3(x), \\
Q_3(c x) = & ~ (c - 4 c^{-1}) Q_0(x) + (4 c^{-1}) Q_1(x) + (-4 c^{-1}) Q_2(x) + (4 c^{-1}) Q_3(x), \\
Q_k(i x) = & ~ (-i) Q_k(x) \text{~for all~} k; \\
\end{split}
\end{equation*}
\item \label{Q(iv)} {\em Adem relations }
\begin{equation*}
\begin{split}
Q_1Q_0(x) = & ~ (-6) Q_0Q_1(x) + 3 Q_2Q_1(x) + (6 h - 54) Q_0Q_2(x) + 18 Q_1Q_2(x) \\
& + (-9) Q_3Q_2(x) + (-6 h^2 + 108 h - 369) Q_0Q_3(x) \\
& + (-18 h + 162) Q_1Q_3(x) + (-54) Q_2Q_3(x), \\
Q_2Q_0(x) = & ~ 3 Q_3Q_1(x) + (-3) Q_0Q_2(x) + (3 h - 27) Q_0Q_3(x) + 9 Q_1Q_3(x), \qquad \qquad \\
Q_3Q_0(x) = & ~ Q_0Q_1(x) + (-h + 9) Q_0Q_2(x) + (-3) Q_1Q_2(x) \\
& + (h^2 - 18 h + 63) Q_0Q_3(x) + (3 h - 27) Q_1Q_3(x) + 9 Q_2Q_3(x);
\end{split}
\end{equation*}
\item \label{Q(v)} {\em Cartan formulas }
\begin{equation*}
\begin{split}
Q_0(xy) = & ~ Q_0(x) Q_0(y) + 3 \big( Q_3(x) Q_1(y) + Q_2(x) Q_2(y) + Q_1(x) Q_3(y) \big) \\
& + 18 Q_3(x) Q_3(y), \\
Q_1(xy) = & ~ \big( Q_1(x) Q_0(y) + Q_0(x) Q_1(y) \big) \\
& + (-h + 9) \big( Q_3(x) Q_1(y) + Q_2(x) Q_2(y) + Q_1(x) Q_3(y) \big) \\
& + 3 \big( Q_3(x) Q_2(y) + Q_2(x) Q_3(y) \big) + (-6 h + 54) Q_3(x) Q_3(y), \qquad \qquad \qquad
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
Q_2(xy) = & ~ \big( Q_2(x) Q_0(y) + Q_1(x) Q_1(y) + Q_0(x) Q_2(y) \big) \\
& + 6 \big( Q_3(x) Q_1(y) + Q_2(x) Q_2(y) + Q_1(x) Q_3(y) \big) \\
& + (-h + 9) \big( Q_3(x) Q_2(y) + Q_2(x) Q_3(y) \big) + 39 Q_3(x) Q_3(y), \\
Q_3(xy) = & ~ \big( Q_3(x) Q_0(y) + Q_2(x) Q_1(y) + Q_1(x) Q_2(y) + Q_0(x) Q_3(y) \big) \qquad \qquad \qquad \\
& + 6 \big( Q_3(x) Q_2(y) + Q_2(x) Q_3(y) \big) + (-h + 9) Q_3(x) Q_3(y);
\end{split}
\end{equation*}
\item \label{Q(vi)} {\em The Frobenius congruence }
\begin{equation*}
Q_0(x) \equiv x^3 ~~{\rm mod}~ 3. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
The relations in \eqref{Q(i)}, \eqref{Q(ii)}, \eqref{Q(iii)}, and
\eqref{Q(v)} follow computationally from the formulas in Corollary
\ref{cor:psi3} together with the fact that $\psi^3$ is a ring homomorphism.
For \eqref{Q(iv)}, there is a canonical isomorphism $C/C[3] \cong C$ of
elliptic curves. Given the correspondence between deformations of
Frobenius and power operations in \cite[Theorem B]{cong}, the
commutativity of \eqref{frob^2} then implies that the composite
\eqref{psi3^2} lands in $A_0$. In terms of formulas, we have
\begin{equation*}
\begin{split}
\psi^3 \big( \psi^3(x) \big) = & ~ \psi^3 \big( Q_0(x) + Q_1(x) \alpha + Q_2(x) \alpha^2 + Q_3(x) \alpha^3 \big) \\
= & ~ \sum_{k = 0}^3 \psi^3 \big( Q_k(x) \big) \big( \psi^3(\alpha) \big)^k \\
= & ~ \sum_{k = 0}^3 \sum_{j = 0}^3 Q_jQ_k(x) \alpha^j (-\alpha^3 + 6 \alpha - h + 9)^k \\
\equiv & ~ \Psi_0(x) + \Psi_1(x) \alpha + \Psi_2(x) \alpha^2 + \Psi_3(x) \alpha^3 ~~{\rm mod}~ \big( w(\alpha) \big)
\end{split}
\end{equation*}
where each $\Psi_i$ is an $E_0$-linear combination of the $Q_jQ_k$'s.
The vanishing of $\Psi_1(x)$, $\Psi_2(x)$, and $\Psi_3(x)$ gives the
three relations in \eqref{Q(iv)}.
For \eqref{Q(vi)}, by \cite[Propositions 3.25 and 10.5]{cong} we have
\[
\psi^3(x) \equiv x^3 ~~{\rm mod}~ 3.
\]
In view of \eqref{Amod3}, the congruence in \eqref{Q(vi)} then follows
from \eqref{Q_k}.
\end{proof}
\begin{ex}
\label{ex}
We have $E^0 S^2 \cong {\mb Z}_9 \llbracket h \rrbracket [u] / (u^2)$. By
definition of $\kappa$ in \eqref{KL}, the $Q_k$'s act canonically on
$u \in E^0 S^2$:
\[
Q_k(u) = \left\{
\begin{array}{ll}
u, & \quad {\rm if}~k = 1, \\
0, & \quad {\rm if}~k \neq 1. \\
\end{array}
\right.
\]
We then get the values of the $Q_k$'s on elements in $E^0 S^2$ from
\q{i}-\eqref{Q(iii)}.
\end{ex}
\subsection{The Dyer-Lashof~ algebra}
\begin{defn}
\label{def:go}
\mbox{}
\begin{enumerate}[(i)]
\item \label{go(i)} Let $i$ be an element generating ${\mb Z}_9$ over
${\mb Z}_3$ with $i^2 = -1$. Define $\gamma$ to be the associative ring
generated over ${\mb Z}_9 \llbracket h \rrbracket$ by elements $q_0$,
$q_1$, $q_2$, and $q_3$ subject to the following relations: the
$q_k$'s commute with elements in
${\mb Z}_3 \subset {\mb Z}_9 \llbracket h \rrbracket$, and satisfy {\em
commutation relations}
\begin{equation*}
\begin{split}
q_0 h = & ~ (h^3 - 27 h^2 + 201 h - 342) q_0 + (3 h^2 - 54 h + 171) q_1 + (9 h - 81) q_2 \\
& + 24 q_3, \\
q_1 h = & ~ (-6 h^2 + 108 h - 334) q_0 + (-18 h + 171) q_1 + (-72) q_2 + (h - 9) q_3, \\
q_2 h = & ~ (3 h - 27) q_0 + 8 q_1 + 9 q_2 + (-24) q_3, \\
q_3 h = & ~ (h^2 - 18 h + 57) q_0 + (3 h - 27) q_1 + 8 q_2 + 9 q_3, \\
q_k i ~ = & ~ (-i) q_k \text{~for all~} k,
\end{split}
\end{equation*}
and {\em Adem relations}
\begin{equation*}
\begin{split}
q_1q_0 = & ~ (-6) q_0q_1 + 3 q_2q_1 + (6 h - 54) q_0q_2 + 18 q_1q_2 + (-9) q_3q_2 \\
& + (-6 h^2 + 108 h - 369) q_0q_3 + (-18 h + 162) q_1q_3 + (-54) q_2q_3, \quad~~ \\
q_2q_0 = & ~ 3 q_3q_1 + (-3) q_0q_2 + (3 h - 27) q_0q_3 + 9 q_1q_3, \\
q_3q_0 = & ~ q_0q_1 + (-h + 9) q_0q_2 + (-3) q_1q_2 + (h^2 - 18 h + 63) q_0q_3 \\
& + (3 h - 27) q_1q_3 + 9 q_2q_3.
\end{split}
\end{equation*}
\item \label{go(ii)} Write $\omega \coloneqq \pi_2 E$ which is the
kernel of $E^0 S^2 \to E^0$ with
$E^0 S^2 \cong {\mb Z}_9 \llbracket h \rrbracket [u] / (u^2)$. Define
$\omega$ as a $\gamma$-module in the sense of \cite[2.2]{h2p2} with one
generator $u$ by
\[
q_k \cdot u = \left\{
\begin{array}{ll}
u, & \quad {\rm if}~k = 1, \\
0, & \quad {\rm if}~k \neq 1. \\
\end{array}
\right.
\]
\end{enumerate}
\end{defn}
\begin{rmk}
\label{rmk:rank}
In \go{i}, an element $r \in {\mb Z}_9 \llbracket h \rrbracket \cong E_0$
corresponds to the multiplication-by-$r$ operation (see
\cite[Proposition 6.4]{cong}), and each $q_k$ corresponds to the
individual power operation $Q_k$ (also cf.~\go{ii} and Example
\ref{ex}). Under this correspondence, the relations in
\q{ii}-\eqref{Q(v)} describe explicitly the structure of $\gamma$ as that
of a {\em graded twisted bialgebra over $E_0$} in the sense of
\cite[Section 5]{cong}. The grading of $\gamma$ comes from the number of
the $q_k$'s in a monomial: for example, commutation relations are in
degree 1, and Adem relations are in degree 2. Under these relations,
$\gamma$ has an {\em admissible basis}: it is free as a left $E_0$-module
on the elements of the form
\[
q_0^m q_{k_1} \cdots q_{k_n}
\]
where $m, n \geq 0$ ($n = 0$ gives $q_0^m$), and $k_i = 1$, 2, or 3.
If we write $\gamma[d]$ for the degree-$d$ part of $\gamma$, then $\gamma[d]$ is of
rank $1 + 3 + \cdots + 3^d$.
\end{rmk}
We now identify $\gamma$ with the Dyer-Lashof~ algebra of power operations on
$K(2)$-local commutative $E$-algebras.
\begin{thm}
\label{thm:gamma}
Let $A$ be a $K(2)$-local commutative $E$-algebra. Let $\gamma$ be the
graded twisted bialgebra over $E_0$ as defined in \go{i}, and $\omega$
be the $\gamma$-module in \go{ii}. Then $A_*$ is an {\em $\omega$-twisted
${\mb Z}/2$-graded amplified $\gamma$-ring} in the sense of
\cite[Section 2]{cong} and \cite[2.5 and 2.6]{h2p2}. In particular,
\[
\pi_* L_{K(2)} {\mb P}_E (\Sigma^d E) \cong \big( F_d \big)_{(3,h)}^\wedge,
\]
where $F_d$ is the free $\omega$-twisted ${\mb Z}/2$-graded amplified
$\gamma$-ring with one generator in degree $d$.
\end{thm}
Formulas for $\gamma$ aside, this result is due to Rezk \cite{cong, h2p2}.
\begin{proof}
Let $\Gamma$ be the graded twisted bialgebra of power operations on $E_0$
in \cite[Section 6]{cong}. We need only identify $\Gamma$ with $\gamma$.
There is a direct sum decomposition $\Gamma = \bigoplus_{d \geq 0} \Gamma[d]$
where the summands come from the completed $E$-homology of
$B\Sigma_{3^d}$ (see \cite[6.2]{cong}). As in Remark \ref{rmk:rank},
we have a degree-preserving ring homomorphism
\[
\phi \colon\thinspace \gamma \to \Gamma, \qquad q_k \mapsto Q_k
\]
which is an isomorphism in degrees 0 and 1. We need to show that
$\phi$ is both surjective and injective in all degrees.
For the surjectivity of $\phi$, we use a transfer argument. We have
\[
\nu_3(|\Sigma_3^{\wr d}|) = \nu_3(|\Sigma_{3^d}|) = (3^d - 1) / 2
\]
where $\nu_3(-)$ is the 3-adic valuation, and $(-)^{\wr d}$ is the
$d$-fold wreath product. Thus following the proof of
\cite[Proposition 3.17]{cong}, we see that $\Gamma$ is generated in degree
1, and hence $\phi$ is surjective.
By Remark \ref{rmk:rank} and (the $E_0$-linear dual of)
\cite[Theorem 1.1]{Str98}, $\gamma[d]$ and $\Gamma[d]$ are of the same rank
$1 + 3 + \cdots + 3^d$ as free modules over $E_0$. Hence $\phi$ is
also injective.
\end{proof}
\section{$K(1)$-local power operations}
\label{sec:K(1)}
Let $F \coloneqq L_{K(1)} E$ be the $K(1)$-localization of $E$. The
following diagram describes the relationship between $K(1)$-local power
operations on $F^0$ (cf.~\cite[Section 3]{hopkins} and
\cite[Section IX.3]{H_infty}) and the power operation on $E^0$ in
Corollary \ref{cor:psi3}:
\begin{center}
\begin{tikzpicture}
\node (LT) at (0, 2) {$E^0$};
\node (RT) at (3, 2) {$E^0 B\Sigma_3 / I$};
\node (LB) at (0, 0) {$F^0$};
\node (MB) at (3, 0) {$F^0 B\Sigma_3 / J$};
\node (RB) at (4.3, 0) {$\cong F^0. $};
\draw [->] (LT) -- node [above] {$\scriptstyle \psi^3$} (RT);
\draw [->] (LT) -- (LB);
\draw [->] (RT) -- (MB);
\draw [->] (LB) -- node [above] {$\scriptstyle \psi_F^3$} (MB);
\end{tikzpicture}
\end{center}
Here $\psi_F^3$ is the $K(1)$-local power operation induced by $\psi^3$, and
$J \cong F^0 \otimes_{E^0} I$ is the transfer ideal
(cf.~\eqref{transfer}). Recall from \isog{i}, \eqref{S_3}, and
Corollary \ref{cor:psi3} that $\psi^3$ arises from the universal degree-3
isogeny which is parametrized by the ring $S^\bullet_3$ with
\[
\big( S_3 \big)_{(3,h)}^\wedge \cong E^0 B\Sigma_3 / I.
\]
The vertical maps are induced by the $K(1)$-localization $E \to F$. In
terms of homotopy groups, this is obtained by inverting the generator
$h$ and completing at the prime 3 (see \cite[Corollary 1.5.5]{hovey}):
\[
E_* = {\mb Z}_9 \llbracket h \rrbracket [u^{\pm1}] \qquad {\rm and} \qquad F_* = {\mb Z}_9 \llbracket h \rrbracket [h^{-1}]_3^\wedge [u^{\pm1}]
\]
with
\[
F_0 = {\mb Z}_9 (\!(h)\!)_3^\wedge = \left.\left\{\sum_{n = -\infty}^{\infty} k_n h^n~\right|~k_n \in {\mb Z}_9, \lim_{n \to -\infty} k_n = 0\right\}.
\]
The formal group $\widehat{C}$ over $E^0$ has a unique order-3 subgroup after
being pulled back to $F^0$ (cf.~Remark \ref{rmk:dmod3}), and the map
\[
E^0 B\Sigma_3 / I \to F^0 B\Sigma_3 / J \cong F^0
\]
classifies this subgroup. Along the base change
\[
E^0 B\Sigma_3 / I \to F^0 \otimes_{E^0} (E^0 B\Sigma_3 / I) \cong (F^0 \otimes_{E^0} E^0 B\Sigma_3) / J \cong F^0 B\Sigma_3 / J,
\]
the special fiber of the 3-divisible group of $\widehat{C}$ which consists
solely of a formal component may split into formal and \'etale
components. We want to take the formal component so as to keep track of
the unique order-3 subgroup of the formal group over $F^0$. This
subgroup gives rise to the $K(1)$-local power operation $\psi_F^3$.
Recall from \eqref{S_3} that $S_3 = S[\alpha] \big/ \big( w(\alpha) \big)$.
Since
\[
w(\alpha) = \alpha^4 - 6 \alpha^2 + (h - 9) \alpha - 3 \equiv \alpha (\alpha^3 + h) ~~{\rm mod}~ 3,
\]
the equation $w(\alpha) = 0$ has a unique root $\alpha = 0$ in ${\mb F}_9 (\!(h)\!)$
(cf.~\eqref{Amod3}). By Hensel's lemma this unique root lifts to a root
in ${\mb Z}_9 (\!(h)\!)_3^\wedge$; it corresponds to the unique order-3
subgroup of $\widehat{C}$ over $F^0$. Plugging this specific value of $\alpha$ into
the formulas for $\psi^3$ in Corollary \ref{cor:psi3}, we then get an
endomorphism of the ring $F^0$. This endomorphism is the $K(1)$-local
power operation $\psi_F^3$.
Explicitly, with $h$ invertible in $F^0$, we solve for $\alpha$ from
$w(\alpha) = 0$ by first writing
\[
\alpha = (3 + 6 \alpha^2 - \alpha^4) / (h - 9) = (3 + 6 \alpha^2 - \alpha^4) \sum_{n = 1}^\infty 9^{n-1} h^{-n}
\]
and then substituting this equation into itself recursively. We plug
the power series expansion for $\alpha$ into $\psi^3(h)$ and get
\[
\psi_F^3(h) = h^3 - 27 h^2 + 183 h - 180 + 186 h^{-1} + 1674 h^{-2} + (\text{lower-order terms}). ~~~
\]
Similarly, writing $h$ as $c^2 + 1$ in $w(\alpha) = 0$, we solve for $\alpha$ in
terms of $c$ and get
\[
\psi_F^3(c) = c^3 - 12 c - 6 c^{-1} - 84 c^{-3} - 933 c^{-5} - 10956 c^{-7} + (\text{lower-order terms}).
\]
|
{
"timestamp": "2012-10-16T02:02:08",
"yymm": "1210",
"arxiv_id": "1210.3730",
"language": "en",
"url": "https://arxiv.org/abs/1210.3730"
}
|
\subsection{Clustered Erd\"{o}s-R\'{e}nyi graphs} \label{sub:structure}
Recall that for an integer $n \ge 1$ and $0 \le p \le 1$, the Erd\"{o}s-R\'{e}nyi graph
$G(n, p)$ is the random graph obtained by starting with vertex set
$V = \{1,2,\ldots,n\}$ and connecting each pair of vertices $u, v \in V$,
independently with probability $p$.
Let $\Pi$ denote a partition $(V_1, V_2, \ldots, V_k)$ of $V$,
let $\pi$ denote the real number sequence $(p_1, p_2, \ldots, p_k)$,
where $0 \le p_i \le 1$ for all $i$ and let $0 \le p' < \min_i \{p_i\}$.
The \textit{clustered Erd\"{o}s-R\'{e}nyi } graph $G(\Pi, \pi, p')$ has vertex set
$V$ and edges obtained by independently connecting each pair of
vertices $u, v \in V$ with probability $p_i$ if $u, v \in V_i$ for some
$i$ and with probability $p'$, otherwise (see Figure~\ref{fig:erdos}).
Thus each induced subgraph $G[V_i]$ is the standard Erd\"{o}s-R\'{e}nyi graph
$G(n_i, p_i)$, where $n_i = |V_i|$.
\begin{figure}[t]
\centering
\fbox{
\begin{tikzpicture}[scale=1]
\draw (2,0) ellipse (1 and 2);
\node[blank] at (2,-2.5) {$V_1$};
\node[cir] at (2.2,1) (u1) {$u_1$};
\node[cir] at (1.8,-0.3) (u2) {$u_2$};
\node[cir] at (2.2,-1.2) (u3) {$u_{n_1}$};
\node[blank] at (2.4,0) {$\vdots$};
\draw (u1) -- (u2) node[above,left,midway]{$p_1$};
\draw (u2) -- (u3) node[above,left,midway]{$p_1$};
\draw (6,0) ellipse (1 and 2);
\node[blank] at (6,-2.5) {$V_2$};
\node[cir] at (5.6,1) (v1) {$v_1$};
\node[cir] at (6,0) (v2) {$v_2$};
\node[cir] at (5.8,-1) (v3) {$v_{n_2}$};
\node[blank] at (6.5,0.5) {$\vdots$};
\draw (v1) -- (v2) node[above,left,midway]{$p_2$};
\draw (v2) -- (v3) node[above,left,midway]{$p_2$};
\node[blank] at (8.5,0) {$\ldots$};
\node[blank] at (8.5,-2.5) {$\ldots$};
\draw (11,0) ellipse (1 and 2);
\node[blank] at (11,-2.5) {$V_k$};
\node[cir] at (11.2,1) (w1) {$w_1$};
\node[cir] at (10.7,-0.4) (w2) {$w_2$};
\node[cir] at (11.2,-1.2) (w3) {$w_{n_k}$};
\node[blank] at (10.4,0.3) {$\vdots$};
\draw (w1) -- (w3) node[above,right,midway]{$p_k$};
\draw (w2) -- (w3) node[above,left,midway]{$p_k$};
\draw (u1) -- (v2) node[below=5pt,left,midway]{$p'$};
\draw (v3) -- (w2) node[below=5pt,right,midway]{$p'$};
\end{tikzpicture}
}
\caption{The clustered Erd\"{o}s-R\'{e}nyi graph. We connect two nodes in the $i$-th ellipse
(i.e., $V_i$) with probability $p_i$ and nodes from different ellipses are
connected
with probability $p'< \min_i\{p_i\}$. \label{fig:erdos}}
\end{figure}
Given that $p' < p_i$ for all $i$, one might view $G(\Pi, \pi, p')$ as having a
natural
community structure given by the vertex partition $\Pi$.
Specifically, when $p'$ is much smaller than $\min_i\{p_i\}$, the
inter-community edge density is
much less than the intra-community edge density and it may be easier
to detect the community structure $\Pi$.
On the other hand as the intra-community probabilities $p_i$ get closer to
$p'$, it may be hard for an algorithm
such as \textsc{Max-LPA} to identify $\Pi$ as the community structure.
Similarly, if an intra-community probability $p_i$ becomes very
small, then the subgraph $G[V_i]$
can itself be quite sparse and independent of how small $p'$ is relative to
$p_i$, any community detection algorithm may end up viewing each $V_i$ as
being composed of several communities.
In the rest of the section, we explore values of the $p_i$'s and $p'$ for which
\textsc{Max-LPA} ``correctly'' and quickly identifies $\Pi$ as the community
structure of
$G(\Pi, \pi, p')$.
\subsection{Analysis}
\label{subsection:analysis}
In the following theorem we establish fairly general conditions on the
probabilities
$\{p_i\}$ and $p'$ and on the node subset sizes $\{n_i\}$ and $n$ under which
\textsc{Max-LPA} converges
correctly, i.e., to the node partition $\Pi$, w.h.p.
Furthermore, we show that under these circumstances just 2 rounds suffice for
\textsc{Max-LPA} to reach convergence!
\begin{lemma}
\label{lemma:probBound}
Let $G(\Pi, \pi, p')$ be a clustered Erd\"{o}s-R\'{e}nyi graph such that $p' <
\min_i\{\frac{n_i}{n}\}$.
Let $\ell_i$ be the maximum label of a node in $V_i$.
Then for any node $v \in V_i$ the probability that $v$ is not adjacent to
a node outside $V_i$ with label higher than $\ell_i$ is at least $1/2e$.
\end{lemma}
\begin{proof}
Let $v'$ be a node in $V \setminus V_i$.
Given that $|V_i| = n_i$ and $\ell_i$ is the maximum label among these $n_i$
nodes, the probability that the label assigned uniformly at random to $v'$ is
larger than $\ell_i$ is $1/(n_i + 1)$.
The probability that $v$ has an edge to $v'$ \textit{and} $v'$ has a higher
label than $\ell_i$ is $p'/(n_i + 1)$.
Therefore the probability that $v'$ has no edge to a node outside $V_i$ with
label larger than $\ell_i$ is
$$\left(1 - \frac{p'}{n_i + 1}\right)^{n - n_i}.$$
We bound this expression below as follows:
$$\left(1 - \frac{p'}{n_i + 1}\right)^{n - n_i} > \left(1 -
\frac{p'}{n_i}\right)^{n} > \left(1 - \frac{1}{n}\right)^{n} > \frac{1}{2e}.$$
\qed
\end{proof}
\begin{theorem}
\label{theorem:ER}
Let $G(\Pi, \pi, p')$ be a clustered Erd\"{o}s-R\'{e}nyi graph.
Suppose that the probabilities $\{p_i\}$ and $p'$ and the node subset sizes
$\{n_i\}$ and $n$
satisfy the inequalities:
$$\mbox{(i) } n_i p_i^2 > 8n p' \qquad\mbox{and}\qquad \mbox{(ii) } n_i
p_i^4 > 1800 c \log n,$$
for some constant $c$.
Then, given input $G(\Pi, \pi, p')$, \textsc{Max-LPA} converges correctly to
node partition $\Pi$ in two rounds w.h.p. (Note that condition (ii) implies for
each $i$, $p_i > \frac{\log n_i}{n_i}$ and hence each $G[V_i]$ is connected.)
\end{theorem}
\begin{proof}
Let $V_i = \{u_1, u_2, \ldots, u_{n_i}\}$ and without loss of generality assume
that $\ell_{u_1}
> \ell_{u_2} > \cdots > \ell_{u_{n_i}}$.
Since all initial node labels are assumed to be distinct, in the first round of
\textsc{Max-LPA}
every node $u \in V$ acquires a label by breaking ties.
Since ties are broken in favor of larger labels, all neighbors of $u_1$ in $V_i$
that have no neighbor outside $V_i$ with a label larger than $\ell_{u_1}$
will acquire the label $\ell_{u_1}$.
Consider a node $v \in V_i$.
Let $\beta$ denote the probability that $v$ has no neighbor outside $V_i$
with label larger than $\ell_{u_1}$.
Note that inequality (i) in the theorem statement implies the hypothesis of
Lemma \ref{lemma:probBound} and therefore $\beta > 1/2e$.
The probability that $v$ is a neighbor of $u_1$ and does not have a neighbor
outside $V_i$ is $\beta \cdot p_i$.
Hence, after the first round of \textsc{Max-LPA}, in expectation, $n_i \cdot
\beta \cdot p_i$ nodes
in $V_i$ would have acquired the label $\ell_{u_1}$.
In the rest of the proof we will use
$$X := n_i \cdot \beta \cdot p_i.$$
Now consider node $u_j$ for $j > 1$.
For a node $v \in V_i$ to acquire the label $\ell_{u_j}$ it must be the case
that $v$ is adjacent to $u_j$, not adjacent to any node in
$\{u_1, u_2, \ldots, u_{j-1}\}$, and not adjacent to any node outside $V_i$
with a label higher than $\ell_{u_j}$.
Since $\ell_{u_j}$ is smaller than $\ell_{u_1}$, the probability that $v$
is not adjacent to a node outside $V_i$ with label higher than $\ell_{u_j}$
is less than $\beta$.
Thus the probability that a node in $V_i$ acquires the label $\ell_{u_j}$ is at
most $p_i (1 - p_i)^{j-1} \cdot \beta < p_i (1 - p_i) \cdot \beta$.
Furthermore, the probability that a node outside $V_i$ will acquire
the label $\ell_{u_j}$ at the end of the first round is at most $p'$.
Therefore, the expected number of nodes in $V$ that acquire the label $u_j$, at
the end of the first round,
is in expectation at most $n_i \cdot p_i (1 - p_i) \cdot \beta + (n - n_i)p'$.
We now use inequality (i) and the fact that $2\beta e > 1$ to upper bound this
expression as follows:
$$n_i \cdot p_i(1 - p_i) \cdot \beta + (n - n_i)p' < n_i \cdot p_i(1 - p_i)
\cdot \beta + \frac{2 \beta e \cdot n_i p_i^2}{8} <
n_i \cdot p_i\left(1 - \frac{3p_i}{4}\right) \cdot \beta.$$
Therefore, the expected number of nodes in $V$ that acquire the label $u_j$, at
the end of the first round,
is in expectation at most
$$Y := n_i \cdot p_i\left(1 - \frac{3p_i}{4}\right) \cdot \beta.$$
It is worth mentioning at this point that $X - Y = n_i p_i^2 \beta/4$.
Note that all the random variables we have utilized thus far, e.g., the number
of nodes adjacent to $u_1$
and not adjacent to any node outside $V_i$ with label higher than $\ell_{u_1}$,
can be expressed as sums
of independent, identically distributed indicator random variables.
Hence we can bound the deviation of such random variables using the tail bound
in (\ref{eqn:chernoff2}).
In particular, let $Y'$ denote $Y + \sqrt{3c Y \log n}$ and $X'$ denote $X -
\sqrt{3c X \log n}$.
With high probability, at the end of the first round of \textsc{Max-LPA}, the
number of nodes in $V_i$
that acquire the label $u_1$ is at least $X'$ and the number of nodes in $V$
that acquire the label $\ell_{u_j}$, $j > 1$,
is at most $Y'$.
Next we bound the ``gap'' between $X'$ and $Y'$ as follows:
\begin{align*}
X' - Y' & = X - Y - \sqrt{3c X \log n} - \sqrt{3c Y \log n}\\
& > \frac{3n_i p_i^2 \beta}{4} - 2 \sqrt{3c X \log n}\\
& > \frac{3n_i p_i^2 \beta}{4} - 2 \sqrt{3c n_i p_i \beta \log n}\\
& > \frac{3n_i p_i^2 \beta}{4} - \frac{3n_i p_i^2 \beta}{5}\\
& = \frac{3 n_i p_i^2 \beta}{20}
\end{align*}
The second inequality follows from $X - Y = 3n_ip_i^2\beta/4$ and $Y < X$,
the third from the fact that $X = n_i p_i \beta$, and the fourth by using
inequality (ii) from
the theorem statement.
We now condition the execution of the second round of \textsc{Max-LPA} on the
occurrence of the two high probability
events: (i) number of nodes in $V_i$ that acquire the label $u_1$ is at least
$X'$ and
(ii) the number of nodes in $V$ that acquire the label $\ell_{u_j}$, $j > 1$,
is at most $Y'$.
Consider a node $v \in V_i$ just before the execution of the second round of
\textsc{Max-LPA}.
Node $v$ has in expectation at least $p_i X'$ neighbors labeled $\ell_{u_1}$ in
$V_i$.
Also, node $v$ has in expectation at most $p_i Y'$ neighbors labeled
$\ell_{u_j}$, for each $j > 1$, in $V$.
Let us now use $X''$ to denote the quantity $p_i X' - \sqrt{3c p_i X' \log n}$
and
$Y''$ to denote the quantity $p_i Y' + \sqrt{3c p_i Y' \log n}$.
By using (\ref{eqn:chernoff2}) again, we know that w.h.p. $v$ has
at least
$X''$ neighbors with label $\ell_{u_1}$ and at most $Y''$ neighbors with a label
$\ell_{u_j}$, $j > 1$.
We will now show that $X'' > Y''$ and this will guarantee that in the second
round of \textsc{Max-LPA} $v$ will acquire the label $\ell_{u_1}$, with high
probability. Since $v$ is an
arbitrary node in $V_i$, this implies that all nodes in $V_i$ will acquire the
label $\ell_{u_1}$
in the second round of \textsc{Max-LPA} w.h.p.
\begin{align*}
X'' - Y'' & = p_i(X' - Y') - \sqrt{3c p_i X' \log n} - \sqrt{3c p_i Y' \log
n}\\
& > \frac{3 n_i p_i^3}{20} - 2 \sqrt{3c p_i X' \log n}\\
& > \frac{3 n_i p_i^3}{20} - 2 \sqrt{3c n_i p_i^2 \beta \log n}\\
& > \frac{3 n_i p_i^3}{20} - \frac{n_i p_i^3 \beta}{10}\\
& = \frac{3 n_i p_i^2}{20}\\
& > 0
\end{align*}
The second inequality follows from the bound on $X' - Y'$ derived earlier and
$Y' < X'$,
the third from the fact that $X' < n_i p_i \beta$, and the fourth by using
inequality (ii) from
the theorem statement.
Thus at the end of the second round of \textsc{Max-LPA}, w.h.p.,
every node in $V_i$ has label
$\ell_{u_1}$. This is of course true, w.h.p., for all of the
$V_i$'s.
Now note that every node $v \in V_i$ has, in expectation $n_i p_i$ neighbors in
$V_i$ and
fewer than $n p'$ neighbors outside $V_i$.
Inequality (i) implies that $n p' < n_i p_i/8$ and inequality (ii) implies that
$n_i p_i = \Omega(\log n)$.
Pick a constant $\epsilon > 0$ such that $n_i p_i (1 + \epsilon)/8 < n_i p_i (1
- \epsilon)$.
By applying tail bound (\ref{eqn:chernoff1}), we see that w.h.p.
$v$ has more than
$n_i p_i (1 - \epsilon)$ neighbors in $V_i$ and fewer than $n_i p_i (1 +
\epsilon)/8$ neighbors
outside $V_i$. Hence, w.h.p. $v$ has no reason to change its label.
Since $v$ is an arbitrary node in an arbitrary $V_i$, w.h.p. there are no
further changes
to the labels assigned by \textsc{Max-LPA}.
\qed
\end{proof}
To understand the implications of Theorem \ref{theorem:ER} consider the
following example.
Suppose that the clustered Erd\"{o}s-R\'{e}nyi graph has $O(1)$ clusters and each cluster had
size $\Theta(n)$.
In such a setting, inequality (ii) from the theorem simplifies to requiring that
each $p_i = \Omega((\log n/n)^{1/4})$
and inequality (ii) simplifies to $p' < p_i^2/c$ for all $i$.
This tells us, for instance, that \textsc{Max-LPA} converges in just two rounds
on a clustered Erd\"{o}s-R\'{e}nyi graph in which
each cluster has $\Theta(n)$ vertices and an intra-community probability of
$\Theta(1/n^{1/3})$ and
the inter-community probability is $\Theta(1/n^{2/3})$.
This example raises several questions.
If we were willing to allow more time for \textsc{Max-LPA} to converge, say
$O(\log n)$ rounds, could we significantly
weaken the requirements on the $p_i$'s and $p'$.
Specifically, could we permit an intra-community probability $p_i$ to become as small as
$c \log n/n$ for some constant $c > 1$?
Similarly, could we permit the inter-community probability $p'$ to come much closer to the smallest
$p_i$, say within a constant factor.
We believe that it may be possible to obtain such results, but only via substantively different
analysis techniques.
\subsection{Preliminaries} \label{sub:pre}
We use $G = (V,E)$ to denote an undirected connected graph (network) of size
$n=|V|$.
For $v \in V$, we denote by $N(v) = \{u : u \in V, (u,v) \in E\}$ the neighborhood
of $v$ in graph $G$, by $deg(v) = |N(v)|$ the degree of $v$, and by
$\Delta(G)=\max_{v\in V} deg(v)$ the maximum degree over all the
vertices in $G$.
A \textit{$k$-hop neighborhood} ($k \geqslant 1$) of $v$ is defined as $N_k(v) = \{w : \mbox{dist}_G(w, v) \le k\} \setminus \{v\}$.
We denote the \textit{closed neighborhood} (respectively,
\textit{closed $k$-hop neighborhood}) of $v$ as $N'(v) = N(v) \cup \{v\}$
(respectively, $N'_k(v) = N_k(v) \cup \{v\}$).
Denote by $\ell_u(t)$ the label of node $u$ just before round $t$.
When the round number is clear from the context, we
use $\ell_u$ to denote the current label of $u$.
Since the number of labels in the network is finite,
LPA will behave periodically starting in some round $t^*$, i.e.,
for some $p \ge 1$, $0 \le i < p$, and $j = 0, 1, 2, \ldots$,
$$\ell_u(t^*+ i)=\ell_u(t^* + i + j \cdot p)$$
for all $u \in V$.
Then we say that \textsc{Max-LPA} has \textit{converged} in $t^*$ rounds.
We now describe \textsc{Max-LPA} precisely (see \textbf{Algorithm 1}).
Every node $v \in V$ is assigned a unique label uniformly and independently
at random.
For concreteness, we assume that these labels come from the range $[0, 1]$.
At the start of a round, each node sends its label to all neighboring nodes.
After receiving labels from all neighbors, a node $v$ updates its label as:
\begin{equation} \label{eqn:label}
l_v \leftarrow \max\left\{\ell \mid \sum_{u \in N'(v)}[\ell_u== \ell] \ge \sum_{u \in N'(v)}[\ell_u== \ell']\mbox{ for all $\ell'$}\right\},
\end{equation}
where $[\ell_u==\ell]$ evaluates to 1 if $\ell_u=\ell$,
otherwise evaluates to 0.
Note that there is no randomness in the algorithm after the initial
assignments of labels.
\begin{algorithm}[t]
\caption{\textsc{Max-LPA} on a node $v$} \label{algo:lpa}
\begin{algorithmic}
\STATE $i=0$
\STATE $l_v[i] \leftarrow$ random(0,1)
\WHILE{true}\label{algo:lpa:while}
\STATE $i++$;
\STATE send $l_v[i-1]$ to $\forall u \in N(v)$
\STATE receive $l_u[i-1]$ from $\forall u \in N(v)$
\STATE $l_v[i] \leftarrow \max\left\{\ell \mid \sum_{u \in
N'(v)}[\ell_u[i-1]== \ell] \ge \sum_{u \in N'(v)}[\ell_u[i-1]== \ell']\mbox{ for
all $\ell'$}\right\}$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
By ``w.h.p.'' (with high
probability)
we mean with probability at least $1-\frac{1}{n^c}$ for some constant
$c\geqslant1$.
In this paper we repeatedly use the following versions of a tail
bound on the probability
distribution of a random variable, due to Chernoff and Hoeffding
\cite{chernoff1952measure, hoeffding1963probability}.
Let $X_1, X_2, \ldots, X_m$ be independent and identically distributed binary
random variables.
Let $X = \sum_{i = 1}^m X_i$.
Then, for any $0 \le \epsilon \le 1$ and $c \geqslant 1$,
\begin{align}
\label{eqn:chernoff1}
\Pr\left[X > (1 + \epsilon) \cdot E[X] \right] & \le
\exp\left(-\frac{\epsilon^2 E[X]}{3}\right)\\
\label{eqn:chernoff3}
\Pr\left[X < (1 - \epsilon)\cdot E[X]\right] & \le
\exp \left(-\frac{\epsilon^2 E[X]}{2}\right) \\
\label{eqn:chernoff2}
\Pr\left[|X - E[X]| > \sqrt{ 3c \cdot E[X] \cdot \log n}\right] & \le
\frac{1}{n^c}
\end{align}
\subsection{Results} \label{sub:result}
As mentioned earlier, the purpose of this paper is to counterbalance the predominantly
empirical line of research on LPA and initiate a systematic analysis of \textsc{Max-LPA}.
Our main results can be summarized as follows:
\begin{itemize}
\item As a ``warm-up'' we prove (Section~\ref{sec:path})
that when executed on an $n$-node path \textsc{Max-LPA}
converges to a cycle of period one in $\Theta(\log n)$ rounds w.h.p.
Moreover, we show that w.h.p. the state that \textsc{Max-LPA} converges to
has $\Omega(n)$ communities.
\item %
In our main result (Section~\ref{sec:er}), we define a class of random graphs
that we call \textit{clustered Erd\"{o}s-R\'{e}nyi graphs}.
A clustered Erd\"{o}s-R\'{e}nyi graph $G = (V, E)$ comes with a node partition $\Pi = (V_1, V_2, \ldots, V_k)$
and pairs of nodes in each $V_i$ are connected with probability $p_i$ and pairs
of nodes in distinct parts in $\Pi$ are connected with probability $p' < \min_i \{p_i\}$.
Since $p'$ is small relative to any of the $p_i$'s, one might view a clustered Erd\"{o}s-R\'{e}nyi graph
as having a natural community structure given by $\Pi$.
We prove that even with fairly general restrictions on the $p_i$'s and $p'$ and
on the sizes
of the $V_i$'s, \textsc{Max-LPA} converges to a period-1 cycle in just 2 rounds, w.h.p.
\textit{and} ``correctly'' identifies $\Pi$ as the community structure of $G$.
\item Roughly speaking, the above result requires each $p_i$ to be
$\Omega\left(\left(\frac{\log n}{n}\right)^{1/4}\right)$.
We believe that \textsc{Max-LPA} would correctly and quickly identify $\Pi$
as the community structure of a given clustered Erd\"{o}s-R\'{e}nyi graph even when the
$p_i$'s are much smaller, e.g. even when $p_i = \frac{c \log n}{n}$ for $c > 1$.
However, at this point our analysis techniques do not seem adequate for situations with
smaller $p_i$ values and so we provide empirical evidence (Section~\ref{sec:erp})
for our conjecture that \textsc{Max-LPA} correctly converges to $\Pi$ in
$O(\mbox{polylog}(n))$ rounds even when $p_i = \frac{c \log n}{n}$ for some
$c > 1$ and $p'$ is just a logarithmic factor smaller than $p_i$.
\end{itemize}
\subsection{Related Work}\label{sub:relwork}
There are several variants of LPA presented in the
literature~\cite{cordasco2010community, gregory2009finding,
subelj2011unfolding, liu2009bipartite}. Most of these are concerned about
``quality'' of the output and present
empirical studies of output produced by LPA.
Raghavan et al.~\cite{raghavan2007near}, based on the experiments, claimed that
at least 95\% of the nodes are classified correctly by the end of 5 rounds of label updates. But the
experiments that they carried out were on the small networks.
Cordasco and Gargano~\cite{cordasco2010community} proposed a semi-synchronous
approach which is guaranteed to converge without oscillations and can be parallelized.
They provided a formal proof of convergence but did bound the
running time of the algorithm. Lui and
Murata~\cite{liu2009bipartite} presented a variation of LPA for
bipartite networks which converges but no formal proof is provided,
neither for the convergence nor for the running time.
Leung et al.~\cite{leung2008towards} presented
empirical analysis of quality of output produced by LPA on larger data sets.
From experimental results on a special structured network they claimed that
running time of LPA is $O(\log n)$.
\section{Introduction}
\input{introduction}
\section{Analysis of \textsc{Max-LPA} on a Path} \label{sec:path}
\input{path}
\section{Analysis of \textsc{Max-LPA} on Clustered Erd\"{o}s-R\'{e}nyi Graphs}
\label{sec:er}
\input{clustered_er}
\section{Empirical Results on Sparse Erd\"{o}s-R\'{e}nyi Graphs}
\label{sec:erp}
\input{sparse_er_arxiv}
\section{Future Work}
\label{section:futureWork}
We believe that with some refinements, the analysis technique used to show
$O(\log n)$-rounds convergence of \textsc{Max-LPA} on paths, can be used
to show poly-logarithmic convergence on sparse graphs in general,
e.g., those with degree bounded by a constant.
This is one direction we would like to take our work in.
At this point the techniques used in Section~\ref{sec:er} do not seem applicable to more sparse
clustered Erd\"{o}s-R\'{e}nyi graphs. But if we were willing to allow more time for
\textsc{Max-LPA} to converge, say
$O(\log n)$ rounds, could we significantly
weaken the requirements on the $p_i$'s and $p'$?
Specifically, could we permit an intra-community probability $p_i$ to become as
small as
$c \log n/n$ for some constant $c > 1$?
Similarly, could we permit the inter-community probability $p'$ to come much
closer to the smallest
$p_i$, say within a constant factor?
This is another direction for our research.
\subsubsection*{Acknowledgments.}
We would like to thank James Hegeman for helpful discussions and for
some
insightful comments.
\subsection{Simulation Setup}
We implemented \textsc{Max-LPA} in a C program and executed on a Linux machine
(with 2.4
GHz Intel(R) Core(TM)2 processor). We examined the number of rounds it takes
and
also number of communities it declares at the end of the execution.
We executed \textsc{Max-LPA} on $G(n,p)$ and on $G(\Pi, \pi, p')$ with
$\Pi = (V_1, V_2)$, $|V_1| = |V_2| = n/2$, $\pi = (p, p)$, $p'=0.6/n$ for
various values of $n$ and $p$.
For each $n$, $p$ combination we ran \textsc{Max-LPA} 50 times. We used $p$
values of
the form $\frac{c\cdot \log n}{n}$ for various values of $c\geq 1$.
\subsection{Results}
We executed \textsc{Max-LPA} using the setup discussed above.
Table~\ref{tab:multi} shows the number of simulations out of 50 simulations per
$n$ and $c$ values for which it ended up in
a single community for each pair of $n$ and $c$. If the input graph is
disconnected then obviously there will be multiple communities. Therefore, we
also noted
number of simulations for which the graph was connected
and this number is shown in the brackets.
\begin{table}[!ht]
\centering
\caption{This table shows simulations on Erd\"{o}s-R\'{e}nyi graphs $G(n, p)$ where
$p = \frac{c \log n}{n}$.
Each entry in the table shows the
number of simulations out of 50 simulations per $n$ and $c$ values in
which a single community is
declared by \textsc{Max-LPA} and number of simulations in which
the graph $G(n,p)$ was connected is shown in brackets.\label{tab:multi}}
\begin{tabularx}{\textwidth}{|X|X|X|X|X|}
\hline
$n$ & $c=1$ & $c=1.2$ & $c=1.5$ & $c=1.7$\\ \hline
1000 & 44 (50) & 47 (47) & 50 (50) & 50 (50)\\ \hline
2000 & 42 (46) & 47 (50) & 47 (50) & 50 (50)\\ \hline
4000 & 45 (47) & 49 (50) & 50 (50) & 50 (50) \\ \hline
8000 & 47 (48) & 50 (50) & 50 (50) & 50 (50) \\ \hline
16000 & 49 (50) & 50 (50) & 50 (50) & 50 (50) \\ \hline
32000 & 49 (50) & 50 (50) & 50 (50) & 50 (50) \\ \hline
64000 & 50 (50) & 50 (50) & 50 (50) & 50 (50)\\ \hline
128000 & 50 (50) & 50 (50) & 50 (50) & 50 (50)\\ \hline
\end{tabularx}
\end{table}
\begin{figure}[!ht]
\centering
\input{fig}
\caption{Number of rounds for \textsc{Max-LPA} when executed on sparse
Erd\"{o}s-R\'{e}nyi (averaged over simulations where it ended with a single
community out of 50 simulations per $n$ and $p$). \label{fig:rounds}}
\end{figure}
It is well known that $p=\frac{\log n}{n}$ is a
threshold for connectivity in Erd\"{o}s-R\'{e}nyi graphs and therefore we are getting few runs
for $c=1$ where the input graph was disconnected. From Table~\ref{tab:multi},
we can say that \textsc{Max-LPA} when executed on Erd\"{o}s-R\'{e}nyi graphs
with $p=\frac{c\log n}{n}$ and $c>1$, with high probability, terminate with one
community. It also seem to be the case that as $c$ increases, we are getting
more single community runs. This is because as $c$ increases, the graph become
more dense.
Figure~\ref{fig:rounds} shows a plot of the number of rounds \textsc{Max-LPA}
takes to converge on $G(n, p)$ as $n$ increases,
averaged over all simulations which resulted in a single community at the
end of the execution. The running time seems to grow in a linear fashion with
logarithm of graph size. Also as $c$ increases the running time decreases,
which implies that as the graph becomes more dense \textsc{Max-LPA} converges
more
quickly to a single community. Our results lead us to conjecture that when
\textsc{Max-LPA} is executed on Erd\"{o}s-R\'{e}nyi graphs $G(n,p)$ with $p=O(\frac{\log
n}{n})$ it will, with high probability, terminate with a single community in
$O(\log n)$
rounds.
Table~\ref{tab:clustered} shows the number of simulations out of
50 simulations per
$n$ and $c$ values for which \textsc{Max-LPA} correctly identified the
partition $\Pi$ when executed on $G(\Pi,\pi,p')$ for $p'=\frac{0.6}{n}$. From
previous results in Table~\ref{tab:multi}, for $c=1.5$ \textsc{Max-LPA}
declared a single community when executed on $G(n,p)$ w.h.p. Therefore in this
experiments we started with $c=1.5$. But for $c=1.5$, the influence from
the nodes from other partition is significant. As $c$ increases this influence
is not significant compared to the influence from nodes within the same
partition.
\begin{table}[!ht]
\centering
\caption{This table shows simulations of \textsc{Max-LPA} on $G(\Pi, \pi, p')$
with $\Pi = (V_1, V_2)$, $|V_1| = |V_2| = n/2$, $\pi = (p, p)$, where
$p = \frac{c \log n}{n}$ and $p' = \frac{0.6}{n}$.
Each entry in the table shows, for particular $n$ and $c$ values, the number of
simulations out of 50
in which \textsc{Max-LPA} identified two communities $V_1$ and $V_2$.
The number of simulations in which
graph was connected is shown in brackets.\label{tab:clustered}}
\begin{tabularx}{\textwidth}{|X|X|X|X|}
\hline
$n$ & $c=1.5$ & $c=2$ & $c=4$ \\ \hline
1000 & 22 (45) & 39 (50) & 50 (50) \\ \hline
2000 & 21 (39) & 40 (50) & 50 (50) \\ \hline
4000 & 22 (36) & 47 (50) & 50 (50) \\ \hline
8000 & 14 (38) & 47 (50) & 50 (50) \\ \hline
16000 & 26 (35) & 49 (49) & 50 (50) \\ \hline
32000 & 17 (33) & 49 (49) & 50 (50) \\ \hline
64000 & 26 (34) & 46 (50) & 50 (50) \\ \hline
128000 & 5 (35) & 47 (47) & 50 (50) \\ \hline
\end{tabularx}
\end{table}
|
{
"timestamp": "2012-10-16T02:02:14",
"yymm": "1210",
"arxiv_id": "1210.3735",
"language": "en",
"url": "https://arxiv.org/abs/1210.3735"
}
|
\section{Introduction}
Young dense star clusters observed in the Milky Way and the Large
Magellanic Cloud (LMC), e.g.,
R136 \citep{1998ApJ...493..180M,2010MNRAS.408..731C},
NGC 3603 \citep{2006AJ....132..253S,2008ApJ...675.1319H},
Westerlund 1
\citep{2005A&A...434..949C,2008A&A...478..137B,2011MNRAS.412.2469G}
and 2 \citep{2007A&A...466..137A,2007A&A...463..981R}, are
good samples for understanding the formation mechanism of dense
star clusters. They are massive ($\sim 10^5M_{\odot}$) and dense
($>10^4M_{\odot}{\rm pc}^{-3}$), and seem to be approaching
(or might have experienced) core collapse although they are
young ($<4$Myr) \citep{2003MNRAS.338...85M}. For example, in R136 in
the LMC, its high core density
($>5\times10^4 M_{\odot}{\rm pc}^{-3}$) \citep{2003MNRAS.338...85M}
and the existence of high-velocity stars (runaway stars) escaping
from the cluster \citep{2007ASPC..367..629B,2010ApJ...715L..74E,
2011A&A...530L..14B,2011MNRAS.410..304G} suggest that it
experienced core collapse \citep{2011Sci...334.1380F}.
If such a young massive cluster experiences core collapse,
repeated collisions (so-called runaway collisions) of stars, and
as a consequence the formation of very massive stars ($>100M_{\odot}$),
are expected \citep{1999A&A...348..117P,2002ApJ...576..899P}.
Such very massive stars form through multiple stellar collisions
could result in the formation of intermediate-mass black holes (IMBHs)
\citep{2001ApJ...562L..19E}.
The formation of IMBHs in dense star clusters via
multiple collisions has been studied
using $N$-body simulations \citep{1999A&A...348..117P,
2004Natur.428..724P,2004ApJ...604..632G,2006MNRAS.368..141F}, and
the results suggest that IMBHs with $10^2-10^3M_{\odot}$
could be formed in such dense clusters.
Including stellar evolution, however, a high
mass-loss rate due to the stellar wind of massive stars prevents the
growth of the massive stars \citep{2007ApJ...659.1576B,2009A&A...497..255G}.
A very high collision rate is required for such very massive stars
to overcome the copious mass-loss and nevertheless leads to the
formation of an IMBH \citep{2008ApJ...686.1082F}.
There are some mechanisms to enhance the growth rate of the
very massive stars, but the most important factor is the moment of
core collapse, $t_{\rm cc}$. This short but high density phase is
necessary for the cluster to become collisionally dominated, which is
critical for the collision rate of stars in the cluster.
Earlier collapse times assist an efficient mass
accumulation because stars can start multiple collisions before
the cluster starts to lose massive stars via stellar evolution.
The core-collapse time is determined by the relaxation time of the
cluster if the cluster is initially virialized, and it is roughly
20\% of the half-mass relaxation time, $t_{\rm rh}$, with a
Salpeter-type power-low mass function
\citep{2002ApJ...576..899P,2003gmbp.book.....H}. For most
massive clusters, therefore, it is difficult to reach
core collapse before the end of main-sequence lifetime of the
most massive stars, which is $\sim 3$ Myr for $> 40M_{\odot}$.
One way to achieve an early core collapse is kinematically cool initial
conditions. A sub-virial cluster
can reach core collapse faster than initially virialized clusters
\citep{2009ApJ...700L..99A}, and that enables an efficient
mass-growth via multiple collisions of stars.
Another way to enhance the stellar-mass growth is by adopting mass
segregated initial conditions.
A mass function causes massive stars to sink to the cluster center
and as a result massive stars pile up in the cluster core.
The mass-growth due to stellar collisions can be quite efficient,
if massive stars concentrate in the core.
Initial mass segregation enhances the growth-rate of the colliding
stars \citep{2008ApJ...682.1195A,2012ApJ...752...43G}, and
cool initial conditions result in a high degree of the mass
segregation in a short time \citep{2009ApJ...700L..99A}.
Another way for star clusters to reduce their core collapse time
is by the assemblage of sub-clusters \citep{1972A&A....21..255A,
2007ApJ...655L..45M,2009MNRAS.400..657M,2011ApJ...732...16Y,
2011MNRAS.416..383S,2012ApJ...753...85F}.
The short relaxation time of sub-clusters compared to initially
more massive clusters causes early mass
segregation and core collapse. Since the memory of such early
dynamical evolution is conserved in the merger remnant
\citep[][hereafter Paper 1]{2007ApJ...655L..45M,2012ApJ...753...85F},
the formation of star clusters by assembling them is also an
effective way for efficient multiple collisions
of stars in young star clusters.
In paper 1, we found that the formation scenario of young dense
star clusters via mergers of ensemble clusters can
successfully explain the mature characteristics of young massive
star clusters such as R136 in 30 Dor region. The age of R136
is only 2--3 Myr, but it shows dynamically mature characteristics,
such as mass segregation, a high core density, and a wealth of high
velocity escaping stars \citep{2003MNRAS.338...85M,2007ASPC..367..629B,
2010ApJ...715L..74E,2011A&A...530L..14B,2011MNRAS.410..304G}.
However, the relaxation time of R136 obtained from its current mass
and radius is $\sim 100$ Myr
\citep{2003MNRAS.338...85M}, which is too long to have
reached core collapse at its current age. In paper 1,
we performed a series of $N$-body simulations of ensemble clusters
and demonstrated that ``ensemble''-cluster models can reproduce
observations such as the core density, the fraction of
high-velocity escapers, and the distribution of massive stars which
experienced collisions, but ``solo''-cluster models, which are
initially spherical and virialized, fail to reproduce these observations.
Furthermore, these
characteristics of the ensemble models are also consistent with the
characteristics of other massive young clusters like
R136 in the LMC and
NGC 3603 in the Milky Way \citep{2010MNRAS.408..731C}.
If young dense clusters formed via assembling sub-clusters
and have experienced core collapse, it is expected that repeating
collisions can lead to the formation of very massive stars and
possibly even IMBHs. In the observed young dense
clusters, however, there is no evidence of IMBHs, but
some very massive stars
with an initial mass of 100--300 $M_{\odot}$ are observed
\citep{2006AJ....132..253S,
2011A&A...530L..14B, 2009Ap&SS.324..321M,
2011MNRAS.412.2469G,2011MNRAS.416..501R}.
In this paper, we perform a series of $N$-body simulations of
solo and ensemble star clusters and demonstrate that the
growth of very massive stars through multiple collisions is
mediated by star cluster complexes. Our simulations show that
the quick dynamical evolution of
ensemble clusters does not always result in the formation of
extremely-high-mass stars.
When the assembling of clusters proceeds after each sub-cluster
experiences core collapse (``late-assembling'' case),
multiple-collision stars that form in each sub-cluster fail
to coalesce to an extremely massive star,
but leads to the formation of several very massive stars.
Some of these very massive stars can escape from the cluster as
high-velocity stars due to the three-body or binary-binary
encounters. When the sub-clusters assemble before they experience
core-collapse (``early-assembling'' case), the collision rate
is enhanced and the assembled cluster forms an extremely massive star
of $\sim 1000 M_{\odot}$.
\section{Method and Initial Conditions}
We performed a series of $N$-body simulations of solo clusters
and ensemble clusters, that merge to a single cluster with a mass
equal to solo clusters. For the ensemble of sub-clusters, we
adopted two models. A:
a King model \citep{1966AJ.....71...64K} with
a dimensionless concentration parameter, $W_0$, of 2 and the
total mass $M_{\rm cl}=6300M_{\odot}$, and B: a King model
with $W_0=5$ and $M_{\rm cl}=2.5\times 10^4 M_{\odot}$.
The half-mass radii, $r_{\rm h}$, of these models are 0.092 and
0.22 pc, and the numbers of particles, $N$, are 2048 (2k) and
8192 (8k), respectively.
The core density is the same for both models
($\rho_{\rm c} \simeq 2 \times 10^6 M_{\odot}\,\mathrm{pc}^{-3}$).
We assumed a Salpeter initial mass function (IMF)
\citep{1955ApJ...121..161S} between 1 and 100
$M_{\odot}$. We call these models 2kw2 and 8kw5.
We distribute 4 or 8 of these sub-clusters in two different
initial configurations: spherical or
filamentary. The former model stems from clumpy star
formation in giant molecular clouds, and the latter is motivated
by star formation in a filamentary gas distribution or shocked region
of colliding gas in the spiral arms of a galactic disk.
The clumpy star formation is initiated by observations of Westerlund 1
\citep{2011MNRAS.412.2469G} and R136 \citep{2012ApJ...754L..37S} and
simulations \citep{2011MNRAS.410.2339B,2011IAUS..270..483S}.
For the spherical models, we adopted 4 or 8 of models 2kw2 as
sub-clusters, and distributed them randomly in a volume with a radius
of $r_{\rm max}$ and with zero velocity.
We varied $r_{\rm max}$ between 1 and 6 pc.
For the filamentary models, we initialized 8 individual 8kw5
model sub-clusters.
We initialized these sub-clusters with
two different initial mean separations (models e8k8f1 and e8k8f2),
but with zero velocity.
The initial positions of the sub-clusters for these models are
illustrated in Figure \ref{fig:init_pos}.
All runs are summarized in table \ref{tb:model}.
For the solo models, we adopted two more initial conditions with
$M_{\rm cl}$ of $5.1 \times 10^4 M_{\odot}$ and $2.0 \times 10^5 M_{\odot}$.
With the same mass function, these models have
16384 (16k) and 65536 (64k) stars and are initialized
using King models with $W_0=6$ and 8, respectively.
In order to obtain the same core density as that of sub-clusters,
their half-mass radii are 0.32 and 1.0 pc.
We call these models as 16kw6 and 64kw8. In Table \ref{tb:model_cl}
we summarize the initial conditions, and we present
their initial density profiles in Figure \ref{fig:cd}.
We performed additional simulations of sub-virial (cold)
initial conditions for 16kw6, and an extra set of simulations
in which we reduced the kinetic energy
(velocity of each particle) to two-thirds and 10\% of the
virialized velocity.
We call these models as s16k-cool and s16k-cold, respectively
(see table \ref{tb:model}).
The $N$-body simulations are performed using the sixth-order Hermite
scheme with individual timesteps with an accuracy parameter
$\eta=$0.15--0.3 \citep{2008NewA...13..498N}.
We adopted the accuracy parameter to balance speed and accuracy,
and the energy error was $< 0.1$ \% for all runs.
Our code does not include special treatment for binaries, but
the sixth-order Hermite scheme can handle hard binaries formed
in our simulations (see section 2 in Paper 1).
We took into account collisions of stars with a sticky-sphere
approach and mass loss due to the stellar wind for stars with
$>100M_{\odot}$ with a rate of
$5.0\times 10^{-7} M_{\odot}$yr$^{-1}$ \citep{2009ApJ...695.1421F}.
We neglected the mass-loss from stars with $<100M_{\odot}$ because
it does not affect the results on the short timescale of
our simulations ($<5$Myr).
The stellar radii are taken from the zero-age main-sequence
for solar metallicity \citep{2000MNRAS.315..543H}.
\begin{table*}
\begin{center}
\caption{Models of single clusters\label{tb:model_cl}}
\begin{tabular}{cccccccccc}\hline \hline
Model & $N$ & $M_{\rm cl}$ & $W_0$ & $r_{\rm h}$ & $\rho_{\rm c}$& $\sigma$ & $t_{\rm rh}$ & $t_{\rm rc}$ &$M_{\rm core}/M_{\rm cl}$ \\
& & ($M_{\odot}$) & & (pc) & ($M_{\odot}$pc$^{-3}$) & (km/s) & (Myr) & (Myr) \\\hline
2kw2 & 2048 & $6.3\times 10^3$ & 2 & 0.097 & $1.7\times 10^6$ & 11 & 0.30 & 0.58 & 0.28 \\
8kw5 & 8192 & $2.5\times 10^4$ & 5 & 0.22 & $1.7\times 10^6$ & 15 & 1.9 & 0.92 & 0.15 \\
16kw6 & 16384 & $5.1\times 10^4$ & 6 & 0.32 & $1.7\times 10^6$ & 17 & 4.4 & 1.1 & 0.12\\
64kw8 & 65536 & $2.0\times 10^5$ & 8 & 1.0 & $1.6\times 10^6$ & 19 & 44 & 1.8 & 0.053 \\ \hline
\end{tabular}
\medskip
\\
$\sigma$ is the velocity dispersion.
\end{center}
\end{table*}
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f1.eps}
\caption{Initial density profiles of single clusters\label{fig:density_init}}
\end{center}
\end{figure}
\begin{table*}
\begin{center}
\caption{Runs\label{tb:model}}
\begin{tabular}{cccccc}\hline
Model & $N_{\rm cl}$ & geometry & $\langle d_{\rm min}\rangle$ (pc) & (sub-)cluster & $N_{\rm run}$ \\ \hline\hline
e2k4r3 & 4 & spherical & 2.5 & 2k2w & 3\\
e2k4r6 & 4 & spherical & 5.1 & 2k2w & 1\\ \hline
e2k8r1 & 8 & spherical & 0.51 & 2k2w & 2\\
e2k8r3 & 8 & spherical & 1.3 & 2k2w & 1\\
e2k8r5 & 8 & spherical & 2.8 & 2k2w & 2\\
e2k8r6 & 8 & spherical & 3.3 & 2k2w & 2\\ \hline
e8k8f1 & 8 & filamentary & 2.8 & 8kw5 & 1\\
e8k8f2 & 8 & filamentary & 4.2 & 8kw5 & 1\\ \hline \hline
s2k & 1 & - & - & 2kw3 & 7 \\
s8k & 1 & - & - & 8kw5 & 6\\
s16k & 1 & - & - & 16kw6 & 6\\
s64k & 1 & - & - & 64kw8 & 2\\ \hline
s16k-cool & 1 & - & - & 16kw6 & 2\\
s16k-cold & 1 & - & - & 16kw6 & 1\\ \hline
\end{tabular}
\medskip
\\
The models are named according to the following rules; ``e'' and ``s'' indicate ensemble and solo models, respectively. For ensemble models, following numbers indicate the number of particles of sub-clusters and the number of sub-clusters. The last part indicates the initial configuration of sub-clusters; ``r'' and the following number mean spherical and the value of the maximum radius, $r_{\rm max}$, ``f'' indicates filamentary initial configurations (see figure \ref{fig:init_pos} for the initial positions of sub-clusters in these models). For solo models, the number indicates the number of particles.
$\langle d_{\rm min}\rangle$ is the averaged distance to the nearest-neighbour sub-clusters, and $N_{\rm run}$ is the number of runs.
s16k-cool and s16k-cold are the same model as s16k, but
the velocity of 67\% and 10\% of s16k, respectively.
\end{center}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics[width=70mm]{f2a.eps}
\includegraphics[width=70mm]{f2b.eps}
\caption{Initial position of ensemble models, e8k8f1 (right) and e8k8f2 (left). We mimicked filamentary star forming regions. \label{fig:init_pos}}
\end{center}
\end{figure*}
\section{Solo-cluster models}
\subsection{Virialized solo-cluster models}
We describe the results of the initially virialized
solo-cluster models, which we will refer as the ``standard'' model.
In Figure \ref{fig:cd} we present the time evolution of the core
density for models s2k, s8k, s16k, and s64k.
The core densities are calculated using the method of
\citet{1985ApJ...298...80C}.
We identify the moment when the cluster reaches the highest
core density as the core-collapse time.
The core-collapse time measured from the simulations is
$t_{\rm cc} = 0.29\pm 0.07$, $0.71\pm 0.11$, $1.2\pm 0.13$, and
$1.8\pm 0.0$ Myr for models s2k, s8k, s16k, and s64k, respectively
(see also Table \ref{tb:results}). The core-collapse time is
consistent with those obtained previous simulations
\citep{2004ApJ...604..632G}, if we take into account the differences
in the mass range of the mass function.
\citet{2004ApJ...604..632G} showed that the core-collapse
time scales with the central relaxation time \citep{2003gmbp.book.....H}:
\begin{eqnarray}
t_{\rm rc} = \frac{0.065 \sigma_{\rm c, 3D} ^3}{G^2 \langle m\rangle \rho_{\rm c} \ln \Lambda}.
\label{eq:tcrlx}
\end{eqnarray}
Here $G$, $\langle m \rangle$, $\sigma _{\rm c}$, and, $\rho_{\rm c}$ are the
gravitational constant, the mean mass of stars, and the central velocity
dispersion and density, respectively.
Here $\ln \Lambda$ is the Coulomb logarithm, and $\Lambda$ is written as a
function
of the number of particles as $\gamma N$ and $\gamma \simeq 0.1$ for
$t_{\rm rh}$ of star clusters \citep{1994MNRAS.268..257G}. We adopted
the number of particles in the core as $N$.
We find that the core-collapse time scales better using our definition
than when we adopt $0.01N$, which is adopted by \citet{2004ApJ...604..632G}.
In our simulations, $t_{\rm cc}/t_{\rm rc}\simeq 1$ for models s8k, s16k,
and s64k, but $t_{\rm cc}/t_{\rm rc}\simeq 0.5$ for model s2k.
For model s2k, however, $t_{\rm rh}$ is shorter than $t_{\rm rc}$ because
the core radius exceeds the half-mass radius. If we adopt
a shorter relaxation time, then $t_{\rm cc}/t_{\rm rh}\sim 1$ for all the
models.
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f3.eps}
\caption{Time evolution of the core densities for solo clusters. The results
are averaged in order to reduce the run-to-run
variations.\label{fig:cd}}
\end{center}
\end{figure}
The core collapse of the cluster initiates a collision runaway
in the cluster core \citep{1999A&A...348..117P}. In Figure
\ref{fig:m_his_single} we present
the merger histories of the multiple-collision stars in the solo-cluster
simulations s2k, s8k, s16k, and s64k. In each model, one primary
collision product
(PCP) per cluster grows through repeated collisions of stars.
In model s2k the mass-loss due to the stellar wind exceeds the
mass-gain by the collisions, and therefore the PCP has lost
all gained mass by the end of the simulation (5Myr).
PCPs grow up to the maximum mass $m_{\rm max}\sim 400 M_{\odot}$ via
repeating collisions, but by the time it explodes the star is
$\sim 100M_{\odot}$. Here we define $m_{\rm max}$ as the
maximum mass of a star reached during its lifetime as a result of
collisions.
PCPs are not the only stars that experienced collisions. In models
s8k, s16k, and s64k, we find secondary collision products (SCPs).
In most cases SCPs experience only one collision
(sometimes a few collisions), but never
grow as massive as PCPs, although SCPs sometimes exceed our adopted
upper-limit to the IMF ($100M_{\odot}$). The SCPs end up merging with PCPs
(see bottom right panel in Figure \ref{fig:m_his_single})
or just lose their mass by stellar evolution (see top right panel
in Figure \ref{fig:m_his_single}). This result agrees with previous
numerical simulations \citep{2006MNRAS.368..141F}.
We also find that the time when the PCPs reach their maximum mass
$m_{\rm max}$,
$t_{\rm max}$, is scaled by $t_{\rm rc}$, and that
$t_{\rm max}/t_{\rm rc}=$ 2.3, 2.2, 2.2, and 2.6 for models
s2k, s8k, s16k, and s64k, respectively.
\begin{figure*}
\begin{center}
\includegraphics[width=70mm]{f4a.eps}
\includegraphics[width=70mm]{f4b.eps}
\includegraphics[width=70mm]{f4c.eps}
\includegraphics[width=70mm]{f4d.eps}
\caption{Mass evolution of PCPs (solid curve) and SCPs (dashed curve) for models s2k, s8k, s16k, and 64k. Dotted line indicates the core-collapse time. Cross indicates the time when the star merged with more massive ones.\label{fig:m_his_single}}
\end{center}
\end{figure*}
\subsection{Cold solo-cluster models}
Sub-virial (cold) initial conditions reach core collapse considerably
earlier than virialized ones. Cold models have therefore been
suggested to explain the dynamically advanced appearance of observed
young star clusters \citep{2009ApJ...700L..99A}.
In Figure \ref{fig:cd_cold} we present the core-density evolution of
models s16k-cool and s16k-cold, which initially have 67\% and 10\% of
the virialized temperature. These models reach core collapse much
earlier than virialized models, and as a consequence multiple collisions
start earlier and proceed at a higher collision rate.
In figure \ref{fig:m_his_cold} we present the mass evolution
of the PCPs for models s16k-cold and s16k-cool.
Colder initial conditions result in a higher $m_{\rm max}$ of the
PCPs. The high $m_{\rm max}$ is a result of the high collision
rate, which is caused by the high density in the core
(see Figure \ref{fig:cd_cold}).
By the time the PCPs leave the main sequence (of $\sim 3$Myr),
their masses have been reduced considerably due to stellar mass-loss,
which competes with the mass gain by collisions.
In model s16k-cold, the PCP grows quickly in the beginning
of the simulation, but after 0.5 Myr the mass-loss rate due to the
stellar wind becomes higher than the mass-growth by stellar collisions.
In model s16k-cool, the PCP stops growing at $\sim$0.5 Myr because the
mass-growth rate balances to the mass-loss rate,
and then the PCP maintains its mass until the end of the simulation (3.5 Myr).
Although the final masses of the PCPs are comparable in both s16k-cool and
s16k-cold models, $m_{\rm max}$ of model s16k-cold is twice as massive as
that of model s16k-cool.
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f5.eps}
\caption{Time evolution of the core density, $\rho_{\rm c}$, for models s16k, s16k-cool, and s16k-cold.\label{fig:cd_cold}}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=84mm]{f6.eps}
\caption{Merger history of PCPs for models s16k, s16k-cool, and s16k-cold.\label{fig:m_his_cold}}
\end{center}
\end{figure}
In Figure \ref{fig:m_max_single} we show $m_{\rm max}$ of the PCPs
for all solo models.
The maximum mass of the PCPs in models s8k, s16k, and s64k is quite
similar ($\sim 400 M_{\odot}$) irrespective of $M_{\rm cl}$.
One might expect that more massive clusters contain a larger number of
massive stars and therefore a more massive cluster can form a more massive
PCP. In our simulation, however, the
number of stars which merged into the PCPs and the mean mass of the
merged stars are quite similar among these models
(see Table \ref{tb:results}).
By comparing models s8k, s16k, and s64k, their $\langle m_{\rm col} \rangle$
and $N_{\rm col}$ are quite similar even though their total cluster masses are
different. If the collisions selectively occur among the most
massive stars and the numbers of collisions are the same, larger clusters
should have a larger mean collision mass $\langle m_{\rm col} \rangle$ because
larger clusters
contain more massive stars. However, the number of massive stars does not
simply follow this relation. In figure \ref{fig:nr_m50} we plot the
cumulative number distribution of massive stars with $m>50M_{\odot}$ at
the moment in which the mass of the PCP reaches $m_{\rm max}$.
The number of stars with $>50M_{\odot}$ within $\sim$0.05 pc are similar
($\sim 20$) among models s8k, s16k, and s64k and slightly smaller for
model s2k. In particular for the models s16k and s64k,
the distribution of massive stars preserves the initial distribution
in the outer part of the cluster because the half-mass relaxation time
exceeds $t_{\rm max}$. The dynamical evolution in these models is driven
on a timescale of $t_{\rm rc}$, and they have similar
core properties: $t_{\rm rc}$, $M_{\rm core}$, and $\rho _{\rm c}$
(see Table \ref{tb:model_cl} and Figure \ref{fig:cd}).
In model s2k, on the other hand, $t_{\rm rc}\sim t_{\rm rh}$ and as a consequence
the dynamical evolution proceeds throughout the
entire cluster. Model s8k shows an evolution similar to that of
model s2k: $m_{\rm max}/M_{\rm cl}$ for model s8k is as high as
that of model s2k. For these models $t_{\rm rh}\sim 2$ Myr, which is
sufficiently short for massive stars in the outer part of the cluster
to join the collisions in the core.
Similar to the model s2k, models s16k-cool and s16k-cold can
also gather massive stars from the entire cluster to the cluster center
irrespective of their initial positions. In addition,
these sub-virial models achieve very
high density (see Figure \ref{fig:cd_cold}), which enhances
the collision rate. The massive stars in model s16k-cold are more
concentrated towards the cluster center compared with model s16k
(see Figure \ref{fig:nr_m50}).
Even though for model s2k the PCP can accumulate stars from
the entire cluster population of massive stars, their total number
and mass still cannot compete with the population of massive stars
in the more massive clusters.
In these latter models, the maximum mass of the PCP is limited
by the reservoir of massive stars, which manages to segregate to the
core by the moment of the core collapse. A larger cluster mass
therefore does not automatically lead to a massive PCP.
As seen in Figures \ref{fig:m_his_single} and \ref{fig:m_his_cold},
the mass evolution of the PCPs in models s2k, s8k, and s16k-cold
show a clear peak in the middle of the simulation. In the later phase,
when the collision rate decays,
their mass-loss rate exceeds their mass-growth rate by stellar
collisions.
In models s16k and s64k, on the other hand, they have not exhausted
their reservoir of massive stars because their half-mass relaxation
time is not shorter than the simulation time and therefore some of
the massive stars still remain in the outer part of the clusters.
We empirically obtained a relation that $m_{\rm max} = 0.02M_{\rm cl}$
(dotted line in Figure \ref{fig:m_max_single}) for the low
cluster-mass models ($M_{\rm cl} < 2\times 10^4M_{\odot}$) and
the cold model.
For massive clusters, however, $m_{\rm max}$ is smaller than that
according to this
relation. For the most massive cluster ($M_{\rm cl} = 2\times 10^5M_{\odot}$),
$m_{\rm max}$ is consistent with the result presented by
\citet{2002ApJ...576..899P}, $m_{\rm max} = 0.002M_{\rm cl}$.
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f7.eps}
\caption{The maximum mass of PCPs for solo models. Filled circles with error bars indicate models s2k, s8k, s16k, and s64k from left to right. Cross and plus indicate s16k-cool and s16k-cold, respectively. Since the error bar for model s16k-cool is smaller than the marker size, we do not plot the error bar. Dotted and dashed lines indicate $m_{\rm max} = 0.02M_{\rm cl}$ and $m_{\rm max} = 0.002M_{\rm cl}$, respectively. \label{fig:m_max_single}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f8.eps}
\caption{Cumulative number distribution of stars with $m>50M_{\odot}$ at $t_{\rm max}$ for models s2k, s8k, s16k, s64k, and s16k-cold. \label{fig:nr_m50}}
\end{center}
\end{figure}
\begin{table*}
\begin{center}
\caption{Summary of the results.\label{tb:results}}
\begin{tabular}{ccccccccc}\hline
Model & $m_{\rm max} (M_{\odot})$ & $t_{\rm max}$ (Myr)& $t_{\rm merge}$ (Myr)& $t_{\rm cc}$ (Myr) & $m_{\rm SCPs} (M_{\odot})$ & $\langle m_{\rm col}\rangle (M_{\odot})$ & $N_{\rm col}$ \\ \hline\hline
e2k4r3-1 & 287 & 0.45 & 0.2--0.87 & $0.29 \pm 0.07$ & 375 & 78.3 & 8 \\
e2k4r3-2 & 260 & 0.78 & 0.03--1.2 & & 454 & 60.6 & 15 \\
e2k4r3-3 & 268 & 0.61 & 0.6--1.3 & & 139 & 56.7 & 12 \\
e2k4r6 & 238 & 0.57 & 2.2--2.7 & & 743 & 46.8 & 16 \\
\hline
e2k8r1-1 & 998 & 0.80 & 0.03--0.38 & $0.29 \pm 0.07$ & 0 & 44.2 & 45 \\
e2k8r1-2 & 667 & 1.35 & 0.03--0.32 & & 160 & 69.4 & 22 \\
e2k8r3 & 530 & 0.86 & 1.0 --0.75 & & 147 & 80.2 & 14 \\
e2k8r5-1 & 334 & 1.11 & 0.47--1.9 & & 1192 & 57.0 & 25 \\
e2k8r5-2 & 486 & 0.78 & 0.03--2.0 & & 651 & 61.2 & 19 \\
e2k8r6-1 & 245 & 0.59 & 0.77--2.4 & & 1367 & 51.0 & 24 \\
e2k8r6-2 & 274 & 0.42 & 0.03--$>3$ & & 970 & 45.5 & 21 \\
\hline
e8k8f1 & 1310 & 1.40 & 0.4--1.2 & $0.71\pm 0.11$ & 268 & 73.0 & 42 \\
e8k8f2 & 659 & 2.28 & 0.8--2.0 & & 995 & 88.4 & 33\\
\hline\hline
s2k & $182 \pm 21$ & $1.3 \pm 0.6$ & - & $0.29 \pm 0.07$ & $16 \pm 40$ &53.3 & 4.6 \\
\hline
s8k & $399\pm 60$ & $2.2\pm 0.2$ & - & $0.71\pm 0.11$ & $149 \pm 115$ & 63.2 & 11.3 \\
\hline
s16k & $431\pm 54$ & $2.6\pm 0.9$ & - & $1.2\pm 0.13$ & $54 \pm 77$ & 65.8 & 13.2\\
\hline
s64k & $488\pm 57$ & $4.4\pm 0.2$ &- & $1.8 \pm 0.0$ & 0 & 66.0 & 15.5 \\
\hline
s16k-cold & 1064 & 0.59 & - & $<0.02$ & 0 & 46.6 & 40\\
s16k-cool & $707\pm 36$ & $2.55\pm 0.75$ & - & $0.325\pm0.075$ & 0 & 51.1 & 28.5 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Ensemble-cluster models}
In section 3 we demonstrated that the results obtained from
our solo-cluster models are consistent with previous numerical studies.
In this section
we present the results of ensemble-cluster models, in which
sub-clusters assemble to finally form one single cluster.
In ensemble-cluster models, sub-clusters collapse on a timescale
shorter than that for solo-clusters with the same total mass.
Their further
evolution is dominated by the dynamical evolution of the sub-clusters
before they merge. The conservation of the dynamical states through
the mergers \citep{2009Ap&SS.324..277V} drives the further evolution
of the cluster merger products. As a result, ensemble
clusters tend to experience core collapse considerably faster than
solo clusters which have initially similar properties to those of the
merger remnant of ensemble clusters.
In paper 1 we already showed that the quicker dynamical evolution of
ensemble clusters can explain the mature characteristics of
young dense clusters such as R136 and NGC 3603. Here we use that
enhanced dynamical evolution to study the PCPs.
The early dynamical evolution of ensemble clusters is similar
to that of cold solo-clusters. One might expect that ensemble
clusters also result in the formation of massive PCPs, but
we will show that the early evolution of
ensemble clusters is somehow more complicated.
In Figure \ref{fig:merger}
we illustrate the schematic evolution of two typical evolutionary
paths of ensemble clusters. We find that
the most important parameter for the evolution of ensemble clusters
is the moment of assembling, $t_{\rm ens}$, compared to $t_{\rm cc}$ of
sub-clusters.
If $t_{\rm cc}>t_{\rm ens}$ (``early assembling''), the PCPs in the
remnant cluster grow efficiently by stellar collisions because
the short relaxation time of the sub-clusters drives
mass-segregation and core collapse faster than solo clusters.
This evolution is similar to that of cold solo-clusters.
If $t_{\rm cc}<t_{\rm ens}$ (``late assembling''), each
sub-cluster experiences core collapse before they assemble and
form a PCP per individual sub-cluster. The mass of each PCP is limited
by the sub-cluster mass as we described in section 3.
After the assembling of two or more sub-clusters,
the PCPs formed in the sub-clusters sink to the center of the remnant
cluster and interact each other. Most of them, however, are
scattered and ejected from the cluster because they tend to reside in
hard binaries with a massive companion.
The PCPs tend to be in the hardest binaries with the
most massive stars when they formed in the
sub-clusters.
In each binary-binary encounter following a sub-cluster merger,
two PCPs may collide although they are also ejected without experiencing
a collision.
Therefore, the majority of the PCP-binaries are scattered or ionized,
and only one PCP-binary survives in the remnant cluster by the time the
assembly is completed. The surviving PCP cannot continue to grow in
mass because by that time the central density of the assembled clusters
has been depleted due to the early dynamical evolution.
\begin{figure*}
\begin{center}
\includegraphics[width=140mm]{f9.eps}
\caption{Schematics picture of two typical assembling processes. Early assembling ($t_{\rm cc}>t_{\rm ens}$): Sub-clusters assemble before they experience core collapse. The merger remnant is more mass-segregated than solo clusters which initially have similar properties to the merger remnant because sub-clusters have a shorter relaxation time than the solo cluster. After their assembling, the remnant cluster collapses and a massive PCP forms. Late assembling ($t_{\rm cc}<t_{\rm ens}$): sub-clusters experience core collapse and form small PCPs before they assemble. After their assembling, however, the PCPs do not grow efficiently because most of them are scattered from the remnant cluster by binary-binary encounters.
\label{fig:merger}}
\end{center}
\end{figure*}
\subsection{Stellar collisions in ensemble clusters\label{sc_ensamble}}
In Figures \ref{fig:m_his_2k_8} and \ref{fig:m_his_8k_8}, we present
the mass evolution of PCPs and SCPs in ensemble clusters. The left and
right panels show early and late assembling models, respectively.
In early assembling models, one massive PCP per remnant cluster grows
after the assembling of sub-clusters. Even though some of the sub-clusters
start forming PCPs before assembling, the PCPs merge
after the host sub-clusters merged.
In late assembling models, on the other
hand, each sub-cluster grows its own PCP, but most of them do not collide
with each other even after the assembling of their host sub-clusters.
\begin{figure*}
\begin{center}
\includegraphics[width=70mm]{f10a.eps}
\includegraphics[width=70mm]{f10b.eps}
\includegraphics[width=70mm]{f10c.eps}
\includegraphics[width=70mm]{f10d.eps}
\caption{Top: Time evolution of the separation between sub-clusters projected onto $x$-axis (full curves) and the collisions of PCPs (black dots) for models e2k8r1 (left) and e2k8r6-1 (right). The positions of the dots show the collision time and the the sub-cluster to which the star initially belongs.
Bottom: Mass evolution of PCPs and SCPs for models e2k8r1 (left) and e2k8r6-1 (right). Crosses indicate the time when the SCPs merged to PCPs. Arrows indicate the time when sub-clusters merged.
In all panels, the shaded region indicates the core-collapse time with error obtained from the simulations of isolated sub-clusters. \label{fig:m_his_2k_8}}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=70mm]{f11a.eps}
\includegraphics[width=70mm]{f11b.eps}
\includegraphics[width=70mm]{f11c.eps}
\includegraphics[width=70mm]{f11d.eps}
\caption{Same as Figure \ref{fig:m_his_2k_8} but for models e8k8f1 (left) and e8k8f2 (right). \label{fig:m_his_8k_8}}
\end{center}
\end{figure*}
We find the reason for the difference between early and late assembling
cases in the density evolution of these clusters.
In figure \ref{fig:n_dens_merger} we show the time evolution of the
maximum number densities for ensemble and solo clusters. Here we
plot the maximum value of the local density, which is calculated
using six nearest neighbours. (Note that the maximum local density
does not trace the density of one individual sub-cluster.)
In early assembling cases, the density increases on
the core-collapse
timescale of the solo sub-cluster (model s2k), but the maximum
density is higher than that of model s2k and rather comparable to
those of the cold models (models s16k-cold, s16k-cool). The evolution
after the core collapse is similar to that of the
cold models. The density gradually decreases and eventually becomes
comparable to that of virialized solo-clusters (model s16k).
The density in late assembling cases also grows on the core-collapse
timescale of the sub-clusters until a peak is reached at $\sim 0.5$ Myr.
The density decreases as quickly as that of the solo sub-clusters
(model s2k), which is different from early assembling cases. By the
end of the simulations, the number density of the late assembling cases
is an order of magnitude lower than in the early assembling
cases. The relatively low density prevents the growth of PCPs
in the late assembled clusters. The effect of the difference in the
density can be seen in the number
of stellar collisions, $N_{\rm col}$ in Table \ref{tb:results}.
In early assembling models (e2k8r1-1 and e8k8f1) and the cold solo model
(s16k-cold), $N_{\rm col}= 42\pm 2$ and $m_{\rm max}= 1100\pm 130$, but
in late assembling models (e2k8r5, e2k8r6, and e8k8f2) $N_{\rm col}=24
\pm 5$ and $m_{\rm max}= 400\pm 150$.
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f12.eps}
\caption{Time evolution of maximum local number density (local densities of six nearest neighbours) for models s16k, s2k, s16k-cool, s16k-cold, e2k8r1 (early assembling), and e2k8r6 (late assembling). \label{fig:n_dens_merger}}
\end{center}
\end{figure}
In late assembling models (for e2k8r5 and e2k8r6), the maximum mass
of the PCPs is 200--400 $M_{\odot}$, but the mass of the PCP is similar
to those of multiple SCPs, which were PCPs in the sub-clusters.
This feature is consistent with young dense
clusters such as R136 in the LMC, which contains
five $>100M_{\odot}$ mass stars
\citep{2010MNRAS.408..731C,2011A&A...530L..14B},
although there is no evidence of any
extremely massive stars with $\sim 1000 M_{\odot}$.
\subsection{Maximum mass of PCPs in ensemble clusters}
As we show in section \ref{sc_ensamble}, early assembling of
sub-clusters results in the formation of a PCP, while late assembling
forms a less massive PCP and multiple SCPs as massive as the PCP.
In figure \ref{fig:m_max_t} we present the relation between
$m_{\rm max}/M_{\rm cl}$ and $t_{\rm enc}/t_{\rm cc}$ of ensemble models,
where $t_{\rm cc}$ is the core-collapse time of the sub-clusters.
Irrespective of the number
of sub-clusters, the maximum mass of the PCPs decreases as the
assembling time is delayed.
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f13.eps}
\caption{The maximum mass of the PCPs scaled by the total mass of the
ensemble clusters as a function of the assembling time scaled by the
core-collapse time of the sub-clusters for all e2k8 and e2k4 models.
Each horizontal line corresponds to one model. Vertical lines and
crosses indicate the individual
merger time of sub-clusters for four and eight sub-cluster models,
respectively. The dotted line indicates $t_{\rm ens}/t_{\rm cc} = 1$.
\label{fig:m_max_t}}
\end{center}
\end{figure}
In the left panel of Figure \ref{fig:m_max}, we show the relation
between $m_{\rm max}$ of the PCPs and $M_{\rm cl}$ for both
solo and ensemble clusters. (Note that for the solo clusters, the data is
the same as that shown in Figure \ref{fig:m_max_single}).
The PCP mass of early assembling models is higher than that of
solo clusters with the same cluster mass and as massive as that of the cold
model. In late assembling
models, the PCPs is almost as massive as those of the solo clusters
with the same mass.
The difference in the maximum mass of PCPs is understood if we
take into account all the PCPs and SCPs in the cluster.
In the right panel of Figure \ref{fig:m_max}, we present the total
mass of all the PCPs and SCPs in the cluster. The total masses are
roughly located on the relation that
$m_{\rm max} = 0.02 M_{\rm cl}$.
This result suggests that the potential maximum mass of the PCPs
is 2\% of the cluster mass, although the value depends on
the initial mass function and the mass-loss rate due to the stellar
wind. The total mass of the SCPs is
summarized in Table \ref{tb:results} as $m_{\rm SCPs}$. These PCPs
fail to merge with the most massive PCP and their mass will be lost
from the cluster by escape or stellar evolution.
\begin{figure*}
\begin{center}
\includegraphics[width=70mm]{f14a.eps}
\includegraphics[width=70mm]{f14b.eps}
\caption{The maximum mass of the PCP, $m_{\rm max}$, in the cluster (left) and the total mass of $m_{\rm max}$ and the sum of the maximum mass of the SCPs, $m_{\rm SCPs}$ (right). \label{fig:m_max}}
\end{center}
\end{figure*}
In Figure \ref{fig:massive_star}, we plot the radial distribution
of PCPs and SCPs, which grows to $>100 M_{\odot}$.
We combine the results from several runs, separating them in the early and
late assembling cases.
While all the PCPs are located in the cluster core in the early assembling
case, $\sim40$\% of the PCPs are ejected from the clusters or located
in the outskirts of the cluster ($>10$ pc) in the late assembling case.
The numbers of PCPs per cluster are on average 1.75 and 5.8 for the
early and late assembling cases, respectively. In Figure
\ref{fig:massive_star} we also present the cumulative number
distribution of stars with $>100M_{\odot}$ in the R136 region
\citep{2010MNRAS.408..731C,2011A&A...530L..14B}. The number of such
massive stars and their distribution imply that R136 experienced
some late assembling, and observationally a sub-cluster has been
found around R136 \citep{2012ApJ...754L..37S}.
\begin{figure}
\begin{center}
\includegraphics[width=84mm]{f15.eps}
\caption{Cumulative distribution of PCPs and SCPs as a function of the distance from the cluster center. Dashed and filled curves indicate early and late assembling models, respectively. For the early assembling models, we combined the data from e2k8r1-1, e2k8r1-2, e2k8r2, and e8k8s1 (4 runs), and the average number of PCPs per run is 1.75. For the late assembling models, we combined the data from e2k8r5-1, e2k8r5-2, e2k8r6-1, e2k8r6-2, and e8k8s2 (5 runs), and the average number of PCPs is 5.8. Squares indicate the distribution of massive ($>100M_{\odot}$) stars observed in R136 region \citep{2010MNRAS.408..731C,2011A&A...530L..14B}. Since the observation is projected distance, we multiplied them by $\sqrt[]{3}$. For both the simulations and observations, we treat stars within 0.1 pc as at 0.1 pc because the distance is affected by the definition of the cluster center. \label{fig:massive_star}}
\end{center}
\end{figure}
\section{Summary and Discussion}
We performed a series of $N$-body simulations of solo
and ensemble star clusters and found that ensemble clusters evolve
through typically two paths depending on their assembling time
compared to the core-collapse time of the sub-clusters.
In the early assembling case ($t_{\rm cc}>t_{\rm ens}$), the remnant
clusters have dynamically mature characteristics
(mass segregation and core collapse) compared to solo-clusters.
The evolution of early assembling clusters is similar to
that of sub-virial solo-clusters.
The early assembling clusters experience mass segregation and
core collapse on the time scale of the sub-clusters, which is
shorter than that of initially large solo clusters, and the short
relaxation time of sub-clusters is conserved in the remnant clusters.
This dynamically early evolution results in efficient multiple
collisions of stars and helps the formation of extremely massive
PCPs with $\sim 1000 M_{\odot}$.
In the late assembling case ($t_{\rm cc}<t_{\rm ens}$), the
dynamically mature characteristics suppress the growth of massive stars
via stellar collisions. In this case,
the sub-clusters experience core collapse individually and form their
own PCPs, but the maximum mass of the PCPs in the sub-clusters is limited
by the total mass of the sub-clusters. Even after the sub-clusters
assemble, the PCPs stop growing because the central density
of the remnant cluster is already depleted due to the quick dynamical
evolution of the sub-clusters. Since the PCPs in sub-clusters form
massive binaries, they interact with each other in the
remnant clusters. Some of them (SCPs) collide, but the others are
scattered from the cluster by three-body or binary-binary
encounters. In our simulations, 40\% of the SCPs are ejected
from the cluster or scattered to the outskirts of the remnant clusters.
The SCPs sometimes escape with a high velocity ($>30$km/s) and reach
$\sim 100$ pc from the cluster within their life time ($\sim 3$ Myr).
The observed massive high-velocity stars such as VFTS 682 might be
formed in this way (see also Paper 1).
We also investigated the maximum mass of the PCPs and found that
in ensemble clusters, the maximum mass depends on the assembling time
of sub-clusters. In the early assembling models, the maximum mass
of the PCPs is comparable to that of sub-virial solo-clusters.
In the late assembling models, however, the maximum mass is similar to
that of the solo sub-clusters. The difference between them is mainly
caused by the number of collisions. In the late assembling
models, a larger number of SCPs are ejected from the cluster
and fail to merge to the PCP than in the early assembling case.
When the collisions of stars proceed most successfully (in early
assembling and cold solo models), we find that the maximum masses
of the PCPs reach $\sim$2\% of the total mass of the clusters
even if we take into account the high mass-loss rate due to the
stellar wind. Assuming an R136-like cluster of
$\sim 5\times 10^4 M_{\odot}$, the expected maximum mass is
$\sim 1000 M_{\odot}$. Such an efficient mass growth might
result in the formation of IMBHs. For lower metalicity, the massive
stars are predicted to collapse directly to IMBHs
\citep{2003ApJ...591..288H}.
In late assembling cases, however, multiple smaller PCPs
($100$--$400 M_{\odot}$) are expected to exist inside or around the remnant
clusters. These stars are in the mass range of
type Ib/c supernovae (SNe) assuming solar metallicity
\citep{2003ApJ...591..288H}.
In recent observations of dense molecular clouds in the central molecular
zone in the Galactic center, several expanding shells were found,
and the estimated total kinetic energy of them is $\sim 10^{52}$ erg.
\citep{2007PASJ...59..323T, 2012ApJS..201...14O}. Especially,
three major shells have a kinetic energy of $\sim 10^{51}$ erg, which
corresponds to a hypernova explosion.
A young dense massive clusters which is similar to our late-merger
models might be embedded in this dense molecular cloud.
Furthermore, escaping PCPs will explode up to $\sim 100$ pc
from the host cluster.
Actually type Ib/c SNe associate with star forming regions
\citep{2010MNRAS.407.2660A,2011A&A...530A..95L,2012MNRAS.424.1372A,
2012arXiv1210.1126C},
and for example Type Ic SN 2007gr is
located at $\sim 7$ pc from a young cluster \citep{2008ApJ...672L..99C}.
\section*{Acknowledgments}
The authors thank Jeroen B\'{e}dorf for the Sapppro2 library, Alex
Rimoldi for careful reading of the manuscript, and Masaomi Tanaka for
fruitful discussion.
This work was supported by the Japan Society for the Promotion of
Science (JSPS) Research Fellowship for Research Abroad, the
Netherlands Research Council NWO
(grants \#643.200.503, \#639.073.803 and \#614.061.608), the
Netherlands Research School for Astronomy (NOVA).
Numerical computations were carried out on the Cray XT4 at the
Center for Computational Astrophysics (CfCA) of the National
Astronomical Observatory of Japan and the Little Green Machine
at Leiden University.
\bibliographystyle{mn}
|
{
"timestamp": "2012-10-16T02:02:10",
"yymm": "1210",
"arxiv_id": "1210.3732",
"language": "en",
"url": "https://arxiv.org/abs/1210.3732"
}
|
\section*{}
\begin{Large}
\begin{center}
Abstract
\end{center}
\end{Large}
\begin{small}
\noindent\noindent
The current attempt is aimed to honor
the first centennial of Johannes
Diderik van der Waals (VDW) awarding
Nobel Prize in Physics. The VDW
theory of ordinary fluids is reviewed
in the first part of the paper, where
special effort is devoted to the equation
of state and the law of corresponding
states. In addition, a few mathematical
features involving properties of cubic
equations are discussed, for appreciating
the intrinsic beauty of the VDW theory.
A theory of astrophysical fluids is
shortly reviewed in the second part of
the paper, grounding on the tensor virial
theorem for two-component systems, and
an equation of state is formulated with
a convenient choice of reduced variables.
Additional effort is devoted to particular
choices of density profiles, namely a simple
guidance case and two cases of astrophysical
interest.
The related macroisothermal curves are
found to be qualitatively similar to
VDW isothermal curves below the critical
threshold and, for sufficiently steep
density profiles, a critical macroisothermal
curve exists, with a single horisontal
inflexion point. Under the working
hypothesis of a phase transition (assumed
to be gas-stars) for astrophysical fluids,
similar to the vapour-liquid phase transition
in ordinary fluids, the location of gas clouds,
stellar systems, galaxies, cluster of galaxies,
on the plane scanned by reduced
variables, is tentatively assigned.
A brief discussion shows how van der Waals' two
great discoveries, namely
a gas equation of state where tidal
interactions between molecules are
taken into account, and the law of
corresponding states, related to
microcosmos, find a counterpart
with regard to macrocosmos. In conclusion,
after a century since the awarding
of the Nobel Prize in Physics, van
der Waals' ideas are still valid and
helpful to day for a full understanding
of the universe.
\noindent
{\it keywords -
galaxies: evolution - dark matter: haloes.}
\end{small}
\end{quotation}
\section{Introduction} \label{intro}
One century ago (1910), the Nobel Prize in Physics was awarded to
Johannes Diderik van der Waals (hereafter quoted as VDW).
In his doctoral thesis (1873) the ideal gas equation of
state was generalized for embracing both the gaseous and
the liquid state, where these two states of aggregation
not only merge into each other in a continuous manner,
but are in fact of the same nature. With respect to ideal
gases, the volume of the molecules and the intermolecular
tidal forces were taken into account.
The VDW equation was later reformulated in terms of reduced
(dimensionless) variables (1880), which allows the description
of all substances in terms of a single equation. In other
words, the state of any substance, defined by the values of
reduced volume, reduced pressure, and reduced temperature, is
independent of the nature of the substance. This result is
known as the law of corresponding states.
The VDW equation of state in dimensional and reduced form,
served as a guide during experiments which ultimately led
to hydrogen (1898) and helium (1908) liquefaction. The
Cryogenic Laboratory at Leiden had developed under the
influence of VDW's theories. For further details on VDW's
biography refer to specialized textbooks (e.g., Nobel Lectures
1967).
The current paper has been written in honor of
the first centennial of VDW awarding Nobel Prize in Physics. The
ideal and VDW equation of state, both in dimensional and
reduced form, are reviewed, and a number of features are
analysed in detail, in Section \ref{vande}.
Counterparts to ideal and VDW equations of state
for astrophysical fluids, or macrogases, are briefly
summarized and compared with the classical formulation
in Section \ref{macro}. The discussion and the
conclusion are drawn in Section \ref{disc}.
\section{Equation of state of ordinary fluids}\label{vande}
Let ordinary fluids be conceived as fluids which
can be investigated in laboratory. The simplest
description is provided by the theory of ideal gas,
where the following restrictive assumptions are made:
(i) particles are identical
spheres; (ii) the number of particles is extremely
large; (iii) the motion of particles is random;
(iv) collisions between particles or with the wall
of the box are perfectly elastic; (v) interactions
between particles or with the wall of the box are
null.
The equation of state of ideal gases may be written
under the form (e.g., Landau and Lifchitz, 1967,
Chap.\,IV, \S42, hereafter quoted as LL67):
\begin{equation}
\label{eq:gid}
pV=kNT~~;
\end{equation}
where $p$ is the pressure, $V$ the volume, $T$ the
temperature, $N$ the particle number, and $k$ the
Boltzmann constant.
In getting a better description of ordinary fluids,
the above assumption (v) is relaxed and tidal interactions
between particles are taken into consideration.
The VDW generalization
of the equation of state of ideal gases, Eq.\,(\ref{eq:gid}),
reads (van der Waals, 1873):
\begin{equation}
\label{eq:VdW}
\left(p+A\frac{N^2}{V^2}\right)(V-NB)=kNT~~;
\end{equation}
where $A$ and $B$ are constants which depend
on the nature of the particles. More specifically,
the presence of an attractive interaction between
particles reduces both the force and the frequency
of particle-wall collisions: the net effect is a reduction
of the pressure, proportional to the square numerical
density, expressed as $A(N/V)^2$.
On the other hand, the whole volume of the box,
$V$, is not accessible to particles, in that they are
conceived as identical spheres: the free volume within the
box is $V-NB$, where $B$ is the volume of a single sphere.
For further details refer to specific textbooks (e.g., LL67,
Chap.\,VII, \S74).
The isothermal ($T=$ const) curves for ideal gases are
hyperbolas with axes, $p=\mp V$, conformly to Eq.\,(\ref
{eq:gid}). In VDW theory of real gases, the isothermal
curves exhibit two extremum points below a threshold,
which reduce to a
single horisontal inflexion point when a critical temperature
is attained, as shown in Fig.\,\ref{f:viso}.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{viso100.eps}
\caption{Isothermal curves related to ideal (left panel)
and VDW (right panel) gases, respectively. Isothermal
curves (from bottom to top) correspond to $T/T_{\rm c}=$
20/23, 20/22, 20/21, 20/20, 20/19, 20/18. No extremum point exists
above the critical isothermal curve, $T/T_{\rm c}=1$.
}
\label{f:viso}
\end{center}
\end{figure*}
Well above the critical isothermal curve, $T\gg T_{\rm c}$,
the trends exhibited by ideal and VDW gases look very
similar. Below the critical isothermal curve, $T<T_{\rm c}$,
the behaviour of VDW gases is different with respect to
ideal gases and, in addition, the related isothermal
curves provide a wrong description within a specific
region where saturated vapour and liquid phases coexist.
Further details are shown in Fig.\,\ref{f:vris}.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{vris100.eps}
\caption{Same as in Fig.\,\ref{f:viso} (right panel), where
the occurrence (within the bell-shaped area bounded by the
dashed curve) of saturated vapour is considered.
Above the critical isothermal
curve $(T=T_{\rm c})$ the trend is similar with respect to
ideal gases. Below the critical isothermal curve and
on the right of the dashed curve, the gas
still behaves as an ideal gas. Below the critical
isothermal curve and on the left of the dashed curve, the
liquid shows little change in volume as the pressure rises.
Within the bell-shaped area bounded by the dashed curve,
the liquid phase is in equilibrium with the saturated vapour
phase. A diminished volume implies
smaller saturated vapour fraction and larger liquid fraction
at constant pressure, and vice versa. The VDW equation of
state is
no longer valid in this region. The dashed curve (including
the central branch) is the locus of intersection between VDW
and real isothermal curves, the latter being related to constant
pressure where liquid and vapour phases coexist. The dotted
curve is the locus of VDW isothermal extremum points.}
\label{f:vris}
\end{center}
\end{figure*}
Above the critical isothermal
curve $(T=T_{\rm c})$ the trend is similar with respect to
ideal gases. Below the critical isothermal curve and
on the right of the dashed curve, the supersaturated
vapour still behaves as an ideal gas. Below the critical
isothermal curve and on the left of the dashed curve, the
liquid shows little change in volume as the pressure rises.
Within the bell-shaped area bounded by the dashed curve,
the liquid phase is in equilibrium with the saturated vapour
phase. A diminished volume implies
smaller saturated vapour fraction and larger liquid fraction
at constant pressure, and vice versa. The VDW equation of
state is
no longer valid in this region. The dashed curve (including
the central branch) is the locus of intersections between VDW
and real isothermal curves, the latter being related to constant
pressure where liquid and vapour phases coexist. The dotted
curve is the locus of VDW isothermal extremum points.
A specific $(T/T_{\rm c}=20/23)$ VDW and corresponding real
isothermal curve, are represented in Fig.\,\ref{f:vrar}.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{vrar100.eps}
\caption{A specific $(T/T_{\rm c}=20/23)$ VDW and corresponding
real isothermal curve. The above mentioned curves
coincide within the range, $V\le V_{\rm A}$
and $V\ge V_{\rm E}$. The VDW isothermal curve exhibits
two extremum points: a minimum, ${\sf B}$, and a maximum,
${\sf D}$, while the real isothermal curve is flat within
the range, $V_{\rm A}\le V\le V_{\rm E}$. Configurations related to
the VDW isothermal curve within the range, $V_{\rm A}\le V\le V_{\rm B}$
(due to tension forces acting on the particles yielding
superheated liquid), and
$V_{\rm D}\le V\le V_{\rm E}$ (due to the occurrence of undercooled
vapour), may be obtained under special conditions, while
configurations within the range, $V_{\rm B}\le V\le V_{\rm D}$, are
always unstable. The volumes, $V_{\rm A}$ and $V_{\rm E}$, correspond
to the maximum value in presence of the sole liquid phase
and the minimum value in presence of the sole
vapour phase, respectively. The regions, {\sf ABC} and
{\sf CDE}, have equal area. For further details refer to
the text.}
\label{f:vrar}
\end{center}
\end{figure*}
The VDW isothermal curve and the
real isothermal curve coincide within the range, $V\le V_{\rm A}$
and $V\ge V_{\rm E}$. The VDW isothermal curve exhibits
two extremum points: a minimum, ${\sf B}$, and a maximum,
${\sf D}$, while the real isothermal curve is flat, within
the range, $V_{\rm A}\le V\le V_{\rm E}$. Configurations related to
the VDW isothermal curve within the range, $V_{\rm A}\le V\le V_{\rm B}$
(due to tension forces acting on the particles yielding
superheated liquid), and
$V_{\rm D}\le V\le V_{\rm E}$ (due to the occurrence of undercooled
vapour), may be obtained under special conditions, while
configurations within the range, $V_{\rm B}\le V\le V_{\rm D}$, are
always unstable. The volumes, $V_{\rm A}$ and $V_{\rm E}$, correspond
to the maximum value in presence of the sole liquid phase
and the minimum value in presence of the sole
vapour phase, respectively.
The surfaces, {\sf ABC} and {\sf CDE}, are equal, as first
inferred by Maxwell (e.g., Rostagni, 1957, Chap.\,XII,
\S19). The VDW and real isothermal curves represented
in Fig.\,\ref{f:vrar} being related to the same temperature,
$T$, the cycle, {\sf ABCDECA}, is completely both isothermal
and reversible, and the work, $W$, performed therein cannot be
positive to avoid violation of the second law of the
thermodynamics. The cycles, {\sf ABCA} and {\sf CDEC},
occurring in counterclockwise and clockwise sense, respectively,
are also completely both isothermal and reversible.
Accordingly, $W_{\sf ABCDECA}=W_{\sf ABCA}-W_{\sf CDEC}\le0$.
A similar procedure, related to the reversed cycle,
{\sf ACEDCBA}, yields $W_{\sf ACEDCBA}=W_{\sf CEDC}-W_{\sf CBAC}
\le0$. Then $W_{\sf ABCDECA}=W_{\sf ACEDCBA}=0$, which implies
$W_{\sf ABCA}=W_{\sf CDEC}=
W_{\sf CEDC}=W_{\sf CBAC}$ and, in turn, the equality between
the related surfaces. For further details refer to specific
textbooks (e.g., LL67, Chap.\,VIII, \S85).
In order to simplify both notation and calculations, it is
convenient to deal with (dimensionless) reduced variables (e.g.,
Rostagni, 1957, Chap.\,XII, \S16; LL67, Chap.\,VIII, \S85).
To this aim, the first step is the knowledge of the parameters
related to the critical point, $V_{\rm c}, p_{\rm c}, T_{\rm c}$. Using the
VDW equation of state, Eq.\,(\ref{eq:VdW}), the pressure and
its first and second partial derivatives, with respect to
the volume, read:
\begin{lefteqnarray}
\label{eq:pW}
&& p=\frac{kNT}{V-NB}-A\frac{N^2}{V^2}~~;\qquad N={\rm const}~~; \\
\label{eq:p1W}
&& \left(\frac{\partial p}{\partial V}\right)_{V,T}=-\frac{kNT}{(V-NB)^2}+
2A\frac{N^2}{V^3}~~; \\
\label{eq:p2W}
&& \left(\frac{\partial^2 p}{\partial V^2}\right)_{V,T}=\frac{2kNT}
{(V-NB)^3}-6A\frac{N^2}{V^4}~~;
\end{lefteqnarray}
where the domain is $V>NB$, $V=NB$ is a vertical
asymptote, and $p=0$ is a horisontal asymptote.
The critical isothermal corresponds to the highest
temperature allowing a liquid phase, which occurs
therein only at the critical point. The critical
isothermal curve exhibits neither a minimum nor a
maximum, which are replaced by a horisontal inflexion
point coinciding with the critical point. Accordingly,
$(\partial p/\partial V)_{V_{\rm c},T_{\rm c}}=0$,
$(\partial^2p/\partial V^2)_{V_{\rm c},T_{\rm c}}=0$, and
$p_{\rm c}=kNT_{\rm c}/(V_{\rm c}-NB)-AN^2/V_{\rm c}^2$. The solution
of the related system is:
\begin{lefteqnarray}
\label{eq:Vc}
&& V_{\rm c}=3NB~~; \\
\label{eq:Tc}
&& T_{\rm c}=\frac8{27}\frac AB\frac1k~~; \\
\label{eq:pc}
&& p_{\rm c}=\frac1{27}\frac A{B^2}~~; \\
\label{eq:Zc}
&& Z_c=\frac{p_{\rm c}V_{\rm c}}{kNT_{\rm c}}=\frac38~~;
\end{lefteqnarray}
where, in general, the compressibility factor,
$Z=pV/(kNT)$, defines the degree of departure
from the behaviour of ideal gases, for which
$Z=1$, according to Eq.\,(\ref{eq:gid}).
For further details refer to specific textbooks
(e.g., Rostagni, 1957, Chap.\,XII, \S20; LL67,
Chap.\,VIII, \S85).
With regard to the reduced variables:
\begin{equation}
\label{eq:rv}
\sV=\frac V{V_{\rm c}}~~;\qquad\sP=\frac p{p_{\rm c}}~~;
\qquad\sT=\frac T{T_{\rm c}}~~;
\end{equation}
the ideal gas equation of state, Eq.\,(\ref{eq:gid}),
and the VDW equation of state, Eq.\,(\ref{eq:VdW}),
reduce to:
\begin{lefteqnarray}
\label{eq:ri}
&& \sP\sV=\frac83\sT~~; \\
\label{eq:rW1}
&& \left(\sP+\frac3{\sV^2}\right)\left(\sV-\frac13\right)=\frac83\sT~~;\qquad
\sV>\frac13~~;
\end{lefteqnarray}
and Eqs.\,(\ref{eq:pW}), (\ref{eq:p1W}), and
(\ref{eq:p2W}), reduce to:
\begin{lefteqnarray}
\label{eq:rW2}
&& \sP=\frac{8\sT}{3\sV-1}-\frac3{\sV^2}~~; \\
\label{eq:rW3}
&& \left(\frac{\partial\sP}{\partial\sV}\right)_{\sV,\sT}=-\frac{24\sT}
{(3\sV-1)^2}+\frac6{\sV^3}~~; \\
\label{eq:rW4}
&& \left(\frac{\partial^2\sP}{\partial\sV^2}\right)_{\sV,\sT}=\frac
{144\sT}{(3\sV-1)^3}-\frac{18}{\sV^4}~~;
\end{lefteqnarray}
where, for assigned $\sT$, the domain of the function,
$\sP(\sV)$, is $\sV>1/3$, $\sV=1/3$ is a vertical
asymptote, and $\sP=0$ is a horisontal asymptote.
In the special case of the critical point, $\sV=1$,
$\sT=1$, $\sP=1$, the partial derivatives are null,
as expected.
The extremum points, via Eq.\,(\ref{eq:rW3}), are
defined by the relation:
\begin{equation}
\label{eq:ext}
f(\sV)=\frac{(3\sV-1)^2}{4\sV^3}=\sT~~;
\end{equation}
which is satisfied at the critical point, as
expected. The function on the left-hand side
of Eq.\,(\ref{eq:ext}) has two extremum points:
a minimum at $\sV=1/3$ (outside the physical
domain) and a maximum at $\sV=1$, where $\sT=1$.
Accordingly, Eq.\,(\ref{eq:ext}) is never
satisfied for $\sT>1$, which implies no extremum
point for related isothermal curves, as expected.
The contrary holds for $\sT<1$, where it can be
seen that the third-degree equation associated
to Eq.\,(\ref{eq:rW3}) has three real solutions,
related to extremum points. One lies outside
the physical domain, which implies $\sV\le1/3$.
The remaining two are obtained as the intersections
between the curve, $f(\sV)$, expressed by Eq.\,(\ref
{eq:ext}), and the straight line, $y=\sT$, keeping
in mind that $f(1/3)=0$, $f(1)=1$, and $\lim_{\sV\to
+\infty}f(\sV)=0$.
The third-degree equation associated to Eq.\,(\ref
{eq:rW3}), may be ordered as:
\begin{leftsubeqnarray}
\slabel{eq:3dea}
&& \sV^3-9a\sV^2+6a\sV-a=0~~; \\
\slabel{eq:3deb}
&& a=\frac1{4\sT}~~;
\label{seq:3de}
\end{leftsubeqnarray}
with regard to the standard formulation (e.g.,
Spiegel, 1968, Chap.\,9):
\begin{equation}
\label{eq:3dx}
x^3+a_1x^2+a_2x+a_3=0~~;
\end{equation}
the discriminants of Eq.\,(\ref{eq:3dea}) are:
\begin{lefteqnarray}
\label{eq:Q}
&& Q=\frac{3a_2-a_1^2}9=a(2-9a)~~; \\
\label{eq:R}
&& R=\frac{9a_1a_2-27a_3-2a_1^3}{54}=\frac{a(1-18a+54a^2)}2~~; \\
\label{eq:D}
&& D=Q^3+R^2=\frac{a^2(1-4a)}4~~;
\end{lefteqnarray}
where $D=0$ in the special case of the critical
isothermal curve $(\sT=1, a=1/4)$, $D<0$ for
$\sT<1$, and $D>0$ for $\sT>1$. Accordingly,
three coincident real solutions exist if $D=0$,
three (at least two) different real solutions
if $D<0$, one real (outside the physical domain)
and two complex coniugate if $D>0$.
The three real solutions $(D\le0)$ may be expressed as
(e.g., Spiegel, 1968, Chap.\,9):
\begin{leftsubeqnarray}
\slabel{eq:rsola}
&& \sV_1=2\sqrt{-Q}\cos\left(\pi+\frac\theta3\right)-\frac13a_1~~; \\
\slabel{eq:rsolb}
&& \sV_2=2\sqrt{-Q}\cos\left(\pi+\frac\theta3+\frac{2\pi}3\right)-
\frac13a_1~~; \\
\slabel{eq:rsolc}
&& \sV_3=2\sqrt{-Q}\cos\left(\pi+\frac\theta3+\frac{4\pi}3\right)-
\frac13a_1~~; \\
\slabel{eq:rsold}
&& \theta=\arctan\frac{\sqrt{-D}}R~~;
\label{seq:rsol}
\end{leftsubeqnarray}
where $a_1=-9a$ and, in the special case of the
critical isothermal curve, $a=1/4$, $Q=-1/16$,
$D=0$, which implies $\sV_0=\min(\sV_1,\sV_2,
\sV_3)$, $\sV_{\rm A}=\sV_{\rm B}=\sV_{\rm C}=\sV_{\rm D}=\sV_{\rm E}=\max
(\sV_1,\sV_2,\sV_3)$. In the special case,
$\sT\to0$, Eq.\,(\ref{eq:3dea}) reduces to a
second-degree equation whose solutions are
$\sV_{01}=\sV_{02}=1/3$, while the related
function is otherwise divergent as $a\to+\infty$.
In general, the extremum points of VDW isothermal
curves $(\sT\le1)$ occur at $\sV=\sV_{\rm B}$ (minimum) and
$\sV=\sV_{\rm D}$ (maximum), $\sV_{\rm B}\le\sV_{\rm D}$. As
$\sT\to0$, $\sV_{\rm B}\to1/3$, $\sV_{\rm D}\to+\infty$,
where, in all cases, $1/3<\sV_{\rm B}\le1\le\sV_{\rm D}$.
The two areas defined by the intersection of a
generic VDW isothermal curve $(\sT\le1)$ and
related real isothermal curves (see Fig.\,\ref
{f:vrar}), are expressed as:
\begin{leftsubeqnarray}
\slabel{eq:S1a}
&& W_1=\int_{V_{\rm A}}^{V_{\rm C}}p_{\rm C}\diff V-\int_{V_{\rm A}}^{V_
{\rm C}}p\diff V=p_{\rm C}V_{\rm C}
\left[\sP_C(\sV_{\rm C}-\sV_{\rm A})-\int_{\sV_{\rm A}}^{\sV_{\rm C}}\sP
\diff\sV\right];\qquad \\
\slabel{eq:S1b}
&& W_2=\int_{V_{\rm C}}^{V_{\rm E}}p\diff V-\int_{V_{\rm C}}^{V_{\rm E}}p_
{\rm C}\diff V=p_{\rm C}V_{\rm C}
\left[\int_{\sV_{\rm C}}^{\sV_{\rm E}}\sP\diff\sV-\sP_C(\sV_{\rm E}-\sV_
{\rm C})\right];\qquad
\label{seq:S1}
\end{leftsubeqnarray}
and the substitution of Eq.\,(\ref{eq:rW2}) into
(\ref{seq:S1}) allows explicit expressions for
the integrals. The result is:
\begin{leftsubeqnarray}
\slabel{eq:S2a}
&& \frac{W_1}{p_{\rm C}V_{\rm C}}=\sP_{\rm C}(\sV_{\rm C}-\sV_{\rm A})-\frac
83\sT\ln\frac{3\sV_{\rm C}-1}{3\sV_{\rm A}-1}+
\frac{3(\sV_{\rm C}-\sV_{\rm A})}{\sV_{\rm A}\sV_{\rm C}}~~; \\
\slabel{eq:S2b}
&& \frac{W_2}{p_{\rm C}V_{\rm C}}=\frac83\sT\ln\frac{3\sV_{\rm E}-1}{3\sV_
{\rm C}-1}-
\frac{3(\sV_{\rm E}-\sV_{\rm C})}{\sV_{\rm C}\sV_{\rm E}}-\sP_C(\sV_{\rm E}-
\sV_{\rm C})~~;
\label{seq:S2}
\end{leftsubeqnarray}
and the condition, $W_1=W_2$, after some algebra reads
(Caimmi 2010, hereafter quoted as C10):
\begin{equation}
\label{eq:S12}
\sP_C=\frac83\frac{\sT}{\sV_{\rm E}-\sV_{\rm A}}\ln\frac{3\sV_{\rm E}-1}{3\sV_{\rm A}-1}-\frac3
{\sV_{\rm A}\sV_{\rm E}}~~;
\end{equation}
where, for a selected isothermal curve, the unknowns
are $\sP_C=\sP_A=\sP_E$, $\sV_{\rm A}$, and $\sV_{\rm E}$.
The reduced volumes, $\sV_{\rm A}$, $\sV_{\rm C}$, $\sV_{\rm E}$,
see Fig.\,\ref{f:vrar}, may be considered as
intersections between a VDW isothermal curve
$(\sT<1)$ and a horisontal straight line, $\sP=\sP_C$,
in the $({\sf O}\sV\sP)$ plane. In other words,
$\sV_{\rm A}$, $\sV_{\rm C}$, $\sV_{\rm E}$, are the real solutions
of the third-degree equation:
\begin{equation}
\label{eq:3Wrr}
\sV^3-\left(\frac13+\frac83\frac{\sT}{\sP_C}\right)\sV^2+\frac3{\sP_C}\sV-
\frac1{\sP_C}=0~~;
\end{equation}
which has been deduced from Eq.\,(\ref{eq:rW2}),
particularized to $\sP=\sP_C$. The related
solutions may be calculated using Eqs.\,(\ref{seq:rsol}).
The last unknown, $\sP_C$, is determined from Eq.\,(\ref
{eq:S12}).
An inspection of Fig.\,\ref{f:vrar} shows that
the points, {\sf A} and {\sf E}, are located
on the left of the minimum, {\sf B}, and on
the right of the maximum, {\sf D}, respectively.
Keeping in mind the above results, the following
inequality holds: $\sV_{\rm A}\le\sV_{\rm B}\le1\le\sV_{\rm D}\le
\sV_{\rm E}$, which implies further investigation on
the special case, $\sV_{\rm C}=1$. The particularization
of the VDW equation of state, Eq.\,(\ref{eq:rW2}), to the
point, ${\sf C}={\sf C_1}$, assuming $\sV_{C_1}=1$,
yields:
\begin{equation}
\label{eq:TVC1}
\sT=\frac{\sP_{C_1}+3}4~~;
\end{equation}
and Eq.\,(\ref{eq:3Wrr}) reduces to:
\begin{leftsubeqnarray}
\slabel{eq:3dba}
&& \sV^3-(1+2b)\sV^2+3b\sV-b=0~~; \\
\slabel{eq:3dbb}
&& b=\frac1{\sP_{C_1}}~~;
\label{seq:3db}
\end{leftsubeqnarray}
with regard to the generic third-degree equation,
Eq.\,(\ref{eq:3dx}), the three solutions, $x_1$,
$x_2$, $x_3$, satisfy the relations (e.g., Spiegel,
1968, Chap.\,9):
\begin{leftsubeqnarray}
\slabel{eq:x123a}
&& x_1+x_2+x_3=-a_1~~; \\
\slabel{eq:x123b}
&& x_1x_2+x_2x_3+x_3x_1=a_2~~; \\
\slabel{eq:x123c}
&& x_1x_2x_3=-a_3~~;
\label{seq:x123}
\end{leftsubeqnarray}
where, in the case under discussion:
\begin{leftsubeqnarray}
\slabel{eq:b123a}
&& a_1=-1-2b~~;\qquad a_2=3b~~;\qquad a_3=-b~~; \\
\slabel{eq:b123b}
&& x_1=\sV_{\rm A}~~;\qquad x_2=\sV_{C_1}=1~~;\qquad x_3=\sV_{\rm E}~~;
\label{seq:b123}
\end{leftsubeqnarray}
and the substitution of Eqs.\,(\ref{seq:b123})
into two among (\ref{seq:x123}) yields:
\begin{leftsubeqnarray}
\slabel{eq:VAEa}
&& \sV_{\rm A}=b-\sqrt{b^2-b}~~; \\
\slabel{eq:VAEb}
&& \sV_{\rm E}=b+\sqrt{b^2-b}~~;
\label{seq:VAE}
\end{leftsubeqnarray}
and the combination of Eqs.\,(\ref{eq:TVC1}),
(\ref{eq:3dbb}), and (\ref{seq:VAE}) produces:
\begin{leftsubeqnarray}
\slabel{eq:VAETa}
&& \sV_{\rm A}=\frac{1-2\sqrt{1-\sT}}{4\sT-3}~~;\qquad\sT\le1~~; \\
\slabel{eq:VAETb}
&& \sV_{\rm E}=\frac{1+2\sqrt{1-\sT}}{4\sT-3}~~;\qquad\sT\le1~~;~~;
\label{seq:VAET}
\end{leftsubeqnarray}
which, together with $\sV_{C_1}=1$, are the abscissae
of the intersection points between a selected VDW
isothermal curve in the $({\sf O}\sV\sP)$ plane and
the straight line, $\sP=\sP_{C_1}$, in the special case
under discussion.
The substitution of Eqs.\,(\ref{seq:VAET}) into
(\ref{eq:S12}), the last being related to the real
isothermal curve, yields:
\begin{equation}
\label{eq:S12C}
\frac{\sT}{\sqrt{1-\sT}}\ln\frac{3-2\sT+3\sqrt{1-\sT}}{3-2\sT-3\sqrt{1-\sT}}
=6~~;
\end{equation}
which holds only for the critical isothermal curve,
$\sT=1$. Accordingly, the abscissa of the intersection
point, {\sf C}, between a selected VDW isothermal curve
and related real isothermal curve, see Fig.\,\ref{f:vrar},
cannot occur at $\sV_{\rm C}=1$ unless the critical isothermal
curve is considered. Then the third-degree equation,
Eq.\,(\ref{eq:3Wrr}), must be solved in the general case
by use of Eqs.\,(\ref{seq:rsol}). The results are shown
in Tab.\,\ref{t:vispo}, where the following parameters
(in reduced variables) are listed for each VDW
isothermal curve, see Fig.\,\ref{f:vrar}: the temperature, $\sT$, the lower
volume limit, $\sV_{\rm A}$, for which the liquid and vapour phase
coexist; the extremum point (minimum) volume, $\sV_{\rm B}$;
the intermediate volume, $\sV_{\rm C}$, for which the pressure equals
its counterpart related to the corresponding lower and
upper volume limit, for which the liquid and vapour phase coexist;
the extremum point (maximum) volume, $\sV_{\rm D}$;
the upper volume limit, $\sV_{\rm E}$, for which the liquid and
vapour phase coexist; the extremum point (minimum) pressure,
$\sP_B$; the pressure, $\sP_A=\sP_C=\sP_E$, related to the
horisontal real isothermal curve; the extremum point (maximum)
pressure, $\sP_D$.
\begin{table}
\caption{Values of parameters, $\sT$, $\sV_{\rm A}$, $\sV_{\rm B}$,
$\sV_{\rm C}$, $\sV_{\rm D}$, $\sV_{\rm E}$, $\sP_B$, $\sP_C$,
$\sP_D$, within the range, $0.85\le\sT\le0.99$, using a step,
$\Delta\sT=0.01$. Additional values are computed near the
critical point, to increase the resolution. The true value of
the reduced temperature on the last row is $\sT=0.9999$ or
$10\sT=9.999$.
All values equal unity at the critical point.
Index captions: A, C, E - intersections
between VDW and real isothermal curves; B - extremum point
of minimum; D - extremum point of maximum. Extremum points
are related to VDW isothermal curves, while their real
counterparts are flat in presence of both liquid and vapour
phase. To save aesthetics, 01 on head columns stands for unity.}
\label{t:vispo}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline
$10\sT$ & $10\sV_{\rm A}$ & $10\sV_{\rm B}$ & $01\sV_{\rm C}$ & $01\sV_{\rm D}$ & $01\sV_{\rm E}$ & $10\sP_B$ &
$10\sP_C$ & $10\sP_D$ \\
\hline
8.50 & 5.5336 & 6.7168 & 1.1453 & 1.7209 & 3.1276 & 0.4963 & 5.0449 & 6.2055 \\
8.60 & 5.6195 & 6.8003 & 1.1337 & 1.6821 & 2.9545 & 1.2750 & 5.3125 & 6.4005 \\
8.70 & 5.7116 & 6.8883 & 1.1225 & 1.6436 & 2.7909 & 2.0346 & 5.5887 & 6.6011 \\
8.80 & 5.8106 & 6.9814 & 1.1116 & 1.6052 & 2.6360 & 2.7752 & 5.8736 & 6.8076 \\
8.90 & 5.9176 & 7.0804 & 1.1009 & 1.5669 & 2.4889 & 3.4965 & 6.1674 & 7.0205 \\
9.00 & 6.0340 & 7.1860 & 1.0905 & 1.5285 & 2.3488 & 4.1984 & 6.4700 & 7.2401 \\
9.10 & 6.1615 & 7.2994 & 1.0804 & 1.4900 & 2.2151 & 4.8807 & 6.7816 & 7.4669 \\
9.20 & 6.3022 & 7.4221 & 1.0706 & 1.4511 & 2.0869 & 5.5430 & 7.1021 & 7.7014 \\
9.30 & 6.4593 & 7.5561 & 1.0610 & 1.4117 & 1.9634 & 6.1849 & 7.4318 & 7.9443 \\
9.40 & 6.6369 & 7.7040 & 1.0516 & 1.3715 & 1.8438 & 6.8058 & 7.7707 & 8.1963 \\
9.50 & 6.8412 & 7.8697 & 1.0425 & 1.3300 & 1.7271 & 7.4049 & 8.1188 & 8.4584 \\
9.60 & 7.0819 & 8.0593 & 1.0336 & 1.2867 & 1.6118 & 7.9811 & 8.4762 & 8.7319 \\
9.70 & 7.3756 & 8.2830 & 1.0249 & 1.2404 & 1.4960 & 8.5328 & 8.8429 & 9.0185 \\
9.80 & 7.7554 & 8.5611 & 1.0164 & 1.1892 & 1.3761 & 9.0576 & 9.2191 & 9.3209 \\
9.90 & 8.3091 & 8.9461 & 1.0081 & 1.1278 & 1.2430 & 9.5510 & 9.6048 & 9.6437 \\
9.95 & 8.7471 & 9.2353 & 1.0040 & 1.0876 & 1.1618 & 9.7830 & 9.8012 & 9.8157 \\
9.98 & 9.1727 & 9.5049 & 1.0016 & 1.0540 & 1.0972 & 9.9158 & 9.9202 & 9.9240 \\
9.99 & 9.4018 & 9.6456 & 1.0008 & 1.0377 & 1.0670 & 9.9585 & 9.9600 & 9.9614 \\
9.9$\bar{9}$ & 9.8035 & 9.8856 & 1.0001 & 1.0117 & 1.0204 & 9.9960 & 9.9960 & 9.9960 \\
\hline
\end{tabular}
\end{center}
\end{table}
The locus of the intersections between VDW and real
isothermal curves is represented in Fig.\,\ref{f:vris}
as a trifid curve, where the left, the right, and the
middle branch correspond to $\sV_{\rm A}$, $\sV_{\rm E}$, and
$\sV_{\rm C}$, respectively. The common starting point
coincides with the critical point. The locus of the
VDW isothermal curve extremum points is represented
in Fig.\,\ref{f:vris} as a dotted curve starting from
the critical point, where the left and the right
branch corresponds to minimum and maximum points,
respectively.
A fluid state can be represented in reduced variables
as ($\sV$, $\sP$, $\sT$), where one variable may be
expressed as a function of the remaining two, by use
of the reduced ideal gas equation of state, Eq.\,(\ref
{eq:ri}), or the reduced VDW equation of state,
Eq.\,(\ref{eq:rW1}). The formulation in terms of
reduced variables, Eqs.\,(\ref{eq:rv}), makes the
related equation of state universal i.e. it holds
for any fluid. Similarly, the Lane-Emden equation
expressed in polytropic (dimensionless) variables,
describes the whole class of polytropic gas spheres
with assigned polytropic index, in hydrostatic
equilibrium (e.g., Chandrasekhar 1939, Chap.\,IV,
\S4).
The states of two fluids with equal ($\sV$, $\sP$,
$\sT$), are defined as corresponding states.
The mere existence of an equation of state
yields the following result.
\begin{trivlist}
\item[\hspace\labelsep{\bf Law of corresponding
states.}] \sl
Given two fluids, the equality between two among
three reduced variables, $\sV$, $\sP$, $\sT$,
implies the equality between the remaining related
reduced variables i.e. the two fluids are in
corresponding states.
\end{trivlist}
The law was first formulated by van der Waals in
1880. For further details refer to specific
textbooks (e.g., LL67, Chap.\,VIII, \S85).
\section{Equation of state of astrophysical fluids}\label{macro}
Let macrogases be defined as two-component fluids
which interact only gravitationally. For assigned
density profiles, the virial theorem can be
formulated for each subsystem, where the potential
energy is the sum of the self potential energy of
the component under consideration, and the tidal
energy induced by the other one. The virial
theorem for subsystem can be expressed as a
macrogas equation of state in terms of dimensionless
variables, $X_V$, $X_p$, $X_T$, related to
axis ratio, mass ratio, virial (i.e. self + tidal)
potential energy ratio, respectively. The result
is (C10):
\begin{leftsubeqnarray}
\slabel{eq:Xa}
&& X_pX_VF_X(X_p,X_V)=X_T~~; \\
\slabel{eq:Xb}
&& X_p=m^2~~;\qquad X_V=\frac1y~~;\qquad X_T=\phi~~;
\label{seq:X}
\end{leftsubeqnarray}
where the function, $F_X$, depends on the selected
density profiles, $m$ is the (outer to inner component)
mass ratio, $y$ is the (outer to inner component)
axis ratio along a generic direction, $\phi$ is the
(outer to inner component)
virial energy ratio, and the density profiles are
restricted to be homeoidally striated. The variables,
$X_V$, $X_p$, $X_T$, play a similar role as the
volume, the pressure, and the temperature, for ordinary
fluids. Accordingly, $X_V$, $X_p$, $X_T$, may
be defined as macrovolume, macropressure, and macrotemperature,
respectively. For further details refer to the parent
paper (C10).
Macroisothermal curves on the $({\sf O}X_VX_p)$
plane exhibit a similar trend with respect to VDW
isothermal curves on the $({\sf O}Vp)$ plane, with
two main differences. First, no critical point
occurs for sufficiently mild density profiles,
where all macroisothermal curves are characterized
by two extremum points, one maximum and one minimum.
Second, a critical macroisothermal curve appears for sufficiently
steep density profiles, above (instead of below)
which macroisothermal curves exhibit extremum points.
For further details refer to the parent paper (C10)
and an earlier attempt (Caimmi and Valentinuzzi 2008).
The last inconvenient may be avoided turning
Eq.\,(\ref{seq:X}) into the following:
\begin{leftsubeqnarray}
\slabel{eq:Ya}
&& Y_pY_VF_Y(Y_p,Y_V)=Y_T~~; \\
\slabel{eq:Yb}
&& Y_p=\frac1{X_p}~~;\qquad Y_V=\frac1{X_V}~~;\qquad Y_T=\frac1{X_T}~~; \\
\slabel{eq:Yc}
&& F_Y(Y_p,Y_V)=F_X(X_p,X_V)~~;
\label{seq:Y}
\end{leftsubeqnarray}
as suggested in the parent paper (C10).
The existence of a phase transition moving along a selected
macroisothermal curve, where the path is a horisontal
line (``real'' macroisothermal curve) instead of a curve including
the extremum points (``actual'' macroisothermal curve),
must necessarily be assumed as a working
hypothesis, due to the analogy between VDW isothermal curves
and macroisothermal curves. Unlike the VDW equation of state,
Eq.\,(\ref{eq:pW}), the theoretical macrogas equation of state,
Eq.\,(\ref{eq:Ya}), is not analytically integrable, which implies
the procedure used for determining a selected
macroisothermal curve, must be numerically performed.
The main steps are (i) calculate the intersections,
$Y_{V_{\rm A}}$, $Y_{V_{\rm C}}$, $Y_{V_{\rm E}}$,
$Y_{V_{\rm A}}<Y_{V_{\rm C}}<Y_{V_{\rm E}}$, between the
generic horizontal line in the $({\sf O}Y_VY_p)$ plane,
$Y_p=$const, and the theoretical macrogas
equation of state, within the range, $Y_{p_{\rm B}}<Y_p<
Y_{p_{\rm D}}$, where B and D denote the extremum points
of minimum and maximum, respectively; (ii) calculate the
area of the regions, ${\sf ABC}$ and ${\sf CDE}$; (iii)
find the special value, $Y_p=Y_{p_{\rm C}}$, which makes
the two areas equal; (iv) trace the real macroisothermal
curve as a horisontal line connecting the points,
$(Y_{V_{\rm A}},Y_{p_{\rm A}})$, $(Y_{V_{\rm C}},Y_{p_{\rm C}})$,
$(Y_{V_{\rm E}},Y_{p_{\rm E}})$, $Y_{p_{\rm A}}=Y_{p_{\rm C}}=
Y_{p_{\rm E}}=Y_{p_c}$. For further details refer to an earlier
attempt (C10).
The procedure related to point (ii) above is rather cumbersome
and should be performed again with the new variables, $Y_{\rm V}$,
$Y_{\rm p}$, and $Y_{\rm T}$, with respect to an earlier attempt
(C10). For this reason, the current paper
shall be restricted to theoretical macroisothermal curves and
related extremum points. In order to preserve the
analogy with ideal and VDW gases, the tidal potential energy shall
be excluded and included, respectively, in the formulation of the
virial theorem and related equation of state. The following
cases shall be dealt with: UU macrogases, where no critical point
occurs; HH macrogases, where the critical point occurs; HN/NH
macrogases, where the critical point occurs.
In presence of the critical point, Eq.\,(\ref{seq:Y}) may be translated
into reduced variables, as:
\begin{leftsubeqnarray}
\slabel{eq:sYa}
&& \sY_p\sY_VF_Y(\sY_p,\sY_V)\frac{Y_{p_c}Y_{V_c}}{Y_{T_c}}=\sY_T~~; \\
\slabel{eq:sYb}
&& \sY_p=\frac{Y_p}{Y_{p_c}}~~;\qquad\sY_V=\frac{Y_V}{Y_{V_c}}~~;
\qquad \sY_T=\frac{Y_T}{Y_{T_c}}~~; \\
\slabel{eq:sYc}
&& F_Y(\sY_p,\sY_V)=F_Y(\sY_pY_{p_c},\sY_VY_{V_c})~~;
\label{seq:sY}
\end{leftsubeqnarray}
where $Y_{p_c}$, $Y_{V_c}$, $Y_{T_c}$, are the values of
the variables related to the critical point. The
counterpart of Eq.\,(\ref{eq:sYa}) for ideal macrogases
reads:
\begin{lefteqnarray}
\label{eq:sYi}
&& \sY_p\sY_VG_Y(\sY_p,\sY_V)\frac{Y_{p_c}Y_{V_c}}{Y_{T_c}}=\sY_T~~;
\end{lefteqnarray}
where $G_Y(\sY_p,\sY_V)$ is the expression of
$F_Y(\sY_p,\sY_V)$ where the interaction terms are
omitted. For further details refer to an earlier
attempt (C10). Accordingly, the equation
of state for ideal macrogases
where $G_Y(\sY_p,\sY_V)Y_{p_c}/$ $(Y_{V_c}Y_{T_c})=3/8$,
coincides with its counterpart related to ideal gases,
conformly to Eq.\,(\ref{eq:ri}).
Macroisothermal curves related to IUU (tidal potential energy
excluded) and AUU (tidal potential energy included)
macrogases, are plotted in Fig.\,\ref{f:uuso}, left and
right panel, respectively, for values of the
macrotemperature, $Y_{\rm T}=20/23$, 20/22, 20/21, 20/20, 20/19,
20/18, from bottom to top. The coordinates, $Y_{\rm V}$,
$Y_{\rm p}$, $Y_{\rm T}$, may be
conceived as normalized to their fictitious critical
counterparts, $Y_{V_{\rm c}}=1$, $Y_{p_{\rm c}}=1$,
$Y_{T_{\rm c}}=1$ (C10). The comparison with ideal and
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{uuso100.eps}
\caption{Macroisothermal curves related to IUU (left panel)
and AUU (right panel) macrogases, respectively.
Macroisothermal
curves (from bottom to top) correspond to $Y_{\rm T}=$
20/23, 20/22, 20/21, 20/20, 20/19, 20/18. No critical
macroisothermal curve exists, above which the extremum
points disappear. The coordinates, $Y_{\rm V}$,
$Y_{\rm p}$, $Y_{\rm T}$, may be
conceived as normalized to their fictitious critical
counterparts, $Y_{V_{\rm c}}=1$, $Y_{p_{\rm c}}=1$,
$Y_{T_{\rm c}}=1$.}
\label{f:uuso}
\end{center}
\end{figure*}
VDW gases, plotted in Fig.\,\ref{f:viso}, shows a
similar trend, except the absence of a critical
macroisothermal curve, above which the extremum
points disappear.
Macroisothermal curves
related to IHH (tidal potential energy excluded)
and AHH (tidal potential energy included)
macrogases, are plotted in
Fig.\,\ref{f:hhso}, left and right panels,
respectively, for
infinitely extended subsystems and
values of the reduced macrotemperature,
$\sY_{\rm T}=Y_{\rm T}/Y_{T_{\rm c}}=$ 20/23, 20/22,
20/21, 20/20, 20/19, 20/18, from bottom to top.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{hhso100.eps}
\caption{Macroisothermal curves
($\sY_{\rm p}=Y_{\rm p}/Y_{p_c}$ vs. $\sY_{\rm V}=Y_{\rm V}/Y_{V_c}$)
related to IHH (left panels)
and AHH (right panels) macrogases, respectively, for
infinitely extended subsystems. Macroisothermal
curves (from bottom to top) correspond to $\sY_{\rm T}=Y_{\rm T}/
Y_{T_{\rm c}}=$20/23, 20/22, 20/21, 20/20, 20/19, 20/18.
The general case of bounded subsystems makes only little changes.}
\label{f:hhso}
\end{center}
\end{figure*}
The general case of bounded subsystems makes only little changes.
The comparison with ideal and VDW gases,
plotted in Fig.\,\ref{f:viso}, shows a similar trend
where macroisothermal curves are more extended along the
horisontal direction with respect to isothermal curves.
Macroisothermal curves
related to IHN/NH (tidal potential energy excluded)
and AHN/NH (tidal potential energy included)
macrogases, are plotted in
Fig.\,\ref{f:hnso}, left and right panels,
respectively, for
infinitely extended subsystems and
values of the reduced macrotemperature,
$\sY_{\rm T}=Y_{\rm T}/Y_{T_{\rm c}}=$ 20/23, 20/22,
20/21, 20/20, 20/19, 20/18, from bottom to top.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{hnso100.eps}
\caption{Macroisothermal curves
($\sY_{\rm p}=Y_{\rm p}/Y_{p_c}$ vs. $\sY_{\rm V}=Y_{\rm V}/Y_{V_c}$)
related to IHN/NH (left panels, to be noted the scale difference)
and AHN/NH (right panels) macrogases, respectively, for
infinitely extended subsystems. Macroisothermal
curves (from bottom to top) correspond to $\sY_{\rm T}=Y_{\rm T}/
Y_{T_{\rm c}}=$23/20, 22/20, 21/20, 20/20, 19/20, 18/20.
The general case of bounded subsystems makes only little changes
for AHN/NH macrogases, while the scale difference tends to disappear
for IHN/NH macrogases.}
\label{f:hnso}
\end{center}
\end{figure*}
The general case of bounded subsystems makes only little changes
for AHN/NH macrogases, while the scale change tends to disappear
for IHN/NH macrogases.
The comparison with ideal and VDW gases,
plotted in Fig.\,\ref{f:viso}, shows a similar trend
where macroisothermal curves are more extended along the
horisontal direction with respect
to isothermal curves, and the occurrence of a scale difference
for ideal macrogases. The last is due to a mass divergence
for infinitely extended N density profiles, which makes tidal
effects higly increase.
The comparison between the VDW critical isothermal
curve and its counterparts related to HH and HN/NH
macrogases is shown in Fig.\,\ref{f:mris}.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.8]{mris100.eps}
\caption{Comparison between VDW critical isothermal curve
(full), HH critical macroisothermal curve (dotted) and
HN/NH critical macroisothermal curve (dot-dashed). With
regard to ordinary fluids, the vapour and the liquid phase
coexist within the bell-shaped region bounded by the dashed
curve and, in addition, $Y_{\rm V}=V$, $Y_{\rm p}=p$. More
extended (along the horisontal direction) bell-shaped regions
are expected for HH and HN/NH macroisothermal curves. The
critical point belongs to all curves. Different letters
denote the expected location of different astrophysical systems.
Caption: EG - elliptical galaxies; S0 - lenticular galaxies;
SG - spiral galaxies including barred; IG - irregular galaxies;
DS - dwarf spheroidal
galaxies; GC - globular clusters; CG - clusters of galaxies;
WC - wholly gaseous clouds i.e. in absence of star formation;
WG - (hypothetical) wholly gaseous galaxies i.e. in absence of
star formation.}
\label{f:mris}
\end{center}
\end{figure*}
The broken curve is the same as in Fig.\,\ref{f:vris}.
Accordingly, the vapour and the liquid phase of
ordinary fluids coexist within the bell-shaped
region bounded by the broken curve.
Both HH and HN/NH macroisothermal curves
are more extended along the horisontal direction
with respect to VDW isothermal
curves, which implies a more flattened counterpart
of the above mentioned bell-shaped region.
The critical point belongs to all curves.
\section{Discussion and conclusion}
\label{disc}
Tidal interactions between neighbourhing
bodies span across the whole admissible
range of lengths in nature: from, say,
atoms and molecules to galaxies and
clusters of galaxies i.e. from micro to
macrocosmos. Ordinary fluids are collisional,
which makes the stress tensor be isotropic
and the velocity distribution obey the
Maxwell's law. Tidal interactions
(electromagnetic in nature) therein act
between colliding particles (e.g., LL67,
Chap.\,VII, \S74). Astrophysical fluids
are collisionless, which makes the stress
tensor be anisotropic and the velocity
distribution no longer obey the Maxwell's
law. Tidal interactions (gravitational
in nature) therein act between a single
particle and the system as a whole (e.g.,
C10).
In both cases, an equation of state can
be formulated in reduced variables: the
VDW equation for ordinary fluids and an
equation which depends on the density
profiles for astrophysical fluids.
For sufficiently mild density profiles,
macroisothermal curves are characterized
by the occurrence of two extremum points,
similarly to isothermal curves where a
transition from liquid to gaseous phase
takes place, or vice versa. For
sufficiently steep density profiles, a
critical macroisothermal curve exhibits
a single horisontal inflexion point,
which defines the critical point.
Macroisothermal curves below and above
the critical one, show two or no extremum
point, respectively, in complete analogy
with VDW isothermal curves. In any
case, the existence of an equation of
state in reduced variables implies the
validity of the law of corresponding
states for macrogases with assigned
density profiles.
For astrophysical fluids, the existence
of a phase transition must necessarily
be assumed as a working hypothesis by
analogy with ordinary fluids. The
phase transition has to be conceived
between gas and stars, and the (${\sf O}
\sY_V\sY_p$) plane may be divided into
three parts, namely (i) a region bounded
by the critical macroisothermal curve on
the left of the critical point, and
the locus of onset of phase transition
on the right of the critical point, where
only gas exists; (ii) a region bounded by
the critical macroisothermal curve on
the left of the critical point,
the locus of onset of phase transition
on the left of the critical point, and
the vertical axis, where only stars exist;
(iii) a region bounded by
the locus of onset of phase transition,
and the horisontal axis, where gas and
stars coexist.
The locus of onset of phase transition,
not shown in Fig.\,\ref{f:mris} for
reasons explained above, is similar to
its counterpart related to ordinary fluids,
represented by the bell-shaped curve in
Fig.\,\ref{f:mris}, but more extended
along the horisontal direction.
In this view, elliptical and S0 galaxies
lie on (ii) region unless hosting hot
interstellar gas, and the same holds for
globular clusters; spiral, irregular,
and dwarf spheroidal galaxies lie on
(iii) region, and the same holds for
cluster of galaxies; gas clouds in
absence of star formation lie on
(i) region, and the same holds for
hypothetic galaxies with no stars.
In conclusion, van der Waals' two
great discoveries, more specifically
a gas equation of state where tidal
interactions between molecules are
taken into account, and the law of
corresponding states, related to
microcosmos, find a counterpart
with regard to macrocosmos.
After a century since the awarding
of the Nobel Prize in Physics, van
der Waals' ideas are still valid and
helpful to day for a full understanding
of the universe.
|
{
"timestamp": "2012-10-16T02:01:28",
"yymm": "1210",
"arxiv_id": "1210.3688",
"language": "en",
"url": "https://arxiv.org/abs/1210.3688"
}
|
"\\section{Introduction}\r\n \r\nSuppose that one has $n$ copies of a quantum system each in the sam(...TRUNCATED)
| {"timestamp":"2013-08-30T02:06:24","yymm":"1210","arxiv_id":"1210.3749","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\r\n\r\nRicci solitons are precisely those Riemannian metrics that are `nice(...TRUNCATED)
| {"timestamp":"2012-10-16T02:00:56","yymm":"1210","arxiv_id":"1210.3656","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\r\n\r\nWe consider spherically symmetric motions of atmosphere governed by (...TRUNCATED)
| {"timestamp":"2013-07-16T02:07:20","yymm":"1210","arxiv_id":"1210.3670","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\\label{intro}\n\nParticle-based simulation methods have a long and succes(...TRUNCATED)
| {"timestamp":"2012-12-14T02:02:49","yymm":"1210","arxiv_id":"1210.3743","language":"en","url":"https(...TRUNCATED)
|
"\\section*{References} \n[1] Brennen, C. E., 1970, “Cavity Surface Wave Patterns and General Appe(...TRUNCATED)
| {"timestamp":"2012-10-30T01:05:10","yymm":"1210","arxiv_id":"1210.3676","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\\label{Intro}\n\nThe topic of vector meson interactions with nuclei has a(...TRUNCATED)
| {"timestamp":"2012-10-16T02:02:16","yymm":"1210","arxiv_id":"1210.3738","language":"en","url":"https(...TRUNCATED)
|
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