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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 3
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 49851)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 3
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
dict |
|---|---|
\section{Introduction}
\label{intro}
Star clusters are a powerful tool in the investigation of Galaxy structure and dynamics, star
formation and evolution processes, and as observational constraints to N-body codes. This applies
especially to the long-lived and populous globular clusters (GCs) that, because of their relatively
compact nature, can be observed in most regions of the Galaxy, from near the center to the remote
halo outskirts.
In general terms, the structure of most star clusters can be described by a rather dense core and
a sparse halo, but with a broad range in the concentration level. In this context, the standard
picture of a GC assumes a isothermal central region and a tidally truncated outer region (e.g.
\citealt{Binney1998}). Old GCs, in particular, can be virtually considered as dynamically relaxed
systems (e.g. \citealt{NoGe06}). During their lives clusters are continually affected by internal
processes such as mass loss by stellar evolution, mass segregation and low-mass star evaporation,
and external ones such as tidal stress and dynamical friction e.g. from the Galactic bulge, disk
and giant molecular clouds (e.g. \citealt{Khalisi07}; \citealt{Lamers05}; \citealt{GnOs97}). Over
a Hubble time, these processes tend to decrease cluster mass, which may accelerate the core collapse
phase for some clusters (\citealt{DjMey94}, and references therein). Consequently, these processes,
combined with the presence of a central black hole (in some cases) and physical conditions associated
to the initial collapse, can affect the spatial distribution of light (or mass) both in the central
region and at large radii (e.g. \citealt{GLO99}; \citealt{NoGe06}).
It is clear from the above that crucial information related to the early stages of Galaxy formation,
and to the cluster dynamical evolution, may be imprinted in the present-day internal structure and
large-scale spatial distribution of GCs (e.g. \citealt{MvdB05}; \citealt{GCProp}). To some extent,
this reasoning can be extended to the open clusters (OCs), especially the young, which are important
to determine the spiral arm and disk structures and the rotation curve of the Galaxy (e.g. \citealt{Friel95};
\citealt{DiskProp}). Consequently, the derivation of reliable structural parameters of star clusters,
GCs in particular, is fundamental to better define their parameter space. This, in turn, may result
in a deeper understanding of the formation and evolution processes of the star clusters themselves
and the Galaxy.
Three different approaches have been used to derive structural parameters of star clusters.
The more traditional one is based on the surface-brightness profile (SBP), which considers
the spatial distribution of the brightness of the component stars, usually measured in circular
rings around the cluster center. The compilation of Harris (1996, and the 2003 update\footnote{\em
http://physun.physics.mcmaster.ca/Globular.html}) presents a basically uniform set of parameters for
150 Galactic GCs. Among their structural parameters, the core (\mbox{$\rm R_c$}), half-light (\mbox{$\rm R_{hL}$}) and tidal
(\mbox{$\rm R_t$}) radii, as well as the concentration parameter $c=\log(\mbox{$\rm R_t$}/\mbox{$\rm R_c$})$, were based mostly on the
SBP database of \citet{TKD95}. SBPs do not necessarily require cluster distances to be known,
since the physically relevant information contained in them is essentially related to the relative
brightness of the member stars. In principle, it is easy to measure integrated light. However, SBPs
are more efficient near the cluster center than in the outer parts, where noise and background
starlight may be a major contributor. Another potential source of noise is the random presence of
bright stars, either from the field or cluster members, especially outside the central region in
the less-populous GCs or most of the OCs. Structural parameters derived from such SBPs would certainly
be affected. One way to minimise this effect is the use of wide rings throughout the whole radius range,
but this would cause spatial resolution degradation on the profiles, especially near the center.
The obvious alternative to SBPs is to use star counts to build radial density profiles (RDPs),
in which only the projected number-density of stars is taken into account, regardless of the
individual star brightness. This technique is particularly appropriate for the outer parts, provided
a statistically significant, and reasonably uniform, comparison field is available to tackle the
background contamination. On the other hand, contrary to SBPs, RDPs are less efficient in central
regions of populous clusters where the density of stars (crowding) may become exceedingly large.
In such cases it may not be possible to resolve individual stars with the available technology.
Finally, a more physically significant profile can be built by mapping the cluster's stellar mass
distribution, which essentially determines the gravitational potential and drives most of the dynamical
evolution. However, mass density profiles (MDPs) not only are affected by the same technical problems
as the RDPs but, in addition, the cluster distance, age and a reliable mass-luminosity relation are
necessary to build them.
In principle, the three kinds of profiles are expected to yield different values for the structural
parameters under similar photometric conditions, since each profile is sensitive to different cluster
parameters, especially the age and dynamical state. Qualitatively, the following effects, basically
related to dynamical state, can be expected. Large-scale mass segregation drives preferentially low-mass
stars towards large radii (while evaporation pushes part of these stars beyond the tidal radius, into
the field), and high-mass stars towards the central parts of clusters. If the stellar mass distribution
of an evolved cluster can be described by a spatially variable mass function (MF) flatter at the cluster
center than in the halo, the resulting RDP (and MDP) radii should be larger than SBP ones. The differences
should be more significant for the core than the half and tidal radii, since the core would contain,
on average, stars more massive than the halo and especially near the tidal radius. Besides, the
presence of bright stars preferentially in the central parts of young clusters (\citealt{DetAnalOCs}
and references therein) should as well lead to smaller SBP core and half-light radii than the respective
RDP ones.
Another relevant issue is related to depth-limited photometry. When applied to the observation
of objects at different distances, depth-limited photometry samples stars with different brightness
(or mass), especially at the faint (or low-mass) end. Thus, it would be interesting to quantify the
changes produced in the derived parameters when RDPs, MDPS and SBPs are built with depth-limited
photometry, as well as to check how the structural parameters derived from one type of profile relate
to the equivalent radii measured in the other profiles.
In the present work we face the above issues by deriving structural parameters of star clusters
built under controlled conditions, in which the radial distribution of stars follows a pre-established
analytical profile, and field stars are absent. Effects introduced by mass segregation (simulated
by a spatially variable mass function), age and structure are also considered. This work focuses
on profiles built in the near-infrared range. The main goal of the present work is to examine
relations among structural parameters measured in the different radial profiles, built under ideal
conditions, especially noise-free photometry and as small as possible statistical uncertainties (using
a large number of stars). In this sense, the results should be taken as upper-limits.
\begin{table*}
\caption[]{Model star cluster specifications}
\label{tab1}
\renewcommand{\tabcolsep}{2.65mm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ccccccccrcccc}
\hline\hline
Model&$R_t/R_c$&c&$\chi_0$&$\chi_t$&Age&\mbox{$\rm [Fe/H]$}&$m_i$&$m_s$&$\langle m\rangle$
&$\rm M_J(TO)$&$\rm M_J(bright)$&$\rm M_J(faint)$\\
& & & & &(Myr)&&(\mbox{$\rm M_\odot$})&(\mbox{$\rm M_\odot$})&(\mbox{$\rm M_\odot$})&(mag)&(mag)&(mag)\\
(1) &(2) &(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10)&(11) &(12) &(13)\\
\hline
GC-A &5&0.7&0.00&1.35&$10^4$&$-1.5$&0.15&1.02&0.43&$+2.86$&$-2.14$&$+9.12$\\
GC-B&20&1.3&0.00&1.35&$10^4$&$-1.5$&0.15&1.02&0.43&$+2.86$&$-2.14$&$+9.12$\\
GC-C&20&1.3&0.00&0.00&$10^4$&$-1.5$&0.15&1.02&0.46&$+2.86$&$-2.14$&$+9.12$\\
GC-D&40&1.6&0.00&1.35&$10^4$&$-1.5$&0.15&1.02&0.43&$+2.86$&$-2.14$&$+9.12$\\
OC-A&15&1.2&0.30&1.35&$10^3$&$~~0.0$&0.15&2.31&0.59&$+0.32$&$-2.68$&$+9.18$\\
OC-B&15&1.2&0.30&1.35&$100$&$~~0.0$&0.15&5.42&0.92&$-1.82$&$-4.82$&$+9.18$\\
OC-C&15&1.2&0.30&1.35&$10$&$~~0.0$&0.15&18.72&1.76&$-4.82$&$-8.82$&$+9.18$\\
\hline
\end{tabular}
\begin{list}{Table Notes.}
\item Col.~3: concentration parameter $c=\log(\mbox{$\rm R_t$}/\mbox{$\rm R_c$})$. Cols.~4 and 5: mass function slopes
at the cluster center and tidal radius. Cols.~8-10: lower, upper and average star mass. Col.~11:
absolute J magnitude at the turnoff (TO). Cols.~12 and 13: absolute J magnitude at the bright and
faint ends.
\end{list}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig1.eps}}
\caption{Model star cluster specifications. Panel (a): a random selection of $n$ in the
range $0\leq n\leq1$ produces King-like RDPs in the range $0\leq R\leq\mbox{$\rm R_t$}$ (see Eq.~\ref{eq2}).
Panel (b): Radially-variable mass function slopes $\left(\frac{dN}{dm}\propto m^{-(1+\chi)}\right)$
used in the models. Panel (c): Padova isochrones used to simulate the mass-luminosity relation
of the star cluster models. The 10\,Gyr, $\mbox{$\rm [Fe/H]$}=-1.5$ metallicity isochrone is adopted in the
globular cluster models. Panel (d): distribution of concentration parameters of the GCs in
H03 with peaks at $c\approx1.6,~1.3,~{\rm and}~0.7$. Panel (e): model fraction of stars
brighter than $M_J=M_{J_{TO}}+\Delta_{TO}$. In all cases, the fraction of stars brighter
than the TO ($M_{J_{TO}}$) is below the $1\%$ level.}
\label{fig1}
\end{figure}
This work is structured as follows. In Sect.~\ref{ModelSCs} we present the star cluster models
and build radial profiles with depth-limited photometry. In Sect.~\ref{Struc} we derive structural
parameters from each profile, discuss their dependence on depth, and compare similar radii derived
from the different types of profiles. In Sect.~\ref{N6397} we compare relations derived from model
parameters with those of the nearby GC NGC\,6397. Concluding remarks are given in Sect.~\ref{Conclu}.
\section{The model star clusters}
\label{ModelSCs}
For practical reasons, the model star clusters are simulated by first establishing the number-density
radial distribution. The approach we follow is to build star clusters of different ages and
concentration parameters, with the spatial distribution of stars truncated at the tidal radius
(\mbox{$\rm R_t$}). Stars are distributed with distances to the cluster center in the range $0\leq R\leq\mbox{$\rm R_t$}$,
with the $R$ coordinate having a number-frequency given by a function similar to a \citet{King62}
three-parameter surface-brightness profile. The mass and brightness of each star are subsequently
computed according to a pre-defined mass function and mass-luminosity relation consistent with the
model age. The last step is required for the derivation of the MDP and SBPs.
We point out that different, more sophisticated analytical models have also been used to fit the SBPs
of Galactic and extra-Galactic GCs, other than \citet{King62} profile. The most commonly used are
the single-mass, modified isothermal sphere of \citet{King66} that is the basis of the Galactic
GC parameters given by \citet{TKD95} and H03, the modified isothermal sphere of \citet{Wilson75},
that assumes a pre-defined stellar distribution function (which results in more extended envelopes
than \citealt{King66}), and the power-law with a core of \citet{EFF87} that has been fit to massive
young clusters especially in the Magellanic Clouds (e.g. Mackey \& Gilmore 2003a,b,c). Each function
is characterised by different parameters that are somehow related to the cluster structure. However,
the purpose here is not to establish a ``best'' fitting function of the structure of star clusters in
general. Instead, we want to quantify changes in the structural parameters, derived from RDPs, MDPs
and SBPs of star clusters with the stellar distribution assumed to follow an analytical function,
under different photometric conditions. We expect that changes in a given parameter should have
a small dependence, if any at all, on the adopted functional form.
The adopted King-like radial distribution function is expressed as
\begin{equation}
\label{eq1}
\frac{dN}{2\pi\,R\,dR}=\sigma_0\left[\frac{1}{\sqrt{1+(R/R_c)^2}} -
\frac{1}{\sqrt{1+(R_t/R_c)^2}}\right]^2,
\end{equation}
where $\sigma_0$ is the projected number-density of stars at the cluster center, and \mbox{$\rm R_c$}\ and
\mbox{$\rm R_t$}\ are the core and tidal radii, respectively. Since structural differences are basically
controlled by the ratio $\mbox{$\rm R_t$}/\mbox{$\rm R_c$}$, we set $\mbox{$\rm R_c$}=1$ in all models. Such a King-like RDP (for
$\sigma_0=1.0$) is obtained by numerically inverting the relation (see App.~\ref{Transf})
\begin{equation}
\label{eq2}
n(R) = \frac{x^2-4u(\sqrt{1+x^2}-1)+u^2\ln(1+x^2)}{u^2\ln{u^2}-(u-1)(3u-1)},
\end{equation}
where $x\equiv R/R_c$ and $u^2\equiv 1+(R_t/R_c)^2$. Thus, a random selection of numbers
in the range $0\leq n\leq1$ produces a King-like radial distribution of stars with the
radial coordinate in the range $0\leq R/R_t\leq1$. The $R/R_t$ curves as a function of
$n$ for the models considered in this work are shown in Fig.~\ref{fig1} (Panel a).
Once a given star has been assigned a radial coordinate, its mass is computed with a
probability proportional to the mass function
\begin{equation}
\label{eq3}
\frac{dN}{dm}\propto m^{-(1+\chi)},
\end{equation}
where the slope varies with $R$ according to $\chi=\chi(R)=\chi_t + (\chi_t-\chi_0)(R/R_t-1)$,
where $\chi_0$ and $\chi_t$ are the mass function slopes at the cluster center and tidal radius,
respectively (Table~\ref{tab1} and Fig.~\ref{fig1}). Thus, the presence of large-scale mass
segregation in a star cluster can be characterised by a slope $\chi_0$ flatter than $\chi_t$.
Mass values distributed according to Eq.~\ref{eq3} are obtained by randomly selecting numbers in
the range $0\leq n\leq1$ and using them in the relation of mass with $n~\rm{and~} \chi$
(App.~\ref{Transf})
\begin{equation}
\label{eq4}
m=\left\{
\begin{array}{lc}
m_i\,(m_s/m_i)^n, & \rm{for~\chi=0.0,}\\
m_s/[(1-n)(m_s/m_i)^\chi+n]^{1/\chi}, & \rm{otherwise},
\end{array}
\right .
\end{equation} where $m_i$ and $m_s$ are the lower and upper mass values considered in the models
(Table~\ref{tab1}).
In what follows we adopt the 2MASS\footnote{\em http://www.ipac.caltech.edu/2mass/releases/allsky/}
photometric system to build SBPs. Finally, the 2MASS \mbox{$\rm J$}, \mbox{$\rm H$}\ and \mbox{$\rm K_s$}\ magnitudes for each star
are obtained according to the mass-luminosity relation taken from the corresponding model
(Table~\ref{tab1}) Padova isochrone (\citealt{Girardi02}). For illustrative purposes the model
isochrones are displayed in Fig.~\ref{fig1} (panel c).
The set of models considered here is intended to be objectively representative of the star cluster
parameter space. For globular clusters we use the standard age of 10\,Gyr and the spatially
uniform metallicity $\mbox{$\rm [Fe/H]$}=-1.5$, which is typical of the metal-poor Galactic GCs (e.g.
\citealt{GCProp}). However, we note that abundance variations have been suggested to occur within
GCs (e.g. \citealt{Gratton04}). Basically, small to moderate metallicity gradients would produce slight
changes in the colour and magnitude of the stars in different parts of the cluster, which has no effect
on the (star-count derived) RDPs and MDPs. The effect on the SBPs may be small as well, provided that
the magnitude bin used to build the SBPs is wide enough to accommodate such magnitude changes.
As for the core/tidal structure
we consider the ratios $R_t/R_c=40,~20,~15,~{\rm and}~5$, or equivalently the concentration parameters
$c=\log{(R_t/R_c)\approx1.6,~1.3,~1.2,~{\rm and}~0.7}$, which roughly correspond to the peaks in the
distribution of $c$ values presented by the regular (non-post core collapse) GCs given in H03
(Fig.~\ref{fig1}, panel d). Models GC-A, B and D take into account mass segregation by means of a
flat ($\chi_0=0.00$) mass function at the center and a Salpeter (1955) IMF ($\chi_t=1.35$) at the
tidal radius. GC-C model is similar to GC-B, except that it considers a uniform, heavily depleted
MF ($\chi_0=0.00$) throughout the cluster. OCs are represented by solar-metallicity models with the
ages 10\,Myr (to allow for the presence of bright stars in young OCs), 100\,Myr (somewhat evolved
OCs) and 1\,Gyr (intermediate-age OCs), $R_t/R_c=15$ ($c\approx1.2$) and a spatially variable MF
(Table~\ref{tab1}). The values of $c$ and the core/halo MF slopes are representative of OCs
(\citealt{DetAnalOCs}). Another effect not considered here is differential absorption.
In principle, low to moderate differential absorption should have a minimum effect on the radial
profiles, because of the same reasons as those given above for the metallicity gradient. High values,
on the other hand, would affect RDPs as well, because of a radially-dependent loss of stars due to
depth-limited photometry. However, inclusion of this effect is beyond the scope of the present work.
As expected, the fraction of stars brighter than the turnoff (TO) in the resulting star cluster
models is significantly smaller than 1\% (Fig.~\ref{fig1}, panel e). Thus, we had to use a total
number of stars of $1\times10^9$ in all models, so that the radial profiles resulted statistically
significant (small $1\sigma$ Poisson error bars) especially at the shallowest magnitude depth.
\subsection{Depth-varying radial profiles}
\label{DeptVP}
The radial profiles were built considering all stars brighter than a given magnitude threshold,
with the TO as reference. At the bright end, statistically significant GC profiles were obtained
for $\mbox{$\rm \Delta_{TO}$}\equiv M_{J,th}-M_{J,TO}=-5$, where $M_{J,th}$ and $M_{J,TO}$ are the threshold
and TO absolute magnitudes in the 2MASS \mbox{$\rm J$}\ band. At the faint end, GC-models have $\mbox{$\rm \Delta_{TO}$}=6.3$.
OC models have $\mbox{$\rm \Delta_{TO}$}=-3~{\rm and~} -4$ at the bright end, and $\mbox{$\rm \Delta_{TO}$}=8.9,~11.0,~{\rm and}~14.0$,
at the faint end.
Starting at the bright magnitude end, RDPs, MDPs and SBPs were built considering stars with
the \mbox{$\rm J$}\ magnitude brighter than a given faint threshold, with the magnitude depth increasing
in steps of $\mbox{$\rm \Delta_{TO}$}=1$, up to the respective faint magnitude end.
Figure~\ref{fig2} displays a selection of profiles corresponding to both extremes in magnitude
depths, for the GC-D and OC-C models. These profiles are representative of the whole set of models,
especially in terms of the small uncertainties associated with each radial coordinate. Reflecting
the large differences in the number of stars at different photometric depths, the central values
of the number and mass densities, and surface-brightness, vary significantly from the shallowest
to the deepest profiles.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig2.eps}}
\caption{A selection of RDPs (top panels), MDPs (middle) and 2MASS \mbox{$\rm J$}\ magnitude SBPs (bottom)
that illustrate structural changes under different magnitude depths. Arbitrary units (au) are
used both for the radial coordinate and projected area.}
\label{fig2}
\end{figure}
\begin{table*}
\caption[]{Model star cluster structural parameters for different photometric depths}
\label{tab2}
\renewcommand{\tabcolsep}{1.3mm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{cccccccccccc}
\hline\hline
&\multicolumn{3}{c}{RDP}&&\multicolumn{3}{c}{MDP}&&\multicolumn{3}{c}{SBP (\mbox{$\rm J$}\ band)}\\
\cline{2-4}\cline{6-8}\cline{10-12}
$\Delta_{TO}$&\mbox{$\rm R_c$}&\mbox{$\rm R_{hSC}$}&\mbox{$\rm R_t$}&&\mbox{$\rm R_c$}&\mbox{$\rm R_{hM}$}&\mbox{$\rm R_t$}&&\mbox{$\rm R_c$}&\mbox{$\rm R_{hL}$}&\mbox{$\rm R_t$}\\
(mag)&(au)&(au)&(au)&&(au)&(au)&(au)&&(au)&(au)&(au)\\
(1) &(2) &(3) &(4) &&(5) &(6) &(7) &&(8) &(9) &(10) \\
\hline
&\multicolumn{11}{c}{Model: GC-A; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=5.0$}\\
\cline{2-12}
$-5.0$&$0.78\pm(\dag)$&$1.02\pm(\dag)$&$4.61\pm0.01$&&$0.78\pm(\dag)$&$1.02\pm(\dag)$&$4.61\pm0.01$&&$0.75\pm0.01$&$1.01\pm(\dag)$&$4.80\pm0.01$\\
$~0.0$&$0.76\pm(\dag)$&$1.02\pm(\dag)$&$4.77\pm0.01$&&$0.76\pm(\dag)$&$1.02\pm(\dag)$&$4.77\pm0.01$&&$0.75\pm0.01$&$1.01\pm(\dag)$&$4.79\pm0.01$\\
$+6.3$&$1.00\pm(\dag)$&$1.19\pm(\dag)$&$5.00\pm0.01$&&$0.92\pm(\dag)$&$1.14\pm(\dag)$&$4.91\pm(\dag)$&&$0.75\pm0.01$&$1.03\pm(\dag)$&$4.80\pm0.01$\\
\hline
&\multicolumn{11}{c}{Model: GC-B; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=20.0$}\\
\cline{2-12}
$-5.0$&$0.87\pm0.01$&$2.03\pm(\dag)$&$17.31\pm0.08$&&$0.87\pm0.01$&$2.03\pm(\dag)$&$17.31\pm0.08$&&$0.86\pm0.01$&$2.04\pm0.01$&$17.82\pm0.05$\\
$~0.0$&$0.83\pm0.01$&$2.03\pm(\dag)$&$18.72\pm0.08$&&$0.83\pm0.01$&$2.03\pm(\dag)$&$18.72\pm0.08$&&$0.86\pm0.01$&$2.03\pm(\dag)$&$17.80\pm0.03$\\
$+6.3$&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$20.00\pm(\dag)$&&$0.95\pm(\dag)$&$2.27\pm(\dag)$&$19.28\pm0.03$&&$0.86\pm0.01$&$2.05\pm(\dag)$&$17.80\pm0.02$\\
\hline
&\multicolumn{11}{c}{Model: GC-C; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=20.0$}\\
\cline{2-12}
$-5.0$&$1.00\pm(\dag)$&$2.38\pm0.01$&$20.02\pm0.03$&&$1.00\pm(\dag)$&$2.38\pm0.01$&$20.02\pm0.04$&&$1.00\pm(\dag)$&$2.38\pm0.01$&$19.94\pm0.06$\\
$~0.0$&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$20.00\pm0.01$&&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$20.00\pm0.01$&&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$19.95\pm0.03$\\
$+6.3$&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$20.00\pm(\dag)$&&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$20.00\pm(\dag)$&&$1.00\pm(\dag)$&$2.39\pm(\dag)$&$19.97\pm0.03$\\
\hline
&\multicolumn{11}{c}{Model: GC-D; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=40.0$}\\
\cline{2-12}
$-5.0$&$0.90\pm0.01$&$2.81\pm0.02$&$33.96\pm0.19$&&$0.90\pm0.01$&$2.81\pm0.02$&$33.96\pm0.19$&&$0.91\pm0.01$&$2.82\pm(\dag)$&$34.18\pm0.05$\\
$~0.0$&$0.86\pm0.01$&$2.82\pm(\dag)$&$37.15\pm0.17$&&$0.86\pm0.01$&$2.82\pm(\dag)$&$37.15\pm0.17$&&$0.91\pm0.01$&$2.82\pm(\dag)$&$34.00\pm0.05$\\
$+6.3$&$1.00\pm(\dag)$&$3.30\pm(\dag)$&$39.99\pm0.01$&&$0.96\pm(\dag)$&$3.14\pm(\dag)$&$38.51\pm0.07$&&$0.91\pm0.01$&$2.82\pm(\dag)$&$34.20\pm0.04$\\
\hline
&\multicolumn{11}{c}{Model: OC-A; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=15.0$}\\
\cline{2-12}
$-3.0$&$0.82\pm0.01$&$1.70\pm(\dag)$&$12.85\pm0.07$&&$0.82\pm0.01$&$1.70\pm(\dag)$&$12.85\pm0.07$&&$0.81\pm0.01$&$1.72\pm(\dag)$&$13.18\pm0.02$\\
$~0.0$&$0.78\pm0.01$&$1.72\pm(\dag)$&$13.78\pm0.06$&&$0.78\pm0.01$&$1.72\pm(\dag)$&$13.78\pm0.06$&&$0.82\pm0.01$&$1.72\pm(\dag)$&$13.19\pm0.02$\\
$+8.9$&$1.00\pm(\dag)$&$2.08\pm(\dag)$&$15.00\pm0.01$&&$0.91\pm(\dag)$&$1.93\pm(\dag)$&$14.43\pm0.03$&&$0.81\pm0.01$&$1.73\pm(\dag)$&$13.20\pm0.01$\\
\hline
&\multicolumn{11}{c}{Model: OC-B; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=15.0$}\\
\cline{2-12}
$-3.0$&$0.72\pm0.01$&$1.61\pm(\dag)$&$13.30\pm0.08$&&$0.72\pm0.01$&$1.61\pm(\dag)$&$13.30\pm0.08$&&$0.76\pm0.01$&$1.61\pm(\dag)$&$12.75\pm0.03$\\
$~0.0$&$0.70\pm0.02$&$1.61\pm(\dag)$&$13.67\pm0.08$&&$0.70\pm0.02$&$1.61\pm(\dag)$&$13.67\pm0.08$&&$0.77\pm0.01$&$1.61\pm(\dag)$&$12.74\pm0.03$\\
$+11.0$&$1.00\pm(\dag)$&$2.08\pm(\dag)$&$15.00\pm(\dag)$&&$0.84\pm0.01$&$1.84\pm(\dag)$&$14.33\pm0.04$&&$0.74\pm0.02$&$1.63\pm(\dag)$&$12.75\pm0.02$\\
\hline
&\multicolumn{11}{c}{Model: OC-C; Input RDP parameters: $\mbox{$\rm R_c$}=1.0$, $\mbox{$\rm R_t$}=15.0$}\\
\cline{2-12}
$-4.0$&$0.62\pm0.02$&$1.49\pm(\dag)$&$13.06\pm0.10$&&$0.62\pm0.02$&$1.49\pm(\dag)$&$13.05\pm0.10$&&$0.71\pm0.01$&$1.49\pm(\dag)$&$11.99\pm0.03$\\
$~0.0$&$0.62\pm0.02$&$1.49\pm(\dag)$&$13.10\pm0.10$&&$0.62\pm0.02$&$1.49\pm(\dag)$&$13.09\pm0.10$&&$0.71\pm0.01$&$1.49\pm(\dag)$&$11.99\pm0.03$\\
$+14.0$&$1.00\pm(\dag)$&$2.08\pm(\dag)$&$15.00\pm(\dag)$&&$0.70\pm0.02$&$1.70\pm(\dag)$&$14.35\pm0.05$&&$0.64\pm0.03$&$1.49\pm(\dag)$&$12.00\pm0.02$\\
\hline
\end{tabular}
\begin{list}{Table Notes.}
\item ($\dag$): uncertainty smaller than 0.01 arbitrary units (au). The half-type radii are half-star
counts (\mbox{$\rm R_{hSC}$}), half-mass (\mbox{$\rm R_{hM}$}) and half-light (\mbox{$\rm R_{hL}$}).
\end{list}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig3.eps}}
\caption{Structural parameters of the GC models. Top panels: Ratio of the tidal radius measured
in profiles with a photometric depth \mbox{$\rm \Delta_{TO}$}\ with respect to that derived from the deepest one, for
the RDPs (left panels), MDPs (vertical-middle) and SBPs (right). Horizontal-middle panels: half-type
radii. Bottom: core radii. TO values are indicated by the dotted line. Except for GC-C (uniform
mass function), the remaining models present changes in radii in the RDPs and MDPs. SBP radii
are essentially uniform.}
\label{fig3}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig4.eps}}
\caption{Same as Fig.~\ref{fig3} for the OC models. For comparison purposes, the y-scale
is the same as in Fig.~\ref{fig3}. Similarly to the GC models, radii changes are conspicuous
in the RDPs and MDPs. }
\label{fig4}
\end{figure}
\section{Structural parameters {\em vs.} photometry depth}
\label{Struc}
The depth-varying model SBPs are fit with the empirical three-parameter function introduced by
\cite{King62} to describe the surface-brightness distribution of GCs, which is characterised
by the presence of the core and tidal radii. For RDPs and MDPs we use the King-like analytical
profile that describes the projected number-density of stars as a function of \mbox{$\rm R_c$}\ and \mbox{$\rm R_t$},
$\sigma(R)=\frac{dN}{2\pi\,R\,dR}$, as given by eq.~\ref{eq1}. We also compute the distances from
the center which contains half of the cluster's total light, stars and mass. The half-star count
(\mbox{$\rm R_{hSC}$}), light (\mbox{$\rm R_{hL}$}) and mass (\mbox{$\rm R_{hM}$}) radii are derived by directly integrating the corresponding
profiles.
A selection of the resulting structural parameters as a function of \mbox{$\rm \Delta_{TO}$}\ is given in
Table~\ref{tab2}. For simplicity we only present the values obtained from the bright and faint
magnitude ranges, as well as for $M_J\leq M_{J,TO}$. The whole set of parameters are contained
in Figs.~\ref{fig3} - \ref{fig6}. At first glance, RDP and MDP radii present a significant
decrease for shallower photometry, with respect to the intrinsic values. SBP radii, on the other
hand, are more uniform. The most noticeable feature is that, except for GC-C (uniform mass function),
RDP and MDP radii tend to become increasingly larger than SBP ones with increasing photometric
depth.
\subsection{Dependence on photometric depth}
\label{DependDepth}
In Fig.~\ref{fig3} we compare the radii measured in GC profiles built with a given photometric
depth (e.g. $\mbox{$\rm R_c$}(\Delta_{TO})$) with the intrinsic ones, i.e. those derived from the deepest
profiles ($R_{c,deep}$). RDP parameters are more affected than the MDP ones, while the SBP ones
are essentially uniform, thus insensitive to photometric depth. Among the radii, RDP and MDP
core are the most affected (underestimated), followed by the half and tidal radii. In the most
concentrated model (GC-A, $c\approx0.7$), measurements or \mbox{$\rm R_c$}\ in the RDP may be underestimated
by a factor $\approx25\%$ in profiles shallower than near the TO, with respect to $R_{c,deep}$,
and $\approx20\%$ in MDPs. The effect is smaller in \mbox{$\rm R_{hSC}$}\ and \mbox{$\rm R_{hM}$}, which may be underestimated
by $\approx15\%$ in the same profiles. The underestimation in the tidal radii is smaller than
$\approx10\%$. As expected, RDP, MDP and SBP radii do not change when the mass function is
uniform (GC-C model).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig5.eps}}
\caption{GC model profiles. Ratio between the same type of radii as measured in RDPs and MDPs
(left panels) and RDPs and SBPs (right panels). From top to bottom: tidal, half and core radii.
TO values are indicated by the dotted line.}
\label{fig5}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig6.eps}}
\caption{Same as Fig.~\ref{fig5} for the OC models. For comparison purposes, the y-scale
is the same as in Fig.~\ref{fig5}.}
\label{fig6}
\end{figure}
Similar radii ratios in the OC models are examined in Fig.~\ref{fig4}. Qualitatively, the
same conclusions drawn from the GC models apply to the OC ones. However, the underestimation
factor of RDP radii increases for younger ages, to the point that \mbox{$\rm R_c$}\ drops to $\approx60\%$
of the deepest value for all profiles shallower than $\approx3$\, mag below the TO in the OC-C
model ($10$\,Myr), and to $\approx70\%$ for OC-B ($100$\,Myr). The respective half-star count
radii are affected by similar, although smaller, underestimation factors. MDP radii are less
affected by cluster age than RDP ones. Similarly to the GC models (Fig.~\ref{fig3}), the three
types of SBP radii are essentially insensitive to photometric depth, within uncertainties. We
note that the presence of bright stars in the central region of young clusters (OC-C) appears
to introduce a small dependence of the core radius on photometric depth (bottom-right panel).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig7.eps}}
\caption{Top panels: relation of the half-type radii with the concentration parameter,
for the RDPs (left panel), MDPs (middle) and SBPs (right). For each model, $R_h$ values
increase for deeper profiles. Dashed line in panels (a) and (b): $R_h\sim c^2$. In
panel (c): $\mbox{$\rm R_{hL}$}\sim c$. Bottom panels: concentration parameter as a function of photometric
depth.}
\label{fig7}
\end{figure}
\subsection{Comparison of similar radii among different profiles}
\label{CompDifProf}
Differences on the same type of radii among the profiles, introduced essentially by a
spatially variable MF, are discussed in Fig.~\ref{fig5} for the GC models. Regardless
of the model assumptions, RDP and MDP radii are essentially the same, except for the
profiles corresponding to deep photometry, for which the RDP radii become slightly larger
than the MDP ones. This occurs basically because of the larger fraction of low-mass stars
at the outer parts of the clusters. Since all stars have equal weight in the building of
the RDPs, the accumulation of low-mass stars at large radii ends up broadening the RDPs
with respect to the MDPs. On the other hand, RDP core and half-star count radii tend to
be larger than the SBP ones for profiles including stars fainter than near the TO. RDP
\mbox{$\rm R_t$}\ may be 10 -- 20\% larger than SBP ones for all depths. As discussed above, the
uniformly-depleted MF of GC-C model produces profiles whose radii are independent of
photometry depth. The RDP to SBP core and tidal radii ratios decrease with concentration
parameter. The RDP to SBP half-type radii ratios do not depend on $c$.
The same analysis applied to the OC models is discussed in Fig.~\ref{fig6}. The presence
of massive stars in young clusters enhances the RDP to MDP radii ratios, especially the
core and to some extent, the half-type radii. This occurs for profiles that contain stars
brighter than $\approx4$\,mag below the TO. For the youngest model (OC-C), the core radius
measured in the RDP may be $\approx40\%$ larger than the MDP one. This effect is enhanced
when RDP radii are compared to SBP ones, again decreasing in intensity from the core to
tidal radii. For OC-C, RDP core, half and tidal radii are $\approx55\%$, $\approx40\%$,
and $\approx25\%$ larger than the equivalent SBP ones. Comparing with the GC models
(Fig.~\ref{fig5}), the presence of a larger fraction of more massive (brighter) stars
towards the center in young clusters tend to enhance radii ratios of RDP with respect to
MDP, and especially, RDP to SBP.
\subsection{Further relations}
\label{FurtRel}
The models discussed in previous sections can be used as well to examine the dependence of
the half-type radii with the concentration parameter, and to test how $c$ varies with
photometric depth. These issues are presented in Fig.~\ref{fig7}.
As already suggested by Figs.~\ref{fig3} and \ref{fig4}, the relation of the half radius
with $c$, in a given model, changes significantly with photometric depth in RDPs (panel a)
and MDPs (panel b). In SBPs, on the other hand, it is almost insensitive to depth (panel c).
From eq.~\ref{eq1}, the half-star count radius is tightly related to the concentration
parameter according to $\mbox{$\rm R_{hSC}$}=(0.69\pm0.01)+(1.01\pm0.01)\,c^2$. This curve fits
well the values measured in the deepest RDP of all GC and OC models alike (panel a). Such
a relation fails for the shallower profiles. A similar, but poorer, relation applies to the
values derived from the deepest MDPs (panel b), $\mbox{$\rm R_{hM}$}=(0.63\pm0.09)+(0.99\pm0.05)\,c^2$.
It fails especially for the young (OC) models. The GC SBPs, on the other hand, can be poorly
fit with the linear function $\mbox{$\rm R_{hL}$}=(-0.9\pm0.1)+(2.4\pm0.1)\,c$ (panel c).
Concentration parameters measured in RDPs and MDPs (panels d and e) change with photometric depth.
Around the TO they reach the maximum value, which corresponds to a star cluster $\approx15\%$ more
concentrated than the pre-established value (Table~\ref{tab1}). At the shallowest profiles $c$
presents a value intermediate between the maximum and the pre-established one, which is retrieved
at the deepest profiles with the inclusion of the numerous low-mass stars. The exception again is
the uniform MF model GC-C, whose $c$ values do not change with $\Delta_{TO}$. $c$ values measured
in SBPs are essentially insensitive to photometric depth (panel f).
\section{NGC\,6397: a test case}
\label{N6397}
We compare the results derived for the model star clusters with similar parameters measured in
the $\mbox{$\rm M_V$}=-6.63$, nearby GC ($\mbox{$\rm d_\odot$}=2.3$\,kpc) NGC\,6397. Being populous is important to produce
statistically significant radial profiles, while the proximity allows a few magnitudes fainter
than the giant branch to be reached with depth-limited photometry.
NGC\,6397 is a post-core collapse GC with evidence of large-scale mass segregation, as indicated
by a mass function flatter at the center than outwards (\citealt{Andreuzzi04} and references
therein).
Additional relevant data (from H03) for the metal-poor ($\mbox{$\rm [Fe/H]$}=-1.95$) GC NGC\,6397 are the
Galactocentric distance $\mbox{$\rm R_{GC}$}=6$\,kpc, half-light and tidal radii (measured in the V band)
$\mbox{$\rm R_{hL}$}=2.33\arcmin$ and $\mbox{$\rm R_t$}=15.81\arcmin$, and Galactic coordinates $\ell=338.17^\circ$,
$b=-11.96^\circ$. Thus, bulge star contamination is not heavy, and cluster sequences can be
unambiguously detected, which is important for the extraction of radial profiles with small
errors (see below). Using SBPs built with 2MASS images and a fit with \citet{King62} profile,
\citet{Cohen07} derived the core radius in the \mbox{$\rm J$}\ band $\mbox{$\rm R_c$}(J)=61.5\arcsec\pm9.3\arcsec$.
However, based on Hubble Space Telescope data and using a power-law plus core as fit function,
\citet{NoGe06} derived $\mbox{$\rm R_c$}=3.7\arcsec$ in the equivalent V band, thus roughly resolving the
post-core collapse nucleus.
The post-core collapse state of NGC\,6397 does not affect the present analysis, since the goal here
is the determination of changes produced in cluster radii derived under the assumption of a King-like
profile (Sect.~\ref{Struc}) applied to RDP, MDP and SBPs built with different magnitude depths. We base
the analysis of NGC\,6397 on \mbox{$\rm J$}, \mbox{$\rm H$}\ and \mbox{$\rm K_s$}\ 2MASS photometry extracted using VizieR\footnote{\em
vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/246} in a circular field of radius $\mbox{$\rm R_{ext}$}=70\arcmin$
centered on the coordinates provided in H03. This extraction radius is large enough to encompass
the whole cluster, allowing as well for a significant comparison field.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig8.eps}}
\caption{Structural analysis of NGC\,6397. Panel (a): decontaminated CMD of a central
($R<5\arcmin$) region. The reference magnitude $\mbox{$\rm J$}=15$ is indicated by the dashed-line.
Shaded region: colour-magnitude filter. Background-subtracted RDPs for stars brighter
than $\mbox{$\rm J$}<15+\Delta_{J15}$, with $\Delta_{J15}=1,~0,~-1,~-2$ are shown in panels (b) to
(d), respectively. The respective King-like fits (solid-line) together with the fit
uncertainty (shaded region) are shown.}
\label{fig8}
\end{figure}
For a better definition of the cluster sequences we apply the statistical decontamination algorithm
described in \cite{BB07}, which takes into account the relative number-densities of candidate cluster
and field stars in small cubic CMD cells with axes corresponding to the magnitude \mbox{$\rm J$}\ and the colours
\mbox{$\rm (J-H)$}\ and \mbox{$\rm (J-K_s)$}. Basically, the algorithm {\em (i)} divides the full range of magnitude and colours of the
CMD into a 3D grid, {\em (ii)} computes the expected number-density of field stars in each cell based on
the number of comparison field stars with magnitude and colours compatible with those in the cell, and
{\em (iii)} subtracts the expected number of field stars from each cell. Typical cell dimensions are
$\Delta\mbox{$\rm J$}=0.5$, and $\Delta\mbox{$\rm (J-H)$}=\Delta\mbox{$\rm (J-K_s)$}=0.25$, which are large enough to allow sufficient star-count
statistics in individual cells and small enough to preserve the morphology of the CMD evolutionary
sequences. The comparison field is the region located between $50\leq R(\arcmin)\leq70$, which is
beyond the tidal radius.
Field-decontaminated CMDs allow for a better definition of colour-magnitude filters, useful to remove
stars (and artifacts) with colours compatible with those of the field which, in turn, improves the
cluster/background contrast in RDPs and SBPs. They are wide enough to accommodate cluster MS and evolved
star colour distributions and dynamical evolution-related effects, such as enhanced fractions of binaries
and other multiple systems (e.g. \citealt{BB07}; \citealt{N188}).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig9.eps}}
\caption{Left panels: RDP and SBP structural radii of NGC\,6397 as a function of $\Delta_{J15}$,
normalised to the values measured in the deepest profile. Right panels: RDP to SBP radii ratios
(similar to Fig.~\ref{fig5}).}
\label{fig9}
\end{figure}
Figure~\ref{fig8} (panel a) displays the decontaminated CMD of a central region of NGC\,6397, with
$R<5\arcmin$, somewhat larger than the half-light radius (Table~\ref{tab3}). We take $\mbox{$\rm J$}=15$ as
reference to extract the depth-variable profiles. RDPs and SBPs are built with colour-magnitude
filtered photometry, with the faint end varying in steps of $\Delta_{J15}=0.5$, with the deepest
(i.e. at the available 2MASS depth) profile beginning at $\mbox{$\rm J$}=16$ and the brightest one ending
near the giant clump at $\mbox{$\rm J$}=13$. The extracted profiles are fitted with the King-like function
discussed in (Sect.~\ref{ModelSCs}). A selection of depth-limited RDPs, together with the respective
fits and uncertainties, is shown in Fig.~\ref{fig8}, and the corresponding RDP and SBP (\mbox{$\rm J$}\ band)
radii are given in Table~\ref{tab3}. Within uncertainties, the present value of the core radius (for
the deepest profile), $\mbox{$\rm R_c$}(\mbox{$\rm J$})=1.4\arcmin\pm0.3\arcmin$, agrees with that derived by \citet{Cohen07},
using the same fit function. The near-infrared half-light radius, on the other hand, is larger than
the optical one (H03), $\mbox{$\rm R_{hL}$}(\mbox{$\rm J$})\approx1.5\mbox{$\rm R_{hL}$}(V)$.
Effects of the varying magnitude depth on the radii of NGC\,6397 are examined in Fig.~\ref{fig9}.
Qualitatively, the resulting curves agree, within uncertainties, with the behaviour predicted by
the GC models (Figs.~\ref{fig3} and \ref{fig5}). Compared to the values measured in the deepest
RDP, the tidal (panel a), half-star counts (b) and core (c) radii decrease for shallower profiles,
especially for $\Delta_{J15}\geq-0.5$, remaining almost uniform for $\Delta_{J15}<-0.5$. In
particular, the core radius measured in shallow RDPs (containing essentially giants) drops to
$\approx45\%$ of its deepest value (which includes stars at the top of the MS). Consistently with
the GC models containing a spatially variable MF (Sect.~\ref{Struc}), the varying depth affects
the tidal, half and core radii, with increasing intensity. SBP radii, on the other hand, remain
essentially uniform with variable depth, consistent with the GC models (Sect.~\ref{Struc}). The
same conclusions apply to the RDP to SBP radii ratio (right panels).
\begin{table}
\caption[]{Radii of NGC\,6397 from RDPs and 2MASS SBPs}
\label{tab3}
\renewcommand{\tabcolsep}{0.9mm}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{cccccccc}
\hline\hline
&\multicolumn{3}{c}{RDP}&&\multicolumn{3}{c}{SBP (\mbox{$\rm J$}\ band)}\\
\cline{2-4}\cline{6-8}
$\Delta_{J15}$&\mbox{$\rm R_c$}&\mbox{$\rm R_{hSC}$}&\mbox{$\rm R_t$}&&\mbox{$\rm R_c$}&\mbox{$\rm R_{hL}$}&\mbox{$\rm R_t$} \\
(mag)&(\arcmin)&(\arcmin)&(\arcmin)&&(\arcmin)&(\arcmin)&(\arcmin)\\
(1)&(2)&(3)&(4)&&(5)&(6)&(7)\\
\hline
$-2.0$&$1.3\pm0.1$&$3.8\pm0.1$&$33\pm5$&&$1.2\pm0.3$&$3.4\pm0.1$&$28\pm5$ \\
$-1.5$&$1.3\pm0.1$&$4.0\pm0.2$&$39\pm8$&&$1.2\pm0.3$&$3.4\pm0.1$&$30\pm5$ \\
$-1.0$&$1.3\pm0.1$&$4.0\pm0.2$&$42\pm8$&&$1.2\pm0.3$&$3.4\pm0.1$&$26\pm5$ \\
$-0.5$&$1.4\pm0.1$&$3.9\pm0.2$&$44\pm7$&&$1.2\pm0.3$&$3.4\pm0.1$&$27\pm8$ \\
$~0.0$&$1.7\pm0.1$&$4.0\pm0.1$&$41\pm4$&&$1.2\pm0.3$&$3.4\pm0.1$&$27\pm6$ \\
$+0.5$&$2.3\pm0.1$&$4.4\pm0.1$&$40\pm4$&&$1.4\pm0.4$&$3.4\pm0.1$&$28\pm8$ \\
$+1.0$&$2.9\pm0.1$&$4.9\pm0.1$&$48\pm3$&&$1.4\pm0.3$&$3.5\pm0.1$&$32\pm2$ \\
\hline
\end{tabular}
\begin{list}{Table Notes.}
\item Core and tidal radii were derived from fits of \citet{King62} functions (Sect.~\ref{Struc})
to the respective profiles. The half-star counts and half-light radii were measured
directly on the profiles.
\end{list}
\end{table}
\section{Concluding remarks}
\label{Conclu}
In this work we simulated star clusters of different ages, structure and mass functions, assuming
that the spatial distribution of stars follows an analytical function, similar to \citet{King62}
profile. The mass and near-infrared luminosities of each star were assigned according to a mass
function with a slope that may depend on distance to cluster center. They form the set of models
from which we built number-density, mass-density and surface-brightness profiles, allowing for a
variable photometric depth. The structural parameters core, half-light, half-mass and half-star
count, and tidal radii, together with the concentration parameter, were measured in the resulting
radial profiles. Next we examined relations among similar parameters measured in different profiles,
and determined how each parameter depends on photometric depth. We point out that the results
should be taken as upper-limits, especially for open clusters, since we have considered noise-free
photometry and a large number of stars, which produced small statistical uncertainties.
With respect to the adopted form of the radial distribution of stars, we note that \citet{King62}
isothermal sphere, single-mass profile has been superseded by more realistic models like those of
\citet{King66}, \citet{Wilson75} and \citet{EFF87}, which have been fit mostly to the SBPs of
Galactic and extra-Galactic GCs (Sect.~\ref{ModelSCs}).
The analytical functions associated with these models are characterised by different scale radii
(among other parameters) that are roughly related to \citet{King62} radii. Thus, it is natural to
extend the scaling with photometric depth undergone by \citet{King62} radii to the equivalent ones
in the other models.
The main results can be summarised as follows.
\begin{itemize}
\item {\em (i)} Structural parameters derived from surface-brightness profiles are essentially
insensitive to photometric depth, except perhaps the cluster radius in very young clusters.
\item {\em (ii)} Uniform mass functions also result in structural parameters insensitive to
photometric depth.
\item {\em (iii)} Number-density and mass-density profiles built with shallow photometry result
in underestimated radii, with respect to the values obtained with deep photometry. Tidal, half-star
count and half-mass, together with the core radii are affected with increasing intensity.
\item {\em (iv)} Because of the presence of bright stars, radii underestimation increases for
young ages.
\item {\em (v)} For clusters older than $\sim1$\,Gyr, number-density and mass-density radii
present essentially the same values; for younger ages, RDP radii become increasingly larger
than MDP ones, especially at the deepest profiles.
\item {\em (vi)} Irrespective of age, profiles deeper than the turnoff have RDP radii
systematically larger than SBP ones, especially the core.
\item {\em (vii)} The concentration parameter also changes with photometric depth, reaching
a maximum around the turnoff.
\end{itemize}
Most of the above model predictions were qualitatively confirmed with radii measured
in ground-based RDPs and SBPs of the nearby GC NGC\,6397.
In principle, working with SBPs has the advantage of producing more uniform structural parameters,
since they are almost insensitive to photometric depth. However, as discussed in Sect.~\ref{intro},
SBPs usually present high levels of noise at large radii. Noise that is also present in SBPs of
clusters projected against dense fields and/or the less populous ones. A natural extension
of this work would be to examine radial profiles built with photometry that includes observational
uncertainties, differential absorption, metallicity gradients, binaries, and star cluster models
with a number of stars compatible with those of open clusters.
As a consequence of the wide range of distances to the Galactic (and especially extra-Galactic)
star clusters, interstellar absorption, and intrinsic instrumental limitations, the available
photometric data for most clusters do not sample the low-mass stars. All sky surveys like 2MASS,
usually are restricted to the giant branch, or the upper main sequence, for clusters more distant
than a few kpc. In such cases, the structural parameters have to be derived from radial profiles
built with photometry that does not reach low-mass stars. The present work provides a quantitative
way to estimate the intrinsic (i.e. in the case of photometry including the lower main sequence)
values of structural radii of star clusters observed with depth-limited photometry.
\begin{acknowledgements}
We thank the anonymous referee for helpful suggestions.
We acknowledge partial support from the Brazilian institution CNPq .
\end{acknowledgements}
|
{
"timestamp": "2007-11-19T13:54:32",
"yymm": "0711",
"arxiv_id": "0711.2919",
"language": "en",
"url": "https://arxiv.org/abs/0711.2919"
}
|
\section{Introduction}
Accretion disks can carry small- and large-scale magnetic fields.
The small-scale field ($\ell\la R$, where $\ell$ is the field
scale length and $R$ measures the radial distance from the disk center)
can be locally generated by the MHD dynamo
(Brandenburg et al. 1995; Stone et al. 1996) supported by
the turbulence, which results from the
magneto-rotational instability (MRI, Balbus \& Hawley 1991).
This field can provide the outward transport of angular momentum
in the bulk of the disk with the help of local Maxwell stresses
(Shakura \& Sunyaev 1973; Hawley, Gammie, \& Balbus 1996).
The large-scale field ($\ell > R$) is unlikely produced in
accretion disks (however, see Tout \& Pringle 1996),
and can either be captured from the environment and
dragged inward by an accretion flow (Bisnovatyi-Kogan \& Ruzmaikin
1974, 1976), or inherited from the past evolution (see \S 4).
The large-scale field can remove the angular momentum from accretion disks
by global Maxwell stresses through the magnetized disk corona (K\"onigl 1989).
A large-scale bipolar field, unlike a small-scale field,
can not dissipate locally due to the magnetic diffusivity
and can not be absorbed by the central black hole.
In the case of inefficient outward diffusion of the bipolar field
through the disk
(see Narayan, Igumenshchev, \& Abramowicz 2003; Spruit \& Uzdensky 2005;
Bisnovatyi-Kogan \& Lovelace 2007; and
for other possibility, see van Ballegooijen 1989;
Lubow, Papaloizou, \& Pringle 1994;
Lovelace, Romanova, \& Newman 1994; Heyvaerts, Priest, \& Bardou 1996;
Agapitou \& Papaloizou 1996; Livio, Ogilvie, \& Pringle 1999),
this field is accumulated in the innermost region of an accretion disk
and forms a ``magnetically arrested disk,'' or MAD
(Narayan et al. 2003).
The MAD consists of two parts: the outer, almost axisymmetric,
Keplerian accretion disk and
the inner magnetically dominated region, in which the accumulated
vertical field disrupts the outer disk at the magnetospheric
radius $R_{\rm m}\sim 8\pi GM\rho/B^2$, where $M$ is the central mass,
$\rho$ is the accretion mass density, and $B$ is the magnetic induction.
It is believed that the large-scale bipolar field in accretion disks
is responsible for the formation of jets
observed in a large variety of astrophysical objects
(e.g., Livio, Pringle, \& King 2003).
The magnetically driven jets can be of two types,
basically depending on a mass load by the disk matter
(e.g., Lovelace, Gandhi, \& Romanova 2005):
Poynting jets and hydromagnetic jets, which
have, respectively, small and large mass loads.
The hydromagnetic jets can be formed by two mechanisms:
the magneto-centrifugal mechanism
(Blandford \& Payne 1982; K\"onigl \& Pudritz 2000)
and the toroidal-field pressure generated by the disk rotation
(Lynden-Bell 2003; Kato, Mineshige, \& Shibata 2004).
The magneto-centrifugal mechanism
produces relatively wide outflows and, to be consistent with observations
of the collimated jets, requires an additional focusing mechanism.
Jets driven by the toroidal-field pressure can have
a high degree of collimation, but these jets are known to be kink
unstable (Eichler 1993; Appl 1996; Spruit, Foglizzo, \& Stehle 1997).
Poynting jets are naturally self-collimated and
expected to be marginally kink stable
(Li 2000; Tomimatsu, Matsuoka, \& Takahashi 2001).
These jets can originate in the innermost region of accretion disks
and powered either by the disks themselves (Lovelace, Wang, \& Sulkanen 1987;
Lovelace et al. 2002)
or by rotating black holes (Blandford \& Znajek 1977; Punsly 2001;
also see,
Takahashi et al. 1990; Komissarov 2005; Hawley \& Krolik 2006; McKinney 2006).
Our study is based on two- and three-dimensional
(2D and 3D, respectively) MHD simulations
and has two main goals.
First, we investigate the dynamics and structure of MAD's.
We show that MAD's are formed in the accretion flows, which carry inward
large-scale poloidal magnetic fields.
Inside the magnetospheric radius $R_{\rm m}$,
the matter accretes as discrete streams and blobs, fighting its way
through the strong vertical magnetic field fragmented in separate bundles.
Because of rotation, the streams take spiral shapes.
Second, we demonstrate a link between the existence of MAD's and production
of powerful Poynting jets.
These jets should always be generated in MAD's
because of the interaction of the spiraling accretion flow
with the vertical magnetic bundles, which, as the result, are twisted around
the axis of rotation.
This paper extends the work of Igumenshchev, Narayan, \& Abramowicz (2003)
by studying in more detail the radiatively inefficient accretion disks
with poloidal magnetic fields.
We employ a new version of our 3D MHD code, which can be utilized in
multi-processor simulations.
The paper is organized as follows:
We describe the solved equations, the numerical method used, and
initial and boundary conditions in \S 2.
We present our numerical results in \S 3, and
discuss and summarize them in \S 4.
\section{Numerical method}
We simulate nonradiative accretion flows around
a Schwarzschild black hole of mass $M$ using the following equations
of ideal MHD:
\begin{equation}
{d\rho\over dt} + \rho{\bf\nabla\cdot v} = 0,
\end{equation}
\begin{equation}
\rho{d{\bf v}\over dt} = -{\bf\nabla} P - \rho{\bf\nabla}\Phi +
{1\over 4\pi}({\bf\nabla}\times{\bf B})\times {\bf B},
\end{equation}
\begin{equation}
{\partial\over\partial t}\left(\rho{v^2\over 2}+\rho\epsilon+
{B^2\over 8\pi}+\Phi\right)=-\nabla\cdot{\bf q},
\end{equation}
\begin{equation}
{\partial{\bf B}\over \partial t} = {\bf\nabla}\times({\bf v}\times{\bf B}),
\end{equation}
where
${\bf v}$ is the velocity, $P$ is the
gas pressure, $\Phi$ is the gravitational potential,
$\epsilon$ is the specific internal energy of gas,
and ${\bf q}$ is the total energy flux per unit square
(see, e.g., Landau \& Lifshitz 1987).
We adopt the ideal gas equation of state,
\begin{equation}
P=(\gamma-1)\rho\epsilon,
\end{equation}
with an adiabatic index $\gamma=5/3$.
We neglect self-gravity of the gas
and employ a pseudo-Newtonian approximation (Paczy\'nski \&
Wiita 1980) for the black hole potential
\begin{equation}
\Phi=-{GM\over R-R_g},
\end{equation}
where $R_g=2GM/c^2$ is the gravitational radius of the black hole.
No explicit resistivity and viscosity are applied in equations (2)-(4).
However, because of the use of the total energy equation (3),
the energy released
due to the numerical resistivity and viscosity is consistently accounted
as heat in our simulations.
The MHD equations (1)-(4) are solved employing the time-explicit
Eulerian finite-difference method, which is an extension
to MHD of the hydrodynamic piecewise-parabolic method by Colella \&
Woodward (1984).
We solve the induction equation (4) using the constrained transport
(Evans \& Hawley 1988; Gardiner \& Stone 2005), which preserves
the $\nabla\cdot{\bf B}=0$ condition.
In our method, we employ the approximate MHD Riemann solver by Li (2005).
Test simulations have shown that this solver is robust and provides
a good material interface tracking.
We use the spherical coordinates $(R,\theta,\phi)$.
Our 3D numerical grid has $182\times 84\times 240$
zones in the radial, polar, and azimuthal directions, respectively.
The radial zones are spaced logarithmically from $R_{in}=2 R_{g}$
to $R_{out}=220 R_{g}$.
Both hemispheres are considered, in which polar cones with the opening
angle $\pi/8$ are excluded. Therefore, the polar domain extends
from $\theta=\pi/16$ to $15\pi/16$.
The grid resolution in the polar direction is gradually changed from
a fine resolution around the
equatorial plane to a coarse resolution near the poles
(with the maximum-to-minimum grid-size ratio $\approx 3$).
The azimuthal zones are uniform and cover the full $2\pi$ range
in $\phi$.
The absorption condition for the mass and
the condition of the zero-transverse magnetic field
are applied in the inner and outer radial boundaries,
providing that the mass and field can freely leave the computational
domain through these boundaries.
In the boundaries around the excluded polar regions,
we apply the reflection boundary conditions,
which means that no streamlines and magnetic lines can go through
these boundaries.
At the beginning of our simulations,
the computational domain is filled with a very low-density,
nonmagnetized material.
The simulations are started in 2D, assuming the axial symmetry,
with an injection of mass in a slender torus, which is
located in the equatorial plane at $R_{inj}=210\, R_{g}$;
i.e., close enough to $R_{out}$.
This mass has the Keplerian angular momentum and specific internal energy
$\epsilon_{inj}=0.045\,GM/R_{inj}$.
After an initial period of simulation
without magnetic fields, the mass forms a steady
thick torus, which has the inner edge at $R\approx 150\,R_{g}$ and
which outer half is truncated at $R_{out}$.
The torus contains a constant amount of mass and is in a dynamic equilibrium:
all the injected mass flows outward through
$R_{\rm out}$ after a circulation inside the torus.
No accretion flow at this point is formed.
This hydrodynamic, steady, thick torus is used as an initial configuration
in our MHD simulations.
The MHD simulations are started at $t=0$ from the steady, thick torus
by initiating the injection of a poloidal magnetic field
into the injection slender torus at $R_{inj}$.
The numerical procedure for the field injection is
similar to that described by Igumenshchev et al. (2003)
except for one modification:
now the strength of the injected field
can be limited by setting the minimum $\beta_{\rm inj}$,
which is the ratio of the gas pressure to
the magnetic pressure at $R_{\rm inj}$.
This modification allows us to better control the rate of field injection.
The entire volume of the thick torus is filled by the field during
about one orbital period, $t_{\rm orb}$, estimated at $R_{\rm inj}$.
Since this moment, $t\simeq t_{orb}$, the formation of accretion flow begins
as a result of redistribution of the angular momentum in the torus due to
Maxwell stresses.
In the following discussion, we will use the time normalized by
the time-scale $t_{orb}$, i.e. $t\rightarrow t/t_{\rm orb}$.
\section{Results}
We present the results of combined axisymmetric 2D and
non-axisymmetric 3D simulations. The initial evolution
in our models has been simulated in 2D.
This allows us to consider longer evolution times
in comparison to those that can be obtained in 3D simulations,
because of the larger requirements for computational resources
in the latter case.
We initiate 3D simulations starting from developed
axisymmetric models. The results
have shown that non-axisymmetric motions are not very important
in the outer parts of the constructed accretion flows and, therefore,
the use of 2D simulations on the initial evolution stages is
the reasonable simplification.
We consider three models, which differ by
the rates of field injection determined by
$\beta_{\rm inj}=10$, 100, and 1000, and we will refer to these models
as Model~A, B, and C, respectively. All other properties of the models,
including the injection radius $R_{inj}$, internal energy $\epsilon_{inj}$,
and numerical resolution, are the same (see \S 2).
\subsection{Accretion flows}
The initial axisymmetric development of the models is
qualitatively similar: the inner edge of the thick torus
is extended toward the black hole, forming relatively
thin, almost Keplerian accretion disks.
The time of the disk development
is varied, depending on the strength of the injected field.
The accretion of the mass into the black hole begins at
$t\approx 0.7$ in Model~A,
$\approx 1.3$ in Model~B, and $\approx 4.2$ in Model~C
(time is given in units of the orbital period at $R_{inj}$, see \S 2).
At this stage, the evolution of the
disks is governed mainly by global Maxwell stresses
produced by the poloidal field component.
This component is advected inward with the accretion flow and,
because of the disks' Keplerian rotation,
generates relatively strong toroidal magnetic fields localized above and
below the mid-plane.
These toroidal fields form a highly magnetized disk corona with a typical
$\beta\sim 0.01$. Model~B and, especially Model~C, demonstrate
the development of 2D MRI.
This development is similar to that
observed by Stone \& Pringle (2001); in particular, in their ``Run C."
We have found the origin of the
channel solution (see Hawley \& Balbus 1992)
in the central regions of Models~B and C.
This solution consists of oppositely directed radial streams
and is the characteristic feature of the axisymmetric
non-linear MRI (Stone \& Norman 1994).
Analysis of the models shows that the channel solution is developed when the
wavelength $\lambda=2\pi V_A/\sqrt{3}\Omega$
of the fastest growing mode of the MRI is
well resolved on the numerical grid,
i.e. $\lambda\ga 5\Delta x$, where $\Delta x$
is the grid size, $V_A$ is the Alfv\'en velocity, and $\Omega$ is
the angular velocity.
Model~A shows no indication of the MRI, which can be attributed
to the strong magnetic fields, which suppress the instability.
In this model, the estimate of $\lambda$ typically exceeds
the disk thickness.
Model~A has some resemblance to nonturbulent
``Run F'' of Stone \& Pringle (2001).
In spite of the mentioned similarities with the results of
Stone \& Pringle (2001),
our models show different behavior on the long evolution times.
Our simulation design with
the permanent injection of mass and magnetic field results in
accretion disks, which accumulate the poloidal field
in the center and form MAD's. The models of Stone \& Pringle (2001; also
De Villiers, Hawley, \& Krolik 2003; Hirose et al. 2004;
McKinney \& Gammie 2004; Hawley \& Krolik 2006)
did not form MAD's and did not show a long-time
accretion history, probably because of the initiation of simulations from
static magnetized tori, which contain
a limited amount of mass and magnetic flux of one sign.
We will concentrate on the results describing the formation, evolution,
and structure of MAD's in the following text.
Other aspects of our results will be reported elsewhere.
Figure~1 shows example snapshots of the axisymmetric density distribution in
Model~B from 2D simulations at two successive moments,
$t=5.1153$ and $5.1458$.
The accretion flow is nonuniform because of the development of turbulence.
The turbulence results from the combined effect
of the MRI and current sheet instability. The latter instability locally
releases heat due to reconnections of the oppositely directed toroidal
magnetic fields.
The reconnection heat makes a significant contribution to the local energy
balance in the central regions of the flow,
because of the relatively high energy density of the field, which
is comparable to the gravitational energy density of the accretion mass.
The thick disk structure observed in Fig.~1 is explained by
convection motions supported by the reconnection heat.
Note that the case, in which the turbulence is
supported by only convection motions
from the reconnections,
without the effects of rotation and MRI,
had been demonstrated in simulations of spherical magnetized
accretion flows (Igumenshchev 2006).
In the case of disk accretion,
almost axisymmetric convection motions, similar to those found here,
had been observed in 3D models
with toroidal magnetic fields (Igumenshchev et al. 2003).
The convection motions in Models~B and C make
these models relevant to
convection-dominated accretion flows (Narayan, Igumenshchev, \&
Abramowicz 2000; Quataert \& Gruzinov 2000).
Our simulations show that the poloidal
field is transported inward in axisymmetric turbulent flows
and accumulated in the vicinity of the black holes. When the central poloidal
field reaches some certain strength (about equipartition with
the gravitational energy of the accreting mass), the accretion flow
becomes unstable (Narayan et al. 2003).
In axisymmetric simulations, the instability takes the form of cycle
accretion, in which the more-extended periods of halted accretion
(see Fig.~1a) are followed by
the relatively short periods of accretion (see Fig.~1b).
In the case of the halted accretion period, the inner accretion disk is
truncated at the magnetospheric radius $R_{m}$, which is
$\approx 15\,R_g$ in Fig.~1a.
The pressure of the strong central vertical field (see Fig.~2a) prevents
the mass accumulated at $R_{m}$ from falling into the black hole.
The accretion begins as soon as the gravity of
the accumulated mass overcomes the magnetic pressure.
During the accretion period, the whole magnetic flux,
which is localized inside $R_{\rm m}$ in the
period of halted accretion, is moved on the black hole horizon
(see Fig.~2b). Note that similar structural features of the inner MHD
flows in accretion disks related to the model of gamma-ray bursts
were discussed by Proga \& Zhang (2006).
Figure~3 illustrates the time dependence of the accretion flow in
Model~B, showing the evolution of the mass accretion rate $\dot{M}_{in}$
and magnetic fluxes
in the midplane inside the five specific radii: $210\,R_g$
($=R_{inj}$), $100\, R_g$, $50\, R_g$, $25\, R_g$,
and $2\, R_g$ ($=R_{inj}$).
This figure shows the evolution, which has been simulated
in 2D from $t=0$ to $2.14$
and in 3D after $t=2.14$.
The vertical dashed line in Fig.~3 indicates the moment of initiation
of the 3D simulations.
The cycle accretion in the 2D simulations begins at $t\approx 1.4$
and is clearly seen as a sequence of spikes
in the time-dependence of $\dot{M}_{in}$ (see Fig.~3a).
Spikes, which are related to the same cycle
accretion, are also observed in the variation of magnetic flux inside
$R=2\, R_g$ (see Fig.~3b).
The magnetic fluxes inside the other selected radii are gradually
increased with time because of the inward advection of the vertical field.
The time dependence of these fluxes is not significantly influenced
by the cycle accretion.
The structure and dynamics of the inner region in Model~B are drastically
changed in the 3D simulations.
Shortly after the initiation of the 3D simulations at $t=2.14$,
the axisymmetric distribution of mass near $R_{\rm m}$
undergoes the Rayleigh-Taylor and, possibly,
Kelvin-Helmholtz instability (see Kaisig, Tajima, \& Lovelace 1992;
Spruit, Stehle, \& Papaloizou 1995;
Chandran 2001; Li \& Narayan 2004)
with the fastest growing azimuthal mode number $m\simeq 50$
(the latter is probably determined by our grid resolution).
As a result,
the empty region inside $R_{\rm m}$ is quickly filled,
on the free-fall time scale estimated at $R_{\rm m}$, with the large
number of density spikes moved almost radially toward the center.
These spikes quickly disappear in the black hole and, at a later time,
the inner disk structure is modified toward establishing a
dynamic quasi-steady state. This state is characterized by
a low $m$-mode ($m\simeq 1$-5) spiral-flow structure,
which results from the interaction of the accreting mass with
the strong vertical magnetic field.
Note that the similar low $m$-mode flow structure
was found in the simulations
of accretion flows onto a magnetic dipole (Romanova \& Lovelace 2006).
The non-axisymmetric inner flow is highly time-variable and experiences a
quasi-periodic behavior.
Figure~4 shows an example of the flow structure inside 50 $R_g$
in Model~B, at two successive moments: $t=2.2767$ and 2.2867.
The flow is essentially 3D inside the magnetically
dominated region limited by the radius $\simeq 35\,R_g$
and remains almost axisymmetric on the outside of this radius.
In the magnetically dominated region,
the flow forms moderately tightened spirals of dense
matter, which are clearly seen in Figs~4a and 4b.
This matter is quickly
accreted into the black hole with the radial velocity, which is
$\sim 0.5$ a fraction of the free-fall velocity.
Such a relatively fast infall is explained by the efficient loss
of the angular momentum by the mass during its interaction with
the vertical field.
The field is distributed nonuniformly in the disk plane,
concentrating in bundles that penetrate through
the plane in very low density,
magnetically dominated (with $\beta\sim 0.01$) regions,
or magnetic ``islands.''
The rotating mass interacts with magnetic bundles
and forces them to twist around the disk's rotational axis.
In the simulations, this twist is observed as the rotation of
magnetic islands around the center in the disk plane.
The rotational velocity of the islands typically has the
reduced rotational velocity by the factor of $\sim0.5-1$
in comparison with the velocity of the surrounding accretion matter.
This can be explained by the resistance
of the large-scale vertical field to such a twist.
The faster rotation of accretion matter and
slower rotation of magnetic islands produces a shear flow.
The shear flow plays two roles in our simulations.
First, it provides the exchange of momentum and energy between
the accreting mass and vertical field.
Second, the shear flow
results in an ablation of the islands caused by magnetic
diffusivity, making each island a temporal structure.
An example evolution of magnetic islands can be seen in Fig.~4:
the magnetic islands observed as low-density spiral arms
above the center in Fig.~4a are observed below the center in Fig.~4b,
after about half a revolution in the clockwise direction.
In the latter figure,
the islands are apparently reduced in size due to the ablation.
The vertical field ablated from magnetic islands
is carried inward by the accretion flow and accumulates on the
black hole horizon. This accumulation results in
quasi-periodic eruptions of the field outward from the horizon
as soon as the field pressure overcomes
the dynamic pressure of the accreting mass.
The eruptions typically take the form of high-velocity narrow streams
(in the equatorial cross-section) of a low-density,
magnetically dominated medium fountained outward from the black hole.
In Fig.~4b, four magnetic islands observed as low-density regions
inside $R\approx 15\, R_g$
result from such eruptions and the eruption of one of these islands
(to the right from the center; see also the steam that produced it)
still continues at the moment shown.
In the consequent evolution, these islands are pushed outward
and stretched in the azimuthal direction by the accretion flow, and
take the spiral shape similar to that shown in Fig.~4a.
Model~A evolves faster and accumulates a larger magnetic flux at the
center in comparison with Model~B.
Figure~5 shows the evolution of the accretion rate $\dot{M}_{\rm in}$
and magnetic fluxes in Model~A.
Qualitatively, the evolution of these quantities is similar to
the evolution of those in Model~B (see Fig.~3).
Quantitatively, however, Model~A demonstrates
significantly larger
time-averaged accretion rates (by about two orders of magnitude) and
longer quasi-periods of the cycle accretion
(represented by the intervals between spikes
in the time-dependence of $\dot{M}_{\rm in}$ in Fig.~5a)
in the 2D simulations.
By the end of the 2D simulations at $t\approx 1.7$, this model has
the maximum $R_m\simeq 30-40\,R_g$.
In the 3D simulations,
Model~A experiences the initial transient period,
similar to the period of the development of
the Rayleigh-Taylor instability in Model~B (see above), in which
the non-axisymmetric, small-scale structures quickly appear
and disappear.
Figure~6 shows an example of the developed low $m$-mode spiral structure
in Model~A obtained after the transient period.
This structure is clearly dominated by the $m=1$ mode.
The magnetically dominated region is extended up to
$R\simeq 70\, R_g$.
Note that the spiral-density arms seen in Fig.~6
are more open than the arms in Model~B
(see Fig.~4a). This could be due to the stronger central field
in Model~A.
Model~C is our slowest evolving model and,
accordingly, shows the slowest rate of accumulation
of the central vertical field.
This model has been calculated only in 2D and
demonstrated the qualitative similarity to the axisymmetric
evolution of Models~A and B.
The cycle accretion, which is caused by the accumulated field,
begins at $t\approx 4.8$ in Model~C.
The model demonstrates more-efficient turbulent motions in the
accretion flow.
This can be attributed to weaker magnetic fields, which
suppress less the MRI
and convection motions. At the end of simulation
at $t\approx 6$, the model has the maximum $R_m\simeq 6\, R_g$.
\subsection{Poynting Jets}
The 3D simulations of Models~A and B
show that the vertical field penetrated the central
magnetically dominated regions in MAD's is twisted around
the axis of rotation by the rotating accretion flows.
The field twist generates electromagnetic perturbations, which
propagate outward and transport the released energy
in the form of a Poynting flux (e.g., Landau \& Lifshitz 1987)
\begin{equation}
{\bf S}={1\over 8\pi}(({\bf v}\times{\bf B})\times{\bf B}).
\end{equation}
The Poynting flux is distributed nonuniformly in the polar angles,
basically showing two components:
a jet-like concentration of the flux near the poles and
a wide-spread distribution of the flux in the equatorial and
mid-polar-angle directions (from $\theta\sim\pi/4$ to $\sim 3\pi/4$).
Figures~7 and 8 show example $\theta$-distributions of the
radial Poynting flux (solid lines),
\begin{equation}
S_R=v_R{B^2\over 4\pi}-{B_R\over 4\pi}({\bf v}\cdot {\bf B}),
\end{equation}
at six different radii, 5 $R_g$, 10 $R_g$, 25 $R_g$, 50 $R_g$,
100 $R_g$, and 220 $R_g$,
which cover most of the radial domain
in Models~B and A, respectively. The shown distributions are averaged in
the azimuthal direction and in time over the interval $\Delta t\simeq 0.05$,
using a set of data files stored during the simulations.
Outside of the
inner magnetically dominated region (at $R\ga 35\,R_g$ in Model~B and
$R\ga 70\,R_g$ in Model~A),
the outward (positive) Poynting flux in the equatorial and
mid-polar-angle directions
is supported mostly by the rotation of the outer,
almost axisymmetric disks, in which the poloidal field component is frozen in.
Such a flux is present in both the axisymmetric 2D and 3D simulations.
The 3D simulations introduce new important features in
the Poynting flux distribution:
an increase of the flux at the equatorial and mid-polar-angle
directions inside the magnetically dominated region, and
at the polar directions outside this region (see Figs~7 and 8).
The equatorial flux inside the magnetically dominated region
is generated due to the twist of the vertical field by the spiraling
non-axisymmetric accretion flows. In Model~B, this flux,
represented by bumps in the $\theta$-distributions, gradually
deviates from the equatorial plane toward the poles as the radial
distance increases (see Figs~7a-7d). At $R\ga 100\,R_g$, the flux
is collimated into bi-polar Poynting jets
(see Figs~7e and 7f).
In the case of Model~A, the process of jet collimation
is less evident and somewhat different; but still, one can observe the
formation of narrow bi-polar Poynting jets starting from $R\simeq 10\,R_g$
and further development of these jets at large radii (see Figs~8b-8f).
Figures~7 and 8 show $\theta$-distributions of the kinetic
flux (dashed lines),
\begin{equation}
F_R=v_R\rho{v^2\over 2},
\end{equation}
for comparison with the Poynting flux.
Typically, the kinetic flux is comparable,
but does not exceed the Poynting flux in the polar jets
(except in the outermost region in Model~A, see Fig.8f).
Accordingly, the jet velocity is mostly sub-Alfv\'enic.
However, the value of the kinetic flux is relatively large and
this is in some contradiction
with our expectations that MAD's can develop Poynting flux dominated jets.
The problem of the excessive kinetic flux in our simulations
can probably be explained
by the action of the numerical magnetic diffusivity (see \S 4),
which results in an unphysically large mass load of the Poynting jets
and the consequent excessive kinetic flux in them.
The Poynting jets are powered by
the released binding energy of the accretion mass.
To estimate quantitatively the amount of energy going into the jets,
we calculate the Poynting jet ``luminosity" $\dot{E}_{jet}$
as a function of the radius R,
\begin{equation}
\dot{E}_{jet}(R)=\int R^2 S_R\,d\Omega,
\end{equation}
where
the integration is taken over the solid angles $\Omega$
occupying the polar regions with $\theta < \pi/4$
and $\theta > 3\pi/4$ (excluding the boundary polar cones, see \S 2).
For comparison, we also calculate the Poynting total luminosity
$\dot{E}_{tot}$, which is defined analogously to $\dot{E}_{jet}$,
but with the integration in eq.~(9) taken over the whole sphere.
Figures~9 and 10 show the radial profiles of the normalized
$\dot{E}_{jet}$ (solid lines) and $\dot{E}_{tot}$ (dashed lines)
in Models~B and A, respectively.
The jet luminosity $\dot{E}_{jet}$ weakly depends on the radius
at $R\ga 50\,R_g$ in both models and equals to
$\approx 1.5\%$ in Model~B and $\approx 0.5\%$ in Model~A.
Here, we quantify the luminosity in the units of
accretion power $\dot{M}_{in}c^2$.
The total luminosity $\dot{E}_{tot}$ includes the flux from
the bi-polar Poynting jets and wide equatorial Poynting outflow.
The latter component of $\dot{E}_{tot}$ exceeds $\dot{E}_{jet}$
by the factor of $\sim 3$ at large radii (see Figs~9 and 10).
This can be the consequence of the employed simulation design
(see \S 2), in which
the disk accretion at outer radii
is mostly provided by the global Maxwell stresses.
The smaller value of the final (at large radii)
relative $\dot{E}_{jet}$ in Model~A,
in comparison with that in Model~B,
can be explained by the different structure of the
inner magnetically dominated region in these models.
Model~A has the less tighten spiral density arms (see Fig.~6),
in which the mass accretes with larger radial velocity and, therefore,
delivers less energy to the field.
Note, also, that the value of $\dot{E}_{jet}$
in Model~A takes the relatively large
finite value right at the inner boundary $R_{in}$
(see Fig.~10), whereas $\dot{E}_{jet}$
in Model~B begins from a small value at $R_{in}$
and gradually increases outward (see Fig.~9).
This difference in the behavior of $\dot{E}_{jet}$
can be attributed to the discussed difference of the innermost structure
in the considered models.
\section{Discussion and Conclusions}
We have performed a numerical study of the formation and
evolution of quasi-stationary MAD, which is
characterized by a strong vertical magnetic field
accumulated at the disk center.
We employ the simulation design, in which the
poloidal magnetic field of one sign is permanently injected into the
computational domain at the outer numerical boundary and
the unipolar vertical field is transported inward by the accretion flow.
The accumulated field has a significant impact on the inner flow structure and
dynamics in both 2D and 3D simulations.
In the axisymmetric 2D simulations,
the field pressure can temporarily halt
the mass falling into the black hole, resulting in
the cycle accretion, in which the longer periods of accumulation
of the mass at the magnetospheric radius $R_m$
are followed by the short periods of accretion.
The 3D simulations have shown, however, that the axisymmetric cycle accretion
is not realized.
Instead, the accumulated field
causes the mass to accrete quasi-regularly in the form of non-axisymmetric
spiral streams and blobs.
We have demonstrated that 3D MAD's can be efficient sources of
collimated, bipolar Poynting jets,
which originate in the vicinity of the central black hole.
These jets develop due to and are powered by
the interaction of the spiral mass inflows
with the central field split into separate magnetic bundles.
The efficiency of conversion of the accretion energy $\dot{M}_{in} c^2$
into the Poynting jet energy is up to 1.5\% in our simulations.
This estimate may not be accurate
(we believe, underestimated) because of the use
of the pseudo-Newtonian approximation (see \S 2).
The better estimate of the efficiency can be obtained using
general relativistic MHD simulations.
We have presented the simulation results from
three models of radiatively inefficient accretion disks,
which differ by the strength of
the injected field. In accordance with the previous studies
(e.g., Stone \& Pringle 2001), the structure of the outer disks
in these models is determined by the field strength.
In Model~A, which has the largest injected field,
the MRI is suppressed and the accretion flow is driven by
global Maxwell stresses, which transport
the excessive angular momentum outward
from the disk through the disk corona.
In Models~B and C, which have the smaller injected fields,
the MRI and turbulent motions are developed.
The turbulence in these models is axisymmetric and
partially supported by the efficient convection motions resulting from
dissipation of toroidal and small-scale, poloidal magnetic fields.
These motions cause the increase of the disk thicknesses
in comparison with non-turbulent Model~A.
The 3D simulations of Models~A and B have demonstrated that
non-axisymmetric motions are not important in the outer parts of the disks,
outside the inner magnetically dominated region (the disks remain
almost axisymmetric),
but very important inside this region,
resulting in the development of the spiral accretion flows
and bi-polar Poynting jets.
Numerical magnetic dissipations and reconnections
result in a magnetic diffusivity, which
influences the structure and dynamics of our models.
The spatial scale, on which the diffusivity
occurs in our simulations,
is determined by the gridsize, which greatly exceeds
the scales of various resistive mechanisms (e.g., Coulomb collisions,
dissipative plasma instabilities)
in the relevant astrophysical conditions.
Therefore, the magnetic diffusivity is significantly overestimated
in our models and results in an excessive
slippage of an accretion flow through magnetic field.
This slippage reduces the ability of the flow to drug
inward the vertical field, but, however, the numerical diffusivity
is not efficient enough
to totally prevent the field accumulation at the disk center.
The numerical diffusivity suppresses the MRI on the scales
of the gridsize, and, therefore, prevents
the development of turbulence in the outer regions of our models,
where the gridsize is increased.
Other effect of the numerical magnetic diffusivity is
the enhancement of the ablation of
magnetic islands, which are found in the 3D simulations (see \S 3.1).
To test the sensitivity of our models to
magnetic dissipations,
we have performed a 2D simulation of the model, which is
similar to Model~A, but has the double number of grid points in
the $R$- and $\theta$-directions.
The simulation has demonstrated the qualitative similarity of
axisymmetric evolution
of the high resolution model and Model~A:
both models show the formation of accretion disks,
accumulation of the vertical field in the disk centers, and
development of the cycle accretion.
The high resolution model forms a thiner laminar accretion disk.
Unfortunately, a more detailed quantitative
comparison of these models meets some difficulties because of
the different properties of the mass
and field injection region (see \S 2),
which are changed with the change of the resolution.
The main results of our study,
the formation of MAD's and Poynting jets,
have been obtained under the assumption of radiatively inefficient flows,
but, we believe that these results
can also be applied to the radiatively efficient, dense
accretion disks (e.g., Kato, Fukue, \& Mineshige 1998).
The formation of MAD's should not be affected by the
radiative cooling as soon as the central field satisfies
the equipartition condition,
\begin{equation}
{B^2\over 8\pi} \sim {GM\rho\over R_g},
\end{equation}
where $\rho$ is the mass density in the innermost region.
The radiative losses results in the higher $\rho$ and, therefore,
the larger B is necessary to obtain radiative MAD's.
We expect that the qualitatively similar spiral
structure of the inner magnetically dominated region,
to that found in our simulations, can be developed
in the case of the radiative disks.
Poynting jets should be a necessary attribute of the radiative MAD's as well.
The formation of MAD's in the radiative disks can be used to explain
the observations of the low/hard state in black hole binaries
(for a review, see Remillard \& McClintock 2006).
Here, we briefly discuss basic moments of this application of MAD's and leave
more quantitative considerations for future works.
We assume, for example, the development of the MAD in the
radiation pressure dominated accretion disk
at the subcritical regime (see Shakura \& Sunyaev 1973).
In such a disk, the radiation diffusion time scale
$t_{rad}\simeq H^2\sigma_T\rho/c$
can significantly exceed the Keplerian time
$t_{K}= 2\pi R^{3/2}/\sqrt{GM}$ at small values of the $\alpha$-parameter, $\alpha \la 0.1$,
in virtu of the relation
\begin{equation}
{t_{K}\over t_{rad}} \simeq 3.4\alpha,
\end{equation}
which follows from the Shakura-Sunyaev solution.
Here, we denote $H$ to be the disk half-thickness and $\sigma_T=0.4$ cm$^2$/g to be
the Thomson scattering cross-section.
As soon as $t_{rad}\gg t_{K}$ in the outer Shakura-Sunyaev disk,
$t_{rad}$ can also significantly exceed the accretion velocity $t_{accr}\sim t_{K}$
in the inner spiral flow in the MAD, in virtu of the relation
\begin{equation}
{t_{accr}\over t_{rad}}\propto R,
\end{equation}
which is satisfied in accretion flows with the scaling law of the accretion velocity
$v_{accr}\propto R^{-1/2}$ and $H\propto R$.
Having $t_{rad}\gg t_{accr}$, one concludes that the radiation is traped
inside the spiral flow on the accretion time scale and, therefore,
this flow is radiatively inefficient.
From the point of view of an observer, which detects
the softer part of the spectrum of outgoing radiation (below $\sim$ eVs),
the MAD will look like a Shakura-Sunyaev disk truncated at the inner radius
$R_{tr}$, which coinsides with the transition radius between the inner
magnetically dominated region and outer axisymmetric accretion disk.
Typically, in observations, $R_{tr}$ is in the interval from a few tens
to hundreds of $R_g$ (e.g., in Cyg X-1, see Done \& Zycki 1999), which
is consistent with that obtained in our Models~A and B.
The observed specta of black hole binaries in the low/hard state
are dominated by the hard x-ray component
(e.g., Done, Gierli\'nski, \& Kubota 2007) and this
can be explained by the radiation from
the hot, optically thin magnetized medium, which surrounds the accreting spiral flows
and in which the binding energy of these flows is released.
The synchrotron radiation from the magnetized medium and Poynting jets in the MAD
(see Goldston, Quataert, \& Igumenshchev 2005)
can be used to explain the observed radio luminosity
in the low/hard state; this luminosity is believed to be due to steady jets
(e.g., Corbel et al. 2003; Gallo, Fender, \& Pooley 2003).
Our simulations assume accretion disks
around black holes and can be relevant to objects with
relativistic jets containing accreting
stellar-mass black holes (e.g., in micro-quasars; black holes resulted from
type Ib/c supernova explosions and mergers of two compact objects) and
supermassive black holes (in galactic centers).
However, we believe that our main results
can also be relevant to nonrelativistic
astrophysical objects, in which accretion disks and jets are observed.
These objects include, for example, young stellar objects
and accreting stars (e.g., white dwarfs) in binary systems.
Qualitatively, we expect that the structure and dynamics of MAD's and
Poynting jets are similar in the both relativistic and nonrelativistic cases.
We expect, however, large quantitative differences
in these two cases
because of the different energy-density scales involved
in the regions of jet formation.
Black holes, which are capable of launching jets almost from
the event horizon, can produce ultra relativistic jets
(e.g., McKinney 2006).
Jets from nonrelativistic objects,
which have the surface radius $R_*\gg R_g$,
are limited by the velocities $v\sim\sqrt{GM/R_*}\ll c$.
For example, the latter formula gives the upper estimate of the jet velocity
$\sim 400\,km/s$
from the solar-type stars.
The problem of inward transport and amplification of the vertical field
in turbulent accretion disks was intensively discussed
in past and recent years (see, e.g., Spruit \& Uzdensky 2005).
The solution of this problem can help to discriminate models of
accretion disks, which are consistent with observations
(e.g., Meier \& Nakamura 2006; Schild, Leiter, \& Robertson 2006).
Our simulation results show that the vertical field is transported inward
and amplified
independent of the disk structure, either laminar or turbulent.
It is worth noting, however,
that magnetic fields in our models are imposed and
relatively large. The assumed strength of these fields
exceeds the possible strength of the self-sustained magnetic fields that
could be developed due to the MRI (Sano et al. 2004).
Therefore, these results should be considered with some caution, because
they do not represent the case
of weak vertical magnetic fields.
The strong vertical field in the center of accretion disks
can be, in principal,
a relic field that is inherited from the previous evolution.
This field can appear, for example, in the merger scenario
(merger of two magnetized neutron stars or
a black hole with a magnetized neutron star, e.g., Berger et al. 2005) or
in the course of
the gravitational collapse of an extended ``proto" object
(e.g., proto-stellar cloud, supernova progenitor),
which produces a significantly more compact object
(protostar, black hole).
In the latter case,
the proto object may contain some amount of the poloidal field,
which will be amplified and accumulated at the center during the collapse.
After the formation of the compact object, the remained
noncollapsed mass can still
move inward, forming an accretion disk and confining
the field in the vicinity of the object.
Depending on the relative strength of this relic field
and the mass accretion rate, MAD's and Poynting jets can be developed.
The considered scenario can be applied
to young stellar objects (T-Tauri stars, e.g., Donati et al. 2007)
and the hyper-accretion model for gamma-ray bursts
(Woosley 1993; Paczy\'nski 1998).
The spiral-flow pattern in MAD's rotates with about
the same angular velocity at all radial distances; i.e.,
it rotates almost as a rigid body.
Such a rotation can result in
quasi-periodic oscillations (QPO's)
in the emitted radiation, if the disk's axis is inclined
to the line of view of an observer
(e.g., Alpar \& Shaham 1985; Lamb et al. 1985; Strohmayer et al. 1996;
Lamb \& Miller 2001; Titarchuk 2003).
The frequency of these QPO's should be related to the
rotation of the spiral pattern,
which angular velocity is defined by the radius
of the magnetically dominated region and can be a
fraction ($\sim 0.5-1$) of the orbital velocity at this radius.
More investigations are required to make quantitative
predictions about QPO's from MAD's.
\acknowledgments
This work was supported by
the U.S. Department of Energy (DOE) Office of Inertial Confinement
Fusion under Cooperative Agreement No. DE-FC52-92SF19460, the
University of Rochester, the New York State Energy Research and
Development Authority.
\clearpage
|
{
"timestamp": "2007-11-28T01:01:40",
"yymm": "0711",
"arxiv_id": "0711.4391",
"language": "en",
"url": "https://arxiv.org/abs/0711.4391"
}
|
\section{Introduction}
The discovery of irregularities in the cosmic ray energy spectrum at
the energy of $\sim 3 \times 10^{15}$~eV (Khristiansen et al.,
1956~\cite{1}) and $\sim 8 \times 10^{18}$~eV (Krasilnikov et al.,
1978~\cite{2,3,4}), the detection of sharp decreases in the cosmic ray
intensity at $E_{0} > 5 \times 10^{19}$~eV (Greisen-Zatsepin-Kuzmin
effect, 1966~\cite{5, 6}) at the EAS arrays in Yakutsk, HiRes (USA),
AUGER (Argentina) are the most important achievement in the
investigation of the superhigh and ultrahigh energy cosmic rays in
recent years. Such a character of spectrum turn out to be associated
directly with processes in interstellar space, namely, with the
origin, acceleration and propagation of cosmic rays in the Galaxy and
beyond. The interpretation of these experimental facts using the
different models of cosmic ray origin still remains to be answered.
In this paper the comparison of the cosmic ray energy spectrum by EAS
Cherenkov light measurements at the Yakutsk array~\cite{7, 8} with the
calculations according to an anomalous diffusion model of cosmic rays
in interstellar space~\cite{9} is performed.
\section{Method to construct the EAS spectrum}
The showers at the Yakutsk array are selected with the central
register by both scintillation and Cherenkov ``masters''~\cite{10,
11}. The all showers registered form the database of the Yakutsk EAS
array.
To construct the spectrum in energy, scattered by EAS particles in the
atmosphere (Cherenkov radiation) the following selection criteria of
showers are used: a) a shower core is to be placed within a perimeter
of the array for the giant showers and near a center of the array for
the showers with $E_{0} < 10^{18}$~eV. The showers whose cores are
near the observation station $R \le 60$~m are excluded from a
sampling: b) the probability to register a shower by Cherenkov photons
is $W_{\text{ch}} \ge 0.9$; c) a zenith angle is less than one-half of
an aperture of Cherenkov detector, i.e. $\theta < 55^{\text{o}}$ in the case
of the detector of the first type and $\theta < 60^{\text{o}}$ for the second
type detector; d) a transmission coefficient of the atmosphere is $\ge
0.60$ for the wave length of $430$~nm.
Thus, more than 60000 showers with $E_{0} \ge 10^{17}$~eV and 300000
showers with $10^{15} \le E_{0} \le 10^{17}$~eV were recorded in the
database. To construct the spectrum, the showers were selected by the
classification parameter $Q(R=150)$, i.e. by Cherenkov light flux
density at a distance 150~m from a core, which was proportional to the
primary shower energy. The measurement accuracy for $Q(R=150)$ in the
individual showers was $\delta = \Delta Q(R=150) / Q(R=150) \ge 15$\%.
The estimation of the shower energy $E_{0}$ is determined by a
quasicalorimetric method which does not depend on the EAS development
model. A basis of the method is experimental data about the Cherenkov
light total flux, $F$, the total number of charged particles,
$N_{\text{s}}$, and the total number of muons with $E_{\text{thr}} \ge
1~GeV$, $N_{\mu}$~\cite{12, 13, 14}. The energy of individual showers
is determined by the following formula:
\begin{equation}
E_{0} = (9.1 \pm 2.2) \cdot 10^{16} \times Q(R=150)^{0.99 \pm0.02}
\label{eq1}
\end{equation}
The intensity of cosmic ray flux in the given interval of EAS
classification parameter is found as a ratio of the number of
registered EAS events to $S_{\text{eff}} \cdot T \cdot \Omega$.
\section{Results and Discussion}
The differential energy spectrum of primary cosmic rays in the
interval of $10^{15} - 5 \times 10^{19}$~eV obtained from a totality
of the all Cherenkov detector measurement data at the Yakutsk EAS
array is shown on the Fig.~\ref{fig01}. Our data confirm an
irregularity of the spectrum of ``knee'' type in the energy range of
$(2-5) \times 10^{15}$~eV discovered in~\cite{1}, and the irregularity
of ``ankle'' type at $E_{0} \sim 8 \times 10^{18}$~eV. It is
established that in the first case the spectrum index is $\gamma = 2.7
\pm 0.1$ below the break and $\gamma = 3.03 \pm 0.05 at E_{0} > 3
\times 10^{15}$~eV, and in the second case, the more sloping spectrum
with $\gamma = 2.6 \pm 0.3$ at $E_{0} > 8 \times 10^{18}$~eV is
observed.
\begin{figure}[ht]
\centering
\includegraphics[width=0.42\textwidth,clip]{fig01}
\caption{Energy spectrum of primary cosmic rays by measurement data of
Cherenkov light at the Yakutsk complex EAS array.}
\label{fig01}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.42\textwidth,clip]{fig02}
\caption{Differential cosmic ray intensity versus the energy. The
points are Yakutsk array data, curves are the calculation
from~\cite{9}.}
\label{fig02}
\end{figure}
For the period of continuous observations of Cherenkov radiation more
than 30 years (10\% relative to one year time of EAS registration with
the scintillation detectors), the showers with $E_{0} > 6 \times
10^{19}$~eV did not detect. This fact confirms once more the GZK
hypothesis~\cite{5, 6} about the sharp break in the cosmic ray energy
spectrum at $E_{0} > 5 \times 10^{19}$~eV.
\subsection*{The galactic model}
The attempt to explain a form of obtained spectrum from the point of
view of cosmic ray anomalous diffusion model and fractality of the
Galaxy's magnetic field was made by Lagutin et al~\cite{9}. The basis
for the cosmic ray propagation in the Galaxy is the following
assumptions: a) after the generation in the sources, the particles
move in fractal interstellar medium by means of two ways: the first
way is ``Levy flights'', the second way is the motion along a spiral
in the nonhomogeneous magnetic field, $b$) the particles exist during
anomalous long time. The lifetime of particles is of a wide
distribution and its tail is described by a power law $q(t) \propto B
\cdot t^{-\beta -1}, t \to \infty, \beta < 1$ (so-called ``Levy
trapping time''). Calculations of the spectrum were separately made
for each of following groups of nuclei: p, He, CNO, N-Si, Fe. The
resulting sum spectrum for the all particles is shown by a solid curve
in Fig.~\ref{fig01}. From the calculations it follows that the
suggested model reproduces the irregularity in the energy spectrum of
the ``knee'' type at $E_{0} \simeq 3 \times 10^{15}$~eV and also the
irregularity of the ``ankle'' type at $E_{0} \simeq 8 \times
10^{18}$~eV. This model does not explain the behavior of a spectrum in
the energy region of $10^{17} - 10^{18}$~eV and the break of the
spectrum at $E_{0} > 6 \times 10^{19}$~eV in more detail. The mass
composition in the energy region of $5 \times 10^{15} - 5 \times
10^{18}$~eV is some heavier than at $E_{0} \simeq 10^{19}$~eV, but
this change is not very significant that is expected from an
experiment (see Fig.~\ref{fig03}).
\subsection*{The galactic model with the
sources of two types}
In contrast to~\cite{9}, in the paper~\cite{15} a scenario is
considered, in which supernovae are the main sources of cosmic rays
and the acceleration up to $E_{\text{max}} \simeq 105 \cdot Z$~GeV
takes place in the shock fronts. The particle spectrum formed in this
case can be presented in the form of $S_{\text{SN}} \sim E^{-2}
\theta (E_{\text{max}} - E)$, where the Heaviside function $\theta(x)$
reflects qualitatively the presence of a sharp cut-off in the spectrum
at $E > E_{\text{max}}$~\cite{16, 17}. The new calculations fulfilled
by the above scenario of particle generation in the sources of two
different types under the assumption of anomalous diffusion of
particles in inhomogeneous medium show that at some parameters the
anomalous diffusion model describes satisfactorily the features of
cosmic ray energy spectrum and mass composition up to $E_{0} \sim
10^{18}$~eV observed in an experiment. First of all, it refers to the
fine structure of cosmic ray intensity change depending on the energy
(see Fig.~\ref{fig01}). By using these calculations, the sharp peaks
in the mass composition depending on energy are also observed (see
Fig.~\ref{fig03}). In this connection, it is of interest to compare
calculations in mass composition with experimental data obtained at
the Yakutsk EAS big and small Cherenkov sub-arrays in recent years.
\subsection*{Mass composition of primary
cosmic rays}
Fig.~\ref{fig03} presents the results in mass composition of primary
cosmic rays of the Yakutsk array. The data were obtained in the
framework of QGSJET-01 model and two-component mass composition
(proton-iron nucleus). The several characteristics corresponding to
the radial and longitudinal development of EAS are used in the
analysis~\cite{18, 19, 20, 21, 22}.
\begin{figure}
\centering
\includegraphics[width=0.42\textwidth, clip]{fig03}
\caption{Mass composition of cosmic rays at superhigh and ultrahigh
energies. The curve is a calculation by Lagutin et al (2001)
according to the anomalous diffusion model for cosmic ray
propogation.}
\label{fig03}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.42\textwidth, clip]{fig04}
\caption{Comparison of the experimental spectrum with the calculated
spectrum from~\cite{23} for the metagalactic protons ($E_{0} > 5
\times 10^{17}$~eV) and galactic iron nuclei ($E_{0} = 10^{15} - 5
\times 10^{17}$~eV).}
\label{fig04}
\end{figure}
The value $\left<\ln{A}\right>$ in each case is determined by using the
interpolation method~\cite{24}. It is seen from Fig.~\ref{fig03} that
the mass composition is varied up to heavy elements in the energy
region of $(2-5) \times 10^{17}$~eV and becomes more light beginning
with $E_{0} \sim 3 \times 10^{18}$~eV.
The lines are the calculations according to the anomalous diffusion
model for the propagation of cosmic rays in the Galaxy
(Fig.~\ref{fig03}, two sources) in the case of inhomogeneous galactic
medium. In the first case, the monotone change in the mass composition
up to $E_{0} \ge 3 \times 10^{18}$~eV is observed, after of which the
mass composition becomes more light. In the second case, the
complicated structure in the dependence of mass composition on the
energy is observed, peaks for the nuclei of different mass in the
energy region of $10^{15} - 10^{17}$~eV are noticeable. According to a
hypothesis~\cite{25} and calculations from \cite{15}, such an
inhomogeneous structure can be formed by a near supernova. Our data
(Fig.~\ref{fig03}) do not contradict to this hypothesis.
Such a sharp change of the mass composition in the energy region of $5
\times 10^{16} - 5 \times 10^{17}$~eV is not explained in the
framework of the galactic model and is likely associated with the
existence of a transition boundary from galactic to metagalactic
cosmic rays. This conclusion is confirmed by calculations
from~\cite{23}, where a scenario of galactic and metagalactic origin
of cosmic rays is considered. These calculations are shown in Fig.4
together with our experimental data. It can be seen from
Fig.~\ref{fig04} that cosmic rays in the energy region of $5 \times
10^{16} - 5 \times 10^{17}$~eV are most likely of galactic origin with
the noticeable portion of heavy nuclei in the total flux.
It should be noted the estimations of cosmic ray mass composition in
the region after the ``knee'', obtained at the compact arrays, agree
well with each other. The same cannot be said of the energy region of
$\sim 10^{18}$~eV (see Fig.~\ref{fig03}) where HiRes array data point
to more speedy enrichment of primary radiation by the light nuclei and
protons as compared with the Yakutsk array data. The Yakutsk EAS
array data, on the contrary, show the gradual change from the heavy to
light composition (protons and He nuclei) at $E_{0} \sim
10^{19}$~eV. In both cases, data point to the existence of the
tendency of ``protonization'' of primary cosmic rays at $E_{0} >3
\times 10^{18}$~eV.
\subsection*{Conclusions}
Direct measurements of the cosmic ray energy spectrum in the region
of ultrahigh energies (in energy scattered by EAS particles in the
atmosphere) have confirmed the complicate form of spectrum. The
spectrum becomes steeper at $E_{0} \ge 3 \times 10^{15}$~eV and more
sloping at $E0 \ge 8 \times 10^{18}$~eV. A character of energy
dependence of $\left<\ln{A}\right>$ by the Yakutsk EAS data point to
the change of the mass composition of primary particles at singular
points of cosmic ray energy spectrum. The value $\left<\ln{A}\right>$
rises with the energy after the “knee” up to its maximum equal to
$3.5$ at $(2-5) \times 10^{17}$~eV and then it begins to
decrease. Such an energy dependence of $\left<\ln{A}\right>$ does not
contradict a hypothesis of cosmic rays propagation according to laws
of the anomalous diffusion model in fractal interstellar medium
(Lagutin et al., 2001). The value $\left<\ln{A}\right>$ at $E_{0} >
10^{18}$~eV decreases gradually and at $E_{0} \sim 10^{19}$~eV the
mass composition consists of He nuclei and protons. The cosmic ray
intensity beyond $E_{\text{thr.}} > 6 \times 10^{19}$~eV decreases
sharply and this effect is not described in the framework of the
galactic model only. Such a character of spectrum does not contradict
to the calculations by Berezinsky et al~\cite{23} for the metagalactic
model, in which the ``ankle'', observed in the experiments on
ultrahigh energy cosmic ray registration, can be produced by the
proton component only arriving from the Metagalaxy. Thereby, the
details of experimental spectrum form, for example, ``dip'', i.e. the
decrease of intensity at $E_{0} \times 10^{19}$~eV, are caused by,
most likely, the interaction of extragalactic protons with a relic
radiation photons ($p + \gamma_{\text{\footnotesize{}CMB}} \to \text{p} + e^{+} +
e^{-}$). As a direct argument of this hypothesis, the anisotropy can
be used which is related to the origin and sources of cosmic
rays. Based on data of~\cite{26, 27, 28}, at $E_{0} \ge 8 \times
10^{18}$~eV the weak correlation in the arrival directions of EAS with
the Galaxy plane and the close correlation with the Supergalaxy plane
are observed and that the quasars can be the possible sources of
cosmic rays.
|
{
"timestamp": "2007-11-16T03:50:04",
"yymm": "0711",
"arxiv_id": "0711.2548",
"language": "en",
"url": "https://arxiv.org/abs/0711.2548"
}
|
\section{Introduction}
Field-induced magnetic ordering (FIMO) in spin-gapped systems, in which
an energy gap exists for low-lying excited states, has been investigated
in a vast number of compounds, particularly in the context of the Bose-Einstein condensation (BEC) of triplet magnons \cite{nikuni}.
The BEC picture is useful for understanding the nature of the FIMO
with the commensurate (C) antiferromagnetic order pependicular to the field direction.
Recently, Suzuki {\it et al.} and Maeshima {\it et al.} have added a new aspect
to the FIMO on the basis of numerical analyses combined with field theories \cite{suzuki, maeshima}.
These authors have predicted that a magnetic field induces a novel
incommensurate (IC) order parallel to the field direction
in $S=1/2$ alternating chains with a frustrated next-nearest-neighbor (NNN)
interaction.
Around the central field of the field-induced Tomonaga-Luttinger liquid (TLL)
phase of this system between the lower and upper critical field, $H_{\rm C1}$ and $H_{\rm
C2}$, frustration changes the dominant spin correlation from C to IC.
If small inter-chain interactions exist,
the dominant IC correlation leads to long-range IC ordering in the field direction.
In the case frustration is not strong enough to stabilize the IC order at high temperatures,
a first order phase transition will happen from the BEC to the IC order at very low temperatures\cite{maeshima, maeshima2}.
The theoretical studies mentioned above \cite{suzuki, maeshima} have been stimulated by experimental works on
the organic radical compound pentafluorophenyl nitroxide (F$_5$PNN) \cite{across, izumi, izumithesis, lt23}.
The magnetism of F$_5$PNN arises from unpaired electrons delocalized around the NO moieties.
Although this compound has a uniform chain structure at room temperature,
the magnetic susceptibility and the magnetization curve at low temperatures
are well reproduced by calculations for an $S=1/2$ alternating chain model
which is described by the spin Hamiltonian \cite{across};
\begin{equation}
H = -2J \sum_{i}^{N/2} (S_{2i-1}\cdot S_{2i}+\alpha S_{2i}\cdot S_{2i+1}).
\end{equation}
Here, $S$ denotes the $S=1/2$ Heisenberg-type spin operator, $N$ is the total number of spins, and
$\alpha$ is the alternation ratio between competing two nearest-neighbor interactions in a one-dimensional chain.
When $\alpha$=1, the system becomes a uniform chain, whereas when $\alpha $=0
the system breaks up into the assembly of isolated dimers.
In Ref. \onlinecite{across},
the alternation ratio $\alpha=0.4$ and exchange interaction $2J/k_\mathrm{B}=-5.6$ K
were obtained for F$_5$PNN.
The lower and upper critical fields of F$_5$PNN are determined to be about $H_\mathrm{C1}=3.0$ T and $H_\mathrm{C2}=6.5$ T
from the magnetization curve.
NMR shows a TLL behavior in spin-lattice relaxation
and provides evidence for a NNN interaction \cite{izumi, suga, izumithesis}.
In previous works,
we observed FIMO by measuring the specific heat of a polycrystalline sample
in magnetic fields up to 8.0 T ( $>H_\mathrm{C2}$ ) \cite{prl}.
Above the critical temperature of the FIMO,
the temperature dependence of the specific heat $C(T)$ in magnetic fields was
in good qualitative agreement with a numerical calculation which assumes the TLL \cite{prl, wang}.
In this paper, we present the $H$-$T$ phase diagrams of a single crystal and powder of F$_5$PNN
obtained from detailed specific heat measurements in magnetic fields.
Reentrant $H$-$T$ phase diagrams for the FIMO phase are obtained for both samples.
However, the shape of the phase boundary depends on the form of the sample.
That of the single crystal is symmetric with respect to a central field of the gapless field region between $H_\mathrm{C1}$ and $H_\mathrm{C2}$,
whereas the powder has a phase boundary which is distorted and pushed to lower temperatures than that of the single crystal.
\section{Experimental procedures}
F$_5$PNN was prepared using the method described in Ref. \onlinecite{sample}.
Specific heat measurements were performed by the adiabatic heat-pulse method using a $^3$He-$^4$He dilution refrigerator.
The powder sample was mixed with Apiezon N grease to ensure good thermal contact,
and was mounted on the sample cell in the refrigerator.
The single crystal sample was attached to the cell with the same grease.
The nuclear contributions of hydrogen and fluorine to the specific heat were subtracted.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{single031107.eps
\end{center}
\caption{(color online) Specific heat of the single crystal
in magnetic fields. Arrows indicate peak temperatures of the FIMO.
Upper panel: 5.0 T $\leq H\leq$ 6.5 T.
Lower panel: 2.5 T $\leq H\leq$ 4.5 T.
}
\end{figure}
\begin{figure}[t]
\vspace{0.18cm}
\begin{center}
\includegraphics[width=8.3cm]{BpowderBUT5p2T031107.eps
\end{center}
\vspace{0.25cm}
\caption{(color online) Specific heat of the powder in magnetic fields.
Arrows indicate peak temperatures of the FIMO.
Upper panel: 4.75 T $\leq H\leq$ 6.75 T.
Lower panel: 2.5 T $\leq H\leq$ 4.5 T.
}
\end{figure}
\section{Results}
Figure 1 shows $C(T)$ of the single crystal in magnetic fields.
A sharp peak due to the FIMO is clearly seen in fields between 3.25 T and 6.0 T.
A small peak is observed at 3.0 T, and an upturn indicating a peak at a lower temperature at 6.25 T.
Also, exponential temperature dependences are observed at 2.5 T and 6.5 T,
indicative of energy gaps for low-lying excitations.
We conclude from these observations that the critical fields of the single crystal are
$H_\mathrm{C1}\simeq 3.0$ T and $H_\mathrm{C2}\simeq 6.25$ T.
A sharp peak in $C(T)$ is clearly observed also in the powder in fields $3.25$ T $\leq H\leq$ 6.0 T as shown in Fig. 2,
and the peak temperatures are in accordance with those of our previous results \cite{lt23, prl}.
However, the field and temperature dependences of the peak are quite different from those in the single crystal.
As the field increases from 3.5 T,
the peak becomes much sharper.
The field dependence of the peak temperature is weaker than for the single crystal.
At 3.0 T, 6.25 T, and 6.5 T,
a sharp upturn is observed indicating a peak at lower temperatures,
and exponential behaviors are observed at 2.5 T and 6.75 T.
Based on these features, the gapless field region of the powder is most likely 3.0 T $\leq H\leq$ 6.5 T.
This field region is wider than that of the single crystal.
Figure 3 is the $H$-$T$ phase diagram of the single crystal and powder obtained from the peaks
in the specific heat.
We note two differences between the $H$-$T$ phase boundaries for the two sample forms.
One is that the peak temperatures are lower for the powder than for the single crystal.
The other is a difference in the shape of the phase boundary between the FIMO and paramagnetic phase.
The phase boundary of the single crystal is symmetric with respect to the central field of the gapless field region
as observed or expected in isotropic spin-gapped compounds investigated so far \cite{FIMOs}.
In contrast,
that of the powder is distorted.
Since the peak in $C(T)$ of the powder is sharp even at 6.0 T in Fig. 1(a),
it is unlikely that the distinct phase boundary of the powder originates from anisotropy effects.
In addition, it is revealed by high-field ESR measurements on the powder sample of this compound
that the g-value is almost 2.0 \cite{ESR}.
\begin{figure}[tbp]
\includegraphics[width=8cm]{pdsmall031107.eps
\caption{(color online) Magnetic field versus temperature phase diagram of the single crystal and powder of F$_5$PNN obtained
from the specific heat measurements.
Open and filled circles are peak positions of the specific heats of the single crystal and
powder, respectively. Solid and broken lines are guides for the eye.}
\end{figure}
\section{DISCUSSION}
The observed distorted phase boundary of the FIMO
is similar to that of $S=1/2$ strongly frustrated alternating chain models \cite{maeshima}.
The models exhibit a first-order phase transition at very low temperatures from a conventional
field-induced antiferromagnetic order of the spin components perpendicular to the external field direction, which is interpreted as
the BEC of triplet magnons,
to an IC order along the field direction around the middle of the gapless field region
where the IC correlation is dominant.
Because frustration suppresses transverse fluctuations, and then decreases the antiferromagnetic ordering temperature in this field region,
the phase boundary for the FIMO is distorted.
To argue the possibility that an IC order is realized in the powder,
we must first examine if frustration is necessary to explain the powder result.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{singleanddmrg2J.eps
\end{center}
\begin{center}
\includegraphics[width=8.5cm]{powderanddmrg2J.eps
\end{center}
\caption{Temperature dependence of the magnetic specific heats $C_\mathrm{m}(T)$ of F$_5$PNN single crystal and powder at zero field,
together with calculated specific heats with three sets of values for the exchange interaction $J/k_\mathrm{B}$ and alternation ratio $\alpha$
based on the finite temperature DMRG.
Upper panel: the $C_\mathrm{m}(T)$ of the single crystal and calculation with $2J/k_\mathrm{B}=-5.6$ K and $\alpha=0.4$.
Lower panel: the $C_\mathrm{m}(T)$ of the powder and calculations with $2J/k_\mathrm{B}=-5.6$ K and $\alpha=0.4$,
$2J/k_\mathrm{B}=-5.6$ K and $\alpha=0.6$, and $2J/k_\mathrm{B}=-6.8$ K, $J'/J=0.2$ and $\alpha=0.7$.
}
\end{figure}
To determine the exchange interactions $J$ and alternation ratios $\alpha$
of the single crystal and powder,
we examine the magnetic specific heat at zero field for both samples.
The lattice contribution to the total specific heat is estimated from the data at zero field so
that the total magnetic entropy for $N$ spins will approach $Nk_\mathrm{B}\mathrm{ln}(2S+1)$ at high temperatures
where the magnetic susceptibility $\chi $ times temperature $T$ approaches the value for an $S=1/2$ system.
The results are compared with numerical calculations
based on the finite temperature density matrix renormalization group (DMRG) \cite{DMRG} as shown in Fig. 4.
The upper panel of Fig. 4 shows the single crystal result and a calculation
with the set of parameters $2J/k_\mathrm{B}=-5.6$ K and $\alpha=0.4$,
which have been obtained from the magnetic susceptibility and magnetization of a single crystal \cite{across}.
The quantitative agreement between the experimental and numerical results
means that frustration in the single crystal is too small to detect in the specific heat if it exists.
In the lower panel of Fig. 4,
we compare the result of the power with numerical calculations
with various parameter sets.
It should be noted that the calculation with $2J/k_{\rm B}=-5.6$ K and $\alpha=0.4$,
which well reproduces the single crystal result,
is largely different from the powder result
implying the parameters of the powder are not equal to those of the single crystal.
Although results for $2J/k_{\rm B}=-5.6$ K and $\alpha=0.6$ are better than those for the
first parameter set, clear differences appear in the both side of the peak temperature ($\sim 2$ K).
Finally, our best result is obtained by assuming a NNN interaction
for $J/k_{\rm B}=-6.8$ K, $\alpha=0.7$ and $J'/J=0.2$.
We note that the enhanced $J$ and $\alpha$ explain the wider gapless field region of the powder
because $J$ and $\alpha$ govern the width of the gapless field region
of $S=1/2$ bond-alternating chains \cite{bonner}.
Also, this agreement rules out the possibility that the distinct phase boundary of the powder is ascribed to
the disappearance of magnetic moments which comes from the sample deterioration.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=8cm]{incommephase2.eps
\end{center}
\caption{Alternation ratio $\alpha$ versus NNN interaction $J'/J$ phase diagram at zero temperature
at the half value of the saturation magnetization,
which is equivalent to the 1/2 plateau phase diagram in Ref. \onlinecite{tonegawa}.
The filled circle denotes the set of parameters obtained for the F$_5$PNN powder in this study.}
\end{figure}
The next thing to do is to check whether the set of parameters for the powder is
comparable to those in which an IC order is theoretically predicted to appear.
Figure 5 shows the different regions of the dominant correlation at the half value of the saturation magnetization
in the frustrated alternating chain model as a function of $J'/J$ and the alternation ratio $\alpha$ at $T=0$.
The IC correlation becomes dominant in the same region where the half-magnetization plateau is stable \cite{maeshima2}.
The set of parameters for the F$_5$PNN powder, $J'/J=0.2$ and $\alpha=0.7$, turns out to be in this region,
shown as a filled circle in this figure.
This result strongly suggests that the IC correlation is dominant in the powder
around the center field of the gapless field region and an IC order exists at very low temperatures.
However, there remains a question why the NNN interaction exists only in the powder.
The large pressure dependences of the magnetic susceptibility and specific heat of F$_5$PNN
reported in previous works give us a possible answer to this question \cite{hoso2, mito}.
According to these works,
$\alpha $ and $J$ increase with increasing external pressure, and even at $P=0$,
mixing powder F$_5$PNN with Apiezon N grease changes these values.
The grease solidifies at low temperatures and gives some stress to the powder inside the solid.
Effective pressure by the solidification of the grease is also reported for the powder of another organic compound \cite{mukai}.
Generally, an external pressure enhances inter-chain interactions which increase the ordering temperature of the FIMO.
Nevertheless, the ordering temperatures of F$_5$PNN is higher for the single crystal than for the powder
which can be under pressure as mentioned above.
The strength of an antiferromagnetic interaction in organic magnetic materials depends on how the molecular orbital of an unpaired electron
overlaps with the others.
Since this orbital spreads rather widely in each molecule,
the small variation in the molecular stacking can change the magnetic property drastically \cite{f2pnnno}.
From this point of view, an external pressure most likely
changes the molecular stacking in F$_5$PNN
so that the frustrated NNN interaction, which suppresses the ordering temperature,
will be enhanced much more than the inter-chain interactions.
Very recently, we have seen a more clearly distorted phase boundary for the FIMO around the central field
in the specific heat measurement of deuterated F$_5$PNN powder sample.
This result will appear somewhere else.
To investigate quantitatively the pressure-induced frustration in this compound,
we have proceeded specific heat measurement in magnetic fields under pressure.
\section{Summary}
We have performed detailed specific heat measurements on the $S=1/2$ alternating chain material F$_5$PNN
in magnetic fields using a single crystal and powder.
The shape of the phase boundary for the field-induced magnetic ordered phases is different between the two sample forms.
We have shown the possibility of the pressure-induced frustration in the powder
which should lead to field-induced incommensurate ordering around the central field besides
the Bose-Einstein condensation of triplet magnons,
by quantitatively comparing zero-field magnetic specific heats of two samples with numerical calculations based on
the finite temperature density matrix renormalization group.
A future challenge is the direct observation of the incommensurate ordering.
\section{Acknowledgements}
We thank Yasumasa Takano for helpful advice and valuable discussions.
We are grateful to Kazuyoshi Takeda, Masaki Mito,
Seiichiro Suga, Takafumi Suzuki, Akinori Tanaka, Toshihiro Idogaki, Kiyohide Nomura, Luis Balicas, and Takahiro Sakurai for helpful comments.
Y.Y. was supported by Japan Society for the Promotion of Science.
|
{
"timestamp": "2007-11-15T06:12:07",
"yymm": "0711",
"arxiv_id": "0711.2336",
"language": "en",
"url": "https://arxiv.org/abs/0711.2336"
}
|
\section{Some features of Ultraperipheral Collisions
(UPC)}
Photon-photon and photon-hadron interactions
can also be studied in hadron-hadron collisions~\cite{url}.
This may be surprising since in general such collisions are
dominated by strong interactions between the hadrons.
However, by choosing collisions with large
impact parameter b (or, equivalently, small momentum transfer)
one can suppress these strong interactions.
The time-dependent electromagnetic field of a fast moving charged particle
can be thought of as a spectrum of (quasireal, or equivalent)
photons~\cite{fermi}, see Figure \ref{Fig:PH}.
The determination of the
equivalent (or Weizs\"acker-Williams) photon spectrum
corresponding to a fast particle moving past an observer on a
straight line path with impact parameter $b$ is a textbook example
~\cite{jac}.
The probability $P(b)$ of a specific
photon-hadron reaction to occur in a collision
with an impact parameter $b$
is given by $P(b)=N(\omega, b) \sigma_{\gamma h}(\omega)$,
where $\sigma_{\gamma h}$ is the corresponding photoproduction
cross section. The equivalent photon spectrum can be calculated
analytically, a useful approximation for qualitative considerations is
\begin{equation}
N(\omega, b)=\frac{Z^2 \alpha}{\pi^2 b^2}
\label{Eq:nb}
\end{equation}
for $\omega<\frac{\gamma}{b}$
and zero otherwise. The nuclear charge is given
by $Z$, heavy ions have particularly high photon fluxes,
however, this is partially offset by the lower ion-ion luminosities,
as compared to the p-p case.
\begin{figure}[h]
\centerline{\includegraphics[width=1.0\columnwidth]{f1.eps}}
\caption{A fast charged particle
moving on a straight line with impact parameter b causes a time-dependent
electromagnetic field at the point of the observer.
This field corresponds to a spectrum of equivalent photons.}\label{Fig:PH}
\end{figure}
The impact parameter b is restricted to
\begin{equation}
b>b_{min} \sim R_1 + R_2
\end{equation}
where $R_1$ and $R_2$ denote the sizes of the hadrons.
For heavy ion scattering the Coulomb parameter
$\eta \equiv \frac{Z^2 e^2}{\hbar v}\sim Z^2/137$ is
much larger than unity and it is in principle possible to determine the
impact parameter
by measuring the angle of Coulomb scattering.
Whereas this is experimentally feasible at lower
($\sim GeV/A$) energies
~\cite{aum}, this angle is too small at collider
energies. So one generally measures quantities
integrated over all impact parameters.
Too small impact parameters are recognized since the event is
dominated by the violent strong interactions.
The photon spectrum Eq.~\ref{Eq:nb} extends up to
a maximum photon energy given by
\begin{equation}
\omega_{max}=\frac{\gamma}{b_{min}} .
\end{equation}
This energy is about 3 GeV at RHIC (Au-Au, $\gamma \sim 100$),
and 100 GeV at LHC (Pb-Pb, $\gamma \sim 3000$) in the collider
system.
\section{Multiphoton processes: a possible trigger on UPC}
For heavy ions the probability of an electromagnetic
interaction in ultraperipheral collisions is especially
large, and multiphoton processes occur, see e.g.~\cite{npa}.
We mention $e^+e^-$ pair production where
the impact parameter dependent total pair
production probability $P(b)$
is of order unity. Multiple pairs can be produced,
however they may be hard to detect due to their low
transverse momentum.
The nuclear giant dipole resonance is excited with
probabilities of order of one third.
In Figure \ref{Fig:mua} one of the
graphs is shown which leads to the electromagnetic
production of a $\rho^0$ along with the excitation
of the giant dipole resonance.
These graphs can conveniently be evaluated in semiclassical or eikonal
theories~\cite{npa}.
\begin{figure}[h]
\centerline{\includegraphics[width=1.0\columnwidth]
{baur_gerhard.fig2.ps}}
\caption{A graph contributing to the simultaneous
production of a $\rho$-meson and the excitation of the
giant dipole resonance (GDR).}\label{Fig:mua}
\end{figure}
The giant dipole resonance decays dominantly into a neutron.
This neutron is detected in the forward direction and can serve as
a trigger on UPC.
\section{UPC at RHIC}
The physics of UPC at RHIC and
results from the STAR detector were covered by J. Seger
in the session on photon- and electroweak boson physics,
from HERA, RHIC and Tevatron to LHC.
A unique feature to photoproduction
in hadron-hadron collisions is an interference effect
~\cite{kn}: a vector meson can be produced by a photon
originating from either of the hadrons.
It was shown in~\cite{kn} that this interference effect
leads to a reduction of the transverse momentum spectrum
of the vector mesons for small transverse momenta.
Another theoretical approach~\cite{hbt}
leads to very similar conclusions. (Preliminary)
experimental results from STAR/RHIC indeed show a dip
for small transverse momenta, see e.g. Ref. \cite{yr}.
\section{Opportunities for UPC at LHC}
The maximum photon energy scales linearly with
the Lorentz factor $\gamma$, see eq. 3. This leads to
a significant widening of the opportunities at LHC as compared to
RHIC. A most promising area is low-x QCD studies.
The experiments at HERA have shown that
photoproduction processes provide a well-understood
probe of the gluon density in the proton. At LHC,
such processes could be extended to invariant
$\gamma p$ energies exceeding the maximal HERA energy
by a factor of 10. This would allow to use dijet (charm, etc.)
production to measure the gluon density in the proton
and/or nucleus down to $x \sim 3 \times 10^{-5}$.
Ultraperipheral collisions would also allow one
to study the coherent production of heavy quarkonia,
$\gamma + A \rightarrow J/\Psi (\Upsilon) + A$
at $x \lessapprox 10^{-2}$, and to investigate the
propagation of small dipoles through the nuclear medium
at high energies, see Ref. \cite{fra}, see also
Refs.~\cite{kopel,macha}.
Dijet production via photon-gluon fusion is
calculated in Ref. \cite{svw}. Very large rates
are obtained that will considerably extend the HERA x range.
In this session plans for
studying UPC physics with heavy ions at the LHC were covered
by J.Nystrand (ALICE), D.D'Enterria (CMS),
and V. Pozdnyakov (ATLAS).
In addition to diffractive processes in proton-proton
collisions at LHC also a rich program of proton-photon
and photon-photon physics can be pursued, see Ref. \cite{cern}.
The photon flux is lower as compared to the
heavy ion case due to the $Z^2$-factor, but this
is at least partly compensated by higher beam luminosities.
The photon spectrum is harder due to the smaller size as compared to
the heavy ions, this leads to a lower value of $b_{min}$ in Eq. 3 .
Possibilities for electroweak physics and
beyond were presented by S.Ovyn ($\gamma p$) and T. Pierzchala
($\gamma \gamma$) in this session.
Tagging on photon energy by measuring the energy loss of
the scattered protons in the forward detector TOTEM
is an important feature.
In this session J.Pinfold reported on
photon-photon, photon-pomeron and double pomeron
production at CDF.
A recent workshop on photoproduction at collider energies
at ECT*/Trento was devoted to UPC, the mini-proceedings
can be found in~\cite{ect}.
The reviews~\cite{ber,soff,bau,bns} and the
most recent preprint~\cite{yr} reflect the gradual progess
of the field.
\section*{Acknowledgments}
I would like to thank Frederic Kapusta
for his kind invitation to this very pleasant
and interesting conference
at such a venerable place.
\begin{footnotesize}
|
{
"timestamp": "2007-11-19T11:01:48",
"yymm": "0711",
"arxiv_id": "0711.2882",
"language": "en",
"url": "https://arxiv.org/abs/0711.2882"
}
|
\section*{Acknowledgments}
We thank P. Minnhagen and K. Schoutens for many enlightening discussions and
for a critical reading of the manuscript. G. N. would like also to thank N.
Kitanine for interesting discussions on a related lattice quantum integrable
model \cite{PhM,Kolya}. G. N. is supported by the ANR programm MIB-05
JC05-52749.
\section*{Appendix: reminder of TM for the JJL}
Here we briefly summarize the main results of our theory, the TM, for the
fully frustrated JJL \cite{noi3}\cite{noi4}. We first construct the bosonic
theory and show that its energy momentum tensor fully reproduces the
Hamiltonian of eq. (\ref{ha3}) for the JJL. That allows us to describe the
JJL excitations in terms of the primary fields $V_{\alpha }\left( z\right) $%
. Then we show that it is possible to construct the $N-$vertices correlator
for the torus topology in $2D$ (basically by letting the edge to evolve in
``time''\ and to interact with external vertex operators placed at different
points). We assume that a suitable correlator is apt to describe the ground
state wave function of the JJL at $T=0$ temperature and then perform an
analysis of the symmetry properties of its center of charge wave function
(conformal blocks), which emerge in the presence of vortices carrying half
quantum of flux ($\frac{1}{2}\left( \frac{hc}{2e}\right) $).
Let us focus on the $m$-reduction procedure \cite{cgm4} for the special $m=2$
case (see Ref. \cite{cgm2} for the general case), since we are interested in
a system with a $Z_{2}$ symmetry and choose the ``bosonic''\ theory \cite
{noi3}\cite{noi4}, which well adapts to the description of a system with
Cooper pairs of electric charge $2e$ in the presence of a topological defect
\cite{noi1}, i.e. a fully frustrated JJL. To each of the two legs (edges) of
the ladder we assign a chirality, so making a correspondence between up-down
leg and left-right chirality states.
Let us now write each phase field as the sum $\varphi ^{\left( a\right)
}\left( x\right) =\varphi _{L}^{\left( a\right) }\left( x\right) +\varphi
_{R}^{\left( a\right) }\left( x\right) $\ of left and right moving fields
defined on the half-line because of the topological defect located in $x=0$.
Then let us define for each leg the two chiral fields $\varphi
_{e,o}^{\left( a\right) }\left( x\right) =\varphi _{L}^{\left( a\right)
}\left( x\right) \pm \varphi _{R}^{\left( a\right) }\left( -x\right) $, each
defined on the whole $x-$axis \cite{boso}. In such a framework the dual
fields $\varphi _{o}^{\left( a\right) }\left( x\right) $\ are fully
decoupled because the corresponding boundary interaction term in the
Hamiltonian does not involve them \cite{affleck}; they are involved in the
definition of the conjugate momenta $\Pi _{\left( a\right) }=\left( \partial
_{x}\varphi _{o}^{\left( a\right) }\right) =\left( \frac{\partial }{\partial
\varphi _{e}^{\left( a\right) }}\right) $\ present in the quantum
Hamiltonian. Performing the change of variables $\varphi _{e}^{\left(
1\right) }=X+\phi $, $\varphi _{e}^{\left( 2\right) }=X-\phi $\ ($\varphi
_{o}^{\left( 1\right) }=\overline{X}+\overline{\phi }$, $\varphi
_{o}^{\left( 2\right) }=\overline{X}-\overline{\phi }$\ for the dual ones)
we get the quantum Hamiltonian (\ref{ha3}) but now all the fields are chiral
ones. Finally let us identify in the continuum such chiral phase fields $%
\varphi _{e}^{\left( a\right) }$, $a=1,2$, each defined on the corresponding
leg, with the two chiral fields $Q^{\left( a\right) }$, $a=1,2$\ of the TM
with central charge $c=2$.
As a result of the $2$-reduction procedure \cite{cgm2}\cite{cgm4} we get a $%
c=2$ orbifold CFT, the TM, whose fields have well defined transformation
properties under the discrete $Z_{2}$ (twist) group, which is a symmetry of
the TM. Its primary fields content can be expressed in terms of a $Z_{2}$%
-invariant scalar field $X(z)$, given by
\begin{equation}
X(z)=\frac{1}{2}\left( Q^{(1)}(z)+Q^{(2)}(z)\right) , \label{X}
\end{equation}
describing the continuous phase sector of the theory, and a twisted field
\begin{equation}
\phi (z)=\frac{1}{2}\left( Q^{(1)}(z)-Q^{(2)}(z)\right) , \label{phi}
\end{equation}
which satisfies the twisted boundary conditions $\phi (e^{i\pi }z)=-\phi (z)$
\cite{cgm2}. More explicitly such a field can be written in terms of the
left and right moving components $\varphi _{L}^{\left( 1\right) }$, $\varphi
_{R}^{\left( 2\right) }$ as we stated above; then the Mobius boundary
conditions given in eq. (\ref{blr}) are described by the boundary conditions
for $\phi $. This will be more evident for closed geometries, i.e. for the
torus case, where the magnetic impurity gives rise to a line defect in the
bulk, so allowing us to resort to the folding procedure and introduce
boundary states \cite{noi1}\cite{noi2}. Such a procedure is used in the
literature to map a problem with a defect line (as a bulk property) into a
boundary one, where the defect line appears as a boundary state of a theory
which is not anymore chiral and its fields are defined in a reduced region
which is one half of the original one. Our approach, the TM, is a chiral
description of that, where the chiral $\phi $\ field defined in ($-L/2$, $%
L/2)$ describes both the left moving component and the right moving one
defined in ($-L/2$, $\ 0$), ($0$, $L/2$) respectively, in the folded
description \cite{noi1}\cite{noi2}. Furthermore to make a connection with
the TM we consider more general gluing conditions:
\[
\phi _{L}(x=0)=\mp \phi _{R}(x=0),
\]
the $-$($+$) sign staying for the twisted (untwisted) sector. We are then
allowed to use the boundary states given in \cite{Affleck} for the $c=1$
orbifold at the Ising$^{2}$ radius. The $X$ field, which is even under the
folding procedure, does not suffer any change in boundary conditions \cite
{noi1} while condition (\ref{blr}) is naturally satisfied by the twisted
field $\phi \left( z\right) $. So topological order can be discussed
referring to the characters with the implicit relation to the different
boundary states (BS) present in the system \cite{noi1}. These BS should be
associated to different kinds of linear defects compatible with conformal
invariance.
The fields in eqs. (\ref{X})-(\ref{phi}) coincide with the ones introduced
in eq. (\ref{ha3}). In fact the energy momentum tensor for such fields fully
reproduces the second quantized Hamiltonian of eq. (\ref{ha3}). The whole TM
theory decomposes into a tensor product of two CFTs, a twisted invariant one
with $c=\frac{3}{2}$ and the remaining $c=\frac{1}{2}$ one realized by a
Majorana fermion in the twisted sector. In the $c=\frac{3}{2}$ sub-theory
the primary fields are composite vertex operators $V\left( z\right)
=U_{X}^{\alpha _{l}}\left( z\right) \psi \left( z\right) $ or $V_{qh}\left(
z\right) =U_{X}^{\alpha _{l}}\left( z\right) \sigma \left( z\right) $, where
\begin{equation}
U_{X}^{\alpha _{l}}\left( z\right) =\frac{1}{\sqrt{z}}:e^{i\alpha _{l}X(z)}:
\label{char}
\end{equation}
is the vertex of the continuous\ sector with $\alpha _{l}=\frac{l}{2}$, $%
l=1,...,4$ for the $SU(2)$ Cooper pairing symmetry used here. Regarding the
other\ component, the highest weight state in the isospin sector, it can be
classified by the two chiral operators:
\begin{equation}
\psi \left( z\right) =\frac{1}{2\sqrt{z}}\left( :e^{i\sqrt{2}\phi \left(
z\right) }:+:e^{i\sqrt{2}\phi \left( -z\right) }:\right) ,~~~~~~\overline{%
\psi }\left( z\right) =\frac{1}{2\sqrt{z}}\left( :e^{i\sqrt{2}\phi \left(
z\right) }:-:e^{i\sqrt{2}\phi \left( -z\right) }:\right) ; \label{neu1}
\end{equation}
which correspond to two $c=\frac{1}{2}$ Majorana fermions with Ramond
(invariant under the $Z_{2}$ twist) or Neveu-Schwartz ($Z_{2}$ twisted)
boundary conditions \cite{cgm2}\cite{cgm4} in a fermionized version of the
theory. The Ramond fields are the degrees of freedom which survive after the
tunnelling and the parity symmetry, which exchanges the two Ising fermions,
is broken. Besides the fields appearing in eq. (\ref{neu1}), there are the $%
\sigma \left( z\right) $ fields, also called the twist fields, which appear
in the quasi-hole primary fields $V_{qh}\left( z\right) $. The twist fields
have non local properties and decide also for the non trivial properties of
the vacuum state, which in fact can be twisted or not in our formalism.
Starting from the primary fields $V_{\alpha }\left( z\right) $ we can now
construct the non perturbative ground state wave function of the JJL system
for the torus topology. It turns out that by construction it results as a
coherent superposition of gaussian states with all the non trivial global
properties of the order parameter.{\bf \ }In fact by using standard
conformal field theory techniques it is now possible to generate the torus
topology, starting from the edge theory, just defined above. That is
realized by evaluating the $N$-vertices correlator
\begin{equation}
\left\langle n\right| V_{\alpha }\left( z_{1}\right) \ldots V_{\alpha
}\left( z_{N}\right) e^{2\pi i\tau L_{0}}\left| n\right\rangle ,
\end{equation}
where $V_{\alpha }\left( z_{i}\right) $ is the generic primary field
representing the excitation at $z_{i}$, $L_{0}$ is the Virasoro generator
for dilatations and $\tau $ the proper time. The neutrality condition $\sum
\alpha =0$ must be satisfied and the sum over the complete set of states $%
\left| n\right\rangle $ is indicating that a trace must be taken. It is very
illuminating for the non expert reader to pictorially represent the above
operation in terms of an edge state (that is a primary state defined at a
given $\tau $) which propagates interacting with external fields at $%
z_{1}\ldots z_{N}$ and finally getting back to itself. In such a way a $2D$
surface is generated with the torus topology. It is interesting to observe
that such a procedure is equivalent to the coherent insertion of correlated
relevant vortices (as provided by the CFT description) at positions $%
z_{1}\ldots z_{N}$, as they appear in the non perturbative ground state of
the physical JJL system.{\bf \ }From such a picture it is evident then how
the degeneracy of the non perturbative ground state is closely related to
the number of primary states. Furthermore, since in this letter we are
interested in the understanding of the topological properties of the system,
we can consider only the center of charge contribution in the above
correlator, so neglecting its short distances properties. To such an extent
the one-point functions are extensively reported in the following.
On the torus \cite{cgm4} the TM primary fields are described in terms of the
conformal blocks of the $Z_{2}$-invariant $c=\frac{3}{2}$ sub-theory and of
the non invariant $c=\frac{1}{2}$ Ising model, so reflecting the
decomposition on the plane above outlined. The characters $\bar{\chi}%
_{0}(0|\tau )$, $\bar{\chi}_{\frac{1}{2}}(0|\tau )$, $\bar{\chi}_{\frac{1}{16%
}}(0|\tau )$ express the primary fields content of the Ising model \cite{cft}
with Neveu-Schwartz ($Z_{2}$ twisted) boundary conditions \cite{cgm4}, while
\begin{eqnarray}
\chi _{(0)}^{c=3/2}(0|w_{c}|\tau ) &=&\chi _{0}(0|\tau )K_{0}(w_{c}|\tau
)+\chi _{\frac{1}{2}}(0|\tau )K_{2}(w_{c}|\tau )\,, \label{mr1} \\
\chi _{(1)}^{c=3/2}(0|w_{c}|\tau ) &=&\chi _{\frac{1}{16}}(0|\tau )\left(
K_{1}(w_{c}|\tau )+K_{3}(w_{c}|\tau )\right) , \label{mr2} \\
\chi _{(2)}^{c=3/2}(0|w_{c}|\tau ) &=&\chi _{\frac{1}{2}}(0|\tau
)K_{0}(w_{c}|\tau )+\chi _{0}(0|\tau )K_{2}(w_{c}|\tau ) \label{mr3}
\end{eqnarray}
represent those of the $Z_{2}$-invariant $c=\frac{3}{2}$ CFT. They are given
in terms of a ``charged''\ $K_{\alpha }(w_{c}|\tau )$ contribution:
\begin{equation}
K_{2l+i}(w|\tau )=\frac{1}{\eta \left( \tau \right) }\;\Theta \left[
\begin{array}{c}
\frac{2l+i}{4} \\[6pt]
0
\end{array}
\right] (2w|4\tau )\,,\qquad \text{with }l=0,1\text{ and }i=0,1\,,
\label{chp}
\end{equation}
and a ``isospin''\ one $\chi _{\beta }(0|\tau )$, (the conformal blocks of
the Ising Model), where $w_{c}=\dfrac{1}{2\pi i}\,\ln z_{c}$ is the torus
variable of the ``charged''\ component while the corresponding argument of
the isospin block is $w_{n}=0$ everywhere.
If we now turn to the whole $c=2$ theory, the characters of the twisted
sector are given by:
\begin{eqnarray}
\chi _{(0)}^{+}(0|w_{c}|\tau ) &=&\bar{\chi}_{\frac{1}{16}}(0|\tau )\left(
\chi _{0}+\chi _{\frac{1}{2}}\right) (0|\tau )\left( K_{0}+K_{2}\right)
(w_{c}|\tau ), \label{tw1} \\
\chi _{(1)}^{+}(0|w_{c}|\tau ) &=&\chi _{\frac{1}{16}}(0|\tau )\left( \bar{%
\chi}_{0}+\bar{\chi}_{\frac{1}{2}}\right) (0|\tau )\left( K_{1}+K_{3}\right)
(w_{c}|\tau ), \label{tw2}
\end{eqnarray}
for the $A-P$ sector and by:
\begin{eqnarray}
\chi _{(0)}^{-}(0|w_{c}|\tau ) &=&\bar{\chi}_{\frac{1}{16}}(0|\tau )\left(
\chi _{0}-\chi _{\frac{1}{2}}\right) (0|\tau )\left( K_{0}-K_{2}\right)
(w_{c}|\tau ), \label{tw3.} \\
\chi _{(1)}^{-}(0|w_{c}|\tau ) &=&\chi _{\frac{1}{16}}(0|\tau )\left( \bar{%
\chi}_{0}-\bar{\chi}_{\frac{1}{2}}\right) (0|\tau )\left( K_{1}+K_{3}\right)
(w_{c}|\tau ), \label{tw4.}
\end{eqnarray}
for the $A-A$ one. Furthermore the characters of the untwisted sector are
\cite{cgm4}:
\begin{align}
\tilde{\chi}_{(0)}^{-}(0|w_{c}|\tau )& =\left( \bar{\chi}_{0}\chi _{0}-\bar{%
\chi}_{\frac{1}{2}}\chi _{\frac{1}{2}}\right) (0|\tau )K_{0}(w_{c}|\tau
)+\left( \bar{\chi}_{0}\chi _{\frac{1}{2}}-\bar{\chi}_{\frac{1}{2}}\chi
_{0}\right) (0|\tau )K_{2}\,(w_{c}|\tau ), \label{vac1.} \\
\tilde{\chi}_{(1)}^{-}(0|w_{c}|\tau )& =\left( \bar{\chi}_{0}\chi _{\frac{1}{%
2}}-\bar{\chi}_{\frac{1}{2}}\chi _{0}\right) (0|\tau )K_{0}(w_{c}|\tau
)+\left( \bar{\chi}_{0}\chi _{0}-\bar{\chi}_{\frac{1}{2}}\chi _{\frac{1}{2}%
}\right) (0|\tau )K_{2}\,(w_{c}|\tau ),
\end{align}
for the $P-A$ sector while for the $P-P$ sector we have:
\begin{align}
\tilde{\chi}_{\alpha }^{+}(0|w_{c}|\tau )& =\frac{1}{2}\left( \bar{\chi}_{0}-%
\bar{\chi}_{\frac{1}{2}}\right) (0|\tau )\left( \chi _{0}-\chi _{\frac{1}{2}%
}\right) (0|\tau )(K_{0}-K_{2})(w_{c}|\tau )\,, \\
\tilde{\chi}_{\beta }^{+}(0|w_{c}|\tau )& =\frac{1}{2}\left( \bar{\chi}_{0}+%
\bar{\chi}_{\frac{1}{2}}\right) (0|\tau )\left( \chi _{0}+\chi _{\frac{1}{2}%
}\right) (0|\tau )(K_{0}+K_{2})(w_{c}|\tau ), \label{vac4.}
\end{align}
and
\begin{equation}
\tilde{\chi}_{\gamma }^{+}(0|w_{c}|\tau )=\bar{\chi}_{\frac{1}{16}}(0|\tau
)\chi _{\frac{1}{16}}(0|\tau )\left( K_{1}+K_{3}\right) (w_{c}|\tau ).
\end{equation}
Let us comment that the above factorization expresses the parity selection
rule ($m$-ality), which gives a gluing condition for the ``charged''\ and
``isospin''\ excitations.
It is worth underlining that in the $P-P$ sector, unlike for the other
sectors, modular invariance constraint requires the presence of three
different characters. The {\it isospin} operator content of the character $%
\tilde{\chi}_{\gamma }^{+}(0|w_{c}|\tau )$ clearly evidences its peculiarity
with respect to the other states of the periodic (even ladder) case. Indeed
it is characterized by two twist fields ($\Delta =1/16$) in the {\it isospin}
components. The occurrence of the {\it double} twist in the state described
by $\tilde{\chi}_{\gamma }^{+}(0|w_{c}|\tau )$ is simply the reason why such
a state is a periodic state. Indeed, being an {\it isospin} twist field the
representation in the continuum limit of a magnetic impurity (a half flux
quantum trapping\ or equivalently a kink), the double twist corresponds to a
double half flux quantum trapping, i.e. one flux quantum, typical of the
periodic configuration.
The above analysis would suggest that the $P-P$ state described by $\tilde{%
\chi}_{\gamma }^{+}(0|w_{c}|\tau )$ embeds in the continuum limit a
kink-antikink excitation, i.e. it represents an excited state in the $P-P$
sector. In this way, as it happens for all the other sectors, the $P-P$
sector is left with just two degenerate ground states ( $\tilde{\chi}%
_{\alpha }^{+}(0|w_{c}|\tau )$ and $\tilde{\chi}_{\beta }^{+}(0|w_{c}|\tau )$%
) and, as expected on a pure topological base, the ground state degeneracy
in the torus topology is the double of that of the disk.
Let us now present the full list of character transformations under the
insertion of a magnetic flux quantum through the hole of the closed ladder.
In the even closed JJ ladder configuration, we have that the two ground
state wave functions of the $P-A$ sector decouple, being
\begin{equation}
{\cal T}_{1/2}\tilde{\chi}_{(0)}^{-}(0|w_{c}|\tau )=0\,,\text{ \ }{\cal T}%
_{1/2}\tilde{\chi}_{(1)}^{-}(0|w_{c}|\tau )=0. \label{t(1/2)-PA}
\end{equation}
Concerning the $P-P$ sector, we have:
\begin{equation}
{\cal T}_{1/2}\tilde{\chi}_{\alpha }^{+}(0|w_{c}|\tau )=0 \label{t(1/2)-PPa}
\end{equation}
and
\begin{equation}
{\cal T}_{1/2}\tilde{\chi}_{\beta }^{+}(0|w_{c}|\tau )=\tilde{\chi}_{\gamma
}^{+}(0|w_{c}|\tau )\,\text{ \ \ \ \ }(\ {\cal T}_{1/2}\tilde{\chi}_{\gamma
}^{+}(0|w_{c}|\tau )=\tilde{\chi}_{\beta }^{+}(0|w_{c}|\tau )\,\text{\ }).
\label{t(1/2)-PPbc}
\end{equation}
Such transformations show the instability of the $P-P$ sector under the
insertion of a flux quantum through the hole of the closed ladder. More
precisely the state $\tilde{\chi}_{\alpha }^{+}(0|w_{c}|\tau )$ decouples
while the state $\tilde{\chi}_{\beta }^{+}(0|w_{c}|\tau )$ gets excited to
the state with a kink-antikink configuration $\tilde{\chi}_{\gamma
}^{+}(0|w_{c}|\tau )$.
Furthermore in the odd closed JJ ladder configuration, we have that the two
ground state wave functions of the $A-A$ sector decouple, being
\begin{equation}
{\cal T}_{1/2}\chi _{(0)}^{-}(0|w_{c}|\tau )=0\,,\text{ \ }{\cal T}%
_{1/2}\chi _{(1)}^{-}(0|w_{c}|\tau )=0. \label{t(1/2)-AA}
\end{equation}
Concerning the $A-P$ sector, we have that the two ground state wave
functions transform as:
\begin{equation}
{\cal T}_{1/2}\chi _{(0)}^{+}(0|w_{c}|\tau )=\chi _{(1)}^{+}(0|w_{c}|\tau
)\,,\text{ \ }{\cal T}_{1/2}\chi _{(1)}^{+}(0|w_{c}|\tau )=\chi
_{(0)}^{+}(0|w_{c}|\tau )\,. \label{t(1/2)-AP}
\end{equation}
Concluding, the full set of transformations, here presented, allows to claim
the following simple and clear picture: {\it the odd closed JJL
configuration is the only one which is stable under the insertion of a
magnetic flux quantum through the central hole; moreover, in such odd JJL
configuration such a magnetic flux insertion simply implements the flipping
process between the two degenerate ground states }$\left| 0\right\rangle $%
{\it \ and }$\left| 1\right\rangle ${\it .}
|
{
"timestamp": "2007-11-27T14:06:24",
"yymm": "0711",
"arxiv_id": "0711.4245",
"language": "en",
"url": "https://arxiv.org/abs/0711.4245"
}
|
"\\section{Introduction}\r\n\r\nLet $G$ be a compact, $1$--connected and simple Lie group, namely, $(...TRUNCATED)
| {"timestamp":"2010-08-31T02:01:14","yymm":"0711","arxiv_id":"0711.2541","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\\label{s4.introduction}\n\nIn the first paper of this series \\citep[][here(...TRUNCATED)
| {"timestamp":"2008-02-09T00:51:29","yymm":"0711","arxiv_id":"0711.3071","language":"en","url":"https(...TRUNCATED)
|
"\n\\section{Introduction}\n\n\n\nUnderstanding the dynamics of pion production in nucleon-nucleon c(...TRUNCATED)
| {"timestamp":"2007-11-17T17:35:45","yymm":"0711","arxiv_id":"0711.2748","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\nIn the present work we deal with pattern avoidance on words. This\ntopic(...TRUNCATED)
| {"timestamp":"2007-11-21T15:53:22","yymm":"0711","arxiv_id":"0711.3387","language":"en","url":"https(...TRUNCATED)
|
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