ReadingTimeMachine/rtm-sgt-ocr-v1 · Datasets at Hugging Face

source stringlengths 1 2.05k ⌀	target stringlengths 1 11.7k
Therefore. one has to focus on CUECRs with higher energies to exaiine cases where 7(£) is shorter than the appareut time interval between bursts or flares occurring in the same aneular patch. AT.	Therefore, one has to focus on UHECRs with higher energies to examine cases where $\tau(E)$ is shorter than the apparent time interval between bursts or flares occurring in the same angular patch, $\Delta T$.
In reality. UHECRs have finite deviation angles due to cosinic magnetic fields. so UIIECTs from a source arrives within a finite solid augle AQ=we? around the source. which can be regarded as the appropriate size of the finite augular patch.	In reality, UHECRs have finite deviation angles due to cosmic magnetic fields, so UHECRs from a source arrives within a finite solid angle $\Delta \Omega = \pi \psi^2$ around the source, which can be regarded as the appropriate size of the finite angular patch.
For a given py. the apparent tine interval between bursts in the region of the sky with AQ is estimated to be where 0s=/5?aud pay=pQ/109 Cpe% t.	For a given $\rho_s$, the apparent time interval between bursts in the region of the sky with $\Delta \Omega$ is estimated to be where $\psi_5 \equiv \psi / 5^{\circ}$and $\rho_{s,0} = \rho_s / 10^0$ $^{-3}$ $^{-1}$.
Wo ake the typical positional correlation scale as ος and mse c005? as a reference choice. which correspouds o Boch’?<2 ut Mpc/7.	We take the typical positional correlation scale as $\psi$, and use $\psi \sim 5^{\circ}$ as a reference choice, which corresponds to $B_{\rm eff} {\lambda_{\rm eff}}^{1/2} \lesssim 2$ nG $^{1/2}$.
This is reasonable. since lis is consistent with the effective. EGAIFs estimated in the last subsection and a current upper limit of the void EGAIF from a plausible cosmological model is 2.5 we for Ay=1 Ape2010).. but more conservative discussions with larger values of c are also possible.	This is reasonable, since this is consistent with the effective EGMFs estimated in the last subsection and a current upper limit of the void EGMF from a plausible cosmological model is 2.5 nG for $\lambda_{\rm v} = 1$ Mpc, but more conservative discussions with larger values of $\psi$ are also possible.
Equation (8)) imuplies that a smaller AQ gives areer AT. but AT should be limited to the burst/flare intermittence m a host galaxy. zΠρι. where ny is the iiuber density of host galaxies of UITECR sources.	Equation \ref{eq:deltat}) ) implies that a smaller $\Delta \Omega$ gives larger $\Delta T$, but $\Delta T$ should be limited to the burst/flare intermittence in a host galaxy, $\approx n_h/\rho_s$, where $n_h$ is the number density of host galaxies of UHECR sources.
Iu other words. AQ smaller than the corresponding lower Πιτ is meanineless, at which one host galaxy should exist in a volume with a solid angle AQ within {ως1).	In other words, $\Delta \Omega$ smaller than the corresponding lower limit is meaningless, at which one host galaxy should exist in a volume with a solid angle $\Delta \Omega$ within $D_{\rm max}(E)$.
We call the case “bursting case” that only one burst or flare contributes to arviving UIIECTis at a time in a direction. Le. TLE)AT.	We call the case "bursting case" that only one burst or flare contributes to arriving UHECRs at a time in a direction, i.e., $\tau(E) < \Delta T$.
Then. the requirement TE)<AT gives a sufficient coucition to apply equation (6)). which leads to with the usage of equation (6)).	Then, the requirement $\tau(E) < \Delta T$ gives a sufficient condition to apply equation \ref{eq:estrate}) ), which leads to with the usage of equation \ref{eq:estrate}) ).
As demonstrated in Figure 2.. the range of (CE). in which equation (6)) can be applied. is extended to larger κ1) at higher energies. because the aller nunmiber of sources contributiug to the observed flux. decreases the probability that two bursts are temporally overlapped in a region of the skv (see also equation (8))).	As demonstrated in Figure \ref{fig:nlim}, the range of $n_s(E)$ , in which equation \ref{eq:estrate}) ) can be applied, is extended to larger $n_s(E)$ at higher energies, because the smaller number of sources contributing to the observed flux decreases the probability that two bursts are temporally overlapped in a region of the sky (see also equation \ref{eq:deltat}) )).
Thus. we especially focus. on cases of E~10°" eV to demonstrate constrünts on psu Sections 3. and L. although discussions are general for other E.	Thus, we especially focus on cases of $E \sim {10}^{20}$ eV to demonstrate constraints on $\rho_s$ in Sections \ref{prop} and \ref{results}, although discussions are general for other $E$.
Note that. although we here fix e even at higher enereies. stualler values of c are expected there. so that the extension of the curve to higher enereies would be more casily justified.	Note that, although we here fix $\psi$ even at higher energies, smaller values of $\psi$ are expected there, so that the extension of the curve to higher energies would be more easily justified.
Ou the other haud. if τε)>AT. another UITECR burst may start to contribute before the eud of the former CHECR burst is observed. aud equation (8)) nuplies. Since n4CE) can be determined by the auto-correlation analysis. equation (6)) enables us to estimate p, from observational quantities. if TLE) cau be evaluated by ECAIF simulations aud observations.	On the other hand, if $\tau(E) > \Delta T$, another UHECR burst may start to contribute before the end of the former UHECR burst is observed, and equation \ref{eq:deltat}) ) implies, Since $n_s(E)$ can be determined by the auto-correlation analysis, equation \ref{eq:estrate}) ) enables us to estimate $\rho_s$ from observational quantities, if $\tau(E)$ can be evaluated by EGMF simulations and observations.
Iuportautly. 7.2) has the characteristic energv dependence. which is demonstrated in Figure 2..	Importantly, $n_s(E)$ has the characteristic energy dependence, which is demonstrated in Figure \ref{fig:nlim}.
Here. the case that oulv the ECAIF in filamentary structures affects the time spread of VITZECR bursts is considered for demonstration. 1.6.. Bade?~03 ας: P.	Here, the case that only the EGMF in filamentary structures affects the time spread of UHECR bursts is considered for demonstration, i.e., $B_{\rm eff} {\lambda_{\rm eff}}^{1/2} \sim 0.3$ nG $^{-3}$ .
Two represeutative cases for p, are shown. Lo. p,=1 yrt and 10? ? lo	Two representative cases for $\rho_s$ are shown, i.e., $\rho_s = 1$ $^{-3}$ $^{-1}$ and $10^2$ $^{-3}$ $^{-1}$ .
One sees that n.(E) changes by more than one order of magnitude when £ increases by the cubic root of ten.	One sees that $n_s(E)$ changes by more than one order of magnitude when $E$ increases by the cubic root of ten.
This mieaus that anisotropy features are different among energies.	This means that anisotropy features are different among energies.
Thus. observations of UITEC's above 107 eX. are crucial to identify this tendency clearly.	Thus, observations of UHECRs above $10^{20}$ eV are crucial to identify this tendency clearly.
Future CIECRexperiments with huge exposures may	Future UHECRexperiments with large exposures may
To acquire precise O-isotope data for Iris. we developed a sample mount consisting of an Al round. which supported the potted butt at a fixed altitude. covered by an Al annulus which both supported a 500nm-thick membrane (made of gold-coated S1; N, with a 3800s ion-millecl hole) and incorporated polished standards surrounding the central hole.	To acquire precise O-isotope data for Iris, we developed a sample mount consisting of an Al round, which supported the potted butt at a fixed altitude, covered by an Al annulus which both supported a 500nm-thick membrane (made of gold-coated $_3$ $_4$ with a $\mu$ m ion-milled hole) and incorporated polished standards surrounding the central hole.
San Carlos olivine was used to standardize Iris olivine. ancl Mivakajima plagioclase and Burma. spinel were used to standardize Iris mesostasis ancl chromite. respectively.	San Carlos olivine was used to standardize Iris olivine, and Miyakajima plagioclase and Burma spinel were used to standardize Iris mesostasis and chromite, respectively.
We measured three oxvgen isotopes in Iris using the University of Hawaii Cameca inis 1280 ion microprobe in multicollection mode (!O on a Faraday cup. YO and PO on electron multipliers).	We measured three oxygen isotopes in Iris using the University of i Cameca ims 1280 ion microprobe in multicollection mode $^{16}$ O on a Faraday cup, $^{17}$ O and $^{18}$ O on electron multipliers).
A 2530 pA primary ion beam was focused to ~2 jan to allow for analvsis of Iris.	A 25–30 pA $^{+}$ primary ion beam was focused to $\sim$ 2 $\mu$ m to allow for single-grain analysis of Iris.
The data was corrected for background. deacltime. detector vield. and interference from MOLL to FO (typically <0.2%.).	The data was corrected for background, deadtime, detector yield, and interference from $^{16}$ $^{-}$ to $^{17}$ O (typically $<$ $\permil$ ).
We corrected measured compositions of (he mineral phases in Iris for instrumental mass fractionation by comparing wilh appropriate mineral standards of known composition.	We corrected measured compositions of the mineral phases in Iris for instrumental mass fractionation by comparing with appropriate mineral standards of known composition.
We measured (he oxvgen isotopic composition ol San Carlos olivine mounted analogously to Iris in order (ο understand instrumental mass fractionation associated with the mounting of the unknown sample relative to (he standards.	We measured the oxygen isotopic composition of San Carlos olivine mounted analogously to Iris in order to understand instrumental mass fractionation associated with the mounting of the unknown sample relative to the standards.
We found (that there was a reproducible instrumental fractionation effect between (he San Carlos olivine surrounding the central hole and the San Carlos olivine in the central hole (the center olivine plotted ος lower in 0O than the surrounding olivine along the terrestrial fractionation line).	We found that there was a reproducible instrumental fractionation effect between the San Carlos olivine surrounding the central hole and the San Carlos olivine in the central hole (the center olivine plotted $\sim$ $\permil$ lower in $\delta^{18}$ O than the surrounding olivine along the terrestrial fractionation line).
We applied this offset to the final Iris oxvgen measurements.	We applied this offset to the final Iris oxygen measurements.
Our conservative estimate of the uncertainties in the final ratios comes [from adding in quadrature the statistical uncertainty of the individual measurements. the standard deviation of the standard measurements. and (he uncertainty in the instrumental mass fractionation olIset.	Our conservative estimate of the uncertainties in the final ratios comes from adding in quadrature the statistical uncertainty of the individual measurements, the standard deviation of the standard measurements, and the uncertainty in the instrumental mass fractionation offset.
Olivine. chromite. and mesostasis in Iris have oxvgen isotopic compositions indistinguishable from terrestrial oxvgen (Figure 2)).	Olivine, chromite, and mesostasis in Iris have oxygen isotopic compositions indistinguishable from terrestrial oxygen (Figure \ref{oiso}) ).
The oxvgen isotopic compositions of the three mineral phases are consistent with co-crvstallization [rom a single melt.	The oxygen isotopic compositions of the three mineral phases are consistent with co-crystallization from a single melt.
Iris has oxvgen-isotope composition higher in both “O/'O and !' O/!O than most chondriules in meteorites and other chondrule-like objects in the Stardust collection (Figure 2)).	Iris has oxygen-isotope composition higher in both $^{18}$ $^{16}$ O and $^{17}$ $^{16}$ O than most chondrules in meteorites and other chondrule-like objects in the Stardust collection (Figure \ref{oiso}) ).
The oxveen isotopes of nebular solids can evolve toward isotopically heavier compositions through processes such as interaction with water (partitioning between liquid ancl solid) or evaporation (Ravleigh-like distillation).	The oxygen isotopes of nebular solids can evolve toward isotopically heavier compositions through processes such as interaction with water (partitioning between liquid and solid) or evaporation (Rayleigh-like distillation).
Iris apparently lormed from a relatively evolved oxygen isotopic reservoir.	Iris apparently formed from a relatively evolved oxygen isotopic reservoir.
The inerease of rogsry in blue earlv-tvpe dwarfs (Fig.	The increase of $r_{e B}/r_{e H}$ in blue early-type dwarfs (Fig.
reflects the existence of a (strong) negative gradient in 11 and in V along the radial coordinate (Fig.	2) reflects the existence of a (strong) negative gradient in $-$ H and in $-$ V along the radial coordinate (Fig.
3).	3).
The data show that this is a necessary but not sullicient condition however.	The data show that this is a necessary but not sufficient condition however.
This trend is not foun in giant I5 and SO galaxies. which span a broad range in r.g/r.g. though their 11 color is almost as red as in red earlv-tvpe diwarfs.	This trend is not found in giant E and S0 galaxies, which span a broad range in $r_{e B}/r_{e H}$, though their $-$ H color is almost as red as in red early-type dwarfs.
Raclial color eracdients observed. in individual giant. elliptical and lenticular galaxies are commonly. interpreted. as a product either of age/moetallicitv gradients of the stellar populations along the radial coordinate or of dust attenuation (cl.	Radial color gradients observed in individual giant elliptical and lenticular galaxies are commonly interpreted as a product either of age/metallicity gradients of the stellar populations along the radial coordinate or of dust attenuation (cf.
Sect.	Sect.
1).	1).
These interpretations do not imply a correlation between Γρ ond BLL. In the Following sections. these three theoretical interpretations will be extended. to our results for carly-type dvarfs and ciscussed individually. in relation with some scenarios of earlv-tvpe dwarf galaxy formation and evolution which may justify the trend seen in Fig.	These interpretations do not imply a correlation between $r_{e B}/r_{e H}$ and $-$ H. In the following sections, these three theoretical interpretations will be extended to our results for early-type dwarfs and discussed individually, in relation with some scenarios of early-type dwarf galaxy formation and evolution which may justify the trend seen in Fig.
2.	2.
The lack of data does not allow us to achieve a direct. proof of the valiclity of any of these three interpretations.	The lack of data does not allow us to achieve a direct proof of the validity of any of these three interpretations.
Moreover. the absence of a one-to-one correlation between reesra and the strength ofthe gradient in LL. i£ not due to errors in data analysis. may indicate that the interpretation of Fig.	Moreover, the absence of a one-to-one correlation between $r_{e B}/r_{e H}$ and the strength of the gradient in $-$ H, if not due to errors in data analysis, may indicate that the interpretation of Fig.
2 is complex.	2 is complex.
The integrated broacd-banc colors of E and SO galaxies become progressively Επομ toward fainter magnitudes (Faber 1973: see however Scodegeio 2001).	The integrated broad-band colors of E and S0 galaxies become progressively bluer toward fainter magnitudes (Faber 1973; see however Scodeggio 2001).
This correlation. known as the colormagniude relation. is universal and very σαι in he optical for ellipticals and lenticulars in clusters al z=0 Bower. Lucey Ellis 1992a.b).	This correlation, known as the color–magnitude relation, is universal and very tight in the optical for ellipticals and lenticulars in clusters at z=0 (Bower, Lucey Ellis 1992a,b).
Relying on the commonly accopled interpretation of the colormagnitude relation (Ixodama Arimoto 1997). Wwe c'onclude that. in blue carby-type diwvarfs. either the averaee metallicity is lower than in red earlv-tvpe. dwarls of je same L-banel luminosity or star formation is still going «on.	Relying on the commonly accepted interpretation of the color–magnitude relation (Kodama Arimoto 1997), we conclude that, in blue early-type dwarfs, either the average metallicity is lower than in red early-type dwarfs of the same H-band luminosity or star formation is still going on.
Here we make the. hypothesis that the blue colors of the VCC sample cwarls are due to the presence of a voung sellar population.	Here we make the hypothesis that the blue colors of the VCC sample dwarfs are due to the presence of a young stellar population.
‘This hypothesis is supported by spectrosco[ow (GOL) for some individual samyge galaxies and is consisten with the results of Terlevich et al. (	This hypothesis is supported by spectroscopy (G01) for some individual sample galaxies and is consistent with the results of Terlevich et al. (
1999).	1999).
IDe.	Fig.
5h srows that most of the B-bancl emission of the chwarls is contributed by the. st«αρ populations distributed wihin an cxponential-cisk c“OMIpeHen.	5b shows that most of the B-band emission of the dwarfs is contributed by the stellar populations distributed within an exponential-disk component.
This disk-componen. M present. is also responside for more than of the tota L-band luminosity (Fig.	This disk-component, if present, is also responsible for more than of the total H-band luminosity (Fig.
5a).	5a).
Lowe make the asstunption that early-type dwarls are rotaionally [lattened (cf.	If we make the assumption that early-type dwarfs are rotationally flattened (cf.
Sect.	Sect.
1). tr¢ wavelength-dependence of their. effective radius. and the relation between total coor and sign and streneth of the color. gradients. may be understood: as	1), the wavelength-dependence of their effective radius, and the relation between total color and sign and strength of the color gradients may be understood as
(e.g.Bellonterαἱ.2001:Tomsick& (e.g.	\citep[e.g.][]{Bel01,Tom01,Rod02}. \citep[e.g.][]{Now99,Mis00,Cui99,Pou99,Nob01}.
in black hole and neutron star systems. can be explainec in a model where the global disk resonates nonlinearly with the stochastic fluctuations (Abramowiezefal.2003).	in black hole and neutron star systems, can be explained in a model where the global disk resonates nonlinearly with the stochastic fluctuations \citep{Aba03}.
. More compelling model independent evidence has been givel by Uttleyοἱal.(2005). (see also Timmereraf.(2000) and Thieleta£. (2001))). who argue that the log-normal distribution of the fluxes and the linear relationship betweer RMS and flux imply that the response of the disk is non-	More compelling model independent evidence has been given by \cite{Utt05} (see also \cite{Tim00} and \cite{Thi01}) ), who argue that the log-normal distribution of the fluxes and the linear relationship between RMS and flux imply that the response of the disk is non-linear.
They show that an exponential response can explaui these observations.	They show that an exponential response can explain these observations.
Earlier Mineshigeefαἱ.(1994)... had suggested that the behavior of the observed fluctuations could be because the accretion disk is in self-organized critical state.	Earlier \cite{Min94}, had suggested that the behavior of the observed fluctuations could be because the accretion disk is in self-organized critical state.
In all these models the temporal behavior of the system is driven by underlying stochastic variations.	In all these models the temporal behavior of the system is driven by underlying stochastic variations.
Moreover. they address the short (<10 sec) time-scale variability of the systems. although many systems also exhibit quite dramatic long time-scale variability.	Moreover, they address the short $ < 10 $ sec) time-scale variability of the systems, although many systems also exhibit quite dramatic long time-scale variability.
Variability of a system may not necessarily be driven by an underlying stochastic variation.	Variability of a system may not necessarily be driven by an underlying stochastic variation.
The system can show complicated temporal behavior if the governing differential equations are non-linear and have unstable steady state solutions.	The system can show complicated temporal behavior if the governing differential equations are non-linear and have unstable steady state solutions.
In other words. although these systems do not have an explicit time dependent term in the equations describing their structure. they exhibit sustained time variability.	In other words, although these systems do not have an explicit time dependent term in the equations describing their structure, they exhibit sustained time variability.
In fact. the standard aceretion disk theory predicts that the disk is unstable when it is radiation pressure dominated and when the viscous stress scales with the total pressure.	In fact, the standard accretion disk theory predicts that the disk is unstable when it is radiation pressure dominated and when the viscous stress scales with the total pressure.
Numerical hydro-dynamie simulations reveal that under such circumstances. the disk would undergo large amplitude oscillations around the unstable solution (Chen&Taam1994).	Numerical hydro-dynamic simulations reveal that under such circumstances, the disk would undergo large amplitude oscillations around the unstable solution \citep{Che94}.
. These variations occur on a viscous time-scale that may be as large as hundreds of seconds.	These variations occur on a viscous time-scale that may be as large as hundreds of seconds.
Specific dynamic models for the temporal behavior of accretion disk. like the Dripping Handrail (Young&Seargle 1996).. have been proposed where the apparent random behavior is actually deterministic.	Specific dynamic models for the temporal behavior of accretion disk, like the Dripping Handrail \citep{You96}, have been proposed where the apparent random behavior is actually deterministic.
The Galactic micro quasar GRS 19154105 1s a highly variable black hole system.	The Galactic micro quasar GRS 1915+105 is a highly variable black hole system.
It shows a wide range of long term variability (Chenefaf1997; which required Bellonieraf(2000) to classify the behavior in no less than twelve temporal	It shows a wide range of long term variability \citep{Che97,Pau97,Bel97a} which required \cite{Bel00} to classify the behavior in no less than twelve temporal
The magnetorotational instability is. one. of the most important instajlities in astrophysical Ες dynamics.	The magnetorotational instability is one of the most important instabilities in astrophysical fluid dynamics.
\|t applies to a οercntially rotating. electrically. conducting Iluid in which t1e angular velocity. decreases in magnitude awav [rom the axis of 1n the presence of a weak magnetic ielel of arbitrary. configuration. such a Low is subject to a dynamical instability. with a growth rate comparable to he shear rate of the low (Velikhoy 1959: Chandrasekhar 1960: Fricke 1969: Acheson LOTS: Balbus Lawley 1991. 1992: Papaloizou Szuszkiewicz 1992: Balbus 1995: lFoglizzo Vagger 1995: Ogilvie Pringle 1996: Terquem Papaloizou 1996).	It applies to a differentially rotating, electrically conducting fluid in which the angular velocity decreases in magnitude away from the axis of In the presence of a weak magnetic field of arbitrary configuration, such a flow is subject to a dynamical instability, with a growth rate comparable to the shear rate of the flow (Velikhov 1959; Chandrasekhar 1960; Fricke 1969; Acheson 1978; Balbus Hawley 1991, 1992; Papaloizou Szuszkiewicz 1992; Balbus 1995; Foglizzo Tagger 1995; Ogilvie Pringle 1996; Terquem Papaloizou 1996).
The xincipal application of the magnetorotational instability is) to accretion discs. in which the profile of angular velocity ds. fixed by Kepler's third law.	The principal application of the magnetorotational instability is to accretion discs, in which the profile of angular velocity is fixed by Kepler's third law.
The non-linear development of the instability leads to sustained magnetohycdrodyvnamuc (MED) turbulence. which ransports angular momentum outwards in a vain attonipt o neutralize the destabilizing eradient of angular. velocity (Llawles. Gammie albus 1995: Brandenburg et al.	The non-linear development of the instability leads to sustained magnetohydrodynamic (MHD) turbulence, which transports angular momentum outwards in a vain attempt to neutralize the destabilizing gradient of angular velocity (Hawley, Gammie Balbus 1995; Brandenburg et al.
1995: Stone et al.	1995; Stone et al.
1996: Dalbus Lawley 1998).	1996; Balbus Hawley 1998).
Owing to the generality of the conditions or instability. however. further applications exist to stellar interiors ancl other astrophysical objects. ancl possibly to	Owing to the generality of the conditions for instability, however, further applications exist to stellar interiors and other astrophysical objects, and possibly to
O.Struecm In every as .r passes through c. the number of sign changes in (fo(r).fiGr))decreases by L if g(«»-Q. and increases by-- \| if g(«0. If e is a root of g;with /=1....A. then it is neither a root of y;, nor a root of g;,. and g;1f«160)«0. by the of the sequence.	0.5truecm In every as $x$ passes through $c$, the number of sign changes in $(f_0(x),f_1(x))$ decreases by $1$ if $g(c)>0$ , and increases by $1$ if $g(c)<0$ If $c$ is a root of $g_i$with $i=1,...k$, then it is neither a root of $g_{i-1}$ nor a root of $g_{i+1}$, and $g_{i-1}(c)g_{i+1}(c)<0$, by the definition of the sequence.
Passing through c does not lead to any modification of the number of sign changes in (f;Gr).fir).LC6r7))in this case.	Passing through $c$ does not lead to any modification of the number of sign changes in $(f_{i-1}(x),f_i(x),f_{i+1}(x))$in this case.
Using g=1 in previous theorem.	Using $g=1$ in previous theorem.
Rees(1988) assumed that the debris is uniformly distributed in mass between —AL and +AR.	\citet{ree88} assumed that the debris is uniformly distributed in mass between $-\Delta E$ and $+ \Delta E$ .
Numerical simulations (Evans&Iwochanek1950:Aval.Livio.Piran2000) have shown (hat this is a reasonable approximation.	Numerical simulations \citep{eva89,aya00} have shown that this is a reasonable approximation.
The bound material then returns to pericenter al the rate (Phinney1959:Evans&lxochanek1959) where {ο is thetime of the initial tidal disruption. AA/ is the actual mass Chat [alls back to pericenter. which is a fraction f of the original mass of the star. ancl ↕∐∐≼↲≼↲⋟∖⊽∎∪↕⋅↕≸↽↔↴↕∐≀↕↴∐⊔⋯⇂≼↲↥⋅⊔∐↲∖∖⊽∐∪↥≼↲⋟∖⊽↥≀↧↴↕⋅↕⋟∖⊽≼∐⋟∖⊽↕⋅∏↕↽≻∩↲≼⇂≀↧↴↕∐⇂↥⋯↴∐⋟⊔∐↲≼⇂≼↲∣↽≻∏⋟∖⊽↕⋟∖⊽∪∐∣↽≻∪∏∐≺⇂∪	The bound material then returns to pericenter at the rate \citep{phi89,eva89} where $t_D$ is thetime of the initial tidal disruption, $\Delta M$ is the actual mass that falls back to pericenter, which is a fraction $f$ of the original mass of the star, and In Rees' original model, the whole star is disrupted and half the debris is on bound orbits, so that $f = 0.5$.
↕⋅∣↽≻∐⋟∖⊽⋅ ⋟∖⊽∪⊔⋯↥∙∕⋮∶∩⋅⇀↱≻⋅∐∪∖∖⊽≼↲∖⇁≼↲↕⋅⋅≀↧↴↕⋅≼↲≺∢≼↲∐↥∐∏∐∐↲↕⋅↕≺∢≀↧↴↥⋟∖⊽↕∐↓∏↥≀↧↴∐∪∐⋟∖⊽∐∪∖∖⇁⋟∖⊽⊔⋯↴↥∐∪↥≀↧↴∐⊔∐↲↕⋅≼↲⊓∐⋅∐↕∐≸≟ ∐↓≀↧↴∩↲↕⋅↕≀↧↴⊔⋟∖⊽≺∢≀↧↴↕↽≻⊓∐⋅≼↲≼⊓↽≻⋡∖↽⊔∐↲∣↽≻↥≀↧↴≺∢↳↽∐∪↥≼↲∶≀↧↴∣↽≻∪⋯⊺⇀↱≻↖∕⋡∣⋪↙∪↓⋟⊔∐↲↕⋅≼↲⋯↕⋅∐↕∐≸≟∐↓≀↕⊍∖⋱∖⊽∣↽≻≼↲≺∢∪∐∐↲⋟∖⊽∏∐∣↽≻∪	However, a recent numerical simulation shows that not all the returning material is captured by the black hole: about $75\%$ of the returning mass becomes unbound following the large compression it experiences on the way back \citep{aya00}.
∏∐≼⇂ ↓≯∪∐∪∖∖↽↕∐≸↽↔↴⊔∐↲↥≀↧↴↕⋅≸↽↔↴≼↲≺∢∪∐↓↕↽≻↕⋅≼↲⋝∖⊽	This gives rise to a smaller $f\approx 0.1$.
⊳∖⇁↕∪∐∐≼↲⇀↸↕↽≻≼↲↕⋅↕≼↲∐≺∢≼↲⊳∖⊽∪∐⊔∐↲∖∖↽≀↧∶∖⇁∣↽≻≀↧↴≺∢↳↽≼⋝↼≚∡∖⇁≀↧↴↥⋅⊔∖↽↥∪⋅≪↽∖↽↥↴↕↕⋅≀↧↴∐∃∪∩∩↕⋝⋅⋅ ↴⊺∐↕⋝∖⊽≸↽↔↴↕∖↽≼↲⋝∖⊽↕⋅↕⋟∖⊽≼↲↥∪≀↧↪∖⊽∐⋯∐≼↲↕⋅∙∕⋅≈∪⋅⊥⋅↼≚∐∪⊔∐↲↕⋅↕↽≻∪⊳∖⊽⊳∖⇁↕∣↽≻∐∐∡∖↽≸↽↔↴↕∖⇁↕∐≸≟↕⋅↕⊳∖⊽≼↲↥∪≀↧↪∖⇁∐⋯∐∙∕⋅↕⊳∖⊽⊔⋯↴↥⊔∐↲ ⋝∖⊽↥≀↧↴↕⋅↥⊳∖⊽∪∐↥∡∖⇁↕↽≻≀↧↴↕⋅∐≀↧↴∐⋡∖↽≼∐⊳∖⊽↕⋅∏↕↽≻∩↲≺⇂∶∐⊳∖⊽≼↲∐∖↽≼↲↥∪↕↽≻≼↲≺∢⋯∏≼⇂∣↽≻≼↲⊳∖⊽∏⋅∏↽≻↕↽≻≼↲≼⇂∣↽≻∡∖↽⊔∐↲∣↽≻↥≀↧↴≺∢↕≶∐∪↥≼↲⋅↥≼↲≀↧↴∖↽↕∐≸↽↔↴∐↓∪⋝∖⊽↥ ol its core nearlv intact. (IRenzini et al.	Another possibility giving rise to a small $f$ is that the star is only partially disrupted: its envelope could be stripped by the black hole, leaving most of its core nearly intact (Renzini et al.
1995: see also 855 below).	1995; see also 5 below).
Here we treat. f/ as a [ree parameter.	Here we treat $f$ as a free parameter.
The gravitational potential energv available from fallback is determined bv the difference between (he specific binding energy of the circularization orbit al r=2rpνε and the specific binding energy of the incoming material.	The gravitational potential energy available from fallback is determined by the difference between the specific binding energy of the circularization orbit at $r = 2 r_P = 2r_T$ and the specific binding energy of the incoming material.
Since Mj;S>AL,. all the bound debris is on hiehlv eccentric orbils wilh a specilic binding energy much smaller than the binding energy ol the final circular orbit.	Since $M_H \gg M_\star$, all the bound debris is on highly eccentric orbits with a specific binding energy much smaller than the binding energy of the final circular orbit.
Thus. assuming that the fallback material racdiates the energy release promptly. the radiation efficiency. € is independent of time during fallback: The huminositv of the fallback process is then given by The luminosity peaks at /=/p+ Afy. ie. when the most bound debris falls back to the pericenter. sowe have	Thus, assuming that the fallback material radiates the energy release promptly, the radiation efficiency $\epsilon$ is independent of time during fallback: The luminosity of the fallback process is then given by The luminosity peaks at $t = t_D + \Delta t_1$ , i.e. when the most bound debris falls back to the pericenter, sowe have
For tvpical parameters (e.g.. o,~LO7) 7; can be measured to &10s (e.g.. Brown et 22001: Holman et 22006).	For typical parameters (e.g., $\sigma_{ph}\sim~10^{-3}$ ), $t_i$ can be measured to $\simeq10$ s (e.g., Brown et 2001; Holman et 2006).
The period can be measured much more accurately (han ἐν rom observations of multiple transits separated by many orbits.	The period can be measured much more accurately than $t_i$, from observations of multiple transits separated by many orbits.

Therefore. one has to focus on CUECRs with higher energies to exaiine cases where 7(£) is shorter than the appareut time interval between bursts or flares occurring in the same aneular patch. AT.

Therefore, one has to focus on UHECRs with higher energies to examine cases where $\tau(E)$ is shorter than the apparent time interval between bursts or flares occurring in the same angular patch, $\Delta T$.

In reality. UHECRs have finite deviation angles due to cosinic magnetic fields. so UIIECTs from a source arrives within a finite solid augle AQ=we? around the source. which can be regarded as the appropriate size of the finite augular patch.

In reality, UHECRs have finite deviation angles due to cosmic magnetic fields, so UHECRs from a source arrives within a finite solid angle $\Delta \Omega = \pi \psi^2$ around the source, which can be regarded as the appropriate size of the finite angular patch.

For a given py. the apparent tine interval between bursts in the region of the sky with AQ is estimated to be where 0s=/5?aud pay=pQ/109 Cpe% t.

For a given $\rho_s$, the apparent time interval between bursts in the region of the sky with $\Delta \Omega$ is estimated to be where $\psi_5 \equiv \psi / 5^{\circ}$and $\rho_{s,0} = \rho_s / 10^0$ $^{-3}$ $^{-1}$.

Wo ake the typical positional correlation scale as ος and mse c005? as a reference choice. which correspouds o Boch’?<2 ut Mpc/7.

We take the typical positional correlation scale as $\psi$, and use $\psi \sim 5^{\circ}$ as a reference choice, which corresponds to $B_{\rm eff} {\lambda_{\rm eff}}^{1/2} \lesssim 2$ nG $^{1/2}$.

This is reasonable. since lis is consistent with the effective. EGAIFs estimated in the last subsection and a current upper limit of the void EGAIF from a plausible cosmological model is 2.5 we for Ay=1 Ape2010).. but more conservative discussions with larger values of c are also possible.

This is reasonable, since this is consistent with the effective EGMFs estimated in the last subsection and a current upper limit of the void EGMF from a plausible cosmological model is 2.5 nG for $\lambda_{\rm v} = 1$ Mpc, but more conservative discussions with larger values of $\psi$ are also possible.

Equation (8)) imuplies that a smaller AQ gives areer AT. but AT should be limited to the burst/flare intermittence m a host galaxy. zΠρι. where ny is the iiuber density of host galaxies of UITECR sources.

Equation \ref{eq:deltat}) ) implies that a smaller $\Delta \Omega$ gives larger $\Delta T$, but $\Delta T$ should be limited to the burst/flare intermittence in a host galaxy, $\approx n_h/\rho_s$, where $n_h$ is the number density of host galaxies of UHECR sources.

Iu other words. AQ smaller than the corresponding lower Πιτ is meanineless, at which one host galaxy should exist in a volume with a solid angle AQ within {ως1).

In other words, $\Delta \Omega$ smaller than the corresponding lower limit is meaningless, at which one host galaxy should exist in a volume with a solid angle $\Delta \Omega$ within $D_{\rm max}(E)$.

We call the case “bursting case” that only one burst or flare contributes to arviving UIIECTis at a time in a direction. Le. TLE)AT.

We call the case "bursting case" that only one burst or flare contributes to arriving UHECRs at a time in a direction, i.e., $\tau(E) < \Delta T$.

Then. the requirement TE)<AT gives a sufficient coucition to apply equation (6)). which leads to with the usage of equation (6)).

Then, the requirement $\tau(E) < \Delta T$ gives a sufficient condition to apply equation \ref{eq:estrate}) ), which leads to with the usage of equation \ref{eq:estrate}) ).

As demonstrated in Figure 2.. the range of (CE). in which equation (6)) can be applied. is extended to larger κ1) at higher energies. because the aller nunmiber of sources contributiug to the observed flux. decreases the probability that two bursts are temporally overlapped in a region of the skv (see also equation (8))).

As demonstrated in Figure \ref{fig:nlim}, the range of $n_s(E)$ , in which equation \ref{eq:estrate}) ) can be applied, is extended to larger $n_s(E)$ at higher energies, because the smaller number of sources contributing to the observed flux decreases the probability that two bursts are temporally overlapped in a region of the sky (see also equation \ref{eq:deltat}) )).

Thus. we especially focus. on cases of E~10°" eV to demonstrate constrünts on psu Sections 3. and L. although discussions are general for other E.

Thus, we especially focus on cases of $E \sim {10}^{20}$ eV to demonstrate constraints on $\rho_s$ in Sections \ref{prop} and \ref{results}, although discussions are general for other $E$.

Note that. although we here fix e even at higher enereies. stualler values of c are expected there. so that the extension of the curve to higher enereies would be more casily justified.

Note that, although we here fix $\psi$ even at higher energies, smaller values of $\psi$ are expected there, so that the extension of the curve to higher energies would be more easily justified.

Ou the other haud. if τε)>AT. another UITECR burst may start to contribute before the eud of the former CHECR burst is observed. aud equation (8)) nuplies. Since n4CE) can be determined by the auto-correlation analysis. equation (6)) enables us to estimate p, from observational quantities. if TLE) cau be evaluated by ECAIF simulations aud observations.

On the other hand, if $\tau(E) > \Delta T$, another UHECR burst may start to contribute before the end of the former UHECR burst is observed, and equation \ref{eq:deltat}) ) implies, Since $n_s(E)$ can be determined by the auto-correlation analysis, equation \ref{eq:estrate}) ) enables us to estimate $\rho_s$ from observational quantities, if $\tau(E)$ can be evaluated by EGMF simulations and observations.

Iuportautly. 7.2) has the characteristic energv dependence. which is demonstrated in Figure 2..

Importantly, $n_s(E)$ has the characteristic energy dependence, which is demonstrated in Figure \ref{fig:nlim}.

Here. the case that oulv the ECAIF in filamentary structures affects the time spread of VITZECR bursts is considered for demonstration. 1.6.. Bade?~03 ας: P.

Here, the case that only the EGMF in filamentary structures affects the time spread of UHECR bursts is considered for demonstration, i.e., $B_{\rm eff} {\lambda_{\rm eff}}^{1/2} \sim 0.3$ nG $^{-3}$ .

Two represeutative cases for p, are shown. Lo. p,=1 yrt and 10? ? lo

Two representative cases for $\rho_s$ are shown, i.e., $\rho_s = 1$ $^{-3}$ $^{-1}$ and $10^2$ $^{-3}$ $^{-1}$ .

One sees that n.(E) changes by more than one order of magnitude when £ increases by the cubic root of ten.

One sees that $n_s(E)$ changes by more than one order of magnitude when $E$ increases by the cubic root of ten.

This mieaus that anisotropy features are different among energies.

This means that anisotropy features are different among energies.

Thus. observations of UITEC's above 107 eX. are crucial to identify this tendency clearly.

Thus, observations of UHECRs above $10^{20}$ eV are crucial to identify this tendency clearly.

Future CIECRexperiments with huge exposures may

Future UHECRexperiments with large exposures may

To acquire precise O-isotope data for Iris. we developed a sample mount consisting of an Al round. which supported the potted butt at a fixed altitude. covered by an Al annulus which both supported a 500nm-thick membrane (made of gold-coated S1; N, with a 3800s ion-millecl hole) and incorporated polished standards surrounding the central hole.

To acquire precise O-isotope data for Iris, we developed a sample mount consisting of an Al round, which supported the potted butt at a fixed altitude, covered by an Al annulus which both supported a 500nm-thick membrane (made of gold-coated $_3$ $_4$ with a $\mu$ m ion-milled hole) and incorporated polished standards surrounding the central hole.

San Carlos olivine was used to standardize Iris olivine. ancl Mivakajima plagioclase and Burma. spinel were used to standardize Iris mesostasis ancl chromite. respectively.

San Carlos olivine was used to standardize Iris olivine, and Miyakajima plagioclase and Burma spinel were used to standardize Iris mesostasis and chromite, respectively.

We measured three oxvgen isotopes in Iris using the University of Hawaii Cameca inis 1280 ion microprobe in multicollection mode (!O on a Faraday cup. YO and PO on electron multipliers).

We measured three oxygen isotopes in Iris using the University of i Cameca ims 1280 ion microprobe in multicollection mode $^{16}$ O on a Faraday cup, $^{17}$ O and $^{18}$ O on electron multipliers).

A 2530 pA primary ion beam was focused to ~2 jan to allow for analvsis of Iris.

A 25–30 pA $^{+}$ primary ion beam was focused to $\sim$ 2 $\mu$ m to allow for single-grain analysis of Iris.

The data was corrected for background. deacltime. detector vield. and interference from MOLL to FO (typically <0.2%.).

The data was corrected for background, deadtime, detector yield, and interference from $^{16}$ $^{-}$ to $^{17}$ O (typically $<$ $\permil$ ).

We corrected measured compositions of (he mineral phases in Iris for instrumental mass fractionation by comparing wilh appropriate mineral standards of known composition.

We corrected measured compositions of the mineral phases in Iris for instrumental mass fractionation by comparing with appropriate mineral standards of known composition.

We measured (he oxvgen isotopic composition ol San Carlos olivine mounted analogously to Iris in order (ο understand instrumental mass fractionation associated with the mounting of the unknown sample relative to (he standards.

We measured the oxygen isotopic composition of San Carlos olivine mounted analogously to Iris in order to understand instrumental mass fractionation associated with the mounting of the unknown sample relative to the standards.

We found (that there was a reproducible instrumental fractionation effect between (he San Carlos olivine surrounding the central hole and the San Carlos olivine in the central hole (the center olivine plotted ος lower in 0O than the surrounding olivine along the terrestrial fractionation line).

We found that there was a reproducible instrumental fractionation effect between the San Carlos olivine surrounding the central hole and the San Carlos olivine in the central hole (the center olivine plotted $\sim$ $\permil$ lower in $\delta^{18}$ O than the surrounding olivine along the terrestrial fractionation line).

We applied this offset to the final Iris oxvgen measurements.

We applied this offset to the final Iris oxygen measurements.

Our conservative estimate of the uncertainties in the final ratios comes [from adding in quadrature the statistical uncertainty of the individual measurements. the standard deviation of the standard measurements. and (he uncertainty in the instrumental mass fractionation olIset.

Our conservative estimate of the uncertainties in the final ratios comes from adding in quadrature the statistical uncertainty of the individual measurements, the standard deviation of the standard measurements, and the uncertainty in the instrumental mass fractionation offset.

Olivine. chromite. and mesostasis in Iris have oxvgen isotopic compositions indistinguishable from terrestrial oxvgen (Figure 2)).

Olivine, chromite, and mesostasis in Iris have oxygen isotopic compositions indistinguishable from terrestrial oxygen (Figure \ref{oiso}) ).

The oxvgen isotopic compositions of the three mineral phases are consistent with co-crvstallization [rom a single melt.

The oxygen isotopic compositions of the three mineral phases are consistent with co-crystallization from a single melt.

Iris has oxvgen-isotope composition higher in both “O/'O and !' O/!O than most chondriules in meteorites and other chondrule-like objects in the Stardust collection (Figure 2)).

Iris has oxygen-isotope composition higher in both $^{18}$ $^{16}$ O and $^{17}$ $^{16}$ O than most chondrules in meteorites and other chondrule-like objects in the Stardust collection (Figure \ref{oiso}) ).

The oxveen isotopes of nebular solids can evolve toward isotopically heavier compositions through processes such as interaction with water (partitioning between liquid ancl solid) or evaporation (Ravleigh-like distillation).

The oxygen isotopes of nebular solids can evolve toward isotopically heavier compositions through processes such as interaction with water (partitioning between liquid and solid) or evaporation (Rayleigh-like distillation).

Iris apparently lormed from a relatively evolved oxygen isotopic reservoir.

Iris apparently formed from a relatively evolved oxygen isotopic reservoir.

The inerease of rogsry in blue earlv-tvpe dwarfs (Fig.

The increase of $r_{e B}/r_{e H}$ in blue early-type dwarfs (Fig.

reflects the existence of a (strong) negative gradient in 11 and in V along the radial coordinate (Fig.

2) reflects the existence of a (strong) negative gradient in $-$ H and in $-$ V along the radial coordinate (Fig.

3).

The data show that this is a necessary but not sullicient condition however.

The data show that this is a necessary but not sufficient condition however.

This trend is not foun in giant I5 and SO galaxies. which span a broad range in r.g/r.g. though their 11 color is almost as red as in red earlv-tvpe diwarfs.

This trend is not found in giant E and S0 galaxies, which span a broad range in $r_{e B}/r_{e H}$, though their $-$ H color is almost as red as in red early-type dwarfs.

Raclial color eracdients observed. in individual giant. elliptical and lenticular galaxies are commonly. interpreted. as a product either of age/moetallicitv gradients of the stellar populations along the radial coordinate or of dust attenuation (cl.

Radial color gradients observed in individual giant elliptical and lenticular galaxies are commonly interpreted as a product either of age/metallicity gradients of the stellar populations along the radial coordinate or of dust attenuation (cf.

Sect.

1).

These interpretations do not imply a correlation between Γρ ond BLL. In the Following sections. these three theoretical interpretations will be extended. to our results for carly-type dvarfs and ciscussed individually. in relation with some scenarios of earlv-tvpe dwarf galaxy formation and evolution which may justify the trend seen in Fig.

These interpretations do not imply a correlation between $r_{e B}/r_{e H}$ and $-$ H. In the following sections, these three theoretical interpretations will be extended to our results for early-type dwarfs and discussed individually, in relation with some scenarios of early-type dwarf galaxy formation and evolution which may justify the trend seen in Fig.

2.

The lack of data does not allow us to achieve a direct. proof of the valiclity of any of these three interpretations.

The lack of data does not allow us to achieve a direct proof of the validity of any of these three interpretations.

Moreover. the absence of a one-to-one correlation between reesra and the strength ofthe gradient in LL. i£ not due to errors in data analysis. may indicate that the interpretation of Fig.

Moreover, the absence of a one-to-one correlation between $r_{e B}/r_{e H}$ and the strength of the gradient in $-$ H, if not due to errors in data analysis, may indicate that the interpretation of Fig.

2 is complex.

The integrated broacd-banc colors of E and SO galaxies become progressively Επομ toward fainter magnitudes (Faber 1973: see however Scodegeio 2001).

The integrated broad-band colors of E and S0 galaxies become progressively bluer toward fainter magnitudes (Faber 1973; see however Scodeggio 2001).

This correlation. known as the colormagniude relation. is universal and very σαι in he optical for ellipticals and lenticulars in clusters al z=0 Bower. Lucey Ellis 1992a.b).

This correlation, known as the color–magnitude relation, is universal and very tight in the optical for ellipticals and lenticulars in clusters at z=0 (Bower, Lucey Ellis 1992a,b).

Relying on the commonly accopled interpretation of the colormagnitude relation (Ixodama Arimoto 1997). Wwe c'onclude that. in blue carby-type diwvarfs. either the averaee metallicity is lower than in red earlv-tvpe. dwarls of je same L-banel luminosity or star formation is still going «on.

Relying on the commonly accepted interpretation of the color–magnitude relation (Kodama Arimoto 1997), we conclude that, in blue early-type dwarfs, either the average metallicity is lower than in red early-type dwarfs of the same H-band luminosity or star formation is still going on.

Here we make the. hypothesis that the blue colors of the VCC sample cwarls are due to the presence of a voung sellar population.

Here we make the hypothesis that the blue colors of the VCC sample dwarfs are due to the presence of a young stellar population.

‘This hypothesis is supported by spectrosco[ow (GOL) for some individual samyge galaxies and is consisten with the results of Terlevich et al. (

This hypothesis is supported by spectroscopy (G01) for some individual sample galaxies and is consistent with the results of Terlevich et al. (

1999).

IDe.

Fig.

5h srows that most of the B-bancl emission of the chwarls is contributed by the. st«αρ populations distributed wihin an cxponential-cisk c“OMIpeHen.

5b shows that most of the B-band emission of the dwarfs is contributed by the stellar populations distributed within an exponential-disk component.

This disk-componen. M present. is also responside for more than of the tota L-band luminosity (Fig.

This disk-component, if present, is also responsible for more than of the total H-band luminosity (Fig.

5a).

Lowe make the asstunption that early-type dwarls are rotaionally [lattened (cf.

If we make the assumption that early-type dwarfs are rotationally flattened (cf.

Sect.

1). tr¢ wavelength-dependence of their. effective radius. and the relation between total coor and sign and streneth of the color. gradients. may be understood: as

1), the wavelength-dependence of their effective radius, and the relation between total color and sign and strength of the color gradients may be understood as

(e.g.Bellonterαἱ.2001:Tomsick& (e.g.

\citep[e.g.][]{Bel01,Tom01,Rod02}. \citep[e.g.][]{Now99,Mis00,Cui99,Pou99,Nob01}.

in black hole and neutron star systems. can be explainec in a model where the global disk resonates nonlinearly with the stochastic fluctuations (Abramowiezefal.2003).

in black hole and neutron star systems, can be explained in a model where the global disk resonates nonlinearly with the stochastic fluctuations \citep{Aba03}.

. More compelling model independent evidence has been givel by Uttleyοἱal.(2005). (see also Timmereraf.(2000) and Thieleta£. (2001))). who argue that the log-normal distribution of the fluxes and the linear relationship betweer RMS and flux imply that the response of the disk is non-

More compelling model independent evidence has been given by \cite{Utt05} (see also \cite{Tim00} and \cite{Thi01}) ), who argue that the log-normal distribution of the fluxes and the linear relationship between RMS and flux imply that the response of the disk is non-linear.

They show that an exponential response can explaui these observations.

They show that an exponential response can explain these observations.

Earlier Mineshigeefαἱ.(1994)... had suggested that the behavior of the observed fluctuations could be because the accretion disk is in self-organized critical state.

Earlier \cite{Min94}, had suggested that the behavior of the observed fluctuations could be because the accretion disk is in self-organized critical state.

In all these models the temporal behavior of the system is driven by underlying stochastic variations.

Moreover. they address the short (<10 sec) time-scale variability of the systems. although many systems also exhibit quite dramatic long time-scale variability.

Moreover, they address the short $ < 10 $ sec) time-scale variability of the systems, although many systems also exhibit quite dramatic long time-scale variability.

Variability of a system may not necessarily be driven by an underlying stochastic variation.

The system can show complicated temporal behavior if the governing differential equations are non-linear and have unstable steady state solutions.

In other words. although these systems do not have an explicit time dependent term in the equations describing their structure. they exhibit sustained time variability.

In other words, although these systems do not have an explicit time dependent term in the equations describing their structure, they exhibit sustained time variability.

In fact. the standard aceretion disk theory predicts that the disk is unstable when it is radiation pressure dominated and when the viscous stress scales with the total pressure.

In fact, the standard accretion disk theory predicts that the disk is unstable when it is radiation pressure dominated and when the viscous stress scales with the total pressure.

Numerical hydro-dynamie simulations reveal that under such circumstances. the disk would undergo large amplitude oscillations around the unstable solution (Chen&Taam1994).

Numerical hydro-dynamic simulations reveal that under such circumstances, the disk would undergo large amplitude oscillations around the unstable solution \citep{Che94}.

. These variations occur on a viscous time-scale that may be as large as hundreds of seconds.

These variations occur on a viscous time-scale that may be as large as hundreds of seconds.

Specific dynamic models for the temporal behavior of accretion disk. like the Dripping Handrail (Young&Seargle 1996).. have been proposed where the apparent random behavior is actually deterministic.

Specific dynamic models for the temporal behavior of accretion disk, like the Dripping Handrail \citep{You96}, have been proposed where the apparent random behavior is actually deterministic.

The Galactic micro quasar GRS 19154105 1s a highly variable black hole system.

The Galactic micro quasar GRS 1915+105 is a highly variable black hole system.

It shows a wide range of long term variability (Chenefaf1997; which required Bellonieraf(2000) to classify the behavior in no less than twelve temporal

It shows a wide range of long term variability \citep{Che97,Pau97,Bel97a} which required \cite{Bel00} to classify the behavior in no less than twelve temporal

The magnetorotational instability is. one. of the most important instajlities in astrophysical Ες dynamics.

The magnetorotational instability is one of the most important instabilities in astrophysical fluid dynamics.

|t applies to a οercntially rotating. electrically. conducting Iluid in which t1e angular velocity. decreases in magnitude awav [rom the axis of 1n the presence of a weak magnetic ielel of arbitrary. configuration. such a Low is subject to a dynamical instability. with a growth rate comparable to he shear rate of the low (Velikhoy 1959: Chandrasekhar 1960: Fricke 1969: Acheson LOTS: Balbus Lawley 1991. 1992: Papaloizou Szuszkiewicz 1992: Balbus 1995: lFoglizzo Vagger 1995: Ogilvie Pringle 1996: Terquem Papaloizou 1996).

It applies to a differentially rotating, electrically conducting fluid in which the angular velocity decreases in magnitude away from the axis of In the presence of a weak magnetic field of arbitrary configuration, such a flow is subject to a dynamical instability, with a growth rate comparable to the shear rate of the flow (Velikhov 1959; Chandrasekhar 1960; Fricke 1969; Acheson 1978; Balbus Hawley 1991, 1992; Papaloizou Szuszkiewicz 1992; Balbus 1995; Foglizzo Tagger 1995; Ogilvie Pringle 1996; Terquem Papaloizou 1996).

The xincipal application of the magnetorotational instability is) to accretion discs. in which the profile of angular velocity ds. fixed by Kepler's third law.

The principal application of the magnetorotational instability is to accretion discs, in which the profile of angular velocity is fixed by Kepler's third law.

The non-linear development of the instability leads to sustained magnetohycdrodyvnamuc (MED) turbulence. which ransports angular momentum outwards in a vain attonipt o neutralize the destabilizing eradient of angular. velocity (Llawles. Gammie albus 1995: Brandenburg et al.

The non-linear development of the instability leads to sustained magnetohydrodynamic (MHD) turbulence, which transports angular momentum outwards in a vain attempt to neutralize the destabilizing gradient of angular velocity (Hawley, Gammie Balbus 1995; Brandenburg et al.

1995: Stone et al.

1995; Stone et al.

1996: Dalbus Lawley 1998).

1996; Balbus Hawley 1998).

Owing to the generality of the conditions or instability. however. further applications exist to stellar interiors ancl other astrophysical objects. ancl possibly to

Owing to the generality of the conditions for instability, however, further applications exist to stellar interiors and other astrophysical objects, and possibly to

O.Struecm In every as .r passes through c. the number of sign changes in (fo(r).fiGr))decreases by L if g(«»-Q. and increases by-- | if g(«0. If e is a root of g;with /=1....A. then it is neither a root of y;, nor a root of g;,. and g;1f«160)«0. by the of the sequence.

0.5truecm In every as $x$ passes through $c$, the number of sign changes in $(f_0(x),f_1(x))$ decreases by $1$ if $g(c)>0$ , and increases by $1$ if $g(c)<0$ If $c$ is a root of $g_i$with $i=1,...k$, then it is neither a root of $g_{i-1}$ nor a root of $g_{i+1}$, and $g_{i-1}(c)g_{i+1}(c)<0$, by the definition of the sequence.

Passing through c does not lead to any modification of the number of sign changes in (f;Gr).fir).LC6r7))in this case.

Passing through $c$ does not lead to any modification of the number of sign changes in $(f_{i-1}(x),f_i(x),f_{i+1}(x))$in this case.

Using g=1 in previous theorem.

Using $g=1$ in previous theorem.

Rees(1988) assumed that the debris is uniformly distributed in mass between —AL and +AR.

\citet{ree88} assumed that the debris is uniformly distributed in mass between $-\Delta E$ and $+ \Delta E$ .

Numerical simulations (Evans&Iwochanek1950:Aval.Livio.Piran2000) have shown (hat this is a reasonable approximation.

Numerical simulations \citep{eva89,aya00} have shown that this is a reasonable approximation.

The bound material then returns to pericenter al the rate (Phinney1959:Evans&lxochanek1959) where {ο is thetime of the initial tidal disruption. AA/ is the actual mass Chat [alls back to pericenter. which is a fraction f of the original mass of the star. ancl ↕∐∐≼↲≼↲⋟∖⊽∎∪↕⋅↕≸↽↔↴↕∐≀↕↴∐⊔⋯⇂≼↲↥⋅⊔∐↲∖∖⊽∐∪↥≼↲⋟∖⊽↥≀↧↴↕⋅↕⋟∖⊽≼∐⋟∖⊽↕⋅∏↕↽≻∩↲≼⇂≀↧↴↕∐⇂↥⋯↴∐⋟⊔∐↲≼⇂≼↲∣↽≻∏⋟∖⊽↕⋟∖⊽∪∐∣↽≻∪∏∐≺⇂∪

The bound material then returns to pericenter at the rate \citep{phi89,eva89} where $t_D$ is thetime of the initial tidal disruption, $\Delta M$ is the actual mass that falls back to pericenter, which is a fraction $f$ of the original mass of the star, and In Rees' original model, the whole star is disrupted and half the debris is on bound orbits, so that $f = 0.5$.

↕⋅∣↽≻∐⋟∖⊽⋅ ⋟∖⊽∪⊔⋯↥∙∕⋮∶∩⋅⇀↱≻⋅∐∪∖∖⊽≼↲∖⇁≼↲↕⋅⋅≀↧↴↕⋅≼↲≺∢≼↲∐↥∐∏∐∐↲↕⋅↕≺∢≀↧↴↥⋟∖⊽↕∐↓∏↥≀↧↴∐∪∐⋟∖⊽∐∪∖∖⇁⋟∖⊽⊔⋯↴↥∐∪↥≀↧↴∐⊔∐↲↕⋅≼↲⊓∐⋅∐↕∐≸≟ ∐↓≀↧↴∩↲↕⋅↕≀↧↴⊔⋟∖⊽≺∢≀↧↴↕↽≻⊓∐⋅≼↲≼⊓↽≻⋡∖↽⊔∐↲∣↽≻↥≀↧↴≺∢↳↽∐∪↥≼↲∶≀↧↴∣↽≻∪⋯⊺⇀↱≻↖∕⋡∣⋪↙∪↓⋟⊔∐↲↕⋅≼↲⋯↕⋅∐↕∐≸≟∐↓≀↕⊍∖⋱∖⊽∣↽≻≼↲≺∢∪∐∐↲⋟∖⊽∏∐∣↽≻∪

However, a recent numerical simulation shows that not all the returning material is captured by the black hole: about $75\%$ of the returning mass becomes unbound following the large compression it experiences on the way back \citep{aya00}.

∏∐≼⇂ ↓≯∪∐∪∖∖↽↕∐≸↽↔↴⊔∐↲↥≀↧↴↕⋅≸↽↔↴≼↲≺∢∪∐↓↕↽≻↕⋅≼↲⋝∖⊽

This gives rise to a smaller $f\approx 0.1$.

⊳∖⇁↕∪∐∐≼↲⇀↸↕↽≻≼↲↕⋅↕≼↲∐≺∢≼↲⊳∖⊽∪∐⊔∐↲∖∖↽≀↧∶∖⇁∣↽≻≀↧↴≺∢↳↽≼⋝↼≚∡∖⇁≀↧↴↥⋅⊔∖↽↥∪⋅≪↽∖↽↥↴↕↕⋅≀↧↴∐∃∪∩∩↕⋝⋅⋅ ↴⊺∐↕⋝∖⊽≸↽↔↴↕∖↽≼↲⋝∖⊽↕⋅↕⋟∖⊽≼↲↥∪≀↧↪∖⊽∐⋯∐≼↲↕⋅∙∕⋅≈∪⋅⊥⋅↼≚∐∪⊔∐↲↕⋅↕↽≻∪⊳∖⊽⊳∖⇁↕∣↽≻∐∐∡∖↽≸↽↔↴↕∖⇁↕∐≸≟↕⋅↕⊳∖⊽≼↲↥∪≀↧↪∖⇁∐⋯∐∙∕⋅↕⊳∖⊽⊔⋯↴↥⊔∐↲ ⋝∖⊽↥≀↧↴↕⋅↥⊳∖⊽∪∐↥∡∖⇁↕↽≻≀↧↴↕⋅∐≀↧↴∐⋡∖↽≼∐⊳∖⊽↕⋅∏↕↽≻∩↲≺⇂∶∐⊳∖⊽≼↲∐∖↽≼↲↥∪↕↽≻≼↲≺∢⋯∏≼⇂∣↽≻≼↲⊳∖⊽∏⋅∏↽≻↕↽≻≼↲≼⇂∣↽≻∡∖↽⊔∐↲∣↽≻↥≀↧↴≺∢↕≶∐∪↥≼↲⋅↥≼↲≀↧↴∖↽↕∐≸↽↔↴∐↓∪⋝∖⊽↥ ol its core nearlv intact. (IRenzini et al.

Another possibility giving rise to a small $f$ is that the star is only partially disrupted: its envelope could be stripped by the black hole, leaving most of its core nearly intact (Renzini et al.

1995: see also 855 below).

1995; see also 5 below).

Here we treat. f/ as a [ree parameter.

Here we treat $f$ as a free parameter.

The gravitational potential energv available from fallback is determined bv the difference between (he specific binding energy of the circularization orbit al r=2rpνε and the specific binding energy of the incoming material.

The gravitational potential energy available from fallback is determined by the difference between the specific binding energy of the circularization orbit at $r = 2 r_P = 2r_T$ and the specific binding energy of the incoming material.

Since Mj;S>AL,. all the bound debris is on hiehlv eccentric orbils wilh a specilic binding energy much smaller than the binding energy ol the final circular orbit.

Since $M_H \gg M_\star$, all the bound debris is on highly eccentric orbits with a specific binding energy much smaller than the binding energy of the final circular orbit.

Thus. assuming that the fallback material racdiates the energy release promptly. the radiation efficiency. € is independent of time during fallback: The huminositv of the fallback process is then given by The luminosity peaks at /=/p+ Afy. ie. when the most bound debris falls back to the pericenter. sowe have

Thus, assuming that the fallback material radiates the energy release promptly, the radiation efficiency $\epsilon$ is independent of time during fallback: The luminosity of the fallback process is then given by The luminosity peaks at $t = t_D + \Delta t_1$ , i.e. when the most bound debris falls back to the pericenter, sowe have

For tvpical parameters (e.g.. o,~LO7) 7; can be measured to &10s (e.g.. Brown et 22001: Holman et 22006).

For typical parameters (e.g., $\sigma_{ph}\sim~10^{-3}$ ), $t_i$ can be measured to $\simeq10$ s (e.g., Brown et 2001; Holman et 2006).

The period can be measured much more accurately (han ἐν rom observations of multiple transits separated by many orbits.

The period can be measured much more accurately than $t_i$, from observations of multiple transits separated by many orbits.