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The corresponding K-band luninositics are 2.1&1073Lg. aud 1.5«1013Lg. in regions οἱ and DB. respectively. | The corresponding K-band luminosities are $2.4 \times 10^{11} \ \rm{L_{K,\odot}}$ and $1.5 \times 10^{11} \ \rm{L_{K,\odot}}$ in regions $A$ and $B$ respectively. |
According to the LAINB huuinosity function we predict 0.8 aud 0.5 resolved LAINBs above the detection threshold in regious A aud DB. respectively. | According to the LMXB luminosity function we predict $0.8$ and $0.5$ resolved LMXBs above the detection threshold in regions $A$ and $B$, respectively. |
Thus. ouly 2:6% of the resolved sources cau be explained by LAINBs. | Thus, only $\approx6\%$ of the resolved sources can be explained by LMXBs. |
The umuber of predicted CXD sources were determined. using two independent approaches. | The number of predicted CXB sources were determined using two independent approaches. |
As a first method. we used the logXNΊου fuuctiou of Morettietal.(2003). | As a first method, we used the $\log N - \log S$ function of \citet{moretti03}. |
. We converted the observed 0.58 keV. baneIL sensitivity limit (1.131«10.Perestem7) tothe 21 τον band asstuine a power law spectral model witὉτ T=1.1 and Calactic column density. which results ier le sensitivity Bit of 1.03510Poresben7. | We converted the observed $0.5-8$ keV band sensitivity limit $1.14 \times 10^{-15} \ \rm{erg \ s^{-1} \ cm^{-2}} $ ) to the $2-10$ keV band assuming a power law spectral model with $\Gamma=1.4$ and Galactic column density, which results in the sensitivity limit of $1.03 \times 10^{-15} \ \rm{erg \ s^{-1} \ cm^{-2}} $. |
Dasec on this value. the surface area of regions A aud D. anc he log—9 function. we predicted 1.9 aud 2. sources mN et and DB regions. respectively. | Based on this value, the surface area of regions $A$ and $B$, and the $\log N - \log S$ function, we predicted $1.9$ and $2.0$ sources in $A$ and $B$ regions, respectively. |
As a second nethod. we used the field of Abcll 191 to determine he average surface density of resolved CNB sources. | As a second method, we used the field of Abell 194 to determine the average surface density of resolved CXB sources. |
Using a circular region with 3" radii. centered iu the CCD. 17 poiut sources were observed. | Using a circular region with $3\arcmin$ radii, centered in the CCD, $17$ point sources were observed. |
Note that in this computation the sources located within regions A and D avo excluded. aloug with the point source associated with the elliptical ealaxy CCGCC. 385-127. | Note that in this computation the sources located within regions $A$ and $B$ are excluded, along with the point source associated with the elliptical galaxy CGCG 385-127. |
Based ou the surface area of regions A aud D. we expect 2.0 and 2.1 sources within these reeious. | Based on the surface area of regions $A$ and $B$, we expect $2.0$ and $2.1$ sources within these regions. |
We stress that the predicted munbers of CNB sources obtained with the two methods are iu excellent aereeimnent with cach other. | We stress that the predicted numbers of CXB sources obtained with the two methods are in excellent agreement with each other. |
Therefore no more than z20% of the detected sources could arise fro resolved CNB sources. | Therefore no more than $\approx20\%$ of the detected sources could arise from resolved CXB sources. |
Based on these. in the first instance. we fud that LMXDs aud CNB sources altogether can be responsible for 24.L sources. whereas the observed umber is 20. iuplviug a 25e excess of poiut sources. | Based on these, in the first instance, we find that LMXBs and CXB sources altogether can be responsible for $\approx5.4$ sources, whereas the observed number is $20$, implying a $>5\sigma$ excess of point sources. |
We mention that Tidaverdietal.(2006). reported au excess of sources in Abell 191 (aud Abell 1060) in the luminosity range of 1079<Eytot!ores3 over the full FOV. | We mention that \citet{hudaverdi06} reported an excess of sources in Abell 194 (and Abell 1060) in the luminosity range of $10^{39.6} \le L_X \le 10^{41.4} \ \rm{erg \ s^{-1}}$ over the full FOV. |
They suggested that these sources. many identified with cluster member galaxies. are AGN within cluster-aucniber galaxies. | They suggested that these sources, many identified with cluster member galaxies, are AGN within cluster-member galaxies. |
The source excess we report has a very different origin since these sources appear to be associated with the massive carly-type galaxies in Abell 191. aud are fainter. typically Ly~107eres1. than those studied by IDucdaverdietal.(2006). | The source excess we report has a very different origin since these sources appear to be associated with the massive early-type galaxies in Abell 194, and are fainter, typically $L_X \sim 10^{39} \ \rm{erg \ s^{-1}}$, than those studied by \citet{hudaverdi06}. |
. The normalization of the LAINB luuinosity function exlibits a scatter of about a factor of two (c.g.Cal 2001). | The normalization of the LMXB luminosity function exhibits a scatter of about a factor of two \citep[e.g.][]{gilfanov04}. . |
Although such variations cannot explain the total uunuboer of excess sources.It may be responsible in part. | Although such variations cannot explain the total number of excess sources,it may be responsible in part. |
The stable shape of the huuinositv function inuples that if its brieht end normalization is higher. then its faint end must also have au elevated level. | The stable shape of the luminosity function implies that if its bright end normalization is higher, then its faint end must also have an elevated level. |
Thus. we can scale the average LDhnuuinositv functio- | Thus, we can scale the average luminosity function |
We acknowledge useful discussions. by Peter Cossins. Giuseppe Locato ancl Yuri Levin. and we thank the referee for their comments on the draft. | We acknowledge useful discussions by Peter Cossins, Giuseppe Lodato and Yuri Levin, and we thank the referee for their comments on the draft. |
llere we try to estimate the realistic cooling time for the eas clouds in the initial collision event. | Here we try to estimate the realistic cooling time for the gas clouds in the initial collision event. |
Clouds moving at à few hundred km/sec would. vield a niaxiniun shock temperature of a few million Ix. Using the optically thin cooling function at these temperatures (dominated. by Menmsstrahlung and metal line cooling) and. initial cloud densities. one finds that the optically thin cooling time is 35 orders of magnitude shorter than dynamical time at Ho25". the latter being Aa=85LO? ves. | Clouds moving at a few hundred km/sec would yield a maximum shock temperature of a few million K. Using the optically thin cooling function at these temperatures (dominated by Bremsstrahlung and metal line cooling) and initial cloud densities, one finds that the optically thin cooling time is $3-5$ orders of magnitude shorter than dynamical time at $R \sim 25''$, the latter being $t_{\rm dyn} = 8 \times 10^3$ yrs. |
With our chosen parameters for the initial conditions. the clouds are . . . ≺∪⊔↓↓≻⋯⊔⊣↓⋯⇍↓∡⊳↓⊳⋖⋅⊳∖↵∿⋜↧⇂⋖⊾∖∖⊽⋖⋅⊔⊳∖∪⇂⋏∙≟≼⇍⊔↓−⊳∐↕∢⋅⊳∖↓↕⋯∼↳ ⋅ ⋅ ⊳↘⊳∖ ∖∖⊽∪⊔↓∠⇂ | With our chosen parameters for the initial conditions, the clouds are Compton-thick, i.e. $\Sigma \sim $ a few tens of g $^{-2}$. |
⋖⊾⊔↓⊲⊔⊳∖∪∐⇀∖−↓⋅⋜↧∙∖⇁⊳∖⇂↓⊔⋅∪⊔⋏∙≟↓↥≩↓⋅∢⊾⊔↓⊳∖⊳∖⇂↓⋅⋜↧↓⊔⋯⋏∙≟⊳∖∖⊽↓↥↕≼⇍↓↕ would have an absorption opacity much higher than that of the Thompson opacity ie. &y—04 cm? |. | The shock would emit soft X-rays through Bremsstrahlung, which would have an absorption opacity much higher than that of the Thompson opacity i.e. $\kappa_{\rm T} = 0.4$ $^2$ $^{-1}$. |
Phese X-rays would therefore be completely absorbed within the colliding clouds. | These X-rays would therefore be completely absorbed within the colliding clouds. |
This would not lead to a significant increase in the cooling time. however. as the absorbed. X-rays would be re-emitted as thermalised. blackbody emission at the elfective temperature of the radiation. | This would not lead to a significant increase in the cooling time, however, as the absorbed X-rays would be re-emitted as thermalised blackbody emission at the effective temperature of the radiation. |
The latter is estimated to be around 10? Is. Phe clouds’ optical depth to such radiation is of the order of a few tens. meaning that the clouds would still be able to cool elfectivelv via blackbody emission. | The latter is estimated to be around $10^3$ K. The clouds' optical depth to such radiation is of the order of a few tens, meaning that the clouds would still be able to cool effectively via blackbody emission. |
The cooling time for the cloud in its entirety would therefore remain short (using the Stefan-Doltzmann law for radiative emission. fog.) would again be ~3 orders of magnitude shorter than Fas). | The cooling time for the cloud in its entirety would therefore remain short (using the Stefan-Boltzmann law for radiative emission, $t_{\rm cool}$ would again be $\sim 3$ orders of magnitude shorter than $t_{\rm dyn}$ ). |
Llowever. as was pointed out to us by Yuri Levin. these considerations completely neglect magnetic fields that are most likely present in the pre-collision clouds. | However, as was pointed out to us by Yuri Levin, these considerations completely neglect magnetic fields that are most likely present in the pre-collision clouds. |
Lf the radiative cooling time is short. then the shock is essentially isothermal. and we can expect the gas to be compressed in the shock ow ἃ [ον orders of magnitude for our ἱνρίσα conditions. | If the radiative cooling time is short, then the shock is essentially isothermal, and we can expect the gas to be compressed in the shock by a few orders of magnitude for our typical conditions. |
A rozen-in magnetic field would then be amplified by similar actors. and magnetic pressure by four to six. orders. of magnitude. | A frozen-in magnetic field would then be amplified by similar factors, and magnetic pressure by four to six orders of magnitude. |
The strongly increased magnetic pressure might »esent another form of energy and pressure support against eravitational collapse. | The strongly increased magnetic pressure might present another form of energy and pressure support against gravitational collapse. |
If magnetic Dux is then removed from he cloud by magnetic buovaney instability. this happens on ime scales longer than cvnamical. ellectively implving large values of ο. | If magnetic flux is then removed from the cloud by magnetic buoyancy instability, this happens on time scales longer than dynamical, effectively implying large values of $\beta$. |
Summuarising. a realistic value of 3 during the initial clouc-cloud. collision would be very small if magnetic fields are not important. or large if strong field amplification occurs due to the shock. | Summarising, a realistic value of $\beta$ during the initial cloud-cloud collision would be very small if magnetic fields are not important, or large if strong field amplification occurs due to the shock. |
Relativity. | Relativity. |
This use of the Mauko-Novikov metric has Όσο put forward in eravitational-wave astroplivsics. namely iu Gainetal.(2008). | This use of the Manko-Novikov metric has been put forward in gravitational-wave astrophysics, namely in \citet{gair}. |
. See Collins&IIughes(2001):VieclandIIushes(2010):ClampedakisBabak(2006) for other metrics which reduce exactly to the Iker αποο when the equivalent of our anomalous quadrupole parameter q is sot to 0. aud which can therefore be used to perform: null experiments testing the Kerr Iu Figs. 5.. 6.. | See \citet{bumpy1,bumpy2,kostas}
for other metrics which reduce exactly to the Kerr metric when the equivalent of our anomalous quadrupole parameter $q$ is set to $0$, and which can therefore be used to perform null experiments testing the Kerr In Figs. \ref{f-mna}, \ref{f-mnb}, |
aud 7.. we show the racial profile of the thin accreticπι disks effective temperature aud the spectimi κ for a few values of a and q. | and \ref{f-mnc}, we show the radial profile of the thin accretion disk's effective temperature and the spectrum $\nu L(\nu)$ for a few values of $a$ and $q$. |
We stillassmme M=10 M... M=1055 &/s aud i=157. | We still assume $M = 10$ $M_\odot$ , $\dot{M} = 10^{18}$ g/s, and $i = 45^\circ$. |
For a given spinparameter. the value of 4 determines the radius «tthe ISCO — sce also Appendix B.. | For a given spinparameter, the value of $q$ determines the radius of the ISCO – see also Appendix \ref{a-isco}. |
Since the temperature of the disk is higher at stuall racii. a non-zero q produces corrections in the hie1 frequency region of the spectu. while at low frequencies there are no changes. | Since the temperature of the disk is higher at small radii, a non-zero $q$ produces corrections in the high frequency region of the spectrum, while at low frequencies there are no changes. |
The effect is cpute small for slow-rotating objects or couuterrotating disks. while it becomes relevant. and actually non-neelieible. for fast-rotatiug bodies and corotating disks. | The effect is quite small for slow-rotating objects or counterrotating disks, while it becomes relevant, and actually non-negligible, for fast-rotating bodies and corotating disks. |
There are two reasons for this: a sanall deviation from 40 produces a larecr variation m the radis of the ISCO for higher spin paramcters — see Fie. | There are two reasons for this: a small deviation from $q \neq 0$ produces a larger variation in the radius of the ISCO for higher spin parameters – see Fig. |
BO iu Appendix D and. because the ISCO is closer to the conipact object as a approaches 1. the spectrin of the disk is more seusitive to siia] deviations in the imultipole moment expansion. | \ref{f-isco} in Appendix \ref{a-isco} – and, because the ISCO is closer to the compact object as $a$ approaches 1, the spectrum of the disk is more sensitive to small deviations in the multipole moment expansion. |
This is the contrary of what happens in the Tomiuatus-Sato spac‘etimes. where for Jaf}»1 all tje solutions reduce to au extreme Nery DIT aud thus deviations from the Nerr metric are more relevaut for low spin ]xuiuneters (Bamihi&Yoshida2010a). | This is the contrary of what happens in the Tomimatus-Sato spacetimes, where for $|a| \rar 1$ all the solutions reduce to an extreme Kerr BH and thus deviations from the Kerr metric are more relevant for low spin parameters \citep{naoki}. |
. As an example of how accretiou-disz thermal spectra can already ptt significant constraiuts ou the deviation g of the nadrupole moment of BIT candidates from that of a Kerr BIT Eq. (2?))). | As an example of how accretion-disk thermal spectra can already put significant constraints on the deviation $q$ of the quadrupole moment of BH candidates from that of a Kerr BH Eq. \ref{qdef}) )), |
we consider the case of N23. N-7. | we consider the case of M33 X-7. |
This object is an eclipsing N-ray binary cosisting of a BIT candidate accreΠιο from a companion star (Pietschctal.2006).. and its orbital parameters and its distance are micasured with the highest accuracy amoung all kuown DIT binaries (seo Table 19). | This object is an eclipsing X-ray binary consisting of a BH candidate accreting from a companion star \citep{M33X7}, and its orbital parameters and its distance are measured with the highest accuracy among all known BH binaries \citep{M33X7accuracy} (see Table \ref{pubres}) ). |
In partiaiar. the BIT cauclidate’s mass is measured to be AF=15.6541.1542... while the disks inclination is /—71.6?41° and the distance is d=810-220 kpe (Oroszctal.2007 ).. | In particular, the BH candidate's mass is measured to be $M=15.65\pm1.45 M_\odot$, while the disk's inclination is $i=74.6^\circ\pm 1^\circ$ and the distance is $d=840\pm 20$ kpc \citep{M33X7accuracy}. . |
The accurate knowledge of M. / aud d allows the coutimuun fitting method (Zhangetal.1997). to extract reliable information on the spin of the BIT candidate. | The accurate knowledge of $M$, $i$ and $d$ allows the continuum fitting method \citep{zhang} to extract reliable information on the spin of the BH candidate. |
By esseutially fitting the Chandra and NMM-Newtou spectra of M33 with a relativistic accretion disk model depeudiug ou the spin e. the Eddington ratio 6=LiofLraq Gvhere Ly.) and Epag=12572«107(A/M..) eve/s ave the bolometric aud Eddington luminosities) aud the lydrogen colunn density Nyy. Liuetal.(2008.2010) measured the spin to be a=Odl+t0.05.5 The Eddington ratio is instead (—0.0989+0.0073 Table I of Linetal.(2008) and Liuctal. (2010))). | By essentially fitting the Chandra and XMM-Newton spectra of M33 X-7 with a relativistic accretion disk model depending on the spin $a$ , the Eddington ratio $\ell=L_{\rm bol}/L_{\rm Edd}$ (where $L_{\rm bol}$ and $L_{\rm Edd}=1.2572\times 10^{38} (M/M_\odot)$ erg/s are the bolometric and Eddington luminosities) and the hydrogen column density $N_{\rm H}$ , \citet{m33x7,m33x7e} measured the spin to be $a=0.84\pm0.05$ The Eddington ratio is instead $\ell=0.0989\pm 0.0073$ Table I of \citet{m33x7} and \citet{m33x7e}) ). |
We notice that the errors ou e aud f oeclude also the (propagated) effect οf the uucertiiuties on AL. d aud / (Liuetal.2008.2010).. | We notice that the errors on $a$ and $\ell$ include also the (propagated) effect of the uncertainties on $M$, $d$ and $i$ \citep{m33x7,m33x7e}. |
Our simple disk model depends on three parameters. a. q aud the Eddinetou ratio 6=Lioi/Lea. | Our simple disk model depends on three parameters, $a$, $q$ and the Eddington ratio $\ell=L_{\rm bol}/L_{\rm Edd}$. |
The latter can be rexpressed as f=M/Mgqate.q). wheye we defiue the Eddington accretion rate as Y=1dEO) beiug the efficicucy of the conversion between rest-luass aud electromagnetic eucrgy. 19723).. | The latter can be rexpressed as $\ell=\dot{M}/\dot{M}_{\rm Edd}(a,q)$, where we define the Eddington accretion rate as $\eta=1-E_{\rm _{\rm{ISCO}}}(r_{\rm _{\rm{ISCO}}})$ being the efficiency of the conversion between rest-mass and electromagnetic energy \citep{GRbook}. |
Ideally we would then have to fit the observed spectrum of A383 X-7 with this 3-parameter model. | Ideally we would then have to fit the observed spectrum of M33 X-7 with this 3-parameter model. |
However. because of the difficulties aud subtleties of analyzing the real Cliaudra aud NMM-Neswtou spectra aud because of the simplified nature of our disk model. we resorted to a simpler approach. | However, because of the difficulties and subtleties of analyzing the real Chandra and XMM-Newton spectra and because of the simplified nature of our disk model, we resorted to a simpler approach. |
While a thorough analysis of thereal data will be needed to determine the precises constraints ou 4. our siuplifted treatment will show that such an analysis is definitely worth beiug done as it would peruüt ruliie out cutive regions of the (0.4) plane. | While a thorough analysis of thereal data will be needed to determine the precises constraints on $q$, our simplified treatment will show that such an analysis is definitely worth being done as it would permit ruling out entire regions of the $(a,q)$ plane. |
Iu particular. instead of comparing our disk model with the raw data. we compare it to the spectrum of a thin disk with (*=0.0989 and inclination /=71.6" around a Kerry DIT with spin «*=OL and mass AS=15.65M. (these are the values iieasured for A133 X-7). | In particular, instead of comparing our disk model with the raw data, we compare it to the spectrum of a thin disk with $\ell^\star=0.0989$ and inclination $i=74.6^\circ$ around a Kerr BH with spin $a^{\star}=0.84$ and mass $M=15.65M_\odot$ (these are the values measured for M33 X-7). |
While meaniusful aud reliable constraints on the parameter e aud q can only be obtained by fitting the original N-rav data. we use here this sinplified approach because ours is a preliminary investigation and our results are only nieant as a qualitative guide for future more rigorous studies. | While meaningful and reliable constraints on the parameter $a$ and $q$ can only be obtained by fitting the original X-ray data, we use here this simplified approach because ours is a preliminary investigation and our results are only meant as a qualitative guide for future more rigorous studies. |
The spectrum is calculated with the standard. Novikov-Thoruc model revsewed in section 3.. | The spectrum is calculated with the standard Novikov-Thorne model reviewed in section \ref{s-kerr}. |
The observational errors on the “measured” spectrum are then mimicked by using theestimated final errors on the spin (d¢= 0.05) and Eddiugtou ratio (06= 0.0073). | The observational errors on the “measured” spectrum are then mimicked by using theestimated final errors on the spin $\delta a=0.05$ ) and Eddington ratio $\delta \ell=0.0073$ ). |
Because the Edeliugton ratio regulates thebolometric huuimositv the normalization of thespectruni) oue las Letati(5)unEXGeatzda.(*|C) and ο(*)>EXGattéa.(* C). | Because the Eddington ratio regulates thebolometric luminosity the normalization of thespectrum) one has $L^{\rm Kerr}(\nu,a^{\star},\ell^{\star})<L^{\rm Kerr}(\nu,a^{\star}\pm\delta a,\ell^{\star}+\delta\ell)$ and $L^{\rm Kerr}(\nu,a^{\star},\ell^{\star})>L^{\rm Kerr}(\nu,a^{\star}\pm\delta a,\ell^{\star}-\delta\ell)$ . |
Tt therefore makes sense to define the error as To determine the values of αν q aud f eiviug the best fit. one would then have to minimize⋅⋅⋅ the reduced 9y. which. we define as | It therefore makes sense to define the error as To determine the values of $a$ , $q$ and $\ell$ giving the best fit, one would then have to minimize the reduced $\chi^2$, which we define as |
significant. scatter in light-element abunclances has been observed on the red giant branch 1981).. and in some cases even down to the main sequence 2004). | significant scatter in light-element abundances has been observed on the red giant branch , and in some cases even down to the main sequence . |
The observed variations in carbon and nitrogen abundance are part of a larger light-element pattern Chat involves enrichment in N. Na. and Meg along with depletion in C. O. and Al. and is often studied in correlated or anticorrelated abundance pairs (C-N. O-N. Mg-Al. Na-O. ete). | The observed variations in carbon and nitrogen abundance are part of a larger light-element pattern that involves enrichment in N, Na, and Mg along with depletion in C, O, and Al, and is often studied in correlated or anticorrelated abundance pairs (C-N, O-N, Mg-Al, Na-O, etc.). |
reviews a munber of these studies. and dramatically increased the number of cluster stars surveved for (hese variations. | reviews a number of these studies, and dramatically increased the number of cluster stars surveyed for these variations. |
There are (wo independent modes of variation in globular cluster light-element abundances: a steady decline in [C/Fe] ancl increase in [N/Fe] as stars evolve along the RGB. and star-to-star variations in the light-element abunclances at a fixed himinosity. ad all evolutionary phases. | There are two independent modes of variation in globular cluster light-element abundances: a steady decline in [C/Fe] and increase in [N/Fe] as stars evolve along the RGB, and star-to-star variations in the light-element abundances at a fixed luminosity, at all evolutionary phases. |
several hypotheses have been proposed to explain these observed anomalies. | Several hypotheses have been proposed to explain these observed anomalies. |
The progressive abundance changes on the RGB are believed to be (he result of deep mixing within individual stars 2003).. beginning al the “bump” in the RGB luminosity [unction 2003). | The progressive abundance changes on the RGB are believed to be the result of deep mixing within individual stars , beginning at the “bump” in the RGB luminosity function . |
. The hvdrogen-burning shell proceeds oulwarel as à star evolves along the RGD. eventually &9encounleringthemolecular—weightdiscontinuilyle [lbehindbytheinwardreacho [theconvectia | The hydrogen-burning shell proceeds outward as a star evolves along the RGB, eventually encountering the molecular-weight discontinuity left behind by the inward reach of the convective envelope during first dredge-up . |
Whenthisoccurs, theshell sprogressisdelayedasits fusionrateadjuststol hes iweigh | When this occurs, the shell's progress is delayed as its fusion rate adjusts to the new chemical abundances, causing a loop in the star's evolution along the RGB. |
lgredientil | In a population of coeval stars, this produces an enhancement in the differential luminosity function. |
experiencesislower. andl heprocessofdeepmiring( begi burningshellandthesur faceandcontinuouslyadjustingsur facecarbonandnitrogenabundances, | Once the shell begins to proceed outward again, the molecular-weight gradient it experiences is lower, and the process of deep mixing begins to operate, transporting material between the hydrogen-burning shell and the surface and continuously adjusting surface carbon and nitrogen abundances. |
Early studies of star-to-star light-element abundance variations suggested that deep mixing might be responsible for the C-N variations al lixed luminosity as well as the progressive abundance changes along the RGB. | Early studies of star-to-star light-element abundance variations suggested that deep mixing might be responsible for the C-N variations at fixed luminosity as well as the progressive abundance changes along the RGB. |
IHowever. | However, |
equation of state is dominated by thermal radiation pressure. | equation of state is dominated by thermal radiation pressure. |
It can also be shown that SADSs are convective (see Loch Rasio 1991. for a simple proof) with coustaut eutropy per barvon. where sy is the barvon density aud e is the radiation density constant. | It can also be shown that SMSs are convective (see Loeb Rasio 1994, for a simple proof) with constant entropy per baryon, where $n_{b}$ is the baryon density and $a$ is the radiation density constant. |
These conditious inply that the structure of a SMS is that of anv —2 polvtrope where (see eq. | These conditions imply that the structure of a SMS is that of an $n=3$ polytrope where (see eq. |
17.2.6 in Shapiro Toukolsky 1983). | 17.2.6 in Shapiro Teukolsky 1983). |
Here. ig is the mass of a hydrogen atom aud A has been evaluated for a composition of pure ionized hvdrogeu. | Here, $m_H$ is the mass of a hydrogen atom and $K$ has been evaluated for a composition of pure ionized hydrogen. |
In οποία theory. the mass of a static. equilibrium à=3 polvtrope is φομ determined by the polvtropic constant Jv alone (the umuerical coefficients fy aud hy. derived from Lauc-Lauden functions. are given in Table 1 below). | In Newtonian theory, the mass of a static, equilibrium $n=3$ polytrope is determined by the polytropic constant $K$ alone (the numerical coefficients $k_1$ and $k_2$, derived from Lane-Emden functions, are given in Table 1 below). |
Iuvertiug this relation. we can write A in terius of M as Since 50-—23 polytropes are. extremely ceutrally condensed. the selt-gravity of the outermost euvelope of a nuiformly rotating configuration can be neglected aud the Roche approximation (see Section 3.1)) cau be applied. | Inverting this relation, we can write $K$ in terms of $M$ as Since $n=3$ polytropes are extremely centrally condensed, the self-gravity of the outermost envelope of a uniformly rotating configuration can be neglected and the Roche approximation (see Section \ref{roche}) ) can be applied. |
Iu Newtonian gravitation. pj=3 polvtropes are eeinaallv stable to radial collapse. | In Newtonian gravitation, $n=3$ polytropes are ally stable to radial collapse. |
However. even verv sinall general relativistic corrections make these polytropes uustable. auc some mechanism has to be invoked to prevent collapse. | However, even very small general relativistic corrections make these polytropes unstable, and some mechanism has to be invoked to prevent collapse. |
In this paper. we focus on rotation. which up to a critical configuration along the mas sequence cau stabilize SMSs. | In this paper, we focus on rotation, which up to a critical configuration along the mass-shedding sequence can stabilize SMSs. |
Iu Section 2. we will identify this critical configuration (which we will call “configuration A7). | In Section \ref{Sec3} we will identify this critical configuration (which we will call “configuration $A$ ”). |
Alternatively. gas pressure may stabilize SMSs in the absence of angular momentum (see Zeldovich Novvikkov 1971: Shapiro Teulkolsky 1983). but even a small degree of rotation will dominate eas pressure (ef | Alternatively, gas pressure may stabilize SMSs in the absence of angular momentum (see Zel'dovich kov 1971; Shapiro Teukolsky 1983), but even a small degree of rotation will dominate gas pressure (cf. |
Section 3.2)). | Section \ref{anal}) ). |
A dark matter backerouud also tends to have a stabilizing effect (Fuller. Woosley Weaver. 1986). but if is also much less imiportaut than rotation (Disnovatvyi-Ikoeau 1998). | A dark matter background also tends to have a stabilizing effect (Fuller, Woosley Weaver, 1986), but it is also much less important than rotation (Bisnovatyi-Kogan 1998). |
Since the ratio between the rotational kinetic aud the potential energy. T/|W |. is always very πα for a uniformly rotating star durius secular evolution along the mass-shedding sequence (see eq. |] | | Since the ratio between the rotational kinetic and the potential energy, $T/|W|$ , is always very small for a uniformly rotating star during secular evolution along the mass-shedding sequence (see eq. \ref{t_ms}] ] |
and Fieure [)). it is uulikelv that nonracial iiodes of iustabilitv are important durius this phase. | and Figure \ref{fig4}) ), it is unlikely that nonradial modes of instability are important during this phase. |
Once the star lias become unstable and. collapses. however. T/|W| can become very large. aud it is likely that such nouradial modes develop (see Section 5)). | Once the star has become unstable and collapses, however, $T/|W|$ can become very large, and it is likely that such nonradial modes develop (see Section \ref{coll}) ). |
Alassive enough stars (M>109A/7.) do not reach sutficicutly lieh temperatures for unclear burning to become miportaut before reaching the onset of instability (sce Section 3.1)}. | Massive enough stars $M \gtrsim 10^6 M_{\odot}$ ) do not reach sufficiently high temperatures for nuclear burning to become important before reaching the onset of instability (see Section \ref{Sec3.4}) ). |
Also. clectrou-positron pairs play a ueeheible role iu this τοσο (Zeldovich Novvilklov 1971). | Also, electron-positron pairs play a negligible role in this regime (Zel'dovich kov 1971). |
The evolution of the SMS alone the mass-shedding sequence is then determined solely by cooling via thermal photon emission aud loss of mwass and angular momentum. | The evolution of the SMS along the mass-shedding sequence is then determined solely by cooling via thermal photon emission and loss of mass and angular momentum. |
As we lave shown in Paper L the luminosity of a rotating supermassive star at mass-shecddine is where the Eddington Iuninositv is The opacity κ is dominated by Thomson scatterius off free clectrous where Vy is the hydrogen lass fraction. | As we have shown in Paper I, the luminosity of a rotating supermassive star at mass-shedding is where the Eddington luminosity is The opacity $\kappa$ is dominated by Thomson scattering off free electrons where $X_H$ is the hydrogen mass fraction. |
For A>LOPAL... temperatures are low cnougl that Ileiu-Nishina corrections can be neglected. (sec. ο,ο,, Fuller. Woosley Weaver 1986). | For $M \gtrsim 10^5
M_{\odot}$ , temperatures are low enough that Klein-Nishina corrections can be neglected (see, e.g., Fuller, Woosley Weaver 1986). |
As we will find in Section L. the evolutionary timescale for secular evolution along the mass-sledding sequence is eiven by where the coefficient has been evaluated near the critical configuration (see eq. (67))). | As we will find in Section \ref{Sec4}, the evolutionary timescale for secular evolution along the mass-shedding sequence is given by where the coefficient has been evaluated near the critical configuration (see eq. \ref{tcrit_value}) )). |
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