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1 of Selvelli and Friedjung (2003)), where we have plotted the Galex values over the spectrum (uncorrected for reddening).
1 of Selvelli and Friedjung \cite{Selv03}) ), where we have plotted the Galex values over the spectrum (uncorrected for reddening).
Of course, we still need a good UV spectrum to assess what are the real line profiles today.
Of course, we still need a good UV spectrum to assess what are the real line profiles today.
We can note, in connection with non-orbital variations of Ha, that Selvelli and Friedjung (2003)) suspected quite rapid non-orbital variations of the and ultraviolet resonance line absorption inDel, similar to the rapid variations of seen for another old nova in Friedjung et al. (1997))
We can note, in connection with non-orbital variations of $\alpha$, that Selvelli and Friedjung \cite{Selv03}) ) suspected quite rapid non-orbital variations of the and ultraviolet resonance line absorption in, similar to the rapid variations of seen for another old nova in Friedjung et al. \cite{Fried97}) )
and in Prinja et al. (2000))).
and in Prinja et al. \cite{Prin00}) )).
Such variations of the ultraviolet resonance absorption components of P Cygni profiles may be expected for the winds of accretion disks of cataclysmic binaries and have been observed (Froning 2005,, Proga 2005)).
Such variations of the ultraviolet resonance absorption components of P Cygni profiles may be expected for the winds of accretion disks of cataclysmic binaries and have been observed (Froning \cite{Fron05}, Proga \cite{Prog05}) ).
The "S wave type" pattern of Ha, displayed in Figs.
The "S wave type" pattern of $\alpha$, displayed in Figs.
10 to 12,, is quite clear on the blue side, but somewhat less so on the red side of the profiles.
\ref{SwaveB} to \ref{Swave}, is quite clear on the blue side, but somewhat less so on the red side of the profiles.
It does however not appear to be a classical cataclysmic binary S wave.
It does however not appear to be a classical cataclysmic binary S wave.
However, it can, as it moves over the central component of the line profile like the classical wave, be understood as being near the outer edge of a rotating disk in the neighborhood of a fixed point in the rotating frame of the binary.
However, it can, as it moves over the central component of the line profile like the classical wave, be understood as being near the outer edge of a rotating disk in the neighborhood of a fixed point in the rotating frame of the binary.
The profile shows a maximum assymmetry towards the violet around Kürrster and Barwig phase 0.75 or at photometric phase 0.33, that is only 0.08 of an orbital period after photometric phase 0.25.
The profile shows a maximum assymmetry towards the violet around Kürrster and Barwig phase 0.75 or at photometric phase 0.33, that is only 0.08 of an orbital period after photometric phase 0.25.
If we interpret the photometic variations as due to varying visibility of the regions of the surface of the mass loser, heated by radiation from the disk and white dwarf (see below), the photometric zero phase is when the mass loser is nearest the observer in inferior conjunction.
If we interpret the photometic variations as due to varying visibility of the regions of the surface of the mass loser, heated by radiation from the disk and white dwarf (see below), the photometric zero phase is when the mass loser is nearest the observer in inferior conjunction.
The phase of maximum assymetry of the profile around photometric phase 0.33, is then not far from quadrature when the spot is near a line perpendicular to the line of sight.
The phase of maximum assymetry of the profile around photometric phase 0.33, is then not far from quadrature when the spot is near a line perpendicular to the line of sight.
An accretion disk should rotate in the same direction as the binary (conservation of angular momentum), so the Ho bright spot should beloser.
An accretion disk should rotate in the same direction as the binary (conservation of angular momentum), so the $\alpha$ bright spot should be.
Such a situation is not classical.
Such a situation is not classical.
That could possibly be related to the large size of the accretion disk found We can emphasize however that the stability of the S wave over the six years of our spectroscopic observations (of the order of 10,000 orbital cycles) indicates that it moves with a period which is quite close to the orbital period of Kürrster and
That could possibly be related to the large size of the accretion disk found We can emphasize however that the stability of the S wave over the six years of our spectroscopic observations (of the order of 10,000 orbital cycles) indicates that it moves with a period which is quite close to the orbital period of Kürrster and
absorption is quite reasonable.
absorption is quite reasonable.
Hence, we favoured the second model, whose Ylv is usually smaller than in the Ny-free case, and that produces results with smaller errors (because of one degree of freedom more).
Hence, we favoured the second model, whose $\chi^2/\nu$ is usually smaller than in the $N_{\rm H}$ -free case, and that produces results with smaller errors (because of one degree of freedom more).
In only two cases (August 23 and September 2) a double power law model with absorption fixed to the Galactic value clearly improved the fit.
In only two cases (August 23 and September 2) a double power law model with absorption fixed to the Galactic value clearly improved the fit.
The photon index I' ranges from 1.92 to 2.25, indicating a spectrum that oscillates from moderately hard to moderately soft.
The photon index $\Gamma$ ranges from 1.92 to 2.25, indicating a spectrum that oscillates from moderately hard to moderately soft.
The average value is 2.07, with standard deviation of 0.08.
The average value is 2.07, with standard deviation of 0.08.
To understand whether these spectral changes correspond to real variations or are owing to noise, we recall the definition of the mean fractional variation Fu,=wvVo2-ó/«f»2001), which is commonly used to characterise variability.
To understand whether these spectral changes correspond to real variations or are owing to noise, we recall the definition of the mean fractional variation $F_{\rm var}=\sqrt{\sigma^2-\delta^2}/{<f>}$, which is commonly used to characterise variability.
Here <f> is the mean value of the variable we are analysing, c? its variance, and ó? the mean square uncertainty.
Here $<f>$ is the mean value of the variable we are analysing, $\sigma^2$ its variance, and $\delta^2$ the mean square uncertainty.
In our case, o?=0.006 is smaller than à?=0.011, so that the result is imaginary; thus we conclude that the variations are consistent with noise rather than source variability.
In our case, $\sigma^2=0.006$ is smaller than $\delta^2=0.011$, so that the result is imaginary; thus we conclude that the variations are consistent with noise rather than source variability.
The 1 keV flux density varies between 1.34 and 2.24wy, with a mean value of 1.75 and standard deviation of 0.24.
The 1 keV flux density varies between 1.34 and $2.24 \, \mu \rm Jy$, with a mean value of 1.75 and standard deviation of 0.24.
In this case Fy,=0.11, and the variations can be considered reliable.
In this case $F_{\rm var}=0.11$, and the variations can be considered reliable.
Multiwavelength light curves of BL Lacertae in the period around the Swift observations are shown in refmulti..
Multiwavelength light curves of BL Lacertae in the period around the Swift observations are shown in \\ref{multi}.
The source behaviour at 1 keV differs from the common trend characterising the UV, optical, and near-IR bands.
The source behaviour at 1 keV differs from the common trend characterising the UV, optical, and near-IR bands.
In particular, the X-ray flux peaks when the near-[R-UV fluxes reach a minimum.
In particular, the X-ray flux peaks when the near-IR–UV fluxes reach a minimum.
However, there are also similarities, like the flux increase at the beginning of the common observing period, and the final decrease.
However, there are also similarities, like the flux increase at the beginning of the common observing period, and the final decrease.
This may indicate that the 1 keV flux behaviour sometimes is related to the brightness changes that occur at lower wavelengths, while in other cases another variability mechanism prevails.
This may indicate that the 1 keV flux behaviour sometimes is related to the brightness changes that occur at lower wavelengths, while in other cases another variability mechanism prevails.
Indeed, according to this frequency domain receives the variable contribution of two different emission components (see also refsec,,odel)).
Indeed, according to this frequency domain receives the variable contribution of two different emission components (see also \\ref{sec_model}) ).
The X-ray Multi-Mirror Mission (XMM) - Newton satellite observed the source during revolution 1545, on 2008 May 17, with a total exposure of ~134ks.
The X-ray Multi-Mirror Mission (XMM) - Newton satellite observed the source during revolution 1545, on 2008 May 16--17, with a total exposure of $\sim 134 \, \rm ks$.
Data were processed with the Science Analysis System (SAS) package version 9.0.
Data were processed with the Science Analysis System (SAS) package version 9.0.
The Optical Monitor onboard XMM-Newton is a 30-cm telescope carrying six optical/UV filters, and two grisms.
The Optical Monitor onboard XMM-Newton is a 30-cm telescope carrying six optical/UV filters, and two grisms.
BL Lacertae observations in May 2008 consisted of 10 subsequent exposures in UVW1, followed by 9 in M2, and then 8 in UVW2.
BL Lacertae observations in May 2008 consisted of 10 subsequent exposures in $W1$, followed by 9 in $M2$, and then 8 in $W2$.
All exposures were ~4000s long.
All exposures were $\sim 4000 \, \rm s$ long.
We used the SAS task to reduce the data and the tasks and to derive the source magnitude.
We used the SAS task to reduce the data and the tasks and to derive the source magnitude.
The error on the aperture photometry is 0.03, 0.04, and 0.09 mag for the UVW1, UVM2, and UVW2 filters, respectively.
The error on the aperture photometry is 0.03, 0.04, and 0.09 mag for the $W1$, $M2$, and $W2$ filters, respectively.
The resulting light curves are shown in refom+epic;; average magnitudes are UVW1=15.45, M2=16.25, and UVW2=16.54.
The resulting light curves are shown in \\ref{om+epic}; ; average magnitudes are $W1=15.45$, $M2=16.25$, and $W2=16.54$.
To obtain flux densities for further analysis, OM magnitudes were corrected for the Galactic extinction calculated according to the laws at the effective wavelengths of theOM filters (2910, 2310, and 2120 ffor the UVW1, UVM2, and UVW2 filters, respectively).
To obtain flux densities for further analysis, OM magnitudes were corrected for the Galactic extinction calculated according to the laws at the effective wavelengths of theOM filters (2910, 2310, and 2120 for the $W1$ , $M2$ , and $W2$ filters, respectively).
where |y. denotes the mass of the progenitor He-star.
where $M_{He}$ denotes the mass of the progenitor He-star.
The newly formed low-mass black hole is (vpically kicked out of the central high-density region into surrounding; regions core-collapse is completed.
The newly formed low-mass black hole is typically kicked out of the central high-density region into surrounding lower-density regions core-collapse is completed.
It continues to grow oll-center by accretion ol relatively low-density matter a high-density accretion disk never forms.
It continues to grow off-center by accretion of relatively low-density matter – a high-density accretion disk never forms.
With low but non-zero probability. the black hole has small recoil. remains centered and surges into a hieh mass black hole surrounded by a high-density torus.
With low but non-zero probability, the black hole has small recoil, remains centered and surges into a high mass black hole surrounded by a high-density torus.
After nucleation of the black hole. an accretion disk may form provided Chat thespecilic angular momentum j,, of infalling matter exceeds that of the inner most stable circular orbit (ISCO). on the angular momentum ο of a black hole of mass. M.
After nucleation of the black hole, an accretion disk may form provided that thespecific angular momentum $j_m$ of infalling matter exceeds that of the inner most stable circular orbit (ISCO), on the angular momentum $J_H$ of a black hole of mass $M$.
Here. /(a/M) denotes the dimensionless specific angular momentum of matter in circular orbits on (he ISCO. where denotes (he specilic angular momentum of the black hole.
Here, $l(a/M)$ denotes the dimensionless specific angular momentum of matter in circular orbits on the ISCO, where $a=J_H/M$ denotes the specific angular momentum of the black hole.
Explicitly. we have (BarcleenPress&Teukolsky1972) /=(2/34/32)(1+2/32—2) in terms of 223) with Z,2120-74)?[d+gilt and q=a/AM.
Explicitly, we have \citep{bar70,bar72} $l=({2}/3\sqrt{3})\left(1+2\sqrt{3z-2}\right)$ in terms of $z={r_{ISCO}}/{M}=3+Z_2-\left[(3-Z_1)(3+Z_1+2Z_2)\right]^{1/2}$ with $Z_1=1+(1-q)^{1/3}\left[ (1+q)^{1/3} + (1-q)^{1/3} \right],~~ Z_2=\left(3q^2+Z_1^2\right)^{1/2}$ and $q=a/M$.
Notice that 2/V3<1<2V3. between an extremal black hole (a=M.2 1) and a non-rotating black hole (a=0.2: 6).
Notice that $2/\sqrt{3}\le l \le 2\sqrt{3}$, between an extremal black hole $(a=M,~z=1$ ) and a non-rotating black hole $(a=0,~z=6$ ).
The evolution of the newly nucleatec black hole continues to be governed by angular momentum loss of the surrounding malter. until the inequalitv in (4)) is reversed.
The evolution of the newly nucleated black hole continues to be governed by angular momentum loss of the surrounding matter, until the inequality in \ref{EQN_JL}) ) is reversed.
The black hole rapidly erows uninhibited. while the inequality (4)) isreversed Me).
The black hole rapidly grows uninhibited, while the inequality \ref{EQN_JL}) ) isreversed $j_m < l(a/M) GM/c)$ .
This continues until once again (4)) holds.
This continues until once again \ref{EQN_JL}) ) holds.
In dimensionless form. (4)) beconies where in terms of the dimensionless integrals j(s)=4xfjps!ds and m(s)=4x[>ps?ds of (he normalized Lane-Emden density distribution with p=I at the origin and the zero p=0 al sy= 6.89685.
In dimensionless form, \ref{EQN_JL}) ) becomes where in terms of the dimensionless integrals $j(s)=4\pi\int^s_0 \hat{\rho} s^4 ds$ and $m(s)=4\pi\int_0^s \hat{\rho} s^2 ds$ of the normalized Lane-Emden density distribution with $\hat{\rho}=1$ at the origin and the zero $\hat{\rho}=0$ at $s_0=6.89685$ .
Here. (F4.45)=(1.1) in evlindrical geometry for which j,,= «7. and (hy.hs)=(5/3.2/3)/CM in spherical geometry [or which j,,= (2/317: D; denotes the binary period in davs. fy denotes the radius in units of (he solar radius 6.96x10 "cm 1990).. and. Mj; the mass of the progenitorIe-star.
Here, $(k_1,k_2)=(1,1)$ in cylindrical geometry for which $j_m=\omega r^2$ , and $(k_1,k_2)=(5/3,2/3)$ in spherical geometry for which $j_m=(2/3)\omega r^2$ ; $P_d$ denotes the binary period in days, $R_1$ denotes the radius in units of the solar radius $6.96\times 10^{10}$ cm \citep{kip90}, , and $M_{He}$ the mass of the progenitorHe-star.
considerations for example by ? and ??..
considerations for example by \citet{Norman_88} and \citet{Scoville_88,Scoville_95}.
They find that stellar processes play an important role for the fuelling of the central black hole and the envelopes of giant stars might as well correspond to the clouds in the Broad Line Regions (BLR) of galactic nuclei.
They find that stellar processes play an important role for the fuelling of the central black hole and the envelopes of giant stars might as well correspond to the clouds in the Broad Line Regions ) of galactic nuclei.
Observations of nearby Seyfert galaxies indeed find evidence for young and massive nuclear star clusters and a tentative connection with the onset of nuclear activity (?)..
Observations of nearby Seyfert galaxies indeed find evidence for young and massive nuclear star clusters and a tentative connection with the onset of nuclear activity \citep{Davies_07}.
?? are able to confirm this idea with the help of detailed hydrodynamical simulations.
\citet{Schartmann_09,Schartmann_10} are able to confirm this idea with the help of detailed hydrodynamical simulations.
During theBranch (AGB) phase of the evolution of the nuclear star cluster, slow stellar winds provide enough low angular momentum fuel, which can be accreted towards the central region to explain their observed core luminosities, amongst other observational properties (?)..
During the (AGB) phase of the evolution of the nuclear star cluster, slow stellar winds provide enough low angular momentum fuel, which can be accreted towards the central region to explain their observed core luminosities, amongst other observational properties \citep{Schartmann_10}.
The typical outcome of such a simulation is a two-component structure: (i) a filamentary or clumpy stream of gas, which feeds clumps towards the centre from the tens of parsec scale vicinity of the black hole and (ii) a geometrically thin accretion disc around the SMBH on sub-parsec to parsec scale (?).
The typical outcome of such a simulation is a two-component structure: (i) a filamentary or clumpy stream of gas, which feeds clumps towards the centre from the tens of parsec scale vicinity of the black hole and (ii) a geometrically thin accretion disc around the SMBH on sub-parsec to parsec scale \citep{Schartmann_09}.
With the help of a one-dimensional effective treatment of the central few parsecs including the effects of rotation, viscosity, mass inflow from large scales and star formation, ? are able to show that a significant amount of matter in the disc can be accreted towards the centre.
With the help of a one-dimensional effective treatment of the central few parsecs including the effects of rotation, viscosity, mass inflow from large scales and star formation, \citet{Schartmann_10} are able to show that a significant amount of matter in the disc can be accreted towards the centre.
Finally reaching the vicinity of the black hole, this will lead to the formation of a hot inner accretion disc and the birth of the AGN.
Finally reaching the vicinity of the black hole, this will lead to the formation of a hot inner accretion disc and the birth of the AGN.
The outer parsec-sized disc of gas and dust (potentially already puffed up to a toroidal shape by a thus far unknown physical process) will shield part of the radiation.
The outer parsec-sized disc of gas and dust (potentially already puffed up to a toroidal shape by a thus far unknown physical process) will shield part of the radiation.
A transition between a completely shadowed region (behind the torus midplane) and a region exerted to full radiation pressure from the source (around the polar axis) is expected (as sketched in Fig. 1)).
A transition between a completely shadowed region (behind the torus midplane) and a region exerted to full radiation pressure from the source (around the polar axis) is expected (as sketched in Fig. \ref{fig:clumpy_torus_sketch}) ).
The 3D models in ?? which solely cover the pre-active phase, where radiative pressure forces from the central source are negligible, produce clouds in both of these two regions.
The 3D models in \citet{Schartmann_09,Schartmann_10} which solely cover the pre-active phase, where radiative pressure forces from the central source are negligible, produce clouds in both of these two regions.
To investigate the feedback properties transmitted by the radiation pressure of the hot accretion disc in the active phase and how it affects infalling clouds is the subject of this work (see Fig. 1)).
To investigate the feedback properties transmitted by the radiation pressure of the hot accretion disc in the active phase and how it affects infalling clouds is the subject of this work (see Fig. \ref{fig:clumpy_torus_sketch}) ).
In Sect. 2,
In Sect. \ref{sec:mod_num},
we describe the numerical model and physical setup of our simulations and explain the test problems we used to assess their accuracy.
we describe the numerical model and physical setup of our simulations and explain the test problems we used to assess their accuracy.
Sect.
Sect.
3 describes our simulations and the main results of our parameter studies.
\ref{sec:results} describes our simulations and the main results of our parameter studies.
It is followed by a critical discussion in Sect. 4,,
It is followed by a critical discussion in Sect. \ref{sec:discussion},
before we conclude in Sect. 6..
before we conclude in Sect. \ref{sec:conclusions}.
Even with today's computational power the simultaneous solution of the hydrodynamical evolution and the time-dependent radiative transfer equation is impractical for high-resolution multi-dimensional simulations.
Even with today's computational power the simultaneous solution of the hydrodynamical evolution and the time-dependent radiative transfer equation is impractical for high-resolution multi-dimensional simulations.
Severe simplifications have to be applied, depending on the problem under investigation.
Severe simplifications have to be applied, depending on the problem under investigation.
In the simulations shown here, we explore the situation of a strong point source, illuminating a spatially confined cloud, immerged at a distance of several tens of dust sublimation radii in initially dust-free, low density gas.
In the simulations shown here, we explore the situation of a strong point source, illuminating a spatially confined cloud, immerged at a distance of several tens of dust sublimation radii in initially dust-free, low density gas.
The spectral energy distribution of the central source peaks in the ultra-violet wavelength regime, where also the mass absorption coefficient of the adopted dust model shows a prominent maximum.
The spectral energy distribution of the central source peaks in the ultra-violet wavelength regime, where also the mass absorption coefficient of the adopted dust model shows a prominent maximum.
Therefore, radiation can only penetrate a relatively short distance into the cloud, before the high energy UV-photons get absorbed and re-emitted in the infrared wavelength regime.
Therefore, radiation can only penetrate a relatively short distance into the cloud, before the high energy UV-photons get absorbed and re-emitted in the infrared wavelength regime.
By consequence, the surface of the cloud in the direction of the central source will receive almost all of the radiative acceleration.
By consequence, the surface of the cloud in the direction of the central source will receive almost all of the radiative acceleration.
Given the steep drop of the dust temperature distribution at this rim and the fact that the clouds are far away from the sublimation radius, secondary infrared radiation pressure effects are of minor importance for the dynamical evolution in our infalling clump scenario.
Given the steep drop of the dust temperature distribution at this rim and the fact that the clouds are far away from the sublimation radius, secondary infrared radiation pressure effects are of minor importance for the dynamical evolution in our infalling clump scenario.
A second consequence is that radiation pressure effects predominantly act in radial direction and are dynamically unimportant in vertical direction, as long as the dust temperatures are low and the optical depth is not too high.
A second consequence is that radiation pressure effects predominantly act in radial direction and are dynamically unimportant in vertical direction, as long as the dust temperatures are low and the optical depth is not too high.
Hence, a one-dimensional treatment of the radiative transfer problem is reasonable here.
Hence, a one-dimensional treatment of the radiative transfer problem is reasonable here.
Furthermore, we are mainly interested in the dynamical evolution and not in the detailed thermodynamics of the dust distribution.
Furthermore, we are mainly interested in the dynamical evolution and not in the detailed thermodynamics of the dust distribution.
The issues raised above justify the following simplified approach: We treat the central accretion disc as an isotropically radiating point-source with a spectral energy distribution as shown in Fig.
The issues raised above justify the following simplified approach: We treat the central accretion disc as an isotropically radiating point-source with a spectral energy distribution as shown in Fig.
3b of ?,, but normalised to correspond to of the Eddington luminosity for the case of the nearby 22 galaxy 11068.
3b of \citet{Schartmann_05}, but normalised to correspond to of the Eddington luminosity for the case of the nearby 2 galaxy 1068.
The radiation is divided into 54 wavelength bins and propagated along radial rays outwards, where we take geometrical dilution and absorption and reemission by the dust grains in each cell into account.
The radiation is divided into 54 wavelength bins and propagated along radial rays outwards, where we take geometrical dilution and absorption and reemission by the dust grains in each cell into account.
Scattering is neglected.
Scattering is neglected.
A full radiative transfer calculation within each time step is done.
A full radiative transfer calculation within each time step is done.
We further make a one fluid assumption and fully couple gas and dust dynamically.
We further make a one fluid assumption and fully couple gas and dust dynamically.
The reason for this coupling is that dust grains are charged due to the UV and X-ray radiation and couple to the gas with the help of magnetic fields.
The reason for this coupling is that dust grains are charged due to the UV and X-ray radiation and couple to the gas with the help of magnetic fields.
The effects of grain charging and gas-dust-coupling have
The effects of grain charging and gas-dust-coupling have
reactions heat the core on a timescale Ii this equation. C is the specific heat per baryon. AM, is approximately tie number of baryous in thestar. aud AZ/AL is the tinmescale lor the mass to increase [from accretiou.
reactions heat the core on a timescale In this equation $C$ is the specific heat per baryon, $M/m_n$ is approximately the number of baryons in thestar, and $M/\dot{M}$ is the timescale for the mass to increase from accretion.
If the heat is stored iu he electrous (as would be the case if the ueutrous were superlltid). the heat content per baryon (e.g...Landau&Lifshitz1950) is CTzxeyFeT/Ey)Ex. whe| Eye is the electro1 Fe‘nu energy.
If the heat is stored in the electrons (as would be the case if the neutrons were superfluid), the heat content per baryon \citep[e.g.,][]{landau80:_statis_physic} is $CT \approx \pi^2 \kB T (\kB T/\EF)\ll E_N$, where $\EF$ is the electron Fermi energy.
The heat content per baryou is similar if tlie neuLOLs are nornal (Lamb&Laiib1f)r8)..
The heat content per baryon is similar if the neutrons are normal \citep{lamb78:_nuclear_x}.
Equation (1)) shows hat over timescales necessary to establish a thermal steady-state. μοιass ol the star cliauges ouv slightly. aud so the hydrostatic ecuatious need nol be solved similtaeously with the thermal eqlatious.
Equation \ref{eq:heating-timescale}) ) shows that over timescales necessary to establish a thermal steady-state, the mass of the star changes only slightly, and so the hydrostatic equations need not be solved simultaneously with the thermal equations.
Using a fixed byclrostatie st"ueture sliiplifies the tlermal calculalioi.
Using a fixed hydrostatic structure simplifies the thermal calculation.
To caleuale the temperaure as a fuuction of radius requires integrating tle heat transport equalious over the star.
To calculate the temperature as a function of radius requires integrating the heat transport equations over the star.
The strong graviatioual fielcl inocdifies tle heat low.
The strong gravitational field modifies the heat flow.
In ar1 isothermal star. Wwlere there is no heat flow. tle teliperature is constaut. while tle proyer (as leastwed by a ocal thermoueter) temperature iicreases as Olle Inoves Owl he stellar ceuter.
In an isothermal star, where there is no heat flow, the temperature is constant, while the proper (as measured by a local thermometer) temperature increases as one moves toward the stellar center.
Because ue neutrino eimissivity is a stroug function of temperature. the thermal trausport equations must ac'couut lor gravitational effects.
Because the neutrino emissivity is a strong function of temperature, the thermal transport equations must account for gravitational effects.
The appropriate equations. solved for stellar mass aud. EOS. are ue post-Newtonian stellar structure equatious (Thorne1977). lor the radius. gravitational mass. potenial. and. pressure: Iu these equatious the Lagrangian variable e is the total number of baryous iuside a sphere of area. πι. and. p is the mass density.
The appropriate equations, solved for stellar mass and EOS, are the post-Newtonian stellar structure equations \citep{thorne77} for the radius, gravitational mass, potential, and pressure: In these equations the Lagrangian variable $a$ is the total number of baryons inside a sphere of area $4\pi r^2$, and $\rho$ is the mass density.
The potential ® appears in the time-time component. of the metric as eT (it goverus the redshift of photous and neutriuos: 1973)) and satisfies the boundary condition that at the stellar surface e??ejpop—1—2CM/ Bic. where M aud. Ez7> are the total gravitational. mass aud surfaceJ area of" the neutron star.
The potential $\Phi$ appears in the time-time component of the metric as $e^{\Phi/c^2}$ (it governs the redshift of photons and neutrinos; \citealt*{misner73:_gravit}) ) and satisfies the boundary condition that at the stellar surface $e^{2\Phi/c^2}|_{r=R} = 1-2G\Mg/Rc^2$ , where $\Mg$ and $4\pi R^2$ are the total gravitational mass and surface area of the neutron star.