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The next step is to use the 1o upper lait of 0.0L for Το(0) (Cdalloneo. Cristiani Trevese 1992). | The next step is to use the $1 \sigma$ upper limit of 0.04 for $\tau_{{\rm GP,HI}}(3)$ (Giallongo, Cristiani Trevese 1992). |
Since it follows that so that Dijc1.5«1012secl | Since it follows that so that $\Gamma_{\rm HI}\ge 1.5 \times 10^{-12}\; {\rm sec}^{-1}$. |
This lower limit is colpatible with the upper huit for Dig (~3.1027 see!) proposed iu Sec. | This lower limit is compatible with the upper limit for $\Gamma_{\rm HI}$ $\sim 3\times 10^{-12}$ $^{-1}$ ) proposed in Sec. |
3.2. | 3.2. |
This comparison is not strictly seltf-cousisteut because. by choosin,oea value of Py exeater than that due to QSOs. a disturbance has heen introduced iuto the caleulatiou of the opacity of the universe.since the ionisation state of the absorbers would be affected. | This comparison is not strictly self-consistent because, by choosing a value of $\Gamma_{\rm HI}$ greater than that due to QSOs, a disturbance has been introduced into the calculation of the opacity of the universe,since the ionisation state of the absorbers would be affected. |
This disturbance would be reduced if in fact Dia from QSOs had to be increased by a factor of the same order as he ratio Typ/Toso~3. suce ΠΕ Cua would not be uch altered. | This disturbance would be reduced if in fact $\Gamma_{\rm HeII}$ from QSOs had to be increased by a factor of the same order as the ratio $\Gamma_{\rm HI}/\Gamma_{\rm QSO} \sim 3$, since then $\Gamma_{\rm HI}/\Gamma_{\rm HeII}$ would not be much altered. |
Au increase in the ΠΟΠ - ionising power of QSOs over that arising from the usual power-law spectrmu has alveady been proposed by Sciama (1991). who needed to cusure that an increase in Dy duc to decay photons would not drive the universe to become completely opaque at the Hell edge. | An increase in the HeII - ionising power of QSOs over that arising from the usual power-law spectrum has already been proposed by Sciama (1994), who needed to ensure that an increase in $\Gamma_{\rm HI}$ due to decay photons would not drive the universe to become completely opaque at the HeII edge. |
This xoposal was based on existing observational hiuts that nanny QSOs possess a soft x-ray excess qm their spectra. | This proposal was based on existing observational hints that many QSOs possess a soft x-ray excess in their spectra. |
This excess was attributed to a Caulbert-Rees (1988) thermal dump with T~50 eV. resulting from the reprocessing of harder x-rays from the central regions of QSOs by optically thick cold material. | This excess was attributed to a Guilbert-Rees (1988) thermal bump with $T\sim50$ eV, resulting from the reprocessing of harder x-rays from the central regions of QSOs by optically thick cold material. |
Since 199| further observational evidence has acciunnulated for the prevalence of a soft x-ray excess in the spectra of QSOs. | Since 1994 further observational evidence has accumulated for the prevalence of a soft x-ray excess in the spectra of QSOs. |
This evidence has been reviewed by Coudhalekar. Rouillou-Folev Wellett (1996). | This evidence has been reviewed by Gondhalekar, Rouillon-Foley Kellett (1996). |
It is also noteworthy tha a 50 eV. Inuup would fit nicely in the gap (due to ealactic absorption) iu the composite spectrun shown in fig.6 of Laor ct al (1997). | It is also noteworthy that a $50$ eV bump would fit nicely in the gap (due to galactic absorption) in the composite spectrum shown in fig.6 of Laor et al (1997). |
On the theoretical side it has been found recently that a sn accretion disk around a black hole at the centre of a QSO would produce a soft x-rav excess without amy Couilbert-Roees reprocessing (Shimura Takahara 1995. Szuszkiewicz 1996. Szuszkiewicz. Alalkau Abramowiez 1996). | On the theoretical side it has been found recently that a slim accretion disk around a black hole at the centre of a QSO would produce a soft x-ray excess without any Guilbert-Rees reprocessing (Shimura Takahara 1995, Szuszkiewicz 1996, Szuszkiewicz, Malkan Abramowicz 1996). |
These aremuenuts have recently been streugtheued by the considerations of Iworista. Ferland Daldwiu (1997) who pointed out that a QSO emission spectrum without a bump at 50 ον would not account. (via excitation effects) for the observed strengths of the Πο cinission lines in QSO spectra. | These arguments have recently been strengthened by the considerations of Korista, Ferland Baldwin (1997) who pointed out that a QSO emission spectrum without a bump at 50 ev would not account (via excitation effects) for the observed strengths of the HeII emission lines in QSO spectra. |
These authors sugeested hat either the QSOs have a suitably complicated geometry. or that their enissjon spectrüni contains a sieuificant bump in the vicinity of the Well ionisation edge at 51.1ev. | These authors suggested that either the QSOs have a suitably complicated geometry, or that their emission spectrum contains a significant bump in the vicinity of the HeII ionisation edge at 54.4ev. |
A rough estimate for the resulting increase in Eg led to a factor3.6 (Sciaina 199D. | A rough estimate for the resulting increase in $\Gamma_{\rm HeII}$ led to a factor$\sim3.6$ (Sciama 1994). |
Π the acual factor werecloser to 2 the absorption analysis would still not be much changed. while the lower Limit ou Py would be increased | If the actual factor werecloser to 2 the absorption analysis would still not be much changed, while the lower limit on $\Gamma_{\rm HI}$ would be increased |
Another instance is a high initial ratio of molecular to atomic hvdrogen. | Another instance is a high initial ratio of molecular to atomic hydrogen. |
Following a sugeestion by the reviewer of (his paper. our program was run (after minor changes) lor the case where ny(0)=ngs(0)I0!"cm 7 and ne(0)=no(0)3109 7". | Following a suggestion by the reviewer of this paper, our program was run (after minor changes) for the case where $n_{H}(0)=n_{H2}(0)=10^{10}$ $^{-3}$ and $n_{C}(0)=n_{O}(0)=3\,\,10^{6}$ $^{-3}$. |
Comparing with Fig. | Comparing with Fig. |
1 to 3. the final value of ne4, was found to increase to 10° from 10°6 3 and the final value of no to decrease to 1.8109 [rom 2.2410° 7. | 1 to 3, the final value of $n_{Cgr}$ was found to increase to $\,\,10^{6}$ from $\,\,10^{6}$ $^{-3}$, and the final value of $n_{O}$ to decrease to $1.8\,\,10^{6}$ from $2.24\,\,10^{6}$ $^{-3}$. |
This notable increase in grain abundance in the presence of a high density. of II» is due to reactions 48 ancl 237 which generate Clls aud Coll». the progenitors of grains. | This notable increase in grain abundance in the presence of a high density of $_{2}$ is due to reactions 48 and 237 which generate $_{2}$ and $_{2}$ $_{2}$, the progenitors of grains. |
The range 55 to 144 for 105 IL atoms. deduced [rom our model. is consistent with spectroscopic observations of a couple of stars as reported by Snow and Wilt (1995).. in a compilation of measurements. | The range 55 to 144 for $10^{6}$ H atoms, deduced from our model, is consistent with spectroscopic observations of a couple of stars as reported by Snow and Witt \cite{sno}, in a compilation of measurements. |
While more astronomical observations could help. the present work should be an encouragement to further study simple chemical reactions between neutrals. especially reaction kl. so as to restrict (he range of uncertainty of the model. | While more astronomical observations could help, the present work should be an encouragement to further study simple chemical reactions between neutrals, especially reaction k1, so as to restrict the range of uncertainty of the model. |
Such reactions have become more amenable to measurement with the development of specialized apparata. e.g. the CRESU supersonic-[iow machine (see lor instance Howe (2000))). | Such reactions have become more amenable to measurement with the development of specialized apparata, e.g. the CRESU supersonic-flow machine (see for instance Rowe \cite{row}) ). |
aand =I-m class ground-based telescopes suggests that the fraction of dust-obscured GRBs is likely low. and the afterglow recovery rate for bbursts may approach unity. | and $\gtrsim 1$ -m class ground-based telescopes suggests that the fraction of dust-obscured GRBs is likely low, and the afterglow recovery rate for bursts may approach unity. |
The BBurst Alert Telescope (BAT) localized this burst on 2004. December 23.5877 UT to a 7' radius error circle (Tuellerefcf.2004;Markwardteraf... 2004). | The Burst Alert Telescope (BAT) localized this burst on 2004, December 23.5877 UT to a $7'$ radius error circle \citep{gcn2898,gcn2909}. |
. A series of XRT observations was initiated on December 23.780 UT. and a fading source was detected at n20640"49,2, 6=-37°04/21.5" (J2000) with an uncertainty of about 15" radius (Burrowseraf.2004). | A series of XRT observations was initiated on December 23.780 UT, and a fading source was detected at $\alpha$, $\delta$ (J2000) with an uncertainty of about $15''$ radius \citep{gcn2901}. |
. The spectral energy index was 3,=—1.02+£0.13 and the temporal decay rate was about a,2—1.70.2 (Fx ftv) with a flux of 6.5«107 ere em s! (0.5—10 keV) about 6.2 hr after the burst (Table 1:: Burrowsetafl. 2005a)). | The spectral energy index was $\beta_x=-1.02\pm 0.13$ and the temporal decay rate was about $\alpha_x=-1.7\pm 0.2$ $F_\nu\propto t^\alpha\nu^\beta$ ) with a flux of $6.5\times 10^{-12}$ erg $^{-2}$ $^{-1}$ $0.5-10$ keV) about 6.2 hr after the burst (Table \ref{tab:swift}; \citealt{bhc+05}) ). |
Following our discovery of the optical transient. the XRT position was revised to (Tagliaferrierαἱ.2004). a=0640"47.4>, 82—37704/22,3" (J2000). within about 17 of the optical afterglow position. | Following our discovery of the optical transient, the XRT position was revised to \citep{gcn2910} $\alpha$ =, $\delta$ (J2000), within about $1"$ of the optical afterglow position. |
Ground-based observations commenced on December 24.185 UT (14.4 hours after the burst) using the Swope 40-in telescope at LCO (Berger.Krzeminski&Hamuy2004). | Ground-based observations commenced on December 24.185 UT (14.4 hours after the burst) using the Swope 40-in telescope at LCO \citep{gcn2902}. |
. We imaged the entire 7’ radius BAT error circle in the r-band for a total of 20 min. | We imaged the entire $7'$ radius BAT error circle in the $r$ -band for a total of 20 min. |
The data were bias-subtracted. flat-fielded. and combined using standard IRAF routines. | The data were bias-subtracted, flat-fielded, and combined using standard IRAF routines. |
Astrometry was performed relative to the USNO-B catalog using 200 stars in common to thetwo frames. | Astrometry was performed relative to the USNO-B catalog using 200 stars in common to thetwo frames. |
The resulting rms positional uncertainty was 0.15”. | The resulting rms positional uncertainty was $0.15''$. |
Astationary source not present in the Digital Sky Survey (DSS) was detected at 1=06"40"47.323>. d= (J2000) with a magnitude of rz2]£0.15. | Astationary source not present in the Digital Sky Survey (DSS) was detected at $\alpha$, $\delta$ = (J2000) with a magnitude of $r\approx 21\pm 0.15$. |
This position was 7.5" outside of the initial XRT error circle. but only I” from the revised nominal XRT position. | This position was $7.5''$ outside of the initial XRT error circle, but only $1''$ from the revised nominal XRT position. |
A field centered on the position of the afterglow of 0041223 is shown in Figure |.. and the observations are summarized in Table 2.. | A field centered on the position of the afterglow of 041223 is shown in Figure \ref{fig:041223}, and the observations are summarized in Table \ref{tab:gb}. |
Additional observations with the Swope 40-in telescope were obtained starting on December 25.15 UT in the » and i bands. | Additional observations with the Swope 40-in telescope were obtained starting on December 25.15 UT in the $r$ and $i$ bands. |
A total of | hr was obtained in each filter. | A total of 1 hr was obtained in each filter. |
A comparison of the first and second epoch indicated that the afterglow had faded by 1.2 mag. corresponding to a decay rate of az—1.1. | A comparison of the first and second epoch indicated that the afterglow had faded by $1.2$ mag, corresponding to a decay rate of $\alpha\approx -1.1$. |
We subsequently observed the position of the afterglow with the Low-Resolution Imager and Spectrograph (LRIS: Okeeral, 1995)) mounted on the Keck-I 10-m telescope on 2005. January 8.34 UT. | We subsequently observed the position of the afterglow with the Low-Resolution Imager and Spectrograph (LRIS; \citealt{occ+95}) ) mounted on the Keck-I 10-m telescope on 2005, January 8.34 UT. |
We obtained R-band observations for a total of 70 min. | We obtained $R$ -band observations for a total of 70 min. |
The data were reduced and analyzed in the manner described above. | The data were reduced and analyzed in the manner described above. |
These observations reveal a faint source at the position of the afterglow with R~24.540.3 mag. | These observations reveal a faint source at the position of the afterglow with $R\approx 24.5\pm 0.3$ mag. |
An extrapolation of the afterglow flux at f=1.56 d to the epoch of the LRIS observation suggests that this object is most likely the afterglow. although any steepening in the afterglow evolution (e.g.. jet break) would mean that the emission is dominated by the host galaxy. | An extrapolation of the afterglow flux at $t=1.56$ d to the epoch of the LRIS observation suggests that this object is most likely the afterglow, although any steepening in the afterglow evolution (e.g., jet break) would mean that the emission is dominated by the host galaxy. |
Late-time observations were obtained with the Near Infra-Red Camera (NIRC: Matthews&Soifer 1994)) mounted on the Keck-I telescope in the A,-band on 2005. January 25.33 UT. | Late-time observations were obtained with the Near Infra-Red Camera (NIRC; \citealt{ms94}) ) mounted on the Keck-I telescope in the $K_s$ -band on 2005, January 25.33 UT. |
A total of sixty-two 50-s images were collected. | A total of sixty-two 50-s images were collected. |
The individual images were dark-subtracted. flat-fielded. and corrected for bad pixels and cosmic rays. | The individual images were dark-subtracted, flat-fielded, and corrected for bad pixels and cosmic rays. |
We then created object masks. which were used to construct improved flat fields for a second round of reduction. | We then created object masks, which were used to construct improved flat fields for a second round of reduction. |
The data were finally registered. shifted. and co-added. | The data were finally registered, shifted, and co-added. |
Photometry was performed relative to three 2MASS sources in the field. and no object was detected at the position of the afterglow to a 3c limit of K,=22.0 mag. | Photometry was performed relative to three 2MASS sources in the field, and no object was detected at the position of the afterglow to a $3\sigma$ limit of $K_s=22.0$ mag. |
Finally. we obtained spectroscopic observations using LRIS with a 400-line grating on the red side (dispersion of 1.86 A//pix) and a 600-line grism on the blue side (dispersion of 0.63 A//pix). | Finally, we obtained spectroscopic observations using LRIS with a $400$ -line grating on the red side (dispersion of 1.86 /pix) and a $600$ -line grism on the blue side (dispersion of 0.63 /pix). |
Two 2400 s exposures were obtained with a 1.5" slit. | Two 2400 s exposures were obtained with a 1.5" slit. |
The data were bias-subtracted and flat-fielded using IRAF. | The data were bias-subtracted and flat-fielded using IRAF. |
Rectification and sky subtraction were performed using the method and software described in Kelson(2003). | Rectification and sky subtraction were performed using the method and software described in \citet{kel03}. |
. We detect weak continuum emission. but no obvious emission lines in the range ~3500—9500A. | We detect weak continuum emission, but no obvious emission lines in the range $\approx 3500-9500$. |
. This burst was localized by the BAT on 2005. January 17.5365 UT to a 4/ radius error circle (Sakamotoeral...2005;Barthelmyetaf...2005). | This burst was localized by the BAT on 2005, January 17.5365 UT to a $4'$ radius error circle \citep{gcn2952,gcn2962}. |
. XRT observations revealed a fading source at 4=23"53"53.0°. d= (2000) with an uncertainty of 15" radius (Hilleta£...2005b). | XRT observations revealed a fading source at $\alpha$, $\delta$ = (J2000) with an uncertainty of $15''$ radius \citep{gcn2955}. |
. We note that the location of 0050117a less than 4 away from the Galactic plane results in large extinction. £(B—V)=1.75 mag (Schlegel.Finkbeiner&Davis 1998).. which severely hampered optical searches. | We note that the location of 050117a less than $4^\circ$ away from the Galactic plane results in large extinction, $E(B-V)=1.75$ mag \citep{sfd98}, which severely hampered optical searches. |
We observed the XRT position of 0050117a with the Wide Field Infra-red Camera (WIRC) mounted on the Palomar Hale 200-1n telescope on January 18.146 UT (14.6 hrs after the burst; Fox.Cenko&Murphy 2005)). | We observed the XRT position of 050117a with the Wide Field Infra-red Camera (WIRC) mounted on the Palomar Hale 200-in telescope on January 18.146 UT (14.6 hrs after the burst; \citealt{gcn2960}) ). |
A total of 32 min were obtained in the K; band. | A total of 32 min were obtained in the $K_s$ band. |
Several 2MASS and DSS sources were detected within and near the XRT position. | Several 2MASS and DSS sources were detected within and near the XRT position. |
A field centered on the XRT error circle of 0050117a is shown in Figure 2.. | A field centered on the XRT error circle of 050117a is shown in Figure \ref{fig:050117}. |
At the present no afterglow candidate is identified. | At the present no afterglow candidate is identified. |
We observed the field with the on 2005.January 19.08 and 24.14 UT (1.54 and 6.60 days after the burst. respectively) at a frequency of 8.46 GHz (Frail2005:Soderberg&Frail2005b). | We observed the field with the on 2005,January 19.08 and 24.14 UT (1.54 and 6.60 days after the burst, respectively) at a frequency of 8.46 GHz \citep{gcn2963,gcn2980}. |
. No source was detected within the error circle to a 30 limit of 98 (Jan. 19.08) and 84 (Jan. 24.14) gy. | No source was detected within the error circle to a $3\sigma$ limit of 98 (Jan. 19.08) and 84 (Jan. 24.14) $\mu$ Jy. |
This burst was localized by the BAT on 2005. January 24.4792 UT to a 6’ radius error circle (Markwardtοἱαἱ.2005;Cummingsetal. 2005). | This burst was localized by the BAT on 2005, January 24.4792 UT to a $6'$ radius error circle \citep{gcn2972,gcn2973}. |
. An XRT observation was initiated on January 24.607 UT (3.1 hr after the burst). and ground analysis revealed a source at a21230.4, 6=+13°02'39.0" (02000). with an uncertainty of 8 (Pagani51"οἱαἱ. 2005).. | An XRT observation was initiated on January 24.607 UT $3.1$ hr after the burst), and ground analysis revealed a source at $\alpha$, $\delta$ (J2000), with an uncertainty of $8''$ \citep{gcn2974}. . |
The flux of the source was 2«107 erg em s! (2—10 keV). | The flux of the source was $2\times 10^{-12}$ erg $^{-2}$ $^{-1}$ $2-10$ keV). |
We observed the XRT 8" error circle with NIRC in the / and Κ. bands starting on January 25.501 (24.5 hrs after the burst: Berger&Kulkarni 2005a)). | We observed the XRT $8''$ error circle with NIRC in the $J$ and $K_s$ bands starting on January 25.501 (24.5 hrs after the burst; \citealt{gcn2978}) ). |
A total of 15 min were obtained in each band. | A total of $15$ min were obtained in each band. |
Within the XRT error circle we detected a single point source. located at a=12"51"30.35°. 6=+13°02/41.3” (J2000). | Within the XRT error circle we detected a single point source, located at $\alpha$ , $\delta$ (J2000). |
The astrometry was performed relative to an image of the field from the Palomar 60-in telescope with an rms positional uncertainty of 0.2". | The astrometry was performed relative to an image of the field from the Palomar 60-in telescope with an rms positional uncertainty of $0.2''$ . |
The NIR afterglow position is only 2.4" away from the nominal XRT position. | The NIR afterglow position is only $2.4''$ away from the nominal XRT position. |
Follow-upobservations with NIRC on January 26.471 (47.8 hours after the burst) in the / (13.3 min) and Κι (14.2 min) bands revealed a clear fading of the point source confirming its identification | Follow-upobservations with NIRC on January 26.471 (47.8 hours after the burst) in the $J$ (13.3 min) and $K_s$ (14.2 min) bands revealed a clear fading of the point source confirming its identification |
The above shows that a stationary state of suspeuced accretion iu the presence of gravitational wave-etissious is facilitated by maguetolycrodyuamical viscosity. | The above shows that a stationary state of suspended accretion in the presence of gravitational wave-emissions is facilitated by magnetohydrodynamical viscosity. |
Note that uo specific instability mechanism is identified which is to account [or the required non-axisyuunetric delormatious of the torus. | Note that no specific instability mechanism is identified which is to account for the required non-axisymmetric deformations of the torus. |
It would be of interest to study this by utumerical simulatious. | It would be of interest to study this by numerical simulations. |
The GRB-alterglow emissious define the isotropic equivalent. luminosity of the black hole iu the present black hole-torus model. | The GRB-afterglow emissions define the isotropic equivalent luminosity of the black hole in the present black hole-torus model. |
Given the uniform magnetization of the horizon. the collateral interaction onto the torus is of similar inteusity per uuit opening auele. | Given the uniform magnetization of the horizon, the collateral interaction onto the torus is of similar intensity per unit opening angle. |
It follows tliat eiven that gravitational-wave emission is essentially unbeaumied aud assuimiug that the larger fraction of the magnetic field-lines threacdiug the black hole conuect to the torus. | It follows that given that gravitational-wave emission is essentially unbeamed and assuming that the larger fraction of the magnetic field-lines threading the black hole connect to the torus. |
Loug duratiou continuous emissiou with the predicted linear ο) is best detected using matched filtering. | Long duration continuous emission with the predicted linear chirp is best detected using matched filtering. |
Takine into account. therefore. the expected gain by a factor i iu sensitivity. where » is the inunber of cycles iu the emission. the ellective amplitude of the gravitational radiation at a distance D satisfies for a net [hence Feyy in gravitational waves. | Taking into account, therefore, the expected gain by a factor $\sqrt{n}$ in sensitivity, where $n$ is the number of cycles in the emission, the effective amplitude of the gravitational radiation at a distance $D$ satisfies for a net fluence $F_{GW}$ in gravitational waves. |
With a fraction of order unity of the black radiated olf in gravitational waves. derived [rom its spin-enerey of about one-third its otal thass. this poiuts towards GRBs as potentially the most powerful LIGO/VIRGO burst sources in the Universe. | With a fraction of order unity of the black hole-luminosity radiated off in gravitational waves, derived from its spin-energy of about one-third its total mass, this points towards GRBs as potentially the most powerful LIGO/VIRGO gravitational-wave burst sources in the Universe. |
A geometrical beaming factor of 100—200 gives rise to one event yer vear within a distance D~ I00Mpc with fry~10ος | A geometrical beaming factor of $100-200$ gives rise to one event per year within a distance $D\sim 100$ Mpc with $h_{eff}\sim 10^{-20}$. |
"Their approximately monochromatic enissions may have interesting cosmological applications. assuming uo cosmological evolution iu he GRB parameters. | Their approximately monochromatic emissions may have interesting cosmological applications, assuming no cosmological evolution in the GRB parameters. |
This research is supported by NASA Cirant 5-7012. au MIT C.E. Reed Fund aud à NATO Collaborative Linkage Grant. | This research is supported by NASA Grant 5-7012, an MIT C.E. Reed Fund and a NATO Collaborative Linkage Grant. |
The author thanks S. Ixulkarni aud R. Weiss for stimulating cliscussious. | The author thanks S. Kulkarni and R. Weiss for stimulating discussions. |
Assuming the axisvmmetirv of the svstem in (his paper. we numericallv study. the erowth of the non-spherical instability in (he accretion flow through the shock wave onto the protoneutron star. | Assuming the axisymmetry of the system in this paper, we numerically study the growth of the non-spherical instability in the accretion flow through the shock wave onto the protoneutron star. |
The unperturbed steady accretion [lows ancl (he shock waves are assumed to be spherically svinmetric. | The unperturbed steady accretion flows and the shock waves are assumed to be spherically symmetric. |
We take into account the heating and cooling of accreting matter via neutrino absorptions ancl emissions by free nucleons. | We take into account the heating and cooling of accreting matter via neutrino absorptions and emissions by free nucleons. |
Only (he region outside (he neutrino sphere is considered. | Only the region outside the neutrino sphere is considered. |
The basic evolution equations are written as follows. where p.v.6.P.Y.d are density. velocity. internal energy. pressure. electron [raction. and gravitational potential of the central object. respectively. | The basic evolution equations are written as follows, where $\rho, \mbox{\boldmath$ $}, e, P, Y_{\rm e}, \Phi$ are density, velocity, internal energy, pressure, electron fraction, and gravitational potential of the central object, respectively. |
The sell-gravitv of matter in the accretion flow is ignored. | The self-gravity of matter in the accretion flow is ignored. |
(Qj: and Qux are related with the interactions will neutrinos and are explained in more detail in the next section. | $Q_{\rm E}$ and $Q_{\rm N}$ are related with the interactions with neutrinos and are explained in more detail in the next section. |
We denote the Lagrangian derivative as dfdl and ris the radius. | We denote the Lagrangian derivative as $d/dt$ and $r$ is the radius. |
The numerical code for hydrodyvnamic computations emploved in (his paper is based on the ZEUS-2D (Stone&Norman1992).. which is an Eulerian code based on the method and employs an artificial viscosity of von Neumann and Richtaiver tvpe to capture shocks. | The numerical code for hydrodynamic computations employed in this paper is based on the ZEUS-2D \citep{stone}, which is an Eulerian code based on the finite-difference method and employs an artificial viscosity of von Neumann and Richtmyer type to capture shocks. |
We have mace several major changes to the base code to include the microphvsies as described in the following sections. | We have made several major changes to the base code to include the microphysics as described in the following sections. |
First. we have added the equation [ου electron Iraction (Eq. (4))). | First, we have added the equation for electron fraction (Eq. \ref{eq:ye_flow}) )), |
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