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shows that the above relation is satisfied with an accuracy of ~1% for models with central energy densities smaller than the energy density at which the pure quark-core appears. | shows that the above relation is satisfied with an accuracy of $\sim 1$ for models with central energy densities smaller than the energy density at which the pure quark-core appears. |
After the appearance of the pure quark core. numerical errors result in huge values of A. when using the original tabulated EOS. | After the appearance of the pure quark core, numerical errors result in huge values of $\lambda$, when using the original tabulated EOS. |
With the use of the refined tabulated/analytic EOS. these errors are reduced significantly to an acceptable level and diminish as the resolution ts increased. | With the use of the refined tabulated/analytic EOS, these errors are reduced significantly to an acceptable level and diminish as the resolution is increased. |
The above check confirms that the refined EOS indeed produces physically acceptable results. while the original tabulated EOS produces results that are dominated by numerical error in the region in question, | The above check confirms that the refined EOS indeed produces physically acceptable results, while the original tabulated EOS produces results that are dominated by numerical error in the region in question. |
If a compact star 1s detected as a pulsar and one can measure the first two time-derivatives €). © of its angular velocity. then one can define an observational braking index In the Newtonian. slow-rotation limit. the spin-down of a pulsar is usually modeled in the form of a power law (see Shapiro Teukolsky. | If a compact star is detected as a pulsar and one can measure the first two time-derivatives $\dot \Omega$, $ \ddot \Omega$ of its angular velocity, then one can define an observational braking index In the Newtonian, slow-rotation limit, the spin-down of a pulsar is usually modeled in the form of a power law (see Shapiro Teukolsky. |
1983). namely assuming that only kinetic energy is lost and that the rate of loss is proportional to some power of the angular velocity of the star. 1.8. where K«0 and &>O are usually assumed to be real constants. | 1983), namely assuming that only kinetic energy is lost and that the rate of loss is proportional to some power of the angular velocity of the star, i.e. where $\kappa<0$ and $\alpha>0$ are usually assumed to be real constants. |
With these assumptions. the braking index is equal to=a|. | With these assumptions, the braking index is equal to $n=\alpha-1$. |
For example. it is assumed that. for magnetic braking. T—ΩΙ. which yields an expected braking index η=3. while. for gravitational wave emission. T~Q°. which yields n—3. | For example, it is assumed that, for magnetic braking, $\dot T \sim \Omega^4$, which yields an expected braking index $n=3$, while, for gravitational wave emission, $\dot T \sim \Omega^6$, which yields $n=5$. |
For slowly rotating pulsars. the above considerations are. in most cases. appropriate. | For slowly rotating pulsars, the above considerations are, in most cases, appropriate. |
However. for rapidly rotating pulsars one has to take into account the rotational flattening of the star. | However, for rapidly rotating pulsars one has to take into account the rotational flattening of the star. |
To a first approximation. this can be done considering rotational effects up to order O(Q7). | To a first approximation, this can be done considering rotational effects up to order $O(\Omega^2)$. |
Glendenning (1997) gives the rotationally corrected Q and braking index 2(Q) for a sequence of uniformly rotating stars. assuming the spin-down law of Equation (7)). | Glendenning (1997) gives the rotationally corrected $\dot \Omega$ and braking index $n(\Omega)$ for a sequence of uniformly rotating stars, assuming the spin-down law of Equation \ref{dT1}) ). |
As we will show here. the expressions given in Glendenning (1997) are incomplete. in the sense that the derivation is not fully consistent to O(Q7). but misses additional contributions of the same order. | As we will show here, the expressions given in Glendenning (1997) are incomplete, in the sense that the derivation is not fully consistent to $O(\Omega^2)$, but misses additional contributions of the same order. |
The energy lost in the form of electromagnetic. or gravitation radiation is not only to the expense of the star's kinetic energy (which would be the case only in the O(Q) slow- approximation) but to the expense of the star's total | The energy lost in the form of electromagnetic or gravitation radiation is not only to the expense of the star's kinetic energy (which would be the case only in the $O(\Omega)$ slow-rotation approximation) but to the expense of the star's total |
As shortly described iu Sect. | As shortly described in Sect. |
1. both Zola (1996) aud Dacins (1998) lave analyzed the πο curves of W Cru with a disk model. | 1, both a (1996) and Daems (1998) have analyzed the light curves of W Cru with a disk model. |
Zola used mean light curves coustructed from Marino et ((1988). and Pazzi (1993) photoclectzic photometry. respectively. | a used mean light curves constructed from Marino et (1988), and Pazzi (1993) photoelectric photometry, respectively. |
Dacius study was based on then unpublished Ceneva photometry. | Daems study was based on then unpublished Geneva photometry. |
Both studies have arrived to a quite consistent set of the paralcters. | Both studies have arrived to a quite consistent set of the parameters. |
While im our initial caleulations we lave started with the parameters which have cover very broad range in the parameter space. it was munediatelv clear hat solutions will be in the narrow range arotnd t values specified by Zola. aud Daeuis. respectively. | While in our initial calculations we have started with the parameters which have cover very broad range in the parameter space, it was immediately clear that solutions will be in the narrow range around the values specified by a, and Daems, respectively. |
Since we are looking for the optimal set of t xumnueters in a imultidinensional parameter sace is recommended to fix as mauy paraneters as possib | Since we are looking for the optimal set of the parameters in a multi-dimensional parameter space it is recommended to fix as many parameters as possible. |
Therefore. we assigned theoretical values to eyavifv xiehtening. hb darkening. and albedo cocfiicicu | Therefore, we assigned theoretical values to gravity brightening, limb darkening, and albedo coefficients. |
ST1e effective temperature of the G supergiaut. only visible stellar componcut. has been derived by Dacus (1998) fro111 he spectral cnerey distribution iu the broad wavelenghi span. froii IUE UV to optical aud IR photometry obtained at ESO. | The effective temperature of the G supergiant, only visible stellar component, has been derived by Daems (1998) from the spectral energy distribution in the broad wavelength span, from IUE UV to optical and IR photometry obtained at ESO. |
Ie has arrived to Τομ=5500 Is. | He has arrived to $T_{2,\rm eff} =
5500$ K. |
Cartesian frame (v4.x3). | Cartesian frame $(x_1,x_2)$. |
Following again Refregier(2003)... we define the polar shapelet basis functions às where 5 and s, are the left-handed and right-handed modes. respectively. and Bernstein&Jarvis(2002) showed that. for, «n,. one can relate to the associated Laguerre polynomial Note the similarities between the Cartesian basis and the polar basis(??=: Both share a Gaussian weighting function and are intrinsically circular. | Following again \citet{Refregier03.1}, we define the polar shapelet basis functions as where $n_l$ and $n_r$ are the left-handed and right-handed modes, respectively, and \citet{Bernstein02.1} showed that, for $n_l<n_r$, one can relate to the associated Laguerre polynomial Note the similarities between the Cartesian basis and the polar basis: Both share a Gaussian weighting function and are intrinsically circular. |
As Baan, is a complex function. if an,τσa. it is computationally more efficient to decompose the galactic shape into Cartesian shapelets according to and then to perform a coordinate transformation T*7 from Cartesian to polar shapelet space (cf.Eq.(69)inRefregier2003).. We will do that for the tests in the following sections. | As $B_{n_r,n_l}$ is a complex function if $n_r\neq n_l$, it is computationally more efficient to decompose the galactic shape into Cartesian shapelets according to and then to perform a coordinate transformation $\mathbf{T}^{c\rightarrow p}$ from Cartesian to polar shapelet space \citep[cf. Eq. (69) in][]{Refregier03.1}, We will do that for the tests in the following sections. |
Performing this transformation enables us to form a conceptionally different family of shear estimators. | Performing this transformation enables us to form a conceptionally different family of shear estimators. |
By defining nSΠΕΠ and 1=Πρ1. we can see from(10).. that 5,,, behaves like a field with spin 7. | By defining $n\equiv n_r + n_l$ and $m\equiv n_r - n_l$, we can see from, that $B_{n,m}$ behaves like a field with spin $m$. |
Since the shear behaves like à spin-2 field. its action on a circular source can be described by the v7=2 modes of a polar shapelet model. | Since the shear behaves like a spin-2 field, its action on a circular source can be described by the $m=2$ modes of a polar shapelet model. |
Thus. the most basic one of these estimators. uses only the #7=2 mode of any even polar order of C' (Masseyetal.2007)05i | Thus, the most basic one of these estimators, uses only the $m=2$ mode of any even polar order $n$ of $\tilde{G}^\prime$ \citep{Massey07.2}. |
This estimator must be normalized by radial modes gn.oP? obtained from unlensed sources. | This estimator must be normalized by radial modes $g^p_{n,0}$ obtained from unlensed sources. |
For the tests discussed here. we provide the ulensed image G such that those coefficients can be measured from images having the correct unlensed shape. | For the tests discussed here, we provide the unlensed image $G$ such that those coefficients can be measured from images having the correct unlensed shape. |
This approach is drastically different from measuring the quadrupole moment — in the sense that it depends only on a single lensed and two radial unlensed coefficients — and thus expected to behave in a differentmanner. even though the decomposition is done with Cartesian shapelets in both cases. | This approach is drastically different from measuring the quadrupole moment – in the sense that it depends only on a single lensed and two radial unlensed coefficients – and thus expected to behave in a differentmanner, even though the decomposition is done with Cartesian shapelets in both cases. |
G' is decomposed into Cartesian shapelets of maximum order Hoy€18.12}. which is typical given the significance of images (cf.Kuijken 2006 | $G^\prime$ is decomposed into Cartesian shapelets of maximum order $n_{max}\in\lbrace8,12\rbrace$, which is typical given the significance of weak-lensing images \citep[cf.][]{Kuijken06.1}. |
).At first. we investigate the modeling fidelity visually. | .At first, we investigate the modeling fidelity visually. |
In Fig. l.. | In Fig. \ref{fig:models}, , |
we give four examples of Sérrsic-type galaxy shapes | we give four examples of Sérrsic-type galaxy shapes |
1.17+0.08&107 erg em sc. | $1.17\pm 0.08\times
10^{-8}$ erg $^{-2}$ $^{-1}$. |
The best-fit Gaussian indicates a broad line. shifted from the neutral Fe Ka line energy of 6.40 keV: ΕΞ5.3/5! keV. FWHM=3.27558 keV. and W=250£50 eV. The edge is measured to be at ΕΞ6.84:0.3 keV. with a depth of 720.5+0.1. | The best-fit Gaussian indicates a broad line, shifted from the neutral Fe $\alpha$ line energy of 6.40 keV: $=5.3^{+0.1}_{-0.3}$ keV, $=3.2_{-0.6}^{+0.8}$ keV, and $=250\pm 50$ eV. The edge is measured to be at $=6.8\pm 0.3$ keV, with a depth of $\tau=0.5\pm
0.1$. |
However. significant residuals remain in the Fe Ko. line region with this fit. due in part to the non-Gaussian nature of the line profile (see Figure 1). | However, significant residuals remain in the Fe $\alpha$ line region with this fit, due in part to the non-Gaussian nature of the line profile (see Figure 1). |
Moreover. on a broader energy range (one that includes the 20-30 keV "Compton hump" seen in many sources). a reflection model is often required to fit the spectra of stellar-mass black holes (see. e.g.. Gierlinski et al. | Moreover, on a broader energy range (one that includes the 20–30 keV “Compton hump” seen in many sources), a reflection model is often required to fit the spectra of stellar-mass black holes (see, e.g., Gierlinski et al. |
1999). | 1999). |
Broad Gaussian and smeared edge components are merely an approximation to a full reflection model in the 10.0 keV band. | Broad Gaussian and smeared edge components are merely an approximation to a full reflection model in the 0.5--10.0 keV band. |
Therefore. we now focus on the results of fitting more sophisticated. physically-motivated reflection models. | Therefore, we now focus on the results of fitting more sophisticated, physically-motivated reflection models. |
These replace the hard power-law. Gaussian. and smeared edge components discussed above. | These replace the hard power-law, Gaussian, and smeared edge components discussed above. |
Anticipating a highly-ionized accretion disk. we made fits with the "constant density ionized disk" reflection model (hereafter. CDID: Ross. Fabian. Young 1999). | Anticipating a highly-ionized accretion disk, we made fits with the “constant density ionized disk” reflection model (hereafter, CDID; Ross, Fabian, Young 1999). |
This model measures the relative strengths of the directly-observed and reflected flux. the accretion disk ionization parameter Ly /nR?. where Ly is the X-ray luminosity. 1 is the hydrogen number density. and R is radius). and the photon index of the illuminating power-law flux. | This model measures the relative strengths of the directly-observed and reflected flux, the accretion disk ionization parameter $\xi = L_{X}/nR^{2}$ , where $L_{X}$ is the X-ray luminosity, $n$ is the hydrogen number density, and $R$ is radius), and the photon index of the illuminating power-law flux. |
Fe Ko line emission and line broadening due to Comptonization in an ionized disk surface layer are included m this model. | Fe $\alpha$ line emission and line broadening due to Comptonization in an ionized disk surface layer are included in this model. |
The fit obtained with this model is shown in the top panel of Figure 2. | The fit obtained with this model is shown in the top panel of Figure 2. |
The photon index of the irradiating power law is measured to be ΓΞ 2.08407.The ionization parameter is high: C=1.3%)«IO ergem s! and the relative strength of reflected flux is measured to be f£=UR (where ρω Εκ). | The photon index of the irradiating power law is measured to be $\Gamma=2.08^{+0.02}_{-0.04}$ .The ionization parameter is high: $\xi=1.3^{+0.7}_{-0.1}\times 10^{4}$ erg cm $^{-1}$, and the relative strength of reflected flux is measured to be $f=0.5_{-0.1}^{+0.7}$ (where $F_{total} = F_{direct} + f\times
F_{refl.}$ ). |
While the shape of the Fe K edge is reproduced by this model. the width and shape of the Fe Ka line ts not. and a statistically-poor fit is obtained (4=399.7 for 231 d.o.f.). | While the shape of the Fe K edge is reproduced by this model, the width and shape of the Fe $\alpha$ line is not, and a statistically-poor fit is obtained $\chi^{2}=399.7$ for 231 d.o.f.). |
The ionization parameter obtained with this model corresponds to a mixture of helium-like and hydrogenie ion species of Fe (Kallman and MeCray 1982). | The ionization parameter obtained with this model corresponds to a mixture of helium-like and hydrogenic ion species of Fe (Kallman and McCray 1982). |
As Doppler shifts and general relativistic smearing may be expected for lines produced in an aecretion. disk close to the black hole. we next made fits after convolving (or. ~blurring”) the CDID model with the line element expected near à Kerr black hole. | As Doppler shifts and general relativistic smearing may be expected for lines produced in an accretion disk close to the black hole, we next made fits after convolving (or, “blurring”) the CDID model with the line element expected near a Kerr black hole. |
We assumed an inner disk radius of 1.24 R,. an outer line production radius of 400 R,. and an inclination of /2457 (in fits with this and other models. intermediate inclinations were marginally preferred in terms of \7). | We assumed an inner disk radius of 1.24 $R_{g}$ , an outer line production radius of 400 $R_{g}$, and an inclination of $i=45^{\circ}$ (in fits with this and other models, intermediate inclinations were marginally preferred in terms of $\chi^{2}$ ). |
With this blurred model. we obtained parameter constraints which differed marginally from the previous fit: ¢10 erg em s!. f=0.625. and P2 | With this blurred model, we obtained parameter constraints which differed marginally from the previous fit: $\xi=2.5^{+5.5}_{-0.1}\times 10^{4}$ erg cm $^{-1}$, $f=0.6^{+0.6}_{-0.1}$, and $\Gamma=1.96^{+0.04}_{-0.06}$. |
The shape and strength. of the Fe Καὶ line are not fit 1.96550.adequately: the fit is slightly worse than the un-blurred model (4=407.8. 231 d.o.f.). | The shape and strength of the Fe $\alpha$ line are not fit adequately; the fit is slightly worse than the un-blurred model $\chi^{2}=407.8$, 231 d.o.f.). |
Finally. we constructed a model which allows the Fe Ka line and reflection components to be treated separately. | Finally, we constructed a model which allows the Fe $\alpha$ line and reflection components to be treated separately. |
We fit the line with the "Laor" line model (Laor 1991). and the power-law and reflection continuum (minus the line) with the “pexriv™ model (Magdziarz Zdziarski 1995). | We fit the line with the “Laor” line model (Laor 1991), and the power-law and reflection continuum (minus the line) with the “pexriv” model (Magdziarz Zdziarski 1995). |
With pexriv. f=| corresponds to a disk which intercepts half of the incident power-law flux. | With pexriv, $f=1$ corresponds to a disk which intercepts half of the incident power-law flux. |
It should be noted that this model was also used by Wilms et al. ( | It should be noted that this model was also used by Wilms et al. ( |
2001) in fits to the XMM-Newton//EPIC spectrum of the Seyfert galaxy MC6G-6-30-15. allowing for a direct comparison. | 2001) in fits to the /EPIC spectrum of the Seyfert galaxy MCG–6-30-15, allowing for a direct comparison. |
For the Laor line. we initially fixed the inner disk edge at 1.24 Ας. the outer line production region at 400 ΑΟ. and the inclination at /2457. | For the Laor line, we initially fixed the inner disk edge at 1.24 $R_{g}$, the outer line production region at 400 $R_{g}$, and the inclination at $i=45^{\circ}$. |
The line energy. emissivity profile (e—7: we fit for J). and intensity were allowed to vary. | The line energy, emissivity profile $\epsilon \sim r^{-\beta}$; we fit for $\beta$ ), and intensity were allowed to vary. |
The MCD and pexriv reflection components were blurred as before. | The MCD and pexriv reflection components were blurred as before. |
We found that the data could not simultaneously constrain f. &. and the disk surface temperature with pexriv (an additional parameter for this model). | We found that the data could not simultaneously constrain $f$, $\xi$, and the disk surface temperature with pexriv (an additional parameter for this model). |
We therefore fixed the tonization parameter at £=2.0«107 erg em s7!. and the disksurface temperature at AT=1.3 keV (as per Ross. Fabian. Young 1999 in fits to Cygnus X-1. wherein a similarly low MCD disk temperature but similarly high values of¢ are reported). | We therefore fixed the ionization parameter at $\xi=2.0\times 10^{4}$ erg cm $^{-1}$, and the disksurface temperature at $kT=1.3~$ keV (as per Ross, Fabian, Young 1999 in fits to Cygnus X-1, wherein a similarly low MCD disk temperature but similarly high values of $\xi$ are reported). |
The fit obtained with this model is shown in the bottom panel of Figure 2. | The fit obtained with this model is shown in the bottom panel of Figure 2. |
Statistically. this model represents a significant improvement (4=319.9 for 229 d.o.f.). | Statistically, this model represents a significant improvement $\chi^{2}=319.9$ for 229 d.o.f.). |
A power-law index of P22.047548 is obtained. | A power-law index of $\Gamma=2.04^{+0.03}_{-0.02}$ is obtained. |
We measure the Fe Ka line to be centered at ΕΞ0.811 keV. likely due to a blend of Fe XXV and Fe XXVI (helium-like and hydrogenie Fe) and consistent with the high 10nization parameters previously measured. | We measure the Fe $\alpha$ line to be centered at $=6.8^{+0.2}_{-0.1}$ keV, likely due to a blend of Fe XXV and Fe XXVI (helium-like and hydrogenic Fe) and consistent with the high ionization parameters previously measured. |
The line is strong. with an equivalent width of W=350") eV and a flux of 2.2+0.3«1079 erg em? sc! (3.2044107 ph em s) | The line is strong, with an equivalent width of $=350^{+60}_{-40}$ eV and a flux of $2.2\pm 0.3\times
10^{-10}$ erg $^{-2}$ $^{-1}$ $3.2\pm 0.4\times
10^{-2}$ ph $^{-2}$ $^{-1}$ ). |
When the inner disk edge is allowed to vary. an inner radius of 1.24 Αι (the limit of the Laor model. corresponding to «= 0.998) is preferred over an inner radius of 6.0 R, (the marginally stablecircular orbit around a Schwarzschild blackhole) at the 60 level of confidence. | When the inner disk edge is allowed to vary, an inner radius of 1.24 $R_{g}$ (the limit of the Laor model, corresponding to $a=0.998$ ) is preferred over an inner radius of 6.0 $R_{g}$ (the marginally stablecircular orbit around a Schwarzschild blackhole) at the $\sigma$ level of confidence. |
A steep emissivity is suggested via the Laor line model: }=5.4+0.5, | A steep emissivity is suggested via the Laor line model: $\beta=5.4\pm
0.5$ . |
This emissivity 15 preferred over that for a standard accretion disk (3= 3.0) at the 5.60 level of confidence. | This emissivity is preferred over that for a standard accretion disk $\beta=3.0$ ) at the $\sigma$ level of confidence. |
The pexriv reflection "fraction" is f= 0.6*5:1. This is consistent with the | The pexriv reflection “fraction” is $f=0.6_{-0.1}^{+0.3}$ This is consistent with the |
where f. D. S. 7 denote the time. the rest-mass deusity. the moment density. aud the energy density in a fixed fine where the £uid moves at speed v. respectively. | where $t$ , $D$ , $\mathbf{S}$, $\tau$ denote the time, the rest-mass density, the momentum density, and the energy density in a fixed frame where the fluid moves at speed $\mathbf{v}$, respectively. |
These variables are related to quantities in the local rest frame of the fluid through where p. p. Wand / denote the proper rest-iiass deusity. the pressure. the fluid Lorentz factor. aud the specific euthalpy. respectively. | These variables are related to quantities in the local rest frame of the fluid through where $\rho$, $p$, $W$ and $h$ denote the proper rest-mass density, the pressure, the fluid Lorentz factor, and the specific enthalpy, respectively. |
The specific euthalpy is giveu by where ¢ is the specific internal enerex. | The specific enthalpy is given by where $\varepsilon$ is the specific internal energy. |
We can describe these equations with a Lagraugian coordinate i. | We can describe these equations with a Lagrangian coordinate $m$. |
Since we are concerned with spherically απλο] explosions. we will rewrite them iun the dimensional spherical coordinate svsten. | Since we are concerned with spherically symmetric explosions, we will rewrite them in the 1-dimensional spherical coordinate system. |
where edenotes the radial velocity. G the eravitational constant. M, the mass meluded within the radius kc. 1/D.s5-—VS-.r/r (v: the racial vector) aud The Lagrangian coordinate i) is rolated to r through The gravity is included in the weak limit where uo eeneral relativistic effect is prominent. | where $v$denotes the radial velocity, $G$ the gravitational constant, $M_r$ the mass included within the radius $r$, $V=1/D$ , $s=V\mathbf{S}\cdot\mathbf{r}/r$ $\mathbf{r}$: the radial vector) and The Lagrangian coordinate $m$ is related to $r$ through The gravity is included in the weak limit where no general relativistic effect is prominent. |
The pressure is described as the sui of the radiation pressure and the eas pressure: where «@ denotes the radiation constant. TZ the temperature. & the Boltzmann constaut. jp the mean molectlar weight. aud i0, the mass of proton. | The pressure is described as the sum of the radiation pressure and the gas pressure: where $a$ denotes the radiation constant, $T$ the temperature, $k$ the Boltzmann constant, $\mu$ the mean molecular weight, and $m_p$ the mass of proton. |
The specific internal cuerey 5 is given by Tere we will iutroduce two kinds of adiabatic iudices. 54 aud 5» defined by The difference scheme of equations (8)) (10)) cau be written in the following form: We use the Codunov method to αποήσαν solve equations (17)) distinguishing these two adiabatic mdices in our Rien solver. | The specific internal energy $\varepsilon$ is given by Here we will introduce two kinds of adiabatic indices, $\gamma_1$ and $\gamma_2$ defined by The difference scheme of equations \ref{massl}) \ref{energyl}) ) can be written in the following form; We use the Godunov method to numerically solve equations \ref{deff2}) ) distinguishing these two adiabatic indices in our Riemann solver. |
The νο of zones are listed in Table ὃν, | The number of zones are listed in Table \ref{tbl-model}. |
To obtain accurate enerev cdistributious of SN ejecta above the threshold euergies for cosmic-ray spallation reactions. our models have zones i the outermost lavers with iiasses well below 107AL... | To obtain accurate energy distributions of SN ejecta above the threshold energies for cosmic-ray spallation reactions, our models have zones in the outermost layers with masses well below $10^{-9}\ \Msun$. |
When we calculate the explosion of a star. we necd to set the boundary conditious at the outer edge of the star as follows: where nes is the zone number corresponding to the outermost laver. | When we calculate the explosion of a star, we need to set the boundary conditions at the outer edge of the star as follows; where $imax$ is the zone number corresponding to the outermost layer. |
At the center. which corresponds to the zone interface (=1/2. we consider an mias zone /=0. where the physical quautities are eiven by to keep the svuuuetry with respecto to the ceuter. | At the center, which corresponds to the zone interface $i=1/2$, we consider an imaginary zone $i=0$, where the physical quantities are given by to keep the symmetry with respect to the center. |
We consider four stellar models manediatelv before the core collapse as the initial conditions. | We consider four stellar models immediately before the core collapse as the initial conditions. |
These stars are originated from 12.~LOAL. main-sequence stars. | These stars are originated from $12 - \sim 40\Msun$ main-sequence stars. |
Three of thei are thought to have undergone intense stellar winds aud lost their T-rich aud Πο cuvelopes. | Three of them are thought to have undergone intense stellar winds and lost their H-rich and He envelopes. |
As a result. the stellar surfaces αλα] consist of carbou aud oxvecu at explosion. | As a result, the stellar surfaces mainly consist of carbon and oxygen at explosion. |
These three models have corresponding real type Ic supernovae as shown in Table 3.. | These three models have corresponding real type Ic supernovae as shown in Table \ref{tbl-model}. |
These stars have become fairly compact with radii less than the solar radius. | These stars have become fairly compact with radii less than the solar radius. |
This compactucss results iu higher pressures at the shock breakout compared with explosion of a star with an extended envelope if the explosion euergies per unit ejecta mass are the same. | This compactness results in higher pressures at the shock breakout compared with explosion of a star with an extended envelope if the explosion energies per unit ejecta mass are the same. |
The other star is thought to e the progenitor of SN 1987À. one of the best. studied supernovac. | The other star is thought to be the progenitor of SN 1987A, one of the best studied supernovae. |
We have calculated the explosion for this supernova aud compare the result with the previous work o check our numerical code. | We have calculated the explosion for this supernova and compare the result with the previous work to check our numerical code. |
To initiate explosions. we first replace the central Fe core with the poiut mass located at the center. release he cnerev in theimucrimost severalzones in the foxiu of hermal cuerey (or pressure). aud trace the evolution of physical quantities. | To initiate explosions, we first replace the central Fe core with the point mass located at the center, release the energy in theinnermost severalzones in the form of thermal energy (or pressure), and trace the evolution of physical quantities. |
The calculations are stopped when the ejecta keep expanding homologously. | The calculations are stopped when the ejecta keep expanding homologously. |
The parameters of each model axe tabulated in Table 3.. | The parameters of each model are tabulated in Table \ref{tbl-model}. . |
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