6 Heating and cooling

To discuss energy deposition and emission of neutrinos exterior to the neutrinosphere, one starts with the energy equation for $\nu_{\rm e}$ plus $\bar\nu_{\rm e}$, which is

$${1\over 4\pi\,r^2}\,{\partial L_{\nu}\over \partial r}\,=\,-\,Q_{\nu}^+ + Q_{\nu}^- \ ,$$ (68)

where $L_{\nu} = L_{\nu_{\rm e}}+L_{\bar\nu_{\rm e}}$ and $Q_{\nu}^+$ and $Q_{\nu}^-$ are the heating and cooling rates of the stellar medium as given by Eqs. (28) and (31), respectively. In writing Eq. (68), stationarity was assumed for the neutrinos, which is justified because the neutrino emission of the accreting neutrino star changes on a timescale which is typically longer than other relevant timescales of the discussed problem. From Eq. (68) the net effect of heating or cooling in a layer between radii r₁ and r₂ can be deduced as

$$\int\limits_{r_1}^{r_2}{\rm {d}}r\,4\pi\,r^2\left( Q_{\nu}^+ - Q_{\nu}^- \right) \,=\, L_{\nu}(r_1)\,-\,L_{\nu}(r_2)\ .$$ (69)

Refering to Eqs. (23) and (27), a suitable spectral and flavor average for the absorption coefficient of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ can be defined as

$$\langle \kappa_{\rm {a}} \rangle \,=\,{Q_{\nu}^+\over L_{\nu}/ (4\pi\,r^2\langle \mu_{\nu} \rangle)} \ ,$$ (70)

when $\langle \mu_{\nu_{\rm e}} \rangle \approx \langle \mu_{\bar\nu_{\rm e}} \rangle \equiv \langle \mu_{\nu} \rangle$is used. Plugging this into Eq. (68) gives

$${\partial L_{\nu}\over \partial r}\,=\,-\,\langle \kappa_{\rm... ...u}\over \langle \mu_{\nu} \rangle}\,+\,4\pi\,r^2 Q_{\nu}^- \ .$$ (71)

The neutrino luminosity as a function of radius $r \ge r_0$ is the general solution of Eq. (71):

 

$$L_{\nu}(r) \exp\left\lbrace -\int\limits_{r_0}^r {\rm {d}}r'\, {\langle \kappa_{\rm {a}} \rangle\over \langle \mu_{\nu} \rangle} \right\rbrace\, L_{\nu}(r_0)$$ =

$$\int\limits_{r_0}^r {\rm {d}}r'\,4\pi\,(r')^2 Q_{\nu}^- \exp\left... ...le \kappa_{\rm {a}} \rangle\over \langle \mu_{\nu} \rangle} \right\rbrace \cdot$$ + (72)

The first exponential factor represents the absorption damping of the luminosity in the shell between r₀ and r, the second exponential factor the reabsorption of neutrinos emitted at r' in the layer enclosed by radii r' and r.

   
6.1 Heating and cooling between $\mathsfsl{R}_{\nu}$ and $\mathsfsl{R_{g}}$

Here the lower boundary of the considered volume is the neutrinosphere at radius $r_0 = R_{\nu}$. Since both $\langle \kappa_{\rm {a}} \rangle\propto \rho(r)$ (cf. Eqs. (70) and (28)) and $Q_{\nu}^-\propto \rho(r)$ (cf. Eq. (31)) are steep functions of the radius in the region between $R_{\nu }$ and $R_{\rm {g}}$, where the density drops exponentially, most of the absorption and emission occurs in the immediate vicinity of the neutrinosphere. Therefore the neutrino luminosity at the gain radius, $L_{\nu}(R_{\rm {g}})$, can be approximated by the limit for $r\to \infty$ of Eq. (72), and the integral $\int_{r'}^r{\rm {d}}r''\,\langle \kappa_{\rm {a}} \rangle/\langle \mu_{\nu} \rangle$can be replaced by $\int_{R_{\nu}}^\infty{\rm {d}}r\,\langle \kappa_{\rm {a}} \rangle/\langle \mu_{\nu} \rangle$. This leads to

 

$$L_{\nu}(R_{\rm {g}}) \approx \exp\left\lbrace -\int\limits_{R_{\nu}}^\infty {\rm {d}}r\, {\langle \kappa_{\rm {a}} \rangle\over \langle \mu_{\nu} \rangle} \right\rbrace$$

$$\phantom{\times} \quad \times \left\lbrack L_{\nu}(R_{\nu})\,+\,\int\limits_{R_{\nu}}^\infty{\rm {d}}r\, 4\pi\,r^2 Q_{\nu}^- \right\rbrack\, \cdot$$ (73)

To evaluate the exponential damping factor, $\langle \kappa_{\rm {a}} \rangle$ is expressed by Eq. (70), making use of Eq. (28) and $L_{\nu} = 2\,L_{\nu_{\rm e}}$. The neutrino spectrum is assumed not to change outside the neutrinosphere. With Eq. (50) the integral over the density profile becomes $\int_{R_{\nu}}^\infty{\rm {d}}r\,\rho(r)\approx \rho_{\nu}h$. Employing Eqs. (51) and (52), one finds

$$\int\limits_{R_{\nu}}^\infty{\rm {d}}r\, {\langle \kappa_{\rm... ...2\over \widetilde{\langle \mu_{\nu} \rangle}}\,\equiv\, a \ ,$$ (74)

where $\widetilde{\langle \mu_{\nu} \rangle}$ denotes a radial average of the flux factor $\langle \mu_{\nu} \rangle$ in the layer between $R_{\nu }$ and $R_{\rm {g}}$. The energy loss integral is calculated with Eq. (31) where $T(r)\approx T_{\nu_{\rm e}}$ is used near the neutrinosphere. With $\int_{R_{\nu}}^\infty {\rm {d}}r\,r^2\rho(r)\approx \rho_{\nu}h \left\lbrack h^2+(R_{\nu}+h)^2 \right\rbrack\approx \rho_{\nu}R_{\nu}^2h$ (because $h\ll R_{\nu}$, cf. Eq. (51)) and Eqs. (51) and (52) this leads to:

$$\int\limits_{R_{\nu}}^\infty\! {\rm {d}}r\,4\pi\,r^2 Q_{\nu}^... ...t\lbrack {{\rm {erg}}\over {\rm {s}}} \right\rbrack \equiv b .$$ (75)

Equation (73) now becomes

$$L_{\nu}(R_{\rm {g}})\,\approx\,{\rm {e}}^{-a} \left\lbrack L_{\nu}(R_{\nu})\,+\,b \right\rbrack\, ,$$ (76)

and the total energy exchange between $R_{\nu }$ and $R_{\rm {g}}$ according to Eq. (69) therefore is

$$L_{\nu}(R_{\rm {g}})-L_{\nu}(R_{\nu})\equiv L_{\rm {acc}} \approx ({\rm {e}}^{-a}\!-1)L_{\nu}(R_{\nu}) + {\rm {e}}^{-a}b .$$ (77)

Since $R_{\rm {g}}$ separates the layer of neutrino cooling from the one of neutrino heating, the region between $R_{\nu }$ and $R_{\rm {g}}$ must lose energy by neutrino emission. Therefore the neutrino luminosity at $R_{\rm {g}}$must be larger than $L_{\nu}(R_{\nu})$, and $L_{\rm {acc}}$represents the luminosity associated with the accretion of matter through the gain radius onto the surface of the nascent neutron star. The requirement $L_{\rm {acc}} \ge 0$ constrains the luminosity of the neutron star core relative to the product $R_{\nu}^2(kT_{\nu_{\rm e}})^4$ by the inequality

$$L_{\nu}(R_{\nu})\,\le\,b\,\left( {\rm {e}}^a-1 \right)^{-1}\,.$$ (78)

Provided the core luminosity can be expressed in terms of blackbody emission of temperature $T_{\nu_{\rm e}}$,

 

$$L_{\nu}(R_{\nu}) \approx 4\pi R_{\nu}^2\,{c\over 4}\,{7\over 8}\,a_{\gamma} (kT_{\nu_{\rm e}})^4$$

$$\approx 2.9~10^{51}\,R_{\nu,6}^2\left( {kT_{\nu_{\rm e}} \over 4\,{\rm {MeV}}} \right)^{\! 4}\ \ \left\lbrack {{\rm {erg}}\over {\rm {s}}} \right\rbrack\,,$$ (79)

the consistency condition translates into the relation

$$\widetilde{\langle \mu_{\nu} \rangle}\,\mathrel{\mathchoice {... ...{\hfil$\scriptscriptstyle ...$$ (80)

which is satisfied above the neutrinosphere in the layer between $R_{\nu }$ and $R_{\rm {g}}$.

   
6.2 Heating and cooling between $\mathsfsl{R_{g}}$ and $\mathsfsl{R_{s}}$

For reasons of simplicity it will be assumed that in the layer bounded by $R_{\rm {g}}$ and $R_{\rm {s}}$ nuclei are completely disintegrated into free nucleons. Disregarding the occurrence of $\alpha$ particles, in particular, is certainly an approximation which becomes invalid when the temperature drops below about 1$\,$MeV, i.e., when the shock is at large radii, typically around 300$\,$km (see Bethe 1993, 1995, 1996a-c, 1997). The presence of $\alpha$ particles reduces the neutrino heating, because electron neutrinos and antineutrinos are absorbed only on nucleons, but energy released by the recombination of $\alpha$'s during shock expansion supports the shock at a later stage and contributes to the energy budget of the explosion. Since in the context of this paper we do not attempt to calculate the explosion energy, but are interested in a qualitative discussion of the revival phase of the stalled shock, the recombination of nucleons to $\alpha$ particles is probably not a crucial issue.

As will be demonstrated below, the optical depth between $R_{\rm {g}}$ and $R_{\rm {s}}$ is small such that . Therefore the reabsorption probability of emitted neutrinos is also small and an approximation to the solution of Eq. (72) at the shock position is

$$L_{\nu}(R_{\rm {s}})\,\approx\, \left( \! 1-\! \int\limits_{R... ...{\rm {g}}}^{R_{\rm {s}}} \! {\rm {d}}r\, 4\pi\,r^2 Q_{\nu}^- .$$ (81)

The net energy deposition is found to be

 

$$L_{\nu}(R_{\rm {g}})-L_{\nu}(R_{\rm {s}}) \approx \int\limits_{R_{\rm {g}}}^{R_{\rm {s}}} \! {\rm {d}}r\, {\langle ... ...) - \int\limits_{R_{\rm {g}}}^{R_{\rm {s}}} \! {\rm {d}}r\, 4\pi\,r^2 Q_{\nu}^-$$

$$\equiv {\cal H}-{\cal C} \, .$$ (82)

Since in the layer bounded by $R_{\rm {g}}$ and $R_{\rm {s}}$ neutrino heating takes place, $L_{\nu}(R_{\rm {g}}) \ge L_{\nu}(R_{\rm {s}})$(this will also be verified below). Expanding the exponential damping factors only to lowerst order in the exponent implies a slight overestimation of the energy input into the stellar medium by neutrinos (because the luminosity entering at $R_{\rm {g}}$ is assumed to decay linearly through the layer), but also the energy loss by neutrinos is overestimated, because the reabsorption of emitted neutrinos is not included.

Using $Q_{\nu}^+$ from Eq. (28) in Eq. (70), $L_{\nu} = 2L_{\nu_{\rm e}}$, and the density profile from Eq. (63), one finds for the first integral in Eq. (82):

 

$${\cal H} \,\approx\, 4.9~10^{50}\, {L_{\nu,52}(R_{\rm {g}})\over ... ...2}-1 \right\rbrack\ \ \left\lbrack {{\rm {erg}}\over {\rm {s}}} \right\rbrack ,$$ (83)

where $kT_{\nu_{\rm e}}$ was again treated as a constant, $\langle \mu_\nu \rangle^\ast$defines an average value of the flux factor in the layer between $R_{\rm {g}}$ and $R_{\rm {s}}$, and $\rho_{{\rm {s}},9}$ is the density behind the shock in units of $10^9\,$g$\,$cm^-3. The second integral in Eq. (82) is evaluated with $Q_{\nu}^-$from Eq. (31) and the temperature and density relations from Eq. (63):

 

$${\cal C} \,\approx\, 2.9~10^{50}\, (kT_{{\rm {s}},2})^6 \rho_{{\r... ... \right\rbrack\ \ \left\lbrack {{\rm {erg}}\over {\rm {s}}} \right\rbrack \cdot$$ (84)

Employing the gain condition, Eq. (33), and again making use of Eq. (63), one gets

$$\left( {kT_{\rm {s}}\over 2\,{\rm {MeV}}} \right)^{\! 6}\,\ap... ...^{\! 2} \left( {R_{\rm {g}}\over R_{\rm {s}}} \right)^{\! 6} ,$$ (85)

which serves to rewrite Eq. (84) as

 

$${\cal C} \,\approx\, 1.6~10^{50} {L_{\nu,52}(R_{\rm {g}})\over \l... ...ight\rbrack\ \, \left\lbrack {{\rm {erg}}\over {\rm {s}}} \right\rbrack\! \cdot$$ (86)

Combining Eqs. (83) and (86) gives the net energy transfer to the stellar medium in the gain region:

$${\cal H} - {\cal C} \,\approx\, {\cal H}\! \times \! \left\lb... ...R_{\rm {s}}} \right)^{\! 4} \right\rbrack\right\rbrace \!\cdot$$ (87)

Since $\langle \mu_\nu \rangle^\ast/\langle \mu_\nu \rangle_{\rm {g}}\sim 1$ and $R_{\rm {s}} > R_{\rm {g}}$, typically $R_{\rm {s}}\sim 2R_{\rm {g}}$, we verify that ${\cal H}-{\cal C}>0$and therefore $L_{\nu}(R_{\rm {s}}) < L_{\nu}(R_{\rm {g}})$, as expected for the neutrino heating region. For $\beta = \rho_{\rm {s}}/\rho_{\rm {p}} \approx 7$ (Eq. (41)) and $\alpha = 1/\sqrt{2}$, Eq. (44) yields for the postshock density

$$\rho_{\rm {s}}\,\approx\, 3~10^9\,\,{(-\dot M)\over M_\odot /... ...left\lbrack {{\rm {g}}\over {\rm {cm}}^3} \right\rbrack\,\cdot$$ (88)

Using this and $R_{\rm {s}}\sim 2R_{\rm {g}}$ in Eq. (83) leads to

 

$$\int\limits_{R_{\rm {g}}}^{R_{\rm {s}}} {\rm {d}}r\, {\langle \kappa_{\rm {a}} \rangle\over \langle \mu_{\nu} \rangle} {{\cal H}\over L_{\nu}(R_{\rm {g}})}$$ =

$$\sim {0.44\over \langle \mu_\nu \rangle^\ast}\, {(kT_{\nu_{\rm e},4})^... ...{s}}}\, \left( {\widetilde{M}\over M_\odot} \right)^{\!-{1\over 2}}\!\!\! \cdot$$ (89)

For sufficiently small accretion rates $\vert\dot M\vert$ this is less than about 0.5, and the assumption made before Eq. (81) is verified, i.e., the reabsorption of neutrinos emitted in the gain region can be neglected.