9 Evolution of shock radius and shock velocity

The model developed in the preceding sections allows one to study the behavior of the supernova shock in response to the processes that play a role in the collapsed stellar core. The physics between the neutron star surface and the shock is constrained by the energy influx from the neutrinosphere on the one hand and the mass accretion into the shock front on the other. Equations (97), (104) (in combination with (105)), and (39) determine the shock radius $R_{\rm {s}}$, the shock velocity $U_{\rm {s}}$, and the postshock pressure $P_{\rm {s}}$. The state of the matter immediately behind the shock and that at the gain radius are related via Eqs. (59)-(61) and (111), the gain radius $R_{\rm {g}}$ is given by Eq. (66), the postshock temperature by $kT_{\rm {s}} = \left\lbrack 3 P_{\rm {s}}/(f_{\rm {r}}g_{\rm {r}}a_{\gamma}) \right\rbrack^{1/4}$(Eq. (56)), and the postshock density as $\rho_{\rm {s}} = \beta\rho_{\rm {p}}$ with $\rho _{\rm {p}}$from Eq. (44).

The mass accretion rate $\dot M$ into the shock is a fixed parameter of the problem (in Eq. (46) it is expressed in terms of the constant H which is linked to the structure of the progenitor star). The rate of mass advection into the neutron star, $\dot M'$, can be calculated from Eq. (93). The radius $R_{\nu }$ and mass M of the neutron star, the neutrinospheric luminosity $L_\nu$, and the spectral temperature of the emitted electron neutrinos $T_{\nu_{\rm e}}$(assumed to be roughly equal to the temperature $T_{\nu}$ of the stellar gas at the neutrinosphere) are also input parameters. The discussion takes into account the effects of neutrino losses in the cooling region, expressed by Eqs. (74)-(77), and of neutrino heating in the gain region as given by Eqs. (82), (83), and (87).

The time dependence of the considered model requires as initial conditions the values $\Delta M_{\rm {g}}^0$ and $\Delta E_{\rm {g}}^0$ for the initial mass and energy in the gain region. This couples the subsequent evolution in $\Delta M_{\rm {g}}$ and $\Delta E_{\rm {g}}$, which can be followed with Eqs. (102) and (109), respectively, to the situation that exists right after core bounce. Knowing $\dot M'(t)$ allows one to include also the changes of the neutron star mass.

   
9.1 Shock expansion and acceleration

Combining Eqs. (104) and (106) and using Eq. (60) for $P_{\rm {g}}$ in terms of $P_{\rm {s}}$with $K^{3/4} = P_{\rm {s}}^{3/4}/\rho_{\rm {s}}$, one gets the relation

 

$$(x-x_0)\,x^3\,\approx\,\left( \!{R_{\rm {g}}\over R_{\rm {s}}}\! \right)^{\! 3}\! \Biggl\lbrack \,x\! \!\! \left( \!{R_{\rm {s}}\over R_{\rm {g}}}- 1\! \right) \Biggr\rbrack^4$$ +

$$\! 4\,(3\Gamma-4)\left( \!{R_{\rm {s}}\over R_{\rm {g}}} - 1\! \right) x^3 .$$ + (113)

Here x and x₀ were defined as

  

$$\equiv P_{\rm {s}} \left( {G\widetilde{M}\rho_{\rm {s}}\over 4 R_{\rm {s... ... U_{\rm {s}}\,\sqrt{{R_{\rm {s}}\over G\widetilde{M}}} \,\, \right)^{\!\! 2}\,,$$ x (114)

$$\equiv 3(\Gamma-1)\,{\Delta E_{\rm {g}}\over 4\pi R_{\rm {s}}^3} \left( ... ...opto\,-\,{\Delta E_{\rm {g}}\over \dot M\sqrt{R_{\rm {s}}\widetilde{M}}}\,\cdot$$ x₀ (115)

The proportionality relations can be verified by using Eqs. (39), (43) (with $\alpha = 1/\sqrt{2}$) and (44).

Equation (113) is the key equation to understand the behavior of the supernova shock under the influence of accretion and neutrino heating. Typically, $\Delta E_{\rm {g}} < 0$ during the shock stagnation phase, and therefore x₀ < 0. Equation (113) depends on two variables which constrain the conditions at the shock front, namely on x > 0 and on $y \equiv R_{\rm {s}}/R_{\rm {g}}$, for which $y \ge 1$ holds. Fixing the parameters $\dot M$, ${\widetilde M}$ and $R_{\rm {s}}$, one can show that a larger value of x and thus a larger $U_{\rm {s}}$requires that x₀ and therefore $\Delta E_{\rm {g}}$ is bigger (i.e., less negative). Physically, this corresponds to the case where neutrino energy deposition leads to a rising postshock pressure $P_{\rm {s}}$(compare Eq. (114)), which accelerates the shock front. On the other hand, if $\vert U_{\rm {s}}\vert\ll \vert v_{\rm {p}}\vert \approx (G\widetilde{M}/R_{\rm {s}})^{1/2}$ the quantity x is essentially constant, and $y \propto R_{\rm {s}}^{7/16}$ (cf. Eq. (99)) is the variable which reacts to changes of x₀. The corresponding discussion is more transparent when Eq. (113) is rewritten in the following form:

$$\left\lbrack x\! -\! x_0\! +\! 4(3\Gamma\!-\!4) \right\rbrack x^3y^3 \approx (x+y-1)^4\! + 4(3\Gamma\!- 4)x^3y^4 .$$ (116)

For x₀ < 0 this equation has a solution $\hat y$ which shrinks, if $\Delta E_{\rm {g}}$ and thus x₀ is larger (i.e., less negative). Therefore the radius $R_{\rm {s}}$ of the shock, which is compatible with the assumptions, is smaller. Inversely, if $\Delta E_{\rm {g}}$ and x₀ are lower (more negative), $R_{\rm {s}}$ will be larger. This behavior can be explained by the observation that $\Delta E_{\rm {g}}$ decreases when matter with negative specific energy is accumulated in the gain region. Such an accumulation of mass will cause a growth of the shock radius. It should be noted that for x of order unity (i.e., $\vert U_{\rm {s}}\vert\ll \vert v_{\rm {p}}\vert$) a solution $\hat y \ge 1$of Eq. (116) exists only in case of $x_0 \le 0$. A positive value of $\Delta E_{\rm {g}}$, on the other hand, is compatible only with a sufficiently large shock velocity $U_{\rm {s}}$.

The situation is graphically illustrated in Fig. 3, where $\Delta M_{\rm {g}}$ from Eq. (97) and $\Delta E_{\rm {g}}$from Eqs. (104) and (105) are plotted as functions of $U_{\rm {s}}$ for different choices of the shock radius $R_{\rm {s}}$ (with parameters: $\widetilde{M} = 1.25\,$$M_{\odot }$, $\dot M = 0.3\,$ $M_{\odot }\,$s^-1, $\Gamma = \gamma = 4/3$, $\beta = 7$, $\alpha = 1/\sqrt{2}$, $g_{\rm {r}} = 6.41$, $f_{\rm {r}} = 1.16$). Figure 3 (or Eq. (100)) shows that a growth of the mass $\Delta M_{\rm {g}}$in the gain region will cause an increase of $R_{\rm {s}}$. This means that ${\rm {d}}R_{\rm {s}}/{\rm {d}}t \ge 0$ can be ensured if

$${{\rm {d}}\over {\rm {d}}t}(\Delta M_{\rm {g}})\, \ge\, 0\ .$$ (117)

This, however, is not a sufficient criterion for a continued outward motion of the shock. The dotted arrows in Fig. 3 indicate a situation where the decrease of energy in the gain region is so large that the increase of the shock radius implies a deceleration of the shock. For moving along the path marked by solid arrows, i.e., for obtaining stable shock expansion with ${\rm {d}}U_{\rm {s}}/{\rm {d}}t \ge 0$, a necessary condition is

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}})\, \ge\, U_{\... ...ver \partial R_{\rm {s}}} \right\rbrack_{U_{\rm {s}}} \ \cdot$$ (118)

The right hand side of Eq. (118) cannot be zero in general, because ${\rm {d}}(\Delta E_{\rm {g}})/{\rm {d}}t > 0$ can also be associated with a shrinkage of $R_{\rm {s}}$ if more matter (with negative total energy) is lost from the gain region by advection through the gain radius than is resupplied by gas falling into the shock. The combined conditions of Eqs. (117) and (118) guarantee that $R_{\rm {s}}$ and $U_{\rm {s}}$ grow at the same time. Applied to a stalled shock, in which case $U_{\rm {s}} = 0$, Eq. (117) together with Eq. (118) can therefore be considered as "shock revival criterion'', which states that for an expansion and acceleration of the shock front to occur, the energy in the gain region should increase and simultaneously the mass in the gain region should not decrease.

Figure 3: Mass $\Delta M_{\rm {g}}$ and energy $\Delta E_{\rm {g}}$ in the gain region as functions of shock velocity $U_{\rm {s}}$ for different shock radii $R_{\rm {s}}$. The values of neutron star mass ${\widetilde M}$ and mass infall rate $\dot M$ into the shock are fixed. Two possible cases for a transition from initial shock radius $R_{\rm {s}} = 100$ km to final shock radius $R_{\rm {s}} = 200$ km are indicated. The path marked by solid arrows corresponds to stable shock expansion, the situation marked by dotted arrows means a slow-down of the shock

   
9.2 Shock revival criterion

If the conditions between neutrinosphere and shock vary slowly with time, $\dot R_{\rm {g}} \approx 0$ is a good assumption. Since $v_{\rm {p}} \neq 0$, Eq. (109) can then be written in the form

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}}) \approx -\dot... ...ver v_{\rm {p}}} + \dot M' l_{\rm {g}} + {\cal H} - {\cal C} .$$ (119)

Replacing $l_{\rm {s}}$ in the bracket on the right hand side of Eq. (119) by Eq. (110) and using Eq. (39) for $P_{\rm {s}}/\rho_{\rm {p}}$, one derives in case of $\vert U_{\rm {s}}\vert\ll \vert v_{\rm {p}}\vert$ the expression

 

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}}) \,\approx\, \,\dot M\,l_{\rm {s}} - \dot M \!\left( 2\!-\! {1\over \beta}\! -... ...er \beta^2}\,{\Gamma\over \Gamma\! -\! 1} \right)\! v_{\rm {p}}\dot R_{\rm {s}}$$ -

$$\dot M'\,l_{\rm {g}} + {\cal H} - {\cal C}\, .$$ + (120)

This equation is correct to first order in $\vert\dot R_{\rm {s}}/v_{\rm {p}}\vert\ll 1$. From the discussion in Sect. 8.2 follows that for an outward acceleration of a stalled shock ( $\dot R_{\rm {s}} = U_{\rm {s}} = 0$), a necessary condition is (see Eq. (118)):

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}}) \,=\, -\,\dot... ...m {s}} + \dot M'\,l_{\rm {g}} + {\cal H} - {\cal C} \,>\,0 \ .$$ (121)

Note that neutrino heating (${\cal H}$) and cooling (${\cal C}$) in the gain region as well as the mass accretion rate $\dot M$ through the shock have a direct influence. But also neutrino losses below $R_{\rm {g}}$ have an effect by determining $\dot M'$, and a more indirect one by causing additional energy deposition in the gain region, where the neutrino energy extracted from the cooling layer is partly reabsorbed.

The terms proportional to $\dot M = 4\pi R_{\rm {s}}^2\rho_{\rm {p}}v_{\rm {p}}$account for the so-called ram pressure of the infalling matter, which is proportional to $\rho_{\rm {p}}v_{\rm {p}}^2$ and damps shock expansion, because the accretion of matter through the shock yields a negative contribution to the right hand sides of Eqs. (119)-(121). A comparison of Eqs. (120) and (121) shows that the onset of shock expansion enhances this and therefore Eq. (121) gives a minimum requirement.

Instead of just an outward acceleration of the shock, a positive postshock velocity, i.e., $v_{\rm {s}} > 0$, may be considered as a stronger criterion for the possibility of an explosion. With $\beta = \rho_{\rm {s}}/\rho_{\rm {p}}$ and Eq. (36) one derives $v_{\rm {s}} = \beta^{-1}(v_{\rm {p}}-U_{\rm {s}}) + U_{\rm {s}}$, which means that $v_{\rm {s}} > 0$ translates into $U_{\rm {s}} > -\,v_{\rm {p}}/(\beta-1)$. Since $\beta \gg 1$ (Eq. (41)), this condition is fulfilled while $\vert U_{\rm {s}}/v_{\rm {p}}\vert\ll 1$ still holds. Using this more rigorous criterion will therefore affect the details of the discussion, but will not change the picture qualitatively.

${\rm {d}}(\Delta E_{\rm {g}})/{\rm {d}}t > 0$ can be achieved by strong neutrino heating (${\cal H}$ large), but can also result if $\dot M'l_{\rm {g}} > \dot Ml_{\rm {s}}$. For $l_{\rm {g}} = l_{\rm {s}}$, which is true when $\Gamma = \gamma$(see Eq. (112)), this is equivalent to $\dot M' < \dot M$, i.e., when less mass is accreted through the shock than is lost from the gain region into the neutron star (note that $l_{\rm {s}} < 0$; Eq. (110)). As a consequence, however, the mass between $R_{\rm {g}}$ and $R_{\rm {s}}$and therefore the shock radius will decrease, in conflict with the demand for shock expansion (see Sect. 9.1). To make sure the shock expands, also Eq. (117) has to be fulfilled. In case of $U_{\rm {s}} = 0$, $\dot R_{\rm {g}} = 0$, Eq. (102) yields:

$${{\rm {d}}\over {\rm {d}}t}(\Delta M_{\rm {g}}) \,=\, -\,\dot... ...\,\ge\,0 \ \ \Longleftrightarrow\ \ \dot M \,\le\,\dot M' \,.$$ (122)

Both Eqs. (121) and (122) constrain the parameters for which a revival of the stalled shock can occur.

Figure 4: Conditions for shock revival by neutrino heating for different shock stagnation radii $R_{\rm {s}}$. The lines labeled with O $_{\rm {E}}$and O $_{\rm {M}}$ connect the roots of ${\rm {d}}(\Delta E_{\rm {g}})/ {\rm {d}}t$ and ${\rm {d}}(\Delta M_{\rm {g}})/{\rm {d}}t$, respectively, in the plane defined by the mass accretion rate into the shock, $\dot M$, and the neutrinospheric luminosity $L_\nu$. The curves with labels Li( $i \in \{1,...,5\}$) and $\vert{\rm {d}}M_j/{\rm {d}}t\vert$ ( $j \in \{1,2\}$) correspond to the constraints (i)-(vi) listed in Sect. 9.3. They represent warning flags indicating that the assumptions of the discussion may need to be generalized. The hatched areas mark the regions where the conditions are favorable for a neutrino-driven explosion, because Eqs. (121) and (122) are both satisfied such that the supernova shock expands and accelerates. Below the curve O $_{\rm {M}}$ the rate of mass loss from the gain layer to the neutron star exceeds the mass accretion rate $\dot M$ and therefore ${\rm {d}}(\Delta M_{\rm {g}})/{\rm {d}}t$ is negative. Above the curve O $_{\rm {E}}$ the energy deposition by neutrino heating cannot compensate for the accumulation of mass with negative total energy in the gain region and therefore ${\rm {d}}(\Delta E_{\rm {g}})/ {\rm {d}}t$ is negative

   
9.3 Conditions for shock revival

The properties of Eq. (121) together with Eq. (122) will now be discussed in more detail. For chosen fixed values of the shock stagnation radius, those combinations of mass accretion rate $\dot M$ and neutrinospheric luminosity $L_\nu$ will be determined which allow for an outward acceleration of the shock front. For these conditions an explosion driven by neutrino energy deposition may develop.

Assuming $U_{\rm {s}} = 0$ the gain radius is given by Eq. (99). For the neutrino luminosity $L_{\nu} = 2L_{\nu_{\rm e}}$ will be taken again. The accretion rate $\dot M'$ of Eq. (93) can be calculated by using Eqs. (94) and (95). Neutrino effects are evaluated from Eqs. (74)-(77) and Eqs. (83) and (87) with Eq. (88) for the postshock density $\rho _{\rm {s}}$.

Several consistency constraints have to be taken into account to make sure that the assumptions of the analytic model developed in the preceding sections are fulfilled:

(i)

For the gain radius (Eq. (99)) must hold. Here h is the scale height of the exponential neutron star atmosphere, Eq. (51). The left inequality constrains the neutrinospheric luminosity to be , where the limit L₁ depends on the accretion rate $\dot M$. The right inequality, on the other hand, requires $L_{\nu} \ge L_2(\dot M)$;

(ii)

Since the neutrinospheric luminosity $L_\nu$ and temperature $T_{\nu}$ are not coupled here by the assumption of blackbody emission, Eq. (78) must be satisfied to have $L_{\rm {acc}} \ge 0$, i.e., to have a cooling layer outside of the neutrinosphere. This translates into a condition $L_{\nu} \le L_3$;

(iii)

The definition of $R_{\rm {g}}$ implies that neutrinos transfer energy to the stellar gas for $R_{\rm {g}} \le r \le R_{\rm {s}}$. Therefore Eq. (87) has to fulfill the condition ${\cal H}-{\cal C} \ge 0$, corresponding to $L_{\nu} \ge L_4(\dot M)$. This constraint is similar to the one which follows from the requirement that $R_{\rm {g}}\le R_{\rm {s}}$, but somewhat stronger, depending on the value of the ratio between $\langle \mu_{\nu} \rangle_{\rm {g}}$ and $\langle \mu_\nu \rangle^\ast$;

(iv)

Equation (89) (with ${\cal H}$ taken from Eq. (83)) has to be less than about 0.5 to justify the disregard of reabsorption of neutrinos emitted from the gain layer. This limits the neutrinospheric luminosity to ;

(v)

The postshock temperature must be MeV because the matter behind the shock is assumed to be completely disintegrated into free nucleons, and $\alpha$ particles therefore do not exist. For this to hold, the absolute value of the mass accretion rate must exceed some lower limit, , where $\dot M_1$ depends on the shock radius and the effective mass ${\widetilde M}$ of the remnant;

(vi)

Since self-gravity of the gas between neutrinosphere and shock was neglected, the total mass there must be much smaller than the mass of the neutron star. This requirement leads to an upper limit for the rate of mass accretion: .

While conditions (i)-(iii) ensure the logical coherence of the model, a violation of conditions (iv)-(vi) would just reduce the accuracy of the discussion. For example, the framework developed in the previous sections can be generalized such that the (partial) recombination of nucleons to $\alpha$ particles or heavy nuclei at temperatures below about one MeV is taken into account. Item (vi) implies

 

$$\int\limits_{R_{\nu}}^{R_{\rm {s}}}\!\!{\rm {d}}r\,4\pi r^2\rho(r) \int\limits_{R_{\nu}}^{R_{\rm {eos}}}\!\!{\rm {d}}r\,4\pi r^2 \rho(r) + \int\limits_{R_{\rm {eos}}}^{R_{\rm {s}}}\!\!{\rm {d}}r\, 4\pi r^2\rho(r)$$ =

$$\int\limits_{R_{\nu}}^\infty \!\!{\rm {d}}r\,4\pi r^2\rho_1(r) + ... ...\int\limits_{R_{\nu}+h}^{R_{\rm {s}}}\!\!\!\!{\rm {d}}r\, 4\pi r^2\rho_2(r) \,,$$ (123)

where $\rho_1(r)\equiv \rho_{\nu}\exp\left\lbrack -(r\!-\!R_{\nu})/h \right\rbrack$ (Eq. (50) with h from Eq. (51)) and $\rho_2(r) \equiv \rho_{\rm {s}} (R_{\rm {s}}/r)^3$ (Eq. (63)). A reasonable upper bound for this mass integral is

$$\int\limits_{R_{\nu}}^{R_{\rm {s}}} {\rm {d}}r\,4\pi r^2\rho(... ...tstyle ...$$ (124)

which limits the allowed accretion rate according to condition (vi).

The sequence of plots in Fig. 4 shows the results of an evaluation of Eqs. (121) and (122) together with the constraints (i)-(vi) for different shock stagnation radii: $R_{\rm {s}} = 100, 150, 200, 250$ and 300 km, respectively. The numerical values chosen for the other parameters were: $\widetilde{M}= 1.25$ $M_{\odot }$, $R_{\nu} = 50$ km, $kT_{\nu} = 4$ MeV, $\beta = 7$, $\Gamma = \gamma = {4\over 3}$, $f_{\rm {g}} = 1.25$(corresponding to $\eta_{\rm e} Y_{\rm e} \approx 1$ at the neutrinosphere), $q_{\rm {d}} = 8.5~10^{18}$ erg$\,$g^-1, $\langle \mu_{\nu} \rangle^\ast = 0.7$, $\langle \mu_{\nu} \rangle_{\rm {g}} = 0.6$, and $\widetilde{\langle \mu_{\nu} \rangle} = 0.4$.

The roots of ${\rm {d}}(\Delta E_{\rm {g}})/ {\rm {d}}t$ and ${\rm {d}}(\Delta M_{\rm {g}})/{\rm {d}}t$ are represented by the lines labeled with O $_{\rm {E}}$ and O $_{\rm {M}}$, respectively. These lines separate regions in the $\vert\dot M\vert$-$L_\nu$ plane, within which the collapsed stellar core reacts differently to the mass inflow through the shock and to the irradiation by neutrinos emitted from the neutrinosphere (and from the cooling layer). In this respect the plots of Fig. 4 can be considered as "phase diagrams'' for the post-bounce evolution of the supernova. Within the hatched areas both Eqs. (121) and (122) are simultaneously fulfilled. Additional lines correspond to constraints (i)-(vi). They are displayed as warning flags that the assumptions of the treatment may need to be generalized. Left of the vertical dotted line, which corresponds to constraint (v), $\alpha$ particles and heavy nuclei in the postshock medium would have to be taken into account, and the analysis performed here is not very accurate. The vertical dashed line marks the boundary right of which Eq. (124) and thus constraint (vi) is violated.

Figure 5: Left: conditions for shock revival by neutrino heating for shock stagnation radii $R_{\rm {s}} = 150\,$km (cross-hatched area) and 250$\,$km (hatched area). Right: the lines O $_{\rm {M}}$ and O $_{\rm {E}}$ which connect the roots of Eqs. (121) and (122), respectively, for shock radii 300 km, 400 km, 500 km, 600 km, and 700 km. Different from Fig. 4, the gain radius and the integrals for neutrino heating and cooling in the gain layer were evaluated by using the exact solution for hydrostatic equilibrium instead of approximate power-law profiles. In addition, the disappearance of free nucleons and therefore the quenching of neutrino absorption and emission below a temperature of 1 MeV were taken into account. Above the line labeled with L3 the accretion luminosity $L_{\rm {acc}}$ (Eq. (77)) becomes negative. The lines corresponding to constraints (v) and (vi) are omitted for reasons of clarity

Two of the simplifications that entered the analysis for Fig. 4 can be easily removed. On the one hand, the gain radius $R_{\rm {g}}$ and heating and cooling in the gain layer can be calculated more accurately, when the density and temperature profiles of Eqs. (59) and (61) instead of the power-law approximation of the hydrostatic atmosphere [Eq. (63)] are used. In this case $R_{\rm {g}}$ must be numerically determined as the root of Eq. (33) (with $L_{\nu_{\rm e}} = {1\over 2}L_{\nu}(R_{\rm {g}})$as given by Eq. (76)), and the integrals for ${\cal H}$ and ${\cal C}$ in Eq. (82) can also be evaluated numerically. On the other hand, the recombination of free nucleons to $\alpha$ particles and heavy nuclei at low temperatures can roughly be taken into account concerning its effects on the neutrino interaction in the gain region. Provided that above a certain temperature, say 1 MeV, all nuclei are disintegrated into free nucleons and below this temperature all nucleons are bound in nuclei, neutrino absorption and emission reactions will not take place outside of the corresponding radius $R_{\alpha}$. The latter can also be calculated from the temperature profile of Eq. (61). The heating and cooling integrals are then performed with the upper integration boundary being chosen as the minimum of $R_{\rm {s}}$ and $R_{\alpha}$. The recombination of nucleons to $\alpha$particles releases a sizable amount of energy, about 7 MeV per nucleon. This additional energy source in the gain region was not included in the discussion here, because it requires a detailed modelling of the composition history of the postshock medium (considering the different degree of disintegration of nuclei during infall and recombination of nucleons during later expansion in different volumes of matter).

The results of this more general treatment are displayed in Fig. 5 for shock radii $R_{\rm {s}} = 150\,$km and 250$\,$km. The quantitative changes are significant: Compared to Fig. 4 the different value for the ${\cal H}-{\cal C}$ term moves the O $_{\rm {M}}$-line slightly upward and the O $_{\rm {E}}$-line more strongly downward. A similar effect is associated with a moderate reduction of the shock radius in Fig. 4. The gain radius obtained by the exact calculation can also shrink with growing shock radius, different from the approximate representation of Eq. (99). On the other hand, the outer boundary of the gain layer is defined by the recombination radius $R_{\alpha}$ of $\alpha$ particles instead of the possibly larger shock radius. Both effects combined, the total heating rate in the gain layer is similar. Therefore the qualitative picture remains unchanged.

The hatched areas in the plots of Figs. 4 and 5 include those combinations of parameters for which the conditions of Eqs. (121) and (122) are both satisfied and therefore the initially stagnant shock will expand and will be accelerated. Below the O $_{\rm {M}}$ line neutrino cooling outside of the neutrinosphere is very efficient and the neutron star swallows matter faster than gas is resupplied by accretion through the shock. Therefore ${\rm {d}}(\Delta M_{\rm {g}})/{\rm {d}}t$ is negative. When the mass accretion rate $\vert\dot M\vert$ drops below this critical line, shock expansion can be supported only by an increasing core luminosity, because a larger value of $L_\nu$ reduces the neutrino losses from the cooling region. Otherwise advection through the gain radius and thus into the neutron star extracts mass from the neutrino-heating region and the shock retreats. Figures 4 and 5 show that for given rate $\vert\dot M\vert$ there is a lower limit of the neutrinospheric luminosity $L_\nu$, which must be exceeded when shock expansion and acceleration shall occur.

Above the O $_{\rm {E}}$ line neutrino heating (represented by the ${\cal H}-{\cal C}$ term in Eq. (121)) cannot compete with the accumulation of matter with negative total energy in the gain layer. In this case ${\rm {d}}(\Delta E_{\rm {g}})/ {\rm {d}}t$ is negative. $R_{\rm {s}}$ can nevertheless grow for such conditions, simply because gas piles up on top of the neutron star. This pushes the shock farther out, but does not allow positive postshock velocities to develop. Since the postshock matter is gravitationally bound ( $\Delta E_{\rm {g}} < 0$), an explosion, however, requires sufficiently powerful energy input by neutrinos. The position of the O $_{\rm {E}}$ line shifts with changing shock radius. For discussing the destiny of the shock this change of the overall situation associated with the shock motion therefore has to be taken into account. This can be done by solving the equations of the toy model for time-dependent information about the shock radius and the shock velocity (see Sect. 9.4).

The O $_{\rm {E}}$ line is very sensitive to the shock position, whereas the O $_{\rm {M}}$ line is only weakly dependent (Fig. 5). On the one hand, a high core luminosity $L_\nu$ reduces the downward advection of gas through the gain radius. On the other hand, the neutrino heating in the gain layer increases with larger shock radius. Both effects determine the slopes and positions of the critical lines. The distance between the O $_{\rm {M}}$ and O $_{\rm {E}}$ lines in Figs. 4 and 5 grows for larger shock radii, and the hatched area expands. This is caused by an increase of the ${\cal H}-{\cal C}$ term in Eq. (121).

Acceleration is easier for a shock which has stalled at a large distance from the center, i.e., the same core luminosity can then ensure favorable conditions already for a higher value of $\vert\dot M\vert$. Besides stronger neutrino heating in the more extended gain region, another effect contributes to this. The increase of the postshock pressure which is necessary to accelerate the standing shock to a positive velocity $U_{\rm {s}} \ll \vert v_{\rm {p}}\vert$ is given by

 

$$\Delta P_{\rm {s}} P_{\rm {s}}(U_{\rm {s}}) - P_{\rm {s}}(0)$$ =

$$\cong -2\left( \! 1\! -\! {1\over \beta} \right) \rho_{\rm {p}}v_{\rm {... ...\! {1\over \beta} \right) {\dot M U_{\rm {s}}\over 2\pi\, R_{\rm {s}}^2}\ \cdot$$ (125)

This pressure increase is lower when the shock radius $R_{\rm {s}}$ is big.

The O $_{\rm {M}}$ line defines a critical curve for the shock evolution, whose slope and position are hardly dependent on the shock radius. It can be approximated analytically by solving Eq. (122) in case of $\dot M' = \dot M$ for the critical core luminosity $L_{\nu}^\ast$ as a function of $\dot M$. One derives

$$L_{\nu}^\ast(\dot M)\,\approx \, {{\rm e}^{-a}b(1 - 0.1\omega... ...l_{\nu}+q_{\rm {d}}) \over 1 - {\rm {e}}^{-a}(1-0.1\omega)}\ ,$$ (126)

where $l_{\nu}$ is defined as the quantity $l = (e+P)/\rho - G\widetilde{M}/r$at the neutrinosphere (cf. Eqs. (93) and (95)), a and b were introduced in Eqs. (74) and (75), respectively, and $\omega$ is given by

$$\omega\,\equiv\,{(kT_{\nu_{\rm e},4})^2\over \langle \mu_{\nu... ...}\over R_{\rm {g}}} \right)^{\!\! 2} \!\!-\! 1 \right\rbrack .$$ (127)

To obtain Eq. (126) use was made of Eqs. (66), (76), (77), (83), (87), (88), (93), (94), and (99). The term in the curly brackets of Eq. (87) was assumed to be equal to ${2\over 3}$, and the rather weak dependence of the gain radius in Eq. (127) on the neutrino luminosity was ignored in writing $L_{\nu}^\ast$ in the explicit form of Eq. (126). In the latter equation $\omega$ also depends on the mass infall rate $\dot M$. For representative shock radii and accretion rates, $\omega$ is found to be of order unity: $\omega \sim 1$. With this, $L_{\nu}^\ast$ becomes a simple linear function of $\dot M$. Inserting the parameter values used for Figs. 4 and 5 (listed after Eq. (124)), one ends up with

$$L_{\nu}^\ast(\dot M)\,\approx \, \left( 5.6 - 3.3\,{(-\dot M)... ...m {s}}} \right)~10^{52} \,\,{{\rm {erg}}\over{\rm {s}}}\ \cdot$$ (128)

This expression has a root for $\dot M = -1.7$  $M_{\odot}\,{\rm {s}}^{-1}$ and fits the O $_{\rm {M}}$ lines in Figs. 4 and 5 reasonably well.

Figure 6: Shock radius (left) and specific energy in the gain layer (right) as functions of time for different neutrinospheric luminosities (measured in units of 1052 erg$\,$s-1) and for an initial shock stagnation radius of 150 km [with $U_{\rm {s}}(t = 0) = 0$]. The structural polytropic index of the gain layer was chosen to be $\gamma = 4/3$, the mass accretion rate into the shock $\vert\dot M\vert = 0.3$ $M_{\odot }\,$s-1, the neutron star mass $\widetilde{M} = 1.25\,$$M_{\odot }$, and the neutrinospheric radius and temperature $R_{\nu } = 50\,$km and $T_{\nu } = 4\,$MeV, respectively. The gain radius and the integrals for neutrino heating and cooling in the gain layer were evaluated by using the exact solution for hydrostatic equilibrium. The disappearance of free nucleons and therefore the quenching of neutrino absorption and emission below a temperature of 1 MeV were also taken into account. The dashed lines correspond to the case where the density jump in the shock was set to be large (see text)

   
9.4 Time-dependent solutions

The equations of the toy model developed in this paper can be solved for the shock radius $R_{\rm {s}}(t)$ and the shock velocity $U_{\rm {s}}(t)$ as functions of time. For this purpose the mass and energy in the gain layer have to be evolved according to the conservation laws of Eqs. (102) and (109). Together with $U_{\rm {s}} = {\rm {d}}R_{\rm {s}}/{\rm {d}}t \equiv \Delta R_{\rm {s}}/\Delta t$ these equations were integrated implicitly in time, with the velocity of the gain radius given by $\dot R_{\rm {g}} \equiv \Delta R_{\rm {g}}/\Delta t$for time step $\Delta t$. The mass in the gain layer, $\Delta M_{\rm {g}}$, and the corresponding total (internal plus gravitational) energy, $\Delta E_{\rm {g}}$, were initially calculated from Eq. (97) and Eqs. (104) and (105), respectively. The gain radius $R_{\rm {g}}$ and the heating and cooling integrals for the gain layer were evaluated using the exact solution of hydrostatic equilibrium (Eqs. (59)-(61)) with the option to chose an arbitrary value for the structural polytropic index $\gamma$(Eq. (62)). The quenching of neutrino absorption and emission reactions by the recombination of free nucleons to $\alpha$ particles and heavy nuclei below a temperature around 1 MeV was taken into account.

The postshock density is related to the preshock density by $\rho_{\rm {s}} = \beta\rho_{\rm {p}}$, and the postshock pressure is given by Eq. (39). The density contrast $\beta$ as well as the pressure jump at the shock are affected by the conditions in the gain layer. The latter is assumed to be in hydrostatic equilibrium with mass inflow from the infall region and additional gain or loss by mass exchange with the neutron star. Therefore simultaneous conservation of mass and energy requires that $\beta$ is allowed to float, just as $P_{\rm {s}}$is a degree of freedom which adjusts in response to the energy input due to the heating by neutrinos. This means that $\beta$ is also considered as a variable which the set of equations is solved for.

Although generalization is straightforward, the neutron star mass ${\widetilde M}$, the mass accretion rate into the shock, $\dot M$, and the neutrinospheric parameters ($L_\nu$, $T_{\nu}$, $R_{\nu }$) were kept constant with time for reasons of simplicity. Supernova calculations show that during a transient, but rather short period of several 10 ms up to about 100 ms after bounce, $L_\nu$ and $\dot M$ decrease from very high values to a much lower level, and lateron change only slowly with time (cf., for example, Fig. 2 in Rampp & Janka 2000). A discussion of the subsequent destiny of the supernova shock should not be affected by this variation, because the shock expansion turns out to occur on a significantly shorter timescale (see below). The ongoing contraction of the neutron star and a corresponding change of the neutrinospheric temperature and luminosity, however, were found to have considerable influence (see Janka & Müller 1996). For the exemplary purpose of the calculations reported on below, the introduction of additional, model-dependent degrees of freedom will nevertheless be abstained from.

9.4.1 Results for different $\mathsfsl{L_{\nu}}$ and fixed $\dot M$

The shock radius $R_{\rm {s}}(t)$ and the specific energy in the gain layer, $\Delta E_{\rm {g}}/\Delta M_{\rm {g}}$, are shown as functions of time in Fig. 6 for different neutrinospheric luminosities $L_\nu$. The structural polytropic exponent $\gamma$ was set equal to the adiabatic index $\Gamma$ of the equation of state, both chosen to be ${4\over 3}$. The other parameters of the evaluation were $\widetilde{M}= 1.25$ $M_{\odot }$, $R_{\nu} = 50$ km, $T_{\nu} = 4$ MeV, and $\vert\dot M\vert = 0.3$  $M_{\odot }\,$s^-1. The initial shock radius was set to 150 km with $U_{\rm {s}}(t = 0) = 0$ and $\dot R_{\rm {g}}(t = 0) = 0$.

The solid lines display the case where $\beta$ was allowed to vary in all equations. Only for sufficiently high neutrinospheric luminosity the shock is able to expand to large radii. For lower $L_\nu$ the specific energy per nucleon in the gain layer begins to drop again at some stage of the evolution, and continued shock expansion is not possible, because the postshock pressure is not large enough for driving the shock out. The sudden positive acceleration of the shock front towards the end of the solid lines for these unsuccessful cases is a mathematical artifact, which occurs in response to the rapid decrease of the factor $(1-\beta^{-1})$ in Eq. (39) as $\beta$ falls to unphysical values near unity. For a given value of $P_{\rm {s}}$, the decay of the term $(1-\beta^{-1}) \to 0$ is attempted to be compensated by a catastrophic increase of the factor $(v_{\rm {p}}-U_{\rm {s}})^2$in Eq. (39). In contrast to the solid lines, the dashed curves were obtained by explicity setting $(1-\beta^{-1}) \approx 1$ in Eq. (39). Thus assuming that the density jump in the shock is large, the shock velocity is solely determined by the value of the pressure $P_{\rm {s}}$ behind the shock. Therefore the dashed lines show the breakdown of the shock expansion more clearly than the solid lines. They confirm that shock recession is correlated with a decrease of the specific energy in the gain layer.

Figure 7: Same as Fig. 6, but for a structural polytropic index $\gamma = 1.45$ in the gain layer

The properties of the time-dependent solutions for the shock radius agree with the discussion of Sects. 9.1-9.3. Keeping $\dot M$ fixed, there is a threshold value for the core luminosity above which the shock runs out to large radii and the energy per baryon in the gain layer becomes positive. A high neutrinospheric luminosity has two favorable effects: On the one hand the neutrino heating in the gain layer is larger, on the other hand the energy loss by neutrino emission in the cooling layer is lower, thus reducing the mass accretion into the neutron star and the mass loss from the gain layer. The case with $L_{\nu} = 4~10^{52}$ erg$\,$s^-1 is near the borderline between successful shock expansion and failure for the chosen set of parameters (compare also Fig. 5): The shock is already very weak when it has reached a radius of about 400 km, which is clearly visible from the dashed lines.

The mass accretion rates, $\dot M'$, of the nascent neutron star, which correspond to increasing values of the neutrinospheric luminosity $L_\nu$ in Fig. 6, are all negative, with values: $\vert\dot M'\vert = 1.15,\,0.84,\,0.68,\,0.53,$ and 0.21, and the $\nu_{\rm e}$plus $\bar\nu_{\rm e}$ luminosities at the gain radius for these cases are: $L_{\nu}(R_{\rm {g}}) = 5.0,\,5.3,\,5.5,\,5.7,$ and 6.0 10⁵² erg$\,$s^-1. Because of the contribution from the accretion luminosity, $L_{\nu}(R_{\rm {g}})$ shows much less variation than the core luminosity $L_\nu$, and the neutrino heating in the gain layer is also similar. The accretion component is not dominant when the shock moves out. The breakdown of shock expansion is therefore associated with a low neutrinospheric luminosity which causes high mass loss from the gain layer, leading to a decrease of the pressure support behind the shock. At the same time the width of the gain layer shrinks, its optical depth drops, and the neutrino energy deposition decreases. This leads to a negative feedback and the shock recession accelerates dramatically.

The optical depth for $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ absorption in the gain layer is given by Eq. (89), $\tau_{\rm {a}} \equiv {\cal H}/L_{\nu}(R_{\rm {g}})$. Its value depends on the particular conditions, the shock position, mass infall rate into the shock, and the neutrinospheric luminosity, temperature and radius, which influence the position of the gain radius. For the models shown in Figs. 6 and 7 the initial value is between 0.15 and 0.18. In case of shock expansion the optical depth increases for an intermediate period of time by up to 50 per cent due to the growth of the gain region. This improves the conditions for ongoing neutrino heating and leads to a rapid rise of the energy in the gain layer. The positive feedback also causes a sharp bifurcation in the behavior of cases of failing and successful shock expansion. Because neutrinos are not only absorbed, but also reemitted, the net effect of neutrino energy deposition is lowered somewhat. It scales with $({\cal H}-{\cal C})/L_{\nu}(R_{\rm {g}})$, which has typically only about half the value of $\tau_{\rm {a}}$. When the gain layer expands, the temperature decreases and the reemission of neutrinos is reduced.

9.4.2 Results for different $\mathsfsl{\dot M}$

When $\vert\dot M\vert$ lies below the O $_{\rm {M}}$ line of Figs. 4 and 5, the shock expansion is suppressed. But also high mass infall rates damp the shock expansion, because a larger increase of the postshock pressure is needed for shock acceleration (see Eq. (125)) and the optical depth of the gain layer decreases (because the gain radius is farther out). For high $\vert\dot M\vert$ the shock therefore gains speed more slowly.

In case of very high mass infall rates $\vert\dot M\vert$ and small shock radii a gain layer does not exist. Provided the neutrino luminosity is sufficiently large such that $\dot M' > \dot M$ (which is easily fulfilled for high $\vert\dot M\vert$), the shock is slowly pushed outward by the gas that stays in the layer between the neutron star and the shock. Eventually the postshock temperature will be low enough for a gain layer to form. With neutrino-heated gas accumulating above the gain radius (i.e., $\Delta M_{\rm {g}}$ increases) the shock moves even farther out, but the total energy in the gain layer decreases because the neutrino heating cannot compensate the negative binding energy of the growing gas mass. Only when the shock has reached a sufficiently large radius the situation becomes favorable for an explosion because then ${\rm {d}}(\Delta E_{\rm {g}})/{\rm {d}}t > 0$(i.e., the conditions are now left of the corresponding O $_{\rm {E}}$ line in Fig. 5). If this radius is very far out, because $\vert\dot M\vert$ is very high, the energy deposited in the gain layer may not be sufficient to produce a positive total energy in the gain layer. The gas behind the shock will stay bound and an explosion is not possible. The critical accretion rate for this to happen depends on the neutrinospheric parameters ($R_{\nu }$, $T_{\nu}$ and $L_\nu$), to some degree also on the structural polytropic index $\gamma$. For the parameters and neutrino luminosities considered in the present discussion this value is found to be around 4  $M_{\odot }\,$s^-1.

Even for somewhat smaller absolute values of the accretion rate and a positive total energy in the gain layer explosions might not occur. The question, however, whether an outward running supernova shock will reach the stellar surface and what amount of matter it is able to eject, requires a global treatment of the problem, including the possible energy release by nuclear burning and recombination of nucleons, and including the energy which will be spent on lifting the stellar mantle and envelope in the gravitational field of the star. This is far beyond the limits of the current treatment, which focusses on a discussion of the conditions that are necessary for reviving a stalled shock and for pushing it out to a radius of 1000 km by the neutrino heating mechanism.

9.4.3 Thermodynamic conditions

The entropy per nucleon in the gain layer, where relativistic electrons, positrons and photons as well as nonrelativistic nucleons and nuclei contribute to the pressure, is given by

$$s\, =\, {\varepsilon + P\over kT\,(\rho/m_{\rm {u}})} - \sum_i \eta_i Y_i \, ,$$ (129)

where the sum runs over all kinds of particles i. When nuclei are fully dissociated, in which case $Y_{\rm p} = Y_{\rm e}$ and $Y_{\rm n} = 1-Y_{\rm e}$, this sum can be written as

 

$$\sum_i \eta_i Y_i Y_{\rm e}(\eta_{\rm e} + \eta_{\rm p} - \eta_{\rm n}) + \eta_{\rm n}$$ =

$$\approx Y_{\rm e}\eta_{\rm e} + Y_{\rm e}\ln\!\left( {Y_{\rm e}\over 1-Y_{\rm e}} \right)$$

$$\phantom{\approx \eta_{\rm e}} + \ln\!\left( 1.27~10^{-3}\,{(1-Y_{\rm e})\rho_9\over (kT)^{3/2}} \right) \,\cdot$$ (130)

Since typically $Y_{\rm e}\sim 0.2$-0.3 and around the gain radius, the combined terms scaling with $Y_{\rm e}$ are negligibly small. Using $P = (\Gamma-1)\varepsilon$, Eq. (129) can be evaluated for the models plotted in Figs. 6 and 7. Near the gain radius characteristic initial values of the entropy per nucleon are found to be between 10 and 14, with a contribution from relativistic degrees of freedom of $s_{\rm {r}}\sim 2$-3. This is in good agreement with results from detailed hydrodynamical models (see, e.g., Rampp & Janka 2000). In case of successful shock expansion, the total entropy per nucleon increases to values between 25 and 30 towards the end of the computed evolution.

Clearly, neither the entropy nor the pressure are dominated by radiation and leptons, but baryons play an important role, at least at the beginning of shock expansion. Nevertheless, the description in Sect. 5 of the gain layer as being a "radiation-dominated'' region remains justified, although in a generalized sense. While in the region around the neutrinosphere baryons (and possibly degenerate electrons) yield the major contribution to the pressure and internal energy, the importance of electron-positron pairs and photons increases at lower densities. In Sect. 5.2 (Eqs. (56)-(58)) it was argued that for $r > R_{\rm {eos}}$ both the pressure contributions from relativistic and non-relativistic particles can be written as $P \propto T^4$, provided that the electron fraction $Y_{\rm e}$ and the electron degeneracy parameter $\eta_{\rm e}$ do not vary strongly. In the gain layer this is fulfilled, because the electron degeneracy is typically small, i.e., , and electron-positron pairs are abundant (see the detailed discussion by Bethe 1993, 1996b). Indeed, the hydrodynamical simulation of Rampp & Janka (2000) shows that $\eta_{\rm e}$ in the gain layer changes only between about 1.5 and 3 during the interesting phase of the post-bounce evolution.

Despite of the considerable contribution to the pressure which is provided by nonrelativistic baryons, also the use of $\Gamma = (\partial \ln P/\partial \ln\rho)_{\rm s} = {4\over 3}$ for the adiabatic index of the equation of state in the postshock region is justified, although the calculations in this paper are not constrained to this specific choice. At the conditions present between the gain radius and the shock (density between a few 10⁸ g$\,$cm^-3 and several 10⁹ g$\,$cm^-3 and temperatures between roughly 1 MeV and 2 MeV), a finite mass fraction of $\alpha$ particles is still present ( ). The disintegration of these $\alpha$'s at nuclear statistical equilibrium around 1-1.5 MeV and the growing importance of ${\rm e^+e^-}$ pairs and photons for higher temperatures produce $\Gamma$ values between 1.3 and 1.4. This can, for example, be verified by an inspection of the equation of state of Lattimer & Swesty (1991).

9.4.4 Steady-state conditions

Steady-state conditions are realized when $\dot R_{\rm {s}} = \dot R_{\rm {g}} = 0$, which in general requires that ${\rm {d}}(\Delta M_{\rm {g}})/{\rm {d}}t = {\rm {d}}(\Delta E_{\rm {g}})/{\rm {d}}t = 0$. From Eq. (122) one gets $\dot M' = \dot M$, which yields

$$L_{\rm {acc}}\,=\,\dot M\,(l_{\nu} + q_{\rm {d}}) + {\cal H}-{\cal C} \ ,$$ (131)

when Eqs. (93) and (94) and the definition $l = (e+P)/\rho - G\widetilde{M}/r$ are used (note that ${1\over 2}\rho v^2 \ll \varepsilon$ at the neutrinosphere). Neutrino heating and cooling in the gain layer scale with $L_{\nu}(R_{\rm {g}})$ and thus $L_{\nu}(R_{\nu})$ (Eq. (76)), and depend on $R_{\rm {s}}$, directly as well as via $R_{\rm {g}}$. Also $L_{\rm {acc}}$ depends on the neutrinospheric luminosity $L_\nu$ (Eq. (77)). Equation (121), on the other hand, yields

$$\dot M\,=\,{{\cal H}-{\cal C} \over l_{\rm {s}}-l_{\rm {g}}}$$ (132)

for $l_{\rm {s}} < l_{\rm {g}}$, which is fulfilled if $\Gamma < \gamma$. [ $\Gamma = \gamma$ requires ${\cal H}-{\cal C} = 0$ and therefore $R_{\rm {g}} = {\rm {min}}(R_{\rm {s}}, R_{\alpha})$.] From the combined Eqs. (131) and (132) the values of the shock radius $R_{\rm {s}}$ and the neutrinospheric luminosity $L_\nu$ can be determined which correspond to steady-state conditions for a given value of $\dot M$. The solution for $L_\nu$ lies on the critical O $_{\rm {M}}$line displayed in Figs. 4 and 5, where $\dot M'$ and $\dot M$ are equal. The core luminosity has to satisfy this constraint, because the neutrinospheric conditions and the temperature in the cooling layer above the neutrinosphere are assumed to be regulated by the interaction with the high neutrino fluxes from the hot, neutrino-opaque neutron star. This inner boundary condition differs from the one used for discussing ordinary steady-state accretion onto neutron stars (Chevalier 1989). There the temperature of the optically thin medium at the base of the atmosphere is assumed to adopt a value which ensures that photon or neutrino losses carry away the binding energy of the matter which is accreted at a given constant rate. This requirement then yields a condition for calculating the steady-state position of the accretion shock.

   
9.5 Convective energy transport

The simplified analytic model described in this paper can certainly not account for the detailed effects associated with convective overturn in the neutrino-heated layer between gain radius and shock. This overturn is an intrinsically multi-dimensional phenomenon where low-entropy downflows and hot, rising bubbles of neutrino-heated gas coexist in the same region of the star. Therefore the mixing achieved by the gas motions is not complete even on a macroscopic scale. Nevertheless, some consequences and fundamental effects associated with the existence of convective energy transport in the gain region can be figured out.

The described analytic model distinguishes between the adiabatic index $\Gamma$ of the equation of state in the gain layer, and the structural polytropic index $\gamma$. The gain layer is subject to non-adiabatic changes, because energy deposition by neutrino heating takes place. Also, the gain layer is not necessarily isentropic. Using the developed framework of equations with $P = K\rho^\gamma$, $\gamma = \Gamma = {\rm {const.}}$, and $K = {\rm {const.}}$ for the equation of state in the gain region - which is the default setting for the analyses in Sect. 9 -, Eq. (112) yields the same value for the total specific energies at $R_{\rm {g}}$ and $R_{\rm {s}}$. For a gas with adiabatic index $\Gamma$ this also means that isentropic conditions are realized. This, therefore, corresponds to the case where convection is very efficient in carrying energy from the gain radius, close to which neutrino heating is strongest, to directly behind the shock. Chosing instead $\gamma > \Gamma$ yields $l_{\rm {g}} > l_{\rm {s}}$, a result which is more characteristic of the situation without convection. Here the energy deposition by neutrinos establishes negative gradients of the entropy and specific energy between gain radius and shock.

Repeating the derivations of Sects. 5-9 with $\gamma > {4\over 3}$, reveals, on the one hand, that the gain radius $R_{\rm {g}}$ is smaller and therefore the optical depth and the net heating in the gain region, ${\cal H}-{\cal C}$, are somewhat larger than for the "standard'' case of $\gamma = {4\over 3}$ (because the hydrostatic density and temperature profiles are flatter behind the shock). On the other hand, however, a less efficient energy transport from the gain radius to the shock has a severe disadvantage: The gas which is advected inward through $R_{\rm {g}}$, carries away a large fraction of the energy absorbed from neutrinos before. In Eq. (121) the term $\dot M' l_{\rm {g}}$ yields a smaller positive or even a negative contribution when $\dot M' < 0$ and $l_{\rm {g}}$ is negative or positive, respectively. This reduces the net effect of neutrino heating and is harmful for shock expansion and acceleration.

A comparison of Figs. 6 and 7 demonstrates the differences. The time-dependent solutions for shock radius and specific energy in the gain layer were obtained with $\Gamma = {4\over 3}$ in both cases, but in Fig. 7 $\gamma = 1.45$ was chosen instead of $\gamma = {4\over 3}$. For $\gamma > \Gamma$ the shock expansion is weaker and the specific energy in the gain layer stays lower. The effect is particularly obvious for the core neutrino luminosity of $L_{\nu} = 4~10^{52}$ erg$\,$s^-1. Figure 6 shows a marginal success for this case, whereas in Fig. 7 the shock expansion fails.

These findings are confirmed by an inspection of the spherically symmetric simulation of the collapse and post-bounce evolution of a 15 $M_{\odot }$ progenitor star published recently by Rampp & Janka (2000). After an expansion to more than 350 km, the shock in this model finally recedes to a much smaller radius and fails to produce an explosion. The shock recession is caused (or accompanied) by a rapid decrease of the mass in the gain region, because more matter is flowing through the gain radius than is resupplied by accretion through the shock. In the hydrodynamical simulation one finds that $\Delta E_{\rm {g}}$ also decreases during this phase, an effect which should not occur if $l_{\rm {g}} = l_{\rm {s}} < 0$ (compare Eq. (121)).

This discussion emphasizes the importance of convective energy transport between the gain radius and the shock. Postshock convection reduces the mass loss as well as the energy loss from the gain region, which are associated with the continuous inward advection of neutrino-heated gas during the phase of shock revival. Also an increase of the core luminosity can diminish the accretion of gas into the neutron star by suppressing the net neutrino emission from the cooling layer. Both effects have been demonstrated in numerical simulations to be helpful for an explosion.

   
9.6 Limits of the toy model

Employing idealized and in many respects simplifying assumptions, a toy model was developed here on grounds of an approximate solution of the hydrodynamic equations, Eqs. (2)-(4). By reducing the complexity, hopefully without sacrificing fundamental properties, the model is intended to help discussing the principles and the essence of the neutrino-driven mechanism. It is, however, not meant to compete with detailed hydrodynamical simulations, where usually a lot more refinements concerning the description of the stellar fluid and of the neutrino transport are included. The stellar structure outside of the forming neutron star at the center is considered to consist of three layers: The cooling layer, from where neutrino loss extracts energy, is bounded by the neutrinosphere on the one side and the gain radius on the other; the heating layer extends between gain radius and shock and receives net energy deposition by neutrinos; in the infall region exterior to the shock, matter of the progenitor star moves inward at a significant fraction of the free-fall velocity. The evolution of the shock depends on the conditions in the gain layer. Assuming the radial structure of this layer to be given by hydrostatic equilibrium one can discuss the behavior of the shock in response to integral properties of the heating region. The total mass and energy of the gain layer change due to inflow and outflow of gas, neutrino heating, and a possible shift of the boundaries, and are therefore sensitive to the rate of mass accretion by the shock on the one hand, and the irradiation by the neutrinospheric luminosities on the other.

9.6.1 Temperature in the cooling layer

The present work concentrates on a discussion of the phase of shock revival due to neutrino energy deposition, where the infall of the stellar gas is inverted to outflow. This implies that the gas flow through the gain layer cannot be stationary, i.e., the mass infall rate into the shock, $\dot M$, is in general different from the mass accretion rate $\dot M'$by the nascent neutron star. Here $\dot M'$ in dependence of the physical conditions near the neutron star surface is estimated by assuming that the temperature at and just outside the neutrinosphere is governed by the interaction of the stellar medium with the neutrinos streaming out from deeper layers. This temperature determines the energy loss from the cooling layer around the neutron star, the efficiency of which then drives the mass exchange (inflow or outflow) with the gain layer farther out.

This picture is certainly a simplification of the real situation. For example, when outward motion of the postshock gas sets in, the advection of gas through the gain radius may be quenched as found in spherically symmetric simulations. In the described model this effect could be reproduced if the temperature in the cooling layer would drop. The current set of equations, however, does not allow one to calculate this effect because it does not include how the temperature in the cooling layer depends on the expansion or compression of the neutron star atmosphere. On the other hand, in the three-dimensional situation downflows and rising bubbles can coexist when convective overturn is present in the gain layer, as suggested by two-dimensional hydrodynamical simulations (Herant et al. 1994; Burrows et al. 1995; Janka & Müller 1996). In this case accretion does not need to stop even when the shock is accelerated outward (Bethe 1993, 1995; however, Burrows et al. 1995 see a decrease of the neutrino luminosity associated with a reduced accretion rate when the explosion sets in). Convective overturn and thus accretion might in fact continue until the shock has reached a radius above 1000 km (Bethe 1997). It is not easy to estimate the fraction of the gas which stays in the gain layer relative to the part which is advected inward to the neutron star. The ansatz described here may be considered as a crude attempt to do so.

9.6.2 Hydrostatic equilibrium

The stellar structure in the gain layer was calculated as a solution of the equation of hydrostatic equilibrium. The latter is derived from Eq. (3) combined with Eq. (2). When the velocity-dependent terms can be neglected, this yields

$${\partial v\over \partial t} + v\,{\partial v\over \partial r... ...over \partial r} - {\partial \Phi\over \partial r} \,=\, 0 \,.$$ (133)

Hydrostatic equilibrium holds in the regime where the fluid velocity vis much smaller than the local sound speed $c_{\rm {s}}$. This is well fulfilled when the postshock gas settles inward, and is also well fulfilled when it starts moving outward behind an accelerating shock. Of course, omitting the $v(\partial v/\partial r) = \partial(v^2/2)/\partial r$ term from Eq. (133) implies that the discussed toy model will not reproduce the solutions for a stationary wind, which has a critical point where $v = c_{\rm {s}}$. This is not a handicap during the shock revival phase and the onset of shock expansion, because the gas behind the (slowly propagating) shock front moves (relative to the shock) with subsonic velocities. It means, however, that the treatment is not sufficiently general to follow the explosion to large radii and high shock velocities, which is the evolutionary phase when the expanding medium around the nascent neutron star forms a neutrino-driven wind. When the shock is far out and the gain layer has expanded by a large amount, the assumption of hydrostatic equilibrium therefore becomes inadequate and the corresponding structure of the gain layer is not sufficiently accurate any longer. Unless the gas velocity approaches $c_{\rm {s}}$, the modification of the stellar density and pressure profiles due to velocity effects is small. In fact, when the time-dependent calculations of Sects. 9.4 and 9.5 were terminated at t = 0.1 seconds, the specific kinetic energy of the gas immediately behind the shock was only a minor fraction of the specific internal energy, at most about 20%. The integral kinetic energy of the gain layer was even smaller compared to the total internal energy.

Considering hydrostatic conditions means that the fluid velocity is assumed not to be relevant for the structure of the neutron star atmosphere. In fact, an accretion or outflow velocity field in the gain layer was not derived (and was not of direct relevance for the discussion), although the toy model employs the mass accretion rates $\dot M$ and $\dot M'$. The fluid velocity immediately above the shock is given by the infall velocity of the gas, and at the gain radius it must be equal to $v_{\rm {g}} = \dot M'/(4\pi R_{\rm {g}}^2\rho_{\rm {g}})$. Different rates for the mass accretion into the shock and the mass flow through the gain radius imply that the mass of the gain layer can grow or drop due to active mass inflow or outflow (in addition to the motion of the shock and of the gain radius as the outer and inner boundaries, respectively, of the considered stellar shell). Therefore the conditions in the gain layer are non-stationary, i.e., $\partial\rho/\partial t = 0$ cannot be true in general.

Deriving a velocity profile for the gain layer requires solving Eq. (2) with non-vanishing time derivative of the density and the lower boundary condition being given by $v(R_{\rm {g}}) = v_{\rm {g}}$. Doing so, the velocity jump at the shock is also influenced by the physical conditions around the gain radius. This is analogue to the postshock density, which is sensitive to the mass accumulated in the gain layer, and is similar to the postshock pressure, which varies with the integral value of the energy deposited by neutrinos in the gain layer. Mass and energy loss or gain thus affect the whole heating layer simultaneously. Assuming hydrostatic equilibrium implies that the physical state of the gas behind the shock front is coupled to the conditions at the gain radius, because the sound crossing timescale is considered to be small compared with all other relevant timescales of the problem. Therefore the Rankine-Hugoniot relations for the density jump and the velocity jump at the shock front cannot be satisfied exactly, which reflects the approximative nature of the hydrostatic structure. The violation of the Rankine-Hugoniot conditions (for specific values of the EoS parameters), however, is usually small and the overall properties of the calculated solutions should be close to the true ones, in particular at some distance behind the shock front.

9.6.3 Equation of state and convection

The equation of hydrostatic equilibrium in the gain layer was solved assuming that gas pressure, density and temperature are related by $P = K\rho^\gamma = A(kT)^4$ (Eq. 56)) with $\gamma$, K and A being constants. As discussed in Sects. 5.2 and 9.4, this is reasonably well fulfilled in the gain layer, where the stellar gas consists of a mixture of relativistic electrons, positrons and photons and nonrelativistic nucleons and nuclei. On the one hand the electron degeneracy is low ( ) and both $\eta_{\rm e}$ and the electron fraction $Y_{\rm e}$ do not vary strongly (the variation of $Y_{\rm e}$is at most a factor of about two, roughly between 0.25 at the gain radius and 0.5 at the shock). For these reasons fermion captures on nucleons were ignored concerning their effect on the $Y_{\rm e}$ profile in the gain layer. On the other hand, heavy nuclei are mostly disintegrated into free nucleons with some admixture of $\alpha$ particles. Therefore also the sum of the nuclear number fractions $\sum_iY_i$ is roughly constant; in fact, $\sum_i Y_i \approx 1$ is a good approximation. Because there is a finite mass fraction of $\alpha$ particles, , nuclear statistical equilibrium yields an adiabatic index of $\Gamma = 1+P/\varepsilon\approx {4\over 3}$ for the equation of state at typical postshock conditions, although nucleons are responsible for a large fraction of the pressure. None of these assumptions, however, is rigorously fulfilled in the medium of the gain layer at all radii and at all times. But making use of these assumptions simplifies the discussion considerably because the equation of state can be treated analytically. Since the overall properties of the stellar gas are accounted for, it is very unlikely that the conclusions on the qualitative level of this paper are affected when more refinements and complications are added. This might change details, but should not modify the essence of the toy model.

The energy input to the gain layer by neutrino heating is accounted for in the model. This energy gain from neutrinos means that the changes in a fluid element are non-adiabatic. Therefore the structural polytropic index $\gamma$ can be different from the adiabatic index $\Gamma$of the EoS. Varying $\gamma$ allows one to mimic additional processes which might affect the evolution and behavior of the gain layer. Chosing $\gamma = \Gamma = {4\over 3}$ implies that the gain layer is considered to be isentropic, i.e., the energy deposited by neutrinos is assumed to be efficiently (and instantaneously) redistributed such that the entropy is roughly equal everywhere and $l_{\rm {s}} = l_{\rm {g}}$ holds (Eq. (112)). Since neutrino heating is strongest near the gain radius, this means that energy has to be transported from smaller radii to positions closer to the shock. Such an effect is realized by the strong postshock convection seen in multi-dimensional hydrodynamic simulations (e.g., Herant et al. 1994; Burrows et al. 1995; Janka & Müller 1996). Using $\gamma > \Gamma$ a situation is described where more of the deposited energy stays near the gain radius, corresponding to less efficient energy transport by convection. The toy model confirms that this has a negative influence on the possibility of shock expansion.

9.6.4 Neutrino processes

Processes different from $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ absorption and emission by nucleons were not taken into account for the neutrino heating and cooling above the neutrinosphere. Both neutrino-electron/positron scattering and neutrino-pair annihilation have much smaller reaction cross sections than the baryonic processes and are less efficient in transferring energy to the stellar medium. For this reason they do not play a crucial role for the explosion mechanism (Bethe & Wilson 1985; Bethe 1990, 1993, 1995, 1997; Cooperstein et al. 1987).

Effects due to muon and tau neutrinos and antineutrinos ($\nu_x$) were completely ignored in the discussions of this paper. Because muons and tau leptons cannot be produced in the low-density medium above the neutrinosphere, $\nu_{\mu}$ and $\nu_{\tau}$do not interact with nucleons via charged-current reactions and therefore couple to the gas less strongly than electron neutrinos and antineutrinos. Energy exchange by neutral-current scatterings off nucleons contributes in shaping their emission spectra near the neutrinosphere (Janka et al. 1996; Burrows et al. 2000) and might also be relevant for the heating in the gain layer. Although the recoil energy transfer per scattering is reduced by a factor $\epsilon/(mc^2)$ relative to the absorption of neutrinos with energy $\epsilon$(m is the nucleon mass), the cross sections of both processes are similar and all flavors of neutrinos and antineutrinos participate in the neutral-current reactions with neutrons as well as with protons. Using Eq. (10) for the nucleon scattering opacity and the mean energy exchange per reaction as given by Tubbs (1979), one can estimate the importance of nucleon scattering for the energy transfer to the medium relative to $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ absorption as:

 

$${Q_{\nu,{\rm {sc}}}^+\over Q_{\nu,{\rm {abs}}}^+} \sim {{1+3\alph... ..._{\nu_{\rm e}})^2} {kT_{\nu_x}-kT\over mc^2}\sim 30{kT_{\nu_x}-kT\over mc^2} \,$$ (134)

where $Q_{\nu,{\rm {abs}}}^+$ was taken from Eq. (28) and T is the gas temperature. $T_{\nu_{\rm e}}$ and $T_{\nu_x}$ are the spectral temperatures of electron neutrinos and heavy-lepton antineutrinos, respectively, for which $T_{\nu_{\rm e}}^2 \approx {1\over 2} T_{\bar\nu_{\rm e}}^2\approx {1\over 4}T_{\nu_x}^2$ was assumed again (compare Sect. 3). Moreover, the estimate was obtained by using $L_{\nu_{\rm e}}\approx L_{\bar\nu_{\rm e}}\approx L_{\nu_x}$ and $Y_{\rm n} + 2Y_{\rm p}\approx 1$. The spectral average of the third power of the neutrino energy, $\langle \epsilon_{\nu_x}^3 \rangle$, was defined in analogy to Eq. (25). Since the spectral temperatures and therefore the scattering cross sections of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ are smaller than those of $\nu_x$, electron neutrinos plus antineutrinos were given roughly the same weight as one of the heavy-lepton neutrinos. For typical values $kT_{\nu_x} \sim 8$ MeV and $kT\sim 2$ MeV one therefore derives a relative contribution of scattering processes to the neutrino heating in the gain layer of about 20%.

Apart from this moderate amplification of the heating, muon and tau neutrinos have other effects on the shock propagation during the post-bounce evolution of a supernova. Within the first tens of milliseconds after shock formation, muon and tau neutrino pairs are produced by ${\rm e^\pm}$annihilation in the heated matter immediately behind the shock. In addition to the disintegration of nuclei and the emission of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$, this extracts energy from the shock-heated layers and weakens the prompt bounce shock. Somewhat later, between several ten milliseconds and a few hundred milliseconds after bounce, most of the muon and tau neutrinos come from the hot mantle layer of accreted material below the neutrinosphere of the forming neutron star. Since $\nu_{\mu}$ and $\nu_{\tau}$ pairs now carry away energy which otherwise would be radiated in electron neutrinos and antineutrinos and would thus be more efficient for the heating behind the shock, this will have a negative effect on the possibility of shock rejuvenation. During the following phase of the evolution, when the deleptonization of the neutron star advances to deeper layers and the neutron star enters the Kelvin-Helmholtz cooling stage (Burrows & Lattimer 1986), muon and tau neutrinos are mostly produced at higher densities. Being less strongly coupled to the nuclear medium, they diffuse to the surface more rapidly than $\nu_{\rm e}$ and $\bar\nu_{\rm e}$. This helps keeping the neutrinospheric layer hot, where electron neutrinos and antineutrinos take over a larger part of the energy transport. During this late phase of the evolution, $\nu_{\mu}$ and $\nu_{\tau}$ might thus even support higher $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ fluxes.