8 Mass and energy conservation in the gain region

Mass and energy conservation in the gain region between $R_{\rm {g}}$ and $R_{\rm {s}}$ determine the early postbounce evolution of the supernova shock. For example, the shock is pushed outward when the matter that falls through the shock stays hot and piles up on top of the neutron star, forming an extended envelope instead of being accreted into the dense core quickly after efficient energy loss in the neutrino cooling layer below $R_{\rm {g}}$. Similarly, strong neutrino heating in the gain region causes an increase of the postshock pressure and thus drives an expansion of the shock. On the other hand, enhanced neutrino emission will extract mass and/or energy from the layer which supports the supernova shock. The consequence will be a retraction of the shock in radius. These effects need to be accounted for by an appropriate discussion of the delayed explosion mechanism. A steady-state picture is certainly not adequate.

   
8.1 Mass in the gain region

The mass $\Delta M_{\rm {g}}$ in the gain region can be calculated as volume integral over the density:

$$\Delta M_{\rm {g}} \,=\,\int\limits_{R_{\rm {g}}}^{R_{\rm {s}}} {\rm {d}}r\,4\pi r^2\,\rho(r)\ .$$ (96)

with the density $\rho(r)$ given by Eq. (59). Alternatively, since the latter equation is the exact solution for hydrostatic equilibrium, one can use $\rho(r) = -r^2({\rm {d}}P/{\rm {d}}r)/(G\widetilde{M})$ with P(r)from Eq. (60). Defining the coefficients $c_1 \equiv \rho_{\rm {s}}^{1/3}-G\widetilde{M}/(4KR_{\rm {s}})$ and $c_2\equiv G\widetilde{M}/(4K)$, one finds:

 

$$\Delta M_{\rm {g}} 4\pi\, \Biggl\lbrack {1\over 3}\,c_1^3\left( R_{\rm {s}}^3-R_{\rm... ...\right) + c_2^3\ln\! \left( {R_{\rm {s}}\over R_{\rm {g}}} \right)\Biggr\rbrack$$ =

$$4\pi\, \Biggl\lbrack {1\over 3}\left( R_{\rm {s}}^3\rho_{\rm {s}}... ...+ c_2^3\ln\! \left( {R_{\rm {s}}\over R_{\rm {g}}} \right)\Biggr\rbrack \,\cdot$$ = (97)

In deriving the second form of Eq. (97), use was made of $\rho(r) = (c_1 + c_2/r)^3$. Moreover, with $\rho = (P/K)^{3/4}$ the density in Eq. (97) can be substituted by the pressure P. Note that the quantities $\rho_{\rm {g}} = \rho(R_{\rm {g}})$ and $P_{\rm {g}} = P(R_{\rm {g}})$ at the gain radius must be expressed by the exact relations of Eqs. (59) and (60), respectively. They depend on the postshock state of the matter as do the coefficients c₁ and c₂. The gain radius $R_{\rm {g}}$ is given by Eq. (66). It is also a function of the conditions immediately behind the shock. Writing the postshock temperature in terms of the postshock pressure via Eq. (56), $kT_{\rm {s}} = \left\lbrack 3 P_{\rm {s}}/(f_{\rm {r}}g_{\rm {r}}a_{\gamma}) \right\rbrack^{1/4}$, and using Eqs. (39), (43), (44) and (53)-(55) with typical values $\beta \sim 7$, $\alpha\sim 1/\sqrt{2}$, $s_{\gamma}\sim 4$, and $\eta_{\rm e}\sim 2$(the exact values of these parameters are not essential for the discussion and affect the result rather insensitively), one gets in case of $\vert U_{\rm {s}}\vert = \vert\dot R_{\rm {s}}\vert \ll \vert v_{\rm {p}}\vert$:

$$kT_{\rm {s}}\,\approx\,2 \,\, R_{{\rm {s}},7}^{-{5\over 8}} \... ...{\! {1\over 8}} \ \ \left\lbrack {\rm {MeV}} \right\rbrack\,.$$ (98)

Inserting this in Eq. (66) and using $L_{\nu} = 2L_{\nu_{\rm e}}$ yields

 

$$R_{{\rm {g}},7}\,\approx\,1.13\,\,R_{{\rm {s}},7}^{9\over 16} \,\... ... {3\over 8}} \left( {\widetilde{M}\over M_{\odot}} \right)^{\! {3\over 16}}\! ,$$ (99)

where the neutrino luminosity at the gain radius, $L_{\nu}(R_{\rm {g}})$, is given by Eq. (76).

Instead of the exact expression of Eq. (97) an approximation for $\Delta M_{\rm {g}}$ is sometimes preferable. Performing the integration of Eq. (96) with the approximate density profile of Eq. (63), one finds

$$\Delta M_{\rm {g}}\,\approx\,4\pi\,\rho_{\rm {s}} R_{\rm {s}}... ...{3/2} \ln\!\left( {R_{\rm {s}}\over R_{\rm {g}}} \right) \cdot$$ (100)

Here $\rho _{\rm {s}}$ was written in terms of $R_{\rm {s}}$ by making use of $\rho_{\rm {s}} = \beta\rho_{\rm {p}}$ and Eq. (44). Moreover, from Eq. (99) one can deduce that $R_{\rm {s}}/R_{\rm {g}}\propto R_{\rm {s}}^{7/16}$ for $\vert U_{\rm {s}}\vert\ll \vert v_{\rm {p}}\vert$. An increase of the shock radius therefore means that $\Delta M_{\rm {g}}$ will also grow.

The rate at which the mass in the gain region changes in time due to a shift of the upper and lower boundaries of this region but also due to a variation of the density of the stellar medium, is determined as the total time derivative of Eq. (96):

$${{\rm {d}}\over {\rm {d}}t}(\Delta M_{\rm {g}})\,=\, 4\pi R_{... ...m {s}}}\!{\rm {d}}r 4\pi r^2 {\partial \rho\over \partial t} ,$$ (101)

where $\dot R_{\rm {s}}\equiv {\rm {d}}R_{\rm {s}}/{\rm {d}}t = U_{\rm {s}}$ is the shock velocity and $\dot R_{\rm {g}}\equiv {\rm {d}}R_{\rm {g}}/{\rm {d}}t$the velocity of the gain radius. When the integration in Eq. (101) is carried out to a radius infinitesimally smaller than $R_{\rm {s}}$with the help of Eq. (2), one obtains

 

$${{\rm {d}}\over {\rm {d}}t}(\Delta M_{\rm {g}}) 4\pi R_{\rm {s}}^2\rho_{\rm {s}}(\dot R_{\rm {s}}-v_{\rm {s}}) - 4\pi R_{\rm {g}}^2\rho_{\rm {g}}(\dot R_{\rm {g}}-v_{\rm {g}})$$ =

$$4\pi R_{\rm {s}}^2\rho_{\rm {p}}\dot R_{\rm {s}} - 4\pi R_{\rm {g}}^2\rho_{\rm {g}}\dot R_{\rm {g}} - \dot M + \dot M'\ ,$$ = (102)

with $v_{\rm {g}}$ and $v_{\rm {s}}$ being the velocities of the stellar medium at the gain radius and just behind the shock, respectively. The lower expression was derived by using the shock jump condition for the mass flow, Eq. (36), and the definitions $\dot M = 4\pi R_{\rm {s}}^2\rho_{\rm {p}}v_{\rm {p}}$ and $\dot M' = 4\pi R_{\rm {g}}^2\rho_{\rm {g}}v_{\rm {g}} = 4\pi R_{\nu}^2\rho_{\nu}v_{\nu}$ as introduced in Sect. 7. Equation (102) shows that the mass in the gain region can change because of inflow and outflow of gas but also due to the motion of the boundaries $R_{\rm {g}}$ and $R_{\rm {s}}$. Knowing the initial mass in this layer, $\Delta M_{\rm {g}}^0$, Eq. (102) allows one to calculate the value at later times.

   
8.2 Energy in the gain region

Since the postshock matter is effectively in hydrostatic equilibrium (see Sect. 5) the kinetic energy is negligible compared to the internal energy and the gravitational potential energy, and the total energy in the gain region is therefore given by

$$\Delta E_{\rm {g}}\,=\, \int\limits_{R_{\rm {g}}}^{R_{\rm {s}... ...ilon(r)-{G\widetilde{M}\over r}\,\rho(r) \right\rbrack \ \cdot$$ (103)

To evaluate the right hand side, one substitutes $\varepsilon = P/(\Gamma -1)$, which relates the internal energy density $\varepsilon$ and the pressure Pfor an ideal gas, with the adiabatic index $\Gamma$ being typically between 4/3 and 5/3, depending on whether relativistic or nonrelativistic particles, respectively, dominate the pressure. In addition making use of hydrostatic equilibrium (Eq. (47)) or, alternatively, applying the virial theorem, one finds

 

$$\Delta E_{\rm {g}}\,=\, {4\pi \over 3(\Gamma -1)}\, (P_{\rm {s}}R... ...er 3(\Gamma-1)} \int\limits_{R_{\rm {g}}}^{R_{\rm {s}}}{\rm {d}}r\, r\rho(r)\ .$$ (104)

The second term is the gravitational potential energy times $(\Gamma-{4\over 3})/(\Gamma-1)$. An exact expression for the integral is obtained when Eq. (59) is used for $\rho(r)$:

 

$$\int\limits_{R_{\rm {g}}}^{R_{\rm {s}}} {\rm {d}}r\,r\rho(r)\,=\,... ...{g}}) + 3\,c_1c_2^2\, \ln \left( {R_{\rm {s}}\over R_{\rm {g}}} \right) \,\cdot$$ (105)

The coefficients c₁ and c₂ were already defined in Sect. 8.1. For the following discussion an approximation of this integral is sufficient. It can be derived by employing the approximate power law profile for the density, Eq. (63):

$$\int\limits_{R_{\rm {g}}}^{R_{\rm {s}}}\! {\rm {d}}r\,r \rho(... ..._{\rm {s}}^2\over R_{\rm {g}}}\, (R_{\rm {s}}-R_{\rm {g}}) \ .$$ (106)

The rate at which the total energy in the gain region changes with time can be calculated as the time derivative of Eq. (103). With the definition $l \equiv (\varepsilon + P)/\rho - G\widetilde{M}/r$one finds

 

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}}) 4\pi R_{\rm {s}}^2 \rho_{\rm {s}} l_{\rm {s}}\dot R_{\rm {s}} - 4\pi R_{\rm {g}}^2 \rho_{\rm {g}} l_{\rm {g}}\dot R_{\rm {g}}$$ =

$$- 4\pi R_{\rm {s}}^2 P_{\rm {s}} \dot R_{\rm {s}} + 4\pi R_{\rm {g}}^2 P_{\rm {g}} \dot R_{\rm {g}}$$

$$+ \int\limits_{R_{\rm {g}}}^{R_{\rm {s}}}{\rm {d}}r\, 4\pi r^2\le... ...rtial t} - {G\widetilde{M}\over r}\,{\partial \rho\over \partial t} \right) \,,$$ (107)

where $\dot R_{\rm {s}}$ and $\dot R_{\rm {g}}$ have the same meaning as in Eq. (101). The partial derivatives in the integral can be substituted by Eqs. (2) and (4). Making additional use of Eqs. (5) and (6) and of ${1\over 2}\rho v^2 \ll \varepsilon$ yields

 

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}}) = 4\pi R_{\rm {s}... ... \!\! \int\limits_{R_{\rm {g}}}^{R_{\rm {s}}}\!\! {\rm {d}}r 4\pi r^2 Q_{\nu} .$$ (108)

Now employing the continuity equation for the mass flow across the shock, Eq. (36), and replacing the integral for the energy exchange with neutrinos between $R_{\rm {g}}$ and $R_{\rm {s}}$ by ${\cal H}-{\cal C}$ as given in Eqs. (82) and (87), one ends up with

 

$${{\rm {d}}\over {\rm {d}}t}(\Delta E_{\rm {g}}) 4\pi R_{\rm {s}}^2 \rho_{\rm {p}} l_{\rm {s}} (\dot R_{\rm {s}} -... ...-4\pi R_{\rm {g}}^2 \rho_{\rm {g}} l_{\rm {g}} (\dot R_{\rm {g}} - v_{\rm {g}})$$ =

$$\phantom{=} -\,\, 4\pi R_{\rm {s}}^2 P_{\rm {s}} \dot R_{\rm {s}} + 4\pi R_{\rm {g}}^2 P_{\rm {g}} \dot R_{\rm {g}} + {\cal H} - {\cal C}$$

$$4\pi R_{\rm {s}}^2 \rho_{\rm {p}} l_{\rm {s}} \dot R_{\rm {s}} - ... ... l_{\rm {g}} \dot R_{\rm {g}} - 4\pi R_{\rm {s}}^2 P_{\rm {s}} \dot R_{\rm {s}}$$ =

$$\phantom{=} +\,\, 4\pi R_{\rm {g}}^2 P_{\rm {g}} \dot R_{\rm {g}} - \dot M\,l_{\rm {s}} + \dot M'\,l_{\rm {g}} + {\cal H} - {\cal C} .$$ (109)

The mass accretion rates $\dot M$ and $\dot M'$ account for the inflow of matter into the gain region through the shock and for the mass that is advected through the gain radius, respectively (see Sect. 7 and discussion after Eq. (102)). Equation (109) means that the total energy in the gain region changes due to active mass motions, $p{\rm {d}}V$ work associated with these mass motions, the movement of the boundaries, and neutrino heating.

Making use of $\varepsilon_{\rm {s}} = P_{\rm {s}}/(\Gamma -1)$, $\rho_{\rm {s}}/\rho_{\rm {p}} = \beta$, and Eq. (39), one finds for $l_{\rm {s}} = (\varepsilon_{\rm {s}}+P_{\rm {s}})/ \rho_{\rm {s}}-G\widetilde{M}/R_{\rm {s}}$:

$$l_{\rm {s}} \,\approx\, -\left\lbrack 1-{\Gamma\over \Gamma\!... ...)^{\! 2}\, \right\rbrack {G\widetilde{M}\over R_{\rm {s}}} \,,$$ (110)

where Eq. (43) with $\alpha \approx 1/\sqrt{2}$ was employed for $v_{\rm {p}}^2 \approx G\widetilde{M}/R_{\rm {s}} \approx 1.3~10^{19} R_{{\rm {s}},7}^{-1}(\widetilde{M}/M_{\odot})$ erg$\,$g^-1. Because of hydrostatic equilibrium a simple relation exists between $l_{\rm {g}}$ and $l_{\rm {s}}$. With $\varepsilon = P/(\Gamma -1)$ and Eqs. (56) and (59) one obtains

$$l_{\rm {g}}\,=\,l_{\rm {s}} - {3\Gamma - 4\over 4(\Gamma-1)}... ...{\rm {s}}} \left( {R_{\rm {s}}\over R_{\rm {g}}}-1 \right) \,.$$ (111)

Using the more general density-pressure relation $P = K\rho^\gamma$ instead of Eq. (56), and the corresponding hydrostatic density profile of Eq. (62), leads to

$$l_{\rm {g}}\,=\,l_{\rm {s}} - \left( 1 - {\Gamma\over \Gamma\... ...{\rm {s}}} \left( {R_{\rm {s}}\over R_{\rm {g}}}-1 \right) \,.$$ (112)

For $\Gamma = \gamma$, this gives $l_{\rm {g}} = l_{\rm {s}}$.