4 The characteristic radii

The neutrinospheric radius $R_{\nu }$, the gain radius $R_{\rm {g}}$ and the transition radius of the EoS properties, $R_{\rm {eos}}$, will be formally defined below. They are characteristic of the atmospheric structure in the postshock region, which determines, together with the infall region ahead of the shock, the shock radius $R_{\rm {s}}$ and the shock velocity $U_{\rm s} \equiv \dot R_{\rm s}$.

   
4.1 The neutrinosphere

The neutrinosphere relevant for the discussion in the following sections is the "energy-sphere'', where neutrinos decouple energetically from the stellar background. It usually does not coincide with the sphere of last scattering, the so-called "transport-sphere'', outside of which the neutrino distribution becomes strongly forward peaked (for a detailed discussion, see Janka 1995). Only inside their energy-sphere neutrinos can be considered to be roughly in thermodynamic equilibrium with the stellar medium. Besides neutrino-nucleon scattering, which is important for all neutrinos, electron neutrinos $\nu_{\rm e}$ and electron antineutrinos $\bar\nu_{\rm e}$interact via frequent charged-current absorption and emission reactions with nucleons, whereas muon and tau neutrinos and antineutrinos do not. Therefore the energy-spheres of electron neutrinos and antineutrinos are typically located farther out in the star at larger radii than those of muon and tau neutrinos.

The energy deposition in the gain region, however, is clearly dominated by $\nu_{\rm e}$ and $\bar\nu_{\rm e}$. For this reason one can concentrate on their transport properties and neglect muon and tau neutrinos and antineutrinos in the discussion. Scattering off nucleons acts on all neutrinos equally. The charged-current absorption reactions of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ on neutrons and protons, respectively, yield an even larger contribution to the total opacity. The opacities of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ are nearly equal, because $\bar\nu_{\rm e}$ absorption and emission (Eq. (18)) is similarly frequent as $\nu_{\rm e}$ absorption and emission (Eq. (17)) as long as positrons are abundant, i.e., the stellar atmosphere is hot and electrons are not very degenerate. Therefore the transport-spheres and energy-spheres of electron neutrinos and antineutrinos are all close together and it is justified to consider only one, "the'', neutrinosphere at radius $R_{\nu }$. Of course, the real situation is more complex and there is no definite radius interior to which neutrinos are in equilibrium at the local thermodynamical conditions and diffuse, and exterior to which they are decoupled from the background and stream freely. The transition between these two limits is continuous and in case of neutrinos, whose reaction rates are strongly energy dependent, it is also a function of the neutrino energy.

The spectral temperature of electron neutrinos will be taken equal to the gas temperature at the assumed neutrinosphere, $kT_{\nu_{\rm e}} = kT(R_{\nu})$. Detailed simulations of neutrino transport show that electron antineutrinos have somewhat more energetic spectra. A typical result (e.g., Bruenn 1993; Janka 1991a) is $kT_{\bar\nu_{\rm e}}\approx 1.5\,kT_{\nu_{\rm e}}$, which will be used below. The fact that $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ spectra are found to be different in detailed models is an indication that the picture drawn above is overly simplified. Nevertheless it is sufficiently accurate for the analysis in this paper. Note that in general the neutrino luminosity $L_\nu$ can not be related to the neutrinospheric temperature by the Stefan-Boltzmann law for blackbody emission of a sphere with radius $R_{\nu }$, $L_{\nu} = \pi R_{\nu}^2\,{7\over 8}ac (kT_{\nu})^4$. This formula is frequently taken for the combined luminosity of neutrinos plus antineutrinos, assuming their chemical potentials to be zero. However, the effective temperature $kT_{\rm {eff}}$, which should be used in the Stefan-Boltzmann law, is typically not equal to the spectral temperature $kT_{\nu}$ (for a discussion, see Janka 1995). Transport simulations show that due to non-equilibrium effects the difference can be quite significant. For this reason two parameters, $T_{\nu}$ and $L_\nu$, will be retained here to describe the spectrum and the luminosity of the neutrinos emitted from the neutrinosphere. Moreover, the radius of the neutrinosphere will be considered as the position in the star where the mean value of the cosine of the neutrino propagation angle relative to the radial direction has a value of 0.25 (see Eq. (26) and Janka 1991a,b, 1995).

Keeping in mind the simplifications associated with the concept of the neutrinosphere, the radius $R_{\nu }$ can be defined by the requirement that the effective optical depth to energy exchange for neutrinos with average energy is

$$\tau_{\rm {eff}}\,=\,\int_{R_{\nu}}^\infty{\rm {d}}r\,\kappa_{\rm {eff}}(r) \,=\,{2\over 3}$$ (13)

(Suzuki 1989). The effective opacity $\kappa_{\rm eff}$ in case of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ is defined from the scattering opacity $\kappa_{\rm sc}$ and the absorption opacity $\kappa_{\rm a}$ as

$$\kappa_{\rm eff}\,=\,\sqrt{\kappa_{\rm a}\left( \kappa_{\rm a}+\kappa_{\rm sc} \right)}$$ (14)

(Rybicki & Lightman 1979; Shapiro & Teukolsky 1983; Suzuki 1989). Using Eqs. (10) and (11) one obtains for the effective opacity, again averaged over the spectrum of the energy flux which is supposed to have Fermi-Dirac shape with zero degeneracy, in case of electron neutrinos the expression

 

$$\kappa_{{\rm eff},\nu_{\rm e}} 1.62\, {\sigma_0\langle \epsilon_{\nu_{\rm e}}^2 \rangle\over (m_... ...}\,{\rho\over m_{\rm u}}\, Y_{\rm n}\,\sqrt{1+0.21\,{Y_{\rm p}\over Y_{\rm n}}}$$ = (15)

$$2.2~10^{-7}\rho_{10}\, \left( {kT_{\nu_{\rm e}}\over 4\,{\rm MeV}} \right)^{\! 2}Y_{\rm n}\,\sqrt{1+0.21\,{Y_{\rm p}\over Y_{\rm n}}} \ ,$$ =

and in case of electron antineutrinos the analogue result with $Y_{\rm n}\sqrt{1+0.21\,Y_{\rm p}/Y_{\rm n}}$ being replaced by $Y_{\rm p}\sqrt{1+0.21\,Y_{\rm n}/Y_{\rm p}}$. Since nuclei are nearly completely dissociated into free nucleons and $Y_{\rm n}$ is larger than $Y_{\rm p}$, i.e., $Y_{\rm n} \sim 0.8$ and $Y_{\rm p}\sim 0.2$, but $kT_{\nu_{\rm e}}$ is usually somewhat lower than $kT_{\bar\nu_{\rm e}}$, i.e., $kT_{\nu_{\rm e}}\approx 4~$MeV and $kT_{\bar\nu_{\rm e}}\approx 6\,$MeV, we verify the above statement that the effective opacities of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ are approximately equal. Assuming equal luminosities, $L_{\nu_{\rm e}} = L_{\bar\nu_{\rm e}}$, a suitable average value for the effective opacity therefore is

 

$$\langle \kappa_{\rm eff} \rangle = {1\over 2}\,\kappa_{{\rm eff},... ...10^{-7}\rho_{10}\, \left( {kT_{\nu_{\rm e}}\over 4\,{\rm MeV}} \right)^{\! 2} ,$$ (16)

where the composition dependent term in the weighted average has been approximated by $\sqrt{2}$. Knowing the density profile $\rho(r)$, the density $\rho_{\nu}$ at the neutrinosphere can be determined by using Eqs. (16) in (13).

   
4.2 The gain radius

Heating and cooling of the gas outside the neutrinosphere mainly proceed via the charged-current absorption and emission processes of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ (Bethe & Wilson 1985; Bethe 1993, 1995, 1997):

  

$$\nu_{\rm e} + {\rm n} \longleftrightarrow {\rm p} + {\rm e}^- \ ,$$ (17)

$$\bar\nu_{\rm e} + {\rm p} \longleftrightarrow {\rm n} + {\rm e}^+ \ .$$ (18)

To leading order in the particle energies, the cross sections for neutrino and electron/positron absorption, respectively, are

  

$$\sigma_{{\rm a},\nu_{\rm e}} \approx \sigma_{{\rm a},\bar\nu_{\rm e}}\,\approx\, {3\alpha^2+1\over 4}\,\sigma_0\,\left( {\epsilon_{\nu}\over m_{\rm e}c^2} \right)^{\! 2}\ ,$$ (19)

$$\sigma_{{\rm a},{\rm e}^-} \approx \sigma_{{\rm a},{\rm e}^+} \,\approx\, {3\alpha^2+1\over 8}\,\sigma_0\,\left( {\epsilon_{\rm e}\over m_{\rm e}c^2} \right)^{\! 2}\ \cdot$$ (20)

At the considered densities and temperatures, fermion phase space blocking and dense-medium effects can be safely ignored, and electrons are relativistic ( ). The heating rate $Q^+_{\nu_i}$ of the stellar medium by neutrinos $\nu_i$ is given by

$$Q^+_{\nu_i} \,=\, {3\alpha^2+1\over 4}\,{\sigma_0 c\, n_j\ove... ...}\over {\rm d}\epsilon_{\nu}{\rm d}\mu}\, \epsilon_{\nu}^3 \ ,$$ (21)

where n_j is the number density of the target nucleons ($j = p,\,n$), and ${\rm d}^2 n_{\nu_i}/({\rm d}\epsilon_{\nu}{\rm d}\mu)$ the neutrino distribution,

$${{\rm {d}}^2 n_{\nu_i}\over {\rm {d}}\epsilon_{\nu}{\rm {d}}\... ... (hc)^3}\,f_{\nu_i}(\epsilon_{\nu},\mu)\, \epsilon_{\nu}^2 \ ,$$ (22)

with $f_{\nu_i}(\epsilon_{\nu},\mu)$ being the neutrino phase space occupation function at some radius r, which depends on the neutrino energy $\epsilon_{\nu}$ and the cosine of the angle of neutrino propagation relative to the radial direction, $\mu = \cos\theta$. In Eq. (22) the factor h in the denominator is Planck's constant. Introducing Eq. (22) into Eq. (21) and performing the phase space integration over all energies and angles yields

$$Q^+_{\nu_i} \,=\, {3\alpha^2+1\over 4}\,\sigma_0\,n_j\, {\lan... ...} {L_{\nu_i}\over 4\pi r^2 \langle \mu_{\nu_i} \rangle}\ \cdot$$ (23)

Here the neutrino luminosity $L_{\nu_i}$, the average squared neutrino energy $\langle \epsilon_{\nu_i}^2 \rangle$, and the mean value of the cosine of the propagation angle, $\langle \mu_{\nu_i} \rangle$, are calculated from the neutrino phase space occupation function $f_{\nu_i}(\epsilon_{\nu},\mu)$by

   

$$L_{\nu_i} 4\pi r^2c\,{2\pi \over (hc)^3} \,\int\limits_0^\infty{\rm {d}}\ep... ...{-1}^{+1}{\rm {d}}\mu\, \mu\,\epsilon_{\nu}^3\,f_{\nu_i}(\epsilon_{\nu},\mu)\ ,$$ = (24)

$$\langle \epsilon_{\nu_i}^2 \rangle \int\limits_0^\infty {\rm {d}}\epsilon_{\nu}\int\limits_{-1}^{+1}... ...its_{-1}^{+1}{\rm {d}}\mu\, \epsilon_{\nu}^3\,f_{\nu_i} \right)^{\!\! -1}\!\! ,$$ = (25)

$$\langle \mu_{\nu_i} \rangle \int\limits_0^\infty {\rm {d}}\epsilon_{\nu}\int\limits_{-1}^{+1}... ...its_{-1}^{+1}{\rm {d}}\mu\, \epsilon_{\nu}^3\,f_{\nu_i} \right)^{\!\! -1}\!\! .$$ = (26)

The quantity $\langle \mu_{\nu} \rangle$ is also called flux factor and can be understood as the ratio of the neutrino energy flux, $L_{\nu}/(4\pi r^2)$, to the neutrino energy density times c. Typically, it is close to 0.25 near the neutrinosphere of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ and approaches unity when the neutrino distribution get more and more forward peaked in the limit of free streaming with increasing distance from the neutrinosphere (Janka 1991b, 1992, 1995). The total heating rate $Q^+_{\nu}$ is the sum of the contributions from $\nu_{\rm e}$and $\bar\nu_{\rm e}$:

$$Q^+_{\nu} \,=\, Q^+_{\nu_{\rm e}}+Q^+_{\bar\nu_{\rm e}}\ .$$ (27)

To derive a simple expression, one can again assume that $L_{\nu_{\rm e}} \approx L_{\bar\nu_{\rm e}}$, $\langle \epsilon_{\bar\nu_{\rm e}}^2 \rangle\approx 2 \langle \epsilon_{\nu_{\rm e}}^2 \rangle$, and that the $\nu_{\rm e}$ spectrum has Fermi-Dirac shape with zero degeneracy, i.e., $\langle \epsilon_{\nu_{\rm e}}^2 \rangle\approx 21(kT_{\nu_{\rm e}})^2$. In addition, the equality $\langle \mu_{\nu_{\rm e}} \rangle = \langle \mu_{\bar\nu_{\rm e}} \rangle \equiv \langle \mu_{\nu} \rangle$ is reasonably well fulfilled because the opacities of electron neutrinos and antineutrinos are very similar and therefore the neutrinospheres of both of them are nearly at the same radius. Putting everything together, the heating rate per unit volume is derived as

 

$$Q^+_{\nu} {3\alpha^2+1\over 4}\,{\sigma_0\langle \epsilon_{\nu_{\rm e}}^2 \... ...\nu_{\rm e}}\over 4\pi r^2\langle \mu_{\nu} \rangle}\, (Y_{\rm n} + 2Y_{\rm p})$$ =

$$\approx 160\,\,{\rho\over m_{\rm u}}\,{L_{{\nu_{\rm e}},52}\over r_7^2\la... ...}} \right)^{\! 2}\ \ \left\lbrack {{\rm MeV}\over {\rm s}} \right\rbrack\ \cdot$$ (28)

The numerical factor gives the rate in MeV per baryon, r₇ is the radius in $10^7\,$cm, and $L_{{\nu_{\rm e}},52}$ the $\nu_{\rm e}$ luminosity normalized to 10$^{52}\,$erg/s. In the layers where most of the heating and cooling between neutrinosphere and shock take place, nuclei are nearly fully dissociated into free nucleons (Bethe 1993, 1995, 1997; Thompson 2000) and $Y_{\rm p} < Y_{\rm n}$, therefore using $Y_{\rm n} + 2Y_{\rm p}\approx 1$ in the last expression is a reasonable approximation.

The cooling rate of the stellar gas by emission of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$is calculated as

$$Q^-_{\nu}\,=\, {3\alpha^2\!\! +\!\! 1 \over 8}\,{\sigma_0\, c... ...n} + n_n {{\rm d}n_{\rm e^+}\over {\rm d}\epsilon}\! \right) ,$$ (29)

where use was made of Eq. (20), and the distributions of relativistic electrons and positrons are given by

$${{\rm d}n_{\rm e^\pm}\over {\rm d}\epsilon} \,=\, {8\pi \over... ...lon^2 \over 1 + \exp (\epsilon/kT - \eta_{\rm e^\pm})} \ \cdot$$ (30)

T(r) is the local gas temperature and $\eta_{\rm e^\pm}$ the degeneracy parameter of electrons or positrons, defined as the ratio of the chemical potential to the temperature. A factor of 2 was taken into account as the statistical weight for positive and negative spin states. Inserting Eq. (30) into Eq. (29) one gets for the cooling rate per unit volume

 

$$Q^-_{\nu} (3\alpha^2\! +\! 1)\, {\pi\,\, \sigma_0\, c\,\, (kT)^6 \over (hc)^3 (m_{\rm e}c^2)^2}\, {\rho \over m_{\rm u}}$$ =

$$% \times \left\lbrack Y_{\rm p}{\cal F}_5(\eta_{\rm e}) + Y_{\rm n}{\cal F}_5(-\eta_{\rm e}) \right\rbrack$$

$$\approx 145\,\,{\rho\over m_{\rm u}}\, \left( {kT\over 2\,{\rm {MeV}}} \right)^{\! 6} \ \ \left\lbrack {{\rm MeV}\over {\rm s}} \right\rbrack \ ,$$ (31)

where the numerical factor is the rate in MeV per nucleon when $Y_{\rm n}+Y_{\rm p} \approx 1$ and the equilibrium relation $\eta_{\rm e^-} = -\eta_{\rm e^+}\equiv \eta_{\rm e}$ with $\eta_{\rm e}\approx 0$ are used. The latter approximation is good in the shock-heated layers because the electron fraction $Y_{\rm e} = n_{\rm e}/n_{\rm b}$ and thus the electron degeneracy is rather low and ${\rm e^\pm}$ pairs are abundant. ${\cal F}_5(\eta)$ is the Fermi integral for relativistic particles,

$${\cal F}_j(\eta) \,=\, \int_0^\infty{\rm {d}}x\, {x^j\over 1 + \exp(x-\eta)} \ ,$$ (32)

with ${\cal F}_5(0)\approx 118$. (Useful formulae for sums and differences of these Fermi integrals can be found in Bludman & Van Riper 1978, and simple approximations in Takahashi et al. 1978.)

Heating balances cooling at the gain radius, i.e., the gain radius $R_{\rm {g}}$has to fulfill the condition $Q^+_{\nu} = Q^-_{\nu}$ by definition. With Eqs. (28) and (31) one obtains the following relation:

$$R_{{\rm g},7}\left( {kT_{\rm g}\over 2\,{\rm MeV}} \right)^{\... ...}} \left( {kT_{\nu_{\rm e}}\over 4\,{\rm MeV}} \right) \ \cdot$$ (33)

$R_{{\rm g},7}$ is the gain radius in units of 10⁷ cm and $T_{\rm g} = T(R_{\rm g})$ the temperature at the gain radius. Depending on the position of the gain radius, $\langle \mu_{\nu} \rangle_{\rm {g}}$ is a factor somewhere between 0.25 (value at the neutrinosphere) and unity (limit for $r\to \infty$).

   
4.3 The EoS transition radius

It is interesting to consider the conditions for which the pressure is dominated by nonrelativistic nucleons or radiation plus relativistic ${\rm e^\pm}$ pairs ( MeV). In the first case $P \approx P_{\rm {b}} = kT\rho/m_{\rm u}$, if nuclei are fully dissociated into free nucleons. In the latter case $P \approx P_{\rm r} = P_{{\rm e}^\pm}+P_{\gamma}\approx{11\over 12}a_{\gamma}(kT)^4$, when $\eta_{\rm e}\approx 0$ is again assumed for the electron degeneracy and the constant is $a_{\gamma} = 8\pi^5/\left\lbrack 15(hc)^3 \right\rbrack \approx 8.56~10^{31}\,$MeV^-3cm^-3. Setting $P_{\rm b}$ equal to $P_{\rm r}$ gives

$${(kT)^3\over \rho} \,=\, {12\over 11}\,\,{1\over m_{\rm u}a_{\gamma}}$$ (34)

or, using the temperature $kT\approx kT_{\nu_{\rm e}}\approx 4\,$MeV (compare Fig. 2),

$$\left( {kT\over 4\,{\rm {MeV}}} \right)^{\! 3}\rho_{10}^{-1} \,\cong\, 1.2 \ .$$ (35)

This means that the transition from the baryon-dominated to the radiation-dominated regime occurs at a density significantly below that of the neutrinosphere. The latter is typically above $10^{11}\,$g/cm³. When the electron degeneracy is negligibly small, the contributions of relativistic and nonrelativistic gas components to the pressure are equal for a value of the radiation entropy per nucleon of $s_{\rm r} = s_{\rm e^\pm} + s_{\gamma} \approx (\varepsilon_{\rm r} + P_{\rm r})/(kT\rho/m_{\rm u}) = 4P_{\rm r}/P_{\rm b} = 4$. Since the energy density of relativistic particles is $\varepsilon_{\rm {r}} = 3P_{\rm {r}}$, whereas $\varepsilon_{\rm {b}}=3P_{\rm {b}}/2$for nonrelativistic particles, the relativistic electrons-positron pairs and photons dominate the energy density at such conditions. The main contribution to the entropy, however, then still comes from nucleons and nuclei (cf. Fig. 8 in Woosley et al. 1986).

   
4.4 The shock radius and infall region

Conservation of the mass flow, momentum flow and energy flow across the discontinuity of the shock front is expressed by the three Rankine-Hugoniot conditions

$$\rho_{\rm {p}}u_{\rm {p}} = \rho_{\rm {s}}u_{\rm {s}} \ ,$$ (36)

$$P_{\rm {p}} + \rho_{\rm {p}}u_{\rm {p}}^2 = P_{\rm {s}} + \rho_{\rm {s}}u_{\rm {s}}^2 \ ,$$ (37)

$${1\over 2}\, u_{\rm {p}}^2 + w_{\rm {p}} - q_{\rm {d}} = {1\over 2}\, u_{\rm {s}}^2 + w_{\rm {s}} \ ,$$ (38)

where the indices p and s denote quantities just ahead and behind the shock, respectively (see Fig. 2), $w = (\varepsilon + P)/\rho$ is the enthalpy per unit mass, $q_{\rm {d}}$ the nuclear binding energy per unit mass absorbed by photodisintegration of nuclei within the shock front, and $u = v - U_{\rm {s}}$ the fluid velocity relative to the shock when $U_{\rm {s}} = \dot R_{\rm {s}}$ is the shock velocity and v the gas velocity relative to the center of the star. Note that in the infall region v has negative sign.

With the definition $\beta \equiv \rho_{\rm {s}}/\rho_{\rm {p}}$, Eq. (36) gives $u_{\rm {s}} = u_{\rm {p}}/\beta$, which can be used to eliminate $u_{\rm {s}}$ from Eq. (37). For a strong shock, i.e., $P_{\rm {s}}\gg P_{\rm {p}}$, this yields

$$P_{\rm {s}}\,\approx\, \left( 1-{1\over \beta} \right) \rho_{\rm {p}}\left( v_{\rm {p}}-U_{\rm {s}} \right)^2 \ .$$ (39)

Combining Eqs. (36)-(38) one further finds

$$w_{\rm {s}}-w_{\rm {p}} \,=\, {1\over 2}\left( P_{\rm {s}}-P_... ..._{\rm {s}}} + {1\over \rho_{\rm {p}}} \right) - q_{\rm {d}}\ .$$ (40)

With $P_{\rm {p}}\ll P_{\rm {s}}$, $w_{\rm {p}}\ll w_{\rm {s}}$and $w_{\rm {s}} \approx 4P_{\rm {s}}/\rho_{\rm {s}}$ for the radiation-dominated gas in the postshock region, Eq. (40) can be rewritten as

$${\rho_{\rm {s}}\over \rho_{\rm {p}}}\,\approx\,7 + {2\,q_{\r... ...7 \over 4 - 3\sqrt{1 + 14\,q_{\rm {d}}/(9\,u_{\rm {p}}^2)}}\ ,$$ (41)

where in the second transformation Eq. (39) was used to replace $P_{\rm {s}}/\rho_{\rm {s}}$. This shows that for a relativistic gas the density jump in a strong shock is a factor of 7. Energy consumed by photodissociation of nuclei increases the density contrast between preshock and postshock region (Thompson 2000). In a more general treatment, retaining $w_{\rm {p}}$ and taking into account the (subdominant) contributions from nonrelativistic nucleons to the gas pressure behind the shock (but still using $P_{\rm {p}} \ll \rho_{\rm {p}}u_{\rm {p}}^2$in the infall region) one also derives the right hand side of Eq. (41), now with the expression

$$q^\ast \,\equiv\, q_{\rm {d}}-w_{\rm {p}}-{3\over 2\, m_{\rm {u}}}\, \sum_iY_i\,kT_{\rm {s}}$$ (42)

instead of $q_{\rm {d}}$ in the denominator. This means that the density discontinuity is also affected by the preshock enthalpy and the thermal pressure of nucleons and nuclei behind the shock (the nuclear composition is accounted for by the sum of the number fractions, $\sum_iY_i$). Considering $q_{\rm {d}}$ to be several MeV/ $m_{\rm {u}}$, $kT_{\rm {s}}\sim 1\,$MeV, and the preshock medium to be dominated by relativistic, degenerate electrons in which case $w_{\rm {p}}\approx \zeta_{\rm e} Y_{\rm e}/m_{\rm {u}}$ with an electron chemical potential $\zeta_{\rm e} = \eta_{\rm e} kT$ of a few MeV, one can see that all terms in Eq. (42) are of the same order and therefore equally important.

The preshock region is not affected by the postshock conditions. Because the shock moves supersonically relative to the medium ahead of it, sound waves cannot transport information in this direction. The matter there falls into the shock with a significant fraction of the free-fall velocity,

$$v_{\rm {p}}\,=\,-\,\alpha\,\sqrt{{2\,G {\widetilde M}\over R_{\rm {s}}}}$$ (43)

with $\alpha\sim 1/\sqrt{2}$ (Bethe 1990, 1993; Bruenn 1993). Ahead of the shock free nucleons are absent and therefore $\nu_{\rm e}$ and $\bar\nu_{\rm e}$absorption does not play a role, but neutrinos interact with nuclei by coherent scatterings. The opacity of the latter reaction scales roughly with N²/A when Nis the neutron number and A the mass number of the nuclei, and the total neutrino opacity of the preshock medium turns out to be close to the result of Eq. (12). Therefore the momentum transfer by neutrinos was again taken into account by using ${\widetilde M}$ instead of M in Eq. (43). Plugging Eq. (43) into the rate at which mass falls into the shock, $\dot M = 4\pi R_{\rm {s}}^2\rho_{\rm {p}}v_{\rm {p}}(< 0)$, gives the density just above the shock:

$$\rho_{\rm {p}}\,=\,-\,{\dot M \over 4\pi\,\alpha\, \sqrt{2\,G {\widetilde M}}\,R_{\rm {s}}^{3/2}} \ \cdot$$ (44)

On the other hand, if the original presupernova material has a density distribution $\rho_0(r_0) = H\,r_0^{-3}$ with H being a constant, then mass conservation yields a density at the footpoint of the shock at time t after the start of the collapse of

$$\rho_{\rm {p}}\,=\,{2\over 3}\,{H \over \alpha\,\sqrt{2\,G {\widetilde M}}}\,\, t^{-1}R_{\rm {s}}^{-3/2}$$ (45)

(Bethe 1990, 1993; see also Cooperstein et al. 1984). Comparing Eqs. (44) and (45) one finds that the rate at which mass crosses the shock in this case is

$$\dot M\,=\,-\,{8\pi\over 3}\,{H\over t}\ ,$$ (46)

which depends on the structure of the progenitor star through the constant H and decreases with time.