7 Mass accretion onto the neutron star

The shock accretes mass at a rate $\dot M \equiv 4\pi R_{\rm {s}}^2\rho_{\rm {p}}v_{\rm {p}}$ as determined by the conditions in the core of the progenitor star (see Sect. 4.4). In a stationary state, this rate is equal to the rate at which matter is advected inward from the shock to the neutrinosphere to be finally added into the neutron star. The rate at which matter can be absorbed by the neutron star, however, depends on the efficiency by which neutrinos are able to remove the energy excess of the infalling material relative to the energy of the strongly bound matter in the neutron star surface layers. For the large accretion rates typical of the collapsed stellar core right after bounce, the density is so high that the infalling matter becomes opaque to neutrinos. In this case the efficiency of the energy loss is reduced. When the gas is hotter, the neutrino opacity increases (because of the energy dependence of the neutrino cross sections), and the neutrinosphere moves to a larger radius. Due to this regulatory effect, the neutrinospheric temperature is a rather inert quantity and, e.g., turns out to be very similar in different numerical models. Therefore it is not a steady-state mass accretion rate which governs the temperature at the base of the "atmosphere'' (as for accretion in optically thin conditions), but the "surface'' of the nascent neutron star forms where the temperature is sufficiently high for neutrino opaqueness to set in. When neutrino cooling is not efficient enough, the advection of matter through the neutrino cooling region is reduced compared to the accretion into the shock, and matter piles up on top of the neutron star. Similarly, strong neutrino heating in the gain region can reduce the inflow of matter. The transition from accretion to an explosion is characterized by an inversion of infall to outflow. For this reason the analysis of the conditions for shock revival requires the inclusion of this sort of time-dependence in the discussion. In the simplified model considered here, the mass accretion rate is allowed to change between $R_{\rm {s}}$ and $R_{\rm {g}}$. Matter advected through $R_{\rm {g}}$ at a rate determined by the efficiency of neutrino cooling is then assumed to be added into the neutron star (compare Fig. 2). Using Eqs. (2) and (6) and the definition $\dot M(r) = 4\pi r^2\!\rho v$, Eq. (4) can be rewritten in the following form:

 

$${\partial \over \partial r}\left\lbrack \dot M \left( {e\! +\! P\over \rho}- {G\widetilde{M}\over r} \right) \right\rbrack 4\pi r^2 (Q_{\nu} + Q_{\rm {d}})$$ =

$$- \, 4\pi r^2\!\rho\, {\partial \over \partial t}\left( {e\over \rho} \right)$$

$$+ \left( {e\over \rho} - {G\widetilde{M}\over r} \right){\partial\dot M \over \partial r} \ ,$$ (90)

where $Q_{\nu} = Q_{\nu}^+ - Q_{\nu}^-$ is the net rate (per unit volume) of energy transfer between neutrinos and the stellar medium and $Q_{\rm {d}}$ denotes the energy consumed or released by the photodisintegration of nuclei. The latter term has to be introduced in the equation when rest-mass contributions from nucleons and nuclei are not included in the internal energy density $\varepsilon$(Eq. (5)). The nuclei present in the accretion flow through the shock are assumed to be dissociated to free nucleons within the shock front (cf. Eq. (38)). Therefore the rate $Q_{\rm {d}}$ in terms of the (positive) nuclear binding energy per unit mass, $q_{\rm {d}}$, is

$$Q_{\rm {d}}\,=\,\rho\, v\, q_{\rm {d}}\,\delta(r-R_{\rm {s}})\,.$$ (91)

Here $\delta(x)$ is the delta function. For v < 0, which is true in case of accretion, energy is extracted from the stellar medium, i.e., $Q_{\rm {d}} < 0$. Now integrating Eq. (90) between $R_{\nu }$ and a radius r that is infinitesimally larger than $R_{\rm {s}}$ gives

 

$$\dot M \left\lbrack {e + P\over \rho} - {G\widetilde{M}\over r} \... ...d}}\dot M \left( q_{\rm {d}} + {e\over \rho} - {G\widetilde{M}\over r} \right),$$ (92)

where $\partial M/\partial r = 4\pi r^2\!\rho$ was used. The mass accretion rate through the shock was defined as $\dot M = 4\pi R_{\rm {s}}^2\rho_{\rm {p}}v_{\rm {p}}$ and the corresponding accretion rate through the neutrinosphere as $\dot M' \equiv \dot M(R_{\nu}) = 4\pi R_{\nu}^2\rho_{\nu} v_{\nu}$. The term for the rate of energy consumption by nuclear dissociation was split into two parts according to $\dot M(r) = \dot M' + \int_{\dot M'}^{\dot M(r)}{\rm {d}}{\dot M}$.

From Eq. (92) an approximation for $\dot M'$ can be derived by taking into account that $\vert Q_{\nu}/\rho \vert \gg \partial (e/\rho)/ \partial t$ in the region between $R_{\nu }$ and $R_{\rm {s}}$, where strong neutrino heating and cooling occurs. Moreover, the integrand of the last term on the right hand side of Eq. (92) is usually small, because $q_{\rm {d}}$corresponds to about 8-9 MeV per nucleon for complete disintegration of nuclei into free nucleons, $G\widetilde{M}/R_{\rm {s}}\sim 14\,(\widetilde{M}/M_{\odot})/R_{{\rm {s}},7}$ MeV per nucleon, and $e/\rho \approx {1\over 2}v_{\rm {p}}^2 \sim {1\over 2} G\widetilde{M}/R_{\rm {s}}$ immediately above the shock, where the infall velocity $v_{\rm {p}}$ is given by Eq. (43) and the specific internal energy is typically much smaller than the specific kinetic energy. For the same reason, the first term on the left hand side of Eq. (92) is much smaller than the second term when $\dot M$ and $\dot M'$ are of the same order. With all this one gets

$$\dot M'\,\approx\, -\int\limits_{R_{\nu}}^{R_{\rm {s}}}{\rm {... ...M}\over R_{\nu}} + q_{\rm {d}} \right)^{\!\! -1} \!\!\! \cdot$$ (93)

Because of the large gravitational binding energy of matter at the neutrinosphere, the term in brackets in Eq. (93) is negative. The integral adds up the contributions from neutrino cooling between $R_{\nu }$ and $R_{\rm {g}}$ and from neutrino heating between $R_{\rm {g}}$ and $R_{\rm {s}}$. If cooling is stronger (which is the case in the first second after bounce), the integral is negative and $\dot M' < 0$, i.e., the neutron star accretes matter. If neutrino heating dominates, there is mass outflow, $\dot M' > 0$.

Such mass loss takes place during the later phase of the neutrino-cooling evolution of the nascent neutron star, where a baryonic wind, the so-called neutrino-driven wind, is blown off the neutron star surface due to neutrino energy deposition just outside the neutrinosphere (Qian & Woosley 1996). The transition from accretion to mass outflow and the onset of mass loss can be discussed with the formulae presented here. A description of the wind regime (where the fluid velocity v approaches the local speed of sound), however, is beyond the scope of the present work, because it requires retaining the velocity gradient in the momentum equation, Eq. (3). Assuming steady-state conditions, this leads to the well known set of dynamic wind equations which can also be discussed by analytic means (see Qian & Woosley 1996, and references therein). In contrast, the toy model developed in this paper does not make use of steady-state assumptions for the mass flow through the gain layer, i.e., it is allowed that $\dot M \not= \dot M'$ in general.

The integral in Eq. (93) was evaluated in Sect. 6:

$$\int\limits_{R_{\nu}}^{R_{\rm {s}}}{\rm {d}}r\,4\pi r^2\,Q_{\nu} \,=\,-\,L_{\rm {acc}} + {\cal H} - {\cal C} \,.$$ (94)

Equation (77) gives the net energy exchange between neutrinos and stellar medium in the layer $[R_{\nu},R_{\rm {g}}]$, Eq. (87) the corresponding result for the interval $[R_{\rm {g}},R_{\rm {s}}]$, when ${\cal H}$ is taken from Eq. (83) and the neutrino luminosity $L_{\nu}(R_{\rm {g}})$ from Eq. (76) with a and b provided by Eqs. (74) and (75), respectively. Plugging in numbers representative for the early post-bounce evolution, $L_{\nu}\approx 5~10^{52}$ erg$\,$s^-1, $R_{\nu} \approx 50$ km, $\widetilde{M}\approx 1$ $M_{\odot }$, $a\sim 1$, one finds $L_{\nu}(R_{\rm {g}})\approx 6.3~10^{52}$ erg$\,$s^-1 and $\int_{R_{\nu}}^{R_{\rm {g}}}{\rm {d}}r\,4\pi r^2 Q_{\nu} = -L_{\rm {acc}} = L_{\nu}(R_{\nu})-L_{\nu}(R_{\rm {g}}) \approx -1.3~10^{52}$ erg$\,$s^-1, and using $R_{\rm {s}} \approx 2R_{\rm {g}} \approx 200$ km, $\langle \mu_{\nu} \rangle^\ast \sim 1$, $\langle \mu_{\nu} \rangle_{\rm {g}}\sim 0.75$, yields $\int_{R_{\rm {g}}}^{R_{\rm {s}}}{\rm {d}}r\,4\pi r^2Q_{\nu}= {\cal H} -{\cal C} \approx 7.7~10^{51}$ erg$\,$s^-1. The gravitational energy at the neutrinosphere at 50 km is about -28 MeV per nucleon, $q_{\rm {d}}$ is roughly 8 MeV per nucleon, and the internal energy plus pressure account for typically $\sim$10 MeV per nucleon:

$$\left( {e+P\over \rho} \right)_{\! R_{\nu}} \approx \left( {5... ...{u}}} \,\approx\, {7\over 2}\,{kT_{\nu}\over m_{\rm {u}}} \, ,$$ (95)

where $e = \varepsilon$ has been applied because ${1\over 2}\rho v^2 \ll \varepsilon$ at the neutrinosphere. Therefore the sum of the terms in the denominator of Eq. (93) can be estimated to be about -10¹⁹ erg$\,$g^-1. This leads to a mass accretion rate of the neutron star of $\dot M'\sim -0.3$ $M_{\odot }\,$s^-1, a value which is in the range of the results of detailed numerical simulations and is of the order of the mass infall rate on the shock, $\dot M$.