Explosion of a Minimum-Mass Neutron Star within Relativistic Hydrodynamics

Abstract

The relativistic hydrodynamics equations are adapted for the spherically symmetric case and the Lagrangian form. They are used to model the explosive disruption of a minimum-mass neutron star—a key ingredient of the stripping model for short gamma-ray bursts. The shock breakout from the neutron star surface accompanied by the acceleration of matter to ultrarelativistic velocities is studied. A comparison with the results of previously published nonrelativistic calculations is made.

APPENDIX A

ENERGY CONSERVATION IN A STAR

The relativistic equations derived by us look unusual with regard to how the terms with the artificial viscosity $Q$ appear in them. Usually, the introduction of the latter is actually reduced to an additive to the pressure: $P\rightarrow P+Q$. In our case, however, there is also an additional contribution to the equation of motion (the last turn in (31)). Furthermore, the right-hand side of the energy equation (24) is not reduced to the form $Q/\rho_{\mathrm{b}}^{2}(d\rho_{\mathrm{b}}/dt)$ because of the term $r^{3}$ under the logarithm. Let us demonstrate that, nevertheless, our formulas are quite self-consistent. We will work in the nonrelativistic limit. Let us multiply Eq. (31) by $v$ and integrate over $m_{\mathrm{b}}$ throughout the star. On the right we will obtain the term

$$\frac{1}{2}\int\frac{dv^{2}}{dt}dm_{\mathrm{b}}=\frac{dE_{\mathrm{kin}}}{dt},$$

(A.1)

i.e., the total time derivative of the kinetic energy. The gravitational acceleration (20) in the limit under consideration is simply $a_{\mathrm{G}}={-}Gm_{\mathrm{b}}/r^{2}$. Its integral is

$$-\int\!\frac{vGm_{\mathrm{b}}dm_{\mathrm{b}}}{r^{2}}=\int\!\frac{d}{dt}\!\left(\frac{1}{r}\right)Gm_{\mathrm{b}}dm_{\mathrm{b}}={-}\frac{dE_{\mathrm{grav}}}{dt},$$

(A.2)

where $E_{\mathrm{grav}}$ is the gravitational energy. We integrate the next term in (31) by parts:

$$-\int\!4\pi r^{2}v\frac{\partial(P+Q)}{\partial m_{\mathrm{b}}}dm_{\mathrm{b}}=$$

(A.3)

$${}=\int\!(P+Q)\frac{d}{dt}\!\left(\!\frac{1}{\rho_{\mathrm{b}}}\!\right)dm_{\mathrm{b}}.$$

The energy equation (24) can be rewritten in an equivalent form:

$$(P+Q)\frac{d}{dt}\!\left(\!\frac{1}{\rho_{\mathrm{b}}}\!\right)={-}\frac{dE}{dt}+\frac{3Qv}{r\rho_{\mathrm{b}}}.$$

(A.4)

Substituting this expression into (A.3), we will find that the second term on the right in (A.4) is canceled out with the last term in the equation of motion (31) during its integration. There remains only the total derivative of the star’s internal energy $E_{\mathrm{int}}=\int Edm_{\mathrm{b}}$. Thus, we obtained the law of conservation of the star’s total energy in the form $dE_{\mathrm{tot}}/dt=0$, where $E_{\mathrm{tot}}=E_{\mathrm{kin}}+E_{\mathrm{grav}}+E_{\mathrm{int}}$.