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Astron. Astrophys. 341, 499-526 (1999)

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Appendix A: The particle setup

In the initial setup of the SPH-particles we adopt an intermediate approach between the extremes of total randomness and the well defined order of a lattice. In a first step we create a large set of spherical shells, containing initially randomly distributed particles which have been relaxed according to some Coulomb-type force law. The so constructed shells are then joined in a concentric way, and the particles are assigned masses that, together with the smoothing length, reproduce the desired density profile. This configuration is finally relaxed with the SPH-code.

To construct a single shell of a given particle number we distribute the particles randomly on the surface of a unit sphere. We assume some repulsive, dissipative Coulomb-type force law:

[EQUATION]

where [FORMULA] is some dissipation parameter, [FORMULA] is the difference vector of particle i and j, [FORMULA], [FORMULA] and [FORMULA] are particle position and velocity vectors. In the following relaxation process only the force component tangential to the spherical surface was used in order to constrain the movement to a constant radius. After each integration time step the particle positions were projected back onto the surface.

To be able to vary the particle number density with the radius we introduce a parameter [FORMULA]. The distance to the new shell [FORMULA] is then calculated from the previous one [FORMULA] via

[EQUATION]

Denoting the distance from the center to the first shell by [FORMULA] and requiring that

[EQUATION]

[FORMULA] is the radius of our star, we find for a given number of shells [FORMULA] and given [FORMULA]:

[EQUATION]

Now the question arises of how to choose the particle number in a given shell. We have as a constraint that typical particle separations within shell n should be the same as typical distances between shells [FORMULA]. We consider now a sphere of radius R with [FORMULA] SPH-particles distributed on its surface. If we calculate the typical distance of two particles within one shell and equate it to [FORMULA] we find for the particle number in shell n

[EQUATION]

In the case of [FORMULA] we have [FORMULA] and can therefore expand the square root in the denominator:

[EQUATION]

The same result is recovered by assigning to each particle the surface of a circle and setting this equal to [FORMULA].

To find the optimal relative orientation of the shells we go back to the idea of the Coulomb-type forces.

The innermost shell is kept fixed. Then the torque exerted from shell one to shell two is calculated and shell two is rotated (according to the corresponding Eulerian angles) until the torque vanishes. Fixing the second shell at the new position, the torque on shell three from the inner shells is calculated and so on.

The profiles [FORMULA], [FORMULA] and [FORMULA] define our initial neutron star model. They are calculated using a Newtonian 1D stellar structure code and a self-consistent table of [FORMULA], [FORMULA] and u generated using the LS-EOS. For the calculation of [FORMULA], we use (C6) (at [FORMULA] K) and the [FORMULA]-equilibrium condition for the chemical potentials:

[EQUATION]

where Q is the mass difference of neutron and proton times [FORMULA] and the difference in the chemical potentials of neutron and proton (without rest masses), [FORMULA], is provided by the LS-EOS. Since we want to construct a cold neutron star where no neutrinos are present, we assume their chemical potentials to vanish,

[EQUATION]

The [FORMULA]-profiles are for numerical reasons restricted to values [FORMULA] 0.05.

After values of [FORMULA] and u have been assigned to the particles according to the profiles, the masses still have to be distributed. In principle

[EQUATION]

has to be solved, where the components of the vectors are the values at the particle positions (density and mass) and W is the kernel matrix.

In order to enforce sphericity, we assign one single value for all the masses [FORMULA] and one for all the smoothing lengths [FORMULA] to all the particles in a given shell. We prescribe an allowed range of neighbours for the mean neighbour number in each shell (the neighbours are found by traversing the tree of our SPH-code) and adjust the smoothing lengths correspondingly (typical neighbour numbers are [FORMULA]). Starting with the masses from the density self-contribution of each particle

[EQUATION]

where the spherical spline kernel (see Benz (1990)) has been used, the masses are iterated until

[EQUATION]

where [FORMULA] is the mean value of the density in shell j ([FORMULA] was typically a few times [FORMULA]).

This particle setup was then finally relaxed with the SPH-code where a velocity dependent additional force term was applied:

[EQUATION]

The parameter [FORMULA] gives a characteristic damping time scale [FORMULA] To obtain an efficient convergence towards the equilibrium solution it is important to choose [FORMULA] to be slightly over-critical, i.e. [FORMULA], where the typical oscillation period [FORMULA] is approximately given by the sound crossing time of the neutron star [FORMULA]. Some properties of the relaxed star are shown in Fig. 27.

[FIGURE] Fig. 27. Properties of the configuration after relaxation. The upper left (upper right, lower right) panel shows the distribution of the particle masses (densities, smoothing lengths) with radius, the lower left panel shows the neighbour numbers of each particle.

Appendix B: The equation of state

For all our calculations the default set of nuclear parameters (see Lattimer & Swesty (1991)) with [FORMULA] MeV is used.

Most of the scheme that we are going to describe here has been developed for version 2.6 of the LS-EOS. We encountered several regions (especially for low [FORMULA] and densities where large amounts of the heavy nuclei are present) where the Newton-Raphson iteration to solve the set of equilibrium equations did not converge for the default set of guess values (nucleon number density inside nuclei [FORMULA], nucleon degeneracy parameters [FORMULA] and [FORMULA]) due to very small convergence radii.

To find accurate guess values [FORMULA] in the above mentioned critical regions we use the fact that the [FORMULA] are slowly varying functions of [FORMULA]. We chose a fixed, uncritical value of [FORMULA], [FORMULA], for which reference surfaces of the guess values [FORMULA] over the [FORMULA]-plane, n is the nucleon number density, T the temperature, were calculated

[EQUATION]

To patch the surface over the critical region for some given [FORMULA], the corresponding rectangular piece in the reference surface was cut out and matched to the hole in the surface of the desired [FORMULA]. Let

[EQUATION]

be the region where the code does not converge for a given [FORMULA]. We then calculate multipliers [FORMULA] for each guess value [FORMULA]

[EQUATION]

with j labeling the edge point, in such a way that the edge points in the reference surface coincide with those for the critical [FORMULA] when multiplied with the corresponding [FORMULA]:

[EQUATION]

Then a multiplier [FORMULA] for the points within the critical region is calculated by means of linear interpolation:

[EQUATION]

where [FORMULA] and [FORMULA] and the edge points are numbered starting with ([FORMULA]) in a counter-clockwise sense. The guess value variables in the critical region are then approximated by

[EQUATION]

Using this scheme accurate tables for the guess values have been calculated. Using these tables the code converges rapidly and safely everywhere.

In our hydrodynamic calculations we use a tabular form of the LS-EOS (version 2.7), where the above described scheme has been used. Our table contains 153 entries in [FORMULA], 121 entries in [FORMULA], and 25 entries in [FORMULA]. For our calculations we need to tabulate the pressure, the temperature and the difference in the nucleon chemical potentials, [FORMULA]. The abundances are difficult to tabulate since they may vary enormously from grid point to grid point. Therefore, every time that abundances are needed (e.g. to calculate the total nuclear binding energy of the system), the original LS-EOS is used.

Since we use the specific internal energy as independent variable, we performed a bisection iteration to find the corresponding temperatures. If non-monotonic values in T were encountered, the iteration used u values that were averaged over neighboring values. Note, however, that apart from that, no other manipulation of the EOS-data has been performed (Ruffert et al. (1996) used smoothed EOS-tables).

The interpolation in the table is a delicate topic in itself since thermodynamic consistency for the values between the tabulated grid points is not guaranteed. An inconsistent interpolation, i.e. an interpolation that does not fulfill the constraints posed by the Maxwell relations of thermodynamics, will lead to an artificial buildup of entropy, or temperature. Swesty (1996) has proposed an interpolation scheme insuring thermodynamic consistency as well as the continuity of the derivatives of the thermodynamic functions. However, this approach applied to our 3D-input parameter space would require 216 terms and the evaluation of fourth order derivatives for each table call. Clearly, this procedure is - at least at present - computationally prohibitive. Considering the other approximations in our model, we think that we can justify just a linear interpolation in our table.

Appendix C: Neutrino emission rates

Starting from the electron capture cross section (see Tubbs and Schramm (1975)), which reads with our approximations

[EQUATION]

where [FORMULA], [FORMULA] is the energy of the emitted neutrino,

[EQUATION]

and [FORMULA] the Fermi constant, the number of electron captures per time and volume, [FORMULA], is given by ([FORMULA])

[EQUATION]

The corresponding energy loss in neutrinos per time and volume is

[EQUATION]

Here [FORMULA], with the [FORMULA] denoting neutron and proton abundances, [FORMULA] the corresponding chemical potentials (without rest mass) and [FORMULA] Avogadro's constant. The factor [FORMULA], derived by Bruenn (1985) under the assumption that 4-momentum transfer between leptons and nucleons is negligible ("elastic approximation"), takes into account effects resulting from the phase space restrictions of the final state nucleons. [FORMULA] is the usual Fermi integral given by

[EQUATION]

[FORMULA] is the electron degeneracy parameter, [FORMULA]. The electron chemical potential is calculated according to

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA] have been introduced (see Baron (1985)).

Assuming [FORMULA] the corresponding formulae for the positron captures read

[EQUATION]

and

[EQUATION]

where [FORMULA] is found from [FORMULA] by interchanging the indices. For simplicity we assume here that matter consists only of protons and neutrons:

[EQUATION]

The rate of change in the electron fraction is given by

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998
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