Appendix A: Numerical calculation of G($\sigma$,$\mu _*$)

The function $G(\sigma,\mu_*)$ is computed using numerical quadrature. Firstly, the integral over x can be efficiently computed using a Hermite quadrature formula (cf. Ralston 1965). Quadrature weights and knots were computed using a subroutine IQPACK which is an implementation of method described by Kautsky & Elhay (1982). Satisfactory approximation can be obtained using 20 quadrature points.

For an angle integration we used Legendre quadrature formula with 5 quadrature points. Again, quadrature weights and knots were computed using subroutine IQPACK (Kautsky & Elhay 1982).

Finally, the integration over y was performed using Simpson quadrature rule. The quadrature integral $(0,\infty)$ was approximated by (10^-3,10⁵) and the Simpson integration was divided into subintervals of power 10, with 10 quadrature points in each of them.

Numerical tests showed that temperature computed with described approximation of the GO heating has an error less than 1%.