## Appendix A. Technical detailsThe details are given to some of the methods used in this paper. Note that by following the authors quoted the notation is conflicting with the notation used above. ## A.1. View factors between trianglesThe view factors for two finite surfaces can be calculated from Eq. (3) by integrating over their respective surfaces. It is very useful to convert this quadruple integral into two curve integrals by the use of Stokes' theorem twice (Sparrow, 1963): where is a point on the curve , which in turn outlines the boundary of facet , and . The orientation of the curves has to be in the positive direction with respect to the normal of each triangle. For a triangle the curve is divided into three lines described by where the endpoints of the side are given by and . By this the view factor becomes Thus the integration is taken over nine pairs of sides. If the
indices If is not parallel to , the inner integral can be computed analytically by first assuming that where Defining , the integral evaluates to If, however, is parallel to , the degree of the polynomial reduces from 2 to 1. The integral is then evaluated into If coincide with one of the endpoints of , the integrand approaches a singularity there. The integral still exists, and in Eq. (A1) this handled by , where , or . There are a few more special cases but for the surfaces considered here, only one of them needs to be discussed further. If the triangles share one side, the singularity will appear on the interval , which will require the integral to be divided into two subintervals to avoid the singularity. This procedure leads to In most cases, however, the resulting outer integral over is evaluated numerically. The reduction of the quadruple integral into a single integral improves the accuracy and computational efforts substantially. ## A.2. Generating Gaussian random surfacesThe full details behind the process of generating the surfaces are given by Muinonen & Saarinen (1997). In brief, the height above the mean plane is , and the roughness is described by a correlation function . The first step is to consider the 2-dimensional Fourier expansion of the surface: where in the limiting case, and
if is the period in the
Assuming a Gaussian correlation function: in which is the correlation length. The 2-dimensional cosine series coefficients of are then given by, where , and is the Kronecker delta. The real and imaginary parts of the Fourier coefficients are taken to be independent normal distributed random variables, with zero mean and variances given by To make the heights real valued, it is furthermore required that
. This produces a surface
, where Gaussian random surfaces are conveniently generated by the use of
the 2-dimensional inverse Fast Fourier Transform (FFT), in Eq. (A2).
For a finite surface the correlation length should be taken to be
small compared to the period , i.e. a small
. This gives over a
square grid in the ## A.3. Orienting the ellipsoid
The techniques for determining spin vectors and shapes of asteroids
from visual lightcurves have proven to be quite successful (Magnusson
et al., 1989). The low order measure of the shape is often to use the
ellipsoid. The shape is described by the axis ratios
and , where
. The spin vector is usually assumed to be
parallel to the One angle is remaining in order to specify the orientation of the ellipsoid at a given rotational phase. This is achieved by (Magnusson, private communication) defining where is the rotational angle at the time . At this time the there unit vectors parallel to the three axis of the ellipsoid are given in an ecliptic coordinate system by At a time © European Southern Observatory (ESO) 1998 Online publication: March 30, 1998 |