Appendix A. Technical details
The 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 triangles
The 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 k and l are dropped for simplicity, the contribution from one pair is
If is not parallel to , the inner integral can be computed analytically by first assuming that
Defining , the integral evaluates to
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 surfaces
The 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 x and y -directions.
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 z is normal distributed with zero mean and standard deviation . The roughness of the surface is described by , which is the r.m.s. of the slopes and .
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 xy -plane, with points along each side. Since the periodicity of the surface is used, the first row and column of the grid is repeated to complete the period. The resulting side has a length and consists of points. The example surface in Fig. 1 was generated using , , and .
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 c -axis, and its orientation in space is given by the ecliptic coordinates and (assuming a right handed system).
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 t these vectors has to be transformed by
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998