4. Fourier coefficients
In Russell's expansion of the lightcurves in spherical harmonics the term has a factor
with (2n-3)!! = (2n-3) (2n-5)... . This term is 0.001 for n = 10 but it would be erroneous to conclude that a 20th-degree fit with 40 harmonic coefficients would reproduce lightcurves to 0.001 mag. It is often noted, for particular objects, that lightcurves have sharp angles at minima; this was for instance the case for 51 Nemausa in 1983. The discontinuity in the first derivative is related to Gibbs' phenomena (Carslaw p.289-310) and gives a slow convergence. This may be illustrated by a rotating, flat plate with the lightcurve
The maximum errors occur at the minima , and may be expressed in terms of the asymptotic expression for the digamma-function (Abramowitz 6.3.18)
which illustrates the very slow convergence at the singular points.
Most published lightcurves are affected by discontinuities at the beginning and end of the nights caused by a change of comparison stars and errors in the correction for phase angle and extinction. Let a small error be introduced into the composite lightcurve from phase P1 to P2. The Fourier expansion will then contain terms like
where is the unit step function
The decrease of observed Fourier coefficients to the limit given by observational errors and the convergence of the series are thus in practice less than expected by equation (5). It is difficult in practice to decide how many harmonics should be used to represent lightcurves. In retrospect, it was an error to use only 10th degree for the 1990 lightcurve. Figure 5 in Kristensen & Gammelgaard (1993) shows that this number of harmonics does not reproduce the 0.009 mag. hump (a fourth maximum) at P = 0.23 which is so clearly shown in 1983.
Another drawback of a Fourier expansion is connected to the statistical analysis of errors. If the weights of the observations are not uniformly distributed in phase the gaps will induce large correlations between the parameters. This can only be handled if we know a priori that the formula used to represent the observations is practically exact. The statistical difficulties are connected with the simultaneous treatment of all observations. In the next section we shall avoid this by analyzing small groups of observations independently.
The Fourier coefficients are, however, useful for the pole determination (Kristensen (1993)) and Table 4 gives these coefficients to degree n=20 in the lightcurve and degree 2 in the phase factor curve. The 41+5=46 unknowns are solved for rigorously by the method of least squares. Weights were given to the individual observations on the basis of their internal scatter.
Table 4. Fourier coefficients for y in units of mag. The formal mean error is of order to . The epoch of zero phase is 1994 Sept. 26.336 ET corrected for light-time. The position of the bisector is (1950).
At airmass X 1.5 the comparison stars indicated so irregular fluctuations in u and v that the observations had to be rejected. Due to the approximate 3.0 rev/day rotation frequency this rejection occured around the same rotational phase and resulted in gaps in coverage. This again gave large correlations between the Fourier coefficients in u and v. The numbers of observations retained were 310, 324, 327 and 327 respectively in uvby and among the correlation coefficients the number exceeding 0.50 were respectively 219 (the largest being +0.88 and -0.75), 21, 8 and 8.
In y and b the 8 correlations occured between the harmonics n = 13 to 15. This is connected with the accidental grouping of the observations around the 28 nearly equidistant phases as apparent in Fig. 1.
In b the coefficient to sin(20P) is 0.0006 which is - formally - significant. A fast decrease of the higher harmonics is a delusion. The reason to include harmonics to order 20 was to represent the sharp minimun at P = 0.25; to degree n=10 a much smoother lightcurve was obtained. It is important, however, that the Fourier coefficients of low order to be used for pole determination were identical within 0.0010 mag.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998