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Astron. Astrophys. 322, 679-686 (1997)

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6. The phase factors

Table 5 gives our main results which are the normal points for all 28 groups. The table gives the color (index 1..4 is uvby), the phase factor and its mean error, the phase [FORMULA] and the magnitude and its mean error. For groups 1, 23 and 28 the phase factor could not be derived directly so for color y the estimates from the table above were used.

A statistical "analysis of variance" shows that the values in Table 5 can not be regarded as an observation of the same phase factor. The variance between groups is significantly larger than the variance inside groups. The variance ratio is: F = 9.10, with 24 degrees of freedom between groups and 51 inside groups; the 1% limit is 2.17 . This highly significant variation between groups is the phase factor variation found earlier in the 1990 and 1991 oppositions (Kristensen & Gammelgaard (1993)) and shown on Fig. 3. The correlation coefficient between [FORMULA] and [FORMULA] is r = 0.90 and is highly significant; the 1% limit is 0.505.

[FIGURE] Fig. 3. The 25 phase factors [FORMULA] and [FORMULA] plotted over two revolutions in order to see better the 3 maxima. The third plot is the 28 differences between b and y normal points.

Fig. 3 shows the phase factors [FORMULA] and [FORMULA] from Table 5. The quite independent values in y and b obtained by two different photomultipliers seem to support each other and indicate a structure with three maxima. The observed phase factors are too large to be explained by an effect of small errors in rotational phases in the subtracted magnitudes (dy/dP=1.0 mag/rev at max., with errors [FORMULA] 0.01 in P it only may explain 10% of the phase factors).

In Fig. 4 the 1983 lightcurve at phase angle [FORMULA] =1.84 is compared with the 1994 curve at phase [FORMULA] at nearly the same aspect. The V magnitudes in 1983 are interpolated from table 8 in Astron. Nachr. 306 (1985) 246 and the figure gives c=(V(1983)-y(1994))/ [FORMULA]. Due to undetermined constants in V and y (resulting in values [FORMULA]) the phase factor can only be determined apart from a constant. We note, however, the three maxima around 0.2, 0.5 and 0.8 and the two minima 0.4 and 0.65.


[TABLE]

Table 8. The maxima of [FORMULA] (P) for four different phase factor curves. For comparison, the maxima of the 3P-component in the lightcurves are given and denoted Max. 3P. The aspect dependent 3P feature dominates the variation in shape of the lightcurves.


[FIGURE] Fig. 4. Phase factors computed as c=(V(1983)-y(1994))/0[FORMULA]898. V(1983) is observed at [FORMULA] = 1[FORMULA]84 and y(1994) is observed at [FORMULA] = 0[FORMULA]942 at nearly same aspect.

The phase factor variation found earlier is confirmed and the more numerous data now available seem to indicate the presence of three maxima in the phase factor curve. Let us assume that the non-linear phase curve y(G, [FORMULA]) depends on a slope parameter G(P) which is a function of rotational phase P and assume that for all [FORMULA] and [FORMULA] the difference [FORMULA]) is a monotonous function of [FORMULA]. Then all phase factor curves dy/d [FORMULA] are similar in the sense that if for a given [FORMULA] we have

[EQUATION]

then the same relation should be valid for all values of [FORMULA]

[EQUATION]

This means that the phase factor curves should have extrema at the same rotational phases.

We note from figs. 3 &  4 that local minima occur at 0.35 and 0.75, maxima at 0.55 and 0.9. The two minima are, however switched, that is primary and secondary are interchanged when compared with Fig. 3 in Kristensen & Gammelgaard (1993). In both y and b minimum occur at phase 0.37 rev and maximum at 0.91. The average value of [FORMULA] differs from the constant =0.0909 mag/deg found by the Fourier series in Table 4. The two quantities refer to different averages, one integrates all phase and the other one only a part where the higher values around phase 0 is excluded.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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