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

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5. Normal points

The observations of Nemausa were given in series of 5 planet observations interrupted by an approximately constant time interval for the star observations. By chance these groups happened to have nearly the same phase on the 3 nights and this is the reason for the grouping of the observations in phase as is visible on Fig. 1. This is illustrated by group No. 25 in Table 6 which has 5 observations from each night from phase 0.859 to 0.886. Before and after this group are gaps in phase from 0.847 to 0.859 and from 0.886 to 0.893.


Table 5. Contains normal points from 28 groups in rotational phase P. A line gives group number, P, and in order phase factor with mean error, magnitude with mean error. All multiplied by [FORMULA]. The lightcurve refers to 0.942 deg. The epoch of zero phase is 1994 Sept. 26.336 ET corrected for light-time. The position of the bisector is [FORMULA] (1950).


Table 6. The 15 observations in group 25 from rotational phase 0.8590 to 0.8863. The columns are: rotational phase P, night number from 1 to 3, solar phase angle in units of degrees, airmass X, observed differential magnitude b, adoped weight [FORMULA], [FORMULA] residual, observed differential magnitude y, adopted weight [FORMULA], and [FORMULA] residual. Equation (10) with [FORMULA] is solved by least squares. The most accurate normal point is obtained by (11) at [FORMULA] and gives [FORMULA] and [FORMULA]. The phase factors are [FORMULA] mag/deg and [FORMULA]. These values are transferred to Table 5.

In the small phase intervals of each group we can assume that the lightcurve can be approximated by a straight line and that the phase factors ([FORMULA] or [FORMULA]) are constants. The observational equations in the y magnitude are then


which should be solved for the 3 parameters y0, y [FORMULA] and [FORMULA] by least squares. It is only possible to use the fundamental equation (10) because the rotational phase P is regarded known as a function of time and the rotation period need not be solved for. Due to the phase factor variation the concept of a lightcurve makes only sense when referring to a definite value of [FORMULA]. The groups Nos. 1, 23 and 28 have only 5 observations from the third night and can not be reduced to other phase angles. The mean [FORMULA] of the respective phases 0.952, 0.928 and 0.947 is adopted as standard phase.

The observations of each individual group are best represented as a normal point [FORMULA] at phase [FORMULA]. The epoch [FORMULA] is determined such that the mean error of the magnitude is a minimum. If [FORMULA] is the covariance (reciprocal) matrix corresponding to the normal equations the most accurate magnitude is obtained for


Due to the dependence on the phase angle, [FORMULA] differs a little from the mean value of the epoch. These deviations were in the present cases very small. Introducing (11) into the normal equations and solving, we obtain a simple expression for the best magnitude


where the observations of uvby are denoted o, [FORMULA] is substituted for [FORMULA] and the brackets ( ) is Gauss' notation for weighed sums. We note that the expression (12) is independent of the distribution of the phases (does not contain sums like (Po), (PP) etc.) The mean error [FORMULA] of the best magnitude is given by


where [FORMULA] is the observational mean error of unit of weight. The highest accuracy is obtained if ([FORMULA]) = 0, that is at the mean value of the phase angle.

Formulae (11) and (12) define weighted averages of P and o, respectively. This average of 1 and [FORMULA] is 1 and 0, respectively. Averaging the observational equations (10) we obtain the normal point [FORMULA] at phase [FORMULA] with the smallest possible variance (13) or the highest possible accuracy.

Table 6 shows the O-C residuals for the 15 y and b observations in group 25. The most accurate value is y = -0.0128 [FORMULA] 0.0008 at phase P = 0.8745 for the here adopted phase angle: [FORMULA]. The phase factor is [FORMULA]. This large value can only be due to the small phase angle. The observational mean error per unit of weight is [FORMULA] 0.0055 by f = 15 - 3 = 12 degrees of freedom. On basis of all 327 observations in 25 groups this mean error is accurately determined to [FORMULA] 0.0059 by f = 327 - 3 [FORMULA] 25 = 252 degrees of freedom and is used in the error estimates of y and [FORMULA] above. The average weight of the observations in group 25 is 3.4, giving the small observational error [FORMULA] 0.0032 mag.

The approximation involved by regarding the lightcurve as linear in the interval 0.8590 [FORMULA] can easily be estimated by the use of the analytical form of the y-curve given in Table 4. The y-curve is linear within [FORMULA] 0.001 mag., which is accceptable in view of the observational errors. The non-linearity is mainly due to the higher harmonics (n=11 to 20 incl.) and the interpolation error is


where the root-square-sum of [FORMULA] is 0.75 mag. For [FORMULA] the non-linearity is given by [FORMULA] /16, for [FORMULA] it is given simply by [FORMULA].

If two nights with different phase angles are available then the constant difference between these two nights and hence the phase factor can be determined. The mean error in the determination of the constant difference between two series of respectively n and [FORMULA] equally good observations around mean phases [FORMULA] and [FORMULA] is given by


Here [FORMULA] is the component of the covariance matrix corresponding to the phase factor [FORMULA].

This error may be large if the distance between the two groups is large compared to the number of observations and their scatter around their mean phases. However, if only a single observation from one night is situated between two observations from the other night then the critical, phase dependent, term in the radical in (15) has the upper limit 2. Hence, in this case it will always be possible to derive an approximate phase factor if the difference between the phase angles ([FORMULA]) is large enough. In the present situation there were 7 groups (viz: 2, 3, 11, 21, 22, 26 and 27) which had observations from two nights only and which did not overlap. These groups were, however, near enough to give useful results. Three groups (1, 23 and 28) contained only observations from a single night and no phase factor could be derived. In such case the normal points on the lightcurve can not strictly be reduced to a standard phase angle. To minimize this inconvenience the adopted phase angle is chosen to be the [FORMULA] which is mean of the 3 phase angles. In this case we take the average values given in Table 7.


Table 7. Mean values of y and b magnitudes observed only one night. The rotational phase (P) and the phase angle ([FORMULA]) are averaged with the same weights as y and b. The variation of [FORMULA] a single night is too small for a determination of the phase factor. Instead the value c = [FORMULA] mag/deg was adopted and used for the correction (corr) to phase angle 0.942 deg. The resulting normalpoint is transferred to Table 5.

With c in the range 0.05-0.13 mag/deg we can estimate the correction to the standard phase [FORMULA]. For group 23 the correction is of order: [FORMULA] 0.0006 mag. The last column in the above table gives these corrections.

The observational mean errors of unit of weight were also obtained for the individual groups and were consistent with the following adopted mean values for uvby:


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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998