## 5. Normal pointsThe 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.
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 ( or ) are constants. The observational equations in the y magnitude are then which should be solved for the 3 parameters y The observations of each individual group are best represented as a normal point at phase . The epoch is determined such that the mean error of the magnitude is a minimum. If 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, 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, is substituted for 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 of the best magnitude is given by where is the observational mean error of unit of weight. The highest accuracy is obtained if () = 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 is 1 and 0, respectively. Averaging the observational equations (10) we obtain the normal point at phase 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 0.0008 at phase P = 0.8745 for the here adopted phase angle: . The phase factor is . This large value can only be due to the small phase angle. The observational mean error per unit of weight is 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 0.0059 by f = 327 - 3 25 = 252 degrees of freedom and is used in the error estimates of y and above. The average weight of the observations in group 25 is 3.4, giving the small observational error 0.0032 mag. The approximation involved by regarding the lightcurve as linear in the interval 0.8590 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 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 is 0.75 mag. For the non-linearity is given by /16, for it is given simply by . 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 Here is the component of the covariance matrix corresponding to the phase factor . 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 () 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 which is mean of the 3 phase angles. In this case we take the average values given in Table 7.
With c in the range 0.05-0.13 mag/deg we can estimate the correction to the standard phase . For group 23 the correction is of order: 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:
© European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |