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

## 3. The ephemeris for physical observations

From a certain level of precision it is necessary to know which side of the body is being observed so an ephemeris for physical observations is indispensable.

A convenient form of ephemerides for physical observations, which separates long and short period terms, was proposed by Kristensen (1991). An excerpt of this ephemeris based on the rotational elements by Kristensen (1993) is given in Table 3.

Table 3. The table is a small extract of the physical ephemeris and gives the solar phase angle , the equation of time E, the aspect and Julian Date. The ephemeris separates the long period orbital terms from the short period rotation and give the rotational phase P or central meridian as:
P = 3.08366928 + E/360 revolutions.

The (solar) phase angle is denoted , the aspect and E is the "equation of time". The latter is defined analogous to the familiar concept. It is the planetocentric right ascension of the bisector between Sun and Earth minus the right ascension of a fictitious mean Sun moving uniformly in the equatorial plane of the planet. The main inequalities in E are 2e 7 6 due to the orbital eccentricity, tan2 ( /2) 3 2 due to the obliquity (26 6) and - in addition to these familiar terms - we have the angle /2 from the Sun to the bisector. Extreme values of E due to phase angle occur at quadratures and are +12 5 before and -12 5 after opposition, - the signs being determined by the rotation being retrograde. The rotational phase P is given by

where T* is E.T. corrected for light-time. The expression (2) defines the central meridian. The great advantage of using the synodic, rather than the sidereal rotation period, is that we can obtain the time of lightcurve features within a quarter of an hour by simply ignoring E in (2).

All quantities in the physical ephemeris depend on the slow orbital motions and may be tabulated and interpolated at large intervals. Due to the here nearly stationary value of interpolation should be in and to second order in time.

To illustrate the use of the physical ephemeris let us compute the rotational phase t days from the epoch September 26.336 E.T.= 244 9621.836. Interpolation gives, in units of revolutions,

which inserted into (2) gives

The rotational phase is practically zero at the adopted epoch. As E = 0 the apparent rotation is very uniform and the observations can be reduced by a constant period 7.78511 hours. It is a very great advantage that we do not need to solve for the rotational period.

The aspect at the epoch is = 108 5. Table 1 in Kristensen (1993) gives the aspects of earlier oppositions. Aspects are = 112 7 in 1983 and = 116 9 in 1990 so these oppositions should be directly comparable with the present one. Opposite lightcurves have aspect and occurred in 1989 and 1991 .

The 1983 lightcurve (Fig.4 in AN 306(1985)) has sharp minima at 0.24, 0.59 and 0.82 but the origo adopted in this figure has phase +3 11. The same minima should then occur in 1994 at phase 0.249, 0.599 and 0.829, in good accordance with 0.25, 0.59 and 0.83, in the present Fig. 1. The good consistency may be due to the similar values of , respectively 1 85 and 0 94. Important is also that the small (0.01 mag.) peak (the fourth maximum) around 0.20 is confirmed. Fig.2 p.347 in Kristensen & Gammelgaard (1993) gives minima at phase 0.26 and 0.87 in 1990 at . With the correction -7 47 in phase this corresponds to 0.239 and 0.849, or -0.011 and +0.019 rev. relative to the small phase angles above. This may be compared with the bisector angle /2 = 0.018 rev. and may be regarded as a confirmation of the usefulness of the bisector. Phases over 11 years are thus consistent with mean errors of order 0.01 revolutions.

 Fig. 1. Plot of all y observations reduced to absolute y magnitude with 25 phase factors to = 0:O942 corresponding to September 26.3.
 Fig. 2. Plot of the 28 normal points in Strömgren y absolute magnitude reduced with 25 phase factors to 942 corresponding to September 26.3. The solid curve corresponds to a Fourier series of order 10 and the dotted to a Fourier series of order 20. The 1983 and 1990 lightcurves with the same aspect are given for comparison. The 1983 lightcurve at 85 is from Kristensen & Gammelgaard (1985) but with the rotational phase corrected by rev. The 1990 lightcurve at 01 from Kristensen & Gammelgaard (1993) has rotational phase corrected by -0.021 rev. The curve is given by 30 normal points and a Fourier series fit to order 10. Note that the small peak (a fourth maximum) at rev. can not be reproduced by Fourier terms to order 10. The 1990 and 1994 curves may be different because of the large () phase difference. The 1983 and 1994 oppositions were within in the sky, but the equality of the 1990 and 1994 aspects is derived from the adopted pole solution.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998