Precession and Nutation

Up until now it has been assumed that the planes of the ecliptic and the equator are fixed with respect to the stellar background, in the sense that the right ascensions and declinations of the stars referred to the equator and the vernal equinox (one of the two points where equator and ecliptic intersect) do not change. Due to the gravitational attractions of Sun and Moon on the aspherical Earth, however, the Earth’s axis of rotation precesses, so that the north celestial pole P describes a small circle of radius (= ) about the pole of the ecliptic K in a period of about 26 000 years. The ecliptic remains fixed and the vernal equinox moves backwards along it (that is, in a direction such that the celestial lon-gitudes of stars increase) at a rate of about 50 per annum. This is called the luni-solar precession.

It is seen fromfigure 3.2that in general, due to luni-solar precession, the celestial latitude of a star (given by BX) will not change, but that its celestial longitude B will change, increasing by about 50 per annum. Both right ascension and declination, A and AX respectively, will alter in a manner depending upon the star’s present^RAand^DEC. It is easily shown (Smart 1956) that ifθ is the luni-solar precession for one year, a star’s^RAand^DECwill change in that time to (α₁,δ₁) where

Figure 3.2

It is to be noted that these formulae are obtained under the assumption that the changes in the co-ordinates are small.

A further effect due to the Sun and Moon is called nutation, a complicated oscillation of the pole P about the position it would occupy if precession alone acted. Nutation may be broken up into a series of periodic terms depending upon the elements of the orbits of the Sun and Moon about the Earth, their periods being small in comparison with that of the luni-solar precession. In addition, due to nutation, the value of the obliquity of the ecliptic oscillates about a mean value.

The planets themselves affect the Earth’s orbit, resulting in a slow change in the orientation of the ecliptic. This so-called planetary precession decreases the right ascensions of all stars by about 0·13 per annum.

General precession may now be defined as the combination of luni-solar precession and planetary precession. Due to general precession the ecliptic and equator and the vernal equinox will change. If their positions are taken at, say, the beginning of 1950 (1950·0) they may be regarded as fixed planes of reference. Their changed positions in 1951·0, due to general precession, are called the mean eclip-tic, mean equator and mean equinox for 1951·0.

The valueχ of the general precession in longitude and the obliquity of the ecliptic at an epoch t years after 1900 are given by

and

The mean position of a star is its^RAand^DECreferred to the mean equator and equinox of a speci-fied time for a heliocentric celestial sphere (that is, no notice is at present being taken of nutation, aber-ration, stellar parallax or the star’s proper motion, the latter three quantities being defined below).

Equations (3.4) and (3.5) are now generalized to include planetary precession, which decreases right ascension by l (= 0·13 ) in one year and has no effect on declination.

We obtain for the changes in right ascension and declination in one year due to general precession

Putting we obtain

Both m and n vary slowly with time. Thus

For periods longer than 5 years, equations (3.6) and (3.7) are inadequate and a quantity called the an-nual variation is introduced. If the year is taken as the unit, and dα/dt denotes the rate of change of α due to precession, then from equation (3.6) we have

The rate of change of dα/dt per century is defined as the secular variation s in right ascension. Then, neglecting changes in s itself, we have

where the suffix zero denotes evaluation at the earlier epoch, and t as before is in years.

Also, Similarly

where s is the secular variation in declination given by

In the principal star catalogues are given, together with the secular variations, quantities called the annual variation in right ascension and declination. These latter quantities are the annual precessions dα/dt and dδ/dt plus the star’s proper motion (section 3.6).

The true position of a star at any time is its heliocentric right ascension and declination referred to the true equator and equinox of that date. By applying nutation, the mean position computed for that date may be converted to the true position at that date. It has been seen that nutation changes the

lon-gitude of a star and also the obliquity of the ecliptic. If ∆ψ and ∆ denote these changes for the date in question, they may be computed. The change ∆₁α due to ∆ψ and ∆ is then given by

with a similar expression for the change in declination due to nutation at that time.

But the change in^RAdue to precession from the beginning of that year to the present date (a frac-tionτ of a year) is ∆₂α where, using equation (3.6),

Combining ∆₁α with ∆₂α and remembering that

we obtain

If we now express m and n in seconds of time, and l, ∆ψ, θ and ∆ in seconds of arc, and introduce quantities A, B, E, a and b defined by

then

with the right-hand side expressed in seconds of time.

Similarly it is found that

where a = n cosα, b = − sin α, and n is in seconds of arc.

The quantities A, B, E are not functions of the star’s position, and are tabulated in the almanacs for every day of the year under the heading Bessel’s day numbers (or star numbers). The quantities a, b, a , b can be computed for the star concerned.

The procedure to obtain the true position of a star at a given epoch (a date in a particular year) from its mean position in a catalogue of epoch 1950·0 is thus as follows:

(i) Calculate the mean coordinates at the beginning of the year in which the date occurs.

(ii) Change these mean coordinates to the true coordinates for the date in question.

There remains one final correction: namely, to change the origin from the Sun’s centre to the Earth’s centre. This gives the apparent place of the star at that instant which is the position on the geocentric

celestial sphere with respect to the true equinox and equator at that time. The difference between ap-parent place and true place is due to aberration and annual stellar parallax (sections 3.5and3.7). An-ticipating, it is found that except for a very few near stars parallax can be ignored, while the correction due to aberration is of the form

where C and D are tabulated in the almanacs and c, d, c and d are functions of the star’s position.

The star’s geocentric apparent position is now known for the time of observation, in terms of^RAand

DECreferred to the true equator and equinox at that date.

The reverse procedure is adopted when the positions of the brighter stars are measured. By apply-ing the correction for refraction, the star’s geocentric apparent position is found. The application of equations (3.14), (3.15), (3.16) and (3.17) gives the mean coordinates referred to the mean equator and equinox at the beginning of the year in which the observation took place. By applying equations (3.8)–(3.13) the star’s mean coordinates can be obtained relative to the equator and equinox of the epoch of the star catalogue in which it appears. Information concerning its proper motion (section 3.6) can then be obtained.

Photography is employed for the measurement of the positions of the fainter stars. On any photo-graphic plate there are usually a number of stars whose coordinates have been determined and cata-logued already. They can be used as reference stars with which to obtain the positions of the faint stars.

In practice measurements are made from the negatives on various types of plate-measuring engines, since making a positive inevitably introduces some blurring. The measurements made are of the x and y coordinates of the image with respect to a set of rectangular axes Ox and Oy.

In theory, these axes are chosen such that:

(i) the origin lies on the optical axis of the telescope which corresponds to a given^RAand^DECreferred to the mean equator and equinox of, say, 1950·0.

(ii) the y axis is the projection of the great circle through the north celestial pole for 1950·0 and the point towards which the telescope is pointing, and

(iii) the x axis is drawn at right angles to the y axis.

In practice, errors enter due to bad orientation, scale error, nonperpendicularity of axes, wrong cen-tre and tilt of the photographic plate’s plane to the plane perpendicular to the optical axis. In addition, refraction and aberration produce their effect. Two sets of coordinates are therefore distinguished; the measured coordinates x and y of the star image, and the standard coordinatesξ and η that have to be found, free of the above sources of error. Fortunately, they are connected by the simple equations

Only in special cases (see Smart 1956) do quadratic terms in x and y have to be introduced. The quan-tities a, b, c, d, e and f are called the plate constants and have to be calculated.

On the plate will appear a number n of stars whose standard coordinates (ξ, ηi) (i = 1, 2, 3… n) are already known since they are already catalogued. If their measured coordinates (x_i, y_i) are obtained

from the plate, the plate constants can then be computed by the method of least squares or a similar process from the set of equations

The standard coordinates (ξ, η) of the star in question can then be calculated from equations (3.18) and (3.19). These can now be transformed into equatorial coordinatesα and δ with respect to the ob-server for the vernal equinox and equator involved, this vernal equinox and equator being the one the reference stars’ coordinates are themselves referred to.

The formulae involved in this process are

In these equations, A and D are the right ascension and declination of the theoretical plate centre.

For an object within the Solar System, the star is replaced by the object (planet, satellite, spacecraft), but the principles outlined in this section and in section 3.3 are changed only in detail. It is to be noted that where the instrument used gives the object’s altitude and azimuth, the right ascension and decli-nation obtained from these quantities, corrected for refraction, are with respect to the true equator and equinox of the time of observation.

3.5 Aberration

Due to the finite velocity of light, an apparent angular displacement of a star towards the direction of the observer’s own motion relative to the star takes place. Thus the annual revolution of the Earth in its orbit produces an annual displacement on the celestial sphere of each star in an ellipse of major axis κ = 20·47 . It has been seen that this effect is taken care of in the measurement of stellar image posi-tions on a photographic plate.

For stars individually observed, the aberrational displacements in (i) equatorial and (ii) ecliptic co-ordinates are given as follows, the suffix 1 denoting the star’s coco-ordinates affected by aberration:

(i) where

The quantities C and D, functions only of the Sun’s longitude , are given as log C and log D (Bessel’s day numbersor Besselian star numbers) in the almanacs.

(ii)

where stands for the longitude of the Sun.

In the case of an object in the Solar System, the relative velocity with respect to the observer’s sition produces an aberrational effect. In general this is different from stellar aberration so that the po-sition given by its image on a photographic plate must be corrected. If the approximate distance and velocity of the object are known, as is usually the case, this correction may easily be made.

It can be shown (Smart and Green 1977) that ifν is the relative velocity of the observer and the ob-ject, c being the velocity of light, then

whereθ is the angle between the direction of the object as viewed by the observer and the direction in which the observer is travelling relative to the object. ∆θ is the shift due to aberration in seconds of arc, and k = 206 265(ν/c).

Thus, in figure 3.3, the object’s true direction OV is displaced by aberration to an apparent direc-tion OV , where at the moment of observadirec-tion the observer O is travelling with velocity ν towards A, relative to V. The velocity ϖ is compounded of the object’s velocity relative to the Earth’s centre and the observer’s rotational velocity on the Earth’s surface relative to the same centre.

The shift ∆θ produces shifts ∆α and ∆δ which can then be computed from the geometry of the sit-uation, though it should be noted that k cannot be simply inserted in place ofκ in the above equations.

Figure 3.3