Geocentric Parallax

Theoretically the direction of a celestial object as seen from a station on the Earth’s surface (its topocen-tric direction) is not the same as the geocentopocen-tric direction of the object. In practice, if the object is a star the directions are indistinguishable; if the object is the Sun, the angle between them can be as great as 8·8 ; for the nearest planet the angle can amount to about 32 , while its value for the Moon can be about 1°. For a close artificial satellite, the direction as seen from a station on the Earth’s surface can be the best part of 90° different from the satellite’s geocentric direction.

The topocentric equatorial coordinates of the object must be transformed now to the centre of the Earth to get rid of this geocentric parallax due to the finite size of the Earth.

Infigure 3.5the observing station at O on the Earth’s surface, distanceρ from the Earth’s centre C, tracks a satellite V, distance r from O and r from C. The meridian from the Earth’s North pole P through O meets the terrestrial equator A in A where is the direction of the vernal equinox. The direction of as seen from O is O parallel to C . The geocentric and astronomical latitudes of O are O A (φ ) and O A (φ) respectively.

Now let angle A =θ. As the Earth rotates and carries the observer round with it, angle A increases. But angle A is the LST of the observer; therefore

If a set of non-rotating rectangular axes C , CY and CP are taken as shown, then the coordinates of O are given by

whereθ is given by equation (3.23).

Figure 3.5

If the semimajor and semiminor axes of the elliptic cross-section of the Earth (an arc of which is PÔA) are a and b respectively, then it may be shown (Smart and Green 1977) that

and that

where e²= 1 − b²/a².

It should be noted that the distance ρ refers to sea level. If the station O is at height h above sea level thenρ should be increased to (ρ + h).

The instantaneous rectangular coordinates of the station can now be computed.

The observed data are the apparent right ascensionα and declination δ of the vehicle (that is, with respect to a celestial sphere with the observer as origin). The distance r is not in general known, ex-cept approximately, unless range measurements are also being made.

It is desired to obtain the geocentric right ascensionα, declination δ and distance r of the vehicle by removing the effects of geocentric parallax. The problem is seen to be analogous to parts (ii) and (iii) of example 3, chapter 2, section 2.9.2.

Take a set of rectangular axes O , OY , OP through O, parallel to the axes C , CY, CP respec-tively and let the rectangular coordinates of V relative to the set of axes through O be x , y and z .

Then

If the geocentric rectangular coordinates of V are x, y and z, then

Obviously

Hence, substituting equations (3.24), (3.25) and (3.26) into equation (3.27), the resulting relations can be solved to giveα, δ and r in terms of α , δ and r .

Also involved will be the known values ofρ, φ and θ. The three equations are

In practice it is often more convenient to compute (α − α) and (δ − δ).

Multiplying (3.28) by sin α and (3.29) by cos α and subtracting gives Multiplying (3.28) by cos α and (3.29) by sin α and adding gives Dividing (3.31) by (3.32) gives

Putting in equation (3.32) and using equation (3.31) we obtain,

after a little reduction

Let the quantities m and γ be defined by

Then

and by equation (3.30)

Multiplying (3.34) by sin δ , (3.35) by cos δ , and subtracting gives Multiplying (3.34) by cos δ , (3.35) by sin δ and adding gives

Hence, from equations (3.36) and (3.37) we have or

where

In a similar fashion using equations (3.34) and (3.35) we may obtain

The four equations (3.33), (3.39), (3.40) and (3.41) are rigorous and give the corrections for geo-centric parallax. Several cases may be considered:

(i) Object at distances well beyond the Moon’s distance (for example, an interplanetary probe). The corrections (α − α) and (δ − δ) are much less than 1°, since ρ/r is much less than 1/60.

Then, if (α − α) is expressed in radians, equation (3.33) may be written, to sufficient accuracy,

Similarly, equation (3.39) may be written as

whereγ is given by Also, from equation (3.41)

To use these equations, the value of r , as well as values of α and δ must be known. This is usu-ally satisfied in practice.

(ii) Object at lunar distances (for example, an artificial lunar satellite). Again the quantities involved, namely (α − α), (δ − δ) and (r − r) are small corrections. The angles are of order 1° or less, while the quantity (r − r) is of order 1/60 of the vehicle distance or less. The angles α and δ are meas-ured easily and accurately; the range r is also accurately measmeas-ured by radar. Hence equations (3.42)–(3.45) may be used as in (i), though the rigorous equations (3.33), (3.39), (3.40) and (3.41) should be used for objects moving between Earth and lunar orbit distance.

(iii) Object at distances similar to the radius of the Earth (for example, an Earth satellite). The rigorous equations must be used.

The quantities (α − α), (δ − δ) and (r − r) are no longer small. The range r may be either meas-ured directly by high-accuracy radar or, if the satellite is in an established orbit, may be known ap-proximately. If neither of these criteria is satisfied then the corrections for geocentric parallax cannot be applied so simply. Observations from at least two places on the Earth’s surface are required to ob-tain a measure of the distance. If two stations O and O observe the satellite simultaneously then they each obtain its apparent position. Let these positions be given by (α , δ ) and (α , δ ). Its geocentric position is (α, δ). If its distances from O , O and the Earth’s centre are r , r and r, then there are five unknown quantitiesα, δ, r , r and r. Equations (3.28), (3.29) and (3.30), applied first to O and then to O , give six equations in the five unknowns so that they can be determined. In practice, it is unlikely that observations are made simultaneously; the reduction is therefore rather more complicated.