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Memorandum N 5
01.07.997
Topocentric (astronomic) coordinates of planets

Stellar and planetary aberration. In principal we do not intend to take into account the aberration correction in the calculations, i.e. we use the simultaneous coordinates of planets, such as they are in, for example, DE405/LE405. I have at hand the algorithm of the stellar and planetary aberration calculation made directly in the framework of the special theory of relativity (one time I made it for Tarasov having lost hope to understand how traditional astronomers do it). I'll publish it later, as there is no need to use it in astrology (maybe). For the precision calculation of the aberration correction one have to use the general theory of relativity (a little number of people throughout the world knows how to effectuate the corresponding caculations; I can give the reference upon request).

Coordinates of planets relatively to the chosen centre. DE405/LE405 gives the coordinates of planets relatively to the baricentre (mass centre) of the solar system. It is clear that for this it is necessary to recalculate TT into TCB, since the ephemeris has been calculated exactly in the baricentric time (TCB). However in the Interoffice Memorandum on DE403/LE403 we have not found a word on this. In the corresponding program in fortran TT is mentionned as an input argument. Even though we mistaken in this, the error in the position of planets, due to the mistake in the time or timelike input argument, will be much less than the internal error of the ephemeris calculation (unverified statement). So we'll use TT as an input argument in DE405/LE405. It is necessary for the calculation, for instance, of the geocentric vector of a planet to subtract from its baricentric vector the baricentric vector of the Earth; for the calculation of the topocentric vector it is necessary to subtract the baricentric vector of the observer's position on the Earth surface. To calculate the angular coordinates (spherical coordinate system) of a planet it is necessary to take the corresponding trihedron and to compute the polar and azimutal angles (see the Memorandum N2).

Coordinates of the Earth and Moon. The ephemeris LE 405 gives us the coordinate of the Moon relatively to the baricentre Earth-Moon. The cartesian coordinates of the Earth-Moon baricentre are given in the ephemerides DE 405. For the calculation of the position of the Earth's centre of the Land relatively to the baricentre Earth-Moon it is necessary to take with the inverse sign the coordinate of the Moon relatively to the baricentre Earth-Moon and divide the result by the ratio of masses Earth/Moon, equal to 81.300585 (see IERS Technical Note 21, 1996).

The residuary part of the memorandum is devoted to the calculation of the vector of position of the observer on the Earth's surface relative to the center of gravity of the Earth.

Input of the coordinates on the Earth's surface.
Here we do not discusse the possibility to have as a source information the 3 cartesian coordinates from a GPS (or GLONASS) receiver. Let us assume that we have geographical, or astronomical (according the plumb line) angular coordinates of a point on the surface of the Earth, as well as the excess over the quasigeoid, which is taken out, as well as the angular components, from geographical maps. Both geographical and astronomical coordinates are elliptical angular coordinates. The coordinates on the Earth's surface are needed for 2 things: (a) for the determination of the geocentric vector of an observer on the Earth's surface and (b) for the determination of the gravitational vertical line or the vertical line of gravity in the observer's location. To complete the description it is necessary to add both to the geographical and astronomical coordinates the deviation of the plumb line in the considered point. To reconstruct from the geodetic (geographical or astronomical) coordinates the geocentric vector it is necessary to know in what geodetic system they are obtained. Usually this information is marked on the map. Maps (at least the majority of the available maps) were charted in the presatellite epoch.

Let us understand under the geodetic system a combination of trihedron and 2 parameters of a reference-ellipsoid (in fact it is necessary to add the scale factor). As it turned out, the trihedrons of different geodetic systems are rotated relative to each other, their origins are shifted. The scales are also different. The problem of the mutual reduction of different local geodetic system was solved by the american and soviet military departments in the eighties. At present this information is open. It turned out that real geodetic systems contain the nonlinear distortions, which limit the real accuracy of coordinates in some local geodetic system, interpreted as ideal geodetic (i.e. ellipsoidal + excess over reference-ellipsoid), by 3-4 metres. Local geodetic systems are reduced to the WGS-84 system. This system complies with ITRF-94, which is the last realization of the TRF, appearing in the theories of the precession and nutation (see Memoranda NN 3 and 4), with the accuracy approximately up to 0.5 metre.

Transformation of geographical coordinates into the cartesian coordinates of a local geodetic system. Let we have three ellipsoidal coordinates (H, B, L), where H - an excess of a point over the reference-ellipsoid, B - ellipsoidal latitude, counted out from the equator, and L - a longitude of this point.
H=H0+h, where h - the excess of a point over the geoid (more exactly quasigeoid), and H0- the excess of the quasigeoid over the reference-ellipsoid. Namely h can be taken from the topographic map. One of the following memoranda will be devoted to the calculation of H0.
X1=(N+H)cosBcos
X3=(N+H)cosBsinL
X3=[N(1e2)+H]sinB, where
N=ae(1e2sin2B)1/2
ae and e are the equatorial semi-axis of the reference-ellipsoid and its eccentricity, accordingly. The last parameter e is usually given by the compression coefficient f according the formula e2=2ff (see file ellips.ini). There is an inverse non-iterative transformation (Bowring 1985) . I have it at hand, but what is the need, unless for the dialogue window.

Vertical line of the gravity force and the plumb deviation. By the plumb line instead of pair of the geodetic (geographical) coordinates B,L the astronomical coordinates (j,l) are determined, which are required for fixation of the vertical line of the gravity force. They are connected with geographical coordinates by formulae
x=jB, h=(lL)cosj.
h=(lL)cosj.
It is necessary to assign both parameters of the plumb deviation (x,h) in the dialogue window together with (h, B, L). Accepted names: x- component of the plumb deviation in the meridian plane, h - component of the plumb deviation in the plane of the first vertical.

Transformations between different geodetic systems. The transformation between two geodetic systems is affine and depends on 7 parameters: 3 parameters of shift, homothety (isotropic dilatation) and 3 rotations. Traditionally such transformation in the cosmic geodesy notation is written, in view of smallness of the rotations and small difference of scale factor d from unity (d=1+D, where D - differential scale change), in following form:

Indices in this formula comment coefficients in the file geod.ini, which accompanies this Memorandum. The part of the ini-file refering to this section keeps the parameters of the transition from WGS-84 into any other. However if we do a transformation there-back according to this formula, we do not obtail the initial values. Therefore we modify the algorithm. As a result we exceed the accuracy, but obtain the one-to-one correspondence. The following section describes the rules of transformations which garantee the immutability of input values under repeated transformations from one system into another.
The problem is that in our concept of calculations there is no place for affine transformations. Our reference frame is given by trihedron. So let us adopt in our calculations that a geodetic system is a trihedron (quasi-system) + ellipsoid + scale factor D (with respect to WGS-84) + shift of the geodetic system from the centre of mass of the Earth.

Transformation rules between different geodetic systems.
The flow of time TT on the surface of the Earth corresponds in general theory of relativity to the scale of the metre SI (doubtful statement). Having effectuated the transformations from ICRF to ITRF (precession/nutation) we have got the trihedron, which it is necessary to name as linking for the sake of comfort of programming. Today we consider that it is ITRF-942 . Tomorrow it will probably be another system connected with ITRF-94 by the non-identity mapping. Therefore in the file geod.ini this system (linking) will appear under this name unlike the others, and its true name will be reported in comments. Only this system, from our point of view has a true scale of distance. However all scale factors are given relative to WGS-84. Due to this in the future, when passing to the new linking system, no need to rewrite the whole block of data. It will be only necessary to add a new line, name it as linking, and give back to the old linking system its true name from the old comment.

Let us introduce the notion of quasi-system. Example: quasi WGS-84. This system has its centre in the centre of mass of the Earth and the correct scale with the same orientation of axes as the true WGS-84. The only system coinciding with its quasi-system is the linking system.
1. The consecution of transformations from the linking system (quasi-system) into any other quasi-system is defined by following expression
[1:2:3]quasi-system=R1(r1+ r1linking)R2(r2+ r2linking)R3(r3+ r3linking) [1:2:3]linking
2. The consecution of transformations from any geodetic quasi-system into linking system is inverse in order to ensure the one-to-one correspondence of transformations.

The calculation of cartesian "components" of a geocentric vector in some geodetic system. Having got the trihedron of a given quasi-system, we calculate the components of the considered vector. Then we add to these components the components of the centre of gravity of the Earth (the origin of the linking system) in this geodetic system, which are equal to t"tlinking (where t=(t1; t2; t3)) and multiply the obtained components by 1+D"Dlinking

The reconstruction of a geocentric vector from its cartesian "components" in some geodetic system. Having got the "components" of the considered vector in a given geodetic system, we divide them by 1+D"Dlinking and subtract from the obtained components the components of centre of gravity of the Earth (the origin of the linking system) in this geodetic system (t"tlinking). Then we multiply the 3 basis vectors of the trihedron of the given quasi-system by these three just obtained components correspondingly and add together the 3 vectors.

Agreement. As a result of the precession/nutation unity work we get the orientation of the trihedron of the linking system. This system, by agreement, is a single system, coinciding with its quasi-system (i.e. its D 0 by definition).

To pass to the "components" in another geodetic system it is necessary to perform 2 steps: first to reconstruct the geocentric vector using the linking system, second - to calculate the cartesian "components" in the new geodetic system.