The Computation of Orbit Parameters from Interferometer and Doppler Data

E. G. C. Burt
1958 Proceedings of the Royal Society A  
48 Precision op observation required The following is a very rough guide to the kind of precision required in order to obtain useful data from the comparison of theory and observation. I t is clear th a t the radio observations are affected by the conditions in the ionosphere, so th at an independent ephemeris to relatively low accuracy, which can be derived from observations with a precision of about 1° or Is, will be useful in their interpretation. For prediction purposes observations to a
more » ... ater precision than 0°-l or 0sT can not at present be fully used, but such observations are of great value. Observations less precise than 5° or 5s are of little use, unless they are isolated in a period devoid of more precise observations. For periodic effects due to the departure of the gravitational field of the earth from spherical symmetry, the orbital longitude must be known to within a few seconds of arc; observations must be better than 0o,01 or 0S-01. No such precision is necessary for the secular effects which are cumulative with time. The requirements for geodetic work are even more severe, and precisions of 0°-001 or 0S-001 are desirable. The effects of atmospheric drag can be studied from even low-precision observa tions, especially in the final stages when the satellite is spiralling inwards rapidly. Introduction During the lifetime of its transm itter many transits of the first earth satellite were recorded at the Royal Aircraft Establishment and its out-stations, using both interferometric and Doppler methods. Similar measurements were made on the second satellite, and more recently kinetheodolite observations have been obtained. So far only preliminary estimates of the orbit parameters have been derived, but it is hoped th at a more thorough analysis, particularly of the kinetheodolite data, will yield useful information on the effects of the earth's oblateness and of its atmosphere. Interferometer measurements Although the instrument described by Mr Beresford can give a continuous indication of phase difference, we have so far used only the times at which the traces cross the zero levels, corresponding to successive increments of half a wave length in the path difference between the satellite and the two aerials of a pair. For a distant transm itter such as the satellite, the ratio of the path difference to the aerial spacing gives the direction cosine of the satellite position vector with respect to the line joining the aerials. Because a path difference of an integral number of wavelengths is not detected by this form of instrument, the phase measurement defines increments in the direction cosine from an arbitrary zero rather than the direction cosines themselves. The increments between successive zeros on the records is A/2 d,where A is the wavelength and d the aerial spacing. For Sputnik I the spacing was about four wavelengths (at 40 Mc/s), giving an increment of oneeighth ; advantage was taken of the lull between the failure of Sputnik I 's trans m itter and the appearance of Sputnik II to increase the spacing to 10 wavelengths, for which the increment is one-twentieth. There are two pairs of aerials at right angles, so that, apart from uncertain additive constants, the records permit plotting two direction cosines against time. These alone are insufficient to determine the height and track of the satellite, and some assumption must be made about the orbit. Preliminary estimates were obtained by assuming th at in the neighbourhood of the interferometer, over perhaps 400 miles of track (100 s), the path of the satellite is along a horizontal straight line at a constant speed. If lv l2, l3 denote the direction cosines of the satellite position vector p with respect to the east, north and vertical axes at the interferometer, then the height h is given by h - The eastwards component of the satellite's velocity is The computation of orbit parameters 49 or Since vE is assumed to be constant, a graph of 1^1% should yield a straight line of slope vE/h. A similar relation holds for the northwards component of velocity, giving vN/h from the graph of Z 2/Z3. If the orbital period is known, a further relation between v and h obtains for any assumed orbit. The height, speed and bearing of the track can then be found, and also the passing distance and time of latitude transit. Figure 39 shows the interferometer data plotted against time for a particular tra n sit; it will be seen th at the points lie on a smooth curve with very little scatter. Both the sine and cosine of the phase difference were recorded for the east-west pair of aerials, but only the sine for the north-south: for this reason there are twice as many points on the east-west curve. Tables 9 and 10 give some of the results for Sputnik I derived by this method of analysis. The mean height for the morning transit is 250 nm, with a standard deviation of 6 nm. Since the rotation of the major axis is very slow for the par ticular orbital inclination chosen (it is zero when the inclination is 63*4°), the measurements can be regarded as independent estimates of the same height. 4 Vol. 248. A. 50 Although the method gives results fairly quickly, it is difficult to estimate the errors introduced by the various assumptions made, particularly in the case of more distant transits. Accordingly, a program was prepared for a digital computer, using the full equations for a curved orbit over a rotating, spherical earth. The E. G. C. Burt (Discussion Meeting) time (see) Figure 39. Direction cosines from interferometer data, for transit at 21.00 h on 18 October 1957. lx direction cosine with respect to east-west aerials. l2 direction cosine with respect to north-south aerials. object of this was twofold: first, to verify the validity of the above method, and secondly to provide less laborious but more accurate means of height estimation. Figure 40 shows the result of calculating the ratio of direction cosines on the computer for a particular transit, using the full equations. I t is clear th at the assumption of linearity of these functions against time is valid only over the central portion, for perhaps 100 s, so th at the simple method is likely to fail if pressed too far outside this range. In the computer program for extracting orbital d ata trial values of height, bearing, passing distance and time of latitude transit are inserted, and the com puter calculates a succession of values for the direction cosines. These are compared The computation of orbit parameters 51 T a b l e 9. M o r n in g t r a n s it s nominal passing track tim e of transit date tim e height distance bearing
doi:10.1098/rspa.1958.0222 fatcat:xaw244f55vcqfnd3kddxwcxx4q