Letter to the Editors
The Earth’s Gravitational Potential, deduced from the
Orbits of Artificial Satellites
In a paper with the above title in the GeophysicalJournal last April (King-Hele
1961), values of the coefficients Jn of the zonal harmonics were obtained and
compared with previous determinations. Subsequently, several other papers
on this topic have appeared and they are briefly reviewed below.
Michielsen (1961) has obtained values of JZ ...Js,though without any errorestimates, from Vanguard I, Transit IB rocket and Sputnik 4; Newton, Hopfield
& Kline (1961) have evaluated J3, J5 and J7 from Vanguard I, Transit IB and
Transit 2A; and Smith (1961) has determined Jz,J4 and J6 from Sputnik 3,
Transit IB rocket, Tiros I and Vanguard 2. The results of these and the more
complete of the previous determinations are summarized in Table I on page 271.
The errors given by the authors in Table I are those estimated from the observational data, and do not include the unknown errors arising from neglecting the
higher Jn. In these circumstances it seemed best to give at the end of Table I
an unweighted arithmetic mean of the individual values. The number in square
brackets beside each average value is the r.m.s. deviation of the separate values
which contribute to it, and is unrelated to the authors’ error-estimates.
The various values of J3 ...J7 in Table I agree surprisingly well, especially
when it is remembered that no two authors used the same set of satellites: none
of the values is more than 0.4 x 10-6 from the appropriate average value. The
spread among the values of Jz is at first sight disappointingly large, but the two
low values, those of O’Keefe and Kozai, might well have been different if the
apparently rather large J6 term had been taken into account: for example, Smith
found that, on neglecting J6, his value of J2 changed by 0.35 x 10-6. Since the
satellites used in deriving the values in Table I have a wide range of orbital
inclinations, and since the coefficients of the higher-order Jn terms in equations
such as that for 0 usually change sign when i changes from a near-equatorial to
a near-polar value (see King-Hele 1961, equation 5), the errors in the values of
the earlier Jn in Table I caused by neglecting higher-order Jn may well be distributed in a fairly random fashion. If so, the average values in Table I are probably
nearer the truth than any of the individual setsalthough each author will no
doubt continue to prefer his own values.
In the previous paper (King-Hele 1961, equation 2) it was assumed that the
potential U was independent of longitude. Lately, however, four authors have
determined the variation of U with longitude from satellite orbits. It is not
possible here to describe their results in detail, since a re-definition of U,in a
form deriving from equation (I) of the previous paper rather than ( z ) , is required.
The four determinations have been made by Izsak (1961), using Vanguards 2
and 3; by Kaula (1961), using Vanguard I ; by Newton (1961), using Transit
+;
270
1083.15(0.2)
1082'79 (0.15)
1082.7 L0.31
Smith (1961)
King-Hele (1961)
Average
O'Keefe & others (1959) 10825 (0.1)
-
1082.7
Michielsen (1961)
Newton & others (1961)
1082.19(0.03)
Ja x 10'
Kozai (1961a)
Source
(0.02)
-2.4 [o-II
-
-2.4 (0.3)
-2.36 (0.14)
-2.5
-2-29
Ja x 106
(0'1)
(0.2)
-1'7 10.31
-1.4
-1'4 (0.3)
-1.7 (0.1)
-
-1.7
-2.1
J4 X 10'
-0.1
-0.1
[0*21
0.8 [O'I]
0.9 (0.8)
-0.4
-
-
L0.21
-
-
-0.28 (0.11)
-0.6
-
J7 X 10'
-
'7
-
0
10'
-
x
0.7 (0.6)
(0'1)
(0.10)
(0.02)
J0
-
-0.19
0 '3
-0.23
Js x 10'
Table I
Values of J z ...J s determined from satellite orbits
(with authors' estimated standard errors in brackets)
0'1
-
-
-
0'1
-
Ja x 10'
8
E
B
B
c
272
Letter to the Editore
and by Kozai (1961b), using Sputnik 3, Vanguard 2 and Vanguard 3. The values
obtained by the first three for the ellipticity of the equator B = (max. - min. equat.
diam./mean equat. diam.), and for the longitude A, of the major axis of the equator,
are given in Table 2, with the authors' error-estimates in brackets.
Table 2
Values of equatorial ellipticity B, and A,
Source
p
x I06
M O W )
Izsak (1961)
3 '2 (0 '3)
33'2 (0.5)
Kaula (1961)
1'1 (0.5)
36 (16)
Newton (1961)
2'4 (0.5)
11
(6)
Of these values, Newton's is obtained in the most direct manner, from the alongtrack discrepancies in the orbit of Transit @, which show a regular sinusoidal
oscillation of period about 12 hours and amplitude 900 metres. Transit @ thus in
effect provides twice-daily determinations of the equatorial ellipticity. Kozai
(1961b) makes the most thorough analysis of the gravitational potential, but does
not give values of 8.
D. G. KING-HELE
Royal Aircraft Establishment,
Farnborough, Hunts.
1962 January 10.
References
Izsak, I. G., 1961. A determination of the ellipticity of the Earth's equator. Space
Research 11. North-Holland Publishing Co., Amsterdam. p. 352.
Kaula, W. M., 1961. Analysis of satellite observations for longitudinal variations
of the gravitational field. Space Research 11. North-Holland Publishing
Co., Amsterdam. p. 360.
King-Hele, D. G., 1961. The Earth's gravitational potential, deduced from the
orbits of artificial satellites. Geophys.J . 4, 3-16.
Kozai, Y., 1961a. The gravitational field of the Earth derived from the motions
of three satellites. Astronom.J . 66, 8-10.
Kozai, Y., 1961b. Tesseral harmonics of the potential of the Earth as derived
from satellite motions. Smithsonian Astrophys. Obs. Sp. Rpt. 72.
Michielsen, H. F., 1961. The odd harmonics of the Earth's gravitational field.
Advances in Astronautical Sciences 6. Plenum Press, New York.
Newton, R. R., 1961. Ellipticity of the equator deduced from the motion of
Transit +4. (To be published.)
Newton, R. R., Hopfield, H. S. & Kline, R. C., 1961. Odd harmonics of the
Earth's gravitational field. Nature, 190,617-618.
O'Keefe, J., Eckels, A. & Squires, R. K., 1959. The gravitational field of the
earth. Astronom. J., 64, 245-253.
Smith, D. E., 1961. Determination of the Earth's gravitational potential from satellite orbits. Planet. Space Sci., 8, 43-48.