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The orbital periods of revolution of a satellite
Planet. Space Sri.. 1965. Vol. 13. pp. 1283 to 1288. Perg-
Pma Ltd. Printed in Northern hland
THE ORBITAL PERIODS OF REVOLUTION
OF A SATELLITE
D. E. SMITH S.R.C. Radio and Space Research Station, Ditton Park, Slough, Bucks. (Received 5 August 1965) Abstract-The relationship between the anomalistic and nodal periods of revolution of a satellite is investigated and it is shown that the anomalistic period of revolution can change even in a conservative system of forces and it is suggested that the mean nodal period represents the energy of the satellite. INTRODUCTION
The two main periods of revolution of a satellite in its orbit are the anomalistic period, the time between consecutive passages of the satellite through the perigee position, and the nodal period, the time between successive passages of the satellite through the ascending node position. In studies of the upper atmosphere from the decay of a satellite orbit the anomalistic period is normally assumed to be a measure of the energy of the satellite so that any change in this quantity is attributable to the effects of atmospheric drag. It is the purpose of the present paper to investigate the relationship between the anomalistic and nodal periods of revolution and to show that even in a conservative system of forces, where the energy of the satellite remains constant, the anomalistic period can change. Further, it is suggested that the mean nodal period represents the energy of the satellite. THE ANOMALISTIC AND NODAL PERIODS OF REVOLUTION One of the major orbital perturbations by the Earth’s oblateness is the rotation of the perigee position in the plane of the orbit. If during one nodal revolution of the satellite the argument of perigee increases by Aa, then by applying the normal laws of elliptic motion(l) we can show that the nodal period, TN, is given by
1
Acu (1 - ea)sle 277 (1 + e cos co)*
TN = TA 1 - -
(1)
where TA is the anomalistic period, e is the orbital eccentricity and o is the argument of perigee. If, T,, Ahwand e are constant equation (1) may be rewritten to the first order in e and Aw as follows: where
cos 0
(3)
TN would therefore vary cyclically with respect to o about the central value TN, which we shall call the mean nodal period. The change in argument of perigee due to the Earth’s non-sphericity depends on Ja, the first even harmonic in the Earth’s potential function and also on (Ja sin cu)/e where J, is the first odd harmonic(*) so that for very small eccentricities this latter term can be important even though 1J,/J,I - 2-5 x 1O-s. Thus in practice Ao is not constant, varying from one 1283
1284
D. E. SMITH
revolution to the next as sin UJchanges. Accordingly, if equation (2) is used directly to calculate TN from known values of TN, e, Ao and CO,it would be expected that the values of TN so derived at different epochs would be different. If TA is constant, the derived values of G would be proportional to the respective values of [l - (Ao/277)]. Since this relation is not always observed, as will be shown later, it leads to a natural questioning of the assumption of a constant anomalistic period. The nodal period, however, can be obtained without recourse to the anomalistic period from a consideration of the angular momentum of the satellite. The angular momentum of the satellite is a vector quantity acting through the centre of the Earth and perpendicular to the plane containing the instantaneous orbit. Due to the oblateness of the Earth the instantaneous orbital plane precesses about the Earth’s axis and consequently the angular Northyok
FIG. 1. THEOREXTOFAN
EARTHSATELLITE.
momentum changes. The forces which cause the orbital plane to precess, however, are perpendicular to the orbital plane and are therefore parallel to the angular momentum vector. Consequently, the angular momentum vector is only changed in direction, its magnitude remaining constant. The magnitude of the angular momentum per unit mass of a satellitet3) in orbit about the Earth, as shown in Fig. 1, is = const. = K where r is the radial distance of the satellite from the Earth’s centre, u is the argument of latitude, the angle between the ascending node of the instantaneous orbit and the satellite, Q is the right ascension of the ascending node and i the inclination of the instantaneous. orbital plane to the equator. Kis a constant provided the system of forces under which the satellite is moving is conservative and K = ba(l
-
e2)li2]
when the force system is the Earth’s gravitational attraction, where ~1is the product of theEarth’s mass and gravitational constant.
THE ORBITAL
PERIODS
OF REVOLUTION
OF A SATELLITE
1285
If it is assumed that dQ/dt is constant during one nodal period of revolution of the satellite equation (4) can be integrated to give the nodal period as follows T,=%dt=f~~r2(du+ficosidt) =-
(6)
p2 U-2rrdu + h cos i dt K s U=o (1 + e cos Q2
(7)
wherep = a( 1 - es) and 13is the true anomaly of the satellite. Since u increases by 27rduring one nodal period of revolution and fi dt changes by less than 1 degree it may be assumed that jdul > I$] cos i jdtj The r.h.s. of equation (7) may be expanded by successive substitution for dt from equation (4) to give (1 + ed”cos 0)2 + ($~ficosis:_:02’(l
+ :,,
e)4
If we consider only small eccentricities so that terms in e2 may be neglected equation (8) becomes ~“=g[l--
~)ticosi]-‘[2?r-Ze(l
u=o+e
Now,
and hence
- @S’lcosi~~~cosBdu]
cos 0
du = sin 8 f
s
cos 8
s
(9)
(10) do = sin 8 + ci, s
cos 8
dt
(11)
where 8 is assumed constant during one nodal period of revolution. Now
s
COS 8
dt = P” F 9%
s cos
t9(du + fi cos i dt) (1 + ecos ey
[I - (%) hcosi]-‘/cosBdu
(12)
after successive substitution for dt from equation (4) and when all terms involving e are neglected. The neglect of terms in e is permissible since the term in which the integral appears in equation (9) is already of order e. From equation (11) and (12) we obtain u-2n s u-o
cos 8 du =
[sin
u=2n
e] u=o
I-@ti[I--($)Clcosi]” =-
Ao cos cu (13)
1286
D. E. SMITH
where w is the mean value of the argument of perigee during the revolution and Ahwis the change in w during the revolution. Hence
is an expression for the nodal period of revolution which does not depend directly on the anomalistic period or on the assumption that it is constant. Since !G!>dQ -x dt equation (4) shows that K/p2 is of the order du/dt and since, also
equation (14) can be simplified for the following discussion by neglecting the terms in ci, and 0, to give eAo TAv= 27~ ; 1+ cos W ( >[
7r 1
(1%
when terms of order e2 are neglected. If a, K, e and Aw are all constant then TN will oscillate with constant amplitude about a central value of TN given by T,=2r
a2 0K
W-5)
which is independent of Am and is a constant. Further, TN itself is largely independent of any changes in Aw because of the factor e occurring in the second term of equation (15). This effect is thought to be observed and an example of the independence of the nodal period of A.w is shown in Fig. 2 for Echo 2. Figure 2 shows the eccentricity, argument of perigee and nodal period of Echo 2 during the period August to December 1964. During this period the eccentricity decreased to a minimum in mid-October under the action of solar radiation pressure. Due to the smallness of the eccentricity the rate of change of argument of perigee changed due to the importance of the (J, sin o)/e gravitational term and to the contribution to ci, by solar radiation pressure which is also proportional to l/e.f4) Because the solar radiation pressure contribution was positive the decrease of argument of perigee slowed down until the positive solar radiation pressure contribution cancelled the negative gravitational contribution and produced a minimum in the argument of perigee around the beginning of October. Thereafter the argument of perigee increased due to the dominance of the solar radiation pressure effect and continued to increase until the eccentricity had increased and the solar radiation pressure contribution began decreasing. A maximum in the argument of perigee was reached in early November when the solar radiation pressure and gravitational contributions again cancelled and subsequently the argument of perigee began decreasing as the gravitational contribution became dominant.
THE ORBITAL PERIODS OF REVOLUTION
OF A SATELLITE
1287
In the nodal period curve in Fig. 2 the continuous line is the theoretical nodal period given by equation (14) (constant mean nodal period) when account has been taken of the direct effect of radiation pressure on the period of revolution and a constant drag term of -7.7 x 10-T days/days has been included. The dashed line is the theoretical nodal period given by equation (1) (constant anomalistic period) when the same solar radiation pressure and drag terms have been included. The points are observed values of the nodal period. 0.015-
e
l
l
l
OOIO-
l
. l
l
.
.
l
0005 -
.
b.0
.
l
*
o-
30 360 s
l
.
. .
330
.
.
l
.
. l
.
l
5
l
.
*
*
300 ;
Oct.
sept
FIG. 2. THE ECCENTRICITY,
Nov.
ARGUMENT OF PERIGEE AND NODAL PERIOD OF ECHO 2 DURING
AucMhc.
1964.
It was necessary to allow for the e&cts of solar radiation pressure and air drag on the period of revolution because these perturbations made the force system under which the satellite was moving no longer conservative and hence K was not constant. The effect of solar radiation pressure on the period was computed from an equation given by Cookc4) and the magnitude of the drag effect was chosen so that when added to the values of nodal period given by equation (14) gave values which closely agreed with the observed nodal period. The drag correction was found to be considerably greater than the solar radiation pressure perturbation and accounted for nearly all the drop in the observed nodal period shown in Fig. 2. The deviations of the observed points from the curve given by equation (14) are assumed to be due to a varying atmospheric drag perturbation and not a constant one as was assumed. Equations (3) and (16) are two expressions for the mean nodal period 6 from which we can obtain an expression for the anomalistic period T,, viz. =L:
T
A
wJ21K) 1 - At0/2?r
(17)
1288
D. E. SMITH
The above expression, however, shows that TA is only constant if Ao is constant, and consequently the anomalistic period cannot represent the energy of the satellite. Further, the independence of the mean nodal period to changes in Aw, as shown by equation (16), suggests that this may be a better measure of the satellite energy. SUMMARY AND CONCLUSIONS
Two expressions have been given for the nodal period of revolution. Both of these expressions show that the nodal period oscillates with an amplitude of (e Acu/~)~~ about a central value TN, the mean nodal period. One expression, however, suggests that if the anomalistic period is constant g will vary if the rate of change of argument of perigee varies whilst the other expression indicates that the mean nodal period is unaffected by variations in dco/dt. An example is given that indicates agreement with the latter suggestion. The only way in which the two expressions can be reconciled so that the mean nodal period remains constant in both expressions is by removing the restriction that the anomalistic period must remain constant. If this is permitted under a conservative system of forces then the anomalistic period can no longer be a measure of the energy of the satellite and it seems logical to suggest that this is represented by the mean nodal period. The above analysis and example are not intended as a proof of the concept of a constant mean nodal period but to demonstrate that the anomalistic period of a satellite can change in the absence of air drag in an otherwise conservative system of forces and that allowance must be made for this variation in atmospheric drag studies when changes in A.ware important. For most satellites in orbit whose eccentricity is not small Aw will remain virtually constant but for balloon type satellites for which the argument of perigee is perturbed by solar radiation pressure the change in Aw may materially affect the anomalistic period. In such cases it is suggested that the mean nodal period should be obtained if variations of atmospheric drag are being investigated. Acknowledgements-The author would like to thank several members of the Radio and Space Research Station and Mr. D. G. King-Hele of the Royal Aircraft Establishment, Famborough for many useful discussions held during the course of this work. The above work has been carried out as part of the programme of research of the Radio and Space Research Station and is published by permission of the Director.
1. 2. 3. 4.
REPERENCRS W. M. SMART, Spherical Astronomy. Cambridge University Press (1956). R. H. kfImsoN, Geuphys. J. R. Astr. Sot. 4, 17 (1961). F. TB.SERAND,Traite de M&unique Celeste, Vol. 1. (1889). G. E. COOK, Geuphys. J. R. Ah. Sot. 6,271 No. 3 (1962).
PesIoMeHccnefiyeTca nepaoAahiu
B
o6oporax
cooT3oruerrue hlemxy arior4a.rr~cT39ecKmfH npaKorir5recK3u cnyrnnna
H yuaabmaercsf,
UTO
anoManncrri~ecmiti
nepaog
060po~o~ nowepraevx ua MeHemm game npx yMepemoti cmxerde cm. CymecmyeT npennonomeme, wo, x3cpeRneM, ~paKomirecr-&i nepuog npeAcraanfier co602t aaeprmo CnJ'THUKa.
Pma Ltd. Printed in Northern hland
THE ORBITAL PERIODS OF REVOLUTION
OF A SATELLITE
D. E. SMITH S.R.C. Radio and Space Research Station, Ditton Park, Slough, Bucks. (Received 5 August 1965) Abstract-The relationship between the anomalistic and nodal periods of revolution of a satellite is investigated and it is shown that the anomalistic period of revolution can change even in a conservative system of forces and it is suggested that the mean nodal period represents the energy of the satellite. INTRODUCTION
The two main periods of revolution of a satellite in its orbit are the anomalistic period, the time between consecutive passages of the satellite through the perigee position, and the nodal period, the time between successive passages of the satellite through the ascending node position. In studies of the upper atmosphere from the decay of a satellite orbit the anomalistic period is normally assumed to be a measure of the energy of the satellite so that any change in this quantity is attributable to the effects of atmospheric drag. It is the purpose of the present paper to investigate the relationship between the anomalistic and nodal periods of revolution and to show that even in a conservative system of forces, where the energy of the satellite remains constant, the anomalistic period can change. Further, it is suggested that the mean nodal period represents the energy of the satellite. THE ANOMALISTIC AND NODAL PERIODS OF REVOLUTION One of the major orbital perturbations by the Earth’s oblateness is the rotation of the perigee position in the plane of the orbit. If during one nodal revolution of the satellite the argument of perigee increases by Aa, then by applying the normal laws of elliptic motion(l) we can show that the nodal period, TN, is given by
1
Acu (1 - ea)sle 277 (1 + e cos co)*
TN = TA 1 - -
(1)
where TA is the anomalistic period, e is the orbital eccentricity and o is the argument of perigee. If, T,, Ahwand e are constant equation (1) may be rewritten to the first order in e and Aw as follows: where
cos 0
(3)
TN would therefore vary cyclically with respect to o about the central value TN, which we shall call the mean nodal period. The change in argument of perigee due to the Earth’s non-sphericity depends on Ja, the first even harmonic in the Earth’s potential function and also on (Ja sin cu)/e where J, is the first odd harmonic(*) so that for very small eccentricities this latter term can be important even though 1J,/J,I - 2-5 x 1O-s. Thus in practice Ao is not constant, varying from one 1283
1284
D. E. SMITH
revolution to the next as sin UJchanges. Accordingly, if equation (2) is used directly to calculate TN from known values of TN, e, Ao and CO,it would be expected that the values of TN so derived at different epochs would be different. If TA is constant, the derived values of G would be proportional to the respective values of [l - (Ao/277)]. Since this relation is not always observed, as will be shown later, it leads to a natural questioning of the assumption of a constant anomalistic period. The nodal period, however, can be obtained without recourse to the anomalistic period from a consideration of the angular momentum of the satellite. The angular momentum of the satellite is a vector quantity acting through the centre of the Earth and perpendicular to the plane containing the instantaneous orbit. Due to the oblateness of the Earth the instantaneous orbital plane precesses about the Earth’s axis and consequently the angular Northyok
FIG. 1. THEOREXTOFAN
EARTHSATELLITE.
momentum changes. The forces which cause the orbital plane to precess, however, are perpendicular to the orbital plane and are therefore parallel to the angular momentum vector. Consequently, the angular momentum vector is only changed in direction, its magnitude remaining constant. The magnitude of the angular momentum per unit mass of a satellitet3) in orbit about the Earth, as shown in Fig. 1, is = const. = K where r is the radial distance of the satellite from the Earth’s centre, u is the argument of latitude, the angle between the ascending node of the instantaneous orbit and the satellite, Q is the right ascension of the ascending node and i the inclination of the instantaneous. orbital plane to the equator. Kis a constant provided the system of forces under which the satellite is moving is conservative and K = ba(l
-
e2)li2]
when the force system is the Earth’s gravitational attraction, where ~1is the product of theEarth’s mass and gravitational constant.
THE ORBITAL
PERIODS
OF REVOLUTION
OF A SATELLITE
1285
If it is assumed that dQ/dt is constant during one nodal period of revolution of the satellite equation (4) can be integrated to give the nodal period as follows T,=%dt=f~~r2(du+ficosidt) =-
(6)
p2 U-2rrdu + h cos i dt K s U=o (1 + e cos Q2
(7)
wherep = a( 1 - es) and 13is the true anomaly of the satellite. Since u increases by 27rduring one nodal period of revolution and fi dt changes by less than 1 degree it may be assumed that jdul > I$] cos i jdtj The r.h.s. of equation (7) may be expanded by successive substitution for dt from equation (4) to give (1 + ed”cos 0)2 + ($~ficosis:_:02’(l
+ :,,
e)4
If we consider only small eccentricities so that terms in e2 may be neglected equation (8) becomes ~“=g[l--
~)ticosi]-‘[2?r-Ze(l
u=o+e
Now,
and hence
- @S’lcosi~~~cosBdu]
cos 0
du = sin 8 f
s
cos 8
s
(9)
(10) do = sin 8 + ci, s
cos 8
dt
(11)
where 8 is assumed constant during one nodal period of revolution. Now
s
COS 8
dt = P” F 9%
s cos
t9(du + fi cos i dt) (1 + ecos ey
[I - (%) hcosi]-‘/cosBdu
(12)
after successive substitution for dt from equation (4) and when all terms involving e are neglected. The neglect of terms in e is permissible since the term in which the integral appears in equation (9) is already of order e. From equation (11) and (12) we obtain u-2n s u-o
cos 8 du =
[sin
u=2n
e] u=o
I-@ti[I--($)Clcosi]” =-
Ao cos cu (13)
1286
D. E. SMITH
where w is the mean value of the argument of perigee during the revolution and Ahwis the change in w during the revolution. Hence
is an expression for the nodal period of revolution which does not depend directly on the anomalistic period or on the assumption that it is constant. Since !G!>dQ -x dt equation (4) shows that K/p2 is of the order du/dt and since, also
equation (14) can be simplified for the following discussion by neglecting the terms in ci, and 0, to give eAo TAv= 27~ ; 1+ cos W ( >[
7r 1
(1%
when terms of order e2 are neglected. If a, K, e and Aw are all constant then TN will oscillate with constant amplitude about a central value of TN given by T,=2r
a2 0K
W-5)
which is independent of Am and is a constant. Further, TN itself is largely independent of any changes in Aw because of the factor e occurring in the second term of equation (15). This effect is thought to be observed and an example of the independence of the nodal period of A.w is shown in Fig. 2 for Echo 2. Figure 2 shows the eccentricity, argument of perigee and nodal period of Echo 2 during the period August to December 1964. During this period the eccentricity decreased to a minimum in mid-October under the action of solar radiation pressure. Due to the smallness of the eccentricity the rate of change of argument of perigee changed due to the importance of the (J, sin o)/e gravitational term and to the contribution to ci, by solar radiation pressure which is also proportional to l/e.f4) Because the solar radiation pressure contribution was positive the decrease of argument of perigee slowed down until the positive solar radiation pressure contribution cancelled the negative gravitational contribution and produced a minimum in the argument of perigee around the beginning of October. Thereafter the argument of perigee increased due to the dominance of the solar radiation pressure effect and continued to increase until the eccentricity had increased and the solar radiation pressure contribution began decreasing. A maximum in the argument of perigee was reached in early November when the solar radiation pressure and gravitational contributions again cancelled and subsequently the argument of perigee began decreasing as the gravitational contribution became dominant.
THE ORBITAL PERIODS OF REVOLUTION
OF A SATELLITE
1287
In the nodal period curve in Fig. 2 the continuous line is the theoretical nodal period given by equation (14) (constant mean nodal period) when account has been taken of the direct effect of radiation pressure on the period of revolution and a constant drag term of -7.7 x 10-T days/days has been included. The dashed line is the theoretical nodal period given by equation (1) (constant anomalistic period) when the same solar radiation pressure and drag terms have been included. The points are observed values of the nodal period. 0.015-
e
l
l
l
OOIO-
l
. l
l
.
.
l
0005 -
.
b.0
.
l
*
o-
30 360 s
l
.
. .
330
.
.
l
.
. l
.
l
5
l
.
*
*
300 ;
Oct.
sept
FIG. 2. THE ECCENTRICITY,
Nov.
ARGUMENT OF PERIGEE AND NODAL PERIOD OF ECHO 2 DURING
AucMhc.
1964.
It was necessary to allow for the e&cts of solar radiation pressure and air drag on the period of revolution because these perturbations made the force system under which the satellite was moving no longer conservative and hence K was not constant. The effect of solar radiation pressure on the period was computed from an equation given by Cookc4) and the magnitude of the drag effect was chosen so that when added to the values of nodal period given by equation (14) gave values which closely agreed with the observed nodal period. The drag correction was found to be considerably greater than the solar radiation pressure perturbation and accounted for nearly all the drop in the observed nodal period shown in Fig. 2. The deviations of the observed points from the curve given by equation (14) are assumed to be due to a varying atmospheric drag perturbation and not a constant one as was assumed. Equations (3) and (16) are two expressions for the mean nodal period 6 from which we can obtain an expression for the anomalistic period T,, viz. =L:
T
A
wJ21K) 1 - At0/2?r
(17)
1288
D. E. SMITH
The above expression, however, shows that TA is only constant if Ao is constant, and consequently the anomalistic period cannot represent the energy of the satellite. Further, the independence of the mean nodal period to changes in Aw, as shown by equation (16), suggests that this may be a better measure of the satellite energy. SUMMARY AND CONCLUSIONS
Two expressions have been given for the nodal period of revolution. Both of these expressions show that the nodal period oscillates with an amplitude of (e Acu/~)~~ about a central value TN, the mean nodal period. One expression, however, suggests that if the anomalistic period is constant g will vary if the rate of change of argument of perigee varies whilst the other expression indicates that the mean nodal period is unaffected by variations in dco/dt. An example is given that indicates agreement with the latter suggestion. The only way in which the two expressions can be reconciled so that the mean nodal period remains constant in both expressions is by removing the restriction that the anomalistic period must remain constant. If this is permitted under a conservative system of forces then the anomalistic period can no longer be a measure of the energy of the satellite and it seems logical to suggest that this is represented by the mean nodal period. The above analysis and example are not intended as a proof of the concept of a constant mean nodal period but to demonstrate that the anomalistic period of a satellite can change in the absence of air drag in an otherwise conservative system of forces and that allowance must be made for this variation in atmospheric drag studies when changes in A.ware important. For most satellites in orbit whose eccentricity is not small Aw will remain virtually constant but for balloon type satellites for which the argument of perigee is perturbed by solar radiation pressure the change in Aw may materially affect the anomalistic period. In such cases it is suggested that the mean nodal period should be obtained if variations of atmospheric drag are being investigated. Acknowledgements-The author would like to thank several members of the Radio and Space Research Station and Mr. D. G. King-Hele of the Royal Aircraft Establishment, Famborough for many useful discussions held during the course of this work. The above work has been carried out as part of the programme of research of the Radio and Space Research Station and is published by permission of the Director.
1. 2. 3. 4.
REPERENCRS W. M. SMART, Spherical Astronomy. Cambridge University Press (1956). R. H. kfImsoN, Geuphys. J. R. Astr. Sot. 4, 17 (1961). F. TB.SERAND,Traite de M&unique Celeste, Vol. 1. (1889). G. E. COOK, Geuphys. J. R. Ah. Sot. 6,271 No. 3 (1962).
PesIoMeHccnefiyeTca nepaoAahiu
B
o6oporax
cooT3oruerrue hlemxy arior4a.rr~cT39ecKmfH npaKorir5recK3u cnyrnnna
H yuaabmaercsf,
UTO
anoManncrri~ecmiti
nepaog
060po~o~ nowepraevx ua MeHemm game npx yMepemoti cmxerde cm. CymecmyeT npennonomeme, wo, x3cpeRneM, ~paKomirecr-&i nepuog npeAcraanfier co602t aaeprmo CnJ'THUKa.