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Densities deduced from perturbations at high altitudes
Planet. Space Sci. 1973, Vol. 21, pp. 1705 to 1712.
Person
Press. Printed in Northern Ireland
DENSITIES DEDUCED FROM PERTURBATIONS AT HIGH ALTITUDES M. ROUSSEAU Observatoire de Bordeaux, 33 Floirac, France (Received 12 3awary 1973)
Abstract-Atmospheric densities between 1500 and 3000 km have been determined from observations of satellites with large surface to mass ratios. The analysis has been made for optimum periods where the satellite is continuously in sunlight and when the effects of Earth albedo are minimum. Twenty six values of density obtained from 1965 to 1970 have been analysed. 1. lNTRODUCI‘ION
The study of satellite orbits for atmospheric density at high altitudes is a relatively recent field of investigations (for example, Fea;u) Fea and Smith;(2) Prior;(s) Rousseau(4)). We have systematically computed the densities at high altitudes using satellites with large area to mass ratios (A/m) for the optimum conditions when the satellite is continuously in sunlight and when the Earth’s albedo effect is minimum or vanishes. Mean densities over a period of one or two months can only be obtained because the accuracy of the orbital elements is insufficient; the standard deviation of the orbital elements must be smaller than the variations in these elements caused by the air drag. 2. ORBITAL BEGS AND THEIR ACCWR.ACY Three satellites whose perigees are greater than 1200 km have been selected: 1966~56A (Pageos), 1963-3OD (Dash 2), 1963-14B. We have used orbital elements or optical data provided by the Smi~sonian Astrophysical Observatory (S.A.O.) by the U.S. coast and geodetic survey and by the United States Air Force (USAF) Space Track 5 card elements. We have also computed elements using a classical differential orbital improvement (DOI) program (Barlier, c5)1965) from visual and photographic observations made at three French stations (Besanc;on, Bordeaux and Meudon). In the DOI program, perturbations caused by the Earth’s albedo effect and by terms of high degree and order terms in the geopotential have been ignored. We can simulate by numerical integration the effect of this neglect and of the fact that the distribution of the observations is not very good. We find that the systematic errors so introduced are of the same magnitude as the random errors. The semi-major axis has been computed for every three or four days depending on the length of the trajectories and on the distribution of the observations. To assess the accuracy of the five card elements provided by the USAF we have made some comparisons with our results for those periods when there is an overlap and we found that the accuracies of the two types of elements are of the same order of magnitude. We prefer to analyse the variations of the semi-major axis not for every three or four days but rather for periods of one or two months. The total variations are then greater than the standard deviations of the elements and the satellite’s rotation around its center of mass smoothes out some of the ignored effects, for, as shown by Kissel and Smith,@) this rotation significantly modifies the ratio A/m. We have not been able to take this effect into account because of absence of information on the satellite rotation. In order to deduce atmospheric densities from the observed variations in the semimajor axis we must subtract all variations due to known perturbing forces. 1705
1706
M. ROUSSEAU
There are three important perturbations: (1) Solar radiation pressure (SRP for convenience). (2) Earth albedo radiation pressure (EARP for convenience). (3) Air drag, The effect of magnetic induction has been neglected. This effect is not quite negligible for Pageos (Prior(s)) but smaller than the uncertainty in the aerodynamic coefficient C, or in the ratio A/m. 3. SRP EFFECTS
When the orbit of the satellite is in complete sunlight the perturbation in the semi-major axis due to the SRP is generally considered to vanish over a complete revolution. This is not entirely correct due to the motions of perigee and the Sun. Starting with a set of initial orbital elements, we computed the oscillating elements by numerical integration at the end of a complete revoktion of w + M for two cases: with SRP and without SRP o is the argument of perigee; &f is the mean anomaly. The short periodic perturbations due to the Earth’s flattening are the same in both cases. Differences in the elements were found and these are attributed to some coupling effect between the SRP and other perturbations. For Pageos we found a difference of one meter after one revolution. Recently P. Lalao) of the Ondrejov Observatory studied this problem theoretically and has confirmed our general results and found the same variations in the semi-major axis. The effect is not negligible. 4. w EFE’EXTS From the theory of Wyatt@) we write for the EARP effect: AU - =: --&I -A es a
sin i’ sin /3
where 3, = 1.4 x IO-‘; A is the mean albedo, 0.4; e is the eccentricity, major axis. Over each period, this formula can be written as
a is the semi-
Aa = k sin i’ sin /I
k=iawU,$--.
with
- ea
The definitions of the angles i’ and /3 are given in Fig. 1. This formula is based on a simple of the Earth’s albedo as proposed by Wyatt. Therefore we have only chosen periods where the angle becomes zero (sin changes its sign) and where the theoretical variation vanishes.
FIG.
f . P
ORBITAL
PLANE;
0
DIRECTION
OF THJ3 SUN;
-?r PERIGEE.
DENSlTIES DEDUCED FROM PERTU~ATIONS
AT HIGH ALTITUDES
1707
The integration is made symmetrically around this epoch and the total variation is zero or very small. For this effect a comparison with the results of Prior for the satellite 63-30D is satisfactory, the difference not exceeding a few per cent. 5. AIR DENSITIES After correcting for the SRP and EARP effects, the remaining variation in the semimajor axis is due to the air drag; from this, the densities can be deduced. In order to do this, we compute a theoretical value of the variation da of the semi-major axis for each revolution, extrapolating from Jacchia’s 1970@)model assuming that the condition of diffusive equilibrium is satisfied. The effect of the winter helium bulge is also included as proposed by Keating and Prior.(lo)
s 227
da=
-
0
(I i_ e cos E)3/2 (1 - e cos E)lJ2
where E is the eccentric anomaIy. D
l
_
D(1 i- ecosE)
(1 -
ti
e cos E) 1
0 cos i = n(l - @)1/2’
n is the mean motion;
w is the speed of the rotation of the upper atmosphere of the Earth; i is the inclination of the orbit. The density has an important effect near perigee. Therefore knowing the variation du observed and the variation da computed, it is possible to deduce the density at perigee. In fact, the problem is more intricate for the density is not always rn~irn~ at perigee but depends also upon its latitude and the local hour angle. However, if we plot the theoretical variation da as a function of the theoretical value of the variable density at perigee according to the model for the period considered, we find approximatively a linear function (Fig. 2). In this way, it is possible to define a mean
Fm.2.
EACHP~~NT~P~~E~STHE~RETICALVARIATIONOFT~SEMI-~ORAX~SUASA P~~ONOFTHETHEoREfIcALVALuEOFTHEDEN~~
De&cing the mean value of a from the observations, the corresponding mean value of p can be deduced.
M. ROUSSEAU
1708
4-
0” 3-
2000
3ooQ
h, FIG. 3. VARIATIGNSASSUMED
OF
km
THE AERODYNAMIC COEFFICIENT C, HEIGHT A.
AS
FUNCTION OF THE
density corresponding to a mean value of da and we can associate with it a mean latitude, a mean local hour. It is clear that the definition of this density is not very precise in time, in space and in local time. It is a mean value. On the other hand, at very high altitudes like 3000 km, and as emphasized by Prior, @)it has been shown after Brinton’s measurements that ions are important constituents and the drag effect of Hf is non negligible. We can obtain only the sum of two contributions. Conventional values must also be adopted for A/m. For 63-30D, an A/m of 3.62 m2/kg has been adopted after the work of Smith and Fea. ca) An A/m of 13.5 m2[kg for Pageos, A/m of 151 m2/kg for 1963-14B have also been adopted. The drag cofficient is assumed to be a function of height (Fig. 3). 6. RESULTS
The results are given in Table 1. For each value, latitude of perigee, local hours of perigee and density at standard altitude of 2300 km are given. As the scale height is often of the same order as the difference between the apogee and the perigee, the value of the density is a mean value for all latitudes; the density is also a mean value for several weeks. The quality of results is not the same for the three satellites. For the satellite 63-14B we have only the orbital elements of the USAF and the orbit must be considered less precise than the other two satellites. The results obtained with the Pageos data seem to be more reliable. However a statistical agreement exists between all the data. The obtained densities are plotted as a function of time in Fig. 4. For the period 19671970, corresponding to the solar activity maximum, it is possible to define a mean value of about 2 x lo-l9 kg/m3 for the global density at 2300 km. This corresponds to a mean value of 1050°K for the exospheric temperature. In Fig. 5, the different values of the density are plotted against the different months of the year. For the months March, ApriI, August and September the density values are
24-02-G e-07-65 19-06-66 23-05-67 4-11-67 2-05-68 20-10-68 26-03-69 2-09-69 25-12-65 8-05-66 7-02-69 3-01-70 S-12-70 14-11-66 2-11-67 21-03-68 9-09-68 30-01-69 M-07-69 18-12-69 22-05-70 7-l l-70 14-04-71
25-12-64 20-05-65 25-04-66 28-03-67 3P-09-67 22-02-68 16-08-68 4-02-69 12-08-69 S-10-65 31-03-66 10-12-68 17-11-69 7-10-70 29-09-66 3-09-67 28-02-68 27-07-68 28-12-68 13-05-69 15-10-69 16-03-70 20-08-70 30-01-71
3072 2885 2587 2358 2221 2088 1932 1741 1546 3132 3044 2657 2644 2669 3435 2384 2123 2093_ 2148 2310 2412 2535 2653 2850
2G 19h5 19hS 9h7 9”4 7h7 13h4 17h9 20h4 20”6 2OK5 20r’6 2Oh2
h
;:;
7h6 gh6 9h8 h &!: 9h9 8h6 I’“7
2”9 5h7 !P4 Sk3 5”s 4hl 3h7 3h7 3”6 3”O S”4 4”3 4”6 4h6 6”4 4hS 5”7 4h5 5”6 4’19 S”4 9h3 PO 13”6
Variations of the local hour from to
+77 -69 t-l8 -65 +s3 -18 +ss -54 -72 -72 -64 -33 -69 +68 -85 t-75 -4s +36 -26 +19 -16
+72 +76
-51
---Z -18 -46 -59 -65 -16 -18 t-48 -48 +70 -71 +78 -85 +8S -78
+16 +29 -t-24 +21 -22 t-4 i-6 -3
Variations of the latitude (deg) from to -0*3f -O*lO -0*14 -0.70 -0.69 -0.80 -0.69 -1.50 -2.07 -0.07 -0.25 -O*lO -0.09 -0.07 -050 -177 -5.14 -3.05 -2.73 -1.21 -1.36 -1.67 -2.26 -1.94
Mean variations ofa (m/rev)
+ Computed with the scale height of the extrapolated model (about 550 km). t Computed with the scale height of 800 km. $ Densities of the preceding column corrected for the semi-annual effect.
to
from
Epoch
Height of the
126.; 49.4 31.8 29,7 11.2 12.7 14.7 19.8 18.2
::5” 24.1 21.9 28.1 251 52.8 90.0 2.9 12.7 IO.7 6.4 4.3
S-6
;u:
:x
21 22 11
z
: 25 2s 11 19 20
3; 17 10 7 10
14 2 3 26 20 22 16 26 35
19 16 14 24 25 10 If 13 27 18 29 18 17 17 19 20 27 18 25 25 25 16 30 26
Density at Density at Density at Density at perigee 2300km 2300 km 2300km lo-17* 10-l’ 10-q lo-“$
TABLE 1. THE OBTAINED DENSF~~ES
1963-3023 1963-30D 1963-301) 1963-30D 1963-30D 1963-300 1963-30D 1963-30D 1963-30D 1963-14B 1963-14B 1963-14B 1963-14B 1963-14B 1966-56A 196656A 1966S6A 1966~56A 196656A 1966~56A 1966-56A 1966-56A 1966-56A 1966-56A
Satellite
z
5 X
5 5 z
I!! t: c Iz
P
$
@I
e
!z P-l 8
3
1710
M. ROUSSEAU x 1963-148 Al963-300 0 lS66-56A
40-
F 0
A
i?
0
A
0
0
A0
Oo
0
A
0 AX
a: IO-
A
I 1965
x A
Ax'
I 1966
*
I 1966
I 1967
x
8 I969
x
0
! 1970
I 1971
I 1972
FIG. 4. VARIATIONS OF THEDENSITY AT 2300 km AS FUNCTION OF TIME.
x
5C
1963-146
A 1963-30D co
o 1966-56A A
X
!z
b
0
x
0
A
A A
cl A
0
X
A0
0
0
IC
A X
0 XX
I
I
I
I
‘,A,
F
M
A
M
J
J
,
,
,
,
,
,
A
S
0
N
D
J
FIG. 5. VARIATION OFTHEDENSITY AT 2300 km AS A FUNCTION OFTHEMONTHS OFTHEYEAR,t.
higher and for the months May, June and January they are lower, showing a semi-annual effect whose maximum value is about 4 or 5 times the minimum. 7. COMPARISONS
WITH OTHER DATA
Some values can be compared with those obtained by other authors. From Prior’s paper on 1963-30D, the effect due to the drag on the mean motion is about 0.54 X 10W5 rev/day2 between 17 March and 9 June 1967. For the same period approximatively, we have found 0.72 x 1O-6rev/day 2. The difference is greater than the probable error. However, we think, it is quite possible to attribute this difference to a too optimistic estimation of errors. The difference in density obtained by Prior and by us presents a greater problem. Our value of the density is about twice as great as Prior’s value. A possible interpretation is that in our adopted reference model, the scale height is about 550 km but Prior adopted 950 km from simultaneous observations of Pageos and 1963-30D. Unfortunately, for this period our observations of Pageos were too poor to be useful and a more detailed comparison is not possible. However if we plot the densities as a function of height, we can define a mean scale height and it appears that the values of 750 or 800 km are adequate on the average (Fig. 6). Therefore, our value of the density at perigee is possibly too high and that of Prior too low. A more precise computation shows that our value could be too high by about 20 or 30 per cent. The remaining discrepancy can also be partially attributed to the difference in the observed variation of the semi-major axis and the drag effect of H+.
DENSITIES DEDWCED FROM PERTURBATIONS
Fm.6, VARIATI~NSOFTHE~~ARITHMOFTBE
DENSITYLOG
1711
AT HIGH ALTITUDES
p ASFUNCTIONOFTHEALTITUDE
h.
The importance of the choice of the scale height and of the model must be emphasized, for another consequence of this choice is the error introduced when extrapolating the values of the densities at perigee to the reference altitude (2300 km). If the perigee is relatively low, the error so introduced in the density can be 50 or 100 per cent. Therefore in Table 1 we have computed the densities corresponding to the same variations of the semi-major axis using two different scale heights h. The difference can be seen. For one period in 1964, we can also make a comparison with the results obtained by Fea and Smith.(2) The agreement is quite good; Fea and Smith obtained I.8 x 1O-2o g/cm3 at 3500 km compared with our value of 2.3 x lO-2o g/cm3. Generally our values are, on the average, 3.5 times greater than the theoretical values of Nicolet’s model of 1962,“rr) as the values predicted by the extrapolated model of Jacchia 1970. However as the values of our scale heights are possibly too low, the theoretical values of Nicolet’s model needs only be m~ti~iied by a factor 3.
From the present set of data, it is possible to deduce a mean value of the density. A variation with the season is shown but no too precise conclusion can be drawn due to the necessity of averaging the density in time and space in order to obtain a signi~~ant value. However the coherence between the values presented here and those obtained by other authors are an indication of the validity of the results. Ae~owIe~geme~t~-We are grateful for very useful discussions with Drs. F. Bar&x and C. Berger and for the assistance of J. P. Perk. We thank also the Smithsonian Astrophysical Observatory, the U.S. coast and geodetic survey and the United States Air Force which provided some orbital elements.
f . K. H. FEA, Planet. Space Sci. X3,1289 (1965). 2. K. H. -A and D. E. SN(ITW, Some further studies of perturbations of satellites at great altitude. Planet. Space Sci. l&1499 (1970).
3. E. J. PRIOR, Observed effects of earth reflected and hydrogen drag on the orbital energies of balloon satellites. Symposium on the use of artificial satellites for geodesy, Washington, D.C. (1971).
1712
M. ROUSSEAU
4. M. ROUSSEAU,Etude de la pression de radiation solaire et du frottement atmosphkique sur les satellites 1966-56A, 1963-3OD, 1963-14B. Con@ Interobs, Bucarest (1970). 5. F. BARLIER,Recherches sur ia d&termination et I’utilisation des elements osculateurs dans la mkanique des satellites proches. Bull. Astr. Tome XW, Fast. 4. 326. 6. D. E. SMITHand K. E. ICISSEL, Anomalous accelerations of the Pageos Spacecraft (G.S.F.C. aotit 1971). 7. P. LALA, Semi-analytical theory of solar pressure perturbations of satellite orbits during short time intervals. Bull. astr. Insts. Cd. 22, 63 (1971). 8. S. P. WYATT, The effect of terrestrial radiation pressure on satellites orbits. Dynamics of SatdUes (Ed. M. Roy), p. 180. Springer, Berlin (1963). 9. L. G. JACCHIA,New static models of the thermosphere and exosphere with empirical temperature profiles. Smithson. Astrophys. Special Report 313 (1970). 10. G. M. KEATING and E. J. PRIOR,The winter helium Bulge. Space Research, Vol. VIII, p. 982 (1968). 11. G. KOCKARTS and M. NICOLET,LR problbme a6ronamique de I’hklium et de I’hydrog&ne neutres. Annls GPophys. 18,269 (1962).
Person
Press. Printed in Northern Ireland
DENSITIES DEDUCED FROM PERTURBATIONS AT HIGH ALTITUDES M. ROUSSEAU Observatoire de Bordeaux, 33 Floirac, France (Received 12 3awary 1973)
Abstract-Atmospheric densities between 1500 and 3000 km have been determined from observations of satellites with large surface to mass ratios. The analysis has been made for optimum periods where the satellite is continuously in sunlight and when the effects of Earth albedo are minimum. Twenty six values of density obtained from 1965 to 1970 have been analysed. 1. lNTRODUCI‘ION
The study of satellite orbits for atmospheric density at high altitudes is a relatively recent field of investigations (for example, Fea;u) Fea and Smith;(2) Prior;(s) Rousseau(4)). We have systematically computed the densities at high altitudes using satellites with large area to mass ratios (A/m) for the optimum conditions when the satellite is continuously in sunlight and when the Earth’s albedo effect is minimum or vanishes. Mean densities over a period of one or two months can only be obtained because the accuracy of the orbital elements is insufficient; the standard deviation of the orbital elements must be smaller than the variations in these elements caused by the air drag. 2. ORBITAL BEGS AND THEIR ACCWR.ACY Three satellites whose perigees are greater than 1200 km have been selected: 1966~56A (Pageos), 1963-3OD (Dash 2), 1963-14B. We have used orbital elements or optical data provided by the Smi~sonian Astrophysical Observatory (S.A.O.) by the U.S. coast and geodetic survey and by the United States Air Force (USAF) Space Track 5 card elements. We have also computed elements using a classical differential orbital improvement (DOI) program (Barlier, c5)1965) from visual and photographic observations made at three French stations (Besanc;on, Bordeaux and Meudon). In the DOI program, perturbations caused by the Earth’s albedo effect and by terms of high degree and order terms in the geopotential have been ignored. We can simulate by numerical integration the effect of this neglect and of the fact that the distribution of the observations is not very good. We find that the systematic errors so introduced are of the same magnitude as the random errors. The semi-major axis has been computed for every three or four days depending on the length of the trajectories and on the distribution of the observations. To assess the accuracy of the five card elements provided by the USAF we have made some comparisons with our results for those periods when there is an overlap and we found that the accuracies of the two types of elements are of the same order of magnitude. We prefer to analyse the variations of the semi-major axis not for every three or four days but rather for periods of one or two months. The total variations are then greater than the standard deviations of the elements and the satellite’s rotation around its center of mass smoothes out some of the ignored effects, for, as shown by Kissel and Smith,@) this rotation significantly modifies the ratio A/m. We have not been able to take this effect into account because of absence of information on the satellite rotation. In order to deduce atmospheric densities from the observed variations in the semimajor axis we must subtract all variations due to known perturbing forces. 1705
1706
M. ROUSSEAU
There are three important perturbations: (1) Solar radiation pressure (SRP for convenience). (2) Earth albedo radiation pressure (EARP for convenience). (3) Air drag, The effect of magnetic induction has been neglected. This effect is not quite negligible for Pageos (Prior(s)) but smaller than the uncertainty in the aerodynamic coefficient C, or in the ratio A/m. 3. SRP EFFECTS
When the orbit of the satellite is in complete sunlight the perturbation in the semi-major axis due to the SRP is generally considered to vanish over a complete revolution. This is not entirely correct due to the motions of perigee and the Sun. Starting with a set of initial orbital elements, we computed the oscillating elements by numerical integration at the end of a complete revoktion of w + M for two cases: with SRP and without SRP o is the argument of perigee; &f is the mean anomaly. The short periodic perturbations due to the Earth’s flattening are the same in both cases. Differences in the elements were found and these are attributed to some coupling effect between the SRP and other perturbations. For Pageos we found a difference of one meter after one revolution. Recently P. Lalao) of the Ondrejov Observatory studied this problem theoretically and has confirmed our general results and found the same variations in the semi-major axis. The effect is not negligible. 4. w EFE’EXTS From the theory of Wyatt@) we write for the EARP effect: AU - =: --&I -A es a
sin i’ sin /3
where 3, = 1.4 x IO-‘; A is the mean albedo, 0.4; e is the eccentricity, major axis. Over each period, this formula can be written as
a is the semi-
Aa = k sin i’ sin /I
k=iawU,$--.
with
- ea
The definitions of the angles i’ and /3 are given in Fig. 1. This formula is based on a simple of the Earth’s albedo as proposed by Wyatt. Therefore we have only chosen periods where the angle becomes zero (sin changes its sign) and where the theoretical variation vanishes.
FIG.
f . P
ORBITAL
PLANE;
0
DIRECTION
OF THJ3 SUN;
-?r PERIGEE.
DENSlTIES DEDUCED FROM PERTU~ATIONS
AT HIGH ALTITUDES
1707
The integration is made symmetrically around this epoch and the total variation is zero or very small. For this effect a comparison with the results of Prior for the satellite 63-30D is satisfactory, the difference not exceeding a few per cent. 5. AIR DENSITIES After correcting for the SRP and EARP effects, the remaining variation in the semimajor axis is due to the air drag; from this, the densities can be deduced. In order to do this, we compute a theoretical value of the variation da of the semi-major axis for each revolution, extrapolating from Jacchia’s 1970@)model assuming that the condition of diffusive equilibrium is satisfied. The effect of the winter helium bulge is also included as proposed by Keating and Prior.(lo)
s 227
da=
-
0
(I i_ e cos E)3/2 (1 - e cos E)lJ2
where E is the eccentric anomaIy. D
l
_
D(1 i- ecosE)
(1 -
ti
e cos E) 1
0 cos i = n(l - @)1/2’
n is the mean motion;
w is the speed of the rotation of the upper atmosphere of the Earth; i is the inclination of the orbit. The density has an important effect near perigee. Therefore knowing the variation du observed and the variation da computed, it is possible to deduce the density at perigee. In fact, the problem is more intricate for the density is not always rn~irn~ at perigee but depends also upon its latitude and the local hour angle. However, if we plot the theoretical variation da as a function of the theoretical value of the variable density at perigee according to the model for the period considered, we find approximatively a linear function (Fig. 2). In this way, it is possible to define a mean
Fm.2.
EACHP~~NT~P~~E~STHE~RETICALVARIATIONOFT~SEMI-~ORAX~SUASA P~~ONOFTHETHEoREfIcALVALuEOFTHEDEN~~
De&cing the mean value of a from the observations, the corresponding mean value of p can be deduced.
M. ROUSSEAU
1708
4-
0” 3-
2000
3ooQ
h, FIG. 3. VARIATIGNSASSUMED
OF
km
THE AERODYNAMIC COEFFICIENT C, HEIGHT A.
AS
FUNCTION OF THE
density corresponding to a mean value of da and we can associate with it a mean latitude, a mean local hour. It is clear that the definition of this density is not very precise in time, in space and in local time. It is a mean value. On the other hand, at very high altitudes like 3000 km, and as emphasized by Prior, @)it has been shown after Brinton’s measurements that ions are important constituents and the drag effect of Hf is non negligible. We can obtain only the sum of two contributions. Conventional values must also be adopted for A/m. For 63-30D, an A/m of 3.62 m2/kg has been adopted after the work of Smith and Fea. ca) An A/m of 13.5 m2[kg for Pageos, A/m of 151 m2/kg for 1963-14B have also been adopted. The drag cofficient is assumed to be a function of height (Fig. 3). 6. RESULTS
The results are given in Table 1. For each value, latitude of perigee, local hours of perigee and density at standard altitude of 2300 km are given. As the scale height is often of the same order as the difference between the apogee and the perigee, the value of the density is a mean value for all latitudes; the density is also a mean value for several weeks. The quality of results is not the same for the three satellites. For the satellite 63-14B we have only the orbital elements of the USAF and the orbit must be considered less precise than the other two satellites. The results obtained with the Pageos data seem to be more reliable. However a statistical agreement exists between all the data. The obtained densities are plotted as a function of time in Fig. 4. For the period 19671970, corresponding to the solar activity maximum, it is possible to define a mean value of about 2 x lo-l9 kg/m3 for the global density at 2300 km. This corresponds to a mean value of 1050°K for the exospheric temperature. In Fig. 5, the different values of the density are plotted against the different months of the year. For the months March, ApriI, August and September the density values are
24-02-G e-07-65 19-06-66 23-05-67 4-11-67 2-05-68 20-10-68 26-03-69 2-09-69 25-12-65 8-05-66 7-02-69 3-01-70 S-12-70 14-11-66 2-11-67 21-03-68 9-09-68 30-01-69 M-07-69 18-12-69 22-05-70 7-l l-70 14-04-71
25-12-64 20-05-65 25-04-66 28-03-67 3P-09-67 22-02-68 16-08-68 4-02-69 12-08-69 S-10-65 31-03-66 10-12-68 17-11-69 7-10-70 29-09-66 3-09-67 28-02-68 27-07-68 28-12-68 13-05-69 15-10-69 16-03-70 20-08-70 30-01-71
3072 2885 2587 2358 2221 2088 1932 1741 1546 3132 3044 2657 2644 2669 3435 2384 2123 2093_ 2148 2310 2412 2535 2653 2850
2G 19h5 19hS 9h7 9”4 7h7 13h4 17h9 20h4 20”6 2OK5 20r’6 2Oh2
h
;:;
7h6 gh6 9h8 h &!: 9h9 8h6 I’“7
2”9 5h7 !P4 Sk3 5”s 4hl 3h7 3h7 3”6 3”O S”4 4”3 4”6 4h6 6”4 4hS 5”7 4h5 5”6 4’19 S”4 9h3 PO 13”6
Variations of the local hour from to
+77 -69 t-l8 -65 +s3 -18 +ss -54 -72 -72 -64 -33 -69 +68 -85 t-75 -4s +36 -26 +19 -16
+72 +76
-51
---Z -18 -46 -59 -65 -16 -18 t-48 -48 +70 -71 +78 -85 +8S -78
+16 +29 -t-24 +21 -22 t-4 i-6 -3
Variations of the latitude (deg) from to -0*3f -O*lO -0*14 -0.70 -0.69 -0.80 -0.69 -1.50 -2.07 -0.07 -0.25 -O*lO -0.09 -0.07 -050 -177 -5.14 -3.05 -2.73 -1.21 -1.36 -1.67 -2.26 -1.94
Mean variations ofa (m/rev)
+ Computed with the scale height of the extrapolated model (about 550 km). t Computed with the scale height of 800 km. $ Densities of the preceding column corrected for the semi-annual effect.
to
from
Epoch
Height of the
126.; 49.4 31.8 29,7 11.2 12.7 14.7 19.8 18.2
::5” 24.1 21.9 28.1 251 52.8 90.0 2.9 12.7 IO.7 6.4 4.3
S-6
;u:
:x
21 22 11
z
: 25 2s 11 19 20
3; 17 10 7 10
14 2 3 26 20 22 16 26 35
19 16 14 24 25 10 If 13 27 18 29 18 17 17 19 20 27 18 25 25 25 16 30 26
Density at Density at Density at Density at perigee 2300km 2300 km 2300km lo-17* 10-l’ 10-q lo-“$
TABLE 1. THE OBTAINED DENSF~~ES
1963-3023 1963-30D 1963-301) 1963-30D 1963-30D 1963-300 1963-30D 1963-30D 1963-30D 1963-14B 1963-14B 1963-14B 1963-14B 1963-14B 1966-56A 196656A 1966S6A 1966~56A 196656A 1966~56A 1966-56A 1966-56A 1966-56A 1966-56A
Satellite
z
5 X
5 5 z
I!! t: c Iz
P
$
@I
e
!z P-l 8
3
1710
M. ROUSSEAU x 1963-148 Al963-300 0 lS66-56A
40-
F 0
A
i?
0
A
0
0
A0
Oo
0
A
0 AX
a: IO-
A
I 1965
x A
Ax'
I 1966
*
I 1966
I 1967
x
8 I969
x
0
! 1970
I 1971
I 1972
FIG. 4. VARIATIONS OF THEDENSITY AT 2300 km AS FUNCTION OF TIME.
x
5C
1963-146
A 1963-30D co
o 1966-56A A
X
!z
b
0
x
0
A
A A
cl A
0
X
A0
0
0
IC
A X
0 XX
I
I
I
I
‘,A,
F
M
A
M
J
J
,
,
,
,
,
,
A
S
0
N
D
J
FIG. 5. VARIATION OFTHEDENSITY AT 2300 km AS A FUNCTION OFTHEMONTHS OFTHEYEAR,t.
higher and for the months May, June and January they are lower, showing a semi-annual effect whose maximum value is about 4 or 5 times the minimum. 7. COMPARISONS
WITH OTHER DATA
Some values can be compared with those obtained by other authors. From Prior’s paper on 1963-30D, the effect due to the drag on the mean motion is about 0.54 X 10W5 rev/day2 between 17 March and 9 June 1967. For the same period approximatively, we have found 0.72 x 1O-6rev/day 2. The difference is greater than the probable error. However, we think, it is quite possible to attribute this difference to a too optimistic estimation of errors. The difference in density obtained by Prior and by us presents a greater problem. Our value of the density is about twice as great as Prior’s value. A possible interpretation is that in our adopted reference model, the scale height is about 550 km but Prior adopted 950 km from simultaneous observations of Pageos and 1963-30D. Unfortunately, for this period our observations of Pageos were too poor to be useful and a more detailed comparison is not possible. However if we plot the densities as a function of height, we can define a mean scale height and it appears that the values of 750 or 800 km are adequate on the average (Fig. 6). Therefore, our value of the density at perigee is possibly too high and that of Prior too low. A more precise computation shows that our value could be too high by about 20 or 30 per cent. The remaining discrepancy can also be partially attributed to the difference in the observed variation of the semi-major axis and the drag effect of H+.
DENSITIES DEDWCED FROM PERTURBATIONS
Fm.6, VARIATI~NSOFTHE~~ARITHMOFTBE
DENSITYLOG
1711
AT HIGH ALTITUDES
p ASFUNCTIONOFTHEALTITUDE
h.
The importance of the choice of the scale height and of the model must be emphasized, for another consequence of this choice is the error introduced when extrapolating the values of the densities at perigee to the reference altitude (2300 km). If the perigee is relatively low, the error so introduced in the density can be 50 or 100 per cent. Therefore in Table 1 we have computed the densities corresponding to the same variations of the semi-major axis using two different scale heights h. The difference can be seen. For one period in 1964, we can also make a comparison with the results obtained by Fea and Smith.(2) The agreement is quite good; Fea and Smith obtained I.8 x 1O-2o g/cm3 at 3500 km compared with our value of 2.3 x lO-2o g/cm3. Generally our values are, on the average, 3.5 times greater than the theoretical values of Nicolet’s model of 1962,“rr) as the values predicted by the extrapolated model of Jacchia 1970. However as the values of our scale heights are possibly too low, the theoretical values of Nicolet’s model needs only be m~ti~iied by a factor 3.
From the present set of data, it is possible to deduce a mean value of the density. A variation with the season is shown but no too precise conclusion can be drawn due to the necessity of averaging the density in time and space in order to obtain a signi~~ant value. However the coherence between the values presented here and those obtained by other authors are an indication of the validity of the results. Ae~owIe~geme~t~-We are grateful for very useful discussions with Drs. F. Bar&x and C. Berger and for the assistance of J. P. Perk. We thank also the Smithsonian Astrophysical Observatory, the U.S. coast and geodetic survey and the United States Air Force which provided some orbital elements.
f . K. H. FEA, Planet. Space Sci. X3,1289 (1965). 2. K. H. -A and D. E. SN(ITW, Some further studies of perturbations of satellites at great altitude. Planet. Space Sci. l&1499 (1970).
3. E. J. PRIOR, Observed effects of earth reflected and hydrogen drag on the orbital energies of balloon satellites. Symposium on the use of artificial satellites for geodesy, Washington, D.C. (1971).
1712
M. ROUSSEAU
4. M. ROUSSEAU,Etude de la pression de radiation solaire et du frottement atmosphkique sur les satellites 1966-56A, 1963-3OD, 1963-14B. Con@ Interobs, Bucarest (1970). 5. F. BARLIER,Recherches sur ia d&termination et I’utilisation des elements osculateurs dans la mkanique des satellites proches. Bull. Astr. Tome XW, Fast. 4. 326. 6. D. E. SMITHand K. E. ICISSEL, Anomalous accelerations of the Pageos Spacecraft (G.S.F.C. aotit 1971). 7. P. LALA, Semi-analytical theory of solar pressure perturbations of satellite orbits during short time intervals. Bull. astr. Insts. Cd. 22, 63 (1971). 8. S. P. WYATT, The effect of terrestrial radiation pressure on satellites orbits. Dynamics of SatdUes (Ed. M. Roy), p. 180. Springer, Berlin (1963). 9. L. G. JACCHIA,New static models of the thermosphere and exosphere with empirical temperature profiles. Smithson. Astrophys. Special Report 313 (1970). 10. G. M. KEATING and E. J. PRIOR,The winter helium Bulge. Space Research, Vol. VIII, p. 982 (1968). 11. G. KOCKARTS and M. NICOLET,LR problbme a6ronamique de I’hklium et de I’hydrog&ne neutres. Annls GPophys. 18,269 (1962).