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The estimation of atmospheric scale heights from the contraction of satellite orbits
Planet. Space Sci. 1963. Vol. 11. pp. 633 to 637.
Pergamon Press Ltd.
Printed in Northern Ireland
THE ESTIMATION OF ATMOSPHERIC SCALE HEIGHTS FROM THE CONTRACTION OF SATELLITE ORBITS B. R. MAY D.S.I.R., Radio Research Station, Ditton Park, Slough, Bucks. (Received 5 March 1963) Abstract-A numerical investigation is carried out into the behaviour of the ratio Ar,,/Ae, where Ar, and Ae are the changes in the perigree distance and eccentricity per revolution due to air drag. It is shown that Ar,,/Ae can be related to both the density scale height H, and the pressure scale height h, of the upper atmosphere. The results are presented in a graphical form and are used to obtain H and h values from the observed changes in satellite orbits for which any reasonable orbital data are available. Estimates are also made for the temperature/molecular weight ratio and the temperature in the 180-240 km height region.
1.BASIC THEORY
The theory of the contraction of a satellite orbit due to air drag in a spherically symmetrical atmosphere has been treated by Cook et al.(l), who have given expressions for the change in one revolution in a, the semi-major axis and in x = ae where e is the eccentricity. From these, the corresponding changes per revolution in the perigee distance r,, and in e itself, can be readily derived, and are as follows: -FSC m
Ar, = -----“az(l
-FSC, Ae = -.a(1 m where S C, F m E p
= = = = = =
The ratio
of these
-e)
(1)
-e2)
(2)
mean cross-sectional area of the satellite, drag coefficient, factor correcting for the easterly rotation of the Earth’s atmosphere, mass of satellite, eccentric anomaly, air density. changes
4
xi=&
is c2” ; S(
+=;;;)‘.(I-cosE).p.dE
2n l+ecosE 0 S( l-ec0s.E
* ) ’
(3) cosE.p.dE
and is thus a function only of a, e and the distribution of p with z, and is independent of S, C, and m which may not be well known. The ratio At-,/he was chosen for study because it turns out to be almost completely independent of a, and also because the same factor ((1 + e cos E)/(l - e cos E))* occurs in both integrals in (3), thus facilitating the numerical integration. 2. THE RELATION OF Ar,/Ae TO ATMOSPHERIC SCALE HEIGHT In the integration of equation (3) it was first assumed that the density scale height H, defined by H = -p . dzldp, was constant, so that the density varies exponentially with 4
633
634
B. R. MAY
height. The variation of density with E is then given in terms of its value at perigee, ps, by P = pz, exp.
-ae(l
- cos E) H
(
1’
Equation (3) was integrated numerically for a range of values of e from 0.001 to 0.3, a from 6600 to 8000 km and Hfrom 10 to 100 km. The results for a = 7000 km are shown in Fig. 1, and may be taken to apply for all values of a in the range used, the maximum error in H for a given Ar,/Ae being less than 2 km. Thus for the values of e and Ar,/Ae for a satellite orbit, the curves of Fig. 1 enable H to be found, assuming this to be independent of height. o=?OOOkm
t;_-2g.ool
Cl I.0
I
I 4.0
I 3.0
2.0
log,,
(Ar,o/Ae) FIG.
I 5.0
- (Arp/Ae),
6.0
km
1
There is, however, good evidence that the scale height increases with height@), and a better approximation is to assume a linear increase, i.e. H = H,, + B(z - z,)
(4)
where H,, is the value of H at the height of perigee, z = z,, and @is a numerical coefficient, whose value is between 0 and +O*5(3) in the appropriate height range. The variation of density corresponding to (4) is P = PP
1 + pae(l - cos E) -$ (
1
H,
*
(5)
A linear variation of density scale height H implies a linear variation of the pressure scale height h, defined by h=
-*_d’ dP
where p is the air pressure, h is of value in upper atmosphere studies because it is directly related to the absolute temperature T and the molecular weight M by h =-
kT M?
where k = Boltzmann’s constant and g = gravitational acceleration.
(6)
THE ESTIMATION
OF ATMOSF?HERIC SCALE HJXIGH’IS
635
220 >O*I 200
-
180 -
160 -
E x ii
140 -
120 -
q 100 -
80
-
If the variation of g with height is neglected, h and H are related by
(1+;
h=H
and
is found
the variation H
h with
(7)
corresponding to
H
h=(l)=(-_)+(l B
is
(z - ZJ.
Equation 3 was integrated numerically assuming p to vary as in (5), for values of a = 7000 km, e from O*OOlto 0.3, H, from 10 to 100 km and /I from +0-l to +05. Any given values of Ar,/Ae and e now corresponds to an infinite number of combinations of H, and B, so these cannot be found separately. However, an effective value of H, say L, can be found from Fig. 1 and this will be the value of H at some level, say Az~ above the height of perigee suchthat L = H, -I- ,6. AZ,. (8) L will also be the value of h at some other level, say AZ,, above perigee where L = (H, + B . AzJ/(l -
/OS>.
(9)
From (8) and (9), we have AZ, - AZ, = L.
(10) Now it is found from the numerical integrations that for the range of parameters used, AZ, and hence AZ,,, depend only on L and e, and are independent of HP and b. Thus the value of L, obtained from Fig. 1, together with the known values of e and the height of
perigee above the Earth’s surface are sufficient to obtain the levels at which the scale heights H and h respectively have the value L. Figure 2 shows how AzH varies with L for different
636
B. R. MAY
values of e. For the large values of e it is seen that the curves approach a straight line of slope +1*5, as has been found anal~ically by King-Helet@. Having obtained AZ, from Fig. 2, AZ, is found from equation (10). This latter relation is particularly valuable because Az~ is always less than AZ,, and in fact negative for orbits of low eccentricity. This means that for such orbits the pressure scale height h can be estimated at heights rather lower (actually below perigee) than those for which H can be obtained. 3. THATCH
OF SATELLITE DATA
Orbital data for the following satellites have been used : Sputniks 2 and 3, Atlas (1958 5), Explorer 3, Discoverer 5, 14 and 17 and Discoverer 5 capsule. Before using the observed values of ra and e, these were corrected for the effects of the Earth’s oblateness in the manner described by King-Hele f2), The remaining variations of rz, and e with time were assumed to be due to the effects of air drag, since effects due to charge drag, solar radiation pressure and the gravitational attraction of the Sun and Moon are comparatively small and were neglected. For each satellite, r,, was plotted against e and the slope dr,/de of the curve measured at suitable times. This slope was then identified with the ratio Ar,/Ae, of the changes in rP and e in one revolution-this is justified since the measured value of dr,/de represents a mean over at least IO days, and sometimes over as long a period as several months, owing to the sparseness of reliable orbital data. From the value of dr,lde and the known value of e, L, AZ, and AZ&were obtained as described in Section 2. L then represents the value of H and h at heights AZ, and Azn respectively with respect to the perigee height. From the value of h, the ratio of absolute temperature to mean molecular mass, T/M’, can be estimated using equation (6). For the satellites used here the values of dr,/de are accurate to about 20 per cent-in a typical case this corresponds to an accuracy of 15 km in the value of H or h. 4. RESULTS
Figures 3 and 4 show the values of H and h obtained for the period 1957-1960 where the points have been grouped into their appropriate years. The tendency for both scale heights to decrease during this time is clearly seen, and this can be associated with the decline in 60 -
d
50 -
,/I d/
E t
4o 30-
_&s---& --O-
0. @.'o /) /Me tic -O--O+0 oi957-1958 01959 •I1960
?----"
i 20-
IO -
0 170
I 180
I 190
I 200
I 210
I 220
Height, Fit.
I 230 km
3
I ;40
I 250
I 260
637
THE EXTIMATION OF ATMOSPHERIC SCALE HEIGHTS
210
220
230
240
250
260
270
Height,
280
290
IO
300
km
FIG.4
solar activity since 1957. The approximately linear increase of both H and h with height is also evident and indicates that the coefficient /? is of the order of $0.2. These results show similar trends to those obtained by King-Hele and Rees (2) but the scale heights tend to be 5-10 km lower for the earlier years. Values of T/M obtained from equation (6) are shown in Table 1. Values of M taken from the C.I[.R,A.(*) for 1961 are given and values of T so obtained are also tabulated. These results show a decrease in temperature with declining solar activity and suggest also a positive temperature gradient in the range studied. TABLJJ~.iz~~km, TIN “K,M IN ATOMIC ~JSHT~IT~ 1957-58 Height (km) 180 190 200 210 220 230 240
M (Ref.c4)) 27.5 27-3 270 268 265 26.2 259
h 33.0 35.0 37.0 39.5 425 48.3 60.0
1959
T z
T
36.8 38.9 41-o 43.7 46.8 53-o 65.8
1012 1062 1107 1171 1240 1389 1704
h
34*0 35.0 385
T z
37.6 375 42.3
1960 T
1008 994 1108
T Z
T
33‘4 33.7 33.9 34.2
912 910 PO8 906
h 30.0 30.5 30.7 31.0
Ackmwle@mmzts--The work described here was carried out as part of the programme of the Radio Research board and is published by permission of the Director of Radio Research of the Department of Scientiiic and Industrial Research. REFJIRENCES
1. 2. 3. 4.
G. D. D. H.
E. Coo& D. G. KING-HEWand D. M. C. WALKER,Proc. Roy. Sot. A 257, U4 (1960). G. KING-E&LBand J. M. R.nns,Proc. Roy. Sot. A 270,562 (1962). G. KXSHELB and G. E. COOK,R.A.E. F~borough, Technical note (1962). IL KALLMAN-BUL (Ed.)COSPAR Int. Ref. Atm. (1961).
Pergamon Press Ltd.
Printed in Northern Ireland
THE ESTIMATION OF ATMOSPHERIC SCALE HEIGHTS FROM THE CONTRACTION OF SATELLITE ORBITS B. R. MAY D.S.I.R., Radio Research Station, Ditton Park, Slough, Bucks. (Received 5 March 1963) Abstract-A numerical investigation is carried out into the behaviour of the ratio Ar,,/Ae, where Ar, and Ae are the changes in the perigree distance and eccentricity per revolution due to air drag. It is shown that Ar,,/Ae can be related to both the density scale height H, and the pressure scale height h, of the upper atmosphere. The results are presented in a graphical form and are used to obtain H and h values from the observed changes in satellite orbits for which any reasonable orbital data are available. Estimates are also made for the temperature/molecular weight ratio and the temperature in the 180-240 km height region.
1.BASIC THEORY
The theory of the contraction of a satellite orbit due to air drag in a spherically symmetrical atmosphere has been treated by Cook et al.(l), who have given expressions for the change in one revolution in a, the semi-major axis and in x = ae where e is the eccentricity. From these, the corresponding changes per revolution in the perigee distance r,, and in e itself, can be readily derived, and are as follows: -FSC m
Ar, = -----“az(l
-FSC, Ae = -.a(1 m where S C, F m E p
= = = = = =
The ratio
of these
-e)
(1)
-e2)
(2)
mean cross-sectional area of the satellite, drag coefficient, factor correcting for the easterly rotation of the Earth’s atmosphere, mass of satellite, eccentric anomaly, air density. changes
4
xi=&
is c2” ; S(
+=;;;)‘.(I-cosE).p.dE
2n l+ecosE 0 S( l-ec0s.E
* ) ’
(3) cosE.p.dE
and is thus a function only of a, e and the distribution of p with z, and is independent of S, C, and m which may not be well known. The ratio At-,/he was chosen for study because it turns out to be almost completely independent of a, and also because the same factor ((1 + e cos E)/(l - e cos E))* occurs in both integrals in (3), thus facilitating the numerical integration. 2. THE RELATION OF Ar,/Ae TO ATMOSPHERIC SCALE HEIGHT In the integration of equation (3) it was first assumed that the density scale height H, defined by H = -p . dzldp, was constant, so that the density varies exponentially with 4
633
634
B. R. MAY
height. The variation of density with E is then given in terms of its value at perigee, ps, by P = pz, exp.
-ae(l
- cos E) H
(
1’
Equation (3) was integrated numerically for a range of values of e from 0.001 to 0.3, a from 6600 to 8000 km and Hfrom 10 to 100 km. The results for a = 7000 km are shown in Fig. 1, and may be taken to apply for all values of a in the range used, the maximum error in H for a given Ar,/Ae being less than 2 km. Thus for the values of e and Ar,/Ae for a satellite orbit, the curves of Fig. 1 enable H to be found, assuming this to be independent of height. o=?OOOkm
t;_-2g.ool
Cl I.0
I
I 4.0
I 3.0
2.0
log,,
(Ar,o/Ae) FIG.
I 5.0
- (Arp/Ae),
6.0
km
1
There is, however, good evidence that the scale height increases with height@), and a better approximation is to assume a linear increase, i.e. H = H,, + B(z - z,)
(4)
where H,, is the value of H at the height of perigee, z = z,, and @is a numerical coefficient, whose value is between 0 and +O*5(3) in the appropriate height range. The variation of density corresponding to (4) is P = PP
1 + pae(l - cos E) -$ (
1
H,
*
(5)
A linear variation of density scale height H implies a linear variation of the pressure scale height h, defined by h=
-*_d’ dP
where p is the air pressure, h is of value in upper atmosphere studies because it is directly related to the absolute temperature T and the molecular weight M by h =-
kT M?
where k = Boltzmann’s constant and g = gravitational acceleration.
(6)
THE ESTIMATION
OF ATMOSF?HERIC SCALE HJXIGH’IS
635
220 >O*I 200
-
180 -
160 -
E x ii
140 -
120 -
q 100 -
80
-
If the variation of g with height is neglected, h and H are related by
(1+;
h=H
and
is found
the variation H
h with
(7)
corresponding to
H
h=(l)=(-_)+(l B
is
(z - ZJ.
Equation 3 was integrated numerically assuming p to vary as in (5), for values of a = 7000 km, e from O*OOlto 0.3, H, from 10 to 100 km and /I from +0-l to +05. Any given values of Ar,/Ae and e now corresponds to an infinite number of combinations of H, and B, so these cannot be found separately. However, an effective value of H, say L, can be found from Fig. 1 and this will be the value of H at some level, say Az~ above the height of perigee suchthat L = H, -I- ,6. AZ,. (8) L will also be the value of h at some other level, say AZ,, above perigee where L = (H, + B . AzJ/(l -
/OS>.
(9)
From (8) and (9), we have AZ, - AZ, = L.
(10) Now it is found from the numerical integrations that for the range of parameters used, AZ, and hence AZ,,, depend only on L and e, and are independent of HP and b. Thus the value of L, obtained from Fig. 1, together with the known values of e and the height of
perigee above the Earth’s surface are sufficient to obtain the levels at which the scale heights H and h respectively have the value L. Figure 2 shows how AzH varies with L for different
636
B. R. MAY
values of e. For the large values of e it is seen that the curves approach a straight line of slope +1*5, as has been found anal~ically by King-Helet@. Having obtained AZ, from Fig. 2, AZ, is found from equation (10). This latter relation is particularly valuable because Az~ is always less than AZ,, and in fact negative for orbits of low eccentricity. This means that for such orbits the pressure scale height h can be estimated at heights rather lower (actually below perigee) than those for which H can be obtained. 3. THATCH
OF SATELLITE DATA
Orbital data for the following satellites have been used : Sputniks 2 and 3, Atlas (1958 5), Explorer 3, Discoverer 5, 14 and 17 and Discoverer 5 capsule. Before using the observed values of ra and e, these were corrected for the effects of the Earth’s oblateness in the manner described by King-Hele f2), The remaining variations of rz, and e with time were assumed to be due to the effects of air drag, since effects due to charge drag, solar radiation pressure and the gravitational attraction of the Sun and Moon are comparatively small and were neglected. For each satellite, r,, was plotted against e and the slope dr,/de of the curve measured at suitable times. This slope was then identified with the ratio Ar,/Ae, of the changes in rP and e in one revolution-this is justified since the measured value of dr,/de represents a mean over at least IO days, and sometimes over as long a period as several months, owing to the sparseness of reliable orbital data. From the value of dr,lde and the known value of e, L, AZ, and AZ&were obtained as described in Section 2. L then represents the value of H and h at heights AZ, and Azn respectively with respect to the perigee height. From the value of h, the ratio of absolute temperature to mean molecular mass, T/M’, can be estimated using equation (6). For the satellites used here the values of dr,/de are accurate to about 20 per cent-in a typical case this corresponds to an accuracy of 15 km in the value of H or h. 4. RESULTS
Figures 3 and 4 show the values of H and h obtained for the period 1957-1960 where the points have been grouped into their appropriate years. The tendency for both scale heights to decrease during this time is clearly seen, and this can be associated with the decline in 60 -
d
50 -
,/I d/
E t
4o 30-
_&s---& --O-
0. @.'o /) /Me tic -O--O+0 oi957-1958 01959 •I1960
?----"
i 20-
IO -
0 170
I 180
I 190
I 200
I 210
I 220
Height, Fit.
I 230 km
3
I ;40
I 250
I 260
637
THE EXTIMATION OF ATMOSPHERIC SCALE HEIGHTS
210
220
230
240
250
260
270
Height,
280
290
IO
300
km
FIG.4
solar activity since 1957. The approximately linear increase of both H and h with height is also evident and indicates that the coefficient /? is of the order of $0.2. These results show similar trends to those obtained by King-Hele and Rees (2) but the scale heights tend to be 5-10 km lower for the earlier years. Values of T/M obtained from equation (6) are shown in Table 1. Values of M taken from the C.I[.R,A.(*) for 1961 are given and values of T so obtained are also tabulated. These results show a decrease in temperature with declining solar activity and suggest also a positive temperature gradient in the range studied. TABLJJ~.iz~~km, TIN “K,M IN ATOMIC ~JSHT~IT~ 1957-58 Height (km) 180 190 200 210 220 230 240
M (Ref.c4)) 27.5 27-3 270 268 265 26.2 259
h 33.0 35.0 37.0 39.5 425 48.3 60.0
1959
T z
T
36.8 38.9 41-o 43.7 46.8 53-o 65.8
1012 1062 1107 1171 1240 1389 1704
h
34*0 35.0 385
T z
37.6 375 42.3
1960 T
1008 994 1108
T Z
T
33‘4 33.7 33.9 34.2
912 910 PO8 906
h 30.0 30.5 30.7 31.0
Ackmwle@mmzts--The work described here was carried out as part of the programme of the Radio Research board and is published by permission of the Director of Radio Research of the Department of Scientiiic and Industrial Research. REFJIRENCES
1. 2. 3. 4.
G. D. D. H.
E. Coo& D. G. KING-HEWand D. M. C. WALKER,Proc. Roy. Sot. A 257, U4 (1960). G. KING-E&LBand J. M. R.nns,Proc. Roy. Sot. A 270,562 (1962). G. KXSHELB and G. E. COOK,R.A.E. F~borough, Technical note (1962). IL KALLMAN-BUL (Ed.)COSPAR Int. Ref. Atm. (1961).