- Email: [email protected]
A calculated hydrogen distribution in the exosphere
Planet. Space Sci. 1972, Vol. 20, pp. 1147 to 1162. Persamon Press. Printed in Northern Ireland
A CALCULATED
Laboratoire
HYDROGEN DI§TRI~UTION EXOSPHERE
IN THE
A. VIDAL-MADJAR de Physique Stellaire et PlantUre, Centre National de la Recherche Scientifique, Verrieres-le-Buisson, France and
Service d’A&onomie,
J. L. BERTAUX Centre National de la Recherche Scientiiique, Verri&res-leB&son, France (Received injinal form 15 January 1972)
Abstract-The density distribution in a planetary exosphere is a function of the boundary con~tions, i.e. the density and the temperature ~s~ibutio~ at the exobase. We established, with the use of Liouville’s theorem, that the exospheric density may be calculated with a triple integral. The results of a numerical evaluation of this integral are presented for three different exobase models for the case of atomic hydrogen in the Earth’s exosphere. They all satisfy the zero net ballistic flux condition, but correspond to different levels of solar activity. A constant density level was found at 1.8 earth radii, below which the computed density distribution may be approximated by a local spherical Chamberlain model. Above the constant density level, a simple relation was found to tit the computed density dis~ibution within f5 per cent. INTRODIJCTION
The exosphere is the layer of a planetary atmosphere where the number of atomic collisions per unit time becomes negligible. In order to calculate exospheric density distribution it is generally assumed that the constituents have an isotropic Maxwell-Boltzmann distribution in the diffusosphe~ up to a sharp boundary, called the critical level or the exobase (Chamberlain, 1963; Nicolet, 1964). Above this level, in the collisionless exosphere, particles with an upward velocity will describe ballistic trajectories. According to their velocities at the exobase, they will either describe hyperbolic trajectories and escape from the atmosphere, or elliptic trajectories and return to the diffusosphere at another point on the exobase. opik and Singer (1959, 1960, 1961) gave analytical expressions of exospheric densities of both types of particles, elliptic and hyperbolic. Chamberlain added satellite particles, created by a small number of collisions in the exosphere. He introduced the concept of a satellite critical level R,,, above which no satellite particles pass through perigee, and below which their velocity distribution fills in the velocity distribution of ballistic particles. At a particular point in the exosphere, densities of each type of particle are derived by integration of the distribution fun&ion over the momentum part of the phase space. The integration can be analytically performed only in the case of a uniform exobase (temperature and density constant at the exobase). This case, which was considered by the previous authors, leads to an exospheric density distribution with a spherical symmetry. However, the interpretation of ground based, rocket and satellite experimental results often requires some degree of asy~et~ in the exospheric density dist~bution of hydrogen (Meier, 1969; Metzger and Clark, 1970; Tinsley, 1970). A computer program was constructed to perform the numerical integration necessary to, obtain the exospheric density distribution corresponding to any given temperature and. density distributions at the exobase. Special attention was paid to the case of distributions. with axial symmetry and leading to a zero net ballistic flux (McAfee, 1967; Quessette, 1147
1148
A. VIDAL-MADJAR
and J. L. BERTAUX
1972). The corresponding calculations are presented in this paper, along with a numerical approximation of the density distribution above two earth radii (2 ER). THE EXOBASE MODEL An exobase model is characterized by a temperature and density distribution given at the altitude of the exobase. Calculations were completed for atomic hydrogen in the terrestrial atmosphere. The exobase level was assumed to be at a constant altitude of 500 km; the importance of this assumption is discussed in a later section. The temperature distribution at the exobase was a sinusoidal approximation of CIRA 65 model. Three different levels of solar activity were considered. Even though the CIRA 65 model has no axial symmetry, a small displacement can align on the same axis the maximum temperature point, the center of the Earth and the minimum temperature point. Then, the temperature distribution at the exobase has a symmetry about this axis, and can be approximated by:
T,(Y)=
Tmax
+ 2
Gin + Lax - ‘Kmin cos y. 2
(1)
T,(y) is the temperature at any point of the exobase where the local vertical is at the angle y with the axis of symmetry. For each temperature distribution, Quessette computed the density distribution at the critical level (Fig. 3) leading to a zero net ballistic flux situation, i.e. the outward ballistic flux equal to the inward ballistic flux at any point of the exobase. The calculations of Quessette are more accurate than the McAfee calculations; the relative difference between their respective density distributions may reach 10 per cent. We can point out, since we are dealing with a time-independent problem, that temperature and density distributions connected by the zero net ballistic flux condition are the only ones which do not require any sink or source in the diffusosphere (except for the escape component). Since the temperature distribution is symmetrical about one axis, the density distribution at the exobase leading to a zero net ballistic flux, and the density distribution above the exobase, will be symmetrical about the same axis. For the purpose of exospheric calculations, it is not necessary to relate the position of this axis to the Sun-Earth line. DISTRIBUTION
FUNCTION
AND DENSITY
IN THE EXOSPHERE
At any point in the exosphere, the distribution function (or phase space density) can be calculated with the use of Liouville’s theorem, which states that the value of the phase space remains constant along a dynamical trajectory: f(&
Pi) = fc(&
PC?.
(2)
In this equation, the q* are the spatial coordinates, and the pf are the momentum coordinates. The function f(qf, pi) is the distribution function and has the same value all along the trajectory, particularly at the critical level, indicated by the subscript c. At the point qi in the exosphere, a unique dynamical trajectory is determined by the choice of coordinates (qf, pi). Its intersection with the exobase defines the point (qci, pcf) in a unique way, since we consider only the point from where the particle is coming. Then we can state that all the qic and thepi, are unique functions of the (qi, pi), and rewrite the Equation (2) in the following way: J-G?, Pi) = fc(4:(4i, Pi), P6(4, P”)).
(3)
A CALCULATED
HYDROGEN
DISTRIBUTION
IN THE EXOSPHERE
1149
FIG. 1. COORDINATESYSTEMS RELATED TO THE CENTER OF THE PLANET AND TO THE PARTICLE LOCATED
IN
A.
The volume density N(q”) at the point qi in the exosphere is obtained by integrating the distribution functionf(q’,p”) over the momentum part of phase space which is populated (only the dynamical trajectories reaching the exobase are populated):
IV($) =
s
f(qi, pi) d3p'
(4)
POP
where @pi represents the element of momentum Equation (4) can be rewritten:
space. With the use of Equation (2),
At the point qci of the exobase, the distribution function f,is known, since we have made the hypothesis that it is a Maxwell distribution, and since the temperature and density distribution at the exobase are given. We are left with the problem of calculating the six functions qci(qi, p$) and pc*(qi, pi), in order to derive the exospheric density at the point A of coordinates qi. For the sake of clarity, we have used, instead of the (q5, pi) coordinates, the usual spherical coordinates (I, tc, 13)of the point A (Fig. 1) in the Oxyz system, where 0 is the center of the planet, Ox is the symmetry axis of the density distribution at the exobase, and Oxy is the equatorial plane. A second coordinate system AXYZ is related to A, where AZ is the local vertical oriented upwards, and AX perpendicular to AZ, is in the plane OA-Oz oriented in such a way that the angle Oz-Ax is equal to 6. In this coordinate system, the velocity vector of a particle at point A is determined by (a, 0, $)>,where u is the length of the velocity vector, 0 is the angle between AZ and the velocity vector, and I$ is the angle between AX and the projection AU of the velocity vector on the AXY plane, which determines with AZ the orbital plane of the particle.
1150
A. ~DA~MADJAR
and J. L. BERTAUX
Using the normalized variables :
(6) v=L
(7)
V esc
where r, is the radius of the exobase and veSc= 1/2MG/r, is the escape velocity at the exobase level, the equations of motion of a particle are: v,e - f=V-y V sin 8
-
Y
(8)
(9)
= V, sin BC
in which the subscript c stands for variables defined at the critical level, MTis the mass of the planet and G the gravitational constant. The trajectory of a particle passing through A is then completely determined by the set of variables (y, a, 6, V, 0, gd). In order to determine the coordinates a, and 6, at the exobase of the origin point A, of the trajectory, we have calculated in the orbital plane the angle w, defined as the difference between true anomalies of the origin point A, and the point A, respectively PCand p (Fig. 2). The equation of the trajectory, in polar coordinates, has the following form: w
r=
(IO)
1 + e cos p
w and e are constant along the trajectory and can be expressed as functions of (y, V, 0) by applying the conservation of energy and momentum between the points (y, V, 1!7),apogee and perigee. We find: V2 sin’ 0 w = 2r 0 = O(Y, V, 0) Y2 e=
I _ 4 V2 sin2 8 \i
(Y -
(11),
vz) = e(y, v, 0
Y2
The values of #? and /?, can then be extracted from the combination (12): cos p =
cos /!I@ =
Y~Y,
v, 6) -
of (lo), (11) and
T,
e(y, v,0) W(Y,v, f3 - r, e(y, V, 69
and from these equations it is clear that y which is given by: Y=B-A can be expressed as a function ytly, Y, 0). The spherical coordinates Usand 4, of the origin point in the Oxyz system are obtained through the use of the transformation matrix from
A CALCULATED
FIG. 2. GEOMETRY
HYDROGEN
DISTRIBVFION
FOR THE PARTICLE TRAJECTORY
IN TI-IE EXOSPHEREi
1151
IN ITS ORBITAL PLANE.
the AXYZ system to the 0xy.z system, and may be expressed as functions of #, pp, a and 6: 6,(a, 6, +, y) = Arc sin (cos 6 sin y cos 4 + sin 6 cos y) a,& a,$, lu) = Arc cos
-sinBcosasinycos~+sinasinysin~+cos6cosacos~ cos &(a, 6, +, V)
03)
1*
04)
As S,U is a function of (y, V, O), the two angIes a, and 6, are finaily functions of the six variables (y, a, 6, V, 8, #), as expected. The velocity vector of the particle at the origin point is defined by V, and OC,which are functions of (y, V, 0) through the Equations (8) and (9). Given an exobase model and a set of variables (j, a, S, V; 8, #), the knowledge of a0 and 6, allows one to calculate the values of temperature T,(Gc,,6,) and density N,(a,, 6,) at the origin point, as well as the Maxwell distribution function at this point:
Because the distribution functionf, has a Maxwell form, f,depends only on V,, a, and 8,. With the coordinates that we have used, the element of momentum space in Equation (4) is: d3p, = m3u&,V2sin 0 dV d0 d4.
1152
A. VIDAGMADJAR
and J. L. BERTAUX
The volume density N(y, cc, 6) at the point A is derived from the integral of Equation (4), in which fC has the form given by Equation (15) when Y, is expressed as a function of V and y (Equation 8):
x exp
--x
mMG
V2sin8dVd0d$. To The integration over V, 6’ and r$ must be restricted to the existing trajectories. types of particles are generally considered:
(16) Three
-ballistic particles defined by V, < I or V < 2/;; and coming from the exobase -hyperbolic particles defined by V, > 1 or V > dy -satellite particles created by a small number of collisions in the exosphere, and ionized by the EW flux (we have not considered satellite particles in this study for reasons discussed in a later section). (a) Ballistic particles There is a velocity V, at each distance r such that if V < V,, the corresponding trajectory will intersect the exobase for any direction of the velocity vector. This velocity V, corresponds to a trajectory whose apogee is r and whose perigee is r,: sin 8 = sin 6, = 1 and from (8) and (9) is obtained (17) On the other hand, if Y > V,, all the trajectories do not reach the exobasc. The populated trajectories are within a cone of half-angle @,, whose axis is the local vertical. The perigee of the corresponding trajectory lies at the exobase level; from (8) and (9) is derived: sin 8, = ;jv2+1-y.
(18)
The ascending particles populate the region 0 < 13< 8, whereas the descending particles populate the region r - 8, < 8 < rr. The volume density of ballistic particles, iV,(y, A, S) is the sum of the integrals:
(19)
A CALCULATED
HYDROGEN
DISTR~~ION
IN THE EXOSPmRE
1153
(b) Hyperbolic partides The condition (18) is still valid for hyperbolic particles. The populated trajectories are again within a cone of half-angle B,,,; however, only the particles which escape the planet need to be considered, and are those that populate the region 0 < 0 < f3,,,. The volume density &(y, cc, 6) of hyperbolic particles is: WY,
% 4 = ~34*,
IdiVz
dVlmsin
8 dOrfO
d$.
(20)
The total volume density N(y, a, S) in the exosphere is the sum of N,(y, M, S) and N,(y, a, 6). In the case of the Chamberlain model with a uniform temperature and density distribution at the exobase, integrals (19) and (20) are reduced to simple integrals which can be expressed analytically using the I’ function. In the present case, a triple integration was numerically performed with a 10 points Gauss method for each integral. The whole calculation was checked and its precision estimated by comparison with the case of a uniform exobase, for which integrations over 4 and 0 were done analytically, and the integration over V was done with a Gauss method having a much larger number of points (32). The precision was found to be of the order of 1O-3 or better, and was considered satisfactory. RESULTS
Three different exospheric density distributions of atomic hydrogen were calculated, corresponding to three different levels of solar activity. The maximum and minimum temperature, Tmax and Tmin at the exobase, taken from CIRA 65, are indicated in Table I, TABLE 1. EXOBASE MODELS
Model
Solar activity
:r III
Low Mean High
(gy
pg;
bmx
&f$
688 979 1413
1005 1460 2083
2.35 1.92 1.53
4000 5200 6500
along with the ratio pmax, of maximum to minimum hydrogen densities computed by Quessette (1972) with the zero net ballistic flux condition (Fig. 3). In the CIRA model, the point where tem~rature is maximum is at 1400 local time in the equatorial plane (for equinox conditions). Since this point is on the Ox axis of our coordinate system, which is an axial symmetry axis for the present exospheric distribution, calculations were made only in the equatorial plane and are described as a function of distance r and angle CL Because of the symmetry axis, it is important to note that the two functions describing the exobase level may be expressed as functions of only one angle y as explained in Formula (1). These functions have then the following form: N,(% S,) = N,(y)
and
T,(ccc, 8,) = T,(y)
where
y = Arc cos (cos 01,cos 6,).
In the equatorial plane, where 6 = 0, we have then y = CL We must point out that all hydrogen densities are normalized to 1 at the point of minimum density. This point of minimum density coincides with the point of maximum temperature, a phenomenon radically different from the case of major atmospheric constituents. This characteristic of hydrogen dis~ibution at the exobase has already been found from escape flux considerations (Bates and Patterson, 1961; Kockarts and 3
A. VIDAGMADJAR
1154
0
30
60
and J. L. BERTAUX
90
120
150
180
o(
FIG. 3. RELATIVEDENSITY
DISTRIBUTIONS OF ATOMIC HYDROGEN AT THE EXOBASE YIELDING ZBRO NET BALLISTIC FLUX, COMPUTED BY Qu~ss~ne (1972).
Nicolet, 1963). The introduction of the zero net ballistic flux condition does not change this characteristic of hydrogen distribution. However, the ratio of maximum to minimum density is somewhat smaller in the later case. (a) Density variation with altitude: For each exobase model I, II and III, the variation of density with altitude is shown in Fig. 4 for two typical values of a: a = 0” direction of the point of maximum temperature a =
180” direction of the point of minimum temperature.
For each model, the two curves intersect at an altitude 2, (-1.8 earth radii) indicated in Table I, showing that a uniform density level must exist at this altitude. Above Z,, the two curves are again separated and show that the higher density is to be found on the hot side of the Earth, as in the case of the major atmospheric constituents. (b) Density variation at a constant altitude: Density at a constant altitude as a function of a, is compared with the density for a = 0”. These relative variations, shown in Fig. 5, display more clearly the previous results, i.e. the existence of a nearly constant density level, and the inversion above this level of the density variation. However, if the density at the exobase level changes by a factor of 2, above N 2 ER, this variation remains much smaller (less than 25 per cent for any of the three models). Then it can be said, in a first approximation, that the density above 2 ER is only a function of altitude, as in the case of a spherical Chamberlain model (1963), which is characterized by a uniform exobase and defined by two parameters, the temperature T and
A CALCULATED
HYDROGEN
DIS~IB~ION
1155
IN THE EXOSPHERE
Altitude IO” (4
105
(Km)
and J. L. BERTAUX
A. VIRAL-MA~JAR
Model
30
0
I
60
120
90
150
180
ci
(a)
Model
Altitude
II
(km)
I
0
30
60
120 (4
90
150
a
Altitude
Model III
i
180
(km)
500 1000
8
I
Altitude in E.R.
04 0
FIG. 5. RELATI~ U(cLISTHE
ANGLE
30
60
90
120 (4
150
180
cf.
DENSITY VARIATIONS AT CONSTANT ALTITUDES AS FUNCTIONS OF THE ANOLE BETWEEN
THE
LOCAL
VERTICAL
MAXIMUM
AND THE
TEMPERATURE).
VERTICAL
CONTAINING
THE POINT
OF
A CALCULATED
HYDROGEN
DISTRIBUTION
IN THE EXOSPHERE
the density N at the exobase (when satellite particles are not considered). CM(T, N) such a model, in which N is in fact a multiplicative factor.
1157
Let us call
We have tried to find several approximations of the calculated non-uniform exospheric distribution using combinations of Chamberlain models, and we have estimated the degree of approximation by the comparison of both resulting distributions. (c) Comparison with Chamberlain distributions: For each level of solar activity the model CM@‘, N) taken for comparison was such that T and N were equal to the values of temperature and density in the non-uniform model at the exobase and for u = 90”, i.e. T = T,(90*)
N = N,(90”).
The ratio of densities in the non-uniform model and in the model CM(Z’, N) is shown in Fig. 6. Of course, the approximation is very poor below 2 ER, but is within &15 per cent above that level. For the case of a = 90° the situation is better, since the approximation is within f3 per cent, indicating that the CM model gives the best approximation precisely on the vertical, at which origin T and N were taken. This case may be generalized. For each point A in the exosphere defined by the (r, a) coordinates, there exists a CM distribution defined by N = N,(a) and T = T,(a), density and temperature of the exobase point located below the point A on the same vertical. With the use of an infinite number of CM models (one for each value of a), one can construct a distribution called the local CM model. Under these conditions, the density distribution at the exobase is the same for the non-uniform model and the local CM model. The ratio of densities in both models is shown in Fig. 7 as a function of altitude for three different values of a (O’, 90*, 180’). The ratio is 1 at the exobase, remains between 0.95 and 145 up to cu 1.5 ER, and then increases and decreases more or less quickly above that level. It could have been expected that the local CM model would give a good approximation just above the exobase up to 1.5 ER; within that range, the dominant particles are those coming from just below. The particles coming from other locations on the exobase are weighted less in the density calculation. This last result is quite impo~ant because it shows that in a first approximation the exosphere observed from the exobase level is the local exosphere. DISCUSSION
Above 2 ER, we saw that the CM distribution corresponding to cc = 90” was a valid approximation to our non-uniform density distribution within +15 per cent. We tried to find a ~st~bution called ‘numerical approximation’ which would give a better fit above 2 ER. As the density N, at the exobase is only a multiplicative factor in the CM distribution, we built this numerical approximation by using CM distributions with a variable N,(M) and a constant exospheric temperature T,, independent of a. In Fig. 8 are shown the results obtained with this numerical approximation defined by
where fl, and Fc are the mean values of the functions N,(r) and T,(y) between 0 and 7~.
A. VIDAGMADJAR
and J. L. BERTAUX
Model f
Model III
(4 COMPARISON OF THE NON-UNIFORM DENSITY DISTRIBUTION WITH THE ‘MEAN” Cmw BBRLAIN DENSI’IY DI§TRIBIJTIGN (DEFINED BY THE TEMpERATWRE AND DENSITY AT a = wt”). %I3 RATIO OF THESE TWO DENSITIES IS PLOTTER AS A FUNCTION OF GWJCENIWC DISTANCE r POR
FIG. 6.
THREE TYpICAL SITIJATIONS: a = O”, (L = 90” AND a =
180”.
A CALCULATED
HYDROGEN
Model I
DIST~~ON
Model II
_I_____
Model III
--.
12
3
I 0
4
5
IN THE EXOSPHERE
1159
in Earth Radii 6
7
8
9
m.
7. COMPARW)N OF THE NON-UNXFORMDENSITY DISTRIRUTIONWITH THE %OCAL -RLAIN DENSITY DISTRIRUTION, DEFINED BY A TEMPERATURE T = T*(u) AND A DENSITY N = N&x). -b8 RATIO OF THE% TWO DENSITIES J.SRRPRSSENTED FOR THREE TYPICAL SITUATIONS: o! = o”, M = 90” AND a = 180”
The ratio of the non-uniform distribution density to the ‘numerical approximation’ density remains everywhere between 0.95 and 1.05 above 2 ER for the three exobase models I, II and III. The fit is equally good for any level of solar activity because the constants in equation (22) defining the ‘numerical approximation’ are probably determined by the geometry of the problem, and/or by the exobase distribution condition. The effect of satellite particles was neglected in all the preceding calculations. The concept of a satellite critical level R,, was introduced by Cham~rlain (1963). Below that level, satellite particles fill in the Maxwell distribution between the angles 8, and w - 8,. Above that level, no satellite particles pass through perigee. However, in the case of a non-uniform exobase, the velocity distribution in the exosphere is no longer isotropic, and it would be arbitrary to ‘complete’ it in a different way for each point. More likely, some averaging effect must exist in the density distribution of satellite particles. Their existence, except for an additive constant, would not modify substantially the evidence for the existence of a constant density level, nor the approximation with a local CM model and with the numerical formula (21-22). The exobase was assumed to be at a constant level of 500 km, though we know that its altitude is a function of temperature. Kockarts (1967) and McAfee (1967) calculated that its altitude variation for extreme temperature conditions was of the order of 300 km. In order to evaluate, in our calculations, the effect of the choice of the exobase altitude on the exospheric density dist~bution, we calculated for model I the densities, assuming an exobase altitude of 700 km instead of 500 km.
A. VIDAL-MADJAR
1160
and J. L. BERTAUX
I
Model
I 1
I
2
r
I
3
4
5
in Earth I
6
7
radii
8
9
(a)
r in Earth 1
2
3
4
5
6
mdii 7
8
9
(b)
I*
-5
Model
III
~~=~
!% i
r in Earth 1
2
3
4
5
6
radii 7
L 8
9
OF THE NON-UNIFORM DENSITY DISTRIBUTION APPROXIh3ATION'DENSITYDISTRIBuTION.
WITH
w FIG. 8. COMPARISON
THE
‘NUMERICAL
A CALCULATED
HYDROGEN
DISTRIBUTION
IN THE EXOSPHERE
1161
Model 1
5
r in Earth Radii
z 0 FIG.
9.
INFLUENCE 0~
1
2
3
4
5
THE EXOBASE ALLUDE
6
7”
ON THE
8
9
DENSITIES 0~ THE NON-~NIF~RM
DISTRIBUTION.
The ratio of densities calculated in both cases (Fig. 9), found between 0.98 and 1.02 above 2 ER, demonstrates that the choice of the exobase altitude is not very critical, and this is true in the case of a uniform exobase as well as in the case of a non-uniform exobase. The choice of a constant exobase level is justified a posteriori. CONCLUSION
The Earth’s exosphere may be divided into three regions, in each of which the density of atomic hydrogen, calculated with the use of Liouville’s theorem, may be approximated in a different way : -under 1.5 earth radii, the particles coming directly from below are dominant. The uniform exospheric distribution (Chamberlain model), defined by the local density and temperature at the exobase, is in this case a very good approximation of the density (within f5 per cent). -between 1.5 earth radii and 2 earth radii, a level of uniform density appears all around the Earth. The vertical variation of that uniform density may be approximated by the mean Chamberlain model, defined by the density and temperature of the point located on the exobase at a = 90” (within &lo per cent). -above 2 earth radii, a numerical approximation was developed, in fact under the form of a Chamberlain model with a fixed temperature (7’,(90’)) and a variable density (a simple multiphcative factor) defined by the following formula: N=0_95x
- Nc(4 (No+ tri\, 3.5
1‘
1162
A. VIDAL-MADJAR
Such a model approximates f5 per cent).
and J. L. BERTAUX
quite well the non-uniform distribution calculated (within
Acknowledgements-We wish to thank J. Lemaire for useful discussions. This work was completed the computer facilities of the ‘Center de Calcul de I’INAG’, Meudon, France.
with
REFERENCES BATES,D. R. and PATTERSON,T. N. L. (1961). Hydrogen atoms and ions in the thermosphere and exosphere. Planet. Space Sci. 5,257. CHAMBERLAIN, J. W. (1963) Planetary coronae and atmospheric evaporation. Planet. Space Sci. 11, 901. KOCKARTS,G. (1967). Incursion aeronomique jusqu’8 50000 km, in Ciel et Terre, 9-10. KOCKARTS,G. and NICOLET, M. (1963) Le probleme aeronomique de l’h6lium et de I’hydroghne neutre. AnnIs. Geophys. 18,269. MCAFEE, J. R. (1967). Lateral flow in the exosphere. Planet. Space Sci. 15,599. MEIER, R. R. (1969). Balmer alpha and Lyman beta in the hydrogen geocorona. J.geophys. Res. 74,356l. METZOER,P. H. and CLARK, M. A. (1970). On the diurnal variation of the exospheric neutral hydrogen temperature. J. geophys. Res. 75,5587. NICOLBT,M. (1964). The structure of the upper atmosphere, in Research in Geophysics Vol. 1, p. 243. ~)PIK, E. J. and SINOER, S. F. (1959, 1960, 1961). Distribution of density in a planetary exosphere. Phys. Fluids 2, 653; 3, 486; 4, 221. Qumm~~, J. A. (1972). Atomic hydrogen densities at the exobase. Submitted to J. geophys. Res. TINSLZY, B. A. (1970). Variations of Balmer-emission and related hydrogen distributions, in Space Research X. North-Holland, Amsterdam.
A CALCULATED
Laboratoire
HYDROGEN DI§TRI~UTION EXOSPHERE
IN THE
A. VIDAL-MADJAR de Physique Stellaire et PlantUre, Centre National de la Recherche Scientifique, Verrieres-le-Buisson, France and
Service d’A&onomie,
J. L. BERTAUX Centre National de la Recherche Scientiiique, Verri&res-leB&son, France (Received injinal form 15 January 1972)
Abstract-The density distribution in a planetary exosphere is a function of the boundary con~tions, i.e. the density and the temperature ~s~ibutio~ at the exobase. We established, with the use of Liouville’s theorem, that the exospheric density may be calculated with a triple integral. The results of a numerical evaluation of this integral are presented for three different exobase models for the case of atomic hydrogen in the Earth’s exosphere. They all satisfy the zero net ballistic flux condition, but correspond to different levels of solar activity. A constant density level was found at 1.8 earth radii, below which the computed density distribution may be approximated by a local spherical Chamberlain model. Above the constant density level, a simple relation was found to tit the computed density dis~ibution within f5 per cent. INTRODIJCTION
The exosphere is the layer of a planetary atmosphere where the number of atomic collisions per unit time becomes negligible. In order to calculate exospheric density distribution it is generally assumed that the constituents have an isotropic Maxwell-Boltzmann distribution in the diffusosphe~ up to a sharp boundary, called the critical level or the exobase (Chamberlain, 1963; Nicolet, 1964). Above this level, in the collisionless exosphere, particles with an upward velocity will describe ballistic trajectories. According to their velocities at the exobase, they will either describe hyperbolic trajectories and escape from the atmosphere, or elliptic trajectories and return to the diffusosphere at another point on the exobase. opik and Singer (1959, 1960, 1961) gave analytical expressions of exospheric densities of both types of particles, elliptic and hyperbolic. Chamberlain added satellite particles, created by a small number of collisions in the exosphere. He introduced the concept of a satellite critical level R,,, above which no satellite particles pass through perigee, and below which their velocity distribution fills in the velocity distribution of ballistic particles. At a particular point in the exosphere, densities of each type of particle are derived by integration of the distribution fun&ion over the momentum part of the phase space. The integration can be analytically performed only in the case of a uniform exobase (temperature and density constant at the exobase). This case, which was considered by the previous authors, leads to an exospheric density distribution with a spherical symmetry. However, the interpretation of ground based, rocket and satellite experimental results often requires some degree of asy~et~ in the exospheric density dist~bution of hydrogen (Meier, 1969; Metzger and Clark, 1970; Tinsley, 1970). A computer program was constructed to perform the numerical integration necessary to, obtain the exospheric density distribution corresponding to any given temperature and. density distributions at the exobase. Special attention was paid to the case of distributions. with axial symmetry and leading to a zero net ballistic flux (McAfee, 1967; Quessette, 1147
1148
A. VIDAL-MADJAR
and J. L. BERTAUX
1972). The corresponding calculations are presented in this paper, along with a numerical approximation of the density distribution above two earth radii (2 ER). THE EXOBASE MODEL An exobase model is characterized by a temperature and density distribution given at the altitude of the exobase. Calculations were completed for atomic hydrogen in the terrestrial atmosphere. The exobase level was assumed to be at a constant altitude of 500 km; the importance of this assumption is discussed in a later section. The temperature distribution at the exobase was a sinusoidal approximation of CIRA 65 model. Three different levels of solar activity were considered. Even though the CIRA 65 model has no axial symmetry, a small displacement can align on the same axis the maximum temperature point, the center of the Earth and the minimum temperature point. Then, the temperature distribution at the exobase has a symmetry about this axis, and can be approximated by:
T,(Y)=
Tmax
+ 2
Gin + Lax - ‘Kmin cos y. 2
(1)
T,(y) is the temperature at any point of the exobase where the local vertical is at the angle y with the axis of symmetry. For each temperature distribution, Quessette computed the density distribution at the critical level (Fig. 3) leading to a zero net ballistic flux situation, i.e. the outward ballistic flux equal to the inward ballistic flux at any point of the exobase. The calculations of Quessette are more accurate than the McAfee calculations; the relative difference between their respective density distributions may reach 10 per cent. We can point out, since we are dealing with a time-independent problem, that temperature and density distributions connected by the zero net ballistic flux condition are the only ones which do not require any sink or source in the diffusosphere (except for the escape component). Since the temperature distribution is symmetrical about one axis, the density distribution at the exobase leading to a zero net ballistic flux, and the density distribution above the exobase, will be symmetrical about the same axis. For the purpose of exospheric calculations, it is not necessary to relate the position of this axis to the Sun-Earth line. DISTRIBUTION
FUNCTION
AND DENSITY
IN THE EXOSPHERE
At any point in the exosphere, the distribution function (or phase space density) can be calculated with the use of Liouville’s theorem, which states that the value of the phase space remains constant along a dynamical trajectory: f(&
Pi) = fc(&
PC?.
(2)
In this equation, the q* are the spatial coordinates, and the pf are the momentum coordinates. The function f(qf, pi) is the distribution function and has the same value all along the trajectory, particularly at the critical level, indicated by the subscript c. At the point qi in the exosphere, a unique dynamical trajectory is determined by the choice of coordinates (qf, pi). Its intersection with the exobase defines the point (qci, pcf) in a unique way, since we consider only the point from where the particle is coming. Then we can state that all the qic and thepi, are unique functions of the (qi, pi), and rewrite the Equation (2) in the following way: J-G?, Pi) = fc(4:(4i, Pi), P6(4, P”)).
(3)
A CALCULATED
HYDROGEN
DISTRIBUTION
IN THE EXOSPHERE
1149
FIG. 1. COORDINATESYSTEMS RELATED TO THE CENTER OF THE PLANET AND TO THE PARTICLE LOCATED
IN
A.
The volume density N(q”) at the point qi in the exosphere is obtained by integrating the distribution functionf(q’,p”) over the momentum part of phase space which is populated (only the dynamical trajectories reaching the exobase are populated):
IV($) =
s
f(qi, pi) d3p'
(4)
POP
where @pi represents the element of momentum Equation (4) can be rewritten:
space. With the use of Equation (2),
At the point qci of the exobase, the distribution function f,is known, since we have made the hypothesis that it is a Maxwell distribution, and since the temperature and density distribution at the exobase are given. We are left with the problem of calculating the six functions qci(qi, p$) and pc*(qi, pi), in order to derive the exospheric density at the point A of coordinates qi. For the sake of clarity, we have used, instead of the (q5, pi) coordinates, the usual spherical coordinates (I, tc, 13)of the point A (Fig. 1) in the Oxyz system, where 0 is the center of the planet, Ox is the symmetry axis of the density distribution at the exobase, and Oxy is the equatorial plane. A second coordinate system AXYZ is related to A, where AZ is the local vertical oriented upwards, and AX perpendicular to AZ, is in the plane OA-Oz oriented in such a way that the angle Oz-Ax is equal to 6. In this coordinate system, the velocity vector of a particle at point A is determined by (a, 0, $)>,where u is the length of the velocity vector, 0 is the angle between AZ and the velocity vector, and I$ is the angle between AX and the projection AU of the velocity vector on the AXY plane, which determines with AZ the orbital plane of the particle.
1150
A. ~DA~MADJAR
and J. L. BERTAUX
Using the normalized variables :
(6) v=L
(7)
V esc
where r, is the radius of the exobase and veSc= 1/2MG/r, is the escape velocity at the exobase level, the equations of motion of a particle are: v,e - f=V-y V sin 8
-
Y
(8)
(9)
= V, sin BC
in which the subscript c stands for variables defined at the critical level, MTis the mass of the planet and G the gravitational constant. The trajectory of a particle passing through A is then completely determined by the set of variables (y, a, 6, V, 0, gd). In order to determine the coordinates a, and 6, at the exobase of the origin point A, of the trajectory, we have calculated in the orbital plane the angle w, defined as the difference between true anomalies of the origin point A, and the point A, respectively PCand p (Fig. 2). The equation of the trajectory, in polar coordinates, has the following form: w
r=
(IO)
1 + e cos p
w and e are constant along the trajectory and can be expressed as functions of (y, V, 0) by applying the conservation of energy and momentum between the points (y, V, 1!7),apogee and perigee. We find: V2 sin’ 0 w = 2r 0 = O(Y, V, 0) Y2 e=
I _ 4 V2 sin2 8 \i
(Y -
(11),
vz) = e(y, v, 0
Y2
The values of #? and /?, can then be extracted from the combination (12): cos p =
cos /!I@ =
Y~Y,
v, 6) -
of (lo), (11) and
T,
e(y, v,0) W(Y,v, f3 - r, e(y, V, 69
and from these equations it is clear that y which is given by: Y=B-A can be expressed as a function ytly, Y, 0). The spherical coordinates Usand 4, of the origin point in the Oxyz system are obtained through the use of the transformation matrix from
A CALCULATED
FIG. 2. GEOMETRY
HYDROGEN
DISTRIBVFION
FOR THE PARTICLE TRAJECTORY
IN TI-IE EXOSPHEREi
1151
IN ITS ORBITAL PLANE.
the AXYZ system to the 0xy.z system, and may be expressed as functions of #, pp, a and 6: 6,(a, 6, +, y) = Arc sin (cos 6 sin y cos 4 + sin 6 cos y) a,& a,$, lu) = Arc cos
-sinBcosasinycos~+sinasinysin~+cos6cosacos~ cos &(a, 6, +, V)
03)
1*
04)
As S,U is a function of (y, V, O), the two angIes a, and 6, are finaily functions of the six variables (y, a, 6, V, 8, #), as expected. The velocity vector of the particle at the origin point is defined by V, and OC,which are functions of (y, V, 0) through the Equations (8) and (9). Given an exobase model and a set of variables (j, a, S, V; 8, #), the knowledge of a0 and 6, allows one to calculate the values of temperature T,(Gc,,6,) and density N,(a,, 6,) at the origin point, as well as the Maxwell distribution function at this point:
Because the distribution functionf, has a Maxwell form, f,depends only on V,, a, and 8,. With the coordinates that we have used, the element of momentum space in Equation (4) is: d3p, = m3u&,V2sin 0 dV d0 d4.
1152
A. VIDAGMADJAR
and J. L. BERTAUX
The volume density N(y, cc, 6) at the point A is derived from the integral of Equation (4), in which fC has the form given by Equation (15) when Y, is expressed as a function of V and y (Equation 8):
x exp
--x
mMG
V2sin8dVd0d$. To The integration over V, 6’ and r$ must be restricted to the existing trajectories. types of particles are generally considered:
(16) Three
-ballistic particles defined by V, < I or V < 2/;; and coming from the exobase -hyperbolic particles defined by V, > 1 or V > dy -satellite particles created by a small number of collisions in the exosphere, and ionized by the EW flux (we have not considered satellite particles in this study for reasons discussed in a later section). (a) Ballistic particles There is a velocity V, at each distance r such that if V < V,, the corresponding trajectory will intersect the exobase for any direction of the velocity vector. This velocity V, corresponds to a trajectory whose apogee is r and whose perigee is r,: sin 8 = sin 6, = 1 and from (8) and (9) is obtained (17) On the other hand, if Y > V,, all the trajectories do not reach the exobasc. The populated trajectories are within a cone of half-angle @,, whose axis is the local vertical. The perigee of the corresponding trajectory lies at the exobase level; from (8) and (9) is derived: sin 8, = ;jv2+1-y.
(18)
The ascending particles populate the region 0 < 13< 8, whereas the descending particles populate the region r - 8, < 8 < rr. The volume density of ballistic particles, iV,(y, A, S) is the sum of the integrals:
(19)
A CALCULATED
HYDROGEN
DISTR~~ION
IN THE EXOSPmRE
1153
(b) Hyperbolic partides The condition (18) is still valid for hyperbolic particles. The populated trajectories are again within a cone of half-angle B,,,; however, only the particles which escape the planet need to be considered, and are those that populate the region 0 < 0 < f3,,,. The volume density &(y, cc, 6) of hyperbolic particles is: WY,
% 4 = ~34*,
IdiVz
dVlmsin
8 dOrfO
d$.
(20)
The total volume density N(y, a, S) in the exosphere is the sum of N,(y, M, S) and N,(y, a, 6). In the case of the Chamberlain model with a uniform temperature and density distribution at the exobase, integrals (19) and (20) are reduced to simple integrals which can be expressed analytically using the I’ function. In the present case, a triple integration was numerically performed with a 10 points Gauss method for each integral. The whole calculation was checked and its precision estimated by comparison with the case of a uniform exobase, for which integrations over 4 and 0 were done analytically, and the integration over V was done with a Gauss method having a much larger number of points (32). The precision was found to be of the order of 1O-3 or better, and was considered satisfactory. RESULTS
Three different exospheric density distributions of atomic hydrogen were calculated, corresponding to three different levels of solar activity. The maximum and minimum temperature, Tmax and Tmin at the exobase, taken from CIRA 65, are indicated in Table I, TABLE 1. EXOBASE MODELS
Model
Solar activity
:r III
Low Mean High
(gy
pg;
bmx
&f$
688 979 1413
1005 1460 2083
2.35 1.92 1.53
4000 5200 6500
along with the ratio pmax, of maximum to minimum hydrogen densities computed by Quessette (1972) with the zero net ballistic flux condition (Fig. 3). In the CIRA model, the point where tem~rature is maximum is at 1400 local time in the equatorial plane (for equinox conditions). Since this point is on the Ox axis of our coordinate system, which is an axial symmetry axis for the present exospheric distribution, calculations were made only in the equatorial plane and are described as a function of distance r and angle CL Because of the symmetry axis, it is important to note that the two functions describing the exobase level may be expressed as functions of only one angle y as explained in Formula (1). These functions have then the following form: N,(% S,) = N,(y)
and
T,(ccc, 8,) = T,(y)
where
y = Arc cos (cos 01,cos 6,).
In the equatorial plane, where 6 = 0, we have then y = CL We must point out that all hydrogen densities are normalized to 1 at the point of minimum density. This point of minimum density coincides with the point of maximum temperature, a phenomenon radically different from the case of major atmospheric constituents. This characteristic of hydrogen dis~ibution at the exobase has already been found from escape flux considerations (Bates and Patterson, 1961; Kockarts and 3
A. VIDAGMADJAR
1154
0
30
60
and J. L. BERTAUX
90
120
150
180
o(
FIG. 3. RELATIVEDENSITY
DISTRIBUTIONS OF ATOMIC HYDROGEN AT THE EXOBASE YIELDING ZBRO NET BALLISTIC FLUX, COMPUTED BY Qu~ss~ne (1972).
Nicolet, 1963). The introduction of the zero net ballistic flux condition does not change this characteristic of hydrogen distribution. However, the ratio of maximum to minimum density is somewhat smaller in the later case. (a) Density variation with altitude: For each exobase model I, II and III, the variation of density with altitude is shown in Fig. 4 for two typical values of a: a = 0” direction of the point of maximum temperature a =
180” direction of the point of minimum temperature.
For each model, the two curves intersect at an altitude 2, (-1.8 earth radii) indicated in Table I, showing that a uniform density level must exist at this altitude. Above Z,, the two curves are again separated and show that the higher density is to be found on the hot side of the Earth, as in the case of the major atmospheric constituents. (b) Density variation at a constant altitude: Density at a constant altitude as a function of a, is compared with the density for a = 0”. These relative variations, shown in Fig. 5, display more clearly the previous results, i.e. the existence of a nearly constant density level, and the inversion above this level of the density variation. However, if the density at the exobase level changes by a factor of 2, above N 2 ER, this variation remains much smaller (less than 25 per cent for any of the three models). Then it can be said, in a first approximation, that the density above 2 ER is only a function of altitude, as in the case of a spherical Chamberlain model (1963), which is characterized by a uniform exobase and defined by two parameters, the temperature T and
A CALCULATED
HYDROGEN
DIS~IB~ION
1155
IN THE EXOSPHERE
Altitude IO” (4
105
(Km)
and J. L. BERTAUX
A. VIRAL-MA~JAR
Model
30
0
I
60
120
90
150
180
ci
(a)
Model
Altitude
II
(km)
I
0
30
60
120 (4
90
150
a
Altitude
Model III
i
180
(km)
500 1000
8
I
Altitude in E.R.
04 0
FIG. 5. RELATI~ U(cLISTHE
ANGLE
30
60
90
120 (4
150
180
cf.
DENSITY VARIATIONS AT CONSTANT ALTITUDES AS FUNCTIONS OF THE ANOLE BETWEEN
THE
LOCAL
VERTICAL
MAXIMUM
AND THE
TEMPERATURE).
VERTICAL
CONTAINING
THE POINT
OF
A CALCULATED
HYDROGEN
DISTRIBUTION
IN THE EXOSPHERE
the density N at the exobase (when satellite particles are not considered). CM(T, N) such a model, in which N is in fact a multiplicative factor.
1157
Let us call
We have tried to find several approximations of the calculated non-uniform exospheric distribution using combinations of Chamberlain models, and we have estimated the degree of approximation by the comparison of both resulting distributions. (c) Comparison with Chamberlain distributions: For each level of solar activity the model CM@‘, N) taken for comparison was such that T and N were equal to the values of temperature and density in the non-uniform model at the exobase and for u = 90”, i.e. T = T,(90*)
N = N,(90”).
The ratio of densities in the non-uniform model and in the model CM(Z’, N) is shown in Fig. 6. Of course, the approximation is very poor below 2 ER, but is within &15 per cent above that level. For the case of a = 90° the situation is better, since the approximation is within f3 per cent, indicating that the CM model gives the best approximation precisely on the vertical, at which origin T and N were taken. This case may be generalized. For each point A in the exosphere defined by the (r, a) coordinates, there exists a CM distribution defined by N = N,(a) and T = T,(a), density and temperature of the exobase point located below the point A on the same vertical. With the use of an infinite number of CM models (one for each value of a), one can construct a distribution called the local CM model. Under these conditions, the density distribution at the exobase is the same for the non-uniform model and the local CM model. The ratio of densities in both models is shown in Fig. 7 as a function of altitude for three different values of a (O’, 90*, 180’). The ratio is 1 at the exobase, remains between 0.95 and 145 up to cu 1.5 ER, and then increases and decreases more or less quickly above that level. It could have been expected that the local CM model would give a good approximation just above the exobase up to 1.5 ER; within that range, the dominant particles are those coming from just below. The particles coming from other locations on the exobase are weighted less in the density calculation. This last result is quite impo~ant because it shows that in a first approximation the exosphere observed from the exobase level is the local exosphere. DISCUSSION
Above 2 ER, we saw that the CM distribution corresponding to cc = 90” was a valid approximation to our non-uniform density distribution within +15 per cent. We tried to find a ~st~bution called ‘numerical approximation’ which would give a better fit above 2 ER. As the density N, at the exobase is only a multiplicative factor in the CM distribution, we built this numerical approximation by using CM distributions with a variable N,(M) and a constant exospheric temperature T,, independent of a. In Fig. 8 are shown the results obtained with this numerical approximation defined by
where fl, and Fc are the mean values of the functions N,(r) and T,(y) between 0 and 7~.
A. VIDAGMADJAR
and J. L. BERTAUX
Model f
Model III
(4 COMPARISON OF THE NON-UNIFORM DENSITY DISTRIBUTION WITH THE ‘MEAN” Cmw BBRLAIN DENSI’IY DI§TRIBIJTIGN (DEFINED BY THE TEMpERATWRE AND DENSITY AT a = wt”). %I3 RATIO OF THESE TWO DENSITIES IS PLOTTER AS A FUNCTION OF GWJCENIWC DISTANCE r POR
FIG. 6.
THREE TYpICAL SITIJATIONS: a = O”, (L = 90” AND a =
180”.
A CALCULATED
HYDROGEN
Model I
DIST~~ON
Model II
_I_____
Model III
--.
12
3
I 0
4
5
IN THE EXOSPHERE
1159
in Earth Radii 6
7
8
9
m.
7. COMPARW)N OF THE NON-UNXFORMDENSITY DISTRIRUTIONWITH THE %OCAL -RLAIN DENSITY DISTRIRUTION, DEFINED BY A TEMPERATURE T = T*(u) AND A DENSITY N = N&x). -b8 RATIO OF THE% TWO DENSITIES J.SRRPRSSENTED FOR THREE TYPICAL SITUATIONS: o! = o”, M = 90” AND a = 180”
The ratio of the non-uniform distribution density to the ‘numerical approximation’ density remains everywhere between 0.95 and 1.05 above 2 ER for the three exobase models I, II and III. The fit is equally good for any level of solar activity because the constants in equation (22) defining the ‘numerical approximation’ are probably determined by the geometry of the problem, and/or by the exobase distribution condition. The effect of satellite particles was neglected in all the preceding calculations. The concept of a satellite critical level R,, was introduced by Cham~rlain (1963). Below that level, satellite particles fill in the Maxwell distribution between the angles 8, and w - 8,. Above that level, no satellite particles pass through perigee. However, in the case of a non-uniform exobase, the velocity distribution in the exosphere is no longer isotropic, and it would be arbitrary to ‘complete’ it in a different way for each point. More likely, some averaging effect must exist in the density distribution of satellite particles. Their existence, except for an additive constant, would not modify substantially the evidence for the existence of a constant density level, nor the approximation with a local CM model and with the numerical formula (21-22). The exobase was assumed to be at a constant level of 500 km, though we know that its altitude is a function of temperature. Kockarts (1967) and McAfee (1967) calculated that its altitude variation for extreme temperature conditions was of the order of 300 km. In order to evaluate, in our calculations, the effect of the choice of the exobase altitude on the exospheric density dist~bution, we calculated for model I the densities, assuming an exobase altitude of 700 km instead of 500 km.
A. VIDAL-MADJAR
1160
and J. L. BERTAUX
I
Model
I 1
I
2
r
I
3
4
5
in Earth I
6
7
radii
8
9
(a)
r in Earth 1
2
3
4
5
6
mdii 7
8
9
(b)
I*
-5
Model
III
~~=~
!% i
r in Earth 1
2
3
4
5
6
radii 7
L 8
9
OF THE NON-UNIFORM DENSITY DISTRIBUTION APPROXIh3ATION'DENSITYDISTRIBuTION.
WITH
w FIG. 8. COMPARISON
THE
‘NUMERICAL
A CALCULATED
HYDROGEN
DISTRIBUTION
IN THE EXOSPHERE
1161
Model 1
5
r in Earth Radii
z 0 FIG.
9.
INFLUENCE 0~
1
2
3
4
5
THE EXOBASE ALLUDE
6
7”
ON THE
8
9
DENSITIES 0~ THE NON-~NIF~RM
DISTRIBUTION.
The ratio of densities calculated in both cases (Fig. 9), found between 0.98 and 1.02 above 2 ER, demonstrates that the choice of the exobase altitude is not very critical, and this is true in the case of a uniform exobase as well as in the case of a non-uniform exobase. The choice of a constant exobase level is justified a posteriori. CONCLUSION
The Earth’s exosphere may be divided into three regions, in each of which the density of atomic hydrogen, calculated with the use of Liouville’s theorem, may be approximated in a different way : -under 1.5 earth radii, the particles coming directly from below are dominant. The uniform exospheric distribution (Chamberlain model), defined by the local density and temperature at the exobase, is in this case a very good approximation of the density (within f5 per cent). -between 1.5 earth radii and 2 earth radii, a level of uniform density appears all around the Earth. The vertical variation of that uniform density may be approximated by the mean Chamberlain model, defined by the density and temperature of the point located on the exobase at a = 90” (within &lo per cent). -above 2 earth radii, a numerical approximation was developed, in fact under the form of a Chamberlain model with a fixed temperature (7’,(90’)) and a variable density (a simple multiphcative factor) defined by the following formula: N=0_95x
- Nc(4 (No+ tri\, 3.5
1‘
1162
A. VIDAL-MADJAR
Such a model approximates f5 per cent).
and J. L. BERTAUX
quite well the non-uniform distribution calculated (within
Acknowledgements-We wish to thank J. Lemaire for useful discussions. This work was completed the computer facilities of the ‘Center de Calcul de I’INAG’, Meudon, France.
with
REFERENCES BATES,D. R. and PATTERSON,T. N. L. (1961). Hydrogen atoms and ions in the thermosphere and exosphere. Planet. Space Sci. 5,257. CHAMBERLAIN, J. W. (1963) Planetary coronae and atmospheric evaporation. Planet. Space Sci. 11, 901. KOCKARTS,G. (1967). Incursion aeronomique jusqu’8 50000 km, in Ciel et Terre, 9-10. KOCKARTS,G. and NICOLET, M. (1963) Le probleme aeronomique de l’h6lium et de I’hydroghne neutre. AnnIs. Geophys. 18,269. MCAFEE, J. R. (1967). Lateral flow in the exosphere. Planet. Space Sci. 15,599. MEIER, R. R. (1969). Balmer alpha and Lyman beta in the hydrogen geocorona. J.geophys. Res. 74,356l. METZOER,P. H. and CLARK, M. A. (1970). On the diurnal variation of the exospheric neutral hydrogen temperature. J. geophys. Res. 75,5587. NICOLBT,M. (1964). The structure of the upper atmosphere, in Research in Geophysics Vol. 1, p. 243. ~)PIK, E. J. and SINOER, S. F. (1959, 1960, 1961). Distribution of density in a planetary exosphere. Phys. Fluids 2, 653; 3, 486; 4, 221. Qumm~~, J. A. (1972). Atomic hydrogen densities at the exobase. Submitted to J. geophys. Res. TINSLZY, B. A. (1970). Variations of Balmer-emission and related hydrogen distributions, in Space Research X. North-Holland, Amsterdam.