Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 2

THE UPPER ATMOSPHERE

AND

SATELLITE DRAG

by R. J. STTRTON

SMITHSONIAN INSTITUTION

Washington, D.C.

1960
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The Upper Atmosphere and Satellite Drag

By R. J. STIRTON *

The atmospheric drag force on artificial satellites is of interest both for orbit computations

and for computing the density of the upper

atmosphere from the observations of satellite

motion. Density determination by this means

has been suggested by Davis, Whipple, and

Zirker (1956), Spitzer (1956), and Jones and

Bartman (1956). Harris and Jastrow (1958)

and Sterne, Folkart, and Schilling (1958) have

given values of the density obtained from orbit

data of Satellites 1957 ol, 1957 o2, and 1957 0.

These determinations depend on a knowledge

of the drag coefficient GD, which is defined by

the usual equation for the drag force F,

where p is the atmospheric density, v0 is the

satellite velocity, and A is a characteristic area.

For spherical satellites the value CD—2 (based

on cross-section area) has been used (Davis

et al., 1956; Spitzer, 1956). For other shapes

Sterne and Schilling (1958) suggest that CD=2

is a good approximation if one fourth of the

total area of the body is used for A. Harris

and Jastrow (1958) used CD=2.3 in their determination of air density.

These values of CD are based on the following

assumptions:

(a) The mean free path of the molecules is much

greater than the satellite dimensions ("freemolecule flow").

(b) The satellite velocity »& is much greater than

the mean random velocity C of the gas molecules.

(c) The mechanisms that reflect the impinging

molecules range from specular reflection to

inelastic impact and diffuse evaporation at an

effective surface temperature 7*.

In general, at a given density, the drag coefficient for free-molecule flow depends on C

(or the absolute kinetic temperature T) and the

molecular weight M of the gas.

The drag coefficients of a sphere for specular

reflection C&\ inelastic impact C&, and diffuse evaporation C^ are derived in pages 12-

13 and are graphed in figure 1 as functions of

the parameter a:

2kT>

where m is the molecular mass and k is the Boltzmann constant. It is noteworthy that for a

sphere CJP=CJP, a condition that is unique

to this shape. The derivation is based on the

Maxwell-Boltzmann distribution of velocities of

the gas molecules, but it does not impose condition (b) above. The curve of 0$ in figure 1

is for complete temperature accommodation,

and therefore represents the upper bound of the

drag coefficient for diffuse evaporation (see eq.

18, p. 13).

As a specific example, consider a spherical

satellite with velocity P O =7.5X1O* cm/sec.

The drag coefficients are shown in figure 2 as

functions of the temperature and for various

values of the molecular weight. Even for very

high gas temperatures, and with the low densities in the upper atmosphere, the surface of the

kV \

\

\

ySCALE©

SOMJE®-*

• _

1

 Test Department, U.S. Naval Ordnance Test Station, China Lake,

Callt

OJOS 0.10 0.13 03O O.J

VELOCITY RATIO MMMCTER »'%M

FIGURE 1.—The drag coefficients for a sphere.

9

3.0 ©

0.30 ®
10 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TEMPEMTUDE "K

FIGURE 2.—The drag coefficients as functions of temperature,

for a sphere with velocity Vt=7.5X105 cm/sec.

satellite certainly does not completely accommodate, and the drag coefficient for evaporation

is shown for T'=100° K to 1,000° K only.

Clearly the drag coefficient Ci>1)=Cfij) does

not depart seriously from the limiting value of 2

unless the molecular weight is low and the

temperature is high. One is fully justified in

using the limiting values of CD for the density

determinations cited above. However, it is of

interest to consider possible situations in which

the drag coefficient Ci>° could depart from the

limiting value of CD=2 for the sphere.

Considerable theoretical and experimental

work has been done on the properties of the

10*

NICOLET / .

(1958) < /

XNICOLET-S

I ATMOSPHERE

f (1954)

CHAPMAN

(1957)

^ " ^ CHAPMAN

(IN PRESS)

. 0 ' K>4

ALTITUDE, KM

. 0 *

FIGURE 3.—The temperature as a function of altitude; extrapolated from the work of Nicolet and of Chapman.

» See Bates, 1949,1981,1954; Chapman, 1951,1957; Miller, 1957; Nicolet,

1964,1958: Rocket Panel, 1952: Spitasr, 1949: and Whlpple, 1954. References to other work may be found In these papers.

upper atmosphere.2 Four recent papers are

particularly pertinent to the present discussion.

Miller (1957) has given the temperature and

molecular weight of the atmosphere to 600 km.

Nicolet (1958) has indicated that the densities

derived from observations of satellites (Harris

and Jastrow, 1958; Sterne and Schilling, 1958;

Sterne, Folkart, and Schilling, 1958) require

both higher temperatures and higher densities

for the upper atmosphere than was previously

expected. See for example Nicolet's (1954)

atmosphere. Chapman (1957) has treated the

effect of the solar corona on the upper atmosphere. From these papers it is interesting to

attempt to construct a crude atmosphere

extending above 500 or 600 km. Temperature

and molecular weight are of particular interest

since they can have important effects on the

drag coefficient.

Chapman (1957) suggested that the temperature of the coronal gas in the region of the earth's

orbit may be on the order of 220,000° K. He

later revised the estimate to about 100,000° K

(Chapman, in press), and also suggested that the

earth's atmosphere would merge with the coronal gas at a distance of about 50 earth radii.

Using these data let us extrapolate Nicolet's

(1954) atmosphere to 3X108 km (50 earth

radii), as shown in figure 3. Nicolet (1958)

has recently indicated that the temperature

may be on the order of 3,000° K at 500 km.

An extrapolation from this temperature point

is also shown.

Miller (1957) has given the molecular weight

of the atmosphere to 600 km. We might suggest, as a first guess, that the molecular weight

would continue to fall until the nitrogen and

oxygen were completely dissociated and ionized.

Also the very high regions of the upper atmosphere might be expected to be composed

largely of hydrogen in an ionized state. At

about 30,000° K it should be nearly fully

ionized and therefore the molecular weight

should have a limiting value of M=^ when

the temperature reaches 30,000° K. From

these ideas and the temperature curve of

figure 3, Miller's molecular weight is extrapolated as shown in figure 4.

Now let us consider a family of circular satellite orbits with various radii, and estimate the
"UPPER ATMOSPHERE AND SATELLITE DRAG 11

VMILLER'S

ATMOSPHERE—-\ N

\

s

N

FIGURE 4.—The molecular weight as a function of altitude;

extrapolated from the work of Miller and of Chapman.

drag coefficient as a function of distance above

the surface. If we use the estimated molecular

weight and temperature curves of figures 3

and 4, and an r~* dependence of the satellite

velocity on the radius of the orbit, we can

compute CD\ shown in figure 5 as a function

of the altitude. Under these circumstances

C8} departs from the limiting value of 2

around 103 km.

If, as has been proposed by Jones and Bartman (1956) and the Technical Panel (1958),

large balloon-like satellites are used to measure

atmospheric density at very high altitudes,

precise results require that some consideration

be given to the departure of the drag coefficient

from the limiting value corresponding to

vo«C.

Even with the value of CD based on the discussion given here, a number of effects should

be considered before the density is inferred from

the observed motions of satellites in the very

high atmosphere. The gas is probably not in

.... .

1 ... .

/

/

. . r . . . .

FIGURE 5.—The drag coefficient CD

(i> or CDU) for a spherical

satellite in circular orbits of various radii.

equilibrium in the thermodynamic sense. Solar

activity will affect the amount and depth of

penetration of coronal gas in the earth's atmosphere. The free path is comparable to the

size of the earth, and the particles of the gas

can be in orbital motion around the earth rather

than in the random motion postulated for the

Maxwell-Boltzmann distribution. Ionization

complicates this problem further, and the

region of the so-called ring current (see

Technical Panel, 1958) will certainly not have

random velocity distribution.

The satellite will carry some electrostatic

charge, which may modify the drag. Perhaps

a satellite with an electrometer would yield

some interesting results in this connection.

Eddy currents induced in the conducting parts

of the satellite will result in an energy loss,

which will be a draglike effect.

The molecular-flow drag on a sphere

The problem of the resistance to the motion of

a sphere3 moving through a fluid has been

treated extensively in the literature. Newton

considered the fluid as being composed of mass

points at rest. G. C. Stokes later treated the

problem and obtained the well-known Stokes

law. Others, notably Millikan (1923) and

Epstein (1924), refined the theory to apply to

sphere velocities much smaller than the mean

thermal velocity of the gas molecules. More

general treatments have been carried out by

others. For example, Heineman (1948) considers the sphere and a number of other shapes

for gas densities so low that the mean free path

is much greater than the body size. No

restriction was imposed on the sphere velocity.

In the present paper, the drag on a sphere

is calculated for free-molecule flow. The

method follows the general approach of Epstein

(1924) but removes the restriction of low

velocity of the sphere. In particular, the

drag is found for molecular impact, specular

reflection, and diffuse reflection (or impact

and evaporation) without complete temperature accommodation. These cases seem pertinent to the artificial earth satellite and

other space vehicles. The intermediate results,

* In what follows, vector quantities are Indicated by boldface type: the

magnitudes of vectors and other scalar quantities are Indicated by Italic

type.
12 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

equations (4) and (8), are equivalent to those

obtained by Heineman (1948).

The gas is assumed to be in thermal equilibrium, which the sphere's motion does not

disturb. The Maxwell-Boltzmann distribution

of molecular velocities then may be used.

Consider the sphere at rest and that the gas

flows over it with velocity Vo- Then, the

distribution of velocities relative to the sphere

is that given in equation (1):

], (1)

where N is the number density; X, n, and v are

the direction cosines of v<>; vx, vy, and vt are the

components of the molecular velocity v; and

also where &i=m/2kT, m is the molecular

mass, k is the Boltzmann constant, and T is

the absolute temperature.

The number of molecules with velocity v

striking a surface element ds per second is

••ds, and the momentum transferred per

second to ds is then mvr-ds. Further, by

symmetry arguments, the components of

momentum balance except along v0, so that

the desired momentum component is m(Xt>I +

[iVy-\-Wz)vvds if the surface element ds is

along y. The momentum transferred per

second to ds is given by the equation

Ptds=m

«/—a> */ — a> «/ —

Vids. (2)

By introducing the variables, Z=vx-\-\v0, v=Vp

-\-fiv0, and £=vt-\-Wo, equation (2) may be reduced to

(3)

where a=po/3.

We now integrate equation (3) over the entire

surface of the sphere of radius r. The element

of area is ds=r2 sin 9 do d<t>, where 0 is the

colatitude and <f> is the longitude of v0. It may

be seen that /x=cos 0, and P becomes

cos 0sin !a2

) -f *•*+

fa COB 8 . "\ . , . "I

2 e2dx y +2ae~a cos 5sinO dO.

J

°

 f (4)

By some integration by parts the total drag

force due to molecular impact becomes

(5)

r4a 4+4a 2-l f° _x,

L * Joe c

l+2a2 ,_,

The momentum transfer for a molecule striking ds and reflecting specularly is 2mvv along

ds. Again by symmetry arguments, the component along Vo, 2mvvp, is all that is required.

The momentum transferred to ds, by use of

distribution (1), is

r<*> no r<°

P4s=2m\ I I

(6)

By changing to the variables £, i\, and f as

above, equation (6) becomes

- gi [*•*+**•*+

2M

2

ae-°v+2(M+2Al

3

o

2

) f^ e~*dx] ds. (7)

Integration over the sphere gives the total

drag force F$} for specular reflection:

Nmirr2

| •< 7T* cos 6 sin 9+

la? cos3 0 sin 0+2a cos2 0 sin 0<T°2 cos2 0+

(2 cos 0 sin 0-f 4a2 cos3 0 sin 0) X

| e~

x

dx Yd6; (8)

and again, by some integration by parts, equation (8) reduces to

1+2 *.- l (9)

which is identical with equation (5).
UPPER ATMOSPHERE AND SATELLITE DRAG 13

The third case discussed here is the drag due

to molecular impact plus that due to evaporation of the molecules. Assume that the molecules that impact upon the surface transfer all

their momentum and then evaporate with a

Maxwell distribution of velocities corresponding to a surface temperature T', which is constant over the sphere. This may be regarded

as an assumption of perfect thermal conductivity of the sphere. Current artificial earth

satellites (before the orbits decay into dense

atmosphere) are generally at temperatures on

the order of a few hundred degrees Kelvin

rather than possibly the thousands of degrees

Kelvin corresponding to the kinetic temperature of the gas at very high altitudes. Complete thermal accommodation does not occur

with these artificial satellites, and the energy

balance is maintained largely by radiation

processes.

The distribution of velocities of the evaporated molecules relative to the sphere is given

by the equation

=B exp (10)

where P'*=2kT'/m., and the normalizing factor

B is determined by equating the number of

molecules striking and evaporating from ds.

The number of molecules striking ds per

second with velocity v is, as before, v-ds, and

with distribution (1) the total number striking

ds per second is

X

oo /»0 /*<»

I I (11)

After the introduction of the variables £, 77,

and f and the parameter a=vo{J, equation (11)

reduces to the following:

2/3*-* Jo \

J

(12)

The number of molecules evaporating, nj, from

ds is

Jo J - ^'VvVtV^

v

^

Vtdv

or

pi Bda,

(13)

(14)

The net momentum carried away by a

molecule evaporating from ds is mvt, since by

symmetry the other components balance. The

total momentum carried away per second from

ds is then

J

* CO / * OO / • 09

I ftxvvvzvldvxdvydv:lds=

-co J o J-c o

Bra*3' Aff 2

ds. (15)

Only the component of equation (15) along v0 is

required. Then with the value of B from

equation (14) the drag force Fff due to evaporation is

° = oM> i **«

 co s

 0 sin 0+

43p Jo \.

cos d sin 0e" a W d+

2a cos2 9 siin 0 e~

z

 dx >d6

-

(16)

where C is the average velocity of the

evaporated molecules.

The drag coefficients corresponding to either

equation (5) or (9) and to equation (16) are

readily obtained from the definition of CD in

the equation for the drag force for the sphere,

The factor 2Vm is just the mass density, p, and

the coefficients for the various mechanisms

become the following:

(a) Molecular impact or specular reflection:

(17)

(18)

(b) Evaporation:

(c) Molecular impact and evaporation (diffuse

reflection):

Cg> CS>C%K (19)
14 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Two limiting cases are of interest. First, the

case studied by Epstein (1924) assumed that

vo« C, where O=2/(x*/3) is the average speed of

the gas molecules. It is readily seen that either

equation (5) or (9) approaches Epstein's formula

for the drag force due to molecular impact or

specular reflection, namely, 4/3 Nmirr^voC, and

that equation (16) is the same as his formula for

evaporation drag for the perfect thermal

conductor. Second, Newtonian flow assumes

that vo»C. Equation (17) has the limiting

value of CD=2.

Epstein (1924) considered in some detail

several mechanisms of diffuse reflection or

evaporation, some of which, he and Millikan

(1923) pointed out, violate the second law of

thermodynamics. All of the allowable mechanisms of diffuse reflection lead to formulas

similar to equation (16), except that the factor

of x/6 may be replaced by factors ranging from

3x/16 for the perfect thermal nonconductor, to

16x/27 for "conservation of velocity" (see

Epstein, 1924). The latter value seems physically unlikely, and for the present purpose

equation (16) will be used.

Loeb (1934) discussed specular reflection and

indicated that it can occur at small angles on

polished surfaces. Some specular reflection may

occur with highly polished metallic satellites but

it seems reasonable that most of the re-emission

of molecules occurs by an evaporation mechanism.

Molecular impact without some form of

re-emission may seem unlikely, at first thought.

However, if the satellite consists of a thinwalled plastic balloon, as has been proposed for

measurements of atmospheric density (Jones

and Bartman, 1956; Technical Panel, 1958),

a portion of the more energetic molecules may

in fact penetrate the wall.

Acknowledgment

The author wishes to thank Prof. Sydney

Chapman for reviewing this paper in manuscript

form and for making several helpful suggestions

and comments.

References

BATES, D. R.

1949. The earth's upper atmosphere. Monthly

Notices Roy. Astron. Soc, vol. 109, pp.

215-245.

1951. The temperature of the upper atmosphere.

Proc. Phys. Soc, vol. 64B, pp. 805-821.

1954. The physics of the upper atmosphere. In

Kuiper, ed., The earth as a planet, pp.

576-643.

CHAPMAN, S.

1951. Some phenomena of the upper atmosphere.

Proc. Phys. Soc, vol. 64B, pp. 833-844.

1957. Notes on the solar corona and the terrestrial

ionosphere. Smithsonian Contr. Astrophys., vol. 2, pp. 3-12.

DAVIS, R. J.; WHIPPLB, F. L.; AND ZIRKER, J. B.

1956. The orbit of a small earth satellite. In Van

Allen, ed., Scientific uses of earth satellites,

pp. 1-22.

EPSTEIN, P. S.

1924. On the resistance experienced by spheres in

their motion through gases. Phys. Rev.,

vol. 23, pp. 710-733.

HARRIS, I., AND JASTROW, R.

1958. Upper atmosphere densities from Minitrack

observations on Sputnik I. Science, vol.

127, pp. 471-472.

HEINEMAN, M.

1948. Theory of drag in highly rarefied gases.

Communications on [Pure and] Applied

Mathematics, vol. 1, pp. 259-273.

JONES, L. M., AND BARTMAN, F. L.

1956. Satellite drag and air-density measurements.

In Van Allen, ed., Scientific uses of earth

satellites, pp. 85-98.

LOEB, L. B.

1934. The kinetic theory of gases. Ed. 2, pp.

333-335.

MILLER, L. E.

1957. "Molecular weight" of air at high altitudes.

Journ. Geophys. Res., vol. 62, pp. 351-365.

MILLIKAN, R. A.

1923. The general law of fall of a small spherical

body through a gas, and its bearing upon

the nature of molecular reflection from

surfaces. Phys. Rev., vol. 22, pp. 1-23.

NICOLET, M.

1954. Dynamic effects in the high atmosphere.

In Kuiper, ed., The earth as a planet,

pp. 644-712.

1958. High atmosphere densities. Science, voL

127, pp. 1317-1320.
UPPER ATMOSPHERE AND SATELLITE DRAG 15

ROCKET PANEL

1952. Pressures, densities and temperatures in the

upper atmosphere. Phys. Rev., vol. 88,

pp. 1027-1032.

SPITZEK, L., JR.

1949. The terrestrial atmosphere above 300 km.

In Kuiper, ed., The atmospheres of the

earth and planets, pp. 213-249.

1956. On the determination of air density from a

satellite. In Van Allen, ed., The scientific uses of earth satellites, pp. 99-108.

STERNE, T. E.; FOLKAKT, B. M.; AND SCHILLING, G. F.

1958. An interim model atmosphere fitted to

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STERNE, T. E., AND SCHILLING, G. F.

1958. Some preliminary values of upper atmosphere density from observations of

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TECHNICAL PANEL

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pp. 793-802.

WHIPPLE, F. L.

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above 30 kilometers. In Kuiper, ed.,

The earth as a planet, pp. 491-513.

Abstract

The drag on a sphere moving in the regions of the upper atmosphere is investigated. Calculations show that the

drag coefficient departs from the classical value of CD= 2 for spheres in circular orbits of altitude greater than 10*

km. The drag coefficient may well become much greater than 2 at sufficiently high altitudes.