ARTICLE IN PRESS
Planetary and Space Science 53 (2005) 793–801
www.elsevier.com/locate/pss
Gas–surface interactions and satellite drag coefficients
Kenneth Moe, Mildred M. Moe
Science and Technology Corporation, 23 Purple Sage, Irvine, CA 92603
Received 6 May 2004; received in revised form 26 July 2004; accepted 18 March 2005
Abstract
Information on gas–surface interactions in orbit has accumulated during the past 35 years. The important role played by atomic
oxygen adsorbed on satellite surfaces has been revealed by the analysis of data from orbiting mass spectrometers and pressure
gauges. Data from satellites of special design have yielded information on the energy accommodation and angular distributions of
molecules reemitted from satellite surfaces. Consequently, it is now possible to calculate satellite drag coefficients from basic physical
principles, utilizing parameters of gas–surface interactions measured in orbit. The results of such calculations are given. They show
the drag coefficients of four satellites of different compact shapes in low-earth orbit with perigee altitudes in the range from about
150 to 300 km, where energy accommodation coefficients and diffuse angular distributions have been measured. The calculations are
based on Sentman’s analysis of drag forces in free-molecular flow. His model incorporates the random thermal motion of the
incident molecules, and assumes that all molecules are diffusely reemitted The uncertainty caused by the assumption of diffuse
reemission is estimated by using Schamberg’s model of gas–surface interaction, which can take into account a quasi-specular
component of the reemission. Such a quasi-specular component is likely to become more important at higher altitudes as the amount
of adsorbed atomic oxygen decreases. A method of deducing accommodation coefficients and angular distributions at higher
altitudes by comparing the simultaneous orbital decay of satellites of different shapes at a number of altitudes is suggested. The
purpose is to improve thermospheric measurements and models, which are significantly affected by the choice of drag coefficients.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: Gas–surface interactions; Drag coefficients; Accommodation coefficients; Thermospheric density; Density models
1. Introduction
Early in the space age, most orbit analysts used the
drag coefficient 2.2 for satellites of compact shape when
deducing absolute atmospheric densities from orbital
decay. Much of this work was cited in a review article
(Moe, 1973) that was used in the construction of the US
Standard Atmosphere, 1976. Densities deduced from
orbital decay were often used to construct thermospheric density models (Jacchia, 1964; Harris and
Priester, 1965). The value of 2.2 for the drag coefficient
continued to be used in data analysis in subsequent
years (Marcos et al., 1977, Marcos, 1985). This value
Corresponding author. Tel.: +1 949 509 1955;
fax: +1 949 509 1200.
E-mail address: [email protected] (K. Moe).
0032-0633/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pss.2005.03.005
was an intelligent estimate made by Graham Cook
(1965, 1966) in the days when little was known about
gas–surface interactions in orbit. Theoretical work
interpreting laboratory measurements of gas–surface
interactions (Goodman, 1967; Trilling, 1967; Saltsburg
et al., 1967; Bird, 1994; and numerous others) has led to
more recent calculations of drag coefficients by Gaposchkin (1994), Pardini and Anselmo (2001) and
Zuppardi (2004). All of these authors have mentioned
the uncertainties involved. It is now generally recognized
that realistic in-orbit conditions cannot be obtained in
the laboratory, and that the consequent uncertainty in
the drag coefficient can cause important systematic
errors in the measurement of densities by accelerometers
(Bruinsma et al., 2004), as well as in the measurements
derived from orbital decay.
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Fortunately, considerable information on gas–surface
interactions in orbit has accumulated over the past 35
years. It has been discovered that satellite surfaces are
covered with adsorbed molecules that affect the energy
accommodation and angular distributions of molecules
reemitted from satellite surfaces in the altitude range
150–300 km. The energy accommodation coefficients
have been determined from satellites of special design
which had two different measurable aerodynamic
interactions. This information has now made possible
the calculation of drag coefficients from the physics of
surface–particle interactions up to 300 km. A method of
extending this knowledge to higher altitudes will be
discussed. Drag coefficients above 300 km are needed for
improving lifetime predictions of many satellites including the International Space Station and the Space
Shuttle. Improved drag coefficients can also contribute
to the refinement of thermospheric density models.
2. Drag coefficients, density measurements, and models
Atmospheric densities inferred from measurements by
accelerometers or observations of orbital decay rely on
the relationship of the observed acceleration to characteristics of the satellite body. The in-track component
of acceleration caused by air drag is related to the
satellite characteristics through the fundamental relation
F d ¼ ma ¼ 1=2 r V 2i C d Aref ,
(1)
where Fd is the in-track component of the force of air
drag; a is the corresponding in-track component of
acceleration; m is the mass of the satellite; r is the air
density; Cd is the drag coefficient; Aref is the reference
area, usually taken to be the cross-sectional area of the
satellite projected normal to the velocity vector; and Vi
is the speed of the incident air molecules relative to the
satellite. The values of the mass and the area are often
known within 1%, but the speed and drag coefficient are
more uncertain. Vi is usually approximated by the
orbital velocity. This generally causes negligible error,
except at high latitudes during geomagnetic storms,
where, for example, Feess (1973) has measured winds
that exceeded 1 km/s. If the satellite carries an accelerometer and a mass spectrometer or density gauge, one
can solve for the in-track wind, thus removing this
source of error (Moe and Moe, 1992; Moe et al., 2004).
However, the uncertainties in the drag coefficient are
important under all conditions of geomagnetic activity
and at all latitudes and altitudes at which density is to be
measured.
Most thermospheric density models have been developed from density measurements that assumed a drag
coefficient that was independent of altitude. That
assumption has introduced an altitude bias into these
models. By introducing more realistic drag coefficients,
it was recently reported (Chao et al., 1997; Moe et al.,
2004) that, at altitudes above 250 km, the Jacchia 71
model (which assumed C d ¼ 2:2) overestimates the air
density by as much as 23%, and the MSIS 90 model
overestimates density by as much as 15% .
3. Adsorption and energy accommodation
The term ‘‘adsorption’’ refers to the trapping of
molecules on surfaces. By analyzing data from orbiting
pressure (density) gauges and mass spectrometers, it has
been discovered that satellite surfaces at altitudes of
150–300 km are coated with adsorbed atomic oxygen
and its reaction products (Moe and Moe, 1969; Moe
et al., 1972; Hedin et al., 1973; Offermann and
Grossmann, 1973). The higher the altitude, the lower
the surface coverage of adsorbed atoms and molecules
(Moe et al., 1972; Hedin et al., 1973). The effects of
adsorption on energy accommodation have been studied
in the laboratory since the 1930s (Saltsburg et al., 1967;
Thomas, 1981). When the incoming molecules strike a
clean surface, they are reemitted near the specular angle
with a partial loss of their incident kinetic energy. The
fraction of the incident energy lost depends very much
on the mass of the incoming molecule. However, when
the surface becomes heavily contaminated with adsorbed molecules, the incident molecules are reemitted
in a diffuse distribution, losing a large portion of their
incident kinetic energy. Thus, adsorbed molecules
increase energy accommodation and broaden the
angular distribution of molecules reemitted from surfaces. The accommodation coefficient a is defined as
the ratio
a ¼ ðE i E r Þ=ðE i E w Þ,
(2)
where Ei is the kinetic energy of the incident molecule;
Er is the kinetic energy of the reemitted molecule; and
Ew is the kinetic energy the reemitted molecule would
have, if it left the surface at the surface (wall)
temperature. In other words, the accommodation
coefficient indicates how closely the kinetic energy of
the incoming molecule has adjusted to the thermal
energy of the surface. If the adjustment is complete, a ¼
1:00 which is referred to as ‘‘complete accommodation’’.
The angular distribution of molecules reemitted from
a satellite surface in orbit was measured by Gregory and
Peters (1987): In a circular orbit at an altitude of 225 km,
the distribution was found to be about 97% diffuse and
2–3% quasi-specular. Accommodation coefficients in
orbit have been measured by several investigators by
analyzing data from satellites that had two measurable
aerodynamic interactions: The spin decay and orbital
decay of paddlewheel satellites (Moe, 1966; Karr, 1969;
Beletsky, 1970; Imbro et al., 1975), and the lift and drag
of the S3-1 satellite (Ching et al., 1977). The satellites
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involved in these investigations were originally designed
for purposes other than measuring characteristics of
surface–particle interactions. The data from these
satellites were utilized in retrospect when it was
perceived that valuable information could be gleaned
from them.
Table 1 shows the accommodation coefficients that
have been inferred from these satellites if a completely
diffuse reemission is assumed. For those satellites which
were in low-earth orbit, the accommodation coefficients
varied with altitude from about 1.00 near 200 km to
about 0.86 near 300 km. If instead of being diffusely
reemitted, 10% of the molecules had been reemitted
quasi-specularly from Ariel 2, its average accommodation coefficient would have been 0.88 instead of 0.86
(Imbro et al., 1975). For Explorer 6 in its highly eccentric
orbit, the values would change from 0.65 for completely
diffuse reemission to 0.68 for 10% quasi-specular to 0.70
for 20% quasi-specular reemission (Moe, 1966).
The lower energy accommodation near 300 km in
Table 1 is explained by the lower surface coverage of
adsorbed molecules that pressure gauges and mass
spectrometers measure at the higher altitude. The
accommodation coefficients given in the table for Ariel
2 and Explorer 6 were derived with the assumption that
the angular distribution of reemitted molecules was
diffuse. If, contrary to the measurement of Gregory and
Peters (1987), the angular distribution is assumed to be
entirely quasi-specular, then the accommodation coefficient for Ariel 2 becomes 0.96 (Imbro et al., 1975) and
that for Explorer 6 becomes 0.91 (Moe, 1966). These are
unphysical results which contradict the laboratory
findings of the relationship between energy accommodation and angular distribution. We conclude that the
assumption of predominantly diffuse reemission is the
more physically reasonable choice for satellites with
perigee altitudes below 300 km, even in a highly
eccentric orbit such as that of Explorer 6.
The highly eccentric orbit of Explorer 6 is often
referred to as a Molniya or geocentric transfer orbit.
The lower accommodation coefficient deduced for this
satellite is not surprising because one would expect fewer
adsorbed molecules on the surface and a briefer dwell
time. Oxygen atoms struck Explorer 6 at perigee with
50% more kinetic energy than that with which they
795
struck the satellites in low-earth orbit. Consequently, the
incoming atoms would lose a smaller fraction of their
kinetic energy and would be less likely to be trapped on
the surface.
There is independent evidence that the accommodation coefficients derived in Table 1 (under the assumption of diffuse angular distribution of reemitted
molecules) are reasonable for satellites in low-earth
orbit with perigee altitudes around 200 km: When drag
coefficients calculated from these accommodation coefficients and angular distributions were used to recompute thermospheric densities from drag measurements
by a set of Atmosphere Explorer satellites of compact
shape and from drag measurements by a set of long
cylindrical satellites, the densities agreed within 2%.
However, when drag coefficients calculated from the
assumption of entirely quasi-specular reemission were
used, the densities derived from the two sets of satellites
disagreed by 70% (Moe et al., 1998).
4. Drag coefficients at 150–300 km
The parameters of gas–surface interaction which have
been inferred from measurements in orbit can now be
used to calculate the momentum transfer to a satellite
surface and thereby determine physically reasonable
drag coefficients at altitudes from 150 to 300 km. The
analysis of the momentum transfer to a surface in freemolecular flow has been carried out by many workers.
We have found the analyses by Sentman (1961a) and by
Schamberg (1959a, 1959b) to be particularly useful.
(More readily available papers that use and discuss
Schamberg’s model were published by Moe and Tsang
(1973), and by Imbro et al. (1975)). Sentman’s analysis
incorporates the random thermal motion of the incident
molecules, and assumes that all molecules are diffusely
reemitted. Schamberg uses a simpler analytic representation of the incident stream. His model uses the
Joule gas approximation to represent thermal motions.
However, it has the advantage that it can employ a
wide range of angular distributions of the reemitted
molecules. This feature will be used to estimate the
uncertainties in the drag coefficients calculated from
Sentman’s model.
Table 1
Accommodation coefficients measured assuming diffuse reemission
Satellite
Perigee (km)
Orbital eccentricity
Accommodation coefficient
Source
S3-1
Proton 2
Ariel 2
159
168
290
0.22
0.03
0.07
0.99–1.00
1.00
0.86
Ching et al. (1977)
Beletsky (1970)
Imbro et al. (1975)
Explorer 6
260
0.76
0.65
Moe (1966)
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Schamberg relates the incoming molecular beam to
the central axis of the reemitted beam through the
equation (reflection law)
cos yr ¼ ðcos yi Þn ; nX1.
(3)
Here, yi is the angle which the incoming stream makes
with the tangent to the satellite surface and yr is the
angle which the axis of the reemitted beam makes with
the tangent. The specular and diffuse laws of reemission
correspond to n ¼ 1 and 1, respectively. Schamberg
foresaw that, in some cases, a superposition of different
reflection laws might be needed.
Sentman’s model closely approximates the physical
conditions in the altitude range 150–300 km, since it
accurately describes the distribution of incident molecules by superposing the Maxwellian distribution of
molecular velocities on the incident velocity vector of
the atmosphere relative to the spacecraft. Furthermore,
since the reemitted molecules are highly accommodated
at these altitudes, Sentman’s assumption of diffuse
reemission is appropriate. Therefore Sentman’s model
has been used in conjunction with the measurements of
accommodation coefficients in orbit to calculate the
drag coefficients in this altitude range (Moe et al., 1995,
1996, 1998, 2004).
Sentman’s formulas for the drag coefficients of
various shapes are complex, but his expression for
the drag coefficient of one side of a flat plate can
be expressed analytically: If the incident velocity
vector makes an angle b with the inward normal
to the plate surface, then the in-track component of
the drag force is determined from Eq. (1) by the
expression
C d Aref ¼ P=p1=2 þ gQZ þ ðgVr =2V i Þ½gp1=2 Z þ P.
(4)
2 2
s Þ, Q ¼ 1 þ 1=ð2 s2 Þ,
Here, g ¼ cos b, P ¼ ð1=sÞ expð2g
Rx
Z ¼ 1 þ erfðg sÞ, erfðxÞ ¼ 0 expðy2 Þ dy, and V r ¼
ð2=3Þ1=2 V i ½1 þ aðT w =T i 21Þ1=2 .
Vr is the most probable speed of the reemitted
molecules. The quantity s is called the speed ratio. It is
the ratio of the satellite speed to the most probable
speed of the ambient atmospheric molecules. (Schamberg uses a slightly different speed ratio which is the
ratio of the satellite speed to the rms speed of the
ambient atmospheric molecules.)
Fig. 1 shows the results of our calculations of the drag
coefficients of four satellites of compact shape. This
figure illustrates how much drag coefficients of compact
satellites in low-earth orbit can deviate from the value of
2.2, which has often been used in deducing neutral
densities from drag data. By replacing the drag
coefficient 2.2 which was originally used in analyzing
the accelerometer measurements on the satellite S3-1, it
has been possible to halve the discrepancies among
measurements of density made by an accelerometer, a
density gauge (pressure gauge), and a mass spectrometer
aboard that satellite (Moe et al., 2004). The improved
drag coefficients were 2.24, 2.32, and 2.43, corresponding to the altitudes 160, 200, and 250 km. With these
changes, the maximum discrepancy between the accelerometer measurements and the pressure gauge measurements was reduced from 10% to 3%, and the
maximum discrepancy with the mass spectrometer was
reduced from 17% to 8% (Moe et al., 2004).
Long cylindrical satellites that fly like an arrow have
drag coefficients that are too large to fit in Fig. 1. Their
drag coefficients can range up to 4, depending on their
length to diameter ratio and the atmospheric temperature (Sentman, 1961b). The higher drag on long
cylinders is produced by the many collisions of air
Fig. 1. Drag coefficients have been determined for a sphere, a flat plate at normal incidence, the spinning S3-1 Satellite, and a short cylinder with a
flat plate in front. These curves have been calculated from Sentman’s model, which assumes diffuse reemission, using parameters measured in orbit at
times of low solar activity. In contrast, the vertical line at 2.2 has been widely used as the drag coefficient for all satellites of compact shape.
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molecules with the long sides, as a consequence of the
random thermal components of the incoming velocity
vector.
The computed drag coefficients of compact satellites
have a large dependence on altitude (Fig. 1) because the
speed ratio decreases with increasing altitude, while the
decreasing surface coverage causes the accommodation
coefficients to decrease. Drag coefficients also have a
small dependence on time in the sunspot cycle (Moe et
al., 1995) through the temperature and mean molecular
mass. Table 2 indicates the effects of accommodation
coefficient, local temperature, and mean molecular mass
on the drag coefficient of a short cylinder capped by a
flat plate that faces the airstream. The incident velocity
vector is along the axis of the cylinder which has a length
to diameter ratio of 1. The reemission is assumed to be
entirely diffuse (Moe et al., 1995).
5. Gas–surface interactions at higher altitudes
If we want to improve the calculation of drag
coefficients at higher altitudes, we must have information on how gas–surface interactions change character
as the amount of atomic oxygen adsorbed on satellite
surfaces decreases with increasing altitude. We know
797
from laboratory experiments that as the surface
contamination decreases, the energy accommodation
coefficients decrease and the angular distribution of
reemitted molecules has an increasingly important
quasi-specular component (Saltsburg et al., 1967). An
estimate of the uncertainty in drag coefficient caused by
the quasi-specular component is shown in Fig. 2. This
component was estimated by using Schamberg’s
(1959a, b) model, in which the quasi-specular component was chosen to be twice the quasi-specular
component measured by Gregory and Peters (1987), in
order to obtain an upper bound. However, the
momentum transfer caused by the incoming stream
was still taken from Sentman’s analysis. The completely
diffuse case was calculated entirely from Sentman’s
(1961a) model, as in Fig. 1.
To use a model of drag coefficients above 300 km, we
need data on accommodation coefficients from satellites
at higher altitudes. In a pioneering effort, Harrison and
Swinerd (1995) performed a multiple satellite analysis at
800–1000 km. Using satellites of various shapes and
orientations, they found evidence of lower energy
accommodation and more quasi-specular reemission
than measured at the lower altitudes. In a similar
manner, the data on many satellites such as those
collected by the USAF High-Accuracy Satellite Drag
Table 2
Drag coefficients of a short cylinder capped by a flat plate (L/D ¼ 1)
Local temperature (K)
500
500
1000
1000
1500
1500
Atmospheric mean molecular mass (amu)
18
22
18
22
18
22
Accommodation coefficient
a ¼ 1:00
a ¼ 0:95
a ¼ 0:90
2.33
2.30
2.42
2.38
2.50
2.45
2.55
2.53
2.65
2.61
2.72
2.68
2.68
2.66
2.77
2.74
2.85
2.81
Fig. 2. Uncertainties in drag coefficient caused by quasi-specular reemission. The solid curves represent drag coefficients calculated using Sentman’s
model, which assumes completely diffuse reemission. The dashed curves represent our estimated upper bound on the effect of a quasi-specular
component of reemission on the drag coefficient. As in Fig. 1, 2.2 has been widely used for all compactly shaped satellites.
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Model Program (HASDM) provide a valuable resource
that can be used to understand the processes influencing
drag coefficients at higher altitudes (Bowman and Storz,
2003). If the satellite is above 500 km, it becomes
increasingly important to correct for solar radiation
pressure, earth-reflected radiation, and earth-emitted
infrared radiation. The local albedo can vary from 92%
over large and thick cumulonimbus clouds to 7% over a
cloudless Pacific Ocean (Conover, 1965). The infrared
radiation depends on the depth and character of
the cloud cover, as well as on several other variables.
These corrections will become very large near sunspot
minimum.
Tracking data from satellites of several different
shapes and orientations can be used to determine the
fraction, f, of incident molecules that are diffusely
reemitted. The basis of the method is the use of two
models of drag coefficient: one model represents the
behavior of highly accommodated molecules, such as
those reemitted from contaminated surfaces; the other
model describes the behavior of the remaining portion,
(1–f), which represents the quasi-specular reemission
observed on clean surfaces in the laboratory. This linear
combination is similar to Maxwell’s original model of
gas–surface interactions, but it makes use of 100 years of
laboratory measurements and theoretical analyses that
have been performed since Maxwell’s time. The physical
situation is illustrated by the sketch in Fig. 3. The
diffusely reemitted molecules had initially struck the
surface near an adsorbed molecule. Consequently, they
lost much of the information about their initial energy
and momentum. The molecules reemitted in the quasispecular lobe had initially struck a bare patch of the
surface, thereby retaining much of the information
about their initial energy and momentum.
Sentman’s (1961a) model is ideal for calculating the
part of the drag coefficient caused by the diffuse, highly
accommodated component of the reemitted molecules.
For the quasi-specular component, we suggest using
Schamberg’s model with the accommodation coefficients derived from the static-lattice, hard-sphere model
of Goodman (1967), which was constructed to fit the
laboratory measurements on clean surfaces. Goodman’s
formula for the accommodation coefficient is
a ¼ 3:6 u sin y=ð1 þ uÞ2 ,
Fig. 3. Reemission from a flat plate. Many laboratory experiments
have found that molecules are reemitted from a contaminated surface
in a diffuse, or cosine distribution centered on the normal to the
surface. Molecules are reemitted from a clean surface in a narrow
beam near the specular angle. As the contamination increases, the
quasi-specular fraction is reduced, and the diffuse fraction is increased.
(5)
where u is the ratio of the mass of the incoming molecule
to that of the surface molecule.
Goodman’s values for accommodation coefficients
are given in Table 3 for two masses of atmospheric
molecules (24 and 15 amu) and a number of surface
materials. In Fig. 4, Goodman’s values for a molecule
of average mass (iron) are compared with satellite
Table 3
Theoretical accommodation coefficients of hard spheres striking static lattices
Surface material
Aluminum
Chromium
Iron
Glass
(Solar cell protection)
Paint
(Thermal control)
Surface molecular mass (amu)
Mass of incident molecule
mi ¼ 24
mi ¼ 15
27
52
56
60
0.90 sin y
0.76 sin y
0.74 sin y
0.73 sin y
0.82 sin y
0.61 sin y
0.58 sin y
0.56 sin y
75
95
0.64 sin y
0.56 sin y
0.50 sin y
0.43 sin y
mi ¼ mean molecular mass of the incident atmospheric molecule (amu). y ¼ angle between the incident velocity vector and the tangent to the surface.
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799
Fig. 4. Accommodation coefficients measured in orbit, compared with
theoretical hard-sphere models for iron. Goodman’s hard-sphere
model represents laboratory measurements of energy accommodation
on clean surfaces. It gives values from 0.75 at 150 km to 0.6 at 300 km
for normal incidence on a static lattice under average solar conditions
(T 1 ¼ 1000 K). At angels near grazing incidence, Goodman’s model
gives accommodation coefficients below 0.10. In contrast, satellites in
low-earth orbit measured values around 1.00 near 200 km and 0.90
near 300 km.
measurements of accommodation coefficient at altitudes
up to 300 km. Although the comparison is poor at
the lower altitudes, we expect that it will improve at
higher altitudes as the satellite surfaces become less
contaminated, and the accommodation coefficient falls,
while the angular distribution approaches the quasispecular. The progression of angular distributions
observed in the laboratory and in space is illustrated
in the composite Fig. 5. As the surface contamination
and mass of the adsorbed molecules increase, the
angular distribution progresses from largely quasispecular to nearly diffuse.
The two models of drag coefficient can be used to
solve for the fraction f of the surface that is influenced
by the adsorbed molecules. One can use data from pairs
of satellites of different shapes that flew near each other
with the same perigee altitude. The orbital data can be
processed using Bowman’s energy dissipation rate
method, which has been used with great success in the
HASDM program (Bowman, 2002; Bowman and Storz,
2003; Bowman, et al., 2004). A similar method to
HASDM has been used successfully by Cefola, et al.
(2003). By using many pairs of satellites at altitudes
from 200 to 500 km, and by requiring that the highly
accommodated fraction of molecules vary smoothly
with altitude, one can converge on a physically reasonable description of the behavior of drag coefficients as a
Fig. 5. This composite figure shows how the relative mass, binding
energy, and fractional coverage of adsorbed molecules affect the
angular distribution of reemitted molecules in the plane of incidence.
The vertical arrows mark the specular angle. (a) The laboratory
measurement of O’Keefe and French (1969) shows argon, with a
kinetic energy of 1.35 eV reflecting from the 100 crystal face of
tungsten which is partly coated with weakly bound, lighter molecules.
As the fractional coverage and mass of the physisorbed molecules
increase, the high quasi-specular peak is gradually reduced. (b) In
contrast, the satellite experiment of Gregory and Peteres shows that
when atomic oxygen strikes a carbon surface coated with chemisorbed
oxygen, which has a binding energy comparable with the incident
kinetic energy of 5 eV, the quasi-specular component has been reduced
to 2% or 3%.
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function of altitude, encompassing the dependence on
energy accommodation and angular distribution of
reemitted molecules. The sensitivity of the method has
already been tested near 200 km altitude: Recalling the
study mentioned previously, when densities inferred
from accelerometer measurements on board the compact Atmosphere Explorer satellites were compared with
those on board long cylindrical satellites, they were
within 2% of each other when a diffuse, highly
accommodated reemission was assumed, whereas they
were 70% apart when a quasi-specular reemission was
assumed (Moe et al., 1998).
The method described in the paragraph above can also
be used with tracking data from variously shaped satellites
that pass near the Space Shuttle or are deployed from the
Shuttle, to determine the Shuttle’s drag coefficient in
various orientations at various altitudes. This will extend
the studies of Blanchard (1986) and Blanchard and
Nicholson (1995) on Shuttle aerodynamics. Tracking data
from satellites of simple shapes that fly near the
International Space Station, and from the docking of the
Shuttle and Prognoz supply vehicle with the Space Station,
can be used to infer the drag coefficient of the Space
Station by comparing the observed orbital decays of the
various satellites, without having to perform calculations
on the complicated shape of the Space Station.
6. Summary
Energy accommodation coefficients for gas–surface
interactions in orbit have been inferred by utilizing data
from satellites which had two different aerodynamic
interactions. These coefficients have been used in Sentman’s analysis of free-molecular flow to calculate drag
coefficients of four satellites of compact shape in low-earth
orbit in the altitude range 150–300 km. Schamberg’s
analysis has been used to estimate the uncertainty in the
computed drag coefficients. The dependence on parameters such as local atmospheric temperature and mean
molecular mass is demonstrated. New orbit determination
programs have begun analyzing data from large numbers
of pairs of satellites of different shapes that have had
perigee altitudes near the same location above 300 km.
These data can be used in conjunction with models of
gas–surface interaction to extend our knowledge of
satellite drag coefficients up to 500 km. The improvements
in drag coefficients can help refine thermospheric density
measurements. The improved densities can then be used to
construct more accurate density models.
Acknowledgements
We thank Bruce Bowman and the reviewers for
helpful suggestions for the revision of this paper.
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