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Planetary and Space Science 53 (2005) 793–801

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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 thedrag coefficient 2.2 for satellites of compact shape whendeducing absolute atmospheric densities from orbitaldecay. Much of this work was cited in a review article(Moe, 1973) that was used in the construction of the USStandard Atmosphere, 1976. Densities deduced fromorbital decay were often used to construct thermo-spheric density models (Jacchia, 1964; Harris andPriester, 1965). The value of 2.2 for the drag coefficientcontinued to be used in data analysis in subsequentyears (Marcos et al., 1977, Marcos, 1985). This value

e front matter r 2005 Elsevier Ltd. All rights reserved.

s.2005.03.005

ing author. Tel.: +1949 509 1955;

1200.

ess: kmmoe@att.net (K. Moe).

was an intelligent estimate made by Graham Cook(1965, 1966) in the days when little was known aboutgas–surface interactions in orbit. Theoretical workinterpreting laboratory measurements of gas–surfaceinteractions (Goodman, 1967; Trilling, 1967; Saltsburget al., 1967; Bird, 1994; and numerous others) has led tomore recent calculations of drag coefficients by Ga-poschkin (1994), Pardini and Anselmo (2001) andZuppardi (2004). All of these authors have mentionedthe uncertainties involved. It is now generally recognizedthat realistic in-orbit conditions cannot be obtained inthe laboratory, and that the consequent uncertainty inthe drag coefficient can cause important systematicerrors in the measurement of densities by accelerometers(Bruinsma et al., 2004), as well as in the measurementsderived from orbital decay.

ARTICLE IN PRESSK. Moe, M.M. Moe / Planetary and Space Science 53 (2005) 793–801794

Fortunately, considerable information on gas–surfaceinteractions in orbit has accumulated over the past 35years. It has been discovered that satellite surfaces arecovered with adsorbed molecules that affect the energyaccommodation and angular distributions of moleculesreemitted from satellite surfaces in the altitude range150–300 km. The energy accommodation coefficientshave been determined from satellites of special designwhich had two different measurable aerodynamicinteractions. This information has now made possiblethe calculation of drag coefficients from the physics ofsurface–particle interactions up to 300 km. A method ofextending this knowledge to higher altitudes will bediscussed. Drag coefficients above 300 km are needed forimproving lifetime predictions of many satellites includ-ing the International Space Station and the SpaceShuttle. Improved drag coefficients can also contributeto the refinement of thermospheric density models.

2. Drag coefficients, density measurements, and models

Atmospheric densities inferred from measurements byaccelerometers or observations of orbital decay rely onthe relationship of the observed acceleration to char-acteristics of the satellite body. The in-track componentof acceleration caused by air drag is related to thesatellite characteristics through the fundamental relation

Fd ¼ ma ¼ 1=2 rV2i Cd Aref , (1)

where Fd is the in-track component of the force of airdrag; a is the corresponding in-track component ofacceleration; m is the mass of the satellite; r is the airdensity; Cd is the drag coefficient; Aref is the referencearea, usually taken to be the cross-sectional area of thesatellite projected normal to the velocity vector; and Viis the speed of the incident air molecules relative to thesatellite. The values of the mass and the area are oftenknown within 1%, but the speed and drag coefficient aremore uncertain. Vi is usually approximated by theorbital velocity. This generally causes negligible error,except at high latitudes during geomagnetic storms,where, for example, Feess (1973) has measured windsthat exceeded 1 km/s. If the satellite carries an accel-erometer and a mass spectrometer or density gauge, onecan solve for the in-track wind, thus removing thissource of error (Moe and Moe, 1992; Moe et al., 2004).However, the uncertainties in the drag coefficient areimportant under all conditions of geomagnetic activityand at all latitudes and altitudes at which density is to bemeasured.Most thermospheric density models have been devel-

oped from density measurements that assumed a dragcoefficient that was independent of altitude. Thatassumption has introduced an altitude bias into thesemodels. 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 71model (which assumed Cd ¼ 2:2) overestimates the airdensity by as much as 23%, and the MSIS 90 modeloverestimates density by as much as 15% .

3. Adsorption and energy accommodation

The term ‘‘adsorption’’ refers to the trapping ofmolecules on surfaces. By analyzing data from orbitingpressure (density) gauges and mass spectrometers, it hasbeen discovered that satellite surfaces at altitudes of150–300 km are coated with adsorbed atomic oxygenand its reaction products (Moe and Moe, 1969; Moeet al., 1972; Hedin et al., 1973; Offermann andGrossmann, 1973). The higher the altitude, the lowerthe surface coverage of adsorbed atoms and molecules(Moe et al., 1972; Hedin et al., 1973). The effects ofadsorption on energy accommodation have been studiedin the laboratory since the 1930s (Saltsburg et al., 1967;Thomas, 1981). When the incoming molecules strike aclean surface, they are reemitted near the specular anglewith a partial loss of their incident kinetic energy. Thefraction of the incident energy lost depends very muchon the mass of the incoming molecule. However, whenthe surface becomes heavily contaminated with ad-sorbed molecules, the incident molecules are reemittedin a diffuse distribution, losing a large portion of theirincident kinetic energy. Thus, adsorbed moleculesincrease energy accommodation and broaden theangular distribution of molecules reemitted from sur-faces. The accommodation coefficient a is defined asthe ratio

a ¼ ðEi � ErÞ=ðEi � EwÞ, (2)

where Ei is the kinetic energy of the incident molecule;Er is the kinetic energy of the reemitted molecule; andEw is the kinetic energy the reemitted molecule wouldhave, if it left the surface at the surface (wall)temperature. In other words, the accommodationcoefficient indicates how closely the kinetic energy ofthe incoming molecule has adjusted to the thermalenergy 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 andPeters (1987): In a circular orbit at an altitude of 225 km,the distribution was found to be about 97% diffuse and2–3% quasi-specular. Accommodation coefficients inorbit have been measured by several investigators byanalyzing data from satellites that had two measurableaerodynamic interactions: The spin decay and orbitaldecay of paddlewheel satellites (Moe, 1966; Karr, 1969;Beletsky, 1970; Imbro et al., 1975), and the lift and dragof the S3-1 satellite (Ching et al., 1977). The satellites

ARTICLE IN PRESSK. Moe, M.M. Moe / Planetary and Space Science 53 (2005) 793–801 795

involved in these investigations were originally designedfor purposes other than measuring characteristics ofsurface–particle interactions. The data from thesesatellites were utilized in retrospect when it wasperceived that valuable information could be gleanedfrom them.Table 1 shows the accommodation coefficients that

have been inferred from these satellites if a completelydiffuse reemission is assumed. For those satellites whichwere in low-earth orbit, the accommodation coefficientsvaried with altitude from about 1.00 near 200km toabout 0.86 near 300km. If instead of being diffuselyreemitted, 10% of the molecules had been reemittedquasi-specularly from Ariel 2, its average accommoda-tion coefficient would have been 0.88 instead of 0.86(Imbro et al., 1975). For Explorer 6 in its highly eccentricorbit, the values would change from 0.65 for completelydiffuse reemission to 0.68 for 10% quasi-specular to 0.70for 20% quasi-specular reemission (Moe, 1966).The lower energy accommodation near 300 km in

Table 1 is explained by the lower surface coverage ofadsorbed molecules that pressure gauges and massspectrometers measure at the higher altitude. Theaccommodation coefficients given in the table for Ariel2 and Explorer 6 were derived with the assumption thatthe angular distribution of reemitted molecules wasdiffuse. If, contrary to the measurement of Gregory andPeters (1987), the angular distribution is assumed to beentirely quasi-specular, then the accommodation coeffi-cient for Ariel 2 becomes 0.96 (Imbro et al., 1975) andthat for Explorer 6 becomes 0.91 (Moe, 1966). These areunphysical results which contradict the laboratoryfindings of the relationship between energy accommoda-tion and angular distribution. We conclude that theassumption of predominantly diffuse reemission is themore physically reasonable choice for satellites withperigee altitudes below 300 km, even in a highlyeccentric 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 thissatellite is not surprising because one would expect feweradsorbed molecules on the surface and a briefer dwelltime. Oxygen atoms struck Explorer 6 at perigee with50% more kinetic energy than that with which they

Table 1

Accommodation coefficients measured assuming diffuse reemission

Satellite Perigee (km) Orbital eccentricity

S3-1 159 0.22

Proton 2 168 0.03

Ariel 2 290 0.07

Explorer 6 260 0.76

struck the satellites in low-earth orbit. Consequently, theincoming atoms would lose a smaller fraction of theirkinetic energy and would be less likely to be trapped onthe surface.There is independent evidence that the accommoda-

tion coefficients derived in Table 1 (under the assump-tion of diffuse angular distribution of reemittedmolecules) are reasonable for satellites in low-earthorbit with perigee altitudes around 200 km: When dragcoefficients calculated from these accommodation coef-ficients and angular distributions were used to recom-pute thermospheric densities from drag measurementsby a set of Atmosphere Explorer satellites of compactshape and from drag measurements by a set of longcylindrical satellites, the densities agreed within 2%.However, when drag coefficients calculated from theassumption of entirely quasi-specular reemission wereused, the densities derived from the two sets of satellitesdisagreed by 70% (Moe et al., 1998).

4. Drag coefficients at 150–300 km

The parameters of gas–surface interaction which havebeen inferred from measurements in orbit can now beused to calculate the momentum transfer to a satellitesurface and thereby determine physically reasonabledrag coefficients at altitudes from 150 to 300 km. Theanalysis of the momentum transfer to a surface in free-molecular flow has been carried out by many workers.We have found the analyses by Sentman (1961a) and bySchamberg (1959a, 1959b) to be particularly useful.(More readily available papers that use and discussSchamberg’s model were published by Moe and Tsang(1973), and by Imbro et al. (1975)). Sentman’s analysisincorporates the random thermal motion of the incidentmolecules, and assumes that all molecules are diffuselyreemitted. Schamberg uses a simpler analytic represen-tation of the incident stream. His model uses theJoule gas approximation to represent thermal motions.However, it has the advantage that it can employ awide range of angular distributions of the reemittedmolecules. This feature will be used to estimate theuncertainties in the drag coefficients calculated fromSentman’s model.

Accommodation coefficient Source

0.99–1.00 Ching et al. (1977)

1.00 Beletsky (1970)

0.86 Imbro et al. (1975)

0.65 Moe (1966)

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Schamberg relates the incoming molecular beam tothe central axis of the reemitted beam through theequation (reflection law)

cos yr ¼ ðcos yiÞn; nX1. (3)

Here, yi is the angle which the incoming stream makeswith the tangent to the satellite surface and yr is theangle which the axis of the reemitted beam makes withthe tangent. The specular and diffuse laws of reemissioncorrespond to n ¼ 1 and 1, respectively. Schambergforesaw that, in some cases, a superposition of differentreflection laws might be needed.Sentman’s model closely approximates the physical

conditions in the altitude range 150–300 km, since itaccurately describes the distribution of incident mole-cules by superposing the Maxwellian distribution ofmolecular velocities on the incident velocity vector ofthe atmosphere relative to the spacecraft. Furthermore,since the reemitted molecules are highly accommodatedat these altitudes, Sentman’s assumption of diffusereemission is appropriate. Therefore Sentman’s modelhas been used in conjunction with the measurements ofaccommodation coefficients in orbit to calculate thedrag 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 forthe drag coefficient of one side of a flat plate canbe expressed analytically: If the incident velocityvector makes an angle b with the inward normalto the plate surface, then the in-track component ofthe drag force is determined from Eq. (1) by theexpression

Cd Aref ¼ P=p1=2 þ gQZþ ðgVr=2V iÞ½gp1=2Z þ P�. (4)

Fig. 1. Drag coefficients have been determined for a sphere, a flat plate at no

flat plate in front. These curves have been calculated from Sentman’s model, w

times of low solar activity. In contrast, the vertical line at 2.2 has been wide

Here, g ¼ cos b, P ¼ ð1=sÞ expð2g2 s2Þ, Q ¼ 1þ 1=ð2 s2Þ,Z ¼ 1þ erfðg sÞ, erfðxÞ ¼

R x

0 expð�y2Þdy, and V r ¼

ð2=3Þ1=2V i½1þ aðTw=T i21Þ�1=2.

Vr is the most probable speed of the reemittedmolecules. The quantity s is called the speed ratio. It isthe ratio of the satellite speed to the most probablespeed of the ambient atmospheric molecules. (Scham-berg uses a slightly different speed ratio which is theratio of the satellite speed to the rms speed of theambient atmospheric molecules.)Fig. 1 shows the results of our calculations of the drag

coefficients of four satellites of compact shape. Thisfigure illustrates how much drag coefficients of compactsatellites in low-earth orbit can deviate from the value of2.2, which has often been used in deducing neutraldensities from drag data. By replacing the dragcoefficient 2.2 which was originally used in analyzingthe accelerometer measurements on the satellite S3-1, ithas been possible to halve the discrepancies amongmeasurements of density made by an accelerometer, adensity gauge (pressure gauge), and a mass spectrometeraboard that satellite (Moe et al., 2004). The improveddrag coefficients were 2.24, 2.32, and 2.43, correspond-ing to the altitudes 160, 200, and 250 km. With thesechanges, the maximum discrepancy between the accel-erometer measurements and the pressure gauge mea-surements was reduced from 10% to 3%, and themaximum discrepancy with the mass spectrometer wasreduced 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. Theirdrag coefficients can range up to 4, depending on theirlength to diameter ratio and the atmospheric tempera-ture (Sentman, 1961b). The higher drag on longcylinders is produced by the many collisions of air

rmal incidence, the spinning S3-1 Satellite, and a short cylinder with a

hich assumes diffuse reemission, using parameters measured in orbit at

ly used as the drag coefficient for all satellites of compact shape.

ARTICLE IN PRESSK. Moe, M.M. Moe / Planetary and Space Science 53 (2005) 793–801 797

molecules with the long sides, as a consequence of therandom thermal components of the incoming velocityvector.The computed drag coefficients of compact satellites

have a large dependence on altitude (Fig. 1) because thespeed ratio decreases with increasing altitude, while thedecreasing surface coverage causes the accommodationcoefficients to decrease. Drag coefficients also have asmall dependence on time in the sunspot cycle (Moe etal., 1995) through the temperature and mean molecularmass. Table 2 indicates the effects of accommodationcoefficient, local temperature, and mean molecular masson the drag coefficient of a short cylinder capped by aflat plate that faces the airstream. The incident velocityvector is along the axis of the cylinder which has a lengthto diameter ratio of 1. The reemission is assumed to beentirely diffuse (Moe et al., 1995).

5. Gas–surface interactions at higher altitudes

If we want to improve the calculation of dragcoefficients at higher altitudes, we must have informa-tion on how gas–surface interactions change characteras the amount of atomic oxygen adsorbed on satellitesurfaces decreases with increasing altitude. We know

Table 2

Drag coefficients of a short cylinder capped by a flat plate (L/D ¼ 1)

Local temperature (K) Atmospheric mean molecular mass (am

500 18

500 22

1000 18

1000 22

1500 18

1500 22

Fig. 2. Uncertainties in drag coefficient caused by quasi-specular reemission.

model, which assumes completely diffuse reemission. The dashed curves re

component of reemission on the drag coefficient. As in Fig. 1, 2.2 has been

from laboratory experiments that as the surfacecontamination decreases, the energy accommodationcoefficients decrease and the angular distribution ofreemitted molecules has an increasingly importantquasi-specular component (Saltsburg et al., 1967). Anestimate of the uncertainty in drag coefficient caused bythe quasi-specular component is shown in Fig. 2. Thiscomponent was estimated by using Schamberg’s(1959a, b) model, in which the quasi-specular compo-nent was chosen to be twice the quasi-specularcomponent measured by Gregory and Peters (1987), inorder to obtain an upper bound. However, themomentum transfer caused by the incoming streamwas still taken from Sentman’s analysis. The completelydiffuse 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 satellitesat higher altitudes. In a pioneering effort, Harrison andSwinerd (1995) performed a multiple satellite analysis at800–1000 km. Using satellites of various shapes andorientations, they found evidence of lower energyaccommodation and more quasi-specular reemissionthan measured at the lower altitudes. In a similarmanner, the data on many satellites such as thosecollected by the USAF High-Accuracy Satellite Drag

u) Accommodation coefficient

a ¼ 1:00 a ¼ 0:95 a ¼ 0:90

2.33 2.55 2.68

2.30 2.53 2.66

2.42 2.65 2.77

2.38 2.61 2.74

2.50 2.72 2.85

2.45 2.68 2.81

The solid curves represent drag coefficients calculated using Sentman’s

present our estimated upper bound on the effect of a quasi-specular

widely used for all compactly shaped satellites.

ARTICLE IN PRESSK. Moe, M.M. Moe / Planetary and Space Science 53 (2005) 793–801798

Model Program (HASDM) provide a valuable resourcethat can be used to understand the processes influencingdrag coefficients at higher altitudes (Bowman and Storz,2003). If the satellite is above 500 km, it becomesincreasingly important to correct for solar radiationpressure, earth-reflected radiation, and earth-emittedinfrared radiation. The local albedo can vary from 92%over large and thick cumulonimbus clouds to 7% over acloudless Pacific Ocean (Conover, 1965). The infraredradiation depends on the depth and character ofthe cloud cover, as well as on several other variables.

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.

Table 3

Theoretical accommodation coefficients of hard spheres striking static lattic

Surface material Surface molecular mass (amu)

Aluminum 27

Chromium 52

Iron 56

Glass 60

(Solar cell protection)

Paint 75

(Thermal control) 95

mi ¼ mean molecular mass of the incident atmospheric molecule (amu). y ¼ a

These corrections will become very large near sunspotminimum.Tracking data from satellites of several different

shapes and orientations can be used to determine thefraction, f, of incident molecules that are diffuselyreemitted. The basis of the method is the use of twomodels of drag coefficient: one model represents thebehavior of highly accommodated molecules, such asthose reemitted from contaminated surfaces; the othermodel describes the behavior of the remaining portion,(1–f), which represents the quasi-specular reemissionobserved on clean surfaces in the laboratory. This linearcombination is similar to Maxwell’s original model ofgas–surface interactions, but it makes use of 100 years oflaboratory measurements and theoretical analyses thathave been performed since Maxwell’s time. The physicalsituation is illustrated by the sketch in Fig. 3. Thediffusely reemitted molecules had initially struck thesurface near an adsorbed molecule. Consequently, theylost much of the information about their initial energyand momentum. The molecules reemitted in the quasi-specular lobe had initially struck a bare patch of thesurface, thereby retaining much of the informationabout their initial energy and momentum.Sentman’s (1961a) model is ideal for calculating the

part of the drag coefficient caused by the diffuse, highlyaccommodated component of the reemitted molecules.For the quasi-specular component, we suggest usingSchamberg’s model with the accommodation coeffi-cients derived from the static-lattice, hard-sphere modelof Goodman (1967), which was constructed to fit thelaboratory measurements on clean surfaces. Goodman’sformula for the accommodation coefficient is

a ¼ 3:6 u sin y=ð1þ uÞ2, (5)

where u is the ratio of the mass of the incoming moleculeto that of the surface molecule.Goodman’s values for accommodation coefficients

are given in Table 3 for two masses of atmosphericmolecules (24 and 15 amu) and a number of surfacematerials. In Fig. 4, Goodman’s values for a moleculeof average mass (iron) are compared with satellite

es

Mass of incident molecule

mi ¼ 24 mi ¼ 15

0.90 sin y 0.82 sin y0.76 sin y 0.61 sin y0.74 sin y 0.58 sin y0.73 sin y 0.56 sin y

0.64 sin y 0.50 sin y0.56 sin y 0.43 sin y

ngle between the incident velocity vector and the tangent to the surface.

ARTICLE IN PRESS

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

(T1 ¼ 1000K). 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.

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%.

K. Moe, M.M. Moe / Planetary and Space Science 53 (2005) 793–801 799

measurements of accommodation coefficient at altitudesup to 300 km. Although the comparison is poor atthe lower altitudes, we expect that it will improve athigher altitudes as the satellite surfaces become lesscontaminated, and the accommodation coefficient falls,while the angular distribution approaches the quasi-specular. The progression of angular distributionsobserved in the laboratory and in space is illustratedin the composite Fig. 5. As the surface contaminationand mass of the adsorbed molecules increase, theangular distribution progresses from largely quasi-specular to nearly diffuse.The two models of drag coefficient can be used to

solve for the fraction f of the surface that is influencedby the adsorbed molecules. One can use data from pairsof satellites of different shapes that flew near each otherwith the same perigee altitude. The orbital data can beprocessed using Bowman’s energy dissipation ratemethod, which has been used with great success in theHASDM program (Bowman, 2002; Bowman and Storz,2003; Bowman, et al., 2004). A similar method toHASDM has been used successfully by Cefola, et al.(2003). By using many pairs of satellites at altitudesfrom 200 to 500 km, and by requiring that the highlyaccommodated fraction of molecules vary smoothlywith altitude, one can converge on a physically reason-able description of the behavior of drag coefficients as a

ARTICLE IN PRESSK. Moe, M.M. Moe / Planetary and Space Science 53 (2005) 793–801800

function of altitude, encompassing the dependence onenergy accommodation and angular distribution ofreemitted molecules. The sensitivity of the method hasalready been tested near 200 km altitude: Recalling thestudy mentioned previously, when densities inferredfrom accelerometer measurements on board the com-pact Atmosphere Explorer satellites were compared withthose on board long cylindrical satellites, they werewithin 2% of each other when a diffuse, highlyaccommodated reemission was assumed, whereas theywere 70% apart when a quasi-specular reemission wasassumed (Moe et al., 1998).The method described in the paragraph above can also

be used with tracking data from variously shaped satellitesthat pass near the Space Shuttle or are deployed from theShuttle, to determine the Shuttle’s drag coefficient invarious orientations at various altitudes. This will extendthe studies of Blanchard (1986) and Blanchard andNicholson (1995) on Shuttle aerodynamics. Tracking datafrom satellites of simple shapes that fly near theInternational Space Station, and from the docking of theShuttle and Prognoz supply vehicle with the Space Station,can be used to infer the drag coefficient of the SpaceStation by comparing the observed orbital decays of thevarious satellites, without having to perform calculationson the complicated shape of the Space Station.

6. Summary

Energy accommodation coefficients for gas–surfaceinteractions in orbit have been inferred by utilizing datafrom satellites which had two different aerodynamicinteractions. These coefficients have been used in Sent-man’s analysis of free-molecular flow to calculate dragcoefficients of four satellites of compact shape in low-earthorbit in the altitude range 150–300km. Schamberg’sanalysis has been used to estimate the uncertainty in thecomputed drag coefficients. The dependence on para-meters such as local atmospheric temperature and meanmolecular mass is demonstrated. New orbit determinationprograms have begun analyzing data from large numbersof pairs of satellites of different shapes that have hadperigee altitudes near the same location above 300km.These data can be used in conjunction with models ofgas–surface interaction to extend our knowledge ofsatellite drag coefficients up to 500km. The improvementsin drag coefficients can help refine thermospheric densitymeasurements. The improved densities can then be used toconstruct more accurate density models.

Acknowledgements

We thank Bruce Bowman and the reviewers forhelpful suggestions for the revision of this paper.

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