orbit and attitude perturbations due to

7/28/2019 Orbit and Attitude Perturbations Due To

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ORBIT AND ATTITUDE PERTURBATIONS DUE TO

AERODYNAMICS AND RADIATION PRESSURE

H. Klinkrad1 and B. Fritsche2

1Mission Analysis Section, ESA/ESOC, D-64293 Darmstadt, Germany2Hypersonic Technology Gottingen, D-37191 Katlenburg-Lindau, Germany

ABSTRACT

For satellites which are operated in low-Earth orbits(LEO), gravitational forces due to the non-uniform massdistribution of the Earth are dominating the orbit and at-titude perturbation spectra. Non-gravitational forces aremainly caused by momentum exchange with the space-craft surface, and they are mostly of second order. Themost prominent of these forces originate from the interac-tion of the spacecraft surface with molecules and ions ofthe thermosphere, and from the impact of photons whichcome directly from the sun, which are reflected as albedofrom the illuminated Earth hemisphere, or which are re-emitted by the whole Earth as delayed infra-red (IR) re-

radiation. In contrast with gravitational perturbations, theaerodynamic and radiation pressure effects are difficult tomodel since they require a good knowledge of the space-craft geometry and surface properties, and they also re-quire reliable estimates of the molecule and photon par-ticle flux. The necessary models of the thermosphere,and of the Earth albedo and IR re-radiation distributionsare depending on a large set of parameters, including thespacecraft location, the time, the season (sun position),and solar and geomagnetic activity levels. The variabil-ity of these environment models will be explained, andmathematical models will be described which allow to usethe resulting molecule and photon flux models to com-pute perturbing forces and torques acting on a LEO satel-lite. Examples will be provided for ESAs ERS-1 and EN-

VISAT satellites.

Key words: free-molecular aerodynamics, radiation pres-sure, non-gravitational perturbations.

1. INTRODUCTION

Many space applications require very accurate orbit de-terminations of a satellite to use position fixes of knownprecision as absolute reference for high resolution mea-surements of the geoid, the sea surface topography, or themotions of tectonic plates. With the use of on-board pre-cision tracking aids of ESAs ERS-1 and ERS-2 satellites(laser retro-reflectors LRR, and precise range and range-rate equipments PRARE), and with the use of auxiliarytracking data types (direct altimetry, and altimeter cross-over measurements), the ERS-1 & 2 orbits can be fitted

with a root mean square (rms) error of about 5 cm in ra-dial, 10 cm in cross-track, and 40 cm in along-track po-sition. Such precise fits require an orbit prediction soft-ware with very accurate models of the perturbing gravi-tational and non-gravitational accelerations which affectthe motion of an Earth satellite. Gravitational perturba-tions are dominating the force spectrum for most Earthorbits. They are caused by non-uniform mass distribu-tions inside the Earth, by ocean, atmosphere, and Earthtides, and by third body attraction (Sun, Moon, plan-ets). All of these perturbations can be modeled with ahigh level of confidence, and all of them are conserva-tive (causing only periodic changes in the orbit energy).A complementary class of orbit perturbations is denotedas non-gravitational. This class comprises aerodynamic

forces, direct and indirect radiation pressure effects, ther-mal re-radiation, and charged particle drag. Models ofthese non-gravitational forces are affected by uncertain-ties in the molecule-surface and photon-surface interac-tion processes, in the molecule and photon flux models,and in the solar and geomagnetic activity levels and theireffect on the thermosphere and ionosphere. Some of theseperturbations cause a secular, time-proportional decreaseof the orbital energy, and hence of the orbital altitude.For low-Earth orbits (LEO), these altitude decays mustbe compensated by periodic maintenance manoeuvres.

A detailed review of non-gravitational forces (also de-noted as surface forces) is performed by Rubincam(1982), Klinkrad et al. (1990), and Ries et al. (1992).

Apart from the dominant direct and indirect radiationpressure, thermal re-radiation, and aerodynamic perturba-tions, these papers also analyse secondary effects causedby delayed thermal re-emission due to an Earth shadowtransit (Yarkovsky effect), due to the rotation of a satel-lite (Yarkovsky-Schach effect), or due to a frequencyshift between received and emitted radiation (Poynting-Robertson effect). Antreasian & Rosborough (1992) andPowell & Gaposhkin (1988) focus their analysis on radia-tive forces, including thermal re-radiation, Earth albedoand Earth IR contributions. Aerodynamic forces at LEOaltitudes are addressed in more detail by Marcos et al.(1993) and by Koppenwallner et al. (1995). Marcos et al.(1993) and Klinkrad (1996) also investigate the status ofcontemporary thermospheric models, their uncertainties,and the resulting effects on orbit prediction.

The present paper will describe the nature of surfaceforces, the key environmental models and their param-


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Figure 1. Observed solar activity (in terms of daily 10.7 cm radio flux F10 7, and sun-spot numbers SSN), and geomagneticactivity (in terms of daily Ap indices) during solar cycle 21 and 22.

Figure 2. Temperature and total density altitude profiles and their extremes during a solar cycle according to the CIRA-86model (equivalent to MSIS-90e at thermospheric altitudes).

eters, the mathematical methods to determine moleculeand photon incident fluxes, the physics of the particle-surface interaction, and the derivation of satellite spe-cific coefficients of force and torque for the different non-gravitational perturbation source terms. Results will be

provided for the ERS-1 and ENVISAT satellites, basedon runs with ESAs ANGARA program (Analysis of Non-Gravitational Accelerations due to Radiation and Aerody-namics, Fritsche et al. (1998)).

2. ENERGY INPUT FROM THE SUN

The only significant energy source for the Earth is the so-lar radiation which is emitted by the Sun across a widefrequency spectrum with a peak energy flux in the visiblelight. Its energy distribution can be well approximated bya black body radiator of a mean temperature of 5,785 K,

providing a mean energy flux of 1,370 W/m2, with annual


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variations of

3:

3% due to the small eccentricity of theEarth orbit. When the energy input into the illuminatedhemisphere is averaged over the whole Earth surface, the

mean incident energy flux is 349 W/m2, of which 33%is reflected in the visible light as planetary albedo (26%reflected by the clouds, and 7% reflected from the conti-

nents and oceans). 67% of the incident energy is absorbedby the atmosphere, by soil, and by water. It is then re-emitted mainly as delayed infra-red (IR) radiation whichlets the Earth appear as a black body radiator of a meantemperature of 253 K.

The Suns input into the terrestrial energy balance drivesboth aerodynamic and indirect radiation pressure effects(albedo and IR).

3. AERODYNAMIC PERTURBATIONS

For LEO satellites, aerodynamic perturbations are mostlyof second order. The resulting forces have magnitudes ofless than 1/1,000 of the only first order perturbation, theEarth oblateness. Aerodynamic effects (mainly airdrag)are the dominant surface force contributions up to alti-tudes of 500 km to 600km (depending on the atmosphericconditions). During re-entries (i.e. below 120 km), aero-dynamic forces become first order perturbation quantitieswhich ultimately reach the level of the central attractionterm during the atmospheric flight phase. At altitudesabove 500 km to 600 km direct solar radiation pressureprevails.

3.1. Models of the Earth Atmosphere

While most of the Suns energy is received in the visi-ble frequency bands, with only minor annual variations,the dynamics of the neutral upper atmosphere is mainlydriven by extreme ultra-violet radiation (EUV) and itsabsorption by atomic oxygen, by photo-dissociation andre-combination processes, and by Joule heating fromcharged particles which precipitate into the auroral zones.EUV radiation levels are known to change in 11-year so-lar cycles. They are associated with emissions from Sun-spot areas, and are proportional to the observable num-ber of Sun-spots (see Figure 1). The EUV is well cor-related with the 10.7 cm solar flux which can be mea-sured on ground through one of the atmospheric radio

windows (F10 7 is defined in units of 10,

22 W m,

2Hz,

1).Peaks of Joule heating are often associated with high ge-omagnetic activities. These are commonly measured interms of daily planetary indices Ap, or 3-hourly indiceskp (see Figure 1). Figure 2 shows the effect of low andhigh extremes of solar and geomagnetic activity on thetemperature and air density altitude profile. The temper-ature profile varies only slightly in the homosphere (be-low 120 km), and then follows an exponential increasewhich reaches a limiting value, the so-called exospherictemperature, at the top of the thermosphere. The air den-sity above the turbopause (at 120 km) is determined bythe superimposition of concentration profiles ni h of themajor atmospheric constituents N2, O, He, H, O2, Ar,and N (for i = 1 to 7). The concentration scale heights

Hni , which determine the decrease in number densitieswith altitude, are proportional to Mi = T, where Mi are themolecular masses of the constituents, and T is the ambi-ent temperature. Hence, the heavy species N2 and O tend

Figure 3. Diurnal variations of local temperature T ac-cording to the MSIS-90e model (level line units: 100 K;conditions: summer solstice, 780 km altitude, mean ac-tivities).

Figure 4. Diurnal variations of total density accord-ing to the MSIS-90e model (level line units: 10 , 15 kg

=

m3;conditions: summer solstice, 780 km altitude, mean activ-ities).

to dominate in the lower and middle thermosphere, whilethe lighter species He and H prevail in the upper thermo-sphere and in the exosphere. The altitude region in which

N2, O, He, and finally H dominates is shifted downwardswith deceasing temperature (e.g. decreasing activity lev-els).

Apart from changes with altitude, thermospheric temper-atures, concentrations, total densities, and derived quan-tities (e.g. free mean path lengths) are known to varywith local solar time tlst, geographic longitude , geode-tic latitude , day of the year td, mean solar flux F10 7(averaged over 81 days = 3 solar rotations), actual solarflux F10 7, and current geomagnetic activity Ap. The di-urnal (day/night) variation profiles of T, , nO, and nHeare shown in Figures 3 to 6 for at the orbit altitude ofERS-1 and 2. The temperature peak and the peak of thedominant atomic oxygen concentration closely follow thesub-solar point (which is at tlst = 12 hrs) with a delay of3 to 4 hours. The concentration peak of the light weightspecies helium, however, is closer to the anti-solar point,as a consequence of thermal diffusion. The so-called he-lium bulge on the winter hemisphere is clearly visible.

The thermosphere described by the MSISe-90 model


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Figure 5. Diurnal variations of atomic oxygen concentra-tions according to the MSIS-90e model (level line units:

10+ 11 1=

m3; conditions: summer solstice, 780 km alti-tude, mean activities).

Figure 6. Diurnal variations of atomic helium concentra-tions according to the MSIS-90e model (level line units:

10+ 11 1=

m3; conditions: summer solstice, 780 km alti-tude, mean activities).

(Hedin (1987) and Hedin (1991a)) is based on the as-sumption of mass transport by diffusion processes in equi-librium conditions. According to Hedin (1991b), how-ever, there are also thermospheric winds, with directions

almost exclusively in the horizontal plane, and with ve-locities reaching 500 m/s and more. These wind pat-terns are closely linked with the day/night terminator,with largest magnitudes in polar regions towards the nighthemisphere. Due to the increase of viscosity with alti-tude, the magnitude of these winds reduces towards thethermopause.

3.2. Aerodynamic Coefficients of a Satellite

For a spacecraft which moves through the upper atmo-

sphere with a relative velocity of U and a mean cross-section ofAre f, the encountered aerodynamic force ~Fa canbe computed by a summation of individual contributionsfrom all of the 7 atmospheric constituents (i

=

1 to 7 for

N2, O, He, H, O2, Ar, and N), with partial densities i.

~Fa =7

i= 1

1

2iAre fU

2

~Cai (1)

In this equation, all aerodynamic force characteristics are

concentrated in~

Cai , which is a function of the spacecraftgeometry, its surface properties, and its attitude relativeto the air flow. If one assumes that a fraction d (where0

d 1) of the incoming molecules is diffusely re-

flected according to a Lambert distribution, and that thecomplement is specularly reflected, then the local contri-

bution to ~Cai can be described analytically.

~Cai =1

Are f

Z

A

,

d~ ci; d ~ r + 1 , d ~ ci ; s ~ r

dA (2)

The vector quantities~

ci; d Si ; Tw = T and ~ ci ; s Si are thediffuse and specular reflection coefficients for a surfaceelement dA at the position ~ ron that part of the spacecraft

surface A which is exposed to the airflow.~

ci;

d has compo-nents normal and tangential to the surface element (along~

n and ~ t), where the normal (= pressure) contribution isalso depending on the wall temperature Tw. ~ ci; s only con-tributes to the local normal force.

~

ci; d =

1p

Sni

S2i+

1

2

r

Tw

T

Sni

S2i

!

~

n

+

1p

Sti Sni

S2i

!

~ t (3)

~

ci;

s =2

p

Sni

S2i~

n (4)

The auxiliary quantities

Sni and Sni are defined as

Sni = Sni exp,

,S2ni

+

p

S2ni +1

2

1+

erf

Sni

Sni = exp,

,S2ni

+

p

Sni 1 + erf Sni

Here, Si =p

MiU2 = 2kT is the molecular speed ratio

(free stream velocity U divided by the most probablethermal velocity of the i-th atmospheric species), Mi is themolar mass of a contributing gas species, T is the ambi-ent temperature, and k is the Boltzmann constant. Sni andSti are the normal and tangential components of Si for

a particular element of the exposed spacecraft surface A,over which the forces are integrated.

The ANGARA program (Fritsche et al. (1998)) uses theoutlined method to compute aerodynamic coefficients offorce and torque in an analytical way (so-called IntegralMethod). Alternatively, a numerical Monte-Carlo Test-Particle (MCTP) Method is implemented. Both methodsconsider geometric shadowing in their analysis, but theMCTP also considers multiple reflections, and an alterna-tive, more detailed surface interaction model according toSchaaf and Chambre.

4. RADIATION PRESSURE PERTURBATIONS

Radiation pressure is the dominant non-gravitational per-turbation at satellite altitudes above 500 km to 600 km


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(depending on solar activity). In order of significance,there are four major radiation sources: the direct radia-tion from the Sun, the albedo re-radiation reflected fromthe illuminated Earth hemisphere, the more uniformly re-emitted Earth IR radiation, and directed IR radiation emit-ted from the spacecraft (denoted as thermal thrust). Thesources of these contributions, and the computation oftheir perturbation forces will be addressed hereafter.

Figure 7. Planetary albedo map composed from NOAA-9data (based on observations in March 1985).

Figure 8. Earth IR re-radiation map composed fromERBE data (based on observations in March 1985).

4.1. Models of Radiation Sources

The solar energy flux has a spectral distribution whichclosely follows Plancks law for a black body radiator of amean temperature of T

=

T

=

5;

785 K. The peak of theenergy flux (described by Wiens formula) is reached inthe visible light. Integrating the Suns radiation spectrumover all frequencies, leads to an energy flux of e

r

=

e

on the Suns surface according to the Stefan-Boltzmannlaw

e = T4 (5)

where =

5:

67

10 , 8W m , 2K, 4).

The initial energy flux of e

r

=

e

=

6:

4

107W=

m2

is reduced to a mean value of e

r

=

1370W=

m2 at thedistance r

of the Earth, with annual variations of

3:

3%due to the eccentricity of the Earth orbit. About 33 % ofthe incident energy is reflected in the visible light spec-trum from the top of the cloud cover, from the atmo-sphere, and from the Earth surface. Figure 7 shows the

resulting planetary albedo distribution of the Earth forMarch 1985 (NOAA-9 data, NASA (1996)). The largestalbedos are observed in polar regions, due to the snow andice cover, while equatorial albedos are generally smaller.Re-radiation from the Earth albedo is only effective forthose parts of the surface and atmosphere which are lo-cated in the Sun-illuminated hemisphere.

Another secondary source of re-radiation is the time-delayed and frequency shifted re-emission in the IR wave-lengths of 67 % of the solar energy flux which is ab-sorbed by the Earth atmosphere, the continents, and theoceans. Figure 8 shows the global distribution of the ter-restrial IR energy flux as observed by the ERBE satel-lite (Earth Radiation Budget Experiment, NASA (1996))in March 1985. This re-radiation is almost indepen-dent of the illumination conditions, with a mean level ofe

=

234W=

m2, corresponding to a black body radiatorwith a temperature of T

=

253 K.

When viewed from a mean heliocentric Earth distance of1

:

5

106 km (1 astronomical unit), the Sun covers a solidangle of about 0

:

5 . This light source of finite extensioncauses a core shadow (umbra), and a semi shadow re-gion (pen-umbra) on satellite orbits which pass througheclipse. The edges of the core shadow region also receiveradiation due to atmospheric refraction, which can causedeflections of solar rays by up to 1

:

3 , increasing the geo-metrically defined semi-shadow region (with a cone angleof about 0

:

5 ) by up to a factor 7.

Satellites do not only receive and reflect direct solar ra-diation, but they also heat up due to external and internalenergy inputs. A non-uniform re-emission of this energyover the spacecraft surface (e.g. due to shadowing) cancause a non-zero force (Antreasian & Rosborough (1992)and Powell & Gaposhkin (1988)). Such an effect hasbeen observed for the GPS satellites due to thermal en-ergy emission from their radiator panels.

4.2. Radiation Coefficients of a Satellite

The radiation energyflux e which is intercepted by a satel-lite at a certain frequency corresponds to a photon im-pingement rate of

np = e= h (6)

where h=

6:

625

10, 34Js is Plancks constant. Of the

incoming photons, a fraction is absorbed , a fraction sis specularly reflected, and a fraction d is diffusely re-flected according to a Lambert distribution (cosine law).If one assumes a non-transparent surface, the energy con-servation can be expressed as

+

s + d = 1 (7)

All of these coefficients normally depend on the surfacematerial, its temperature, and the wavelength and inci-dent angle of the photons. Together with the emissivity, which describes the IR re-radiation properties, this setof parameters completely defines the photon-surface in-teraction. The ratio of

=

can vary over a wide range(e.g. 0.98/0.98 for black paint, 0.79/0.81 for solar cells,0.45/0.80 for aluminized Kapton, and 0.07/0.76 for silverTeflon).

If one assumes radiation point sources at an infinite dis-tance from the satellite, then the resulting radiation force


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~Fr can be described as

~Fr =4

n= 1

en

cAre f ~Crn (8)

where n=

1 to 4 represents direct, Earth albedo, EarthIR, and thermal radiation, en is the energy flux at the re-spective wavelength, c is the speed of light, and Are f isa spacecraft reference cross-section. The radiation force

coefficients ~Crn for the different radiation sources can bedetermined by integration over the illuminated surface

An of the satellite (which is different for each radiationsource).

~Crn =1

Are f

Z

An ~

crn ~ r dA (9)

The local contributions~

crn ~ r from a surface element dAat its location

~

rare defined by the unit vectors of the inci-dent direction

~

un relative to the surface normal ~ n, and by

the frequency dependent reflection properties (where nwas expressed in terms ofsn and dn via equation 7).

~

crn ~ r = ~ un~ n 1 , sn ~ un +2

3

cnT

4w ~ n

+ ~

un~ n

2sn ~ un~ n ,2

3dn

~

n (10)

For software implementation purposes, the radiation co-

efficients ~Crn are computed separately for each of the fourmajor sources. For each source, a single, most proba-ble frequency is adopted, and non-uniform radiation in-put distributions (i.e. Earth IR and Earth albedo) are

discretised into a finite number of planar sources, witheach of them emitting parallel beams towards the space-craft, while observing spacecraft-to-ground visibility con-straints (and ground illumination conditions in case ofalbedo). For direct solar radiation and spacecraft thermalradiation the umbra and pen-umbra eclipse conditions areconsidered, including refraction and absorption of visiblelight by the Earth atmosphere.

Similar to the aerodynamic analysis, the ANGARA pro-gram (Fritsche et al. (1998)) determines spacecraft spe-cific coefficients of force and torque for each radiation ef-fect by using either an analytical Integral Method, or a nu-merical Monte-Carlo Test-Particle Method (MCTP). Asbefore, both methods consider geometric shadowing, but

the MCTP method also allows multiple reflections (whichcan be important for torques).

5. DISCUSSION OF RESULTS

Non-gravitational perturbation forces (also denoted assurface forces) can play an important role in satellite op-erations. For the dominant class of low-Earth orbiting ob-

jects (which account for about 85% of all tracked objects)it is mainly the airdrag which due to its energy dissipationaffects orbit decay rates, and hence orbit maintenance fre-quencies in case of controlled satellites (e.g. ERS-1 & 2).At higher orbit altitudes, and in case of highly eccentricorbits (e.g. ISO), solar radiation pressure becomes thedominant surface force which affects orbit and attitudemaintenance cycles.

Figure 9. Geometric models of the ERS-1 satellite (top),and of the ENVISAT satellite (bottom), as generated andused by the ANGARA program (displayed in differentscales).

5.1. Effects of Surface Forces

Abandoned LEO objects ultimately re-enter into thedenser layers of the Earth atmosphere, where they mostlyburn up. Statistically, one object of a radar cross-section

(RCS) larger than 1 m2 decays each week. Occasionally,such uncontrolled re-entries involve spacecraft with largemasses (e.g. 75 t for Skylab-1 and 40 t for Salyut-7), orhazardous payloads (e.g. nuclear reactors on Kosmos-954and 1402), parts of which can reach the ground. Duringthe re-entry prediction campaigns for these high risk ob-

jects, the strong effects of solar and geomagnetic activityfluctuations could be observed (see Figure 1):

11-Jul-1979: Skylab-1 re-enters over the IndianOcean and Australia at the start of the maximum ofsolar cycle 21. Intensive studies of the aerodynamicbehavior allow a forward shift of the impact footprintby inducing a tumble 8 hours before the entry.

07-Feb-1983: The detached reactor of Kosmos-1402re-enters over the South Atlantic, 15 minutes beforereaching Europe. The high level of airdrag at the endof the maximum of solar cycle 21 is further enhancedby a geomagnetic storm of magnitude Ap = 150 onFeb. 5. The remaining lifetime was hereby shortenedby 30%.

07-Feb-1991: Salyut-7 (with Kosmos-1686 at-tached) re-enters over South America. The orbit life-time was strongly reduced by the peak of solar cycle22 (F10 7 = 369 on Jan. 30).


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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Time [in orbit periods]

1.0e-05

1.5e-05

2.0e-05

2.5e-05

3.0e-05

3.5e-05

4.0e-05

4.5e-05

5.0e-05Force Magnitude [N]

0.0e+00

2.0e+11

4.0e+11

6.0e+11

8.0e+11

1.0e+12

1.2e+12

1.4e+12

1.6e+12

1.8e+12

2.0e+12

2.2e+12Concentration [1/m**3]

Specular Diffuse O He

Figure 10. Aerodynamic force magnitude for ENVISATover 2 orbits, for specular and diffuse reflection laws, asa function of the local atmosphere composition.

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00


1.0e-05

2.0e-05

3.0e-05

4.0e-05

5.0e-05

6.0e-05


Aero,W Aero,S IR,W IR,S

Figure 11. Aerodynamic and Earth IR force magnitude

for ERS-1 over 2 orbits, for winter and summer solsticeconditions.

In routine operations the orbit decay of LEO satellites isperiodically adjusted to maintain the orbit altitude. Thefrequency of necessary orbit manoeuvres is dependingon the allowed offsets in altitude and groundtrack pat-tern (for ERS-1 & 2 the groundtrack must be maintainedto within

1 km), and it is also determined by the re-

quired periods of undisturbed payload operation. Sur-face forces also play an important role in precise orbitdetermination of satellite programs such as GPS (Pow-ell & Gaposhkin (1988)), LAGEOS (Rubincam (1982)),TOPEX-POSEIDON (Antreasian & Rosborough (1992)),

and ESAs ERS-1, ERS-2 and ENVISAT spacecraft,where in some cases radial orbit accuracies of a few cen-timeters are required.

5.2. Aerodynamic Forces on ERS-1 and ENVISAT

ESAs operational ERS-1 and ERS-2 satellites, and theplanned ENVISAT mission are using near-circular orbits(e

=

0:

001) of retro-grade inclinations (i=

98:

52 ) at al-titudes near 780 km. For the selected altitude and inclina-tion the orbit planes are rotating Sun-synchronously at arateof 0

:

986 =

day under the influence of the Earth oblate-ness. The descending orbit nodes are thus kept at 10:30mean local solar time. The satellites are maintained in anattitude of local normal pointing along the geodetic verti-cal (roll and pitch control), with a yaw steering such thatthe cross-track radar beams are pointing along the zero

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00


0.0e+00

2.0e-05

4.0e-05

6.0e-05

8.0e-05

1.0e-04

1.2e-04

1.4e-04

1.6e-04

1.8e-04

2.0e-04

2.2e-04

2.4e-04

2.6e-04

2.8e-04


Direct,W Direct,S Self,W Self,S Albedo,W Albedo,S

Figure 12. Direct radiation and thermal radiation forcemagnitude for ERS-1 over 2 orbits, for winter andsummersolstice conditions.

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00


0.0e+00

1.0e-05

2.0e-05

3.0e-05

4.0e-05

5.0e-05

6.0e-05


Aero,Sea

Albedo,Sea

IR,Sea

Aero,Land

Albedo,Land

IR,Land

Figure 13. Aerodynamic, Earth albedo, and Earth IR

force magnitude for ERS-1 over one orbit, for a sea andland groundtrack.

Doppler line. The ERS-1 & 2 altitude decay can rangefrom 0.5 m/d to 10 m/d between very low and extremelyhigh solar activity.

The ANGARA program was used to perform a full aero-dynamic analysis (forces and torques) for ERS-1 andENVISAT. The corresponding surface geometry models,which consist of 11,376 and 7,644 panels, respectively,are shown in Figure 9. For the analysis, two orbits werepredicted with the relevant attitude steering profiles su-perimposed. In Figure 10 the importance of gas-surfaceinteraction processes is highlighted for ENVISAT. Themarked curves show the aerodynamic force magnitude(left axis) in case of perfectly diffuse (d = 1, filled di-amond), and in case of perfectly specular reflection (d =0, hollow diamond). The force history is correlated withthe changing dominance of the prevailing atmosphericconstituents He and O (right axis), with stronger specularreflections from atomic oxygen, and a tendency towardsdiffuse re-emission from helium. This is related to thespecies dependent molecular speed ratio Si

p

Mi (withMi = 4 for He and 16 for O) and its effect on equations 3and 4. In Figure 11 the sensitivity of aerodynamic pertur-bations with seasonal changes of the upper atmosphere isanalysed for ERS-1 at the time of summer solstice (hol-low circle) and winter solstice (filled circle). The signa-ture of the diurnal density bulge, which moves with thesub-solar point, is clearly visible, generating seasonal dif-ferences in the aerodynamic drag level of up to 30% at thesame orbit locations (see also Figure 4).


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5.3. Radiation Forces on ERS-1

The altitude hr

a beyond which radiation perturbationsare dominating over aerodynamic forces can be deter-mined from equations 1 and 8 to be

hr a = ho , Ho ln

2e r

coU2

(11)

where ho is a reference altitude (e.g. 400 km) at which themean air density o and the corresponding density scaleheight Ho are defined. Between low and high extremesof solar activity the mean air density at 400 km increases

by a factor of about 10, from 10 , 12 to 10, 11 kg/m3, lead-ing to low and high extremes of hr

a between 500 and600 km.

In Figures 11 and 12 the radiation perturbation magni-tudes due to direct solar radiation, Earth IR and Earthalbedo re-radiation, and thermal radiation from the space-

craft are plotted over two orbits of ERS-1 at summer sol-stice and winter solstice conditions. A comparison withconcurrent aerodynamic forces in Figure 11 shows that di-rect radiation pressure generates perturbation amplitudeswhich are more than 4 times larger than those from air-drag at mean solar activity levels. The seasonal differ-ences in the radiation force magnitudes are less than 10%and can be attributed to annual changes in the Sun-Earthdistance (for direct radiation), and to annual changes inalbedo and IR re-radiation characteristics of the Earth (forindirect radiation). The thermal radiation emission by thespacecraft itself is almost unaffected by seasonal changes,except for shifts of the 33 minutes eclipse mask near theascending node. This Earth shadow transit also affects thedirect solar radiation with an almost instantaneous on/offswitching, which is only damped by pen-umbra and at-mospheric refraction effects. Likewise, the albedo re-radiation is deactivated while the night hemisphere of theEarth comes into the field of view of the satellite (whichis a more gradual process).

The Earth albedo and Earth IR perturbations are relatedto underlying re-radiation maps in a geographic longi-tude/latitude coordinate system (see Figure 7 and 8). Theeffect of the longitude position of the ascending node ofan ERS-1 orbit on the albedo and IR force profiles is anal-ysed in Figure 13 for a groundtrack with a large land andsea coverage, respectively (airdrag profiles are includedas reference). The maximum difference in albedo and IRperturbation force amplitudes due to groundtrack cover-age is on the order of 10%.

6. CONCLUSIONS

Good models of non-gravitational perturbations are a pre-requisite in precise orbit determination applications, pre-dominantly for navigational constellations (e.g. GPS andGLONASS) and satellite programs with geodetic missionobjectives (e.g. LAGEOS, ERS-1 & 2, and TOPEX-POSEIDON). The dominant surface forces which needto be modeled are due to direct radiation (from the Sun),indirect radiation (from Earth albedo and IR), and space-craft emitted thermal radiation. Aerodynamic perturba-tions are only important for LEO satellites and start to bedominating below 500 to 600 km (depending on solar ac-tivity).

The resulting forces and torques due to photon andmolecule impingement on the spacecraft surface (hencethe term surface forces) can be modeled by meansof semi-analytical integral methods or numerical Monte-Carlo test-particle methods for spacecraft geometries andsurface models of arbitrarily high resolution and fidelity.Computerised non-gravitational force models (e.g. ESAsANGARA program, Fritsche et al. (1998)), are mainlylimited by intrinsic inaccuracies of the environment de-scriptions (e.g. atmosphere, Earth albedo and IR distribu-tion), and by the limited knowledge on the surface inter-action parameters of photons and molecules as a functionof the surface material and incident particle properties.Supporting experimental data in this area will be requiredto further improve theoretical models of non-gravitationaleffects.

REFERENCES

Antreasian, P.G., and Rosborough, G.W., Prediction of

Radiant Energy Forces on the TOPEX/POSEIDONSpacecraft, AIAA Journal of Spacecraft and Rockets,vol. 29, no.1, pp. 8190, Jan/Feb, 1992

Fritsche, B., Ivanov, M., Kashkovsky, A., Koppenwall-ner, G., Kudrayavtsev, A., and Zhukova, G., RadiationPressure Forces on Complex Spacecraft, final report,ESA contract no. 11908/96/D/IM, 1998

Hedin, A.E., MSIS-86 Thermospheric Model, Journal ofGeophysical Research, vol. 92, no. A5, pp. 46494662, 1987

Hedin, A.E., Extension of the MSIS Thermosphere Modelinto the Lower Atmosphere, Journal of Geophys. Res.,vol. 96, no. A2, pp. 11591172, 1991

Hedin, A.E., Biondi, M.A., Burnside, R.G., Hernandez,G., Johnson, R.M., Killeen, T.L., Mazaudier, C., Meri-wether, J.W., Salah, J.E., Sica, R.J., Smith, R.W.,Spencer, N.W., Wickwar, V.B., and Virdi, T.S., Re-vised Model of the Thermosphere Winds Using Satel-lite and Ground-based Observations, Journal of Geo-phys. Res., vol. 96, no. A5, pp. 76577688, 1991

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Klinkrad, H., On the Use of Atmosphere Models in Re-Entry Predictions, ESA SP-392, pp. 287298, 1996

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Marcos, F.A., Baker, C.R., Bass, J.N., Killeen, T.L., andRoble, R.G., Satellite Drag Models: Current Statusand Prospects, AAS/AIAA Astrodynamics SpecialistConference, Victoria/B.C., Aug. 1619, 1993

NASA Langley Research Center, ERBE Monthly Scan-ner Data Products, CD ROM distributed by the NASADistributed Active Archive Center, 1996

Powell, G.E., and Gaposhkin, E.M., Modeling of Non-Gravitational Effects on GPS Satellites, AIAA paperno. 88-4291-CP, 1988

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orbit and attitude perturbations due to

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