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Acta Astronautica

Acta Astronautica 88 (2013) 1–7

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Antenna radiation effects on the orbits of GPSand INTELSAT satellites

A. Heilmann a, L.D.D. Ferreira a, C.A. Dartora b,n, K.Z. Nobrega c

a Departamento de Geomatica, Universidade Federal do Parana, Centro Politecnico, Jd. das Americas, C.P. 19011, 81531-990 Curitiba-PR,

Brazilb Departamento de Engenharia Eletrica, Universidade Federal do Parana, Centro Politecnico, Jd. das Americas, C.P. 19011,

81531-990 Curitiba-PR, Brazilc Departamento de Eletro-Eletronica, Instituto Federal de Educ. Tecnologica do Maranh~ao, Av. Getulio Vargas, 04, Monte Castelo,

65030-005, S ~ao Luiz-MA, Brazil

a r t i c l e i n f o

Article history:

Received 8 May 2012

Received in revised form

13 February 2013

Accepted 15 February 2013Available online 13 March 2013

Keywords:

Artificial satellite

Perturbation

Antenna radiation

GPS

INTELSAT

65/$ - see front matter & 2013 IAA. Publish

x.doi.org/10.1016/j.actaastro.2013.02.009

esponding author. Tel.: þ55 41 33 63 46 9

5 41 33 61 32 28.

ail addresses: armando@facear.edu.br (A. He

ra@eletrica.ufpr.br (C.A. Dartora).

a b s t r a c t

A large number of disturbances add to the main force exerted by Earth’s gravitational

field and affect the actual orbital trajectory of artificial satellites. They possess antennas

with specific purposes, such as telecommunication systems operating at specific ranges

of frequencies and radiated power. For instance, the antennas used in the GPS and

INTELSAT satellites are quadrifilar helix and parabolic reflectors respectively. The

radiation emitted by the antennas produces a radiation reaction force on the satellite

making its orbital elements deviate from their expected values. Using a mathematical

model for the radiation reaction force caused by the antenna, derived from the

electromagnetic theory and the energy-momentum conservation law, the perturbation

effects on the orbits of the GPS and INTELSAT satellites were studied. The numerical

integrator used to solve the satellite equations of motion is based on the Runge–Kutta

method of fourth and fifth orders. The theoretical model of antenna radiation reaction

takes into account the satellite mass, antenna radiated power and maximum gain of the

antenna.

& 2013 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Artificial satellites are subjected to a large number ofperturbations which affect its Keplerian orbit. Among themost important perturbation effects we must mention, inorder of magnitude, the geopotential perturbation, theMoon and Sun gravitational fields and direct solar radia-tion pressure, indirect radiation pressure, relativisticeffects and atmospheric drag [1–7]. The orbital elementsare the semi-major axis a, the eccentricity e, inclination i,right ascension of the ascending node O, argument ofperigee o and mean anomaly M. The effects of antennaradiation reaction are rarely taken into account in the

ed by Elsevier Ltd. All right

1;

ilmann),

orbit propagation but according to Maral [8] the forceexerted by the antenna radiation on the satellite is givenby F ¼�EIRP=c, where EIRP is the effective isotropicradiated power measured in watt and c is the speed oflight in vacuum. He also claims that the perturbation ismost significant for highly directive antennas, such asparabolic reflectors. A complete theoretical model for theacceleration on a satellite caused by radiation reactionoriginated from antenna radiation based on electromag-netic principles and conservation of energy-momentumlaws was derived by Heilmann et al. [9]. A rough estimatehas shown that for typical communication satellites theperturbation on the satellite orbit due to the powerradiated by the transmitting antenna amounts to a non-negligible effect in the order of 10�9 m=s2.

The satellites are usually classified according to thealtitude h, measured from the Earth surface, into fourcategories: (i) Low Earth Orbit (LEO) satellites, for which

s reserved.

Fig. 1. Local reference frame of the antenna with the z-axis pointing to

the center of the Earth. The angles ðy,jÞ are defined according to the

standard spherical coordinate system. y¼ arctanðz=rÞ is the angle

between the position vector r¼ ðx,y,zÞ and the z-axis and the angle

j¼ arctanðy=xÞ is measured from the x-axis and is contained in the

x�y plane.

A. Heilmann et al. / Acta Astronautica 88 (2013) 1–72

hr2000 km; Medium Earth Orbit (MEO) satellites, withorbits between 2000 km to approximately 36,000 km; (iii)the geosynchronous orbit (GEO) satellites have orbitalperiod of 24 h and therefore must be located at an altitudeh� 36,000 km, and (iv) satellites above 36,000 km, knownas High Earth Orbit (HEO). Telecommunication satellitesare distributed over a wide range of altitude, from LEO toHEO (geostationary satellites, for example) [10]. Thepurpose of the present contribution is to study in detailsthe effects of antenna radiation reaction on the orbitalelements of the Global Position System (GPS) satelliteclass BLOCK IIR-11 in a MEO orbit, whose communicationsystem operates with quadrifilar helix antenna arraysand, as a second case, the communication satellite INTEL-SAT 907 in GEO orbit, which uses parabolic reflectors inits communication system. To do that, we will solvenumerically the theoretical model proposed in Ref. [9],which takes into account the satellite mass, antennaradiated power and maximum gain of the antenna.

The content of this paper can be described as follows:in the next section we introduce the basic formulae ofantenna theory used in this paper and briefly review thetheoretical model for the radiation reaction force pro-duced by the transmitting antenna on a satellite in itsinertial system. In Section 3 the main characteristics ofthe satellites GPS and INTELSAT, such as the orbitalelements, mass and antenna parameters, are presented.In Section 4 we present the numerical results and discus-sion, and finally, in the last section a few conclusions andremarks are added.

2. Theoretical framework

The artificial satellites antennas are designed accord-ing to specific purposes of the satellites. Quadrifilar helixantennas are commonly used in GPS satellites and operatebelow 4 GHz using circularly polarized waves and gainaround 14 dBi, in order to cover large areas. By far themost common satellite antennas are the parabolic reflec-tors, which have high gain (14–50 dBi) operating withradiated powers in the range 10–500 W.

For the sake of completeness we will introduce thebasic formulae of antenna theory used in this paper. Thetheory of electromagnetic radiation follows directly fromMaxwell’s equations. We suggest to the interested readerRefs. [11–13] for a more detailed exposition of electro-magnetic radiation and antenna theory.

In Fig. 1 we illustrate the local reference frame of theantenna used in a typical satellite communication system.For the sake of simplicity we choose the z-axis in the localreference frame of the antenna in such a way that itpoints to the center of the Earth. The averaged radiatedpower density, i.e., the component of the averaged Poynt-ing vector along the radial direction is given by

Sradðr,y,jÞ ¼ 12 ReðErad �Hn

radÞ ¼ Sradðr,y,jÞar, ð1Þ

where Erad and Hrad are the electric and magnetic fields ofthe radiated waves, Sradðr,y,jÞ is the modulus of thePoynting vector measured in W/m2, r¼ ðr,y,jÞ are thefield coordinates (see Fig. 1), ar is the radial unit vector inthe antenna coordinate system, with the antenna located

at the center of such system, Reð:Þ denotes the real part ofthe expression between brackets and n denotes complexconjugation. We point out that it is usual in satellitecommunication systems the continuous wave (CW)operation mode with operating angular frequency o[rad/s], for which the radiated Poynting vector in thetime-domain is composed of a mean DC value, which isequivalently given by (1) and a rapid time-varying func-tion of frequency 2o. The net effect of such componentaverages to zero, since the satellite dynamics cannotfollow such a rapid variation. Therefore the only netcontribution to the satellite dynamics must come fromthe averaged Poynting vector. It is also worth mentioningthat besides the antenna radiated fields, depending on thedistance r from the antenna as 1/r, the antenna producesthe so-called near fields varying with the distance r as1=rn, nZ2. The near fields only represent a stored energyaround the satellite, but do not correspond to radiatedpower and do not transport momentum away from thesatellite, and therefore, do not contribute to the radiationreaction force, to be given later in this paper.

The most important parameter of the antenna in ouranalysis is the antenna gain Gðy,jÞ, defined below:

Gðy,jÞ ¼ PEIRPðy,jÞPrad

¼4pr2Sradðr,y,jÞ

Prad, ð2Þ

where PEIRP is the effective isotropic radiated power, givenby PEIRP ¼ 4pr2Sradðr,y,jÞ, which corresponds to the effec-tive power that must be delivered to an isotropic antennato produce the same radiated power density Sradðr,y,jÞgenerated by the antenna under consideration at thecoordinate ðr,y,jÞ, and the total radiated power is givensimply by

Prad ¼

ZO

r2Srad dO, ð3Þ

being dO¼ sin y dy dj the infinitesimal solid angle ele-ment and

RO ¼

R py ¼ 0

R 2pj ¼ 0. This way, we can interpret the

antenna gain as the ratio between the effective powerwhich must be delivered to an isotropic antenna and thetotal power radiated by the antenna under consideration

Fig. 2. Radiation pattern of a quadrifilar helix, resembling a cardiod.

Fig. 3. Typical radiation pattern of a parabolic reflector antenna.

A. Heilmann et al. / Acta Astronautica 88 (2013) 1–7 3

in order to produce the same power density at thecoordinate ðr,y,jÞ. The antenna gain function Gðy,jÞdepends only on the angular coordinates ðy,jÞ, since,based on the requirement of total radiated power con-servation, the radiated power density must satisfy aninverse square law, ðSradp1=r2Þ. From Eq. (2) we get

Sradðr,y,jÞ ¼ Gðy,jÞPrad

4pr2: ð4Þ

Notice that the antenna gain is a non-dimensionalparameter which represents how the antenna distributesthe radiated power over the space. It is usual to build athree dimensional (3D) surface plot of Gðy,jÞ as a functionof coordinates ðy,jÞ in a spherical system, known as theantenna gain pattern. The resulting surface must beclosed, with norm

ROGðy,jÞ dO¼ 4p. A normalized gain

function can be conveniently defined as follows:

gðy,jÞ ¼ Gðy,jÞG0

, ð5Þ

where

G0 ¼max½Gðy,jÞ� ¼ 4pROgðy,jÞ dO

ð6Þ

is the maximum antenna gain.The normalized gain of an ideal quadrifilar helix can be

modeled by a cardiod

gðy,jÞ ¼ 12ð1þcos yÞ: ð7Þ

Fig. 2 illustrates the radiation pattern of a quadrifilarhelix. Notice that the maximum gain occurs for y¼ 0, butthe radiation of a quadrifilar helix presents no depen-dence on the azimuth j and the gain is low.

By contrast, a parabolic dish is a high gain apertureantenna whose normalized gain can be assumed for thesake of simplicity as

gðy,jÞ ¼ AJ1ðka sin yÞ

ka sin y, ð8Þ

where J1ð:Þ is the first-order Bessel function of the firstkind, k¼ 2p=l is the wavenumber associated with thewavelength l used in the communication system, a is theradius of the circular dish and A is a normalizationconstant. For highly directive parabolic antenna, theabove expression can be approximated by

gðy,jÞ ¼ 1, 0ryra0,

gðy,jÞ ¼ 0, a0oy, ð9Þ

where a0 is the beam angle, defined as angular aperture ofthe radiated beam for which the radiated power densitydecays to half maximum. Fig. 3 illustrates the radiationpattern of a parabolic dish.

2.1. Theoretical model for antenna radiation reaction

It follows from classical electrodynamics and the lawsof energy-momentum conservation that electromagneticwaves carry energy and linear momentum as well, suchthat the force acting on the satellite produced by theradiation reaction is given by [9]

Fsat ¼�Prad

4pc

ZO

Gðy,jÞarðy,jÞ dO, ð10Þ

or in terms of an acceleration asat

asat ¼�Prad

4pmc

ZO

Gðy,jÞarðy,jÞ dO, ð11Þ

where Prad is measured in watt, m in kg is the satellitemass, c� 3� 108 m=s, Gðy,jÞ is a non-dimensional quan-tity defined above and asat is measured in m=s2. Now wedecompose the radial unit vector ar referred to theantenna coordinate system in its cartesian components

ar ¼ sin y cos jaxþsin y sin jayþcos yaz, ð12Þ

where ðr,y,jÞ are measured in antenna coordinate systemwith the antenna at the origin, and write the antenna gainas Gðy,jÞ ¼ G0gðy,jÞ, being G0 the maximum gain, toobtain our final formula

asat ¼�G0Prad

4pmc

Z p

y ¼ 0

Z 2p

j ¼ 0gðy,jÞ½sin y cos jax

sin y sin jayþcos yaz� sin y dy dj: ð13Þ

A. Heilmann et al. / Acta Astronautica 88 (2013) 1–74

The above expression will be used to describe the pertur-bation effects caused by antenna radiation reaction on thesatellite equations of motion, determined from Newton’ssecond law

drE

dt¼ vE, ð14Þ

dvE

dt¼�

GMrE

r3E

þasat , ð15Þ

where rE with components ðxE, yE, zEÞ is the satelliteposition vector in the conventional inertial system (CIS)where the zE-direction is along the rotation axis, pointingto the North pole, vE with components ðvxE,vyE,vzEÞ is thesatellite velocity vector, rE ¼ 9rE9 is the instantaneousdistance of the satellite from the Earth center,GM¼ 3:9860047� 1014 m3=s2 is the gravitational para-meter of the Earth. The quantitative prediction of theperturbation effect on the orbital elements is made in twosteps: (i) the orbit is determined solely with the gravita-tional acceleration acting on the satellite and (ii) theadditional disturbances are included (in our case asat isconsidered) and the orbit is recalculated. The differencebetween the non-perturbed and the perturbed orbitsgives the effect of the perturbation. The system of differ-ential equations is solved numerically using the Runge–Kutta method of fourth and fifth orders.

2.2. Definition of orbital elements and deviation in the RNT

coordinate system

The state vector of an artificial satellite is given by itsposition and velocity, rE and vE, respectively, expressed incartesian coordinates relative to CIS. The correspondingorbital elements obtained from the state vector ðrE,vEÞ areillustrated in Fig. 4. For a comprehensive presentation ofthe orbital elements we suggest Refs. [2,6]. The semi-major axis a and the eccentricity e define the orbitaldimension and the shape of the satellite trajectory,respectively, the right ascension of the ascending nodeO, the inclination of the orbital plane i and the argumentof perigee o define, respectively, the position and orien-tation of the orbital plane in space. The parameter f isknown as the true anomaly, allowing one to localize the

Fig. 4. Representation of the Keplerian orbital elements for an artificial

satellite.

satellite in its orbit (see Fig. 4). The usual procedure is tosolve numerically equations of motion (14) and (15) forthe state vector ðrE,vEÞ and, after that, to obtain the orbitalelements at any instant of time.

The perturbation effects can be better visualized in theso-called RNT coordinate system, illustrated in Fig. 5. Inthis system, the radial component R points to the satellite,N is perpendicular to the orbital plane and the transversecomponent T is perpendicular to R and N is also along thevelocity vector [6,14]. The errors produced on the RNTcomponents of the satellite orbit due to perturbationswere calculated using standard definitions

DR¼1

rER½ðxEP�xERÞxERþðyEP�yERÞyERþðzEP�zERÞzER�, ð16Þ

DN¼1

G½ðxEP�xERÞGxþðyEP�yERÞGyþðzEP�zERÞGz�, ð17Þ

DT ¼1

H½ðxEP�xERÞHxþðyEP�yERÞHyþðzEP�zERÞHz�, ð18Þ

where rER ¼ ðxER,yER,zERÞ is the position vector of thereference orbit, obtained solely from the action of theEarth’s gravitational field on the satellite defining aKeplerian motion and rER ¼ 9rER9 is its modulus,rEP ¼ ðxEP ,yEP ,zEPÞ is the position vector of the satelliteincluding the perturbation effects which deviates thesatellite trajectory from its Keplerian motion,G¼ ðGx,Gy,GzÞ ¼ rER � vER is the angular momentum vectornormalized by the satellite mass, G¼ 9G9 is the modulusof the normalized angular momentum and H¼ ðHx,Hy,HzÞ

is obtained from the cross product between rER and G, i.e.,H¼ rER � G and H¼ 9H9.

3. Main parameters of the GPS and INTELSAT satellites

The GPS satellite is in a MEO orbit and its state vector,orbital elements and antenna characteristics are given inTable 1.

The communication satellite INTELSAT 907 is a GEOorbit and its state vector, orbital elements and antennacharacteristics are given in Table 2.

Fig. 5. Representation of the radial R, normal N and transverse Tcomponents of the satellite position perturbation.

Table 1State vector, orbital elements and relevant parameters of the GPS satellite at 01 h21 min23 s (UTC) in December

31st 2004.

Position (m) Velocity (m/s)

xE¼21337664.0000000 vxE ¼�69:6990000000000

yE¼11449299.0000000 vyE ¼ 2730:471000000000

zE¼10970074.0000000 vzE ¼�2742:30100000000

Semi-major axis (km) 26560.761

Eccentricity 0.033753

Inclination (1) 54.98430

Right ascension of the ascending node (1) 226.7274

Argument of perigee (1) 257.6704

Mean anomaly (1) 102.0299

Satellite mass (kg) 759

Antenna Quadrifilar helix

EIRP (W) 436.5

Table 2State vector, orbital elements and relevant parameters of the INTELSAT 907 satellite at 02 h24 min00 s (UTC) in November

29th 2010.

Position (m) Velocity (m/s)

xE ¼�42159069:0000000 vxE ¼ 43:3320000000000

yE ¼�581514:000000000 vyE ¼�3074:57575000000

zE¼5890.000000000000 vzE ¼ 1:88300000000000

Semi-major axis (km) 42166.203

Eccentricity 0.003095

Inclination (1) 0.0056

Right ascension of the ascending node (1) 331.0367

Argument of perigee (1) 284.2406

Mean anomaly (1) 181.1215

Satellite mass (kg) 1900

Antenna Parabolic reflectors

EIRP (W) 200,000

A. Heilmann et al. / Acta Astronautica 88 (2013) 1–7 5

In the next section the above parameters will be usedin order to predict the magnitude of the radiation antennaperturbation on the RNT coordinates of the satellites GPSand INTELSAT 907.

4. Results and discussion

First we will analyze the influence of the antennaradiation reaction on the GPS satellite using a quadrifilarhelix, whose radiation pattern is given by the cardiodfunction. Inserting expression (7) into (13) leads to thefollowing perturbation acceleration:

asat ¼�EIRP

3mcaz: ð19Þ

Notice that az is in the direction of the maximum radiatedpower by the antenna, in its reference frame. The antennamaximum radiation direction can strongly influence the neteffect on the satellite dynamics, but we will focus on themost common situation, i.e., communication and remotesensing satellites, for which the maximum radiated power isdirected to the Earth surface, which allows one to make theidentification az ¼�rE=rE. Considering the typical valuesgiven above in Table 1 we obtain 9asat9¼ 6:389�10�10 m=s2. To appreciate the perturbation effects in moredetail we use the RNT coordinate system, described above.

Variations of the position in the RNT coordinates arecalculated using Eqs. (16)–(18). The variation of the RNTcomponents of the position vector rE is shown in Fig. 6a. Thetransverse component is the most affected, as can be clearlyseen from Fig. 6a, where the deviation amounts to � 1:5 min 48 h, while the radial component deviates around 10 cm.The normal component is not affecting, as is expected forthe effect of a small radial perturbation. In Fig. 6b we showthe variation of the semi-major axis due to the antennaradiation effect. It can be seen that such effect produces avery small oscillation of the semi-major axis, related to thenon-perturbed semi-major axis, with amplitude around0.15 mm. The time interval for the analysis of orbitaldynamics was chosen to 48 h because within this timewindow the relevant effects can be observed already.

Following, we make the same comparisons for theINTELSAT 907 using a parabolic reflector antenna and is ina GEO orbit, which corresponds to a higher altitude withrespect to a BLOCK IIR-11. Inserting (9) into (13) yields

asat ¼�G0Prad

4mcsin2 a0az: ð20Þ

Parabolic reflector antennas are highly directive, thushaving small beam angle, such that sin2 a0 � a2

0 is a validapproximation. It follows from the antenna theory [12,13]that for aperture antennas the maximum gain G0 is

Fig. 6. Perturbation effects produced by antenna radiation on the GPS,

using the data given in Table 1: (a) radial, normal and transverse

component deviations and (b) semi-major axis variation.

Fig. 7. Perturbation effects produced by antenna radiation on the

INTELSAT 907, using the data given in Table 2: (a) radial, normal and

transverse component deviations and (b) semi-major axis variation.

A. Heilmann et al. / Acta Astronautica 88 (2013) 1–76

expressible in terms of the beam angle a0, through therelation written below:

G0 �10

a20

: ð21Þ

It is important to notice that a0 must be measured inradians. The above equation is in good agreement withexperimental values for a0r0:4 radians (or a0r231),corresponding to an antenna gain of at least 18 dBi.Typical gain of parabolic reflectors is in the range of20�60 dBi [13]. Taking into account Eqs. (21) and (20),we obtain

asat ��5Prad

2mcaz: ð22Þ

Considering the typical values given above in Table 2 weobtain 9asat9¼ 3:07� 10�9 m=s2. The variations of theposition in the RNT coordinates are shown in Fig. 7a.Again, the transverse component is the most affected, andthe deviation amounts to � 14 m in 48 h, which issignificant in the correct determination of the satelliteposition, while the radial component deviates around

1.2 m. In Fig. 7b the variation of the semi-major axisdue to the antenna radiation effect is shown and oscillateswith amplitude around 0.4 mm.

From the above examples, we observe that the antennaradiation can produce significant and observable position-ing deviations, which are more important in GPS systems.However, considering two satellites using the same anten-nas with the same amount of radiated power, it is obviousthat the satellite in the higher altitude will be the mostaffected by the perturbation caused by antenna radiation,since asat � EIRP=ðmcÞ and the gravitational effect are givenby an acceleration ag ¼ GM=r2. This way the relativestrength of the perturbation is in the order ofasat=ag ¼ C0r2, where C0 ¼ EIRP=ðGMmcÞ.

5. Conclusion

In summary, in this paper we used a model of radiationreaction effects derived from electromagnetic theory topredict the perturbation caused by antenna radiatedpower on the satellite Keplerian orbital elements. Tworelevant examples were considered: (i) the GPS satellite,

A. Heilmann et al. / Acta Astronautica 88 (2013) 1–7 7

located in a MEO orbit and possessing a moderate gainantenna, the quadrifilar helix; (ii) the INTELSAT 907satellite, used in communication applications, which isin a GEO orbit and uses a high gain parabolic reflectorantenna. For the GPS satellite the perturbation accelera-tion is in the range of 9asat9¼ 6:389� 10�10 m=s2 and forthe INTELSAT 907 it is given by 9asat9¼ 3:07� 10�9 m=s2.Finally, if one considers two satellites having the samemass and use the same antennas with the same amount ofradiated power, the satellite in the higher altitude will bemore affected by the antenna radiation reaction, since theantenna produces an acceleration asat � EIRP=ðmcÞ and thegravitational effect is given by an acceleration ag ¼ GM=r2.This way the relative strength of the perturbation is givenby asat=ag ¼ C0r2, where C0 ¼ EIRP=ðGMmcÞ.

Acknowledgments

C.A. Dartora would like to acknowledge CNPq(Conselho Nacional de Pesquisa e Desenvolvimento) forpartial financial support. A. Heilmann would like to thankAraucaria Educational Faculty (FACEAR) for partial finan-cial support and SIMEPAR for technical support. Theauthors also thank the anonymous referees for usefulsuggestions and comments.

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