Constellation Characteristics and Orbital Parameters

Constellation Characteristics and Orbital Parameters

of the Universe According to Copernicus, the Earth and other planets rotated around the Sun

in circular orbits This was the first significant advancement in astronomy since the drian astronomer Ptolemy in his publication Almagest put forward the geocentric universesometime during the period 100–170 AD Ptolemy theorised that the five known planets at thetime, together with the Sun and Moon, orbited the Earth

Alexan-From more than 20 years of observational data obtained by the astronomer Tycho Brahe,Johannes Kepler discovered a minor discrepancy between the observed position of the planetMars and that predicted using Copernicus’ model Kepler went on to prove that planets orbitthe Sun in elliptical rather than circular orbits This was summarised in Kepler’s threeplanetary laws of motion The first two of these laws were published in his book NewAstronomy in 1609 and the third law in the book Harmony of the World a decade later in 1619.Kepler’s three laws are as follows, with their applicability to describe a satellite orbitingaround the Earth highlighted in brackets

† First law: the orbit of a planet (satellite) follows an elliptical trajectory, with the Sun(gravitational centre of the Earth) at one of its foci

† Second law: the radius vector joining the planet (satellite) and the Sun (centre of the Earth)sweeps out equal areas in equal periods of time

† Third law: the square of the orbital period of a planet (satellite) is proportional to the cube

of the semi-major axis of the ellipse

While Kepler’s laws were based on observational records, it was sometime before theselaws would be derived mathematically In 1687, Sir Isaac Newton published his breakthroughwork Principia Mathematica in which he formulated the Three Laws of Motion:

Law I: every body continues in its state of rest or uniform motion in a straight line, unless

impressed forces act upon it

ISBNs: 0-471-72047-X (Hardback); 0-470-845562 (Electronic)

Law II: the change of momentum per unit time is proportional to the impressed force and

takes place in the direction of the straight line along which the force acts

Law III: to every action, there is always an equal and opposite reaction

Newton’s first law expresses the idea of inertia

The mathematical description of the second law is as follows:

where F is the vector force on mass m1due to m2in the direction from m1to m2; G ¼ 6.672 £

10211 Nm/kg2is the Universal Gravitational Constant; r is the distance between the twobodies; r/r is the unit vector from m1to m2

The Law of Universal Gravitation states that the force of attraction of any two bodies isproportional to the product of their masses and inversely proportional to the square of thedistance between them The solution to the two-body problem together with Newton’s ThreeLaws of Motion are used to provide a first approximation of the satellite orbital motionaround the Earth and to prove the validity of Kepler’s three laws

3.1.2 Equation of Satellite Orbit – Proof of Kepler’s First Law

The solution to the two-body problem is obtained by combining equations (3.1) and (3.2) Inthe formulation, the centre of the Earth is the origin in the co-ordinate system and the radiusvector r is defined as positive in the direction away from the origin Re-expressing equations(3.1) and (3.2) to describe the force acting on the satellite of mass m due to the mass of theEarth, M:

Fm¼ 2GmM r

r3 ¼ 2mm r

wherem ¼ GM ¼ 3.9861352 £ 105km3/s2is Kepler’s constant

The negative sign in equation (3.3) indicates that the force is acting towards the origin.Equation (3.1) and (3.3) gives rise to:

d2r

dt2 ¼ 2m r

The above equation represents the Law of Conservation of Energy [BAT-71]

Cross multiplying equation (3.4) with r:

Since the cross product of any vector with itself is zero, i.e r £ r ¼ 0, hence:

r·dr

dt ¼ 0this implies



ð3:12ÞComparing (3.10) with (3.12) gives:

d2r

dt2 £ h ¼md

dt

rr



ð3:13ÞIntegrating (3.13) with respect to t:

3.1.3 Satellite Swept Area per Unit Time – Proof of Kepler’s Second Law

Referring to Figure 3.2, a satellite moves from M to N in time Dt, the area swept by theposition vector r is approximately equal to half of the parallelogram with sides r and Dr, i.e

Since h is a constant vector, it follows that the satellite sweeps out equal areas in equalperiods of time This proves Kepler’s second law.

3.1.4 The Orbital Period – Proof of Kepler’s Third Law

Figure 3.2 Area swept by the radius vector per unit time

At the perigee and apogee,

where b ¼ a(1 2 e2)1/2is the semi-minor axis

Equating (3.27) with (3.28) when t is equal to T, it follows that:

Integrating (3.34) with respect to t:

3.2.1 Overview

In order to design a satellite constellation for world-wide or partial coverage, a satellite’slocation in the sky has to be determined A satellite’s position can be identified with differentco-ordinate systems, the choice being dependent upon the type of application For example,radio communication engineers prefer to use look angles, specified in terms of azimuth andelevation, for antenna pointing exercises The most commonly used co-ordinate systems aredescribed in the following sections

3.2.2 Satellite Parameters

A set of six orbital parameters is used to fully describe the position of a satellite in a point inspace at any given time:

V: the right ascension of ascending node, the angle in the equatorial plane measured

counter-clockwise from the direction of the vernal equinox direction to that of theascending node;

i: inclination angle of the orbital plane measured between the equatorial plane and the

plane of the orbit;

v: argument of the perigee, the angle between the direction of ascending node and

direction of the perigee;

Figure 3.3 Satellite parameters in the geocentric-equatorial co-ordinate system

the satellite moves upward and downward through the equatorial plane are called ascendingnode and descending node, respectively.

In addition to defining the location of a satellite in space, it is important to determine thedirection at which an Earth station’s antenna should point to the satellite in order to commu-nicate with it This direction is defined by the look angles – the elevation and azimuth angles –

in relation to the latitude and the longitude of the Earth station The following sections discussthe location of the satellite with respect to the different co-ordinate systems Note: theformulation of the satellite location outlined in the following sections assumes that theEarth is a perfect sphere

3.2.3 Satellite Location in the Orbital Plane

The location of a satellite in its orbit at any time t is determined by its true anomaly,q, asshown in Figure 3.4 In the figure, the orbit is circumscribed by a circle of radius equal to thesemi-major axis, a, of the orbit O is the centre of the Earth and is the origin of the co-ordinatesystem C is the centre of the elliptical orbit and the centre of the circumscribed circle E is theeccentric anomaly

Refer back to Figure 3.1 and consider the quadrant containing the points P, B, O, C and D

as shown in Figure 3.4 In order to locate a satellite’s position at any time t, the angularvelocity, 4, and the mean anomaly, M, have to be found By using the perigee as thereference point, the mean anomaly is defined as the arc length (in radians) that a satellitewould have traversed at time t after passing through the perigee at time t0had it proceeded

on the circumscribed circle with the same angular velocity The angular velocity is obtainedfrom (3.29) and is given by:

Figure 3.4 Satellite location with respect to an orbital plane co-ordinate system

satellite moves through the perigee at time t0is given by:

then Area ðQBPÞ ¼ ðb=aÞArea ðDBPÞ ¼ ab½E 2 cosEsinE=2

and Area ðOQBÞ ¼ ðOBÞðQBÞ=2 ¼ ðacosE 2 aeÞðbsinEÞ=2 ¼ absinEðcosE 2 eÞ=2

Therefore

Area ðOQPÞ ¼ Area ðOQBÞ 1 AreaðQBPÞ ¼ ab

2 ½E 2 esinE ¼dA ð3:46ÞEquating (3.43) and (3.46) gives:

q¼ 2tan21 1 1 e

1 2 e

tanE2

ð3:51Þ

The location of the satellite in the orbital plane is then given as:

x0

y0

z0

264

37

37

3.2.4 Satellite Location with Respect to the Rotating Earth

Given the three orbital parameters,V, i, andv, as shown in Figure 3.3, the location of thesatellite in the geocentric-equatorial system in terms of the orbital plane –ordinates system isgiven by [PRA-86]:

cosVcosv2 sinVcosisinv 2cosVsinv2 sinVcosicosv sinVsini

sinVcosv1 cosVcosisinv 2sinVsinv 1 sinVcosicosv 2cosVsini

2

6

4

375

x0

y0

z0

264

375ð3:53Þwhere the first matrix on the right hand side of equation (3.53) is called the rotation matrix.The geocentric-equatorial co-ordinate system (xf, yf, zf) represents a fixed Earth system Inorder to account for the rotation of the Earth, a transformation of co-ordinates from the fixedEarth system to a rotating Earth system is required The relationship between the rotatingEarth system and the geocentric-equatorial system is shown in Figure 3.5

In this figure, the Earth is rotating at an angular velocity4e If Teis the elapsed time sincethe xf-axis and the xr-axis last coincided, then the location of the satellite in the rotating Earthsystem is related to the geocentric-equatorial system as follows [NAS-63]:

xr

yr

zr

264

37

5 ¼

cos4eTe

sin4eTe

02sin4eTe

cos4eTe

0

264

375

xf

yf

zf

264

37

The value of4eTeat any time t in minutes is given by:

whereag,o¼ 99.6909833 1 36000.7689t1 0.00038707t2(degrees);t¼ (JD 2 2415020)/

36525 (Julian centuries); JD ¼ 2415020 1 (Y 2 1899) £ 365 1 Int[(Y 2 1899)/4] 1 Mm1(Dm2 D) 1 [(h 2 12)]/24

JD is the Julian date in year Y, on day D of month m and at h hours (using pm/amconvention) JD is calculated using the reference Julian Day 2415020, which is at noon onDecember 31, 1899 Mmis the number of days between month m and December, the value ofwhich for different m is tabulated in Table 3.1 Dmis the number of days in month m In thecase of a leap year, Dm ¼ 29 in February ag,o is the right ascension of the Greenwichmeridian at 0 h UT at Julian date JD.t is the elapsed time in Julian centuries between 0 h

UT on Julian day JD and noon UT on January 1, 1900

3.2.5 Satellite Location with Respect to the Celestial Sphere

Refer to Figure 3.6 and consider the location of a satellite in its celestial sphere The position

of the satellite can be located by its right ascension angle,a, and its declination angle,d, onthe celestial From the spherical triangle ASN and using the law of sines,

Figure 3.5 The rotating Earth system with respect to the fixed Earth system [PRI-93]

Table 3.1 Value of Mmfor Julian dates tion

sinða2VÞ ¼ cosi

From the spherical triangle ABS and using the law of cosines:

Eliminating cosd from equation (3.56) and (3.57) gives:

Thus,

By the law of sines,

From equations (3.57) and (3.60), it follows that

3.2.6 Satellite Location with Respect to Satellite-Centred Spherical Co-ordinates

Satellite-centred spherical co-ordinates locate a satellite in relation to an Earth station’slatitude, Lg, and relative longitude, lg, as shown in Figure 3.7

Figure 3.6 Celestial sphere co-ordinate system

Using the law of sines in triangle GSM0,

From the spherical right-angle triangle formula,

gis called the tilt angle between the Earth station and the sub-satellite point The tilt anglecan be expressed in terms of Lgand lgas follows:

The above equations are normally used to determine the antenna pointing angles and incalculating the antenna gain toward a specified Earth station.

3.2.7 Satellite Location with Respect to the Look Angles

3.2.7.1 Definitions

Radio communications engineers are more familiar with the elevation and azimuth angles

as shown in Figure 3.8 The azimuth angle,j, is the angle measured North Eastward from thegeographic North at the Earth station, G, to the sub-satellite point The sub-satellite point, S0,

is defined as the point where the line joining the centre of the Earth, O, and the satellite meetsthe Earth’s surface The elevation angle,u, is the angle measured upward from this tangentialplane at the Earth station to the direction of the satellite

In Figure 3.8 (and Figure 3.7), the anglew is called the central angle or the coverageangle at the centre of the Earth, O, formed by lines OG and OS, where G denotes the Earthstation and S is the satellite The angle g is called the tilt angle or the nadir angle at thesatellite, formed by the lines GS and OS Lg and lgrepresent the latitude and the relativelongitude (i.e relative to the longitude of the sub-satellite point) of the Earth station,respectively and Lsis the latitude of the satellite Note: in this co-ordinate system, Northernlatitudes and Eastern longitudes are regarded as positive Furthermore, the latitude of thesub-satellite point is the same as the declination angle,d, of the satellite with respect to thegeocentric-equatorial co-ordinate system, that is Ls¼d The slant range R of the satellite isits distance from the Earth station

3.2.7.2 Elevation Angle

In order to calculate the elevation angleu, the central anglewand the slant range R have to bedetermined The co-ordinates of G in Figure 3.8 are related to its Northern latitude andEastern longitude by the following:

xg

yg

zg

264

37

37

where REis the radius of the Earth (6378 km)

By analogy, the co-ordinates of the sub-satellite point, S0, are related to its Northernlatitude and Eastern longitude by the following:

xs

ys

zs

264

37

37

d can also be obtained from the following equation:

d2¼ ðxg2 xsÞ2

1 ðyg2 ysÞ2

Substituting (3.69) and (3.70) into (3.72) and then equating (3.71) and (3.72) gives:

cosw¼ cosLgcosLscoslg1 sinLgsinLs ð3:73ÞNow consider triangle GOS and using the law of cosines again, the slant range, R, is givenby

By applying the law of sines:

Re1 hsin 908 1ð uÞ ¼

Referring to the spherical triangle GNS0, /GNS0¼ lgand arc(GS0) ¼w Using the law ofsines:

3.2.7.4 Minimum Elevation Angle – Visibility

The condition for a satellite to be visible from an Earth station isu$ 08 From Figure 3.8, forthe condition to be met

From a geometrical point of view, the minimum condition that the satellite is visible from

an earth station isu¼ 08 In practise, however, the minimum value ofu, termed the minimumelevation angle, umin, should be high enough to avoid any propagation factors such as

Table 3.2 Value of the azimuth angle j with respect to the relative position of the sub-satellite point

Sub-satellite point (S0) positionw.r.t to the Earth station (G)

shadowing It is then convenient to expresswin terms ofumin Normally, a minimum tion angle in the range 5–78 is used.

eleva-From Figure 3.8, it can be shown from triangle SGO that:

w¼ cos21 REcosumin

3.2.8 Geostationary Satellite Location

For geostationary orbits, the inclination angle i ¼ 08, eccentricity e ¼ 0 and, since the satellite

is placed in the equatorial plane, the satellite’s latitude, Ls¼ 08 Furthermore, for tionary satellites, RE1 h ¼ 42164 km Bearing this in mind, the central angle,w, in equation(3.81) can be rewritten as:

The elevation angle,u, in equation (3.76) can then be re-expressed as:

26

If the satellite and the Earth station are on the same meridian, it follows that lg¼ 0 Thus,the maximum latitude, Lg,max for which the satellite is visible can be obtained from thefollowing equation:

This implies that Lg,max¼ 81.38

3.3.1 General Discussion

The orbital equations derived in the previous section are based on two basic assumptions:

† The only force that acts upon the satellite is that due to the Earth’s gravitational field;

† The satellite and the Earth are considered as point masses with the shape of the Earth being

a perfect sphere

In practise, the above assumptions do not hold The shape of the Earth is far from spherical

In addition, apart from the gravitational force due to the Earth, the satellite will also ence gravitational fields due to other planets, and more noticeably, those due to the Sun andthe Moon Other non-gravitational field related factors including the solar radiation pressureand atmospheric drag also contribute to the satellite orbit perturbing around its elliptical path

experi-In the past, techniques have been derived to include the perturbing forces in the orbitaldescription By assuming that the perturbing forces cause a constant drift, to the satellite’sposition from its Keplerian orbit, which varies linearly with time, the satellite’s position interms of the six orbital parameters (see Section 3.2.1) at any instant of time t1, can be writtenas:

3.3.2 Effects of the Moon and the Sun

Gravitational perturbation is inversely proportional to the cube of the distance between twobodies Hence, the effect of the gravitational pull from planets, other than the Earth, has amore significant effect on geostationary satellites than that on Low Earth Orbit (LEO) satel-lites Among these planets, the effect of the Sun and the Moon are more noticeable Althoughthe mass of the Sun is approximately 30 times that of the Moon, the perturbation effect of theSun on a geostationary satellite is only half that of the Moon due to the inverse cube law Thelunar–solar perturbation causes a change in the orbital inclination The rate of change in ageostationary orbital inclination due to the Moon is described by the following formula[AGR-86]: