satellite orbits and flight dynamics_final

SATELLITE ORBITS AND

FLIGHT DYNAMICS

BY

DR. ENG. MOHAMED AHMED ZAYAN

2006

Dr. Eng. Mohamed Ahmed Zayan Page ii 2/12/2006

TABLE OF CONTENTS

CHAPTER 1 .............................................................................................................................1

1 INTRODUCTION...................................................................................................................1

1.1 Background ....................................................................................................................................... 1 1.1.1 Satellite Orbits ...................................................................................................................... 1

1.1.1.1 Geostationary Orbits ............................................................................................ 1 1.1.1.2 Polar Orbits .......................................................................................................... 2 1.1.1.3 Inclined Orbits ..................................................................................................... 2

1.1.2 Orbits Determination and Estimation Methods .................................................................... 2

CHAPTER 2 .............................................................................................................................4

2 ORBITAL MECHANICS AND REFERENCE SYSTEMS ...........................................................4

2.1 Kepler Laws ..................................................................................................................................... 4

2.2 Julian Date......................................................................................................................................... 4

2.3 Sidereal and Universal Time............................................................................................................ 5

2.4 Reference Coordinate Systems ........................................................................................................ 6

2.5 The Two-Body Problem ................................................................................................................... 8 2.5.1 Orbital Elements, Energy Integral, and Euler Angles......................................................... 12 2.5.2 Position and Velocity from the Orbital Elements ............................................................... 13 2.5.3 Orbital Elements from the Position and Velocity ............................................................... 14 2.5.4 Keplers Equation and the Time Dependence of the Motion.............................................. 14

2.5.4.1 Solution for Ellipse ............................................................................................ 15 2.5.5 Computation Starting from Time in Orbit .......................................................................... 16 2.5.6 Orbital Variation in Keplerian Elements Format ................................................................ 16 2.5.7 Tangential and Normal Components .................................................................................. 19 2.5.8 Use of Tangential and Normal Component (t, n, z)............................................................ 20 2.5.9 Summary Of Equations In Tangential-Normal (t, n, z) Axes [6]........................................ 21

CHAPTER 3 ...........................................................................................................................22

3 SATELLITE PERTURBATIONS AND LINEARIZATION.........................................................22

3.1 Satellite Perturbations.................................................................................................................... 22 3.1.1 Gravitational Field of the Earth .......................................................................................... 22

3.1.1.1 Expansion of Spherical Harmonics.................................................................... 22 3.1.1.2 Geopotential Gravity Acceleration .................................................................... 23

3.1.2 Perturbation from the Sun and the Moon (Point-Mass)...................................................... 24 3.1.3 Solar Radiation Pressure..................................................................................................... 25 3.1.4 Atmospheric Drag Acceleration ......................................................................................... 25

3.2 Linearization and Variational Equations ..................................................................................... 26 3.2.1 The Differential Equation of the State Transition Matrix................................................... 27 3.2.2 The Differential Equation of the Sensitivity Matrix ........................................................... 27 3.2.3 Form and Solution of the Variational Equations ................................................................ 27

Dr. Eng. Mohamed Ahmed Zayan Page iii 2/12/2006

3.2.4 Partial Derivative of the Earth Geopotential Acceleration ................................................. 28 3.2.5 Partial Derivatives of the Sun and the Moon (Point Mass) Accelerations.......................... 29 3.2.6 Partial Derivative of Solar Radiation Pressure Acceleration .............................................. 29 3.2.7 Partial derivative of the Atmospheric Drag acceleration .................................................... 29 3.2.8 Partial of Measurements with Respect to the State Vector ................................................. 30 3.2.9 Partial with Respect to Measurement Model Parameters ................................................... 31

CHAPTER 4 ...........................................................................................................................33

4 SATELLITE ORBITS ESTIMATION AND DETERMINATION ................................................33

4.1 Satellite Tracking and Observation Models ................................................................................. 33 4.1.1 Angle Measurements .......................................................................................................... 33 4.1.2 Ranging Measurements....................................................................................................... 33

4.2 Maneuver Implementation............................................................................................................. 35 4.2.1 Numerical Integration Methods .......................................................................................... 35 4.2.2 Satellite Orbits Correction .................................................................................................. 35

4.2.2.1 Thrust Forces ..................................................................................................... 36

REFERENCES............................................................................................................................37

CHAPTER 1

1 INTRODUCTION

In little over a third of the 20th century, the launching of a satellite has gone from stopping the

nations' business to guarantee that it runs like clockwork. Today, satellites are commonplace

tools of technology, like clocks, telephones, and computers. Satellites serve us for navigation,

communications, environmental monitoring, and weather forecasting. Appropriately, the word

satellite means an attendant. In 1957, Russian launched the Sputnik satellite. U.S.A sent Alan

Shepard up and down in a Mercury capsule in 1961, as John Glenn circled the globe 3 times in

1962, and when Neil Armstrong set foot on the moon in 1969.

1.1 Background

The job of the satellite control station is to determine and estimate satellite orbits, continuously

execute correction maneuvers necessary to maintain the correct orbit altitude, attitude position,

manage the payload and verify the efficiency of the space segment.

1.1.1 Satellite Orbits

Satellites can operate in several types of Earth orbit. The most common orbits for

environmental satellites are geostationary and polar, but some instruments also navigate in

inclined orbits. Other types of orbits are possible, such as the Molniya orbits commonly used

for Russian spacecrafts.

1.1.1.1 Geostationary Orbits

A geostationary (GEO=Geo-synchronous) orbit is one in which the satellite is always in the

same position with respect to the rotating Earth. The satellite orbits at an elevation of

approximately 35,790 km because that produces an orbital period (time for one orbit) equal to

the period of rotation of the Earth (23 hrs, 56 min, 4.09 sec). By orbiting at the same rate, in the

same direction as Earth, the satellite appears stationary (synchronous with respect to the

rotation of the Earth). Geostationary satellites provide a "big picture" view, enabling coverage a

large area of the Earth and weather events. This is especially useful for monitoring severe local

storms and tropical cyclones. Because a geostationary orbit must be in the same plane as the

Earth's rotation, that is the equatorial plane; it provides distorted images of the Polar Regions

with poor spatial resolution.

1.1 Background

Dr. Eng. Mohamed Ahmed Zayan Page 2 2/12/2006

1.1.1.2 Polar Orbits

Polar-orbiting satellites provide a more global view of Earth, circling at near-polar inclination

(the angle between the equatorial plane and the satellite orbital plane, a true polar orbit has an

inclination of 90 degrees). These satellites operate in a sun-synchronous orbit. The satellite

passes the equator and each latitude at the same local solar time each day, meaning the satellite

passes overhead at essentially the same solar time throughout all seasons of the year. This

feature enables regular data collection at consistent times as well as long-term comparisons.

The orbital plane of a sun-synchronous orbit must also rotate approximately one degree per day

to keep pace with the Earth's surface.

1.1.1.3 Inclined Orbits

Inclined orbits fall between those above. They have an inclination between 0 degrees (equatorial orbit) and 90 degrees (polar orbit). These orbits may be

determined by the region on Earth that is of most interest (i.e., an instrument to

study the tropics may be best put on a low inclination satellite), or by the latitude

of the launch site. The orbital altitude of these satellites is generally on the order

of a few hundred km, so the orbital period is on the order of a few hours. These

satellites are not sun-synchronous, however, so they will view a place on Earth

at varying times.

1.1.2 Orbits Determination and Estimation Methods

It is important to distinguish between preliminary orbits determination (navigation) used for

direct computation of the six orbital elements (position r and, velocity v) with no a priori

knowledge of the spacecraft orbit [1], [2] and orbit estimation used for the improvement of a

priori orbital elements from large set of tracking data [3]. The complex mathematical

formulation of orbit prediction and measurements modeling does not allow a direct inversion

except for the simplified case of Keplerian orbits. In addition, the measurements employed for

an orbit determination cannot be expected to be exact quantities due to inevitable measurement

(and model) errors. A preliminary orbit determination may still be required in the case of

launcher injection errors. Most methods for preliminary orbit determination are based in Gauss

algorithm [1]. The goal in orbit estimation is to determine the satellite orbit that best fits or

matches a set of tracking data [4], and [5]. Tracking data or "observation" data includes any

observable quantities that are a function of the position and/or velocity of a satellite at a point in

time. Examples include range, range rate (Doppler), azimuth, elevation from ground stations of

1.1 Background


known location, range and range rate from other satellites, as well as Global Positioning System

(GPS) data. In theory, the six satellite orbit parameters (position r and, velocity v) can be

determined from a geometric computation based on very few observations. Because actual

observation data includes the effects of unordered or poorly modeled forces as well as random

and systematic noise, it is often necessary to obtain more observations than the theoretical

minimum. A primary goal of orbit estimation schemes is to compute an orbit solution that uses

as much of the information in the tracking data as possible while not being overly influenced by

noise or spurious points. In general, the better the quality of tracking data processed, the more

reliable the orbit solution. Theoretically, any parameter influencing the tracking data can be

estimated. In addition to the satellite orbits themselves, other parameters that can be estimated

include the locations of the ground stations, biases in the tracking data, coefficients of

atmospheric drag, solar radiation pressure on the satellites, and parameters of the Earth's

gravitational field.

There are two major types of state estimation schemes commonly used for orbit

determination: batch and sequential. A batch estimator [5], [3], [15], and [16] determines a state

vector based on a single large set of observation data that, in general, can be taken over a period

of time. While the actual state can change significantly over the span of the observations, the

determined state is valid only for a single point in time. The determined state is that which best

fits the observable over the span of the observations. Batch estimation requires an a priori

estimate or "first guess" of the state, which is iteratively corrected to achieve the final state. The

estimated state from previous orbit estimation is typically used as the a priori value.

In sequential estimation [17], [3] the observations are processed one at a time or in small

groups instead of a single large group. For each group of observations, the a priori estimate is

the determined state from the last group. The final solution for sequential estimation thus

incorporates several intermediate "solutions" from each small group of data. An enhancement to

sequential estimation is the use of extended Kalman filter, which reduces the effects of old

observation data on the current state estimate, thus ensuring that the latest estimate is influenced

most heavily by the latest observation data.

CHAPTER 2

2 ORBITAL MECHANICS AND REFERENCE SYSTEMS

2.1 Kepler Laws

The main features of satellite orbits may still be described by a reasonably simple

approximation, even though elaborate models have been developed to compute the motion of

artificial Earth satellites to the high level of accuracy required for many applications

today. This is due to the fact that the force resulting from the Earths central mass governs the

motion of the satellite and all other forces acting on the satellite, (which are Earths oblateness,

elliptic equatorial cross section and other perturbations forces from the Sun and the Moon), may

be ignored. The word perturbation is used to signify forces other than those due to the

gravitational potential of homogeneous, spherical Earth.

Johann Kepler determined three laws, which were found empirically about 400

years ago may, characterizing orbital motion. These laws can be proven mathematically using

Newton's law of gravitation. These laws apply directly to satellite orbital motion, thus the laws

are from the point of view of an Earth-orbiting satellite.

Kepler's First Law: Satellite orbits are elliptical Paths with the Earth at one focus of the ellipse.

Simply states that orbits are shaped like ellipses (elongated circles). This can be proven

mathematically, once it's understood that the gravitational force between the Earth and the

satellite decreases in proportion to the square of distance between the two.

Kepler's Second Law: A line between the center of the Earth and the satellite sweeps out equal

areas in equal intervals of time. This means that the satellite moves fastest at its lowest altitude

(perigee) and it moves slowest at its highest altitude (apogee), which gives elliptical orbits a

very distinct characteristic,

Kepler's Third Law: The Square of the orbital period is proportional to the cube of the orbit's

semi-major axis. States that you can compute the time it takes the satellite to make one

complete orbit (the period) from the semi-major axis of the orbital ellipse. This is also known as

the harmonic law.

2.2 Julian Date

The material of the following sections are based on [1], [3], and [6]. The civilian calendar is not,

however, well suited to finding the time difference between two dates or advancing a date by a

2.3 Sidereal and Universal Time


certain time increment. To cope with this difficulty, a continuous day count is used in

astronomy and commonly for space missions, which is known as the Julian Date. The Julian

Date (JD) is the number of days since noon January 1, 4713 BC including the fraction of day. It

thus provides a continuous time scale, which for all practical purposes, is always positive.

Counting starts at noon for historical reasons to avoid a change of date in the middle of

astronomical observations. Presently, the Julian Day numbers are already quite large (well over

two millions) and it is desirable to start counting at midnight. Therefore, a Modified Julian Date

(MJD) is defined as:

MJD = JD 2400000.5. ( 2-1)

A table of Modified Julian Dates for the beginning of each month between 1975 and 2020 is

given in [3].

2.3 Sidereal and Universal Time

Today the following time scales are of prime relevance in the precision model of Earth orbiting

satellites:

Terrestrial Time (TT), a conceptually uniform time scale that would be measured by an ideal clock on the surface of the geoids. TT is measured in days

of 86400 SI seconds and is used as the independent argument of geocentric

ephemerides.

International Atomic Time (TAI), which provides the practical realization a uniform time scale based on atomic clocks and agrees with TT except for a

constant offset of 32.184 s and the imperfections of existing clocks.

GPS Time, which likes TAI is an atomic time scale but differs in the chosen offset and the choice of atomic clocks used in its realization. The origin of GPS

was arbitrarily chosen to coincide with UTC on 1980 January 6.0 UTC.

Greenwich Mean Sidereal Time (GMST), the Greenwich hour angle of the vernal equinox.

Universal Time (UT1), today's realization of a mean solar time, which is derived from GMST by a conventional relation.

Coordinated Universal Time (UTC), which is tied to the International Atomic Time TAI by an offset of integer seconds that is regularly updated to keep UTC

in close agreement with UT1.

Greenwich Mean Sidereal Time GMST, also known as Greenwich Hour Angle, denotes the

2.4 Reference Coordinate Systems


angle between the mean vernal equinox of date and the Greenwich meridian. It is a direct

measure of the Earth's rotation and may jointly be expressed in angular units or units of time

with 360 corresponding to 24h. In terms of SI seconds, the length of a sidereal day (i.e. the

Earth's spin period) amounts to 23h56m4s.0910s.005 making it about four minutes shorter than

a 24h solar day. Due to length-of-day variations with amplitude of several milliseconds, sidereal

time cannot be computed from other time scales with sufficient precision but must derived from

astronomical and geodetic observations.

Universal Time UT1 is the presently adopted realization of a mean solar time scale with the

purpose of achieving a constant average length of the solar 24 hours. As a result, the length of

one second of Universal Time is not constant because the actual mean length of a day depends

on the rotation of the Earth apparent motion of the Sun (i.e. the length of the year). Similar to

sidereal is not possible to determine Universal Time by a direct conversion from e.g. atomic

time, because the rotation of the Earth cannot be predicted accurately. Every change in the

Earth's rotation alters the length of the day, and must therefore be taken into account in UT1.

Universal Time is therefore defined as a function of sidereal time; which directly reflects the

rotation of the Earth. For any particular day, 0h UT1 is defined as the instant at which

Greenwich Mean Sidereal Time has the value

GMST (0hUT1) = 24110S.54841 + 8640184s.812866T0

+ 0S.093104 - 020TS.0000062 . ( 2-2) 30T

[54]. In this expression the time argument

T0 = (JD (0hUT1)-241545)/ (36525) ( 2-3)

denotes the number of Julian centuries of Universal Time that have elapse 2000 Jan. 1.5 UT1 at

the beginning of the day. For an arbitrary time of the expression may be generalized to obtain

the relation

GMST = 24110s.54841 + 8640184s.812866T0 + 1.002737909350795 UT1 + 0s.093104T2

0s.0000062T3 , ( 2-4)

where the time argument

T = (JD (UTI) 2451545)/36525 ( 2-5)

specifies the time in Julian centuries of Universal Time elapsed since 2000 Jan. 1.5 UT1.


Two global coordinated systems are intended to be fixed in space, i.e. they are inertial, which

can be defined refer to the orbital plane and the rotation axis of the Earth as shown in Figure2-1.



The first one is the most common coordinate system for describing Earth-bound satellite orbits,

which is called the geocentric equatorial coordinate system (x or i, y or j, z or k). This system

gives the position of a point in space with respect to the Earth's equatorial plane (the plane

perpendicular to the rotation axis) and its origin is the center of the Earth.

The other one refers to the ecliptic (the Earth's orbital plane), which is called the

ecliptic coordinate system (x', y', z'). These planes are inclined at an angle = 23.5 and the

line of intersection is a common axis of both coordinate systems. The x/x'-axis is defined as

being the direction of the vernal equinox or First Point of Aries, designated by . It is

perpendicular to both the North Celestial Pole (the z-axis) and the north pole of the ecliptic (the

z'-axis). A standard reference frame is usually based on the mean equator and equinox of some

fixed epoch, which is currently selected as the beginner of the year 2000. Access to the Earth

Mean Equator and Equinox (mean of the lunisolar precession and Earth rotation axis nutation)

of J2000 (EME 2000) is provided by the FK5 star catalog [55], which provides precise

positions and proper motions of some 1500 stars for the epoch J2000 as referred to the given

reference frame. In 1991 to establish a new International Celestial Reference System (ICRS)

adopt it for use from 1998 onwards [56]. For a smooth transition to the new system, the ICRS

axes are chosen such a way as to be consistent with the previous FK5 system to within the

accuracy of the latter. The practical realization of the ICRS is designated the International

Celestial Reference Frame (ICRF) and is jointly maintained by the IERS and the IAU Working

Group on Reference Frames [57].

Complementary to the ICRS, the International Terrestrial Reference System (ITRS)

provides the conceptual definition of an Earth-fixed reference system (aligned with the

equatorial plane and the Greenwich meridian) [58]. Its origin is located at the Earth's center of

Figure 2-1 Equator and Ecliptic Planes [3]

2.5 The Two-Body Problem


mass and its unit of length is the SI meter.


The basic laws of orbital motion are derived from first principles. For this purpose, a satellite is

considered, whose mass is negligible compared to the Earth's mass and the Earth is

assumed to be spherically symmetric. The equations of motion for two masses m

M

1, m2 at

position r1, r2 with large distance apart compared to their size or have an spherically

symmetrical mass distribution and never touch each other are as follows: 3

122111 /)( rmGmm rrr =&& ( 2-6) 3

212122 /)( rmGmm rrr =&& ( 2-7) 21 rrr = , ( 2-8)

where G is the constant of gravitation and is

)( 21 mmG += . ( 2-9) By measuring the mutual attraction of two bodies of known mass, the gravitational constant G

can directly be determined from torsion balance experiments. Due to the small size of the

gravitational force, these measurements are extremely difficult, however, and G is presently

only known with limited accuracy: 213-11 m 10 0.00085) (6.67259 = skgG ( 2-10)

[59]. Independent of the measurement of G itself, the gravitational coefficient , i.e. the

product of the gravitational constant and the Earth's mass, has been determined with

considerable precision from the analysis of laser distance measurements of artificial Earth

satellites [60].

GM

-23skm001.4405.398600 =GM . ( 2-11) The corresponding value of the Earth's mass is given by

kg 10 5.974 24=M . ( 2-12) The center of mass is at

)/()( 212211 mmmm ++ rr , ( 2-13) by combing the above equations yields

0/ 3 =+ rrr && . ( 2-14) The cross product of the above equation with the position vector r

0/)( 3 == rrrrr && . ( 2-15)



Since the cross product of the vector with it self vanishes. The left hand side may be further

written as

0)( ==+= rrrrrrrr &&&&&&&dtd . ( 2-16)

The angular momentum per unit mass is

dtdrrh == v, rh , ( 2-17)

and it is constant for non-perturbated motion, also the h is normal to the plane of motion and the

absolute value h = |h| is known as areal velocity. By taking the vector product of equation of

motion with h

)()/( 3 rrrhr &&& = r . ( 2-18) Since for any triple vector product

).().()( rrrrrrrrr &&& = ( 2-19) it follows that

)/( rdtd rhr =&& . ( 2-20)

Since h is a constant this equation can be integrated directly to yield

rr /)( erhr += & ( 2-21) where e is a constant of integration and is called eccentricity vector and it is along the vector

to the point of closest approach (perigee). To continue with equation (2-21) using (2-17)

)cos(2 rerh +=== hrrhrr && ( 2-22) where is the angle between the vector r and e. solving for r gives

cos1/2

ehr += ( 2-23)

as the equation of the orbit, which is the standard equation of a conic in polar form with the

origin of coordinates at one focus .The semi-latus rectum is

/2hp = , ( 2-24) and e is the eccentricity. The angle is the true anomaly, the angle from the perigee as shown in Figures 2-2. Figure 2-3 represents the following unit vectors.

1. ir is along the radius vector r, i.e. away from the center of attraction.

2. i is perpendicular to r in the plane of the motion and in the direction of

increasing the true anomaly . 3. iz is the normal to the plane motion.



aFocusCenter of Mass

ae

Satellite

PerigeeApogeeb

E r

a semimajor axisb semiminor axise eccentricity

True anomalyE Eccentric anomalyM Mean anomaly

Figure 2-2 Conical Orbit

ir, i, iy, are in orbit plane. i is perpendicular to r in the direction of increasing . iz is normal to orbit palne.

ix is the intersection of orbit plane with the Equator (i,j) plane.

j

i

ix Asending Node

k

iy ir

i

iz

i

Figure 2-3 The Inertial Geocentric Equatorial (i, j, k), Radial Transverse (i , i , i ), and r z(i , i , i ) Coordinates Systems x y z

Figure 2-4 Perifocal coordinate system (PQW frame)

Figure 2-5 Topocentric-horizon coordinate system (SEZ frame):

(a) overall view; (b) detailed view

2.5The Two-Body Problem


Equation (2-17) can be written

)(

)(

&&

&&Qrr

rr

r

r

iirhiiv+=

+= ( 2-25)

&2rh zz iih == . ( 2-26)

2rh

dtd = , ( 2-27)

Eliminating r between equations (2-23), and (2-27), using (2-24)

23 )cos1(

eddt

p += . ( 2-28)

Integration of this equation gives as a function of time in orbit. 2.5.1 Orbital Elements, Energy Integral, and Euler Angles

Assuming h, , and e are constant, then differentiating equation (2-22) and eliminating using equation (2-27)

&

her /sin=& ( 2-29) The square of the velocity v is given by

222 )()( && rrv += ( 2-30) 2222 )/sin(/ herhv += . ( 2-31)

Eliminating r, using equation (2-23),

]cos21[)/( 222 eehv ++= ( 2-32) Define the semi-major axis of the orbit

))1(/()1/( 222 ehepa == . ( 2-33) So, substituting in equation (1-31) to get vis-viva law

)12(2ar

v = . ( 2-34) The energy law states that the sum of kinetic energy and the potential energy is constant during

motion

massunit per

massunit per

2

)2

(

)(

21

aE

rE

mvE

total

potential

kinematic

=

=

=

. ( 2-35)

It is seen that, the total energy depends on the reciprocal semi-major axis of the orbit. The



parameters a, e and represent a choice of the three orbital elements to define the motion in the plane of the orbit. In order to specify the orientation of the orbital plane in space three other

orbital elements are required as described in the next section.

2.5.2 Position and Velocity from the Orbital Elements

Given the 6 orbital elements (a, e, , i, , ) as shown in Figure 2-3, where i is the inclination, is the right ascension of ascending node, and is the argument of perigee and from equations (2-24), (2-25), and (2-31)

)cos1/( epr += , ( 2-36) the velocity vector v with respect to axes (ir, i, iz) is

iiv iii && rr rzr +=,,)( , ( 2-37) using equations (2-29), (2-24), and (2-36)

])cos1(sin[/)( ,, iiv iiir eep rz ++= ( 2-38) rrzr ir iii =,,)( . ( 2-39)

Referring to Figures 2-2 and 2-3, the transformation from axes (ir, i, iz) to inertial axes (i, j, k)

is obtained by two rotational transformations

zr

zr

iiikji

iiikji

vv

rr

,,12,,

,,12,,

)()(

)()(

==

, ( 2-40)

where 1 is the transformation matrix from (ir, i, iz) to (ix, iy, iz).

++++

=1000)cos()sin(0)sin()cos(

1

( 2-41)

where axes (ix, iy, iz) with respect to axes (i, j, k)

xzy

z

x

iiiiii

ii

==

=)cos,cossin,sin(sin

)0,sin,(cos ( 2-42)

and 2 is the transformation matrix from (ix, iy, iz) to (i, j, k) coordinate.

=ii

iiii

cossin0sincoscoscossin

sinsincossincos

2 . ( 2-43)



2.5.3 Orbital Elements from the Position and Velocity

The position and velocity are given with respect to the inertial axes (i, j, k). Calculating the in

plane orbital elements. With reference to the following equation

)( vrh = , ( 2-44) /2hp = , ( 2-45)

so p is obtained by using the magnitude of h. Also from

)1/( 2epa = , ( 2-46) ape /1= , ( 2-47)

and from equation (2-36)

1)/(cos = rpe . ( 2-48) To have a unique determination of , from equation (2-37) and (2-39), and (2-29)

prej /sin. == vr , ( 2-49) or

rjpe //sin = , ( 2-50) Consequently can be uniquely determined by

)]/(/arctan[ rpps = ( 2-51) The vector iz, 3-dimensional vector, is calculated as the unit form of r v and is given in

equation (2-41) in terms of the elements i and . This leads to

))]2(/()1(arctan[ zz ii = ( 2-52) )]3(arccos[ zi i= , ( 2-53)

where iz(1), iz(2), and iz(3) are 1st,2nd, and 3rd elements of the vector iz, respectively. Equation

(2-53) gives i uniquely because it defined only for positive angles in range (0, ). In order to

derive from Figures 2-2, 2-3, equations (2-41), and (2-52) ri =+ xr )cos( . ( 2-54)

Since

rii =+ xz r )sin( ( 2-55) )()sin( rii =+ xzr . ( 2-56)

Equations (2-54) and (2-56) yield a unique solution for ( + ) and hence . 2.5.4 Keplers Equation and the Time Dependence of the Motion

An integral for the time in orbit was given at equation (2-28) but the solution differs depending



on whether the orbit is an ellipse (e < 1), a hyperbola (e >1), or a parabola (e = 1).

2.5.4.1 Solution for Ellipse

An auxiliary variable E, which is called the eccentric anomaly, is defined as the following

equation

araeE /)cos(cos += ( 2-57) or

)cos1( Eear = . ( 2-58) This can be equated to the conic equation from equations (2-33) and (2-36)

)cos1/()1( 2 eear += , ( 2-59) to get the following identities relating the eccentric anomaly to the true anomaly

)cos1/()(coscos EeeE = ( 2-60) )cos1/()sin1(sin 2 EeEe = . ( 2-61)

By differentiation of equation (2-58)

2

2

)cos1(sin)1(sin

EeEdEed

= , ( 2-62)

or anticipating the time integral in equation (2-28)

dEEee

de )cos1()cos1(

)1(2

2/32

=+

. ( 2-63)

Integration using equation (2.33) the time from perigee tp is related to the eccentric anomaly E

by

ptaEeE3/sin = ( 2-64)

In order to express the angle from the perigee in terms of the eccentric anomaly

sincos1

2tan = ( 2-65)

2tan

11

2tan E

ee

+= . ( 2-66)

Equations (2-64), and (2-66) can be used to calculate time in orbit given the angle from perigee,

or vice versa, the latter requires a numerical iterative solution. The angle on the left-hand side of

equation (2-64) is the mean anomaly M and n is the mean motion

)()(/ 3 pp ttnttaM == . ( 2-67) It changes by 360o during one revolution, but in contrast with the true and eccentric anomaly



increases uniformly with time. Instead of specifying the time of perigee passage to describe the

orbit, it is customary to introduce the value M0 of the mean anomaly at some reference epoch t0.

The mean anomaly at an arbitrary instant of time may be then found from

)(0 pttnMM += , ( 2-68) and the period of the ellipse is

/2 3aT = . ( 2-69) This relation states the third Keplers law.

2.5.5 Computation Starting from Time in Orbit

In order to obtain the position of the satellite at time t one has to know the time of perigee

passage and the semi-major axis to calculate the mean anomaly. Kepler's equation can,

however, be solved by iterative methods only. A common way is to start with an approximation

of

== 00 MorEE ( 2-70) and employ Newton's method to calculate successive refinements Ei until the result changes by

less than a specified amount from one iteration to the next. Defining an auxiliary function

MEeEEf = sin)( ( 2-71) the solution of Kepler's equation is equivalent to finding the root of f(E) for a given value of M.

Applying Newton's method for this purpose; an approximate root Ei of f may be improved by

computing

i

iii

i

iii Ee

MEeEEEfEfEE

cos1sin

)()(

1 ==+ . ( 2-72)

The starting value E0 = M recommended above is well suited for small eccentricities, since E

only differs from M by a term of order e. For highly eccentric orbits (e.g. e > 0.8) the iteration

should be started from E0 = to avoid any convergence problems during the iteration.

2.5.6 Orbital Variation in Keplerian Elements Format

The satellite orbit is an ellipse, parabola or hyperbola if it is influenced only by the gravitational

filed of a point mass or spherical body. The orbit elements can be calculated from position and

velocity vector at any time but these elements will be invariant except the true anomoly.

Practically, the satellite motion is perturbated by different forces and the calculation of the orbit

elements will yield a different set of values over an interval of time. This orbit with varying

parameters is called an osculating orbit. The orbital elements can be treated as the dependent



variables of a set of first order differential equations. Conversely, the position and velocity

vectors can be calculated directly from the set of evolving parameters at any time. The material

of this chapter is based on [52]. In the following analysis [6] let indicate a change in an orbital variable due to the application of a vector f of acceleration other than due to the

spherically symmetrical central gravitational field. The change in the energy per unit mass over

a time interval t is as follows: 2

massunit per 21 vEkinematic = , ( 2-73)

=r

E potential

massunit per , ( 2-74)

=a

Etotal 2 massunit per , ( 2-75)

taaE == fv.

2 2 . ( 2-76)

Referring to the orbital plane axes (ir, i, iz). In the limit this gives the rate of change of energy

as

).(22

fva

dtda = . ( 2-77)

Since

)).((2 vrvr =h , ( 2-78)

hdtdh /)).(( frvr = ( 2-79)

h/).( vrfr = ( 2-80) hr /)].)(.().([ 2 frvrfv = . ( 2-81)

In order to obtain the rate of change of eccentricity (e)

)1( 22 eah = . ( 2-82) By differentiate

).)(.().)([(1 2 frvrfv += rpaaedt

de . ( 2-83)

Now to calculate out- of -plan elements (i, , )

==i

ii

hcos

cossinsinsin

vrh . ( 2-84)



Differentiate this equation and arrange the result in the form

=

hihih

&&&sin

2fr , ( 2-85)

where

=ii

iiii

cossin0sincoscoscossin

sinsincossincos

2 , ( 2-86)

which is an orthogonal matrix, transpose equals the inverse. Therefore

ijkfr )(/

//sin

2 =

T

dtdhdthdi

dtidh. ( 2-87)

Note that the elements i and refer to the same axes as the vectors on the right hand side of the above equation in the inertial Equatorial Axes (i, j, k). The final form depends on the final axes

in use. The total rate of change of the true anomaly d, which consists of the rate of change due free motion of the satellite, and the rate of change due to the external applied forces. The rate of

change of the true anomaly , due only to the external applied acceleration vector f, is , is derived by applying the perturbation to the following equations

~

cos1

2

e

hr += ( 2-88)

)).((2 vrvr =h ( 2-89) /2~sincos hhreer = . ( 2-90)

In the limit

dtdhh

dtder

dtdre 2cos~

sin = ( 2-91)

sin).( erh =vr . ( 2-92) By differentiating

++=dtdh

hhp

dtder

dtdre 1).().(sin

~cos vrfr ( 2-93)

dtdhh

hphp

dtdre

+=

sin2).(cos).(cos)/(~

2vrfr ( 2-94)



( ) hererphhp cos)cos1(sinsin

2).(cos2 ++=

vr ( 2-95)

hrp )(sin += . ( 2-96) And finally

+=dtdhrpp

rehdtd sin)().(cos1

~fr . ( 2-97)

The variation of the argument of perigee is obtained by ( ) )cos(0sincos +== rx rri . ( 2-98)

Differentiation results in

( )

++=

dtd

dtdr

dtd

~)sin(0cossin r . ( 2-99)

Therefore

( ) ( ) rir

)sin(cos00 0cossin .0cossin 12 +=

= r ( 2-100)

+=

dtd

dtd

dtdi

~cos , ( 2-101)

or

dtd

dtdi

dtd ~cos = . ( 2-102)

We now have the required equations for variations of the orbital elements. The final form of the

equations depends on the axes in use.

2.5.7 Tangential and Normal Components

From the orbital plane axes (ir, i, iz) then

rrf=fr. , ( 2-103) and

sin. rv=vr , ( 2-104) where is the angle between the velocity vector v and i measured clockwise from the latter

cos1sintane

e+= , ( 2-105)

also



frhf

he

feefp

r

r

+=++=

sin

])cos1(sin[/.fv. ( 2-106)

From equation (2-87), the required transformation from (i, j, k) axes to (r, , z) axes

ijkfr )(/

//sin

2 =

T

dtdhdthdi

dtidh ( 2-107)

zrfr )(122 = T ( 2-108)

=

rfrf z

0

1 , ( 2-109)

++++

=1000)cos()sin(0)sin()cos(

1

( 2-110)

2.5.8 Use of Tangential and Normal Component (t, n, z)

From equation (2-32)

)cos21( 2eep

v ++= ( 2-111)

sinsin ep

v = ( 2-112)

)cos1(cos ep

v += ( 2-113)

t

nt

nt

vf

fvhf

vhre

ffr

===

.

sin

)cossin(

fv

fr

. ( 2-114)

By means of transformation from t n z axes to r z

=

1000sinsin0cossin

3

. ( 2-115)

+

=

cossinsincos

)(

tn

z

z

tnz

rfrfrfrf

fr , ( 2-116)



+=

cossin

0

//

/sin

1

tn

z

fffr

dtdhdthdi

dtidh. ( 2-117)

2.5.9 Summary Of Equations In Tangential-Normal (t, n, z) Axes [6]

tfva

dtda

22= ( 2-118)

+= nt farfe

vdtde sin)cos(21 ( 2-119)

+= nt fefrp

pvrh

dtdh sin ( 2-120)

+= ( 2-121)

dtdirh

dtd

dtd

dtd

dtd =++= cos/

~2

* ( 2-122)

++= nt faref

evdtd )cos2(sin21

~ ( 2-123)

zfihr

dtd

sin)sin( += ( 2-124)

zfhr

dtdi )cos( += ( 2-125)

dtd

dtdi

dtd ~cos = ( 2-126)

CHAPTER 3

3 SATELLITE PERTURBATIONS AND LINEARIZATION

3.1 Satellite Perturbations

This chapter includes and analysis of Earth satellite perturbations as a result from the non-

spherical shape of the Earth, the influence of the Sun, the Moon and any residual atmosphere.

The material of this chapter is based on [3], and [6].

3.1.1 Gravitational Field of the Earth

The Earth is not a perfect sphere but has the form of an oblate spheroid with an equatorial

diameter that exceeds the polar diameter by about 20 km. The resulting equatorial bulge exerts

a force that pulls the satellite back to the equatorial plane whenever it is above or below this

plane and tries to align the orbital plane with the equator.

3.1.1.1 Expansion of Spherical Harmonics

In order to evaluate of the acceleration vector due to non-spherical Earth the expansion of the

potential function is generalized [32] [63]. The Earths gravity potential function is

)),sin()cos()(sin0 0

mSmCPrR

rU nmnmnm

n

n

mn

n

+= = =

( 3-1)

with coefficients

S)S()cos()(sin)!()!(2 3''0 dmP

Rs

mnmn

MC nmn

nm

nm +=

( 3-2)

S)S()sin()(sin)!()!(2 3''0 dmP

Rs

mnmn

MS nmn

nm

nm +=

, ( 3-3)

which describe the depends of the Earths internal mass distribution. Geopotential Coefficients

with m=0 are called zonal coefficients, since they describe the part of the potential that does not

depend on the longitude, the Legendre polynomials is

nn

n

nn udud

nuP )1(

!21)( 2 = , ( 3-4)

with degree n, and the associated Legendre polynomial of degree n and order m is defined as

)()1()( 2/2 uPduduuP nm

mm

nm = , ( 3-5)

R is the Equatorial radius of the Earth, r is the satellite position, S is point inside the Earth

3.1Satellite Perturbations


where r > S, is the longitude, is the latitude of the satellite at point r, , is the corresponding quantities for S, (S) is the density at point S. Because the internal mass distribution of the Earth is not known, the geopotential

coefficients cannot be calculated from the defining equation, but have to be determined

indirectly by combined use of satellite tracking, terrestrial gravimetry, and altimeter Data [3].

Joint Gravity Model of order and degree 70 (JGM-3) was issued in 1996 [3]. Although JGM-3

is a very elaborate global gravity model for precision orbit determination, new models are

continuously being developed.

3.1.1.2 Geopotential Gravity Acceleration

Several recurrence relations [3] for the evaluation of the Legendre polynomials can be used

where

mPr

RW

mPr

RV

nm

n

nm

nm

n

nm

sin).(sin.

cos).(sin.

1

1

+

+

=

= . ( 3-6)

Satisfy the recurrence relations. The gravity potential may written as

)( nmnmnmnm WSVCRGMU +=

. ( 3-7)

The accelerationr , which is equal to the gradient of U, may be directly calculated from the V&& nm and Wnm as

===mn

nmmn

nmmn

nm zzyyxx,,,

,, &&&&&&&&&&&& , ( 3-8)

with the partial accelerations

}{{

})(.)!(

)!2(

)(.21.

.

1,11,1

1,11,12

0

1,102

)0(

++

++++

>

+

=

++++

=

=

mnnmmnnm

mnnmmnnm

m

nn

m

nm

WSVCmn

mn

WSVCR

GM

VCR

GMx&&

( 3-9)



}{{

})(.)!(

)!2(

)(.21.

.

1,11,1

1,11,12

0

1,102

)0(

++

++++

>

+

=

++++

=

=

mnnmmnnm

mnnmmnnm

m

nn

m

nm

VSWCmn

mn

VSWCR

GM

WCR

GMy&&

( 3-10)

{ })(.)1(. ,1,12 mnnmmnnmnm WSVCmnRGMz ++ +=&& . ( 3-11) The derivations of these equations are given in [64].

The formulas given for yield the acceleration in an Earth-fixed coordinates system. To

obtain the acceleration in an inertial coordinates system some coordinates transformation are

required. Using indices ef and sf to distinguish Earth-fixed from space-fixed coordinates, one has

efT

sfsfef tandt rUrrUr &&&& ).( , ).( == , ( 3-12) where

)(= ZRU ( 3-13) represent the rotation of the inertial system by Greenwich hour angle (GMST) around the z-

axis. And

+++

=1000cossin0sincos

)(zR . ( 3-14)

Neglecting long and short-term perturbations of the Earths axis, known as precession and

nutation.

3.1.2 Perturbation from the Sun and the Moon (Point-Mass)

According to the Newtons law of gravity, the acceleration of a satellite by a point mass M is

given by

3rsrsr

= GM&& , ( 3-15)

where r and s are the geocentric coordinates of the satellite and of M, respectively. To describe

the satellites motion with respect to the center of the Earth, the value r in equation (3-15)

refers to an inertial coordinate system in which the Earth is not at rest, but is itself subject to

acceleration due to M.

&&

3ssr GM=&& . ( 3-16)



Both values have to be subtracted to obtain the second derivative

33 ss

rsrsr

= GM&& ( 3-17)

Of the satellites Earth-centered Vector.

3.1.3 Solar Radiation Pressure

A satellite that is exposed to solar radiation experience a small force that is arises from the

absorption or reflection of photons. In contrast to the gravitational perturbations, the

acceleration due to the solar radiation depends on the satellite mass and surface area. Due to the

eccentricity of the Earths orbit, the distance between an Earth-orbiting satellite and Sun varies

between 147106 km and 152106 km during the course of the year. This results in an annual

variation of the solar radiation pressure by about 3.3%. For typical materials used in the

construction of satellites, the reflectivity lies in the range from 0.2 to 0.9. For many applications, (e.g. satellites with large solar arrays), it suffices to assume that the surface points

in the direction of the Sun one can obtain the following expression for the acceleration of the

satellite due to the solar radiation pressure [3]

23 AU

rr

= rmACP R&& , ( 3-18)

where is the solar radiation pressure, A is surface area of the satellite, m is the satellite mass,

is the geocentric position vector of the Sun, and AU is the astronomical unit (the semi-major

axis of the Earths orbit about the Sun=1.49597910

P

r8 km). CR is the radiation pressure

coefficient stands for

CR = 1 + . ( 3-19) The previous equation is commonly used in orbit determination programs with the option of

estimation of CR as a free parameter. Orbital perturbations due to the solar radiation pressure

may be thus account with high precision, even if no details of the satellite structure, orientation

and reflectivity are known.

3.1.4 Atmospheric Drag Acceleration

Atmospheric forces represent the largest non-gravitational perturbations acting on low altitude

satellites. The dominant atmospheric force acting on low altitude satellites, called drag is

directed opposite to the velocity of the satellite motion with respect to the atmospheric flux,

hence decelerating the satellite. Consider a small element mass m of an atmosphere column

that hits the satellites cross-sectional area A in some time interval t

3.2Linearization and Variational Equations


tAvm r= , ( 3-20) where vr is satellite velocity relative to the atmosphere velocity, is the atmospheric density at

the location of the satellite. The impulse dp exerted on the satellite is then given by

tAvmvp rr == 2 , ( 3-21) which is related to the resulting force F by F=p/t. The satellite acceleration due to the drag

can therefore be written as [3]

vrD vmAC er 2

21 =&& , ( 3-22)

where m is the satellite mass and the drag coefficient CD is dimensionless quantity that

describes the interaction of the atmosphere with the satellites surface material. Typical values

of CD range from 1.5-3.0, and are commonly estimated as free parameters in orbit determination

programs. The direction of the drag acceleration is always anti-parallel to the relative velocity

vector indicated by the unit vector ev= vr/vr As the drag force depends on the atmospheric

density at the satellite location, the modeling of the complex properties and dynamics of the Earths atmosphere is a challenging task of modern precision orbit determination. A Varity of

more or less complicated atmospheric models have been established recently, with typical

density differences for different models of about 20% at a lower of 300 km, even increasing at

higher altitudes. There exist relatively simple atmospheric models that already allow for a

reasonable atmospheric density computation. The algorithm of Harris-Priester [65, 66] is still

widely used as a standard atmosphere and may be adequate for many applications.

3.2 Linearization and Variational Equations

The state vector at some specified epoch at t0 determines the form of the orbit and its orientation

in space. TTT ttt ))(),(()( 000 vry = ( 3-23)

),()()(

0660

tttyty =

( 3-24)

The state transition matrix (t,t0) is described as any change of the initial state vector at t0

results in a change of position and velocity of the two-body at a later epoch t. It is to take into

account at least the major perturbations in the computation of (t,t0). As with the treatment of

the perturbed satellite motion, one may not obtain an analytical solution anymore in this case,

but has to solve a special set of differential equations the variational equations by numerical

method. Aside from the increased accuracy that may be obtained by accounting for



perturbations, the concept of the variational equations offers the advantage that it is not limited

to the computation of the state transition matrix, but may also be extended to the treatment of

partial derivatives with respect to force model parameters.

3.2.1 The Differential Equation of the State Transition Matrix

The differential equation, which describes the change of the state transition matrix with time,

follows from the equation of motion of the satellite. The state transition matrix may therefore be

obtained from [3]

),()(

),,()(

),,(10

),( 066

3333

0 ttt

tt

tttdtd

=

vvrr

rvrr &&&& ( 3-25)

and the initial value (t0, t0)= 166, where is the acceleration vector and r, v are the position

and velocity respectively.

r&&

3.2.2 The Differential Equation of the Sensitivity Matrix

The sensitivity matrix S(t,t0) determines the different forces acting on the satellite.

),()()(

0660

tttt S

py =

, ( 3-26)

where the parameter vector p (pi (i=1,,ni) may contain the drag and the radiation pressure

coefficient (CD,CR), the thrust level of a maneuver or the size of certain gravity coefficients. The

differential equation of the sensitivity matrix that gives the partial derivatives of the state vector

with respect to the force model parameter vector may be obtained in a completely analogous

way, yielding [3]

p

p

n

n

t

ttt

tdtd

+

=

6

33

3333

6

)(),,(

0

)()(

),,()(

),,(10

)(

rpvrr

Sv

pvrrr

pvrrS

&&

&&&&

. ( 3-27)

Since the state vector at t0 does not depend on any force model parameter, the initial value of

the sensitivity matrix is given by S(t0) = 0.

3.2.3 Form and Solution of the Variational Equations

By combining the differential equations for the state transition matrix and the sensitivity matrix

one obtains the form of the variational equations



)6(663

363

66

3333

0

00),(

10),(

p

p

n

n

dtd

+

+

=

prS

vr

rrS &&&&&& , ( 3-28)

which is adequate for use with numerical methods for the solution of second-order differential

equations, by decomposing and S into the variational equation may then be written as

+

+

=

pr

SvrS

rrS r

&&

&&&&&&&&&&&&

630

),(),(),( rrrrr ( 3-29)

=

=

=

=

pv

pr

SS

S

vrv

vrr

)(

)(

)(,(()(

)(,(()(

0)0

0)0

t

t

ttt

ttt

v

r

v

r

. ( 3-30)

3.2.4 Partial Derivative of the Earth Geopotential Acceleration

Due to complex structure of the partial derivative of the Geopotential gravity and a finite

accuracy of the derivative is sufficient it may therefore preferable to replace the rigorous

computation by a simple quotient approximation. This technique is mainly applied to the

computation of the state transition and sensitivity matrix where

rvrrrvrr

rr

+

),,(),,()()( tt

tt &&&&&& , ( 3-31)

where is the geopotential acceleration, r is the satellite position ,v is the satellite velocity in

inertial frame. Good result is obtained by restricting the partial

r&&rr /&& to terms involving the

low-order geopotential coefficient.

Since the acceleration due to the Earths attraction does not depend on the satellites

velocity, the partial derivatives with respect to the position are all that is required to compute

the contribution of the geopotential to the variational equations for the state transition matrix. In

the case of the sensitivity matrix neglecting the influence of Earth rotation parameters on the

acceleration the only model parameters of interest are , CGM nm, and Snm but they are not

considered in most orbit determination programs. This is due to the fact that the estimation of



these force parameters is not possible for individual satellites but requires the simultaneous

consideration of a large set of observations from different satellite orbits.

3.2.5 Partial Derivatives of the Sun and the Moon (Point Mass) Accelerations

According to the perturbation of the Sun and the Moon in an Earth-centered in inertial frame

are given by

33 ss

rssr = rGM&& . ( 3-32)

Only the direct gravitational attraction depends on the satellite coordinates and the partial

derivates of the acceleration with respect to r are therefore given by [3]

=

5333 )(

)()(31srsrsr1

srrr TGM&&

. ( 3-33)

The derivative with respect to the solar or lunar mass M can be computed from

rr &&&&GMGM

1= , ( 3-34)

and are only required in special applications.

3.2.6 Partial Derivative of Solar Radiation Pressure Acceleration

Due to large distance of the Sun the partial derivative of the acceleration with respect to

satellite, position is quite small and may therefore safely be neglected in most applications.

What is more important, however, is the partial derivative

23

1 AUrrr

== rmAP

CC RR&&&& , ( 3-35)

this is required to compute the influence of variation in the radiation pressure coefficient on the

satellite trajectory. This allows the estimation of CR during an orbit determination, which cannot

usually be predicted accurately enough from material constants and the satellite geometry.

3.2.7 Partial derivative of the Atmospheric Drag acceleration

Starting from the basic expression

vrD vmAC er 2

21 =&& , ( 3-36)

for the acceleration due to atmospheric drag the derivative with respect to the drag coefficient is

rrD vmAC vr 2

21 =&& . ( 3-37)

The dependence on the satellite velocity is described by the partial derivatives



)(21 1vv

vr

rr

Trr

D vvmAC +=

&& . ( 3-38)

The partial derivative with respect to position involves a direct term describing the atmospheric

density variations as well as a minor contribution resulting from the changing atmospheric wind

velocity:

rv1vv

rv

rr

+

= r

rr

Trr

DrrD vvmACv

mAC )(

21

21 && . ( 3-39)

The r

describes the dependence of the atmospheric density on the satellite location. Except for the simplistic models like that of Harris-Priester, the complexity of representing atmospheric

density models renders the analytical computation of the density gradient extremely difficult.

3.2.8 Partial of Measurements with Respect to the State Vector

In the computation of partial derivatives that describe the dependence of a measurement on the

instantaneous position and velocity of the satellite one may, to first order, neglect all light-time

effects and consider the geometric measurement equations, only. Both angle and distance

measurements may then be expressed as functions of the topocentric local tangent coordinates

s, which are related to the Earth-centered (geocentric equatorial coordinates), space-fixed

satellite position r and the Earth-fixed station coordinates Ref by

))()(()( efttt RrUEs = , ( 3-40) where U is the matrix describing the transformation from space-fixed to Earth fixed coordinate,

while

+++

=

=

sinsincoscoscoscossinsincossin

0cossin

TZ

TN

TE

eee

E , ( 3-41)

is the orthonormal matrix made by the east, north and zenith unit vectors (local tangential

coordinates), which provide a natural and convenient frame for describing a satellites motion

with respect to an antenna. The mutual conversion between the Cartesian and spherical

coordinates is provided by the relation.

=

=

EEAEA

Z

N

E

sincoscoscossin

sss

s , ( 3-42)

and



)arctan(

),arctan(

22NE

Z

N

E

E

A

sss

ss

+=

=, ( 3-43)

where A and E is the Azimuth and elevation, respectively. The azimuth angle A measures the

longitude in the horizontal plane and is counted positively from North to East. The elevation

angle E specifies the latitude above the horizontal plane and is measured positively to the

zenith.

The partials of a range or angle measurement z may then be expressed as

EUsr d

dzddz = . ( 3-44)

Neglecting the light-time correction and propagation effects, the partial derivative of a range

measurement with respect to the instantaneous position vectors is therefore given by

EUsrs

s

T

= , ( 3-45)

with s = |s|, while the partials with respect to the velocity vanish completely. Using the basic

expression for azimuth and elevation the partial derivatives of azimuth and elevation with

respect to the position vector

EUss

sss

sr

+

+= 0A 2

N2E

E2N

2E

N , ( 3-46)

and

EUs

ss

sss

ss

sss

ssr

+

+

+=

2

2N

2E

2N

2E

2

ZN

2N

2E

2

ZEE . ( 3-47)

As with the range measurements, the geometric angles do not depend on the velocity and the

corresponding partials are equal to zero.

3.2.9 Partial with Respect to Measurement Model Parameters

The precise prediction of an observation for a given satellite position involves various

measurement model parameters like Transponder delay, antenna axis displacement,

measurement biases, station coordinates and others. Since many parameters are of interest only

in specialized applications, the following derivative is restricted to the simple bias values,

which are the most commonly considered measurements model parameters. For measurement

Biases q = z- z*, as defined as the difference between the actual measurement z (affected by the



bias) and the corrected (bias free) value z*, the corresponding partial derivatives

iqz , ( 3-48)

are equal to +1 (if qi = qz is the bias value related to the measurement z) or 0 (if qi is the bias

value of some other measurement type).

CHAPTER 4

4 SATELLITE ORBITS ESTIMATION AND DETERMINATION

4.1 Satellite Tracking and Observation Models

Satellite Orbits determination requires input measurements, the pointing angles and the slant

range, that are related to the satellite's position or velocity. These data are collected by a satellite

tracking system that measures the properties of electromagnetic wave propagation between the

transmitter and the receiver. The transmitter as well as the receiver may be either a ground

station or a satellite. The material of this chapter is based on [36].

4.1.1 Angle Measurements

Antenna auto-track mode may be achieved using the conical scan method, where the antenna

feed performs a slight rotation in such a way that the cone always covers the direction to the

satellite. The amplitude modulation of the received signal leads to an error signal that can be

used to steer the antenna. The mono-pulse technique derives antenna-angle offsets by the

extraction two signals from the satellite beacon. To obtain these signals, the difference signal

and the sum signal single, feed such as a corrugated horn is applied. The sum signal is

essentially applied as a reference for the error signal. The amplitude of the difference signal is

proportional to the amplitude of the antenna-angle offset, while the phase of the difference

signal corresponds to the direction of the offset. The error signal together with the sum signal is

fed to a tracking unit to provide azimuth and elevation error outputs.

In general, angle measurements are severely affected by systemic errors that are due to

calibration deficiencies, thermo-elastic distortions, and wind or snow loads. Within an orbit

determination the systematic angle errors may partially absorbed by the estimation of angle

measurement biases, although the error sources lead, in general, to varying angle errors.

4.1.2 Ranging Measurements

The basic technique to generate ranging signal is common tone-ranging systems. The average

of the uplink and downlink distance This is expressed as an equivalent range value s = (1/2)c,

where c is the speed light, and 2 is the two-way signal travel time. This system modulate the

carrier signal with a sine wave of frequency fo = 100 kHz, which is known as major tone. Upon

reception, the demodulator locks onto the incoming tone and determines its phase with respect

to the outgoing tone. The phase shift is directly proportional to the turn around signal travel

4.1Satellite Tracking and Observation Models


time

02 f = , ( 4-1)

and can be measured with a typical resolution of about = 10-2cyc = 210-2. As a result, the two-way range is obtained with a typical accuracy of = 10-2 c/(2f0) = 15 m. Because the phase shift can only be measured in the interval (0,2), the range measurements suffer from an

in determination or ambiguity of

02 fcs = , ( 4-2)

which amounts to 1500 m in the given example. To overcome this difficulty, the ranging signal

is supplemented by a series of sub-harmonic minor tones, which are derived from the major

tone and coherently modulated on the carrier. A representative sequence of major and minor

tones is given by the frequencies f0 = 100 kHz, f1 = 20 kHz, f2 = 4 kHz, f3 = 800 Hz, f4 = 160 Hz,

f5 == 32 Hz, and f6 = 8 Hz [67]. Here, the turn-around time can uniquely be measured up to a

value of 1/8 s as determined by the lowermost minor-tone frequency.

Figure 4-1 Azimuth and Elevation Angles

Y-axis=North,X-axis= East, Z-axis= Zenuith

4.2Maneuver Implementation


4.2 Maneuver Implementation

4.2.1 Numerical Integration Methods

The high accuracy, which is required in computation of satellite orbit, can only be achieved by

using numerical methods for the solution of the equation of motion. Varieties of methods have

been developed for the numerical integration of ordinary differential equations. Multi-step

methods with the availability of variable-order and step-size are suited for the satellite orbits

from near circular orbits to high eccentricity orbits without any precautions. Due to their

flexibility, variable order and step-size multi-step methods are ideal candidates for use in

general satellite orbit prediction and determination systems.

4.2.2 Satellite Orbits Correction

The three component of a corrective velocity (vn, vt, vz) maneuver affect the 6 orbital elements,

and therefore, it is not common to require the adjustment of all the orbital elements.

Geostationary satellite orbits [26] are assumed to be equatorial orbits with a period equal to the

sidereal day (86164.1 seconds), i.e. corresponding to the daily rotation of the Earth relative to

the stars. A satellite of a circular orbit with radius of approximately 42164 km will appear

stationary to an observer on the earth. Although the perturbations on satellites in geostationary

orbits are very small, they become important due to the tight tolerance arising from the mission

requirements. Station keeping, therefore has to be performed, and the spacecraft is maneuvered

in order to keep it within strict latitude and longitude limit defining a dead-zone. The magnitude

of the dead-zone depends upon the characteristics of the communication antennas and

transponders. It is common with modern communication satellites to require that the satellite

remains stationary relative to the ground within 0.1 degree in both latitude and longitude due

to narrow antenna beam width of the ground transmitter. If the inclination of the orbits drifts

away from the Equator then the satellite will appear to have a daily oscillation in latitude equal

to the magnitude of the non-zero inclination. The changes in the inclination of a geostationary

orbit arise from the effects of the gravitational attraction of the Moon and the Sun. The

perturbations caused by the Sun and the Moon are predominantly out-of-plane effects causing a

change in the inclination and in the right ascension of the orbits ascending node. In-plane

perturbations also occur, but these are second order effects and need to be considered when

extremely tight tolerance, i.e. about 0.03 degree, is required.

4.2Maneuver Implementation


4.2.2.1 Thrust Forces

The maneuver may conveniently be treated as instantaneous velocity increment v occurring at

the impulsive maneuver time tm whenever the thrust duration is small as compared to the orbital

period.

)(v)(v)(v mmm ttt += + . ( 4-3) A substantial amount of propellant is consumed during a single maneuver, which results in

continuous change of the spacecraft mass along the burn. Despite the variety of the spacecraft

propulsion systems, a simple, constant thrust model is sufficient to describe the motion of a

spacecraft during thrust. The propulsion system ejects a mass of propellant per time interval dt

at a velocity ve.

dtmdm &= . ( 4-4) A spacecraft mass m experiences a thrust

em vF &= . ( 4-5) And the acceleration

emm

mvFf

&== . ( 4-6)

Integration over the burn time t, the total velocity increment is given by

0

0)(

1 )(lnvv)(fv0

0

0

0m

ttmdmdttttm

mem

tt

te

+=== ++

( 4-7)

0

1ln(Fvm

tmm

= && . ( 4-8)

Assuming, that a mass has a constant flow rate and making use of the total velocity increment

v, the acceleration may be expressed [3] as

tv

mtmtm

mt

=

0

1ln

1)(

)(f &&

( 4-9)

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1 Introduction 1.1 Background 1.1.1 Satellite Orbits 1.1.1.1 Geostationary Orbits 1.1.1.2 Polar Orbits 1.1.1.3 Inclined Orbits

1.1.2 Orbits Determination and Estimation Methods

2 Orbital Mechanics and Reference Systems 2.1 Kepler Laws 2.2 Julian Date 2.3 Sidereal and Universal Time 2.4 Reference Coordinate Systems 2.5 The Two-Body Problem 2.5.1 Orbital Elements, Energy Integral, and Euler Angles 2.5.2 Position and Velocity from the Orbital Elements 2.5.3 Orbital Elements from the Position and Velocity 2.5.4 Keplers Equation and the Time Dependence of the Motion 2.5.4.1 Solution for Ellipse

2.5.5 Computation Starting from Time in Orbit 2.5.6 Orbital Variation in Keplerian Elements Format 2.5.7 Tangential and Normal Components 2.5.8 Use of Tangential and Normal Component (t, n, z) 2.5.9 Summary Of Equations In Tangential-Normal (t, n, z) Axes [6]

3 Satellite Perturbations and Linearization 3.1 Satellite Perturbations 3.1.1 Gravitational Field of the Earth 3.1.1.1 Expansion of Spherical Harmonics 3.1.1.2 Geopotential Gravity Acceleration

3.1.2 Pe

satellite orbits and flight dynamics_final

Documents

orbital elements

satellite orbits

orbits determination

geostationary orbits

polar orbits

inclined orbits

orbital variation

orbital mechanics