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L. Sowmia NarayananProject Manager, Mission Analysis, PSLV Project

VSSC / ISRO

Earth’s Satellite Orbits and its significance

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v A satellite is any smaller object that travels around a larger object.

v Satellites are either called natural or artificial.

v The Moon is a natural satellite to the Earth just like the Earth is a natural satellite to the Sun.

v Artificial satellites are human-made spacecraft that are built and sent into space by people.

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v Each satellite has a set path in space called an orbit.

v The speed and the angle with which a satellite is injected determine the satellite’s orbit.

v The Soviet Union launched the first artificial satellite, Sputnik 1, in 1957. Since then, the United States and about 40 other countries have developed, launched and operated satellites.

v Today, about 3,500 useful satellites and 6,000 pieces of space junk are orbiting Earth

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Earth’s Artificial Satellites• Communications• Remote sensing / Surveillance• Weather forecasting• Space Explorations

Ø Size / Shape / Orientation of the spacecraft orbit depends up on its applications.

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Presentation Topics

• Fundamental laws of Orbital Mechanics• Types of Orbits • Orbital Elements & Its significance• Special orbits• Hohmann Transfer• Orbital plane change • Orbital Perturbations

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Orbital Mechanicsv Study of the motion of artificial

satellites moving under the influence of forces such as Ø gravityØ atmospheric dragØ thrust etc.,

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Basic orbital parameters

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Review of Conics (Calculus of Trigonometry)

Properties of Conics

• Circular (Eccentricity e = 0)• Elliptic ( 0 < e < 1 )• Parabola ( e = 1 )• Hyperbola ( e > 1 )

Eccentricity is the ratio of Focus off-centre to semi-major axis

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Developed by Kepler based on the precision measurements of TychoBrahe, that the moon and the planets moves around the elliptical orbits

In 1609, Kepler’s 1st and 2nd law of Planetary motion :

• 1st Law : The orbit of a planet is an ellipse with the sun in one focal point

• 2nd Law : A line connecting the sun and a planet sweeps equal areas in equal time intervals.

In 1619, Kepler’s 3rd law of Planetary motion :

• 3rd Law : The squares of the planet’s orbit period is proportional to the mean distance to the sun to the third power.

Kepler’s Laws of Planetary Motion: Johannes Kepler (1571 – 1630)

The orbits of the artificial satellites around earth follow the same fundamental laws

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&

Is a also called Central Force filed if m << M

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Equatorial plane

N

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Vcr

mVC

2

=

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13Sowmia / PSLVCircular orbit

Circular orbit : F1 (Centrifugal force) = F2 (Gravitational Force of attraction)

mV2/r = GMm/r2

µ = GM = 398601 km3/s2 ; r = RE + h = 6378.135 + 622 km

Vc for 622 km orbit is 7.546 km/s

=

rGM

Vc

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For Elliptical orbit, Velocity at any instant is given as

For 180 x 36000 km Elliptical orbita = 24468.135 km [0.5 x ((180+6378.135) + (36000+6378.135))]rPerigee = 6558.135 km rApogee = 42378.135 km

Velocity at Perigee altitude is VP = 10.26 km/sVelocity at Apogee altitude is VA = 1.5877 km/s

o

o

Earth

VVP

VA

Elliptical orbit

r pr a

−=

arV

12µ

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Non-Returning Orbits :v Parabolav Hyperbola

* Semi major axis is unbounded

* Useful trajectories for Inter Planetary Missions

Velocity required at Injection :

Vescape = v2 * Vc (For Parabolic orbit)

Escape velocity from Earth surface

Ve = v2 * 7.910 = 11.186 km/s from surface of Earth (Without Drag Loss).

For 622 km altitude, Ve = 10.672 km/s

V > Ve ; It is a Hyperbolic orbit

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A SATELLITE IS AN OBJECT WHICH

FALLS AROUND THE EARTH

SATELLITES AND ORBITS

ELLIPSE

CIRCLE

HYPERBOLA

PARABOLA

Vcircle ~ 7.8 km/s

Vparabola ~ 11.2 km/s

Vsev ~ 16.7 km/s

SEV-(Solar Escape Velocity)

E

VELLIPSE

Escape Velocity

ELLIPSE

V

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Equation of central Force field wrtInertial frame of reference

022

2

22

2

=+

−===

rdtrd

rGMm

dtrd

mmaF

µ

v The above equation is 2nd order differential equation with 3 degrees of freedom.

v Requires 6 Initial values for propagation

This Equation is valid for uniform gravitational filed and for absence of atmosphere and third body effects

Also, this equation is valid only for m <<M

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To define a orbit at any instant (i.e., Initial values for orbit propagation), Following ANY ONE THE THREE sets are required at specified time instant wrt some Reference Frame.

v 6 components of state vector (3 Position & 3 velocity components : x, y, z, u, v & w) ----- (A)

OR

v Orbital or Kepler’s Elements (a, e, i, O, ω & u) --- (B)

OR

v Flight parameters namely, Altitude, Inertial Velocity, Flight path angle, Velocity Azimuth, Latitude & Longitude (h, V, γ, Az, ? & O) ------ (C)

Note : From given any set values (A), other two set values (B or C) can be derived and Vice Versa.

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Reference Frame for Definition of Orbital (Kepler) Elements

O

X

Z

Y

Towards First

point of Aries (γ) (Vernal Equinox)

Completes Right Handed Triad

Towards North

Note : Inertial Frame of reference frozen at specific time (Epoch)

Origin : At centre of Earth

ECI Frame : Earth Centered Inertial Frame

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Definition of Orbital (Kepler) Elements

v Orbital elements (is also called Keplerian Elements), Defines an Orbit, its Orientation about Earth and place the satellite on the ellipse at a particular time.

v Five Orbital Elements are needed to determine the orbit Geometry (a, e, i, O, ω)

[ a & e : determines Size & Shape of the Orbit

i, O & ω : Determines physical orientation of the orbit in space]

v One more element is needed to calculate the position of satellite in the orbit at any time.

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Defines where the satellite is within the orbit with respect to perigee.True Anomaly

Defines the location of the ascending and descending orbit locations with respect to the Earth's equatorial plane.

Right Ascension of the Ascending Node

Defines where the low point, perigee, of the orbit is with respect to the Earth's surface.

Argument of Perigee

Defines the orientation of the orbit with respect to the Earth's equator.iInclination

Defines the shape of the orbit.eEccentricityDefines the size of the orbit.aSemi-major axis

Orbital Elements

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v Definition of Ascending and Descending node

X

Z

YEquatorial plane

Orbital plane

S/C orbit intersects Eq. plane at two points called NodeAscending Node (when S/C cross Eq. Plane from South to North)Descending Node (When S/C cross Eq. plane from North to South)

??Asc. Node

Desc. Node

• Line joining Asc. & Desc. Node along Eq. plane is called as Nodal Line

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Definition of Orbital (Kepler) Elements

Ø RAAN (O ) – Right Ascension of Ascending Node Determines orientation of the nodal line and Direction of S/C Movement.

X

Z

Y

Equatorial plane

Orbital plane

O

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Definition of Orbital (Kepler) Elements

Ø Argument of Perigee (ω ) – Angle between Line of nodes and Perigee.

Ø RAAN & Arg. of Perigee fixes orientation of the orbit W.R.T. Frame of reference

X

Z

Y

Equatorial plane

Orbital plane

Line of Nodes

Perigee Direction

ω

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Definition of Orbital (Kepler) Elements

Ø Reference Epoch (T0) : At start of Orbit Propagation

Ø True Anomaly at reference Epoch (u0) :Angle from Perigee to the satellite position at T0

X

Z

Y

Equatorial plane

Orbital plane

Line of Nodes

Perigee location

u0

+

S/C Position at Reference Epoch

Perigee location

u0

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Non-Uniform Gravity due to Earth OblatenessEarth Gravity Potential can be expressed by Following Non-linear function as

Gravity at any location (Lat & Long) is dependant on above function as

At Equator : Radius = 6378.135 km ; g = 9.80665 m/s2

At Poles : Radius = 6356.783 km

v Effects of Earth’s Oblateness* Secular variations in ω, Ω* Short and long Periodic variations in all

orbital elements

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J2 : Oblateness term J3 : Pear-Shaped term

Due to J4 Due to J5Form of Earth : Including all Harmonics

Form of Earth due to Harmonics

The fact that the Earth is not a sphere but an ellipsoid causes the orbit of a satellite to be perturbed

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Effect of Earth perturbations on satellite orbits

Secular variations in ω, Ω

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Brief introduction to special orbits:1. Sun synchronous polar orbit (SSPO)

- Orbit precession rate

2. GSO : Geo-stationary orbit

3. GTO : Geo-stationary Transfer Orbit (GTO)- Argument of Perigee constraints

4. Molniya Orbit

- Effect of Argument of perigee Drift

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Sun Synchronous Polar Orbit (SSPO)- Orbit precession rate

• The sun synchronous orbit can be defined as the orbit in which the orbital plane rotates in a year in unison with the one revolution / year apparent motion of the sun.

• The advantage of the sun synchronous orbit is that the observation conditions can be kept with a constant solar incident angle.

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Looking from Top view

Orbital plane

Schematic position of Earth wrt Sun & Polar Orbit with out Earth perturbations. Orbit orientation (RAAN – O) is fixed wrtReference frame.

Effect of non Sun-synchronous Polar orbit

Angle between Sun-Earth & Orbital plane makes different angle at each day. Implies different sun illumination at every pass of satellite & day. Comparison of different day photograph is not feasible.

SUN

Earth Position – P1

P2

P3

P4

Day

Night

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Sun-synchronous Orbit

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Sun-synchronous Orbit

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SUN

Orbital plane

Angle between the satellite orbital plane and the direction to the Earth Sun remains constant

North pole of the Earth

Rotational direction of orbital plane

SSPO ENABLES REPEATED OBSERVATIONS AT THE SAME ILLUMINATION

Diagram of the sun-synchronous orbital plane relative to the direction of the Sun

Sun-synchronous Polar orbit

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Processional rate or Drift of Ascending node

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Precession of Orbital Plane due to Earth’s Oblateness

Prograde orbit

Satellite has Easterly velocity component

Retrograde orbit

Satellite has Westerly velocity component

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GSO : Geo-stationary orbit

Ø The orbit with the same earth rotation rate (Orbital period of 24 Hrs) is called an earth synchronous orbit or geosynchronous orbit.

Ø The geosynchronous orbit with an inclination of zero deg (i = 0 deg) is called a geostationary orbitbecause the satellite looks stationary over the equator from a ground surface view.

Ø A Geostationary satellite is useful for covering wide areas. Many meteorological satellites and communication satellites are geosynchronous types.

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Visibility zones of Satellites

GEO-STATIONARY ORBIT FOR COMMUNICATION

Illustrating global communication with the use of three stationary satellites

Arthur Clarke’s SchemeGeo-

synchronous Orbit

Clarke Orbit

36,000 km

Satellite revolves at same angular speed as the earth

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Location of Geo-stationary Satellites of ISRO

GSAT-2 (480E)

INSAT-3E (550E)

INSAT-3C (740E)Kalpana-1 (740EGSAT-3 (740E)) INSAT-3B (830E)

INSAT-2E (830E)

INSAT-3A (93.50E)

Co-located

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GTO : Geo Synchronous Transfer OrbitLaunch Vehicle will place the communication satellites in GTO.Orbital parameters at Injection :

• Perigee & Apogee Altitude• Inclination & Argument of Perigee

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S/C Inclination for planar trajectory

i = Inclination of the OrbitAz = Launch Azimuth at lift-offL0 = Latitude of the launch site

Cos (i) = sin (Az) cos (L0)

For 102 deg Azimuth & SHAR as Launch site, inclination of the orbitWill be 17.8 deg.

Az

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?V1

?V2Hohmann Transfer

• Walter Hohmann - 1925

• Minimum Energy Transfer

• Change from one circular orbit to another circular orbit using Elliptical transfer.

• Assumption : Instantaneous Velocity (Impulse) addition.

• ?V1 = VEP – VCA

• ?V2 = VCC – VEA

VEP = V Elliptical at perigee altitude BVCA = V Circular for orbit AVCC = V Circular for orbit CVEA = V Elliptical at Apogee altitude of B

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Initial CircularParking Orbit

GTO

GSO

Hohmann Transfer

• To Transfer from initial circular parking orbit to GSO

• Perform first burn to transfer to an elliptical orbit which just touches both circular orbits

• Perform second burn to transfer to final circular GEO orbit

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For 180 x 36000 km Elliptical orbit (GTO)

Velocity at Perigee altitude is VP = 10.26 km/s

Velocity at Apogee altitude is VA = 1.587 km/s

Required circular velocity at Apogee is

VCA = (398601/42378.135)0.5 = 3.067 km/s

Delta-V1 to be added to circularize at Apogee is 1.48 km/s

GTO to GSO

R = 42378 km

?V1

VCA

GTO

GSO

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Finite Burn Losses : Due to Low Thrust / Weight ratio

GTO to GSO using very small thrusters like AOCS thrusters ( 22 N)

Finite burn losses :

• Finite burn losses are significant for ( T / W ) < 0.3 and has to be analyzed by numerical simulations

• Finite burn losses may exceed 20%

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i

X

Z

YGSO plane

GTO plane

GTO : Effect of Inclination Correction :

Delta-V required for plane change alone

?V = 2 Vi sin(a/2)

(Valid only for plane change at Equator)

Vi is the velocity of the spacecraft at the instant of plane change manoeuvre)

a is the Angle of plane change.

This imposes two constraints :1. Minimum velocity point is THE OPTIMUM

Location for plane change

2. Minimum Energy required for plane

change at Equator.

?V required for a = 17.8 deg inclination correction at apogee altitude is 0.491 km/s

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i1 i2i1 i2

X OO1 / O2

Inclination change aloneInclination & RAAN change

A

A

Plane change / Delta-V Effect on i Effect on RAANaddition location

1. At Equator i changes (opt) No RAAN change2. At poles No i change RAAN alone changes3. At any location in orbit Both i & RAAN changes respectively

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Energy saving in combined maneuvers for

GTO to GSO

Circular orbit at GSOVs(3.0747 km/s )

Transfer orbitVa (1.609 km/s ) Combined maneuver

Dv(1.831 km/s)

dv2=Va2+Vs2-2*Va*Vs*cos(i)

i = 28.5 deg

For Separate Maneuvres :1. Plane change manevure (For i Correction) DV1 = 0.791 km/s2. Circularization Maneuver (Perigee raising) DV2 = 1.469 km/s

Total DV = 2.260 km/s ; DV saving is 0.429 km/s

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Molniya OrbitØ Apogee altitude = 39863 kmØ Perigee altitude = 504 kmØ Inclination = 63.4 degØ Period = 43082 s (Half day).

Devised by USSR to provide features of GSO with better coverage of the northern latitudes and without large plane change that would be required from their far northern launch sites

Launch from Plesetsk, Russia in due East direction result in 62.8 deg inclination orbit

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v Viewed from Earth fixed coordinates orbit raises alternately above Eurasian continent & North American continents.

v There is a 8 Hr period over the Eurasian continent each day when North American Continent is also in view.

v During that period, a single spacecraft can serve as the communication link between continents.

v A Constellation of Three spacecrafts would provide a continuous direct link.

Molniya Orbit

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Orbit Perturbations

Ø A satellite is always under the influence of disturbing forces, that tend to deviate the satellite orbit from its true Kepleriannature

Ø The sources of these disturbing forces are many. Some of them are,

v Atmospheric Drag

v Earth’s Oblateness

v Solar Radiation

v Third Body Perturbations

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SVD

CD

F 221−=

221

Vm

SD

C

mD

F

Da ρ

−==

m

SD

C

v Atmospheric Drag

Drag Parameter of the satellite is

Drag Force on the satellite is

Drag Acceleration on the satellite is

Drag force affects all orbital parameters except i and O.

Rate change of semi major axis value determines the life time of the spacecraft

Heavier the spacecraft, less is the drag.

Primary effects: Lowering semi-major axis

• For Elliptical orbits apogee is decreased much more than perigee • For circular orbits, it’s an evenly-distributed spiral

( “Ballistic Coefficient” )

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Dragging Down the ISS

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Cosi

ea

RJ

iSin

ea

RJ

iSin

ea

RJ

2212

2

223

225

22212

2

223

1223

23

212

2

2231

0

−

−=Ω

−

−

=

−

−

−=

η

ηω

ηη

&

&

v Earth’s Oblateness Effects

* Secular variations in ω, Ω (Given below)* Short and long Periodic variations in all

orbital elements

Effect is explained in SSPO

Effect is explained in Molniyaorbit

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• Gravity from additional objects complicates matters greatly– No explicit solution exists like the 2-body problem

• Third body effects for Earth-orbiters are primarily due to the Sun and Moon– Affects GEOs more than LEOs

v Third body effects

v Solar radiation Pressure§ Changes eccentricity of the orbit

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Station keeping in GEOThree Major perturbations are :

1. Lunar – Solar Perturbations2. Earth oblateness effects (Ellipticity of equator )3. Solar radiation pressure

A. North – South Control

Ø Perturbations due to Luni-Solar Gravity Effects :

§ Inclination will change at an initial rate of about 0.85 degrees per year. After 26.5 years the object would have an inclination of 15 degrees, decreasing back to zero after another 26.5 years.

§ A lot of energy has to be spent for maneuvers that compensate this tendency.

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Figure of Eight (8) Ground Trace for an Inclined GSO

Figure-8 Ground trace results from the motion of the satellite around its orbit combined with the rotation of Earth.

Longitude (deg)

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Change of the orbital plane by a north thrust, V>0. The dashed line indicates the orbit before the thrust.

GSO : North – South Control

Inclination variation for first 10 years.

On-Orbit Time (Yrs)

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q Period of satellite operation : up to 15 yearsq Typical Delta-V Required : 46 m/s per year

Ø This implies requirement of Most fuel efficient propulsion system (higher Isp system like Plasma / Ion Thrusters)

Station keeping in GEO (Contd..)B. East - West ControlØ Perturbations due to ellipticity of the Earth equator :

• The ellipticity of the Earth equator is causing an East-West drift if the satellite is not placed in one of the stable (75 degrees longitude east, 105 degrees longitude west) or unstable (11 degrees longitude west, 161 degrees longitude east) equilibrium points.

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GSO : East – West Station keeping:

Change in Longitude drift due to Earth’s gravity field

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GSO : East – West Station keeping:

Change in orbit due to an East burn

East burn with a three axis stabilized spacecraft

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Effect of Solar radiation pressure on a Geo-Synchronous Orbit satellites.

• Induces systematic variations in Orbital eccentricity.

• Effect will be smaller for Heavy satellites.

Eccentricity vector shiftOrbital Eccentricity variations

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51.6~ 6700 to 6800( ~ 320 to 410)

Circular / Elliptical

ISS (International Space Station

e ≅ 05512 hrs26,610 (20,232)Semi-Geo synchronous

Navigation (GPS)

ω= 270 dege = 0.7

63.412 hrs26,571 (rP=7971)(ra=45,170)

MolniyaCommunication / Intelligence

e ≅ 028.5 or 57~90 min~ 6700 (~300)Low-Earth OrbitSpace shuttle

e ≅ 0~ 95~90 min~ 6500-7300 (~150-900)

Sun-synchronousRemote Sensing

e ≅ 0~ 024 hrs42,164 (35,786)Geo-stationaryCommunications

OtherInc. (deg)PeriodSemi-major Axis

(Altitude km)Orbit TypeMission

Comparison of satellite orbits

ISS orbit inclination selected to meet both US Shuttle and Russian Soyuz / Progress capability. (Due East Launch from Baikonur gives 45.6 deg inclination, but range safety (china) violation dictates, 51.6 deg inclination launch only)

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Thank you ALL !

Orbital Motion: Reality is More Complicated Than Two Body Motion

Typical trajectory for Interplanetary MissionSowmia / PSLV