lecture 2 (orbital mechanics)

Florida Institute of technologies

Orbit definition and Properties

Kepler’s laws of planetary / satellite motion

Equation of satellite orbits

Describing the orbit of a satellite

Locating the satellite in the orbit

Outline


Orbit Definition and Properties

An orbit is a stable path around the earth traversed periodically by a satellite above the atmosphere of the earth.

Orbits are elliptical

Orbits have an Eccentricity parameter

Certain orbital properties are described by Keppler’s laws


Axes of Ellipse

An ellipse has two axes: a major axis and a minor axis

b ab

a

a: semimajor axis, an ellipse has two semimajor axesb: semiminor axis, an ellipse has two semiminor axes


Ellipse Properties

The sum of the distances from any point P on an ellipse to its two foci is constant and equal to the major diameter

The eccentricity of an ellipse is the ratio of the distance between the two foci and the length of the major axis


Kepler’s laws of planetary motion

Johannes Kepler published laws of planetary motion in solar system in early 17 th century

Laws explained extensive astronomical planetary measurements performed by Tycho Brahe

Kepler’s laws were proved by Newton’s theory of gravity in mid 18 th century

Kepler’s laws approximate motion of satellites around Earth

Kepler’s laws (as applicable to satellite motion)

1. The orbit of a satellite is an ellipse with the Earth at one of the two foci

2. A line joining a satellite and the Earth’s center sweeps out equal areas during equal intervals of time

3. The square of the orbital period of a satellite is directly proportional to the cube of the semi-major axis of its orbit.

cos1.1

epr

const.2 dtdrr

const.3 3

2

aT

Illustration of Kepler’s law


Kepler’s First law

A satellite, as a secondary body, follows an elliptical path around a primary body (earth).

The center of mass of the two bodies, the barycenter, will be at one of the foci.

For semimajor axis a and semiminor axis b, the orbital eccentricity e is be expressed by,


Kepler’s Second law

A ray from the barycenter to an orbiting satellite will sweep out equal areas in the orbital plane in equal time intervals.


Kepler’s Third Law

The square of the orbital time is proportional to the cube of the mean distance, a, between the two bodies (semimajor axis). For a satellite motion of n radians/sec (orbital period P = 2π/n) and the gravitational parameter of the earth, G*M = μ = 3.986004418E5 km3/s2, then the mean distance, a, is calculated as,

a3 n2

P2

4 2


Derivation of satellite orbit (1) Based on Newton’s theory of gravity and laws of motion

Satellite moves in a plane that contains Earth’s origin

Acting force is gravity

Mass of Earth is much larger than the mass of a satellite

Satellite in Earth’s orbit

3rmGM E rF

Gravitational force on the satellite

Newton’s 2nd law

2

2

dtdmm raF

Combining the two

032

2

rdt

d rr

235

24

2211

/skm10983.3

kg1098.5

/kgNm10672.6

EM

G

Constants

Differential equation that determines the orbit


Derivation of satellite orbit (2)

Solution of the motion differential equation gives trajectory in the form of an ellipse

00 cos1 e

pr

moment angular

tyeccentrici

h

hp

e

2

Coordinate system – rotated so that the satellite plane is the same as (X0,Y0) plane

Not all values for eccentricity give stable orbits

Eccentricity in interval (0,1) gives stable elliptical orbit

Eccentricity of 0 gives circular orbit

Eccentricity = 1, parabolic orbit, the satellite escapes the gravitational pull of the Earth

Eccentricity > 1, hyperbolic orbit, the satellite escapes gravitational pull of the Earth

270

300

330

0

e=0.9e=0.5e=0.2e=0

p = 1;e = 0.2fi = 0:0.01:2*pi;r = p./(1+cos(fi));polar(fi,r)


Orbital Coordinates and Other measurements

•Point O is the center of the earth.•Point C is the center of the elli[se.•The orbital plane may be inclined to the earth’s equator.

Apogee height (radius), ra = a(1+e) Perigee height (radius), rp = a(1-e)The flight path angle, θ is,

ro a(1 e2 )

1 ecoso

ea2 b2

a


Describing the orbit of a satellite (1)

E and F are focal points of the ellipse Earth is one of the focal points (say E) a – major semi axis b – minor semi axis Perigee – point when the satellite is closest to

Earth Apogee – point when the satellite is furthest

from Earth The parameters of the orbit are related Five important results:

1. Relationship between a and p

2. Relationship between b and p3. Relationship between eccentricity,

perigee and apogee distances

4. 2nd Kepler’s law

5. 3rd Kepler’s law

00 cos1 e

pr

aFSES 2

Elliptic trajectory – cylindrical coordinates

Basic relationship of ellipse

•Point E is the center of the earth.•Point C is the center of the elli[se.•The orbital plane may be inclined to the earth’s equator



1. Relationship between a and p

2

0000

12

11

02

ep

ep

ep

rra

2

2

2 1/

1 eh

epa

2. Relationship between b and p0

0 cos1 epr

Consider point P: FP+EP=2a

Since FP=EP , EP=a

From triangle CEP

2

2

2

2

2

2

22

222222

1;1

/

1

111

eabe

h

e

pb

ep

epeaeab

3. Relationship between eccentricity, perigee and apogee distances

ap repEAr

epEB

1;

1

errrr

pa

pa



4. 2nd Kepler’s law The area swept by radius vector

hdtdtdtddt

dsrdtvrdsrdsrdA

21

21

21

,sin21,sin

21

000

0000

rrvr

const hdtdA

21

5. 3nd Kepler’s law

dthdA21

hThdtabT

21

21

0

Integrating both sides

32

32

2

32

2

2

32

2

22222

~

4

/4414

aT

aT

aph

pha

heaaT


Locating the satellite in the orbit (1)

Known: time at the perigee tp Determine: location of the satellite at arbitrary time t>tp

Definitions:

S – satelliteO – center of the EarthC – center of the ellipse and corresponding circle

0r - distance between satellite and center of the Earth

0 - “true anomaly”

E - “eccentric anomaly”

A circle is drawn so that it encompasses the satellite’s

elliptical trajectory

2/3

2/12aT - average angular velocity

pttM - mean anomaly


Locating the satellite in the orbit (2)

Algorithm summary:

1. Calculate average angular velocity:

2. Calculate mean anomaly:

3. Solver for eccentric anomaly:

4. Find polar coordinates:

5. Find rectangular coordinates

2/32/1 / a

pttM

EeEM sin

0

02

100

1cos;cos1er

reaEear

000000 sin;cos ryrx


The satellite NOAA-B (1980-43A) was launched in May 1980 into an orbit with perigee height of 260 km and apogee height 1440 km.

We wish to find the orbital period and the orbital eccentricity.

Data:2a = 2re+hp + ha = 2(6378.14)+260+1440 = 14456.28 km

Calculations:a = 7228.14 kmT = 6115.77 sec/orbite = 1 - (re+hp)/a = 0.0816254


Geosynchrounous Orbit

A geosynchronous orbit is an orbit (usually equatorial) having a period of one sidereal day, 23h 56m 04.0905s (23.9344695833 hours, or 86164.090530833 seconds).

A siderial day is the rotation of the earth in relation to the (relatively fixed) position of the stars. Shorter than solar day.


A geosynchronous orbit has a period of one sidereal day, T = 86164.090530833 seconds

The radius is given by,

So a = 42, 164.17 km


Polar Orbit

A polar orbit is an orbit that passes over (or nearly passes over) both North and South poles.

Can be sun-synchronous (heliosynchronous)

Has a low altitude (800 - 1000 km), that is slightly retrograde, and leads to high resolution images with approximately constant illumination angles

Used for weather, environmental, and spy satellites

lecture 2 (orbital mechanics)

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