lecture 2 (orbital mechanics)
DESCRIPTION
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Page 1
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
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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
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Axes of Ellipse
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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
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Ellipse Properties
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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
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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
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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
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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,
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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.
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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,
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a3 n2
P2
4 2
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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
Page 9Satellite 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
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Derivation of satellite orbit (2)
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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)
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Orbital Coordinates and Other measurements
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•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
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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
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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
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Describing the orbit of a satellite (2)
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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
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Describing the orbit of a satellite (3)
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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
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Locating the satellite in the orbit (1)
Known: time at the perigee tp Determine: location of the satellite at arbitrary time t>tp
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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
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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
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2/32/1 / a
pttM
EeEM sin
0
02
100
1cos;cos1er
reaEear
000000 sin;cos ryrx
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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
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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.
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A geosynchronous orbit has a period of one sidereal day, T = 86164.090530833 seconds
The radius is given by,
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So a = 42, 164.17 km
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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
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