GN/MAE155B 1

Orbital Mechanics Overview 2

MAE 155B

G. Nacouzi

GN/MAE155B 2

Orbital Mechanics Overview 2

• Summary of first quarter overview– Keplerian motion– Classical orbit parameters

• Orbital perturbations

• Central body observation– Coverage examples using Excel

• Project workshop

GN/MAE155B 3

Introduction: Orbital Mechanics• Motion of satellite is influenced by the gravity field of multiple

bodies, however, two body assumption is usually sufficient. Earth orbiting satellite Two Body approach:

– Central body is earth, assume it has only gravitational influence on S/C, assume M >> m (M, m ~ mass of earth & S/C)

• Gravity effects of secondary bodies including sun, moon and other planets in solar system are ignored

• Gravitational potential function is given by: = GM/r

– Solution assumes bodies are spherically symmetric, point sources (Earth oblateness not accounted for)

– Only gravity and centrifugal forces are present

GN/MAE155B 4

Two Body Motion (or Keplerian Motion)

• Closed form solution for 2 body exists, no explicit soltn exists for N >2, numerical approach needed

• Gravitational field on body is given by:Fg = M m G/R2 where,

M~ Mass of central body; m~ Mass of Satellite

G~ Universal gravity constant

R~ distance between centers of bodies

For a S/C in Low Earth Orbit (LEO), the gravity forces are:

Earth: 0.9 g Sun: 6E-4 g Moon: 3E-6 g Jupiter: 3E-8 g

GN/MAE155B 5

Elliptical Orbit Geometry & Nomenclature

Periapsis

ApoapsisLine of Apsides

R

a c

V

Rpb

• Line of Apsides connects Apoapsis, central body & Periapsis• Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature)

S/C position defined by R & , is called true anomalyR = [Rp (1+e)]/[1+ e cos()]

GN/MAE155B 6

Elliptical Orbit Definition

• Orbit is defined using the 6 classical orbital elements:– Eccentricity, – semi-major axis, – true anomaly: position of

SC on the orbit– inclination, i, is the angle

between orbit plane and equatorial plane

– Argument of Periapsis (). Angle from Ascending Node (AN) to Periapsis. AN: Pt where S/C crosses equatorial plane South to North

- Longitude of Ascending Node ()~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending node

i

Vernal Equinox

AscendingNode

Periapsis

GN/MAE155B 7

Sources of Orbital Perturbations

• Several external forces cause perturbation to spacecraft orbit– 3rd body effects, e.g., sun, moon, other planets– Unsymmetrical central bodies (‘oblateness’

caused by rotation rate of body):• Earth: Requator = 6378 km, Rpolar = 6357 km

– Space Environment: Solar Pressure, drag from rarefied atmosphere

Reference: C. Brown, ‘Elements of SC Design’

GN/MAE155B 8

Relative Importance of Orbit Perturbations

• J2 term accounts for effect from oblate earth•Principal effect above 100 km altitude

• Other terms may also be important depending on application, mission, etc...

Reference: SpacecraftSystems Engineering,Fortescue & Stark

GN/MAE155B 9

Principal Orbital Perturbations

• Earth ‘oblateness’ results in an unsymmetric gravity potential given by:

where ae = equatorial radius, Pn ~ Legendre Polynomial Jn ~ zonal harmonics, w ~ sin (SC declination)

• J2 term causes measurable perturbation which must be accounted for. Main effects:– Regression of nodes

– Rotation of apsides

GM

r1

2

n

ae

r

n

Jn

Pn

w( )

Note:J2~1E-3,J3~1E-6

GN/MAE155B 10

Orbital Perturbation Effects: Regression of Nodes

Regression of Nodes: Equatorial bulge causes component of gravity vector acting on SC to be slightly out of orbit plane

This out of orbit plane componentcauses a slight precession of the orbit plane.

The resulting orbital rotation is called regression of nodes andis approximated using the dominant gravity harmonics term, J2

GN/MAE155B 11

Regression of Nodes

• Regression of nodes is approximated by:

td

d

3 n J2

R2 cos i( )

2a2

1 e2 2

Where, ~ Longitude of the ascending node; R~ Mean equatorial radiusJ2 ~ Zonal coeff.(for earth = 0.001082)n ~ mean motion (sqrt(GM/a3)), a~ semimajor axis

Note: Although regression rate is small for Geo., it is cumulative and must be accounted for

GN/MAE155B 12

Orbital Perturbation: Rotation of Apsides

Rotation of apsides caused by earthoblateness is similar to regression ofnodes. The phenomenon is caused bya higher acceleration near the equatorand a resulting overshoot at periapsis.This only occurs in elliptical orbits.The rate of rotation is given by:

td

d3n J

2 R

24 5 sin i( )

2 4a

21 e

2 2

GN/MAE155B 13

Ground Track• Defined as the trace of nadir positions, as a function of time,

on the central body. Ground track is influenced by:– S/C orbit

– Rotation of central body

– Orbit perturbations

Trace is calculated using spherical trigonometry (no perturbances)sin (La) = sin (i) sin ALa

Lo = + asin(tan (La)/tan(i))+Re

where: Ala ~ (ascending node to SC)

~ Longitude of ascending node I ~ Inclination

Re~Earth rotation rate= 0.0042t (add to west. longitudes, subtract for eastern longitude)

GN/MAE155B 14

Example Ground Trace

Ground tracefrom i= 45 deg

GN/MAE155B 15

Spacecraft Horizon• SC horizon forms a

circle on the spherical surface of the central body, within circle:– SC can be seen from

central body

– Line of sight communication can be established

– SC can observe the central body

GN/MAE155B 16

Central Body Observation

From simple trigonometry:sin(h) = Rs/(Rs+hs) Dh = (Rs+hs) cos(h)Sw~ Swath width = 2 h Rs