Orbital MechanicsOrbital Mechanics

Part Part 11

Orbital ForcesOrbital Forces

Why a Sat. remains in Why a Sat. remains in orbit ?orbit ?

BcsBcs thethe centrifugalcentrifugal forceforcecausedcaused byby thethe SatSat.. rotationrotationaroundaround earthearth isis countercounter--balancedbalanced byby thethe Earth'sEarth's PullPull..

Kepler’s Laws Kepler’s Laws The Satellite (Spacecraft) which orbits the The Satellite (Spacecraft) which orbits the

earth follows the same laws that govern the earth follows the same laws that govern the motion of the planets around the sun.motion of the planets around the sun.

J. Kepler (J. Kepler (15711571--16301630) was able to derive ) was able to derive empirically three laws describing planetary empirically three laws describing planetary

motion motion I. Newton was able to derive I. Newton was able to derive Keplers laws from his own laws of Keplers laws from his own laws of

mechanics [gravitation theory]mechanics [gravitation theory]

Kepler’s Kepler’s 11stst Law (Law of Orbits) Law (Law of Orbits)

The path followed by a Sat. (secondary body) The path followed by a Sat. (secondary body) orbiting around the primary body will be an orbiting around the primary body will be an ellipse. The center of mass (barycenter) of a ellipse. The center of mass (barycenter) of a twotwo--body system is always centered on one body system is always centered on one

of the foci (earth center).of the foci (earth center).

Kepler’s Kepler’s 11stst Law (Law of Orbits) Law (Law of Orbits)

The eccentricity (abnormality) e:The eccentricity (abnormality) e:

bb-- semiminor axis ,semiminor axis ,aa-- semimajor axissemimajor axis

abae22

VIN:VIN: e=e=0 0 circular orbit circular orbit 00<e<<e<1 1 ellip. orbitellip. orbit

Orbit CalculationsOrbit CalculationsEllipseEllipse is the curve traced by a point moving in a plane such is the curve traced by a point moving in a plane such

that the sum of its distances from the foci is constant.that the sum of its distances from the foci is constant.

Kepler’s Kepler’s 22ndnd Law (Law of Areas) Law (Law of Areas) For equal time intervals, a Sat. will sweep out equal For equal time intervals, a Sat. will sweep out equal

areas in its orbital plane, focused at the barycenterareas in its orbital plane, focused at the barycenter

VIN:VIN: SS11>S>S2 2 at tat t11=t=t2 2 VV11>V>V2 2 Max(V) at Perigee & Min(V) at ApogeeMax(V) at Perigee & Min(V) at Apogee

Kepler’s Kepler’s 33rdrd Law (Harmonic Law) Law (Harmonic Law) The square of the periodic time of orbit is The square of the periodic time of orbit is

proportional to the cube of the mean distance proportional to the cube of the mean distance between the two bodies. between the two bodies.

nn-- mean motion of Sat. (radian/sec) , mean motion of Sat. (radian/sec) , -- earth geocentric gravitational constant = earth geocentric gravitational constant = 33..986005986005xx10101414

The mean distance is equal to the semimajor axis aThe mean distance is equal to the semimajor axis a

VIN: VIN: With n in rad/sec, the orbital period in sec. : With n in rad/sec, the orbital period in sec. : P=P=22/n/n

23

na

Kepler’s Kepler’s 33rdrd Law (Harmonic Law) Law (Harmonic Law)

In English:In English:Orbits with the same Orbits with the same

semisemi--major axis will major axis will have the same periodhave the same period

ExamplesExamples

ExamplesExamples

Orbital ElementsOrbital Elements

A set of mathematical parameters that enables us to A set of mathematical parameters that enables us to accurately describe satellite motionaccurately describe satellite motion

DefinitionDefinition

Discriminate one satellite from other satellitesDiscriminate one satellite from other satellites Predict where a satellite will be in the future or has Predict where a satellite will be in the future or has

been in the pastbeen in the past Determine amount and direction of maneuver or Determine amount and direction of maneuver or

perturbationperturbation

PurposePurpose

SemiSemi--Major Axis (Size)Major Axis (Size) Eccentricity (Shape)Eccentricity (Shape) Inclination Inclination Right AscensionRight Ascension Argument of PerigeeArgument of Perigee Epoch Time (Location within orbit)Epoch Time (Location within orbit) True Anomaly True Anomaly –– Mean AnomalyMean Anomaly

(Orientation)

Orbital ElementsOrbital Elements

Orbital Plane

Equatorial Plane

Inclination ( i)

InclinationInclination

Equatorial: i= 0 or 180

Polar: i = 90

Prograde: 0 i < 90

Retrograde: 90 i < 180

i

Argument of Perigee ()

Right ascension()

True Anomaly ()

Direction of satellite motion

Locate with Respect to SpaceLocate with Respect to Space

Orbital ElementsOrbital Elements

A set of mathematical parameters that enables us to A set of mathematical parameters that enables us to accurately describe satellite motionaccurately describe satellite motion

DefinitionDefinition

Discriminate one satellite from other satellitesDiscriminate one satellite from other satellites Predict where a satellite will be in the future or has Predict where a satellite will be in the future or has

been in the pastbeen in the past Determine amount and direction of maneuver or Determine amount and direction of maneuver or

perturbationperturbation

PurposePurpose

SemiSemi--Major Axis (Size)Major Axis (Size) Eccentricity (Shape)Eccentricity (Shape) Inclination Inclination Right AscensionRight Ascension Argument of PerigeeArgument of Perigee Epoch Time (Location within orbit)Epoch Time (Location within orbit) True Anomaly True Anomaly –– Mean AnomalyMean Anomaly

(Orientation)

Orbital ElementsOrbital Elements

ExampleExample

ExampleExample

Orbit PerturbationsOrbit Perturbations