Orbital MechanicsOrbital Mechanics

I. Gravitating objects orbit one another in an ellipse, of eccentricity e and semimajor axis length a.

I. Conservation of angular momentum means that objects move faster when they are closest to one focus (e.g. the Sun‏)

I. The square of the orbital period P is propotional to the size of the semimajor axis, a, cubed.

Review: Generalized Kepler’s LawsReview: Generalized Kepler’s Laws

r=a (1−e2 )

1+e cos θ

L=2m πaa2√1−e2

P=mrvθ

P2=4π2 a3

G(M+m )

Circular VelocityCircular Velocity

• A body in circular motion will have a constant velocity determined by the force it must “balance” to stay in orbit.

• By equating the circular acceleration and the acceleration of a mass due to gravity:

v circ=√GMr

where M is the mass of the central body and r is the separation between the orbiting body and the central mass.

• This is convenient because many planet and satellite orbits are close to circular.

Orbital ElementsOrbital Elements

•Semi-major axis (a‏)

•Eccentricity(e‏)

•Inclination (i‏)

•Argument of Pericenter (omega‏)

•Longitude of ascending node

(OMEGA‏)

•True anomaly (nu in this diagram,

also often theta or 'f'‏)

•Periapse – distance of closest

approach between bodies q = a(1-

e‏)

•Apoapse – distance of greatest

separation between bodies Q =

a(1+e‏)

Orbital EnergyOrbital Energy

• In the solar system we observe bodies of all orbital types: planets etc. = elliptical, some nearly circular; comets = elliptical, parabolic, hyperbolic;

e>1E>0v>veschyperbolic

e=1E=0v=vescparabolic

0<e<1E<0v<vescelliptical

e=0E<0v=vcirccircular

eEtotvorbit type

E=−GMm2a

=m2v2−

GMmr

Escape velocityEscape velocity

● Escape velocity is the velocity a mass must have to escape the gravitational pull of the mass to which it is “attracted”.

• We define a mass as being able to escape if it can move to an infinite distance just when its velocity reaches zero. At this point its net energy is zero (E=0‏) and so we have:

GMmr

=12mv

esc2

v esc=√2GMr

Vis-Viva EquationVis-Viva Equation

• This powerful equation does not depend on orbital eccentricity.

• If a new object is observed we measure its current velocity and distance, vis viva allows calculation of the semimajor axis and thus have some idea where it came from.

• Can use energy relations to compute properties of transfer orbits, capture of satellites, speed change needed to go from one orbit to another.

v2(r )=2GM (1r−12a

)

• Since we know the relation between orbital energy, distance, and velocity we can find a general formula which relates them all – the Vis-Viva equation

Vis-viva equation

A meteor is observed to be traveling at a velocity of 42 km/s as it hits

the Earth’s atmosphere. Where did it come from?

Kirkwood gapsKirkwood gaps

• The distribution of asteroid periods (or semi-major axes‏) in the main asteroid belt is not smooth, but shows gaps and peaks

Kirkwood gapsKirkwood gaps

Kirkwood gapsKirkwood gaps

Orbital ResonancesOrbital Resonances(Mean Motion Resonances)(Mean Motion Resonances)

• If the orbital period of a body is a small-integer fraction of another perturbing planet (e.g. Jupiter, Neptune,...etc‏) body’s orbital period, the two bodies are said to be commensurable.

• Some resonances (3:2 resonance of Jupiter‏) actually have a stabilising effect.

• Others may be unstable (see gaps and peaks in the asteroid populations with a‏)

Example:

• An asteroid in a 1:2 resonance with Jupiter completes two revolutions, while Jupiter completes one