Celestial Mechanics

Physics 201 Lecture 10

• Kepler’s three laws:– The orbit of every planet is an ellipse

with the Sun at one of the foci – A line joining a planet and the Sun

sweeps out equal areas during equal intervals of time

– The square of the orbital period is proportional to the cube of the orbital size

• Newton showed how these laws all follow from

Newton’s triumph was to derive Kepler’s three laws of planetary motion

Everything you must know about ellipses – and nothing more

Orbitalparameters

Geometricparameters

Eccentricity

Orbital period

Polar components of kinetic energy lead to the “effective potential”

Kinetic energy

“Effective potential”

Using the effective potential makes this a one-dimensional problem

The trajectory of the particle does not depend on its mass – to make things a bit simpler, both energy and angular momentum will be implicitly divided by mass.

Planning a trip in space? Manage your Δv budget

The Hohmann transfer orbitLongest time of transitMinimum Δv budget

“Get to low-Earth orbit, and you’re halfway to anywhere in the solar system” – Robert A. Heinlein

Gravitational slingshot can be a huge source of Δv – e.g., Cassini probe to Saturn

Venus 1 Flyby Venus 2

Flyby

EarthFlyby

TargetingManeuver

JupiterFlyby

Escape velocity and unbound orbits

• Adding speed increases energy and makes the orbit more linear

• If total energy is positive, the orbit will not return

• Five probes will escape our solar system…

No return

Escape orbit

Elliptical orbit

Circular orbit

Semi-latus rectum

  Launch date

Distance (AU)

Speed (AU/yr)

Working?

Pioneer 10 Mar 1972 107 2.54 no

Pioneer 11 Apr 1973 86.8 2.40 no

Voyager 2 Aug 1977 100 3.26 yes

Voyager 1 Sep 1977 122 3.60 yesNew

Horizons Jan 2006 24.8 3.21 yes

Is there a future in space?

Phase 1 : 1950 – 1980

“The Glory Days”

Phase 2 : 1980 – 2010

“No Nonsense”

Phase 3 : 2010 – 2040

“Divide & Conquer”

http://upload.wikimedia.org/wikipedia/commons/8/82/Orbitalaltitudes.jpg