Class 2 - Celestial Mechanics I

Reading = Chapter 2 of Moons & Planets, Hartmann.pp47-56

Motivation for studying celestial mechnics is

- to understand the architecture of the solar system

- to understand the evolution of orbits in the solar system (e.g. resonances, migration, etc)

- to understand how to get spacecraft to planets (Hohmann transfers, gravity assist, etc)

The approach taken will be from a planetary science perspective (rather than that of a physicist who is more interested in the underlying physics equations) or the astrodynamicist/engineer (or is interested in accurate naviagation of a particular spacecraft to a particular planet). You are not expected to derive everything from first principles, nor expected to calculate the orbit that leaves Earth at 10.00 am today and gets to Mars on March 31st at 6pm etc., etc,

The first 5000 years of astronomy - Intro. Astro sections 5-9 - in particular, make sure you know what the Copernican Revolution was all about. Why is the Copernican Revolution THE archetype of scientific reasoning?

Kepler's Laws of planetary motion and Newton's Laws of motion and gravitation. If you are rusty on these topics, I suggest you revise Introductory Astronomy sections 6-9. In particular, you might want to play with the Java Applets for Kepler's Laws.

Force of gravity Fg = GMm/r2 - between M and m separated by r

accelaration of mass m due to mass M: g=F/m=GM/2

Example 1: g on Jupiter (at cloud tops) compared with Earth's surface

gJ/gE = [GMJ/RJ2] / [GME/RE2] = MJ/ME x (RE/RJ)2 = 320 / 100 = 3.2

So, the gravitational acceration on Jupiter is 3.2 times larger than on Earth.

Example 2:

g in Low Earth Orbit @ about 600km ~ 1/10 RE

gLEO/gsurface = (Re/1.1Re)2 = 0.83

Contrary to myth, the gravity is not 'zero' in orbit - it is only reduced by 17%. Astronauts and spacecraft are not in 'zero g' but 'falling' around the Earth under the influence of the Earth's gravity.

Kepler's Laws of Planetary Motion

I - orbits are ELLIPSES (semi-major axis, eccentricity) (conic sections and eccentricity)

II - orbitER moves faster when closer to the orbitEE, slower when farther - or conservation of angular momentum - page 50.

III - P2 = a3 - for orbiting the Sun - where the orbital period is in years and the semi-major axis is in Astronomical Units (AU).

Newton's Version of Kepler's 3rd Law - most important equation of astrophysics! Used to 'weigh' the Earth, asteroids and black holes. Derived by balancing the centrifugal and gravitational forces on an object in orbit. This is relatively easy for circular orbits - it gets messier with ellipses (Newton's proof was purely geometrical - it is easier with calculus).

NVK3L is used to determine the mass of the orbitEE - when the orbitER is a significant mass compared with the orbitEE, then the combined mass (M+m) is determined (not each separately) - important for binary stars, very heavy moons (e.g. Charon) and extra-solar planets orbiting class to a star.