Dr. Green

Dr. Lea

- The planets move in ellipses, with the sun at one focus.
- A line joining the planet to the sun sweeps out equal areas in equal times.
- The square of the orbital period of each planet is proportional to the cube of the semi-major axis of the orbit. That is, if we compare two planets,

OK. So what's an ellipse, and what's a focus?

An ellipse is a curve which has the property that at any point on it, the sum of the distances from that point to each of two fixed points called the foci is a constant.

In the diagram, r

The eccentricity of the ellipse measures its difference from a circle. The two foci are each distance ae from the geometrical center of the ellipse, and the semi-minor axis b is related to the semi-major axis a by:

A circle is an ellipse with e = 0, and so b = a = the radius of the circle.

Kepler didn't know why the planets moved in ellipses. Newton was
the first to realise that if the
sun pulls on each planet with a force which is proportional to
1/(distance)^{2}, then the
planet will move in an ellipse. Newton also realised that Kepler's 2nd
law must hold if the force on each
planet always points toward the sun.

Most planets have almost circular orbits ( e is very much less than 1). Data for the planets are given in the table below.

Planet | Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune | Pluto |

Semi-major axis in AU |
0.387 | 0.723 | 1.00 | 1.524 | 5.203 | 9.54 | 19.18 | 30.06 | 39.44 |

Period | 88 d | 224 d | 1.0 y | 1.88 y | 11.86 y | 29.46 y | 84.01 y | 164.8 y | 248.6 y |

Eccentricity (e) | 0.21 | 0.007 | 0.017 | 0.093 | 0.048 | 0.056 | 0.047 | 0.009 | 0.249 |

Newtons law of gravity says that the force between two objects
decreases as the inverse square of the
distance between them: F = k/d^{2}, where k is a constant. The
brightness of a star
also decreases as the inverse square of its distance from us. The same
amount of light must pass through
the surface of larger and larger spheres surrounding the star,
spreading out over ever larger areas. The
area of a sphere's surface is 4(pi)r^{2}, where r is the radius
of
the sphere. The brightness
of a star is the light energy per unit area falling on our eyes or on a
photographic film, and thus B = L/r^{2}, where L is a constant
that depends on the kind of star.

It was argued that the universe could not be infinite because if it
were, the sky would be infinitely
bright. This is Olber's paradox . While named for Olber, it was
probably first put forward by Kepler.
If we consider a thin shell of the universe, distance r from us, the number of stars in
that shell is the
number n per unit volume (the
density of stars) times the volume of the shell, which is its area
times its
thickness t: N = 4(pi)r^{2}nt. The brightness of each star is B
=
L/r^{2}. Thus the
total brightness of the shell is:

number of stars x brightness of each star

L

= 4(pi)r^{2}nt × --- = 4(pi)ntL, independent of r!

r^{2}

Since each shell has the same brightness, and there are infinitely many shells in an infinite universe, the sky should be blindingly bright at night.

Well, it's not.
Why?

There are two answers.

- The universe is not infinitely old. Light travels at a finite speed, and so light can only have reached us from a finite distance since the stars were born.
- The universe is expanding. As the galaxies rush away from us, their light is red-shifted, so that the most distant stars would not appear visible at all, their light would have been shifted to the infra-red or beyond.