29:52 Characteristics and Origins of the Solar System

Lecture 7; September 12, 2001

The Motion of the Moon and Kepler’s Laws and Planetary Motion

>> Some preliminary announcements

• Watch the skies early on Friday and Saturday morning. You will see the waning crescent moon pass close to Venus (particularly Saturday morning). Observing the skies those mornings will illustrate a number of things we have talked about in class this semester: the fact that the Moon’s orbit lies very close to the ecliptic, and the relationship between the phases of the Moon and where it is in its orbit.
• In class I talked about the Earth-Moon system (more in a couple of weeks). A neat view of the Earth-Moon system is a picture taken by the Mars Odyssey 2001 when it was two million miles out in space, and turned its cameras back toward Earth. Check out this URL
• http://mars.jpl.nasa.gov/odyssey/gallery/latest/latest2001/01apr23a.html

Kepler’s Laws of Planetary Motion

To really understand how the objects are moving in the solar system, and also to understand about the trajectories spacecraft take in interplanetary travel, we have to talk about the properties of orbits.

Although they are called Kepler’s Laws of Planetary Motion they actually apply to all objects, planets, asteroids, comets, etc, that are in the solar system.

There are three of Kepler’s Laws.

(1) All planets move on elliptical orbits, with the Sun at one focus.

An ellipse is the sort of Figure shown in the black dotted line below:

Characteristics

Of an ellipse are as follows:

(1) Definition of an ellipse in terms of distances from foci.

(2) Major and Minor axes (and semimajor axis)

(3) The eccentricity eÎ 0,1

(2) Equal Area law

A line from the Sun to the planet sweeps out equal areas in equal time intervals.

Þ An object on an eccentric orbit will move through a large angular range when it is close to the Sun, and a small angular range when it is far from the Sun.

Þ An object on an elliptical orbit spends forever near Aphelion, and very little time near Perihelion.

(3) The Harmonic Law

There is a relationship between the size of an orbit and the period of an orbit. If we let

The variables be: a= semimajor axis (units of astronomical units) and P = period in years, the Harmonic Law is

A³ = P²

With Kepler’s Laws we can carry out some neat calculations.

A Trip to Mars

How long does it take for a spaceship to travel to Mars? What kind of orbit do we follow?

Since the spaceship is moving in the solar system, it has to obey Kepler’s Laws. The minimum energy orbit is one which has the Earth on the major axis at the point of closest approach to the Sun (perihelion) and Mars at the most distant point on the major axis (aphelion).

From this drawing we see that the major axis = 1.0 + 1.523 = 2.523 astronomical units.

The semimajor axis therefore has to be 1.262 astronomical units. We can apply Kepler’s Laws to figure out the period of the spaceship.

P² = a³ = (1.262)³ = 2.008

P = 1.417 years (total period).

The time for the cruise from Earth to Mars is obviously half that, = 0.708 years=8.5 months.

The cruise back to Earth would obviously take the same amount of time, so the crew would be in interplanetary space for 1.4 years. Add to this the time spent on Mars waiting for a favorable alignment of the planets, and you have a lot of time drinking Tang.

If time permits, we will talk about the following “cleanup” topics:

(1) Eclipses, solar and lunar

(2) History of Astronomy; Aristarchus of Samos