Kepler's laws

1.  Planetary orbits are ellipses with the sun at one focus.

2.  The radius vector from the sun to the planet sweeps out equal areas in equal times (animated gif courtesy of kepler.nasa.gov):

Here we compare the radius vector sweeping for a perfectly circular orbit (on the right) to the case for an elliptical orbit (on the left)

3.  The square of the orbital period T is proportional to the cube of the semimajor axis a.  If we express T in years and a in AU, this relation becomes

T²  = a³

a is called the semimajor axis of the planet's orbit. It is also the average distance from the planet to the sun.

e is the eccentricity of the orbit. At closest approach to the sun (periapse or perihelion) the planet is a(1-e) from the sun.

The angle from periapse to the planet's position in its orbit is theta, the true anomaly.

The value of a for the Earth defines the distance scale for the solar system. It is called the Astronomical Unit, or AU.

1 AU = 149.6 million kilometers (1.496 X 10⁸ km)

All of the eight planets have prograde orbits.

The Earth's orbit lies in a plane, which is called the ecliptic plane, or just ecliptic.

All planetary orbits are in planes which are close to the ecliptic plane (Pluto's orbit is an exception, which is partly why Pluto was demoted). Thus, the solar system is a disk-shaped system (at least as far as the planets are concerned). This is related to the way that the planetary system formed (leading to coplanar and prograde orbits).

The angle between a planet's orbital plane and the ecliptic plane is called the inclination of the planet's orbit. As we have just said, all planets have small orbital inclinations (non-planet Pluto has a large orbital inclination).
 
 

Hohmann transfers

What is the most efficient way to get from one planet to another?  It's not the way Buck Rogers (or Luke Skywalker) would do it, blasting off in a straight line path. The most efficient transfer orbit was discovered by the German engineer, Walter Hohmann, in 1925.  This diagram shows the way to get from a planet close to the Sun to a planet farther from the Sun with the minimum cost in fuel.  This is how we could get the largest payload from Earth to Mars at the lowest cost.  Click here for a short discussion of why energy is so important for spaceflight!

Initially, our spacecraft is moving with the Earth in its orbit around the sun.  We give it a small additional rocket boost to put it on the red elliptical orbit.  We have to "lead" our target, Mars, so that its position and the spacecraft's position come together at the same point.  When they are close together, we apply another rocket boost to match the spacecraft's orbital velocity to Mars'.  The whole trip to Mars takes about 260 days, doing it this way.

To return to Earth from Mars on a Hohmann trajectory, we just do the same thing in reverse, following the lower part of the red ellipse.  Again, we need to "lead" our target, the Earth.  Because we need to wait for the Earth to be in position to lead properly, we can't go right back to the Earth.  We have to wait 455 days for this, so we're stuck on Mars for more than a year, and the whole round trip takes almost 3 years!  The trip could be speeded up by using more fuel and a non-Hohmann trajectory.