Orbit Elements and State Vector
Considering the two body problem you always find elliptical trajectories for the orbiting body as long as the masses may be treated like point masses.

Kepler (1571-1630) discovered the famous laws:

  • Planets are orbiting the Sun along ellipses. The Sun is situated in one of the two focuses (1609)
  • The radius vector from the Sun to the planet sweeps out equal areas in equal times (1609)
  • The cubes of the semimajor axes are proportional to the squares of the revolution periods (1619)

To describe the location and the form of the ellipse you need six parameters:

  • Size and shape are defined by the semi-major axis and the eccentricity
  • The orientation of the orbital plane in space is defined by the (right ascension of the) ascending node and the inclination
  • The position of the orbiting body is defined by the periapsis (point of closest distance to the central body, counted ccw from the ascending node) and the mean anomaly (counted ccw from the periapsis)

Mean Anomaly does not really represent the planet's or satellite's position in case of non-circular orbits. It corresponds to a body orbiting in a circle and completing one revolution in the same time.
The true angle between the line (center body - periapsis) and the line (center body - orbiting body) is called True Anomaly.
True Anomaly does not change constantly because the body is moving fast in the vicinity of the center body and slowly far away (Kepler's 3rd law).
The relationship is nonlinear and has to be solved iteratively (Kepler's equation).

In case of the simple two body problem the relationship between the 6 Kepler elements and the 6 coordinates of the State Vector (position and velocity) is well defined and may be explored with the applet on this page.

Define the center body by picking one out of the list or entering its product of gravitational constant times mass (don't forget to hit RETURN after having typed in the number).

To start the conversion, leave the cursor in one of the textfields that shall become your input and hit RETURN.

To change the units just click the corresponding radio button.
You may optionally enter the mean motion (degrees per second or degrees per day, depending on the selected units) or the period of revolution (seconds or days, depending on the selected units) instead of the semi-major axis.

The starting values are representing an artificial Earth satellite at about 4000 km altitude.

Check out the following:

  • Choose Sun as center body.
  • Select [AU, AU/d] as units.
  • Enter 1 for the semi-major axis, 0 for all other orbit elements.
  • Leave the cursor in one of the textfields on the left hand side and hit RETURN.
  • Watch the time of revolution. It's slightly bigger than the actual length of one year (tropical year about 365.2422 days). That means, that the semi-major axis of the earth's orbit does not exactly match 1 astronomical unit. To find out how big it is, type 365.2422 into the textfield of the revolution period and hit RETURN. Look at the new semi-major axis to see the actual value.

 

Dieter Egger,  last update 2002-02-26