Orbit Elements and State Vector

As solution to the two body problem you always find elliptical trajectories for the orbiting body, as long as the masses are point masses or show a radially symmetric distribution.

Kepler (1571-1630) found the famous three laws:

Planets orbit 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 ellipse in space and the position of the orbiting body along 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 show the planet's or satellite's position. The true angle between the line (central body - periapsis) and the line (central body - orbiting body) is called True Anomaly. But this angle changes faster when near to the center and slowlier when far from the center (3rd law of KEPLER).
It is much easier to calculate orbits if the angle is changing linearly with time. Indeed there exists such an angle. It is called Mean Anomaly.
Mean Anomaly equals True Anomaly in case of circular orbits. Eccentric orbits are treated like circular orbits (equal semi-major axes) to achieve the desired linear behaviour. Only in two points of the orbit both meet each other: zero degrees and 180 degrees. The relationship is nonlinear and has to be solved iteratively. It's called Kepler's equation.

Planetary Motion is adequately described by giving its 3 spatial coordinates and 3 velocity components. The relationship between the 6 Kepler elements and the 6 coordinates of the State Vector is well defined.

The applet below allows conversions between orbit elements and state vectors. Once you have chosen the central body or entered the product of the gravitational constant times mass of the central body manually (don't forget to hit RETURN after having typed in the number), you may start a conversion. To do so, leave the cursor in one of the textfields on the left (orbit elements) or right (state vector) side and hit RETURN. The corresponding values on the other side will be calculated.

To see the results in different units click the radio button showing the desired unit.
Sometimes it is convenient to enter mean motion (degrees per second or degrees per day, depending on selected units) or period of revolution (seconds or days, depending on selected units) instead of the semi-major axis.

The starting values represent an artificial earth satellite at about 4000 km altitude.

Try the following:

Choose Sun as central 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 and hit RETURN.
Watch the period 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. Semi-major axis' textfield now contains the actual value.

Enjoy the orbit applet.

Dieter Egger, 1997-01-15