Give the position and velocity at one instant and all subsequent motion is fixed. Three numbers for the position, three for the velocity: six in all.
But position and velocity as such do not show the shape of the orbit. A row of figures does not say whether it is nearly circular or long and thin, or which way it faces.
So the same six are rewritten as a different six: two for the shape, three for the orientation, one to fix the timing. These are the orbital elements.
No information is gained or lost. Only the legibility changes.
The semi-major axis sets the size. By the third law, fixing fixes the period, so alone says how many years a lap takes.
The eccentricity sets how squashed it is: for a circle, longer and thinner as approaches .
The two do not interfere. Changing the size leaves the squashing alone, and changing the squashing leaves the size alone.
The nearest and farthest distances follow from the pair: and .
With the shape fixed, which way it faces in space remains. That takes three angles.
First choose a reference plane: the orbital plane of the Earth for the solar system, the equatorial plane for satellites. How far the plane of the orbit is tipped from it is the inclination .
The two planes meet in a line. How far that line is rotated from a reference direction is the longitude of the ascending node .
Finally, the angle from that line round to the periapsis is the argument of periapsis . Tip the plane with , spin it with , then turn the ellipse within it with .
The orbit itself is now settled. What is missing is where the body is at the moment.
Add where it was at one stated time. That time is the epoch. There are several conventions, and the usual one is the mean anomaly at epoch, .
Two bodies on the same orbit differ only in : same shape, same orientation, and only their present place along the ellipse differs.
That completes the six. and for shape, , and for orientation, for timing. Feed in a time and the position comes out by calculation.
Open a catalogue of asteroids and each line carries six numbers. No photograph, no diagram of the orbit. They are not needed.
For Ceres: , , degrees, degrees, degrees, degrees. More than million asteroids are registered in this form.
Artificial satellites use a slightly different convention, packed into two lines and carrying in addition the rate at which drag is lowering the orbit.
The next article shows that the speed follows from just and the current distance. After that we take up the calculation from time to position.