The Geometry of Orbits:
Ellipses, Parabolas, and Hyperbolas

Ellipses

Closed orbits that have a period: eccentricity = 0 to 0.9999999. These are the most common and interesting orbits because one object is 'captured' and orbits another. Planet, minor planets, comets, and binar stars all have this kind of orbit. The velocity in orbit is alway less than that needed to escape from the central object.

Hyperbolas

Open orbits that do not close: eccentricity > 1. These orbits are useful in describing two objects that pass by each other but are not 'captured' one by the other. They are used to explain the slingshot effect. Velocities along the hyperbolic orbit are always greater than the escape velocity.

Parabolas

Also an open orbit but a special boundary case with eccentricity = 1 (exactly). These orbits describe the case when an object falls in towards another from very far away when they have very little velocity initially. Long period comets have this type of orbit.

This diagram shows the parameters we need to talk about orbits.

• The size is determined by the semi-major axis , a. (This is 1/2 of the longes distance across the ellipse.
• The shape is determined by the eccentricity, e = c /a. c = the displacement of the focus from the center of the ellipse.
• Two special points in the orbit (in this case orbit around the Sun)
  • Perihelion: place in the orbit closest to the Sun
  • Aphelion: place in the orbit farthest from the Sun

This Diagram relates the various quantities using the unique properties of ellipse.

The relationship between a, b and c

results from the fact that the length of a line drawn from one focus to the ellipse then back to the other focus, will be equal to the major axis no matter where on the ellipse you draw the line. The special case when that point is at the end of the semi-minor axis is shown here and Pythagorean's theorem gives the formula.

The area

of an ellipse is similar to that of a circle except instead of r*r it is a*b. b can be written in terms of a and e.

The length of the periheilion and aphelion

is a result of the geometry along the major axis. Perihelion = a - c while the aphelion = a + c

The hyperbola has two branches as shown in the diagram but an orbit only uses one of them. There are relation between the dimensions of the hyperbola in the same way as there is for the ellipse. As an object moves along the hyperbolic orbit farther from the focus, it approaches the motion of a straight line, asymptote line.

• Eccentricity = e = c/a (in this case it is greater than 1)
• The angle (delta) of the asymptote line is related to the eccentricity: sin(delta) = a/c = 1/e
• The relationship c² = a² + b² is very similar to that for the ellipse. And in this case
  b = (e² - 1)^1/2 a
• Perihelion distance = r_p = c - a = (e - 1) a

A parbolic looks very similar to a hyperbola but there are no asymptotic lines and there is only one branch. The parabola can be drawn by locating points equal distant, d, from the line and the focus. The formula that describes the shape of a parabola is called a quadratic and is in the form
y = r_p + x²/4r_p.
In the figure to the right, y is measured to the left and x is measured upward. r_p = perihelion distance

The parameter, p = 2 r_p in all parabolas.