The Mathematics of Orbits

Some Properties of Conic Sections

Circle

Transparency Master 		Origin centred circles are described by the function

x2 + y2 = r2

A perfectly symmetrical point-mass in empty space could be orbited by an infinitesimally small mass in a circular orbit.

Real orbits may approach the circular state, but will differ slightly due to mass distribution anomalies in either of the objects, and velocity discrepancies in the trajectory.

Absolutely perfectly circular orbits do not occur in nature.

The eccentricity of a circle is 0.

Ellipse

Transparency Master 		Ellipses are described by the function

x2/a2 + y2/b2 = 1

This is the general form of all orbits.

An object with negligible mass will, under the influence of a single large mass, orbit the large mass in a trajectory that is in the form of an ellipse, with the large mass being at one focus of the ellipse.

All planets and periodic comets have elliptical orbits with the Sun at one focus.

The eccentricity of an ellipse is between 0 and 1.

Parabola

Transparency Master 		The general form of a parabola, symmetrical with respect to the x axis, is described by the function

y= ax2 +bx +c

The open ends of the parabola eventually become parallel (at infinity). One can think of a parabola as one end of an ellipse which has been "stretched" to infinity.

Parabolic orbits, like circular orbits, are rare (or non existent) in nature.

The eccentricity of a parabola is equal to 1.

Hyperbola

Transparency Master 		Hyperbolae are described by the function

x2/a2 - y2/b2 = 1

The open ends of the hyperbola diverge forever.

One can think of an hyperbola as one end of an ellipse which has been "stretched" beyond infinity so that the opposite end of the ellipse "pops" back into the weird mathematical universe on the opposite side of the y-axis.

This is the general form of all open (non-returning) orbits.

A major fraction of all comets which fall into the inner solar system have hyperbolic orbits. They are one-time visitors to the Sun.

The eccentricity of an hyperbola is greater than 1.

Characterizing an Elliptical Orbit

The figure below shows an elliptical orbit tilted at an angle i with respect to a plane defined by the plane of the Earth's orbit (called the ecliptic).

The dotted black line pointing to the left in the ecliptic is defined to be 0o and is the direction to the vernal equinox.


Transparency Master
There are seven elements which characterize an elliptical orbit:

  • a semi-major axis - one half the major axis;
  • e eccentricity - the ratio of the distance from the centre to the focus and the semi-major axis;
  • i inclination - the angle that the orbit is tilted to the plane of the Earth's orbit (the ecliptic);
  • longitude of the ascending node - the angle between the vernal equinox and the ascending node of the orbit (in the plane of the ecliptic);
  • argument of perihelion - the angle (in the plane of the orbit) from the ascending node to perihelion;
  • P sidereal period - the orbital period with respect to the fixed stars;
  • T - time of perihelion passage (self explanatory).

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