The Mathematics of Iridium Flare Prediction

By Ben Rush

Over the course of the next few months I will be breaking down a general-purpose system for calculating the time and position of Iridium flares using the most basic mathematics; in the end the whole algorithm will be completed and possible to place into code form for everyday reuse. The system will not take into account various subtleties such as atmospheric drag (and so, over time, the estimates will become far less useful), but will still be useful enough for common, everyday purposes. The first part to understand however – and the purpose of this first article – is exactly what Iridium satellites are and the definitions of the basic numerical properties assigned to them.

 First, what are they? Iridium satellites are a constellation of satellites owned and operated by Iridium Communications, Inc. whose primary function is to provide voice and data capabilities to satellite phones and various other transceivers across our planet. Collectively the network has been in operation since mid-1997 and, today, consists of 66 active satellites. More interestingly, however, is the side-effect of their unique shape and polished antennae: bright “flares” (upwards of -8 magnitude) of sunlight off their bodies when they fall into the right position between the Earth-based observer and the Sun.

Satellites are assigned a range of unique numerical properties with which tracking is made possible. The traditional six are known as the orbital elements, or the Keplerian elements. They each have rather odd names, but their exact definitions are not difficult to understand. Among these are the satellite’s eccentricity, semi-major axis, inclination, longitude of the ascending node, argument of periapsis, and the mean epoch. The diagram below, courtesy of Wikipedia, shows a few of these elements quite clearly.

To summarize,

1. The eccentricity describes the shape of the orbit’s ellipse relative to a perfect circle,
2. the semi-major axis is one half the major axis (or the longest diameter of the ellipse),
3. Inclination is the tilt or angle between the reference plane and the ellipse,
4. Longitude of the ascending node reflects the point at which the orbit crosses through the reference plane and continues “upwards”,
5. Argument of peripasis describes the direction, relative to the longitude of the ascending node, in which the ellipse is “flattened”,
6. And the epoch describes the position of the object along its orbit at a particular point in time.

In the next article we’ll view these numbers for a particular Iridium satellite and begin to piece together how to use them to accurately describe a satellite’s orbit and position along its orbit for any given time, the first step towards knowing exactly where to look in the sky for the satellite