The Hale-Bopp comet explored with A level mathematics

Extract

On 15 March 1997, when boarding an aeroplane bound for Spain, I was handed a copy of the Independent newspaper. One page of the paper was devoted to the Hale-Bopp comet; from two pieces of information about the comet’s orbit I set out to find out as much as I could about the orbit using A level mathematics, and in some cases further mathematics. I was interested in finding the fastest and slowest speeds of the comet, and the size of its orbit in relation to the planetary orbits. I also found speeds and times at other points on the orbit, where distances from the sun could be calculated. These findings, along with an approximate calendar for the next orbit of the comet, are presented in a diagram at the end of this article. During the course of my working, I made two discoveries of a more general nature. The first of these is that the radius of curvature of a conic at the point nearest to a focus is equal to the semi-latus rectum. The second was one of Kepler’s laws, that for an elliptical orbit T² ∝ a³, where ^T is the time of a complete orbit, and a is the semi-major axis.