# How far away from the earth is Halley's comet?

May 11, 2016

Halley's comet is currently close to aphelion and is about 34AU from Earth.

#### Explanation:

Halley's comet has a semi-major axis $a = 17.8$Au, an eccentricity $e = 0.967$ and a period $P = 75.3$ years. It's last perihelion was on 9 February 1986 and its distance from the Sun was 0.586AU. From this we can calculate its position at any time using Kepler's laws.

First we calculate the number of years $n$ since the last perihelion.

We then calculate the mean anomaly $M = \frac{2 \pi n}{P}$.

Then we calculate the eccentric anomaly $E$ from the transcendental equation $M = E = e \sin E$. This has to be solved numerically using for example Newton's method.

The distance of the comet from the Sun is then $r = a \left(1 - e \cos E\right)$AU.

The distance of the comet from Earth is then $r \pm 0.5$AU, depending on the position of the Earth with respect to the comet.

Solving the equations for 11 May 2016 puts the comet at a distance of about 34AU from the Sun. Its aphelion distance is 35.1AU.

Halley's comet doesn't actually obey Kepler's laws as its orbit will be perturbed by any planets it passes near to at any time during its orbit. It is however a reasonable approximation.