# Hyperbolic, Parabolic or Elliptical Orbit?

1. Aug 31, 2004

### cj

A comet is first seen at a distance of d AUs from the Sun and is travelling with a speed of q times the Earth's speed.

Apparently it can be shown that if q2·d is greater than, equal to, or less than 2, then the comet's orbit will be hyperbolic, parabolic or elliptical respectively.

Any idea how this can be shown??

I know that, in general, ε (eccentricity) is less than, equal to, or greater than 1 for an ellipse, parabola, and hyperbola respectively.

2. Aug 31, 2004

### BobG

Just substitute the heliocentric gravitational constant for the geocentric gravitational constant.

The specific energy of object (energy per unit of mass) is just:

$$\frac{v^2}{2}-\frac{\mu_{sun}}{r}=\varepsilon$$
where v is velocity, r is position and $$\mu_{sun}=1.327124421\times 10^{11}km^3/sec^2$$

If the specific energy is less than 0, the object will orbit the Sun. If equal to 0, the object will follow a parabola. If greater than 1, the object will follow a hyperbola.

'e' is normally used to represent eccentricity (depends on the book you're using)