Hyperbolic, Parabolic or Elliptical Orbit?

[cj]

A comet is first seen at a distance of d AUs from the Sun and is traveling 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.

 

[BobG]

Yes, see post: https://www.physicsforums.com/showthread.php?t=40525

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)