Orbit Characteristics from Energy

Classical Mechanics Level 2

A small cosmic object of mass $m$ a distance $r$ from a planet of mass $M \gg m$ has total mechanical energy:

$E = -\frac{GMm}{3r}.$

Which of the following correctly describes the behavior of the small object?

It is in an elliptical orbit It is in a circular orbit It is in a hyperbolic orbit It is going to collide with the planet

1 solution

Matt DeCross
May 13, 2016

Relevant wiki: Characteristics of Circular Orbits

The fact that the total mechanical energy is negative tells you that the small object is in a bound state about the planet. Thus the orbit is closed (assuming Newtonian gravity), ruling out the hyperbolic orbit which escapes to infinity.

Since the total mechanical energy is not $E = -\frac{GMm}{2r}$ , the object is not in circular orbit, since this is the characteristic energy of circular orbit. The object is at a higher energy than this, not lower, so the object is not going to inspiral and hit the planet. Rather, the object is in a non-circular bound orbit -- aka, an elliptical orbit.

Why not hyperbolic?

Kaushik Chandra - 3 years, 9 months ago

See first line of the above solution - if energy is negative, the object is trapped in a potential well, and cannot escape to infinity like a hyperbolic orbit. A more fundamental way you can think of this: a hyperbolic orbit reaches infinity, and at infinity there is no gravitational potential, but there may be kinetic. So for such an orbit, total mechanical energy is positive. Since total mechanical energy given in the problem is negative, hyperbolic (and parabolic) orbits are ruled out.

Matt DeCross - 3 years, 8 months ago

It is given that the total mechanical energy is negative hence that must be a circular orbit

Swapnil Shrivastava - 2 years, 4 months ago

The circular orbit occurs only at the minimum point of total mechanical energy. Any point above the minimum where it is negative is an elliptical orbit, since it is bound but the radius varies.

Matt DeCross - 1 year, 10 months ago

Orbit Characteristics from Energy

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