Reading:
Chapter 2-3 (ZG4)
Key Question: | What is the escape velocity from a planet? | |
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Key Principle: | The Energy Equation | |
Key Problem: | How are total energy and angular momentum related to orbital parameters? | |
Key Quote: | "Test next week." - S. Myers |
There are 3 conserved quantities in an isolated mutually gravitating two-body system: linear momentum, angular momentum, and total energy. The energy equation is derived by considering the work done while moving a particle on a path near a gravitating mass (the path integral of F · dr). If we use the substitution
we obtain the expression for the kinetic energy of a body of mass m
If, on the other hand, we substitute the gravitational force due to a mass M at distance r
we obtain the expression for the gravitational potential energy
The total energy is given by the sum (minding the signs)
and thus the energy per unit mass is
This is the all-important energy equation for a body in orbit at radius r with velocity v around a dominant gravitating mass M.
The limiting case r -> infinity is particularly interesting. Examination of the energy equation shows that the body will come to rest (v=0) at infinity when E/m = 0. Thus, at some finite radius r, if the velocity is equal to the escape velocity
then E/m = 0 and the body is marginally bound to mass M, and the orbit is a parabola. If the velocity at r is less than the escape velocity at that r, then the orbit is a closed ellipse. For a velocity greater than the escape velocity, we have an unbound hyperbolic orbit.
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smyers@nrao.edu Steven T. Myers