To calculate the period of an orbit, we will derive (and use) Kepler's second law, which states that equal area is swept in equal time. Consider a triangle with one end at the central attractive body and the other two ends separated by an infinitesimally small distance on the ellipse, dx, as shown in Figure 3.10. The area of that triangle to first order in dx is:
where 7 is the flight path angle. The velocity at radius r gives us the rate of change of x:
Recalling the definition of the angular momentum, p, we see that the areal sweep rate is given simply by:
A = nab the time required to complete one circuit is A/(dA/dt) or:
If we go back to Figure 3.1 and apply conservation of energy and conservation of angular momentum at points 1 and 2 in the figure (at apoapsis and periapsis) we can express the angular momentum in terms of r1 and r2. The steps are as follows:
2 r1 r2
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