Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 1 / a

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- Solution a

Solution

For a bound Kepler orbit about mass

M

, the specific orbital energy and specific angular momentum are

E = - \frac{GM}{2 a}, L^{2} = GM a (1 - e^{2}) .

The apsidal radii are

r_{-} = a (1 - e), r_{+} = a (1 + e),

equivalently

a = (r_{-} + r_{+}) /2

and

e = (r_{+} - r_{-}) / (r_{+} + r_{-})

. Combining the formulas also gives

r_{-} r_{+} = a^{2} (1 - e^{2}) = \frac{L ^{2} a}{GM} = - \frac{L ^{2}}{2 E} .

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