An Introduction to Modern Astrophysics · Problem 2.9

The Continuous Spectrum of Light — Carroll & Ostlie

Physics — Astrophysics · time-averaged-potential-energy

Problem

In general, an integral average of some continuous function f(t)f(t) over an interval τ\tau is given by ⟨f(t)⟩=1τ∫0τf(t) dt.\langle f(t) \rangle = \frac{1}{\tau} \int_0^\tau f(t)\, dt. Beginning with an expression for the integral average, prove that ⟨U⟩=−GMμa,\langle U \rangle = -G\frac{M\mu}{a}, a binary system's gravitational potential energy, averaged over one period, equals the value of the instantaneous potential energy of the system when the two masses are separated by the distance aa, the semimajor axis of the orbit of the reduced mass about the center of mass. Hint: You may find the following definite integral useful: ∫02πdθ1+ecos⁡θ=2π1−e2.\int_0^{2\pi} \frac{d\theta}{1 + e\cos\theta} = \frac{2\pi}{\sqrt{1-e^2}}.

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