An Introduction to Modern Astrophysics · Problem 7.2

The Classification of Stellar Spectra — Carroll & Ostlie

Physics — Astrophysics · binary-systems-inclination-weighting

Problem

In Problem 2.9 we discussed integral averages that implicitly assumed a probability distribution (or weighting function) that was constant throughout the interval over which the integral was applied. When a normalized weighting function w(τ)w(\tau) is considered, such that ∫0τw(τ) dτ=1,\int_0^\tau w(\tau)\, d\tau = 1, then the integral average of f(τ)f(\tau) becomes ⟨f(τ)⟩=∫0τf(τ) w(τ) dτ.\langle f(\tau) \rangle = \int_0^\tau f(\tau)\, w(\tau)\, d\tau. Comparison with Problem 2.9 reveals that the weighting function implicitly used in that case was w(τ)=1/τw(\tau) = 1/\tau over the interval 0 to τ\tau. In evaluating ⟨sin⁡3i⟩\langle \sin^3 i \rangle between 0 rad and π/2\pi/2 rad (0∘0^\circ and 90∘90^\circ, respectively) as discussed on page 188, it is more likely that the radial velocity variations will be detected if the plane of the orbit is oriented along the line of sight. The weighting function should therefore take into consideration the projection of the plane of the orbital velocity onto the line of sight. ((a)) Select an appropriate weighting function and show that your weighting function is normalized over the interval i=0i = 0 to π/2\pi/2. ((b)) Prove that ⟨sin⁡3i⟩=3π/16\langle \sin^3 i \rangle = 3\pi/16.

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