Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 3 / ii

Put

m = m_{1} + m_{2}

μ = G m

n = μ / a^{3}

v = na

, and

θ = n t

. Work instantaneously in the inertial frame in which the centre of mass is at rest. The unperturbed circular relative orbit has

r = a (cos θ, sin θ), v = v (- sin θ, cos θ) .

The individual velocities are

v_{1} = - (m_{2} / m) v

and

v_{2} = (m_{1} / m) v

. Linear gas drag gives

F_{i} / m_{i} = - (v_{i} - v_{g} \hat{y}) / t_{i}

, where

t_{i} = t_{s i}

κ = \frac{m _{1} / t _{2} + m _{2} / t _{1}}{m}, Δ = \frac{1}{t _{2}} - \frac{1}{t _{1}} .

f_{x} = κ v sin θ, f_{y} = - κ v cos θ + Δ v_{g} .

Equivalently,

f = - κ v + Δ v_{g} \hat{y}

. If the centre of mass moves,

v_{g} \hat{y}

here is its relative gas velocity, held approximately fixed during one short binary orbit.

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