 Donor-wind period invariant with fixed retention fraction

ID: donor-wind-period-invariant-with-fixed-retention-fraction

In a circular binary star let the donor star lose mass

\dot{M}_{1} < 0

, with constant retained fraction

f

and

\dot{M}_{2} = - f \dot{M}_{1}

\dot{J} / J = (1 - f) \dot{M}_{1} M_{2} / [M_{1} (M_{1} + M_{2})]

. Differentiating

J = G^{2/3} (P_{orb} /2 π)^{1/3} M_{1} M_{2} (M_{1} + M_{2})^{- 1/3}

and using this loss law gives

\frac{P ˙ _{orb}}{P _{orb}} = - 3 f \frac{M ˙ _{1}}{M _{1}} - 3 \frac{M ˙ _{2}}{M _{2}} - 2 \frac{M ˙ _{1} + M ˙ _{2}}{M _{1} + M _{2}} .

For constant

f

this integrates to

P_{orb} M_{1}^{3 f} M_{2}^{3} (M_{1} + M_{2})^{2} = constant

. If

f

varies, the differential relation remains valid but the first factor must be replaced by

exp (3 \int f d lo g M_{1})

; treating a time-dependent exponent as a fixed power introduces an extra

\dot{f} lo g M_{1}

term.

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