Roche Lobe - Geometry of The Roche Lobe

Geometry of The Roche Lobe

The precise shape of the Roche lobe depends on the mass ratio, and must be evaluated numerically. However, for many purposes it is useful to approximate the Roche lobe as a sphere of the same volume. An approximate formula for the radius of this sphere is

$\frac{r_1}{A} = 0.38+0.2\log\frac{M_1}{M_2}$

for $0.3<\frac{M_1}{M_2}<20$

and

$\frac{r_1}{A} = 0.46224\left(\frac{M_1}{M_1+M_2}\right)^{1/3}$

for $\frac{M_1}{M_2}<0.8$

where A is the semi-major axis of the system and is the radius of the Roche lobe around mass . These formulas are accurate to within about 2%.

Another approximate formula by Eggleton is as follows:

$\frac{r_1}{A} = \frac{0.49q^{2/3}}{0.6q^{2/3} + \ln(1 + q^{1/3})}$

where . This formula gives results up to 1% accuracy over the entire range of .

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