AST-19

Roche-lobe radius

r_L / a ≈ 0.49 q^{2/3} / (0.6 q^{2/3} + ln(1+q^{1/3})). Eggleton fit, q = m/M.

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Governing equation

\dfrac{r_L}{a}\approx\dfrac{0.49 q^{2/3}}{0.6 q^{2/3}+\ln(1+q^{1/3})}

$q \in \left[0.05, 10\right]$

where

q: Mass ratio (—)
a: Separation (R_\odot)
r_L: Roche radius (R_\odot)

The figure, part by part

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①

Governing law — red

\dfrac{r_L}{a}\approx\dfrac{0.49 q^{2/3}}{0.6 q^{2/3}+\ln(1+q^{1/3})}

$q \in \left[0.05, 10\right]$

$q = 0.05$ $r_L = 0.673$

Along the curve

②

Domain of the variable — blue

q_{\min} = 0.05,\quad q_{\max} = 10

$q \in \left[0.05, 10\right]$

$q = 0.05$ $q = 0.05$

Along the curve

③

Response curve — green

\dfrac{r_L}{a}\approx\dfrac{0.49 q^{2/3}}{0.6 q^{2/3}+\ln(1+q^{1/3})}

$q \in \left[0.05, 10\right]$

$q = 0.05$ $r_L = 0.673$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}r_L}{\mathrm{d}q}

$q \in \left[0.05, 10\right]$

$q = 0.113$ $r_L^{\prime} = 2.375$

Along the curve

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-19 — Roche-lobe radius) is the form associated with Roche. Working symbols: $q$ , $a$ $\rightarrow$ $r_L$ . Overflow at r_L starts mass transfer in binaries. q is the mass ratio of the donor.

Purpose

Purpose: compute

r_L

from

q

a

in Astrophysics via

\dfrac{r_L}{a}\approx\dfrac{0.49 q^{2/3}}{0.6 q^{2/3}+\ln(1+q^{1/3})}

r_L / a ≈ 0.49 q^{2/3} / (0.6 q^{2/3} + ln(1+q^{1/3})). Eggleton fit, q = m/M. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

q = 0.500\,\mathrm{—}

a = 4.000\,\mathrm{R_odot}

, the governing relation

\dfrac{r_L}{a}\approx\dfrac{0.49 q^{2/3}}{0.6 q^{2/3}+\ln(1+q^{1/3})}

yields

r_L = 1.283\,\mathrm{R_odot}

. Two stars, a figure-8 equipotential. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass ratio q0.5 —Separation a4 R_\odot

Outputs

Roche radius r_L1.283 R_\odot

Reading speed

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Narration of this film

Two stars, a figure-8 equipotential.

Overflow at r_L starts mass transfer in binaries. q is the mass ratio of the donor.

Reading speed

Watch on YouTube