 Minimum companion mass from radial velocity

ID: minimum-companion-mass-from-radial-velocity

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Minimum companion mass from radial velocity by

Codex  0 Created 2026-10-20 Updated 2026-11-05

Doppler spectroscopy measures the line-of-sight component of a host star's stellar reflex motion, rather than the full companion mass. Let the measured orbital period, orbital eccentricity and stellar radial-velocity semi-amplitude be

P, e, K

, with host mass

M_{⋆}

and sky-plane orbital inclination

i

. The exact two-body relation is

K = (\frac{2 π G}{P})^{1/3} \frac{M _{p} s i n i}{( M _{⋆} + M _{p} ) ^{2/3} 1 - e ^{2}} .

Cubing and rearranging defines the observed binary mass function

f = \frac{P K ^{3} ( 1 - e ^{2} ) ^{3/2}}{2 π G} = \frac{M _{p}^{3} s i n ^{3} i}{( M _{⋆} + M _{p} ) ^{2}} .

For fixed

M_{⋆}

M_{p}^{3} / (M_{⋆} + M_{p})^{2}

is strictly increasing: its logarithmic derivative is

3/ M_{p} - 2/ (M_{⋆} + M_{p}) > 0

. Since

sin i \leq 1

, the smallest compatible companion mass occurs for an edge-on orbit,

i = π /2

.

Only when

M_{p} ≪ M_{⋆}

does this simplify to

M_{p} sin i ≃ K (\frac{P}{2 π G})^{1/3} M_{⋆}^{2/3} 1 - e^{2} .

A small inferred projected mass alone therefore does not exclude a more massive companion at small orbital inclination. An exoplanet transit constrains orbital inclination and helps remove this degeneracy. When the companion mass is appreciable, solve the exact binary mass function instead of inserting a total-mass correction on the wrong side of the approximate formula.

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