Stellar radial velocity: curve, semi-amplitude K and minimum mass

The "predict" mode starts from a planet and computes the radial velocity curve of its star and its semi-amplitude K. The "weigh" mode starts from a recorded curve and computes the mass function, then the minimum mass m sin i. The figure draws v_r over one period with the instrument’s precision band and, above it, the reflex orbit of the star.

K is equation 12 of Lovis & Fischer (2010), taken up as equation 2.28 in Perryman (2018). The true anomaly ν comes from Kepler’s equation, solved numerically.

Radial velocity

the component of a body’s velocity along the line of sight, counted positive when it recedes.

Semi-amplitude K

half the gap between the two extremes of the curve, obtained by fitting the measurements. The peak-to-peak amplitude is 2K.

m sin i

the product of the planet’s mass and the sine of its orbital inclination. A radial velocity alone gives only this product.

Mass function

f(m) = (m_p sin i)³/(M + m_p)², computed from P, K and e alone. Converting it into a planetary mass requires the stellar mass.

Argument of periastron (ω)

the angle, in the orbital plane, between the line of nodes and the periastron of the star. It sets the shape of the curve, not its period or semi-amplitude.

Reflex orbit

the small orbit of the star around the barycentre, a scaled copy of the planet’s in the ratio of the masses, about 1 048 for Jupiter and the Sun.

Systemic velocity (γ)

the bulk velocity of the system relative to us, a constant added to the whole curve.

• The signal is the ideal Keplerian signal of a single planet: stellar jitter, photon noise, the sampling of observations and the superposition of several planets are not modelled.
• The band drawn is a published instrumental precision, not a detection threshold. Deciding that a planet is detected requires a number of measurements, an error distribution and a statistical test, none of them computed.
• The mass of the star is an assumed input: a ten per cent error on it gives roughly seven per cent on the minimum mass m sin i.
• The systemic velocity is fixed at zero: an additive constant, it changes neither the shape nor the amplitude of the curve, nor the deduced masses.
• In "weigh" mode, the fixed point takes i = 90 degrees in the total mass term, following the convention of the literature. The result is the minimum mass.

• 5 ≤ incl ≤ 90model · refusal

  The inclination runs from 5 to 90 degrees. At 90 degrees, edge-on, the minimum mass equals the true mass; below 5 degrees, sin i falls under 0.087 and the same curve would require a planet eleven times heavier.

  Instead · For a nearly face-on orbit, astrometry replaces radial velocity.

• 0 ≤ ecc ≤ 0.95model · refusal

  The eccentricity runs from 0 to 0.95, a display bound: beyond it, the curve becomes a spike too narrow for the 181 points of the figure.

  Instead · At zero eccentricity, the curve is a sinusoid; this case is included.

• 0.01 ≤ per ≤ 100000practical · refusal

  The period runs from 0.01 to 100 000 days. These are interface bounds; the real lower bound depends on the radius of the star, which is not entered.

  Instead · For an orbit tighter than the stellar radius, tidal destruction (the Roche limit) applies.

A spectrograph measures only the projection of the star’s motion on the line of sight, a fraction sin i of the motion. The curve contains only the product m sin i, a lower bound on the true mass since sin i is at most 1. A planet of 0.47 Jupiter masses may also be a brown dwarf of several tens of Jupiter masses seen nearly face-on. A transit, which forces an inclination near 90 degrees, or astrometry lifts the degeneracy.

An animated spectral line would show nothing: the Doppler shift of a wobble of a few tens of metres per second is about two hundredths of a pixel on a detector. Spectrographs measure it by correlation over thousands of lines.

The radial velocity is the time derivative of the star’s coordinate along the line of sight. That coordinate is periodic, and the integral of its derivative over one period is zero. Without the constant term e cos ω, the time average of the curve would not be zero. The midpoint of the two extremes of the curve equals K e cos ω; it coincides with the average only for a circular orbit or when periastron is perpendicular to the line of sight. At zero eccentricity, ω loses its meaning (a circle has no periastron) and only shifts the phase of the curve.

The published formulas contain G because they express masses in kilograms. Written with nominal gravitational parameters (solar and Jupiter masses, fixed by the International Astronomical Union in 2015), the semi-amplitude, the mass function and both semi-major axes no longer contain G. Its relative uncertainty, two parts in a hundred thousand, does not enter the calculation.

The form is prefilled with the elements published by Michel Mayor and Didier Queloz (Nature, 1995) for the first planet discovered around a solar-type star: P = 4.2293 days, K = 59 m/s, a circular orbit, a star of one solar mass. The "weigh" mode draws from them m sin i = 0.470 Jupiter masses, the value of the article; the "predict" mode, with that mass and an edge-on orbit, returns K = 59.05 m/s. The reflex orbit of the star has a semi-major axis of 3 434 km, and that of the planet is 2 229 times larger. The example precision, 13 m/s, is that of the ELODIE spectrograph used for the discovery; the signal exceeds it by a factor of four and a half. The period of 4.2308 days sometimes attributed to that article is the value refined by later follow-ups.

1. Lovis, C. & Fischer, D., Radial Velocity Techniques for Exoplanets, 2010.

   Location: in Exoplanets, S. Seager ed., University of Arizona Press, eq. 12: the radial velocity semi-amplitude K as a function of period, minimum mass, total mass and eccentricity · accessed 2026-08-26

2. Perryman, M., The Exoplanet Handbook, 2nd edition, 2018.

   Location: Cambridge University Press, eq. 2.28 for the semi-amplitude and the mass function of a single-planet spectroscopic orbit · accessed 2026-08-26

3. Mayor, M. & Queloz, D., A Jupiter-mass companion to a solar-type star, 1995.

   Location: Nature 378, 355: 51 Pegasi b, period 4.2293 d, semi-amplitude 59 m/s, minimum mass 0.47 Jupiter masses, used as a labelled example only · accessed 2026-08-26

4. International Astronomical Union, Resolution B3 on recommended nominal conversion constants, 2015.

   Location: Nominal mass parameters GM = 1.3271244 x 10^20, 1.2668653 x 10^17 and 3.986004 x 10^14 m^3/s^2 exactly for the Sun, Jupiter and the Earth, as tabulated by Prsa et al. 2016, AJ 152:41, table 1 · accessed 2026-08-26

5. Baranne, A. et al., ELODIE: A spectrograph for accurate radial velocity measurements, 1996.

   Location: Astronomy and Astrophysics Supplement Series 119, 373: instrumental precision of about 13 m/s, used as a labelled example only · accessed 2026-08-26

6. Mayor, M. et al., Setting New Standards with HARPS, 2003.

   Location: The Messenger 114: instrumental precision of about 1 m/s, used as a labelled example only · accessed 2026-08-26

Provisional · Sources and assumptions are provided; the documentary review is incomplete.

Physical model. The result follows the significant figures of the least precise input.

revision 1 · reviewed on 26 August 2026

Revision log

1. revision 1 · 2026-08-26 · scope change

   Creation of the tool: radial velocity curve and semi-amplitude K of a star with one planet (Lovis & Fischer 2010 eq. 12, Perryman 2018 eq. 2.28), and an inverse mode giving the mass function, then the minimum mass m sin i.

   For P = 4.2293 d, K = 59 m/s, e = 0 and one solar mass, m sin i = 0.470 Jupiter masses, the value of Mayor & Queloz 1995; the same minimum mass returns K = 59.050 m/s. Jupiter on the Sun gives 12.474 m/s at e = 0.0489 (12.459 m/s at zero eccentricity), the Earth on the Sun 0.089 458 m/s. The time average of v_r is zero to within 1.3 × 10⁻¹³ m/s over 56 (e, ω) pairs, for K = 59 m/s. The envelope of the sweep is K(1 + e) and not K(1 + e)/√(1 − e²), a bound 3.2 times too wide at e = 0.95; G cancels in every formula written with gravitational parameters.