Astronomy & Celestial Mechanics
the sky, computed
Concept

Minimum mass — where it appears

The quantity M sin i that a radial-velocity orbit measures, because the star's motion is seen only along the line of sight. The true mass is larger by an unknown factor 1/sin i, which is small for most orbits but unbounded for a system seen nearly face-on.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as reflex motion — the same set of essays touches all of them, so they are one junction rather than several.

The wobble of a star with a circular companion. The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.

The star moves, and the mass is a lower bound

A planet is found by watching its star fall towards it. The wobble gives a mass multiplied by the sine of an angle nobody has measured — and the shortfall is not a rounding error.

exoplanets · Reflex velocity One eccentric planet, drawn as two circular ones. Top: the velocity of a star with one companion on an orbit of eccentricity 0.25 (argument of periastron 0.9 radians), in units of its semi-amplitude K, over two orbits — and on top of it the sum of two circular orbits, one at the same period and one at half of it, fitted by taking the first two terms of the exact curve's Fourier series. The second orbit's amplitude comes out at 0.246 of the first, which is the eccentricity to first order. Bottom: the difference between the two, on a scale 12 times magnified; its largest excursion is 0.0889 K and its rms 0.0473 K, second order in the eccentricity. For a planet with K = 30 m/s that is 1.42 m/s rms — below the noise of most instruments — so an eccentric single planet and a pair of planets in a 2:1 period ratio are the same measurement unless the data are good enough to see a term of order e².

One eccentric planet, or two circular ones

The velocity curve of a star with one planet on an eccentric orbit is, to first order in the eccentricity, exactly the curve of two planets on circular orbits with periods in the ratio two to one. The difference between the two readings is a term in the eccentricity squared, and for the modest eccentricities most planets have, it is smaller than the noise.

exoplanets · Reflex velocity

Named alongside it

The objects these essays reach for when they reach for this one.

Radial velocityReflex motionSemi-amplitudeBarycentreDegeneracyDoppler shiftEccentricityFourier seriesHarmonicHot jupiterInclinationPeriodogram

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