Astronomy & Celestial Mechanics
the sky, computed
Concept

Reflex motion — where it appears

The small orbit a star traces about the centre of mass it shares with a planet. It is the planet's own orbit scaled down by the ratio of the masses, and measuring its velocity along the line of sight is how most planets of known mass were found.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Velocities that stop pointing along the stream. The angle between the stream's direction on the sky, as seen from the Galactic centre, and the mean motion of its stars, along the stream, for Large Magellanic Cloud masses of 0.0, 0.5, 1.5, 2.5 × 10¹¹ solar masses. In a static halo the misalignment stays at 1° or below — the small offset any stream has from its progenitor's orbit. With the Cloud it grows, peaking at 4°, 13°, 32° for the three masses, on the stretch of stream that passed nearest the Cloud. The misalignment is a measurement: with the halo's own potential known, its size along the stream gives the Cloud's mass.

A stream that points the wrong way

In a halo that holds still, the stars of a tidal stream move along it. The Large Magellanic Cloud, falling into the Milky Way for the first time, pulls on a passing stream directly and drags the Galaxy's own centre towards itself, and the stream is bent sideways faster than its stars' motions can follow. Their velocities stop pointing along the track — and how far they stop is a measurement of the Cloud's mass.

galaxies · Stellar streams

Named alongside it

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

Minimum massRadial-velocitySemi-amplitudeBarycentreDark matter haloDegeneracyDoppler shiftDynamical frictionEccentricityFourier seriesGalactic potentialHarmonic

All concepts