Astronomy & Celestial Mechanics
the sky, computed
Concept

Radial velocity — where it appears

The component of a body's velocity along the line of sight, measured from the shift of its spectral lines. Reaching a metre a second means removing thirty kilometres a second of the observer's own motion and then confronting a floor that is the star's own surface.

Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.

Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

gravitation · The two-body problem One path, five numbers. Left: the apparent path of a star over 4 years, with a proper motion of 193 mas a year and a parallax of 50 mas, at ecliptic latitude 42°. It is one curve and there is nothing in the sky it can be compared against — the reference stars have paths of their own. Right: the same path with a straight line taken out of it. What is left is an ellipse of semi-major axis 50.0 mas and semi-minor axis 33.5 mas, closed and repeating once a year. The two are separated by their time signatures and by nothing else: proper motion is secular and parallax is annual, in a phase the Earth's position fixes in advance. That is why the five parameters can be told apart at all, and why an astrometric catalogue quotes five rather than two — a position without them is a position at one instant, which is not a direction to anything.

Five numbers from one wiggle

A star's path across a plate is a straight line with a one-year ellipse laid on it. Nothing measures either alone — one fit yields five parameters at once, and they are separable only because their time signatures differ.

sky · Parallax The same spectrum, at rest and at 900 km/s. A set of absorption lines at rest and shifted by a radial velocity of 900 km/s. The displacement is proportional to wavelength, so the reddest line here moves 2.06 nm and the bluest 1.18 nm — which is why the measured quantity is the ratio Δλ/λ and not a distance.

A shift in a line is a speedometer, and it works at any distance

A spectral line has a wavelength fixed by physics. Measuring where it actually arrives gives the source's speed toward or away — and the measurement does not degrade with distance.

starlight · The Doppler effect A 28.7 km/s correction, and a 12.5 m/s planet underneath it. Two years of radial velocities of a star at ecliptic latitude 12°, orbited by a companion whose reflex semi-amplitude is 12.5 m/s — the Sun's own, from Jupiter. The upper panel is what the spectrograph measures: the Earth's motion about the barycentre of the solar system, amplitude 28.72 km/s, which is V⊕ cos β to a fraction of a per cent. The planet is in that curve and is 2,297 times smaller than it, which is a line thinner than the stroke it is drawn with. The lower panel is the same data after the correction, and the correction is not a fit: it is computed from an ephemeris, the observatory's position on a rotating deformable Earth, and the star's own coordinates and proper motion. To leave a centimetre a second it has to be right to one part in 2.9·10⁶ — the light-travel time across the Earth's orbit, the relativistic terms, and the fact that the star moves are all inside that budget. What remains is the planet, at 4333 days, and a scatter of 1.2 m/s that is the star rather than the instrument.

The metre per second that is not the star

A line shift is a speedometer, and the speed it reads is mostly the observer's. Getting to a metre a second means removing thirty kilometres of the Earth's own motion to a part in three million, and then confronting a floor that is the star's own surface rather than the instrument.

starlight · The Doppler effect Leavitt's law, through the Milky Way calibrators. Mean absolute magnitude against the logarithm of the pulsation period, for classical Cepheids with independently known distances, and RR Lyrae itself for comparison. The line is M = −2.81 log P − 1.43: a tenfold longer period is 2.81 magnitudes brighter, a factor of 13 in luminosity. The vertical axis runs the astronomers' way, with brighter upward.

A star that tells its distance by how slowly it blinks

Some stars pulsate, and the slow ones are the bright ones. That single correlation turns a clock into a ruler, and it is how the size of the universe was first measured.

stars · Variable stars Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses.

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

stars · Binary stars 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 A 6.2 m s⁻¹ signal at the rotation period, made by no planet at all. Above: the apparent radial velocity of a star carrying one dark spot over 0.4 per cent of its disc, rotating with an equatorial velocity of 3.2 km s⁻¹, over 3 rotations — and beside it a circular-orbit planet of the same period fitted to the same amplitude, 6.2 metres a second. That amplitude is several times the precision of a modern spectrograph and squarely inside the range in which warm sub-Neptunes are claimed, so the two are competing on equal terms. The spot signal is not a sinusoid: the spot is in view for half the rotation and hidden for the other half, so the curve is truncated, and its power at half the period is 1.16 of its power at the period. A Keplerian orbit at the same period has none there at all. Below: the diagnostic that actually settles it. A planet moves the whole spectrum bodily, so every line keeps its shape and the bisector — the locus of midpoints up a line profile — does not move; a spot removes light from one side of the profile, so the bisector tilts in step with the velocity and against it. The two clouds correlate at -0.92 and 0.14. What the picture cannot show is why this took so long to become routine: measuring a bisector to a few metres a second needs a signal-to-noise ratio of several hundred per spectrum, so for two decades the diagnostic existed and could not be applied to the faint stars the interesting claims were about.

A planet that was the star's own rotation

A dark spot rotating across a star removes light from the approaching limb and then the receding one, and the line centroid moves. That is several metres a second at the rotation period, from no planet at all — and two of the most celebrated nearby planets were withdrawn on exactly this evidence.

exoplanets · Reflex velocity A 27.5 m/s velocity the star does not have. Left: a rotating stellar disc, approaching on one side and receding on the other, with the chord a planet of 0.1 stellar radii takes across it at impact parameter 0.5 and a sky-projected obliquity of 0°. Right: the apparent radial velocity that results, computed by covering the disc cell by cell — the flux hidden at each phase and its mean line-of-sight velocity — rather than from a fitted formula. The star's centre of mass does not move at any point in this: the anomaly is entirely a statement about which parts of the line profile are missing. Its amplitude is 27.5 m/s at v sin i = 4.5 km/s, and the two numbers are related by the depth of the transit, since blocking a fraction f of light of mean velocity v shifts a flux-weighted centroid by f·v. The curve is antisymmetric about mid-transit to 0.00% of its own amplitude, which is what an aligned transit gives: equal time on the blue half and the red. Limb darkening is included at u = 0.6, and it matters: it weights the hidden light towards the centre of the disc, where the rotation velocity is smallest.

A velocity measured from a shape

A transiting planet hides part of a rotating disc, so the star's line profile loses a slice at one velocity and its fitted centroid moves. The star has not moved at all — and the lopsidedness of that motion is the whole measurement of whether the orbit lies in the star's own equatorial plane.

exoplanets · Spin–orbit alignment 4000 lines averaged into one profile, and 2.3 m/s out of it. A cross-correlation function: the average absorption profile obtained by shifting a mask of 4000 line positions across a spectrum and summing what falls under it. The faint curves behind are individual lines, each with its own depth, its own width and its own small offset; the heavy curve is what averaging them produces. The velocity is the position of the peak, and its precision is the width divided by the contrast, the signal-to-noise and the square root of the number of lines — 2.3 metres a second here. Nothing about this construction is a measurement of any one line. It is a measurement of where a weighted average of thousands of them sits, and the weights are a choice: a mask built for one spectral type applied to another weights the disagreement between the lines differently, and moves the peak.

A velocity that is an average of lines that disagree

A radial velocity measured to a metre a second is not measured from a line. It is the position of the peak of a cross-correlation against a mask of thousands of lines, and those lines do not agree with each other by hundreds of metres a second — because each one forms at a different depth in an atmosphere that is boiling.

starlight · The Doppler effect Three perturbing masses drawing one curve. Transit-timing residuals for three systems whose perturbing planets are 12, 8, 5 Earth masses — a factor of 2.4 apart — each given the inner-planet eccentricity that the near-resonant theory says will compensate: 0.0000, 0.0166, 0.0363. The three curves have amplitudes of 1.7, 1.7, 1.7 minutes, within 0 per cent of each other, and they are drawn by three separate integrations that were told nothing about the theory used to pick the eccentricities. The eccentricity enters the near-resonant term divided by Δ, the fractional distance from exact resonance — here 0.0267 — so a hundredth of an eccentricity does the work of a factor of two in mass. This is why a transit-timing mass is not a mass until something else fixes the eccentricity, and why the masses that came out of the first years of such fits were systematically lower than the radial-velocity masses of the same planets.

A mass that is only a mass once the eccentricity is known

The near-resonant part of a transit-timing signal carries the perturber's mass and the pair's free eccentricity in the same bracket, divided by the distance from resonance. A hundredth of an eccentricity therefore does the work of a factor of two in mass, and three quite different systems draw one curve.

exoplanets · Transit-timing Io's eclipses, early at opposition and late at conjunction. The delay in the timing of Io's eclipses by Jupiter's shadow caused by the changing distance between the Earth and Jupiter, against days from an opposition, for circular orbits at 1 and 5.2026 AU. At opposition the two planets are closest and the eclipses arrive 8.3 minutes early against the average; 199 days later, near conjunction, they are furthest apart and arrive 8.3 minutes late. The whole swing, 16.6 minutes, is the time light takes to cross the diameter of the Earth's orbit, and the pattern repeats every 399 days, the synodic period. This is transit timing done on a moon in 1676, with a clock that ran on Io's 42.5-hour orbit and a residual that had nothing to do with Io.

A transit late by the width of an orbit

A transiting planet's clock can run fast and slow for a reason that has nothing to do with gravity acting on the planet. If its star is itself in orbit about a distant companion, each transit's light has further or less far to travel, and the timing wanders by the light-travel time across the star's orbit. It is the measurement that first showed light has a speed, made again on a different kind of clock.

exoplanets · Transit-timing 5 perturbers that draw one 58-day timing signal. Every perturbing planet that gives a 3-day transiting planet the same timing signal — a sinusoid with a 57.8-day super-period and an amplitude of 1.27 minutes — placed wide of the nearest first-order commensurabilities inside and outside its orbit, with its mass found by integrating until the amplitude matched. outside, near 4:3 at 4.070 days needs 7.6 Earth masses and would move the star by K = 3.1 m/s; outside, near 3:2 at 4.620 days needs 12.0 Earth masses and would move the star by K = 4.8 m/s; outside, near 2:1 at 6.329 days needs 25.6 Earth masses and would move the star by K = 9.2 m/s; inside, near 3:2 at 1.966 days needs 6.9 Earth masses and would move the star by K = 3.7 m/s; inside, near 2:1 at 1.462 days needs 45.8 Earth masses and would move the star by K = 26.7 m/s. The period ratio is along the bottom on a logarithmic axis and the required mass up the side. A super-period fixes the distance from some resonance and not which resonance it is, and the amplitude then fixes a mass for each guess — so the timing alone returns a list rather than a planet. The velocity semi-amplitudes differ by a factor of 8.5 across the list, which is one of the two ways the list is shortened.

One timing curve and five planets that could draw it

A transiting planet whose times wander at a 58-day period, by just over a minute, is being pulled by something — but the period says only how far from some resonance the pull comes, not from which. Perturbers inside and outside the orbit, near four different commensurabilities, each with its own mass, reproduce the same curve to a fraction of a per cent. Timing alone returns a list, and even the detail that shortens it hides a coincidence of its own.

exoplanets · Transit-timing A 10 Earth-mass Trojan started 10° from its point, seen in the planet's transits. A Jupiter-mass planet on a 4-day orbit about a 1 solar-mass star, with a 10 Earth-mass companion started 10 degrees beyond the leading Lagrange point of the same orbit, integrated for 160 days. Above, the angle between the companion and the planet as seen from the star: it swings about 60 degrees, between 51.1 and 70.3, with a period of 48.5 days measured off the curve, against 49.8 from the small-amplitude formula P/√(27μ/4). Below, the planet's own transit times minus a straight line: they swing by ±300.0 seconds at the same period, because the planet and the companion orbit their common centre of mass and the companion's libration moves that centre along the orbit. The companion shares the planet's period exactly, so it produces no timing signal at any orbital period of its own and would not transit on a schedule distinct from the planet's; the libration period is the only clock it has.

A companion on the same orbit, seen in the planet's clock

A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve.

exoplanets · Transit-timing Three biases against eccentricity, and they do not agree. Four quantities against orbital eccentricity, each relative to a circular orbit of the same semi-major axis, averaged over the argument of periastron. The transit probability rises as (1 − e²)⁻¹, because an eccentric planet spends part of its orbit inside its own semi-major axis: at e = 0.5 a transit is 1.33 times as likely. The transit duration falls as √(1 − e²), so the event carries less signal-to-noise, and the two together — probability times the square root of the time in transit — come to 1.24 at the same eccentricity. They very nearly cancel, and that is the surprise: a transit survey has almost no eccentricity bias at all. The radial-velocity curve is the one that does. A Keplerian of eccentricity e puts less of its variance in the fundamental and more into harmonics no sinusoidal search is looking at — 68 per cent remains at e = 0.6 and 47 per cent at e = 0.8 — so a velocity survey loses amplitude exactly where a transit survey does not. What no figure here can show is which of these the measured eccentricity distribution is made of, because the correction depends on a detection pipeline rather than on geometry, and the two surveys have to be corrected separately before their answers can be compared.

Every method prefers a circle, and not for the same reason

A transit is more likely on an eccentric orbit and shorter when it happens, and the two very nearly cancel. A velocity curve loses amplitude to harmonics no sinusoidal search is looking at, and that one does not cancel at all.

exoplanets · Detection bias How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly.

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

exoplanets · Detection bias

Named alongside it

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

BarycentreTransit-timing variationDoppler shiftLine bisectorN-body integrationPlanet massSelection effectStellar jitterSuper-periodTransit probabilityAstrometryChopping signal

All concepts