Astronomy & Celestial Mechanics
the sky, computed
Concept

Stellar density — where it appears

A star's mass divided by its volume, delivered independently by a transit's four contact times and by the spacing of a star's own oscillations. It is the one stellar quantity a transit light curve supplies without any assumption about the star, which is why it is used to check a spectroscopic classification.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

stars · Asteroseismology A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the limb-darkened stellar disc over the region the planet covers. The depth is 1.26%, deeper than (Rp/R⋆)² = 0.01055 because the planet crosses a limb-darkened disc whose centre is brighter than its average. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

Four contact points, and what they fix

The depth of a transit gives a radius ratio. The shape gives the impact parameter, and then — through nothing but Kepler's third law — the mean density of the star being crossed.

exoplanets · Transits Two objects, the same 1.055 per cent, and only one of them a planet. A planet of 0.1027 stellar radii transiting at impact parameter 0.3, and a background eclipsing binary of radius ratio 0.62 whose 38 per cent eclipse is diluted by the target's light to 1.055 per cent — the same depth to nine decimal places, because the dilution was chosen to make it so. A blend contributing 2.7 per cent of the light in the aperture can manufacture any planetary depth at all, so the depth is not evidence about what produced it. Two things in the same photometry are. The ingress occupies 20.4 per cent of the planet's transit and 80.3 per cent of the blend's, a factor of 3.9: the shape of the shoulders is set by the radius ratio of whatever is actually eclipsing, and dilution scales a curve without changing its shape. And the duration with the period gives the mean density of the star being crossed — 1.41 g/cm³ here against 0.15, a factor of 9 — so a blend usually implies a host of a completely different kind from the one the spectrum shows. Neither test needs an observation the survey did not already make, and neither of them proves a planet: they reject specific alternatives, and what is left is a probability.

A planet that is never confirmed, only validated

A background eclipsing binary diluted by the target's light reproduces a planetary transit depth exactly, and no amount of better photometry separates the two. Most known planets are therefore the output of a probability calculation rather than a detection, and the honest statement about them is a statement about a false-positive rate.

exoplanets · Transits Lane–Emden solutions for n = 0, 1, 1.5, 3, 4.5, 5, and the one that has no surface. The dimensionless density θ against the dimensionless radius ξ, for polytropic indices 0, 1, 1.5, 3, 4.5, 5. Each curve is the whole structure of a star whose pressure is K times its density to the power 1 + 1/n: the equation of state and hydrostatic equilibrium leave one second-order differential equation, and this is its solution. Every curve starts at θ = 1 with zero slope, because the density is greatest at the centre and has no cusp there. What separates them is where they end. At n = 0 the density is uniform and the surface is at ξ₁ = 2.4495; by n = 3 it has moved out to 6.8968 and the central density is 54.2 times the mean. At n = 5 the curve reaches zero only at infinity — a configuration of infinite radius and, remarkably, finite mass — and every index above it has neither. The three curves that have closed forms, n = 0, 1 and 5, are drawn from the same numerical integration as the rest and agree with those forms to better than two parts in a million, which is what licenses reading the others off the picture. What the figure cannot show is the scale: ξ is radius divided by a length that depends on the central density and on K, so two stars of the same index and wildly different sizes have the same curve here.

An equation of state is already a star

Write down how a gas's pressure depends on its density, insist that the pressure hold the weight up, and everything else follows — the run of density, the fraction of the mass inside each radius, and, at one particular index, a mass that does not care what the radius is.

stars · Polytropes Central pressure bracketed without a model: 6 bodies, 23 decades apart. What can be said about the middle of a body from its mass and its radius alone. The lower end of each bar is GM²/8πR⁴, which follows from hydrostatic equilibrium and nothing else — no equation of state, no composition, no temperature, no assumption whatever about how the density is arranged inside. The upper end costs one more assumption, that the density does not increase outward, and it needs a central density, which is a model output rather than an observation and is why that edge is drawn as the softer one. The dot is what a full structural model gives. For the first five bodies every dot lies inside its bar, and what is worth noticing is how wide the bar is: Sun's rigorous floor is 4.48e+13 pascals against a modelled 2.34e+16, a factor of 522. The bound is true and nearly useless there, because most of a centrally condensed body's pressure comes from the concentration and the derivation deliberately knows nothing about it. The relativistic entry is the exception, and the reason to draw the figure at all. neutron star's modelled central pressure is 8.5 times the Newtonian ceiling — a body no Newtonian arrangement of matter with density falling outward can produce. The floor still holds, and holds for a statable reason: relativity makes the pressure gradient steeper than Newtonian gravity does, so the true central pressure can only exceed what the Newtonian derivation demands. The bracket therefore does more than constrain an interior. Applied at a small enough radius it breaks, and where it breaks is where Newtonian hydrostatics has stopped being the right equation.

A floor under the centre that assumes nothing

There is a lower bound on the pressure at the centre of any body in hydrostatic equilibrium, and it needs no equation of state, no composition and no temperature — only a mass and a radius. For the Sun it is nearly useless. For a neutron star it says which theory of gravity the interior needs.

stars · Hydrostatic equilibrium A transit that lasts 4.0 times longer at one end of the orbit than the other. The duration of a transit, relative to what a circular orbit of the same period around the same star would give, against the orientation of the orbit. A planet transiting near perihelion is moving fastest and its transit is shortest; one transiting near aphelion is slowest and its transit is longest. The two extremes are exact reciprocals — the circular duration is their geometric mean, whatever the eccentricity — and at e = 0.6 they differ by a factor of (1+e)/(1−e), which is 4.0. That is an enormous, easily measured effect, and it means a transit duration is not a stellar density unless the orbit is circular. Turned round, it is a measurement: given a stellar density from asteroseismology or from a parallax and a spectrum, the duration anomaly gives the eccentricity — from photometry alone, with no radial velocities at all.

A duration that measures an eccentricity

A transit's length is a measurement of how fast the planet was moving when it crossed, and that speed depends on where it was on its orbit. For a circular orbit the duration gives the star's density; for an eccentric one it gives the density times a factor of up to four — and if the density is known independently, the factor is the eccentricity.

exoplanets · Transits Four transiting planets, one star, and a density that must agree. The stellar density implied by each of four transiting planets of one simulated star, each fitted as though its orbit were circular, relative to the star's true density, against orbital period, both logarithmic, with one-sigma errors of 8, 10, 14, 20 per cent from the transit fits. Three planets have eccentricities of a few hundredths and agree with one another and with the truth (dashed line: their weighted mean, 0.950). The planet at 15.9 days has e = 0.2 with its periastron 70° from the line of sight, which makes it cross the star faster than a circular orbit would, and its implied density is 2.10 times the star's — 5.2 standard deviations from the others. A single planet's density can be reconciled with any star by choosing its eccentricity; several planets must share one star, and the odd one out is caught.

Every planet of a star must agree on its density

A single transiting planet gives the density of its star only if its orbit is known to be circular, and an eccentric orbit can disguise almost any star as almost any other. A star with four transiting planets has four such measurements of one number. They must agree — and the way they disagree measures how eccentric the orbits are, catches the candidate that orbits some other star, and pins down the star better than its own spectrum does.

exoplanets · Transits Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find.

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

exoplanets · Transits The same data searched with a box and with a sine. The signal-to-noise ratio of the best fit at each trial period, for the light curve of the previous figure, searched two ways: with a box of three trial durations at every phase, and with a sinusoid. The box search peaks at 3.85 days with a signal-to-noise ratio of 7.3, against a true period of 3.85. The sinusoid's best peak anywhere is 3.3, at 1.59 days, and at the true period it reaches only 1.5 — no higher than the peaks the noise makes elsewhere in the search. The planet is a clear detection to one method and invisible to the other, in the same numbers.

A periodogram shaped like a shadow

A periodogram that fits sinusoids sees a transiting planet only through the fundamental harmonic of its box-shaped dip, which for a short transit is a few per cent of the signal. Folding the data at each trial period and fitting a box recovers all of it — five times the signal-to-noise for a transit two per cent of the orbit long, eighteen times for an Earth crossing the Sun. The box pays for its extra parameters in trials, and the price is about one unit of signal-to-noise.

starlight · Periodograms Where each test can name the host. The plane of a pair's separation and brightness difference, divided by which measurement can say which star a 1000-ppm transit crossed. Above Δm ≈ 4 a sub-Neptune on the companion is diluted below detection and there is nothing to assign. Imaging resolves the pair to the right of an illustrative contrast curve, Δm = 5 log(s/0.04″). The centroid's shift, read against the pair's separation and flux ratio, names the host to the right of s = σ(1 + f)/δ for a precision σ of 30 μas, about 0.05″ for near twins. The density from the transit's shape names it between Δm ≈ 1.1, below which the two stars are too alike, and the detection limit — at any separation, but only for strong transits on nearly circular orbits. The corner of close, nearly equal pairs, under about 0.04″ and 1.1 magnitudes, is reached by none.

A transit that says which star it crossed

When a planet transits one star of an unresolved pair, the catalogue usually assumes the brighter star. The light curve itself can say otherwise, in two ways: the transit's shape measures the density of the star crossed, and a red dwarf is three times denser than the Sun; and the centre of the pair's light moves during the transit, away from whichever star was dimmed. Each test works where the other fails, and between them they leave a corner of close, nearly equal pairs that no measurement of the transit reaches.

exoplanets · Detection bias

Named alongside it

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

AsteroseismologyEccentricityHydrostatic equilibriumPhotoeccentric effectTransitArgument of periastronCentral condensationDegeneracy pressureDilutionEquation of stateImpact parameterLikelihood ratio

All concepts