Astronomy & Celestial Mechanics
the sky, computed
Concept

Impact parameter — where it appears

The perpendicular distance from a target to the undeflected line of an incoming body's approach. It is what a flyby is aimed by, and the turning angle depends on it and on the approach speed and on nothing else.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

The aim point and the miss distance are the same number far out and nothing like it close in. Periapsis distance against aim point for a hyperbolic approach to Jupiter at v∞ = 5.6 km/s, both in planet radii. The diagonal is where the two would be equal — where gravity did nothing — and the curve falls below it everywhere, by more the closer in the aim is. A trajectory aimed at 10.68 radii grazes the surface, because gravitational focusing means the planet's effective size is √(1 + v_esc²/v∞²) times its radius. The slope of this curve is what a navigation team cares about: it is 0.208 at an aim of 12 radii and 0.936 at 150, so the same correction manoeuvre changes the periapsis distance by 4.5 times as much at one end of the range as at the other, while changing B by exactly the same amount at both; far outside this plot, where focusing has run out, it reaches 1.000 and the two numbers become the same one. That is why the aim point is the coordinate a manoeuvre is quoted in, why an error ellipse is published in the B-plane, and why the turn angle — 2 arctan(μ/Bv∞²), 156.0° at 12 radii and 77.8° at 70 — is thought of as a function of B rather than of anything the spacecraft does.

Aiming at a plane instead of at a planet

A spacecraft arriving at a planet is not aimed at a periapsis distance. It is aimed at a point in a plane perpendicular to its own incoming asymptote, because that is the one coordinate in which the miss distance responds linearly to a correction — and every navigation product ever published for a flyby is written in it.

orbits · Hyperbolic orbits A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the uniform stellar disc over the region the planet covers. The depth is 1.05%, which is exactly (Rp/R⋆)² = 0.01055. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

A planet measured by the light it removes

A transit gives a depth, and the depth is a ratio of two radii rather than a size. Everything a transit says about a planet is said in units of a star nobody has visited either.

exoplanets · Transits 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 A transit depth of 1.200 per cent for a planet of area 1.000 per cent. Three transits of the same planet across the same star, differing only in how the star's brightness falls toward its edge. A planet of radius ratio 0.1 covers 1.000 per cent of the stellar disc's area, and if the disc were uniformly bright that would be the depth. It is not uniformly bright: a sight line near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer, so the edge is dimmer than the centre, and a planet crossing near the middle blocks light that is brighter than average. The transit drawn with realistic coefficients is 1.200 per cent deep — 20 per cent deeper than the area — and it is also rounder, because the covered brightness changes through the crossing instead of staying flat. The consequence is stated in the numbers beside the curves. Each is a least-squares fit of the radius ratio to the realistic curve, performed with a different assumed limb-darkening law, and the recovered radius moves by up to 3.6 per cent depending on which law is assumed. Fitting with the law the curve was made from returns the input to five figures, which is the control: the bias is the mis-specification and not the fitter. Since the coefficients come from a model atmosphere rather than from the light curve, every published planetary radius carries a systematic from stellar physics that no amount of photometric precision removes — and it is the dominant one for the best-measured planets. The picture holds the impact parameter fixed; a grazing transit is worse, because it samples only the limb, where the disagreement between laws is largest.

The depth is not the area

A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

starlight · Limb darkening Four defensible choices, and a factor of 2.9 between them. The Coulomb logarithm against the ratio of the two impact parameters it is cut off at. The curve is a logarithm, so it is flat — a factor of ten in the ratio buys 2.3 — and that is usually offered as the reason not to worry. The four marked conventions are all in current use and all defensible, and they give ln Λ from 3.4 to 9.9. Since the drag force is proportional to ln Λ and not to its logarithm, that is a factor of 2.9 in every sinking time computed from it. The flatness protects the answer from a wrong guess about the ratio; it does not protect it from there being no correct guess, which is the actual situation.

A drag computed with a logarithm nobody can pin down

Chandrasekhar's drag formula is exact, derived from first principles, and contains a logarithm of a ratio of two lengths that the derivation does not supply. Every sinking time in astronomy is proportional to that logarithm, and the four conventions in current use differ by a factor of three.

gravitation · Dynamical friction 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 A bump that crosses the line from -41 to 25 km/s. The residual of a rotationally broadened line profile during a transit, drawn at five epochs and offset vertically. The planet covers a strip of the stellar disc whose radial velocity is the projected rotation at that position, so it removes light from one velocity and leaves a bump in the residual there. As the planet crosses, the bump travels across the profile — and where it starts and ends is set by the geometry of the chord. An orbit aligned with the star's equator gives a track symmetric about the line centre; this one, tilted by 30 degrees, runs from -41 to 25 kilometres a second and is not. The measurement is of a path rather than of a centroid, which is why it works on rapidly rotating stars where the velocity anomaly is swamped by the line's own width.

A shadow crossing a rotating line

A transiting planet hides a strip of a rotating star, and that strip has a definite velocity. So the planet removes light from one place in the line profile and leaves a bump there — a bump that travels across the line as the transit proceeds, tracing the path the planet took across the disc.

exoplanets · Spin–orbit alignment

Named alongside it

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

DegeneracyLimb darkeningRadius ratioTransitAsteroseismologyEccentricityPhotometryStellar densityTransit depthTransit durationAim pointArgument of periapsis

All concepts