Astronomy & Celestial Mechanics
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Exoplanets

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.

Assumes Detection bias and The ellipse.

18 min read 7 figures Measuring the unreachableEverything is a conic

The eccentricity distribution of planets is one of the most argued-about numbers in the subject, because it is the clearest surviving trace of how a system was assembled. A planet that arrived by high-eccentricity migration must have spent time on a long ellipse; one that spiralled in through a disc need never have been anything but round.

So it matters a great deal whether the measured distribution is the real one. It is not, and the corrections are three separate effects that do not point the same way.

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.
Fig. 1 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(1-e^2)^{-1}; the transit duration falls as 1−e2\sqrt{1-e^2}; their combination — probability times the square root of the time in transit — comes to 1.24 at e=0.5e = 0.5, which is almost no bias at all. The fourth curve is the fraction of a Keplerian velocity curve’s variance left in its fundamental, and it falls to 68 per cent at e=0.6e = 0.6 and 47 at e=0.8e = 0.8.

The probability rises

A planet transits if its orbit carries it across the stellar disc from here, and for a circular orbit the chance of that is R⋆/aR_\star/a — the fraction of viewing directions from which the geometry works.

An eccentric orbit does not have one distance. It spends part of its period inside its own semi-major axis and part outside, and the transit probability depends on the distance at conjunction, which is where the planet happens to be when it crosses the line between here and the star. Averaged over the orientation of the ellipse in its own plane, the probability comes to

Ptr=R⋆a⋅11−e2,P_{\text{tr}} = \frac{R_\star}{a}\cdot\frac{1}{1-e^2},

which is larger than the circular value, and by a factor that runs away as the eccentricity approaches one.

The reason the average goes that way rather than the other is worth pausing on, because the intuition that an eccentric planet spends most of its time far out is correct and points the wrong direction. The planet does spend most of its time near apoapsis — that is Kepler’s second law — but the transit probability does not average over time. It averages over the orientation of the orbit, which is uniform, and the periapsis passage contributes a large probability from a narrow range of orientations.

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 at an argument of periastron of 90°. 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 2.00 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 1.52 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.
Fig. 2 The same three curves at a fixed argument of periastron of 90°, meaning that periapsis points at the observer and the planet is closest to its star exactly when it crosses the disc. The probability rises far faster — exactly twice the circular value at e=0.5e = 0.5 — and the duration falls correspondingly harder, since a planet at periapsis is also moving fastest, so that the two together come to 1.52 rather than the averaged 1.24. The averaged case in the previous figure is not a typical case; it is a mean over two competing extremes, and any individual system sits at one orientation and not at the average.

The duration falls

A transit’s depth is a ratio of areas and does not depend on the eccentricity at all. Its duration does, and it depends on it twice.

An eccentric planet at conjunction is at a distance that differs from aa, so the chord it draws across the star differs in length. And it is moving at a speed that differs from the circular speed at that distance, by the amount vis-viva requires. Combining the two, the duration relative to a circular orbit of the same semi-major axis is

TTcirc=1−e21+esin⁡ω,\frac{T}{T_{\text{circ}}} = \frac{\sqrt{1-e^2}}{1+e\sin\omega},

and averaged over the argument of periastron the denominator drops out, leaving 1−e2\sqrt{1-e^2}.

A shorter transit carries less signal. The detection statistic scales as the square root of the time in transit, so a transit half as long has seventy per cent of the signal-to-noise, and near a threshold that is the difference between a detection and nothing.

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 at an argument of periastron of -90°. 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 0.67 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 0.88 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.
Fig. 3 The opposite orientation: apoapsis towards the observer, so the planet transits at its furthest and slowest. The probability now falls with eccentricity and the duration rises — a transit at apoapsis is long and leisurely, and easier to detect than a circular one of the same semi-major axis. The two orientations bracket everything, and what the averaged figure shows is a mean over a distribution running from one to the other.

Which very nearly cancel

Put the two together — the probability of a transit happening, times the signal-to-noise it delivers when it does — and the product at e=0.5e = 0.5 is 1.24 relative to a circle. At e=0.7e = 0.7 it is 1.35.

That is a remarkably small bias for a quantity that runs over a factor of three in each of its two parts. A transit survey, averaged over orientations, has almost no preference between round orbits and eccentric ones, and the cancellation is not approximate for any deep reason — it happens because one effect goes as (1−e2)−1(1-e^2)^{-1} and the other as (1−e2)1/4(1-e^2)^{1/4} after the square root of the duration is taken, and the residual is mild over the range that matters.

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 1.15 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.
Fig. 4 The same construction with the signal-to-noise taken proportional to the duration rather than to its square root — the regime in which the noise is correlated on the transit timescale rather than white, so that a longer transit buys signal linearly instead of as a root. The cancellation is then nearly exact: the combined curve sits at 1.15 at e=0.5e = 0.5 and stays within fifteen per cent of unity across the whole range drawn. The degree of cancellation depends on the noise the survey is fighting, which is not a property of planets and cannot be corrected for without knowing it.

Why the cancellation is not a coincidence worth trusting

It is tempting to read the near-cancellation as a piece of luck that lets transit surveys off, and the reading is half right.

The two effects have a common origin: both come from the fact that an eccentric orbit’s conjunction distance differs from its semi-major axis. The probability goes as the inverse of that distance and the transit’s chord-crossing speed goes as roughly its inverse square root, so the exponents are −1-1 and +1/2+1/2 before the signal-to-noise takes another root. Written that way the partial cancellation is structural rather than accidental.

What is accidental is how complete it is, and that part depends on the noise. The exponent connecting duration to detectability is 1/21/2 for white noise and 11 for noise correlated on the transit timescale, and real light curves sit between them. So the residual bias is a small number whose size is set by an instrumental property nobody quotes.

A small residual whose magnitude is unknown is a worse situation than a large one that is known. A factor of three can be corrected; a factor of 1.2±0.21.2 \pm 0.2 cannot be, because the uncertainty on the correction is comparable to the correction, and it propagates into every moment of the recovered eccentricity distribution.

The one that does not cancel

Radial velocity is a different situation entirely, and the difference is that the signal is a shape rather than an event.

A circular orbit produces a pure sinusoid in the star’s line-of-sight velocity. An eccentric one does not: the velocity curve is skewed, with a rapid excursion through periastron and a long slow return, and Fourier analysis puts a growing fraction of its variance into the second harmonic and beyond.

A search that fits a sinusoid — which is what a periodogram is — sees only the fundamental. So the amplitude it recovers is smaller than the true semi-amplitude by the square root of the fraction of power remaining in the first harmonic, and that fraction falls steadily with eccentricity.

The fraction is 68 per cent at e=0.6e = 0.6 and 47 per cent at e=0.8e = 0.8, which is a loss of 18 and 31 per cent in recovered amplitude. Near a detection threshold that is decisive, and unlike the transit case there is nothing pushing the other way.

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.
Fig. 5 The same four curves extended to e=0.95e = 0.95. The transit probability diverges as (1−e2)−1(1-e^2)^{-1} and the duration collapses, so their product still rises only mildly — while the power in the fundamental has fallen to a fifth. The two methods’ biases separate completely at high eccentricity, one nearly flat and one severe, so the discrepancy between the eccentricity distributions the two methods report is largest exactly where the astrophysics is most interesting.

A full Keplerian fit does better than a periodogram, because it is fitting the right shape. It pays for that in two extra free parameters — the eccentricity and the argument of periastron — and two extra parameters cost significance in any model comparison. So even a search that fits the correct model requires a higher signal-to-noise to claim an eccentric planet than a circular one of the same amplitude.

It is worth setting the three effects against each other by kind, because only two of them are correctable in principle.

The transit probability is a geometric bias. It depends on nothing but the orbit and the observer’s position, it is exact, and correcting for it needs no knowledge of the instrument at all. Every survey that has ever existed shares it, and it will not change.

The transit duration is an instrumental bias. It becomes a bias only because a shorter event carries less signal, and how much less depends on the noise. A survey with infinite sensitivity would have no duration bias whatever.

The harmonic loss is a methodological bias. It exists because a particular search algorithm looks for a particular shape, and a different algorithm has a different bias or none. A search that fitted full Keplerians on a grid of eccentricity would not lose the harmonics; it would lose significance to the extra parameters instead, which is a different penalty of a similar size.

Only the first is a property of the universe. The other two are properties of a decision somebody made, and they change between surveys, between pipelines, and between reanalyses of the same data — which is why two eccentricity distributions derived from the same catalogue by different groups can differ, without either being wrong about the catalogue.

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.
Fig. 6 The same decomposition carried to twenty harmonics rather than twelve, out to e=0.85e = 0.85. The fundamental’s share is unchanged to the digit — the extra terms carry almost nothing — which is the check that the truncation is not what produces the falling curve. What the higher harmonics do carry is the sharpness of the periastron passage, so a search that used the first three harmonics rather than one would recover most of the loss: the information is in the data and the standard tool discards it.

What the corrected distribution says

Correcting for all of this is what turns the catalogue into a statement about formation, and the statement is worth having.

Giant planets on periods beyond about ten days have a broad eccentricity distribution, running from zero to above 0.9, with a median near 0.25. That is nothing like a disc-driven population, which would be nearly circular, and it is the strongest evidence that gravitational interaction between planets — scattering, or secular exchange with a distant companion — is a common part of a system’s history.

Inside ten days the distribution collapses to zero, and that wall is a tidal clock rather than a formation signature.

And the small planets, the ones the transit surveys find in abundance, are a third case: eccentricities that are small but not zero, typically below 0.1, and systematically smaller in systems with more planets than in systems with one. That last correlation is the interesting one, and it is close enough to the detection biases to be worth suspicion — a survey finds more planets per system when the system is flat and calm, which is a selection effect on multiplicity, and flat calm systems are also the ones with low eccentricities.

What each method can see. Planet mass against orbital distance, both logarithmic, with the detection threshold of each method drawn as the boundary it actually is. Radial velocity at 1 m/s needs mass rising as √a; astrometry at 20 µas needs it falling as 1/a, which is the only method that gets easier further out; a 100 ppm transit is a threshold on radius and so a horizontal line at about 0.3 Earth masses, cut off at 1.02 AU by the need for three transits in 4 years; direct imaging begins outside the diffraction limit, 0.6 AU at 10 parsecs for a 39 m aperture at 10 µm. The solar system is drawn on top: for two decades every one of its planets except Jupiter lay outside every region, which is the whole of what the early census was measuring.
Fig. 7 The detection domains around a 0.6 solar-mass star, for scale. Every boundary here is drawn for circular orbits, which is the convention and is now visibly an approximation: the transit line should be smeared by the duration effect, the velocity line by the harmonic loss, and neither smearing is the same size. A domain diagram is a statement about a circular planet, and the population it bounds is not circular.

An asymmetry nobody put there

There is a third consequence of the orientation dependence, and it is the one that produces an effect in a quantity that ought to have none.

The transit probability depends on ω\omega through 1+esin⁡ω1 + e\sin\omega, and so does the duration. Both favour periastron pointing towards the observer. So a transit survey does not merely over-represent eccentric planets slightly — it over-represents eccentric planets in a particular orientation, and the orientation is defined relative to the line of sight.

That is a selection effect on a quantity that has no physical meaning. The argument of periastron measured from the sky plane is an accident of where the Earth happens to be, and any real distribution of it must be uniform. So a measured distribution of ω\omega that is not uniform is a pure measurement of the selection function, with no astrophysics in it whatever, and it can be used as a check on everything above.

The transiting planets do show it. The distribution of the argument of periastron in the transiting sample is skewed towards periastron-on-the-near-side, by about the amount the geometry predicts, and the agreement is one of the few places where a detection bias has been confirmed rather than modelled.

There is one confounder, and it is the orbital decay of the very hottest planets, where a tidal argument also produces an orientation preference by circularising the systems whose periastra come closest. The two effects are separable by period, and the check is made outside the range where tides act.

Exact geometry, an honest quadrature, and the thing neither supplies

The transit probability and duration expressions are geometry and Kepler’s laws, exact within the two-body problem, with no free parameters. They are not approximations in any sense that matters here.

The harmonic decomposition is computed from the velocity curve itself — Kepler’s equation solved at four thousand mean anomalies, the true anomaly taken from the eccentric one, the resulting curve transformed, and the power in the fundamental compared against the total by Parseval’s relation rather than against a series expansion. At e=0e = 0 it returns exactly one, which is the check that the solver and the transform agree.

What is not measured is the thing the corrections are for, and this is the honest limit of the whole exercise. The detection efficiency of a real pipeline against an eccentric planet is not a formula; it is the same injection experiment that measures a pipeline’s efficiency, run with eccentric signals, and it has been done far less often than the circular version.

Fixed semi-major axis, and what that choice hides

Every curve is drawn at fixed semi-major axis, which is a choice and not the only one. Compared at fixed period the same statements hold, since period and semi-major axis are related by a law with no eccentricity in it. Compared at fixed periastron distance — which is what a tidal argument would want — they do not.

The transit case ignores impact parameter. An eccentric orbit changes the geometry of the crossing as well as its duration, and a grazing transit of an eccentric planet has a different shape from a grazing transit of a circular one. That affects what can be measured from the transit rather than whether it is found.

And nothing here shows the eccentricity distribution itself. The figures give the bias against each eccentricity; converting that into a corrected distribution requires the true distribution’s shape, which is what the exercise is trying to find, so the correction has to be done inside the inference rather than applied to its output.

What a bias in a shape does to a mass

One more consequence deserves separating, because it changes a number rather than a count.

A velocity semi-amplitude gives a minimum mass, and the relation between the two contains the eccentricity: at fixed mass and period, KK rises as (1−e2)−1/2(1-e^2)^{-1/2}. So an eccentric planet produces a larger velocity amplitude than a circular one of the same mass — which is a bias towards detecting it, running opposite to the harmonic loss.

The two do not cancel, because they act at different stages. The amplitude gain is in the signal; the harmonic loss is in the search. A periodogram searching for sinusoids sees the fundamental of a larger curve, and the product of the two factors is (1−e2)−1⋅f1(e)\sqrt{(1-e^2)^{-1}\cdot f_1(e)}, which stays near one up to e≈0.5e \approx 0.5 and then falls.

So radial velocity has its own near-cancellation, over a narrower range, and the interesting regime is above it. Beyond e=0.6e = 0.6 the harmonic loss wins decisively, which is precisely where the most dynamically informative planets are.

And the mass that comes out of a fit in which the eccentricity is poorly constrained inherits that. Fitting a low-signal eccentric orbit with the eccentricity free returns a posterior that is broad in ee and correlated with KK, so the mass and the eccentricity are measured together or not at all — which is the same coupling a transit timing amplitude has, arriving from a completely different observable.

The general shape

The recurring structure is that a selection effect is only correctable when it is known, and the three effects here are known to very different degrees.

Geometry is known exactly. A transit probability is a solid angle and a duration is a chord, and neither depends on anything a survey does. Those corrections can be applied with confidence.

Pipeline response is known only by experiment, and the experiment has usually been run for the circular case alone.

And the correlation between eccentricity and everything else is not known at all. Eccentric planets are found around different stars, at different periods, in different multiplicities — so a correction applied to eccentricity alone is applied to a variable that is not independent of the others. A duration measured against a transit’s expected length gives the eccentricity directly for an individual system, which is the way out, and it works only where the stellar density is independently known.

Why this is not a small correction to a large number

A reader could reasonably conclude from the near-cancellations above that the eccentricity biases are a second-order concern. For the mean of the distribution they nearly are. For its tail they are not, and the tail is the part that carries the physics.

The distinguishing prediction of planet–planet scattering is not a shifted mean; it is the existence of planets at e>0.7e > 0.7, which disc migration cannot produce at all. The number of such planets in a catalogue is small, the correction to it is large, and a factor of two in the velocity method’s efficiency at e=0.8e = 0.8 moves the inferred fraction of scattered systems by a factor of two.

So the quantity most sensitive to the bias is the one the measurement exists to determine, which is the usual arrangement and is worth stating each time it recurs. The safe part of a measurement is the part nobody needs.

There is a partial escape, and it is the reason the two methods are worth running together. The transit sample’s bias at high eccentricity is mild and the velocity sample’s is severe, so a planet population measured both ways gives a consistency check on the correction: if the corrected distributions agree, the corrections were probably right, and if they disagree, the disagreement localises which method’s correction failed. That check has been run, and the two agree within their uncertainties for giant planets and have too few objects in common to compare for small ones.

Still open: the multiplicity that travels with it

The correlation between low eccentricity and high multiplicity is the loose end. It is a real pattern in the catalogue and it has at least two readings, one astrophysical and one instrumental, and they predict the same thing.

The astrophysical reading is that a system with many surviving planets is one that never scattered, so its planets are both numerous and round. The instrumental reading is that a survey finds many planets in a system only when the system is flat — and a flat system is a dynamically cold one, which is also a circular one.

Separating them requires knowing how a survey’s multiplicity is biased, and that turns out to be the hardest quantity a catalogue reports.

Where this goes next

What each method can see. Planet mass against orbital distance, both logarithmic, with the detection threshold of each method drawn as the boundary it actually is. Radial velocity at 1 m/s needs mass rising as √a; astrometry at 20 µas needs it falling as 1/a, which is the only method that gets easier further out; a 100 ppm transit is a threshold on radius and so a horizontal line at about 1.4 Earth masses, cut off at 1.21 AU by the need for three transits in 4 years; direct imaging begins outside the diffraction limit, 0.6 AU at 10 parsecs for a 39 m aperture at 10 µm. The solar system is drawn on top: for two decades every one of its planets except Jupiter lay outside every region, which is the whole of what the early census was measuring. Every survey draws a different sky The first exoplanets found were enormous and impossibly close to their stars. That was not a discovery about planets. It was a measurement of what a 10 m/s spectrograph watching for three years is able to see. 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. What comes back is a ramp, not a threshold. Detection efficiency against signal-to-noise: the fraction of synthetic transits injected into real photometry that the pipeline afterwards finds. The measured curve is a gamma cumulative distribution of shape 4.65 and scale 0.98 beginning at 4.1, which is the form a survey's own injection tests are fitted with; the dashed line is the step at 7.1 that a threshold calculation assumes instead. Half the injections are recovered at 8.33, 1.2 units above the nominal threshold — the ramp is a property of the search and the cut is a separate decision, so the two need not meet anywhere in particular. The rest of the disagreement is the area between the curves. The pipeline does not reach 99 per cent efficiency until 14.9, four units above the threshold, and it recovers 45 per cent one unit above it. Over a population whose signal-to-noise falls as s^-2 — which is what a planet population looks like, because there are far more small planets than large ones — the step function counts 1.21 times as many detections as the ramp does. That factor is not an error bar. It multiplies every occurrence rate computed without it, and it is larger for the small planets than for the large ones, because the small ones live where the ramp is. The threshold that is not a threshold A survey's detection limit is quoted as a number — seven point one — and a pipeline does not behave that way. Half the injected signals come back at the threshold, and full efficiency arrives four units above it.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Argument of periastronDetection thresholdKepler's equationOrbital eccentricityRadial velocitySelection effectSignal-to-noiseSurvey completenessTransit durationTransit probability