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Sandbox Physics
From planet catalogs to populations

AC24 · L2

Which transits survive the survey?

Sandbox Physics ·

A transit reaches a catalog only when the geometry, observing times and noisy measurements permit the same event to be recognized. The useful probability averages that complete sequence over unknown orbital directions and phases. Multiplying unrelated average penalties can miss their dependence.

An exposure averages the missing light

Let δ(t)\delta(t) be the fractional stellar light blocked at time tt. In the uniform-disk model it comes from the actual overlap of a circular planet and star. The planet’s projected centre follows a local straight chord near the transit. Its distance from the stellar centre sets the overlap at every instant.

A detector does not freeze one instant. It collects photons during an exposure of duration Δt\Delta t, in days here. Its predicted deficit is the average of the instantaneous blocked fraction over that interval. Near ingress or egress, the average can differ substantially from the value at the central time.

The exposure integral can be split at the times when the two disks first touch, when the planet enters wholly, when it begins leaving, and when overlap ends. Fully internal pieces have constant depth for a uniform star. Only partial-overlap pieces need numerical integration. The synthetic survey uses thirty-minute exposures and this contact-split calculation.

δˉi=1Δt∫ti−Δt/2ti+Δt/2δ(t) dt\bar\delta_i=\frac{1}{\Delta t}\int_{t_i-\Delta t/2}^{t_i+\Delta t/2}\delta(t)\,dt
The factor 1/Δt1/\Delta t turns the accumulated deficit into an average. Both integration limits and exposure duration use days.

Reading: Christiansen 2017, Pixel-Level Transit Injection Tests of Pipeline Detection Efficiency for DR25 · Burke and Catanzarite 2017, Per-Target Detection Contours for DR25

Ask one prepared template a precise question

For the teaching experiment, assume we already know the period and event times. Prepare a vector sis_i of expected fractional deficits, expressed in parts per million. A positive measurement yiy_i means less light than the baseline. Independent Gaussian noise has mean zero and standard deviation σ\sigma per exposure.

Weight the data by the expected shape. Samples inside a deep part of the transit receive larger weights than samples where almost no light is blocked. Dividing by the noise scale and by the length of the template gives a statistic ZZ whose noise variance is one.

With a true signal matching that template, the mean statistic is ν\nu. Without it, the mean is zero. The probability of passing a threshold z0z_0 is the fraction of a unit Gaussian above that threshold. It increases smoothly with signal strength; it is not a step function of the expected signal.

A complete blind search tries many periods and templates and applies additional rejection tests. Kepler’s measured recovery includes that search and its subsequent vetting. The prepared teaching statistic supplies a tractable probability calculation; the real tutorial calibrates a different conditional response directly from actual injection outcomes.

Z=∑isiyiσ∑isi2,ν=∑isi2σ,P(Z>z0)=Φ(ν−z0)Z=\frac{\sum_i s_i y_i}{\sigma\sqrt{\sum_i s_i^2}},\qquad \nu=\frac{\sqrt{\sum_i s_i^2}}{\sigma},\qquad P(Z>z_0)=\Phi(\nu-z_0)
Φ\Phi is the cumulative probability of a standard normal variable. The synthetic threshold is z0=7.1z_0=7.1, specific to this prepared detector rule.

Reading: Christiansen 2017, Pixel-Level Transit Injection Tests of Pipeline Detection Efficiency for DR25

A season selects orbital phases

Write the centres of successive transits as ϕ+nP\phi+nP, where PP is period, nn is an integer and the unknown phase ϕ\phi is uniform from zero to one period. A viewing season supplies intervals during which observations exist.

The synthetic survey accepts a transit only when its entire finite-exposure template fits inside a valid interval. Shrink each valid interval at both ends by the half-width of that template. An event centre in the remaining interval supplies one complete event. This rule avoids pretending that a transit beside a long gap contains a full set of measurements.

For a worked limit, let events become instant points and keep an uninterrupted thirty-seven-day season. A fifteen-day planet has at least three events only when the first event occurs within seven days of the start. Seven out of fifteen possible phase-days succeed, giving about 0.46670.4667.

The same result follows by counting equally weighted phases. Move the phase continuously from zero to fifteen days. The number of covered events changes only when an event centre passes a season boundary. With many observing intervals, their endpoints modulo the period divide phase into pieces on which the count is constant. Summing the lengths of those pieces computes the exact phase average for this window rule.

W(P)=1P∫0P1[Ncovered(P,ϕ)≥3] dϕW(P)=\frac1P\int_0^P\mathbf1[N_{\rm covered}(P,\phi)\ge3]\,d\phi
The indicator is one for a phase with enough complete events and zero otherwise. For point events and one continuous span TT, this reduces to max⁡(0,min⁡(1,T/P−2))\max(0,\min(1,T/P-2)).

Reading: Burke and Catanzarite 2017, Per-Target Detection Contours for DR25 · Measuring Transit Signal Recovery in the Kepler Pipeline IV

The window and signal belong to the same transit

A grazing event is short and shallow. Its complete template may fit beside a gap more easily than a central transit, yet its weaker signal is harder to detect. Impact parameter therefore affects both the window rule and the Gaussian recovery probability.

Keep those two effects together inside the average over phase and orientation. For a given radius and period, first choose an orientation, calculate its actual template, count how many full copies were observed, and evaluate the probability of passing the detection rule. Only then average over the possible orientation and phase.

If a recovery experiment has already run through a schedule with interruptions, its final recovered fraction already contains those losses. In the thirty-seven-day example, multiplying the measured fraction 7/157/15 by another 7/157/15 would give roughly 0.21780.2178 instead of 0.46670.4667. The second factor would describe an additional independent observing requirement, not the same interruptions.

q(P,R)=∫01dμ 1P∫0PP(pass∣P,R,μ,ϕ) dϕq(P,R)=\int_0^1 d\mu\,\frac1P\int_0^P P(\mathrm{pass}\mid P,R,\mu,\phi)\,d\phi
Here μ=∣cos⁡i∣\mu=|\cos i|. The integrand is zero when there is no disk crossing or too few covered events.

Reading: Burke and Catanzarite 2017, Per-Target Detection Contours for DR25

Check a calculation before trusting its precision

The synthetic calculation has several numerical layers. Agreement after refining the population integral does not test the exposure integral inside its signal model. Each layer needs its own reference.

The delivered checks compare circular overlap with an independent Cartesian strip integral, exposure averaging with an adaptive time integral, and the phase partition with direct event enumeration. An independent conditional Monte Carlo calculation then samples period, radius, phase and impact parameter. Its statistical uncertainty is reported alongside the selection comparison.

The real DR25 source presents a separate warning. A simple depth divided by noise proxy differs from the supplied expected multiple-event statistic. The source’s corrected expected statistic and a single measured search statistic are also different quantities. The tutorial keeps those differences visible and uses actual successful and failed injections to fit its response.

Reading: Christiansen 2017, Pixel-Level Transit Injection Tests of Pipeline Detection Efficiency for DR25 · Burke and Catanzarite 2017, Per-Target Detection Contours for DR25

Exercises and explanations

  1. A continuous forty-day season is used to find fifteen-day point events. What is the probability of observing at least three? What if a one-day-wide complete template must fit?

    Read the explanation

    Point events succeed over ten out of fifteen phase-days, giving two thirds. Requiring the one-day-wide template shrinks the valid centre interval to thirty-nine days, leaving nine out of fifteen phase-days, or three fifths.

  2. A stored injection recovery fraction already includes the actual gaps. Should you multiply by the gap duty fraction again?

    Read the explanation

    Only if it represents a new observing requirement absent from the injection experiment. Reusing the same gaps would count the same missing measurements twice.

Try the prediction

Change the detector and watch the missing planets

  1. Predict whether repeated gaps and a shorter continuous season remove identical planets.
  2. Choose Repeated gaps and compare the detected period-radius points.
  3. Keep those gaps and reduce detector noise. Explain which missing planets return and which still lack three complete events.

Sources and calculation scope

  1. Christiansen 2017, Pixel-Level Transit Injection Tests of Pipeline Detection Efficiency for DR25

    KSCI-19110-001, sections 2–4 and pp. 9–12: ephemeris matching, aperture-corrected depth, expected MES and original table columns.

  2. Burke and Catanzarite 2017, Per-Target Detection Contours for DR25

    KSCI-19111-002, Equations 1–2, sections 3.5–3.6 and 4–7: OSD, windows and target-dependent transfer.

  3. Measuring Transit Signal Recovery in the Kepler Pipeline IV

    Version 2010.04796v1, sections II–IV: pixel-injection design and pooled recovery response.

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