AC24 · The planets a survey misses

Planet Survey Selection

Does a catalog full of large planets mean small planets are rare? Small planets block less starlight, so many disappear into detector noise. Orbital alignment and observing gaps hide others. To estimate how common planets are, keep every searched star and calculate which planets the survey could have found.

Turn the detector dial and extend an observing season. Watch the same planet population pass through orbital alignment, gaps and noisy transit detection.

Detected planets $N_{\rm det}=79$

All searched stars $N_\star=5000$

Effective searched-star exposure $E=121$

Planets per star, median and 95% interval $f=0.651\;[0.518,0.805]$

Find the planets that were missed

Noise per exposure

240\,\mathrm{ppm}

\sigma\;[\mathrm{ppm}]

Observing season

160\,\mathrm d

T\;[\mathrm d]

Gap in each fourteen-day cycle

15\%

g

Generating planets per star

0.600

f_{\rm true}

Drag the detector dial or observing window. Orange marks the generated population; blue marks planets that pass the survey.

Radius errors and the known-truth experiment

Log-radius scatter

8\%

Half the log-radius variance is shared by planets around the same star. The total-count posterior uses all detected planets, so moving between radius bins changes the distribution plot and leaves this total-count analysis intact.

N_{\rm det}\mid f,E\sim\operatorname{Poisson}(fE),\quad f\mid N_{\rm det},E\sim\operatorname{Gamma}(N_{\rm det}+1/2,E+1/2)

The rate parameter is planets per searched star. It conditions on the displayed generating period/radius shape and a Gamma prior with shape and rate one half. These are synthetic complete event templates with a known period, not a blind Kepler pipeline.

Replay the actual DR25 parent sample

Model assumptions

Synthetic Poisson planet population on 5000 identical stars with circular, isotropically oriented orbits. Uniform stellar disks and opaque planets produce the same finite-exposure transit templates shown in the scene. The local transit path is a straight chord. A prepared known-period Gaussian template statistic requires at least three completely covered events. Fourteen-day observing cycles have explicit gaps; the exact phase integral already includes them. The Gamma occurrence posterior conditions on the generating period and radius shape, and contains no false events. Shared-host and individual log-radius errors affect the displayed radius catalog. The real DR25 exercise is separate: fixed source data, a conditional empirical response and a propagated family of unknown-selection bounds. Star-field spacing, telescope scale and time speed are schematic.

The same population before and after the search

Synthetic truth is orange, detections are blue. Only the blue catalog receives the shared-host and individual radius errors.

Period and radius of the generated planets

Radius distribution on the same absolute count scale

The default survey is already complete. Change one setting to rerun it.

Start with the essentials

Focus question: Can the same planet population produce two very different discovery catalogs?
One-sentence intuition: Keep every searched star and average the complete geometry-and-recovery probability before dividing detections by effective exposure.

Core mathematical model

Projected shadow

\delta=\frac{A_{\rm overlap}}{\pi R_\star^2}

A uniform stellar disk loses light in proportion to the actual blocked area. The finite exposure averages the changing overlap.

Whole searched sample

E=\sum_j\int q_j(P,R)h(P,R)\,dP\,dR

The normalized density describes the assumed planet shape inside the chosen period-radius region. Each searched star contributes, including nondetections.

Integer discoveries

N_{\rm det}\mid f,E\sim\operatorname{Poisson}(fE)

In this independent synthetic Poisson population, selection thins the planet count. The generating mean is planets per searched star.

Common difficulties

Keep nondetections

Typical misconceptionOnly known planet hosts belong in the denominator.

Better mental modelThe search also examined stars whose planets were missed. Removing them changes the question to an already selected host sample.

Use a window once

Typical misconceptionEvery measured recovery curve needs another duty-cycle penalty.

Better mental modelA recovery experiment that used the actual interruptions already contains them. Add a factor only for a new requirement absent from that experiment.

Run the experiment

01
Predict the shallow signals
Record the orange population, then turn the detector dial toward lower noise.
What to observe: The generated planets stay fixed and more shallow transits pass.
02
Wait for more events
Restore default noise and extend the observing season.
What to observe: Some long-period planets gain the three complete events the detector rule requires.
03
Compare a real valley with a selected sample
Choose A true radius valley, then increase radius scatter in the advanced group.
What to observe: The orange bimodal truth remains; measured blue radii move between bins and blur its minimum.

Planet Survey Selection

The same population before and after the search

Follow one population through a survey

Core mathematical model

Projected shadow

Whole searched sample

Integer discoveries

Common difficulties

Keep nondetections

Use a window once

Run the experiment

Predict the shallow signals

Wait for more events

Compare a real valley with a selected sample

Related experiments and articles

The planets a survey misses