SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish detected planets from intrinsic occurrence

A detected sample is filtered by geometry, time coverage, signal strength and analysis choices. Transit probability is larger for close orbits when other quantities are fixed, and a finite observing baseline can miss longer-period events. These biases mean a histogram of detections is not automatically a histogram of underlying planet occurrence.

Learning outcome: Reconstruct each selection factor and corrected fraction. Change short completeness without changing the true population; name an uncertainty or false-positive process omitted from the correction.

Starting recap

Keep interior, thermal, surface-history, orbital and detection assumptions distinct before drawing a planetary inference.

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Try

With f=0.3 in each bin and short completeness 0.5, expected short detections are 150 and long detections 15. The raw ratio is ten despite equal underlying occurrence. Dividing by the separate geometry and completeness factors recovers 0.3 for both.

Open the experiment

Predict

Can ten times as many detections coexist with equal occurrence in the two supplied bins?

Your prediction and reason

Experiment

Reconstruct each selection factor and corrected fraction. Change short completeness without changing the true population; name an uncertainty or false-positive process omitted from the correction.

Correct explicit geometry and completeness before comparing occurrence

Worked teaching model · synthetic values · Original 10000-star survey, each star in at most one planet bin; both bins have occurrence f<=0.5. Transit probabilities 0.1/0.02; short completeness variable, long completeness 0.25. Expected detections=Nstar f ptransit completeness. No false positives, multiplicity, uncertain calibration or actual population estimate; bin geometry fixed independently.

Short-period detection completeness: 0.5 fraction True occurrence in each of two disjoint bins: 0.3 fraction

Reset keeps your prediction, captures and practice history.

A detected sample is filtered by orbital geometry and instrument completeness before it represents a population.

The correction recovers truth only because the selection factors are supplied exactly and no false positives or multiplicity are present. Real occurrence estimates need validated completeness, uncertainty, stellar selection and candidate reliability rather than dividing by convenient guessed factors.

Original conditional model ledger
Quantity	Value	Unit
Expected short-bin detections	150	detections
Expected long-bin detections	15	detections
Raw short/long detection ratio	10	ratio
Geometry/completeness-corrected short occurrence	0.3000000	fraction
Geometry/completeness-corrected long occurrence	0.3000000	fraction
True supplied occurrence in each bin	0.3000000	fraction

Original supplied models, not actual planet observations or a unique interior, atmosphere or discovery claim.

A detected sample is filtered by orbital geometry and instrument completeness before it represents a population.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the planetary evidence

2. Connect mechanism and inference

The original survey contains 10,000 stars and two disjoint planet bins, each with true occurrence fraction f≤0.5. Fixed transit probabilities are 0.1 and 0.02; long-period detection completeness is 0.25 while short-period completeness is controlled. Expected detections are star count times occurrence times geometry probability times completeness.

When these selection factors are known and the model excludes false positives, dividing detections by star count, geometry and completeness recovers the stipulated occurrence. Fractional expectations are not guaranteed integer outcomes. An actual estimate would also propagate counting uncertainty and uncertainty in the completeness calibration.

Detection sensitivity may depend jointly on period, radius, host variability and pipeline behavior. Multiplicity and correlated planet architectures can alter the simple disjoint-bin model. A survey claim should state its target population and tested selection function rather than extrapolating an uncorrected sample to every star or claiming an exact current discovery count.

3. Work the supplied case

4. Predict and investigate

Can ten times as many detections coexist with equal occurrence in the two supplied bins?

Reconstruct each selection factor and corrected fraction. Change short completeness without changing the true population; name an uncertainty or false-positive process omitted from the correction.

Apply the evidence to a changed case

Use f=0.5 and short completeness 0.8. Find both expected counts and their raw ratio.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Short detections are 10,000×0.5×0.1×0.8=400; long detections are 25. Their ratio is sixteen, while corrected occurrence remains 0.5 in both disjoint bins.

Source and boundaries

Primary source. Checked 31 October 2026. Original diagrams and supplied model records are not actual planetary observations. No direct solar viewing, instrument operation or physical experiment is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Use f=0.5 and short completeness 0.8. Find both expected counts and their raw ratio.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

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