SCIENCE · INVESTIGATE AND EXPLAIN

Separate a transit radius signal from blended light

A transit reduces observed light when an opaque object crosses the projected stellar disk. For a uniform star and a fully overlapping small planet, the blocked fraction of the host’s light is k², where k is planet radius divided by stellar radius. The measured depth concerns all light in the observing aperture, not necessarily just the host.

Learning outcome: Compare depth and contact duration as separate observables. Change contamination at fixed radius, then radius at fixed contamination; identify what a second measurement constrains and what false-positive alternatives remain.

Starting recap

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

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Try

For k=0.1 and contamination 0.25, depth is 0.0075, or 0.75%. Ignoring dilution gives radius ratio sqrt(0.0075)≈0.0866025. Outer-contact duration is 2sqrt(1.21−0.25)≈1.95959 time units, independent of the added light under this purely geometric model.

Open the experiment

Predict

Can contamination change inferred radius without changing the actual transit chord?

Your prediction and reason

Experiment

Compare depth and contact duration as separate observables. Change contamination at fixed radius, then radius at fixed contamination; identify what a second measurement constrains and what false-positive alternatives remain.

Keep stellar dilution and contact geometry in a transit inference

Worked teaching model · synthetic values · Original opaque planet, uniform-brightness star, full nongrazing overlap at fixed impact parameter 0.5 stellar radii; depth=k²(1−contamination). Projected speed 1 stellar radius/time, straight chord approximation; first-to-fourth contact length2sqrt[(1+k)²−0.5²]. No limb darkening, orbital curvature or false-positive validation.

Planet/star radius ratio: 0.1 ratio Contaminating fraction of total out-of-transit light: 0.25 fraction

Reset keeps your prediction, captures and practice history.

The same apparent dimming can arise from different radius and contaminating-light combinations.

The depth formula assumes full opaque overlap and uniform stellar brightness. Grazing geometry, limb darkening, starspots or an eclipsing blend can break it; a repeating dip is evidence to investigate, not automatic planet validation.

Original conditional model ledger
Quantity	Value	Unit
True planet/star radius ratio	0.1000000	ratio
Intrinsic full-overlap depth	0.01000000	fraction
Observed diluted depth	0.007500000	fraction
Radius ratio inferred ignoring dilution	0.08660254	ratio
First-to-fourth contact duration	1.959592	model time
Fixed impact parameter	0.5000000	stellar radii

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

The same apparent dimming can arise from different radius and contaminating-light combinations.

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 case adds contaminating light fraction c of the total out-of-transit flux. The host supplies 1−c, so measured depth is k²(1−c). Ignoring the contaminant gives inferred radius ratio k sqrt(1−c), smaller than the true supplied k. An observed shallow event can therefore reflect a smaller planet, dilution or other geometry.

The separate contact model fixes impact parameter at 0.5 stellar radii and projected speed at one stellar radius per time. First-to-fourth contact duration is 2sqrt[(1+k)²−0.5²]. This is an outer-contact chord, not the chord of the planet center across only the stellar radius. The supported radii are nongrazing, so the full-overlap depth rule is consistent with this geometry.

Actual signals require limb darkening, orbital curvature, sampling and false-positive alternatives such as an eclipsing binary. Transit timing gives a repeat interval only after identifying which events repeat; depth alone does not prove a planet. Stellar radius and contamination uncertainty propagate into inferred physical size.

3. Work the supplied case

4. Predict and investigate

Can contamination change inferred radius without changing the actual transit chord?

Apply the evidence to a changed case

Use k=0.15 with contamination 0.5. Find depth and the radius ratio inferred by ignoring contamination.

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

Compare a worked response

Depth is 0.0225×0.5=0.01125, or 1.125%. The naive radius ratio is≈0.106066 rather than 0.15. The shallower fractional signal results from extra unblocked light, not a smaller supplied planet.

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 k=0.15 with contamination 0.5. Find depth and the radius ratio inferred by ignoring contamination.

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.

Previous: Infer a surface history without equating craters with age Next: Combine size and reflex motion without dropping finite mass

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