O014 · A planet crosses a star

Eclipses and Transit Photometry

How much can a small dip in starlight tell us about a planet? A transit measures how much stellar light a foreground body blocks. Converting its depth into a size needs the stellar brightness profile, path, exposure and contaminating light; the planet’s physical radius also needs the star’s radius. The same overlap geometry explains eclipse contacts and shadows.

Drag the planet’s path across a glowing star. Compare full and grazing transits, finite exposures and light from a nearby companion, then return to the eclipse-shadow geometry.

Current averaged total flux $0.99$

Mid-transit depth · parts per million $17000$

First-to-fourth contact · hours $3.3$

Circularly inferred density · kilograms per cubic metre $900$

Move the planet’s path

Planet / star radius

0.12

p=R_{\rm p}/R_\star

Path offset

0.35

b\;[R_\star]

Limb-darkening slope

0.45

u_1

Exposure window

120

\Delta t\;[\mathrm s]

Companion / host light

0

c=F_{\rm cont}/F_{\star,0}

Drag the planet up or down to move its path, and sideways to choose a moment.

Orbit, brightness and model limits

Limb-darkening curvature

0.2

u_2

Orbit / star radius

8

a/R_\star

Orbital period

3.2888

P\;[\mathrm d]

The two limb coefficients stay within a positive, darkening brightness profile. Changing the path prepares a new synthetic curve. A circular orbit and negligible planet mass are assumptions behind the displayed density.

Read the transit course and fit real TESS observations

Model assumptions

The transit is a synthetic opaque circular planet on a prescribed circular orbit across a spherical star with quadratic radial limb darkening. Annular integration supplies host flux, a uniform time window supplies finite exposures, and a constant unresolved companion dilutes that same flux. The circularly inferred density assumes negligible planet mass; physical planet radius additionally needs stellar radius. Starspots, atmosphere, orbital eccentricity, detector noise and binary validation are outside the interactive model. Shadow and solar views retain exact apparent-disk overlap and common-tangent geometry with compressed axial distances and enlarged bodies. Real TESS observations are fitted only in the separate fixed-input tutorial, with quality flags, supplied errors and conditional assumptions preserved.

Synthetic circular-orbit transit, finite exposure and constant contaminating light.

Cyan: recorded exposure with companion; amber dashed: instantaneous host-only reference.

Integration and display conventions

The stellar disk follows the same radial brightness law as the flux integral. Its color and outer glow are illustrative. Thin planet outlines mark the distance swept during the exposure, not additional planets. The companion’s angular separation and glyph size are schematic; its light ratio is explicit. The replay follows the visible passage; the chart also includes the full exposure wings and their baseline.

The two geometric eclipse views retain apparent-disk overlap and tangent shadow boundaries. They omit topography, atmospheric refraction, coronal radiance and diffraction. Their axial lengths are compressed and bodies enlarged.

Start with the essentials

Focus question: Does a shallow dip necessarily mean a small planet?
One-sentence intuition: Depth measures blocked brightness; geometry and the source turn it into a size.

Core mathematical model

Uniform centered disk

\delta=p^2,\qquad p=R_{\rm p}/R_\star

This area rule applies to a fully contained planet across a uniformly bright disk.

Brightness weighted by the hidden area

F_\star=1-\frac{\int_{\rm blocked}I\,\mathrm dA}{\int_{\rm disk}I\,\mathrm dA}

The stellar center and limb need not contribute equal light per area.

Averaging an exposure and blending light

\overline F_{\rm obs}=\frac{\Delta t^{-1}\int F_\star(t)\,\mathrm dt+c}{1+c}

The window spans one exposure, and the constant companion ratio refers to unobscured host light.

Common difficulties

A grazing planet

Typical misconceptionEvery planet blocks its full disk area at mid-transit.

Better mental modelA grazing path leaves part of the planet outside the stellar disk, so even its deepest point can be partial overlap.

A successful fit

Typical misconceptionA good planet curve confirms that the signal came from a planet.

Better mental modelBlended eclipsing stars and stellar structure need separate tests. The curve fit assumes its object and source model.

Run the experiment

01
Move the chord
Drag the planet vertically from the stellar center toward its limb.
What to observe: The passage shortens and can become grazing; the same radius need not produce the same depth.
02
Watch a shutter average
Select A long exposure and compare the cyan curve with the amber instantaneous reference.
What to observe: Ingress and egress are rounded by a time window; the outlined swept path shows the same interval.
03
Add a companion
Select A bright companion, then reduce its light ratio without changing the planet.
What to observe: The normalized dip deepens as contaminating light falls.
04
Return to eclipse geometry
Open the shadow and solar views and move the foreground body.
What to observe: The same apparent-disk contacts connect visible flux to tangent shadow boundaries.

Eclipses and Transit Photometry

Read a dimming star in the right order

Core mathematical model

Uniform centered disk

Brightness weighted by the hidden area

Averaging an exposure and blending light

Common difficulties

A grazing planet

A successful fit

Run the experiment

Move the chord

Watch a shutter average

Add a companion

Return to eclipse geometry

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A planet crosses a star