Astronomy · Evidence for planets around other stars
Transits and dips in a light curve
Connect a calibrated light curve with the overlapping silhouettes that can produce it, keeping lost light, radius ratio, timing and competing causes distinct.
Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.
Back to the chapterAfter reading, you can
- Define fractional transit depth and normalized remaining flux.
- Calculate the radius ratio in an explicitly ideal full-overlap model.
- Interpret ingress, dilution and repeated dips without treating a candidate as a confirmed planet.
A light curve records a measured signal rather than a moving outline
A light curve plots measured brightness against time in a specified observing band. For an unresolved stellar system, the detector combines contributions within its extraction region. Calibration, background subtraction, exposure duration and time convention are therefore part of the record.
A transit occurs when a smaller foreground body crosses part of a larger body’s apparent disk. A planet passing in front of its host can block some stellar light without being spatially resolved separately. When the planet passes behind the star, the loss instead concerns the planet’s own emitted or reflected contribution; that event is often called a secondary eclipse or occultation.
Let F_out be a chosen stable out-of-event flux and F_in the flux at a specified point in the event, measured in the same band and units. Fractional lost light is δ = (F_out − F_in)/F_out. Normalized remaining flux is F_in/F_out = 1 − δ. These quantities move in opposite directions during a dip.
A depth of 0.01 means one percent of the baseline flux is lost and 0.99 remains. A plot zoomed around 0.99 can show this event clearly, but its scale must be labelled. The visual height of a dip in a zoom is not the fraction of the star’s total light that disappeared.
Uniform surface brightness makes the area calculation simple
Choose a spherical opaque planet of radius R_p smaller than a spherical star of radius R_star. The star has stable uniform surface brightness across its apparent circular disk. Neglect planetary light, contaminating sources, atmospheric transparency and any other changes during the event.
While the whole projected planet disk lies within the stellar disk, the blocked fraction is the ratio of areas: δ = πR_p²/(πR_star²) = (R_p/R_star)². Write k = R_p/R_star for the dimensionless radius ratio. The model then gives δ = k².
Original ratios k = 0.05, 0.10 and 0.20 give full-overlap depths 0.0025, 0.01 and 0.04 respectively. The corresponding remaining fluxes are 0.9975, 0.99 and 0.96. Doubling the radius ratio quadruples the lost fraction under these stated conditions.
The equation supplies a ratio rather than an absolute radius. A physical planet radius requires a stellar radius estimate: R_p = kR_star. Neither a radius ratio nor a depth alone supplies planet mass or excludes every nonplanetary cause of the event.
| Radius ratio k | Lost fraction δ | Remaining normalized flux |
|---|---|---|
| 0.05 | 0.0025 | 0.9975 |
| 0.10 | 0.01 | 0.99 |
| 0.20 | 0.04 | 0.96 |
Curved silhouettes produce a continuous ingress
For an original geometric time model, take R_star = 1 L₀ and R_p = 0.1 L₀. Put the star at the origin and move the planet centre along (x, y) = (t/T₀, 0) L₀. This stipulates constant projected speed 1 L₀/T₀ across a central straight chord.
The disks first touch at t = −1.1 T₀ and the planet becomes fully superimposed at −0.9 T₀. Full overlap persists until +0.9 T₀; last contact is +1.1 T₀. At first and last external tangency, the overlap area is zero.
At t = −1 T₀, the planet centre lies on the stellar limb. Exact circular overlap gives a lost fraction about 0.00489387, so the remaining flux is about 0.99510613. During ingress and egress, curved overlap changes nonlinearly even though the stipulated centre speed is constant.
The figure pairs this geometric model with its computed light curve. It is not an exact full orbit or a measurement of an orbital period. A straight chord can approximate a short transit when the stellar radius is small relative to orbital scale; interpreting a real event requires the appropriate orbital and exposure model.
| t / T₀ | x / R_star | y / R_star | Separation / R_star | Overlap / R_star² | Lost fraction δ | Remaining normalized flux |
|---|---|---|---|---|---|---|
| -1.2 | -1.2 | 0 | 1.2 | 0 | 0 | 1 |
| -1.1 | -1.1 | 0 | 1.1 | 0 | 0 | 1 |
| -1 | -1 | 0 | 1 | 0.015374547 | 0.00489387 | 0.99510613 |
| -0.9 | -0.9 | 0 | 0.9 | 0.031415927 | 0.01 | 0.99 |
| -0.5 | -0.5 | 0 | 0.5 | 0.031415927 | 0.01 | 0.99 |
| 0 | 0 | 0 | 0 | 0.031415927 | 0.01 | 0.99 |
| 0.5 | 0.5 | 0 | 0.5 | 0.031415927 | 0.01 | 0.99 |
| 0.9 | 0.9 | 0 | 0.9 | 0.031415927 | 0.01 | 0.99 |
| 1 | 1 | 0 | 1 | 0.015374547 | 0.00489387 | 0.99510613 |
| 1.1 | 1.1 | 0 | 1.1 | 0 | 0 | 1 |
| 1.2 | 1.2 | 0 | 1.2 | 0 | 0 | 1 |
Curved overlap, nonlinear lost light
Scroll sideways to see the full diagram. Its explanation is below.
Geometry and stellar structure change the relation to depth
A chord away from the stellar centre is described by an impact parameter: the closest projected centre separation in stellar-radius units. A sufficiently grazing planet never places its entire disk over the star. Its maximum loss is then smaller than k² even for the uniform-brightness star.
Real stellar disks can vary in intensity from centre to limb. A planet blocks local surface brightness rather than an equal share of flux at every location. Starspots and other inhomogeneities can change the curve or its apparent baseline. A suitable light-curve model treats these effects instead of imposing a perfectly flat bottom.
Exposure averaging also matters. A recorded point represents signal integrated over its exposure interval, rather than an exact instantaneous sample. Cadence and gaps can hide or blur contact structure. A precise timing or shape inference must reflect the actual sampling.
Unresolved extra light dilutes a dip. In a same-band original example, a target contributes 10,000 flux units and constant blended light contributes another 10,000. Blocking one percent of the target removes 100 units from a 20,000-unit baseline, so the combined observed depth is 0.005. Ignoring the blend would infer √0.005 ≈ 0.07071 rather than the true model k = 0.10.
Repeated dips provide a hypothesis to test
Recurring events can constrain an orbital period if they belong to the same system and the timing model explains the observations. Sampling gaps and aliases can leave multiple intervals plausible. The interval between two recorded dips is not automatically one complete orbit.
An eclipsing stellar system, possibly blended with other light, can imitate a shallow planet-like event. Detector effects, reduction artefacts and stellar variability also require consideration. The relevant alternatives depend on the actual data and observing setup.
Compare event shape and timing, image positions, spectral behaviour and complementary motion evidence as appropriate. These tests add information rather than supplying a universal rule that a particular number of dips confirms every planet. A candidate, a statistically validated interpretation and a dynamically constrained companion are distinct evidence stages.
The transit method turns a change in combined light into geometric constraints through a model. The next lesson studies a complementary effect: the host star’s own motion around a common centre, measured through line-of-sight velocities and changes of sky position.
Worked examples
Separate lost and remaining light
In the ideal model, what follows from a full-overlap depth δ = 0.01?
- Remaining normalized flux is 1 − 0.01 = 0.99.
- The radius ratio is k = √0.01 = 0.10.
- An absolute radius additionally needs the host-star radius, and mass needs other constraints.
The planet radius is one tenth of the stellar radius under the declared model; 99 percent of the baseline light remains.
Add flux before normalizing
How deep is the combined dip when 100 units are removed from the 20,000-unit target-plus-blend baseline?
- In-event combined flux is 20,000 − 100 = 19,900 units.
- Lost fraction is 100/20,000 = 0.005.
- The source-level one-percent loss has become a half-percent combined loss.
The observed combined depth is 0.005. A radius inference that ignores the contaminating light would be biased in this example.
Try the reasoning
For δ = 0.04, is the remaining normalized flux 0.04?
Show the answer and reasoning
No; it is 0.96.
δ names lost light, while remaining flux is 1 − δ.
Does the ideal depth relation determine planet mass?
Show the answer and reasoning
No.
It constrains a radius ratio under specified conditions; mass requires additional physical information.
Does constant projected centre speed make exact circular-overlap ingress linear?
Show the answer and reasoning
No.
The curved overlap area is not a linear function of centre position.
Does any fixed count of repeated dips exclude all false positives?
Show the answer and reasoning
No.
Timing, sampling and alternative astrophysical or instrumental causes need evidence-specific tests.
What to carry forward
- Define lost-light depth and remaining flux before interpreting a light curve.
- Full-overlap area ratios constrain relative radius only under explicit source and geometry assumptions.
- Shape, contamination, timing and competing hypotheses determine how a dip becomes reliable planetary evidence.
References
Teaching resources and research supporting this lesson’s explanations and examples.