Exoplanet transit laboratory

When a planet passes in front of its star from our viewpoint, it can block a small part of the starlight. This lab connects the geometry to the dip in a light curve, then compares that ideal picture with a short set of processed Kepler observations.

Predict first. In the uniform-brightness model, a planet with one-tenth the star’s radius blocks 1% of its light when fully in front of it. If the planet’s radius doubles, does the depth become 2% or 4%?

Change the model, then read the observations

The first plot is a simulation. The star has uniform brightness, the planet is opaque, and its projected track is straight with constant speed. The radius ratio $k$ compares planet radius to star radius. The impact parameter $b$ is the closest projected center separation in units of the stellar radius.

Try a challenge

Double the planet’s radius. How much more light disappears?

Predict whether the dip doubles or quadruples, then compare A and B. Both planets cross the centre of the same uniformly bright star.

Snapshot A · radius ratio 0.10

At the centre: 99% brightness, a 1% dip.

Snapshot B · radius ratio 0.20

At the centre: 96% brightness, a 4% dip.

The two snapshots use identical horizontal and vertical scales. These are calculated light curves, not the observations below.

Reveal the explanation

The blocked area grows with radius squared. Changing the radius ratio from 0.10 to 0.20 changes the central dip from 1% to 4%: four times as much lost light. This result assumes an opaque planet completely inside a uniformly bright stellar disk.

Explore · an ideal star and planet

The dashed line is the planet’s projected path. The star has uniform brightness; the planet is opaque. The observer is looking toward the screen.

Change the transit

Planet radius / star radius, k: 0.10Closest approach / star radius, b: 0.00Position along the track / star radius, x: 0.00

Load snapshot ALoad snapshot BCentral transitGrazing transitMiss the starReset experiment

The curve uses the actual area of overlap, including entry and exit. The vertical scale adjusts to make small dips visible. The dashed cursor matches the silhouette above. Use the fixed-scale snapshots to compare the 1% and 4% dips.

Model k = 0.10, b = 0.00, x = 0.00 (b and x in stellar radii). At this position, brightness is 99.000% of the unobscured star: a 1.000% dip. The deepest dip is 1.000%. The whole planet fits inside the stellar disk at closest approach.

The static example works without JavaScript. If the controls do not activate, reload the laboratory or continue with the worked activity below.

Lengths are in stellar radii; the horizontal scale is proportional to time, not hours. This model omits limb darkening, blended light, stellar variability and finite exposure time. A dip alone does not prove a planet or measure its mass.

Overlap model: Mandel and Agol (2002), § 2, Eq. (1).

Read the evidence · processed observations of TrES-2

Kepler measured the star TrES-2 (KIC 11446443) in 2009. Dots are pipeline-corrected measurements; bars are reported 1σ errors, mostly smaller than the symbols. Each cadence spans about 29.4 minutes. Changing the model sliders does not change these observations.

Data, uncertainties and processing

We select finite positive PDCSAP_FLUX and PDCSAP_FLUX_ERR values with SAP_QUALITY = 0, from BKJD 124.89 to 125.59. BKJD means barycentric Julian date in TDB minus 2,454,833 days. The origin 125.24 is a rounded display guide, not a fitted mid-transit time.

We divide flux and reported error by the median of the 17 selected measurements at least 0.15 days from that origin: 406,130.375 electrons per second. We do no further detrending, binning, smoothing, phase folding or fitting. Error bars exclude uncertainty in this normalization and correlated systematics.

Source: NASA Kepler mission, NASA Ames pipeline, MAST/STScI, data release 25, quarter 0. Product kplr011446443-2009131105131_llc.fits, retrieved 27 September 2026. See the Thompson and Fraquelli 2014, Kepler Archive Manual, §§ 2.1.2 and 2.3.1 (PDF) and MAST data-use policy. These public mission measurements retain their source credit.

Download all 31 observations (CSV) · Download data and complete provenance (JSON)

Processed observations; dimensionless brightness and 1σ uncertainty

Hours from origin	Relative brightness	1σ error
-8.361	1.000032	0.000042
-7.871	1.000135	0.000042
-7.380	1.000033	0.000042
-6.890	1.000264	0.000042
-6.400	0.999970	0.000042
-5.909	1.000000	0.000042
-5.419	1.000045	0.000042
-4.928	1.000119	0.000042
-4.438	1.000108	0.000042
-3.457	1.000123	0.000042
-2.967	0.999985	0.000042
-2.476	1.000120	0.000042
-1.986	1.000113	0.000042
-1.005	0.998494	0.000042
-0.514	0.988355	0.000042
-0.024	0.986013	0.000042
0.466	0.993492	0.000042
0.957	0.999871	0.000042
1.447	0.999827	0.000042
1.938	1.000137	0.000042
2.428	1.000008	0.000042
2.918	1.000098	0.000042
3.409	0.999852	0.000042
3.899	0.999967	0.000042
4.390	0.999866	0.000042
4.880	0.999879	0.000042
5.371	0.999885	0.000042
5.861	0.999898	0.000042
6.351	1.000086	0.000042
6.842	0.999805	0.000042
7.332	0.999985	0.000042

Start with a central crossing, then increase $b$. Notice the curved ingress and egress as the disks partly overlap. If the planet only grazes the stellar edge, it never covers its full disk area against the star.

Explain the depth

Prediction answer and a worked calculation

Preparation: fractions, circle areas and square roots. A square root reverses squaring: for example, $0.05^2=0.0025$.

The fully overlapping planet blocks area $\pi R_p^2$ from the uniform stellar disk’s area $\pi R_\star^2$. Therefore

$$\delta=\left(\frac{R_p}{R_\star}\right)^2=k^2.$$

At $k=0.1$, the depth is $0.01$, or 1%. At $k=0.2$, it is $0.04$, or 4%. Doubling a radius quadruples a disk area. A depth of $0.0025$ would imply $k=\sqrt{0.0025}=0.05$ under these assumptions and with full overlap.

This relation alone gives a ratio. To infer a planet radius in kilometers, you also need the stellar radius. It does not give the planet’s mass.

The model uses the actual overlap area of two circles rather than attaching an arbitrary dip to the moving planet. The uniform-source case is developed in Mandel and Agol 2002, § 2, Eq. (1).

What the Kepler points contain

The observed excerpt contains 31 measurements of TrES-2, also called Kepler-1, from Kepler quarter 0. These are pipeline-corrected brightness measurements, with the reported uncertainties, not raw detector counts. The controls do not change them or fit a model to them.

We selected quality-zero, finite positive measurements in BKJD 124.89–125.59 and divided the flux and its reported error by the median of 17 specified baseline points. BKJD is a barycentric time coordinate in days; the plot’s zero is a convenient rounded origin, not a fitted transit center. These long-cadence measurements span approximately 29.4 minutes each.

The data notes give the original product, exact selection, normalization, units, and reproduction command. Download the CSV to inspect the numbers. The source columns and pipeline products are documented in Thompson and Fraquelli 2014, §§ 2.1.2, 2.3.1, pp. 12–13, 15–19 (PDF).

The error bars omit uncertainty in our baseline and correlated instrumental or stellar effects. They should not be read as a complete uncertainty budget for a fitted planet radius.

Compare carefully

Real stellar disks are usually darker near their apparent edges: limb darkening changes the transit shape. Blended light, stellar variability, finite exposure duration, and instrumental effects also matter. The ideal model leaves these out. Its horizontal coordinate is proportional to time, but no measured stellar radius or speed sets a physical duration.

A dip resembling a transit can have other explanations, including an eclipsing stellar system. This single excerpt illustrates a real signal from an identified system; it is not an independent confirmation of a planet.

Test your explanation

Set $k=0.1$ and $b=1$. Should the maximum loss of light still be 1%?

Answer

No. The planet’s center passes at the star’s limb, so only part of the planetary disk overlaps the star even at closest approach. The depth is positive but smaller than $k^2=0.01$. Using the full-overlap formula for a grazing transit would overstate the covered area.

For educators

Preparation and objective. Read fractions and a graph; distinguish a model’s radius-ratio inference from a planet confirmation. In ten minutes, compare $k=0.1$ and $0.2$ at $b=0$: the expected depths are 1% and 4%, because area scales as radius squared. In a thirty-minute extension, compare a grazing model with the fixed Kepler excerpt and its downloadable table. Ask learners to identify which features are calculated geometry, which are measurements, and which depend on processing. Expected limitations include limb darkening, exposure averaging and baseline uncertainty; changing a model control does not change the observation. If scripting is unavailable, use the default plot, the examples above and the CSV. Correct the misconception that one small dip establishes either a unique planet radius or the existence of a planet.

Continue with How can we find a planet around another star?.

References

• Mandel, K., and E. Agol. “Analytic Light Curves for Planetary Transit Searches.” The Astrophysical Journal Letters 580 (2002), L171–L175. DOI. Open PDF.
• NASA Kepler mission / MAST. TrES-2, KIC 11446443, quarter 0 long-cadence light curve, data release 25. Original FITS product. Local derivative and credit.
• Thompson, Susan E., and Dorothy Fraquelli. Kepler Archive Manual, KDMC-10008-005, 2014, §§ 2.1.2, 2.3.1, pp. 12–13, 15–19. Open PDF. The archive filename includes a later 2016 date.

The downloadable model and model notes make the ideal calculation reproducible.