--- title: "Find the star that shapes the transit signal" description: "Read fixed stellar-contamination and false-positive papers to separate a spotted light source, a blended eclipse and a conditional planet fit." canonical_url: "https://sandboxphysics.com/stories/transit-evidence/the-star-in-the-signal/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/transit-evidence/the-star-in-the-signal/" last_modified: "2026-10-28" --- [A planet crosses a star](https://sandboxphysics.com/stories/transit-evidence/) AC17 · L4 # Find the star that shapes the transit signal Sandbox Physics · 2026-10-28 A transit fit needs a model of the illuminating star and evidence that the dip belongs to the intended source. Read Rackham and colleagues’ fixed stellar-contamination calculation beside Morton and colleagues’ explicit competing-source hypotheses to see how those two questions are tested. ## Before you begin Use Rackham, Apai and Giampapa, arXiv 1711.05691v3, Section 2.3 and Figures 2, 4 and 6; Morton and colleagues, arXiv 1605.02825v1, Sections 2, 3, 4.2 and 4.7. The previous brightness chapter explains normalized depth and companion light. ## Compare the hidden patch with the whole star Open Rackham’s Section 2.3. Its key comparison is between the spectrum along the transit chord and the spectrum averaged over the full visible star. If those differ, normalizing the blocked light by the whole star imprints stellar structure on the measured depth. For an otherwise immaculate chord and unocculted dark regions, let their disk covering fraction be $f$ and their brightness relative to the ordinary photosphere be $s$ in one band. The disk baseline is reduced by the factor $1-f(1-s)$, while the planet hides the ordinary photosphere. The simple two-component depth becomes: Take a one-percent ordinary depth, ten-percent covering fraction and half-bright regions in that band. The baseline factor is 0.95, so the apparent depth is about 0.01053. A uniform-area radius estimate grows by about 2.6 percent. This direction differs from extra constant companion light, which makes a dip shallower. $$ D_{\rm observed}=\frac{D_{\rm ordinary}}{1-f(1-s)} $$ This is a restricted illustrative form of the paper’s spectral contamination factor. It assumes two brightness components and an unspotted transit chord. Reading: Rackham, Apai & Giampapa 2018, Transit Light Source Effect, v3 · Mandel & Agol 2002, Analytic Lightcurves, arXiv v1 ## Read the variability pictures as model experiments Figure 2 shows an example constructed photosphere and the rotational light curve obtained by summing its visible brightness. Notice which parts rotate into view and how the peak-to-peak variation is measured. It is a model demonstration of what photometry responds to, not an image of a particular observed star. Next compare the spot-size cases in Figure 4. The horizontal axis is covering fraction; the vertical axis is variability amplitude. The shaded spread shows that a given amplitude corresponds to a range of coverage in the simulated distributions. Different spot sizes and arrangements produce different relations. A nearly symmetric distribution can cover considerable area with little rotational variation. Monitoring brightness therefore informs the nonuniform part of the surface, and an apparently quiet light curve need not supply an immaculate reference spectrum. The paper’s M-dwarf temperature contrasts and spectral models should remain attached to its numerical predictions when considering another kind of star. Reading: Rackham, Apai & Giampapa 2018, Transit Light Source Effect, v3 ## Follow the wavelength dependence Figure 6 plots contamination spectra for the specified model photospheres. Read its vertical quantity as a relative change to transit depth. It is not the planet’s own atmospheric absorption spectrum. Features of cooler regions can overlap wavelength bands used to interpret atmospheric molecules. The distinction between a crossed and an uncrossed spot matters. Covering a dark patch during the transit hides less light and can make a bump within the dip. A dark region outside the chord lowers the baseline and can deepen the apparent transit. A broad-band light curve and a transmission spectrum sample these source effects differently. Our fixed TESS fit does not infer a surface map or perform this spectral calculation. Its useful connection is narrower: the assumed chord brightness changes the inferred radius. The retained limb-brightness sensitivity fits show that dependency for one broad band. Reading: Rackham, Apai & Giampapa 2018, Transit Light Source Effect, v3 ## Compare a planet with other eclipsing sources Now open Morton’s Section 2. Its hypotheses include a planet and several configurations of eclipsing stellar companions, including systems blended with the target. A small dip can arise from a deep eclipse whose light is diluted. The probability comparison includes how often each configuration is expected and how well its possible shapes describe the signal. Section 3 lists information beyond the transit depth: stellar properties, allowed sky region for a contaminating source and limits on secondary eclipses. Centroid measurements constrain where the dimming happens; a deep high-resolution image can constrain nearby sources that an ordinary aperture blends together. A good planet-shaped fit supplies one likelihood within this comparison. It does not supply the full list of hypotheses or their prior frequencies. In particular, a planet orbiting an unresolved companion can still be a planet while its inferred radius refers to the wrong host-star scale. Radius accuracy and astrophysical classification are different questions. Reading: Morton et al. 2016, False Positive Probabilities, arXiv v1 · NASA Exoplanet Archive, Kepler FPP Table Description ## Keep the failure tests beside the probability Read Section 4.2 before treating a small false-positive probability as a general reliability score. The calculation selects among specified astrophysical eclipse explanations. It does not include all instrumental artifacts or stellar variability. Separate vetting determines whether that comparison is appropriate. Section 4.7 then lists failed fits, missing inputs and unconverged shape models. A missing probability is not zero. A formally successful calculation with a source position inconsistent with the target also needs care. The official NASA table description preserves these conditions and distinguishes its fixed catalog inputs from later property updates. For the next candidate, make the sequence concrete: identify the changing image location; inspect repeated and secondary events; note the host-star information and allowed blends; then read the probability under its actual hypothesis set. For an already established planet, return to the surface-brightness question when interpreting its radius or spectrum. Reading: Morton et al. 2016, False Positive Probabilities, arXiv v1 · NASA Exoplanet Archive, Kepler FPP Table Description ## Exercises and explanations 1. A star shows almost no rotational modulation. What part of the spot distribution can remain hidden from that observation? Read the explanation A roughly axisymmetric component changes the average spectrum but varies little as the star turns. Low modulation does not uniquely determine a small total covering fraction. 2. Why can an unocculted dark region and an unresolved bright companion shift a radius estimate in opposite directions? Read the explanation The dark region lowers the host baseline for an ordinary bright chord, increasing the fractional dip. The companion adds unblocked baseline light, reducing the fractional dip. 3. Does an astrophysical false-positive probability include the chance of an instrumental false alarm? Read the explanation The cited calculation compares its specified astrophysical hypotheses. Instrumental reliability and eligibility for that comparison need separate vetting; the source paper and archive explicitly state that scope. ## Try the prediction ### [Ask which light changed](https://sandboxphysics.com/labs/shadow-eclipse-lab/) 1. Add constant companion light in the synthetic transit and predict the direction of the apparent size bias. 2. Keep the radius fixed and change the limb brightness, identifying the role of the hidden patch. 3. List an observation outside the synthetic curve that could test whether a blended stellar eclipse caused a real dip. [Previous: Fit a frozen TESS transit without losing its assumptions](https://sandboxphysics.com/stories/transit-evidence/fit-the-tess-transit/) [Next: Read transit duration as an orbital speedometer](https://sandboxphysics.com/stories/transit-evidence/duration-as-a-speedometer/) ## Sources and calculation scope 1. [Mandel & Agol 2002, Analytic Lightcurves, arXiv v1](https://arxiv.org/pdf/astro-ph/0210099v1) 3 October 2002, version 1; ApJ 580,L171–L176. Sections 2–4, uniform overlap, quadratic intensity normalization and Table 1 special cases; Section 5 states the small-planet approximation. The current archive PDF has a later typesetting header; its submission version is pinned here. 2. [Rackham, Apai & Giampapa 2018, Transit Light Source Effect, v3](https://arxiv.org/pdf/1711.05691v3) 24 January 2018, version 3. Section 2.3 equations1–3 and Figures 2,4,6: chord versus disk spectra, modeled variability/coverage distributions and contamination spectra. These are fixed stellar-model calculations, not observed spectra or a spot-map fit to WASP-126. 3. [Morton et al. 2016, False Positive Probabilities, arXiv v1](https://arxiv.org/pdf/1605.02825v1) 10 May 2016, version 1. Sections 2.1–2.2, 3, 4.2, 4.6–4.7 define eclipse hypotheses, ancillary data, astrophysical-only scope, unresolved-host radius questions and failures. No vespa population calculation or planet-validation reproduction is claimed. 4. [NASA Exoplanet Archive, Kepler FPP Table Description](https://exoplanetarchive.ipac.caltech.edu/docs/fpp_TableDescription.html) Official DR25 table notes retain fixed input dates, code commits, necessary prior vetting and the distinction between astrophysical false positives and instrumental false alarms. Used as the interpretation guide, not as a new FPP value for the TESS target.