--- title: "Use the transit clock to measure a chord" description: "Derive circular transit contacts, duration and conditional stellar density, then include exposure averaging and read a boundary-limited fit." canonical_url: "https://sandboxphysics.com/stories/transit-evidence/clocks-chords-and-density/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/transit-evidence/clocks-chords-and-density/" last_modified: "2026-10-28" --- [A planet crosses a star](https://sandboxphysics.com/stories/transit-evidence/) AC17 · L3 # Use the transit clock to measure a chord Sandbox Physics · 2026-10-28 Transit timing measures how a projected orbit crosses the stellar disk. Combined with a radius ratio and period, it can constrain a chord and a circularly inferred stellar density. The exposure and brightness model determine how well those timings can be recovered. ## Before you begin Read Weigh the light hidden by a planet. You need the sine of an angle and speed around a circle. The chapter introduces the fitting objective and explains why a parameter at a boundary has an asymmetric uncertainty. ## Project a circular orbit onto the sky Let $A=a/R_\star$ be orbital semimajor axis in stellar radii. The orbit is circular, has period $P$ and inclination $i$, with an edge-on orbit at a right angle. Set the primary mid-transit time to $t_0$. The phase is $\phi=2\pi(t-t_0)/P$. Choose the horizontal coordinate along the apparent passage and the vertical coordinate perpendicular to it. Rotating the orbital circle by its inclination gives: Close to the midpoint, the projected path looks almost straight and horizontal speed is nearly constant. The equations above keep the circular curvature instead of making that approximation. A measured duration is therefore tied to a declared orbit, not merely an arbitrary sliding disk. $$ x=A\sin\phi,\qquad y=b\cos\phi,\qquad b=A\cos i,\qquad z^2=A^2\sin^2\phi+b^2\cos^2\phi $$ The impact parameter $b$ is the midpoint’s projected distance from the stellar center. Only the front half of the orbit produces a primary transit. Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1 · Dawson & Johnson 2012, Photoeccentric Effect I, v3 ## Locate the four contacts First contact occurs when the disks touch externally: $z=1+p$. Fourth contact is the corresponding departure. If the planet fits fully inside, second and third contact occur at $z=1-p$. A grazing path has no fully contained interval. Rewrite the separation as $z^2=b^2+(A^2-b^2)\sin^2\phi$. Substituting the outer contact condition and solving for the positive phase gives the half-duration. Multiply it by $P/(2\pi)$ to turn phase into time, then double it: The ingress duration is $(T_{14}-T_{23})/2$ in this symmetric circular model. A higher path shortens the full passage but changes the relative importance of ingress and egress. Measuring only a total width leaves more ambiguity than measuring both time scales. $$ T_{14}=\frac P\pi\arcsin\sqrt{\frac{(1+p)^2-b^2}{A^2-b^2}} $$ A transit requires $b<1+p$. For $b<1-p$, replacing the plus sign in the contact radius with a minus sign gives $T_{23}$; otherwise the interior duration is zero. Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1 · Kipping 2010, Finite Integration Time, arXiv v1 ## Work the geometry all the way through Take $p=0.1$, $b=0.5$, $A=10$ and $P=3$ days. The outer-contact ratio under the square root is $0.96/99.75$. Its inverse sine is about 0.0983 radians. Multiplying by $3/\pi$ gives $T_{14}=0.09383$ days, or 2.252 hours. For the inner contacts, use $0.56/99.75$ instead. This gives $T_{23}=1.719$ hours. Each ingress or egress lasts about 16.0 minutes. A thirty-minute exposure spans longer than ingress itself, so an instantaneous curve is an unsuitable description of those measurements. Now move the same planet to $b=1.04$. The outer contacts still exist, but the inner-contact radius is smaller than the path offset. The correct interpretation is grazing, with no full interior interval. Attempting to take a square root of the negative inner-contact expression would be a misuse of the formula. Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1 ## Combine the orbit with gravity Kepler’s third law relates orbital size, period and total mass. If the planet mass is negligible compared with the star, write $M_\star=4\pi^2a^3/(GP^2)$. The mean stellar density divides that mass by the volume $4\pi R_\star^3/3$: This result measures a density scale without separately knowing the stellar radius. It does not give the planet’s physical radius. That still needs $R_{\rm p}=pR_\star$. An eccentric orbit can cross the same chord at a different speed, changing the density inferred from a circular fit. The last source guide develops that degeneracy. The worked geometry above gives a circularly inferred density of about 2100 kilograms per cubic metre. It is a synthetic example with prescribed orbital parameters. The real TESS fit later gives a different conditional value; the two examples are not measurements of the same system. $$ \rho_{\rm circ}=\frac{3\pi}{GP^2}\left(\frac a{R_\star}\right)^3 $$ Use seconds for $P$ and the SI gravitational constant to obtain kilograms per cubic metre. The subscript remembers the circular-orbit assumption. Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3 · Maxted et al. 2016, Five WASP-South Planets, arXiv v1 ## Average the model across each exposure An exposure records a weighted interval around its time stamp. In the simple uniform-window approximation: Numerical integration needs its own resolution check, especially near contacts. The default browser uses a fixed quadrature and saves comparisons against a much denser independent time average. The real-data tutorial separately refines its two-minute window. The TESS live-time and readout metadata are retained because the exact onboard sampling kernel is more complicated than a continuous shutter. Long averaging broadens the apparent shoulders. If it is ignored, a fitter can change the path and orbital scale to mimic that broadening. This is a parameter bias caused by a different measurement model, even when both curves look smooth. $$ \overline F(t_j)=\frac1{\Delta t}\int_{t_j-\Delta t/2}^{t_j+\Delta t/2} F(t)\,\mathrm dt $$ Average the predicted flux before comparing it with a measured point. Changing the model’s sampling density is different from changing the instrument’s exposure window. Reading: Kipping 2010, Finite Integration Time, arXiv v1 · TESS Science Data Products Description, Revision F ## Fit the light and retain the uncertainty shape For observed normalized fluxes $f_j$ with supplied pointwise standard errors $s_j$, a conditional Gaussian comparison minimizes: Radius, path and limb brightness can compensate one another. A profile calculation fixes one impact parameter, refits the others and records the increase in the objective. It displays the ridge of similar shapes and the parameters that vary together. An impact parameter cannot be negative in this symmetric description. When the best fit lies at zero, a symmetric error bar extending equally below zero is misleading. Keep the full profile and state its fixed brightness, dilution and noise assumptions. Residual correlations and alternative source models require further checks before a conditional objective becomes a research uncertainty statement. $$ \chi^2=\sum_j\left[\frac{f_j-\overline F(t_j;\boldsymbol\theta)}{s_j}\right]^2 $$ The parameter vector includes radius ratio, path, orbital scale and timing; local baseline terms also enter the actual tutorial fit. Independent Gaussian errors are an assumption of this objective. Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3 · TESS Science Data Products Description, Revision F ## Exercises and explanations 1. A fitted ingress is sixteen minutes long and exposures last thirty minutes. Is it enough to evaluate the exact instantaneous transit at each reported time? Read the explanation No. Exact orbital geometry does not account for time averaging. Integrate that geometry’s flux over the exposure; then refine the integration until its changes are small relative to the scientific precision required. 2. Double the stellar radius while keeping the measured radius ratio unchanged. What happens to the planet radius? Read the explanation It doubles. The transit ratio has no absolute length unit; the independent stellar radius supplies it. 3. A circular fit gives a density incompatible with an independent stellar estimate. Must the star estimate be wrong? Read the explanation An eccentric crossing speed, brightness/dilution errors or a different source can also change the fitted orbital scale. Compare these assumptions before assigning the discrepancy to the star. ## Try the prediction ### [Compare the two transit clocks](https://sandboxphysics.com/labs/shadow-eclipse-lab/) 1. Move the path toward the limb and predict the change in total duration and the fully contained interval. 2. Select the long exposure, then shorten its window without changing radius or path. Follow the rounded shoulders. 3. Change orbital scale in the advanced controls and explain which circularly inferred density assumption you have changed. [Previous: Weigh the light hidden by a planet](https://sandboxphysics.com/stories/transit-evidence/weigh-the-hidden-light/) [Next: Fit a frozen TESS transit without losing its assumptions](https://sandboxphysics.com/stories/transit-evidence/fit-the-tess-transit/) ## 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. [Kipping 2010, Finite Integration Time, arXiv v1](https://arxiv.org/pdf/1004.3741v1) 21 April 2010, version 1. Section 2 follows contact shifts and biased parameter retrieval; Section 3.3 treats numerical integration errors around curved segments and contact discontinuities. A model window is kept distinct from an exact instrument kernel. 3. [TESS Science Data Products Description, Revision F](https://archive.stsci.edu/files/live/sites/mast/files/home/missions-and-data/active-missions/tess/_documents/EXP-TESS-ARC-ICD-TM-0014-Rev-F.pdf) 11 September 2020. Sections 2.4, 2.7, 5.2.2 and quality Table 32: time standards, frame rejection, light-curve columns and flags. The actual DR42/SPOC source header and values are frozen with the observed input; generic manual examples are not substituted for those values. 4. [Maxted et al. 2016, Five WASP-South Planets, arXiv v1](https://arxiv.org/pdf/1602.01740v1) 4 February 2016, version 1; A&A 591,A55. Sections 2–4 and Table 6: WASP-126 discovery photometry, radial velocities, adopted orbital assumptions and initialization period 3.28880 days. The table was also checked in the original arXiv TeX source. This tutorial fits a separate later TESS product, not the discovery data. 5. [Dawson & Johnson 2012, Photoeccentric Effect I, v3](https://arxiv.org/pdf/1203.5537v3) 2 November 2012, version 3. Section 2 equations3–7 and Sections 3.1–3.2 with Figures 2–3: short-transit speed factor, conditional density, curved eccentricity/orientation posteriors, stellar uncertainty and transit selection. Posterior illustrations and observed applications are labeled separately.