Weigh the light hidden by a planet

Define a disk that the observer sees

Put the stellar center at the origin of the observer’s sky plane. Measure projected distances in stellar radii, so the disk has radius one. Let the opaque planet have radius ratio $p=R_{\rm p}/R_\star$ and projected center distance $z$ . These are apparent projected disks; the planet is in front during the primary transit.

Let $I(r)$ be received brightness per projected area at radial position $r$ . A small area $\mathrm dA$ contributes $I(r)\mathrm dA$ . Adding these contributions over the whole disk gives the unobscured host flux. Adding only the hidden contributions gives the loss.

For constant brightness, it cancels from the ratio. When the planet lies wholly inside the stellar disk, the lost fraction is $p^2$ at any fully contained position. At partial overlap, including a grazing midpoint, the numerator is the lens-shaped intersection.

F_\star(p,z)=1-\frac{\displaystyle\int_{\mathrm{disk}\cap\mathrm{planet}} I(r)\,\mathrm dA}{\displaystyle\int_{\mathrm{disk}} I(r)\,\mathrm dA}

The normalized host flux is one outside transit. The numerator and denominator use the same brightness law and projected area.

Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1

Give the brightness a radial profile

On a spherical photosphere, define $\mu=\sqrt{1-r^2}$ . It is the cosine of the angle between the local surface normal and our line of sight. The quadratic law specifies two dimensionless coefficients:

Expand the brackets into a constant term, a term proportional to $\mu$ and a term proportional to $\mu^2$ . An annulus of radius $r$ and width $\mathrm dr$ has area $2\pi r\mathrm dr$ . This turns the disk integral into a one-dimensional sum of rings.

The three needed integrals are $\int_0^1 r\mathrm dr=1/2$ , $\int_0^1 r\sqrt{1-r^2}\mathrm dr=1/3$ and $\int_0^1 r(1-r^2)\mathrm dr=1/4$ . For the middle one, substitute $q=1-r^2$ , so $\mathrm dq=-2r\mathrm dr$ . Putting the terms together gives:

A physical profile needs positive intensity. A convenient restricted family has nonnegative coefficients whose sum is below one; it remains positive and darkens toward the edge. For more general coefficient pairs, check both positivity and the radial derivative.

I(\mu)=I(1)[1-u_1(1-\mu)-u_2(1-\mu)^2]

The central value is

I(1)

. The coefficients describe how the brightness falls toward the limb; their values depend on the star and observing band.

F_{\star,0}=\pi I(1)\left(1-\frac{u_1}{3}-\frac{u_2}{6}\right)

This normalizes the same quadratic disk before computing any transit depth.

Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1 · batman author tutorial and pinned package 2.5.3

Integrate only the covered part of each ring

At distance $z$ from the stellar center, a planet covers an angular segment of an annulus. Its two circle boundaries meet when the law of cosines is satisfied. For a partially covered ring:

This construction retains the finite planet size, including near contacts. Treating the entire blocked patch as one brightness value is a small-planet approximation. It can be useful, but should not silently replace the full integral for a large or grazing object.

The playable calculation evaluates the annular integral after a change of variable that smooths its contact endpoints. The independent reference uses the published Mandel–Agol analytic cases. Exactly when the planet edge passes through the stellar center, the saved comparison also keeps a separately evaluated special-case elliptic expression and a Cartesian area integral. The raw reference-library failure at that boundary remains in the downloadable results.

\Delta\theta(r)=2\arccos\!\left(\frac{r^2+z^2-p^2}{2rz}\right),\qquad F_{\rm blocked}=\int I(r)r\Delta\theta(r)\,\mathrm dr

Fully covered rings use an angle of

2\pi

; uncovered rings use zero. The partial formula applies only where the two circles intersect.

Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1

Work a centered brightness example

Take $p=0.1$ , $u_1=0.6$ and $u_2=0$ . The mean brightness over the stellar disk is $0.8I(1)$ . The central blocked patch is brighter than this average, so its transit must be deeper than the one-percent area limit.

For a centered planet, integrate complete rings from zero to $p$ . The substitution used above gives:

The default-results table below uses the actual synthetic records at their stated parameters and exposure windows. It lets you compare full and grazing paths and then isolate long exposure or extra light. Its digits describe those numerical examples, not the accuracy of a real stellar atmosphere law.

\delta=\frac{(1-u_1)p^2+\frac{2u_1}{3}[1-(1-p^2)^{3/2}]}{1-u_1/3}=0.0124812

The exact loss is about 1.25 percent. Treating that brightness-weighted depth as a uniform area would give a radius ratio near 0.112 instead of 0.100.

Actual synthetic records: circular period 3.2888 days, orbital axis eight stellar radii; midpoint depths are integrated over the stated window. freeze-synthetic.ts stores full curves and representative current measurements.
Case	Radius ratio	Impact	Companion ratio	Window · seconds	Depth · parts per million	Duration · hours
Limb-darkened chord	$0.12$	$0.35$	$0$	$120$	$1.71\times10^{4}$	$3.35$
Uniform area limit	$0.12$	$0$	$0$	$120$	$1.44\times10^{4}$	$3.53$
Grazing	$0.18$	$1.04$	$0$	$120$	$8050$	$1.77$
Long exposure	$0.07$	$0.78$	$0$	$1800$	$4780$	$2.31$
Companion light	$0.18$	$0.35$	$1$	$120$	$1.92\times10^{4}$	$3.55$

Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1

Add the companion before normalizing

Let a constant unresolved companion contribute a fraction $c$ of the unobscured host light. Add that light to both the in-transit and out-of-transit totals. Then:

For a uniform fully covered transit with $p=0.12$ , the host loss is 0.0144. Equal companion light gives a measured loss of 0.0072. Taking its square root without modeling the companion gives about 0.0849. Keeping the companion in the model recovers 0.12 for this ideal example.

A mission pipeline can already apply a crowding correction. The TESS PDCSAP product in our tutorial does so. Retaining its original metadata is essential: adding the same correction again would change the transit even though no new light was observed.

F_{\rm obs}=\frac{F_\star+c}{1+c},\qquad\delta_{\rm obs}=\frac{\delta_\star}{1+c}

This definition uses companion-to-host light. A catalog crowding fraction defined as host-to-total light is a different quantity.

Reading: MAST TESS Data Product Overview · Morton et al. 2016, False Positive Probabilities, arXiv v1

Weigh the light hidden by a planet

Define a disk that the observer sees

Give the brightness a radial profile

Integrate only the covered part of each ring

Work a centered brightness example

Add the companion before normalizing

Exercises and explanations

Try the prediction

Separate area, brightness and dilution

Sources and calculation scope