Exoplanet transit light-curve simulator

What’s happening

When a planet passes in front of its star, it blocks a small part of the stellar disk and the star’s measured brightness dips. If the planet’s radius is $R_p$ and the star’s is $R_\star$, the fraction of light blocked is roughly the ratio of their areas:

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

How long the dip lasts depends on the orbit. For a circular orbit with period $P$, scaled distance $a/R_\star$ and impact parameter $b$ (how far from the centre of the disk the planet crosses, in units of $R_\star$), the inclination is $\cos i = b\,/\,(a/R_\star)$ and the total transit duration is

$$T_{14} = \frac{P}{\pi}\,\arcsin\!\left(\frac{R_\star}{a}\,\frac{\sqrt{(1+k)^2 - b^2}}{\sin i}\right).$$

Replacing $(1+k)^2$ with $(1-k)^2$ gives $T_{23}$, the time the planet is fully inside the disk. If $b > 1-k$ the planet never fits fully inside the disk: the transit is grazing and $T_{23}$ doesn’t exist.

Assumptions and limits. Circular orbit; the planet is dark; the star is a uniform disk. Real stars are dimmer at the edge (limb darkening), which rounds the bottom of the dip and makes the depth depend slightly on $b$. The interactive version will include limb darkening.

Try this

1. Earth seen from far away. With $R_\oplus = 6371$ km and $R_\odot = 695\,700$ km, what transit depth would an alien astronomer measure for Earth?
   Answer

   $k = 0.00916$, so $\delta = k^2 \approx 84$ ppm — less than 0.01%.
2. A hot Jupiter. Repeat for Jupiter ($R_\mathrm{J} = 71\,492$ km) around the Sun.
   Answer

   $k = 0.103$, $\delta \approx 1.06\%$ — over 100 times deeper than Earth’s transit.
3. Duration. A planet has $P = 3$ d, $a/R_\star = 8.8$, $k = 0.1$ and $b = 0.3$. Find $i$, $T_{14}$ and $T_{23}$.
   Answer

   $i = 88.05°$, $T_{14} \approx 2.76$ h, $T_{23} \approx 2.22$ h.
4. Earth’s transit. Earth orbits at $a/R_\odot \approx 215$. Estimate how long a central ($b = 0$) transit of Earth lasts.
   Answer

   About 13 hours.
5. Grazing. For $k = 0.1$, above which impact parameter does $T_{23}$ stop existing?
   Answer

   $b > 1 - k = 0.9$.