It's a Trap(pist)!

I didn't get around to posting last week, since I was busy moving in with my amazing fiancee, but there was some big news from the astronomical community!  In case you missed it, a number of groups confirmed the existence of seven Earth-size planets in the habitable zone of the star Trappist-1.  I thought I'd talk a little bit about how these planets get detected.

Isolated stars are generally too small and too distant for us to see them as more than single points of light, even with powerful telescopes, so you might wonder how we can find the significantly smaller planets that orbit them.  The key is when the planet passes between the star and our line of sight, it casts a shadow on the star:

We can figure out the total light blocked from the two bodies' apparent size:

where d is the diameter of the body, and D is the distance from Earth to the body.  Since the 39 light-years to Trappist-1 is a lot bigger than any orbit the planets would have, we can assume D is the same for the planets and the star.

Closer to home, the apparent size of the sun and the moon from Earth are approximately equal, which is how we can get a total solar eclipse, even though their diameters differ by a factor of 400.

In the case of Trappist-1 though, the amount of light blocked is approximately the ratio of the cross-sections:

where Rp is the radius of the planet, and Rs is the radius of the star.  That leads to data that look like this:

From Nature paper

That plot shows the drop in intensity as each of the seven planets passes in front of the star.  The height of the drop tells us how big the planet is, but we can also use the duration of the drop to find the orbit.

Assuming a circular orbit, we can find the angular velocity of the planet from Newton's gravitational equation:

where G is Newton's constant, M is the mass of the star, and R is the orbital radius of the planet.  To figure out how long the planet takes to cross in front of the star, a picture helps:

The angular distance over which the planet is crossing the star is

Combing this with the previous equation, the transit time is

We measured the transit time from the dimming of the star, and we can figure out the radius and mass of the star from stellar dynamics, so this gives the orbital radius of the planet!

If you want to know more about the discovery, I highly recommend my friend Josh Sokol's piece at New Scientist.