How do we know the order of the new Trappist-1 planets?

Using transit photometry astronomers were able to discover Trappist b through h. Based on this question we learn that the naming of planets is based on their distance from their star (b being the closest).

But how do we actually determine the distance from the star? From my understanding with transit photometry we basically just see a shadow against the backdrop of the star. That can tell us the planet exists, but how do we know the distance between the planet and the star?

• Actually the answers to the question you link tell you that the planets are labeled chronologically first, and by distance second. Planets discovered by the same study will be labeled by distance, but it's possible other planets may be discovered later which orbit closer than all of those, or between some of them. To keep the literature consistent and not confusing, the labels are not altered every time new things are discovered, and newly discovered objects just get the subsequent letters (though they still go by distance within their own set if multiple ones are discovered by that study). – zibadawa timmy Feb 24 '17 at 15:53
• @zibadawatimmy Yeah, I just wanted to keep it simple. – David says Reinstate Monica Feb 24 '17 at 15:55

Using Kepler's laws.

We can determine the orbital period, $P$, each planet has by just looking at the time between transits for a given planet. Each planet has a distinct transit so its easy to distinguish which planet is transiting and we can calculate a period pretty easily.

Kepler's law then tells us that if you're orbiting with a specific period, you have to be orbiting at a specific distance (based on parameters of the system). The general equation is given by

$$P^2 = \frac{4\pi}{GM_{star}}a^3$$

Where $G$ is just the gravitational constant, $M_{star}$ is of course the mass of the star and fairly well known, and $a$ is the radius at which the planet orbits. Simply calculate $a$ for each of your planets and put them in the correct order. You could even do this yourself using data from the paper.

• Would that still apply of the orbit is highly elliptical? – David says Reinstate Monica Feb 24 '17 at 15:49
• Here's a convenient JPL worksheet that guides you to using Kepler's laws on actual observation data. – zibadawa timmy Feb 24 '17 at 15:55
• @DavidGrinberg None of the orbits appear to be highly elliptical, but yes this equation applies for both circular or elliptical orbits. Technically $a$ is the semi-major axis of the (elliptical) orbit. You can define the closest and farthest points of the orbit by $r_p=(1-e)a$ and $r_a=(1-e)a$, respectively, where $e$ is the eccentricity. All these planets have eccentricities of about 0.08 or less. – zephyr Feb 24 '17 at 16:19