# Distances needed to make this system stable

Let's take the Trappist-1 system, and specifically, I'm thinking about the star itself and Trappist 1e because it's the most habitable of the seven planets:

• Trappist 1 star is approximately 8.98% of the Sun's mass
• Trappist 1e is a rocky planet with approximately 77.2% of the Earth's mass, located about 0.029 AU from its star on average.

Take this system and put it in orbit around a G-type star with the same mass as our sun. How far apart would the two stars have to be for their orbit to be stable and for Trappist 1e's orbit to remain stable?

You'll have to make sure Trappist 1e is within the Hill sphere of Trappist 1, meaning that the planet is primarily orbiting its smaller star and not the bigger star.

The radius of the Hill sphere (assuming a circular orbit) is given as:

$${\displaystyle r_{\mathrm {H} }\approx a{\sqrt[{3}]{\frac {m}{3M}}}}$$

And plugging in our values:

$${\displaystyle 0.029{\mathrm {\ AU} }\approx a{\sqrt[{3}]{\frac {0.0898}{3}}}}$$

Solving for $$a$$ gives:

$${a \approx 0.093 \mathrm {\ AU}}$$

The wikipedia page mentions that for a stable you'd need to be within 1/2 or 1/3 of the radius so you should double or triple that number to be safe. Maybe try to stick Trappist 1 at least 0.3 AU from the star it orbits?

• Thanks! Really helpful answer. So if the G-type star was approximately 1 AU from Trappist, would that be a stable system? Commented Mar 16, 2023 at 22:33
• Yep! It only gets more stable the further away from 0.3 AU it is. Commented Mar 16, 2023 at 23:01