system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a limit to the number of planets that can plausibly form around star systems of two or more stars?
gpt
It depends on how closely the stars orbit each other. 

There are three types of possible orbits for planets in a binary star system: 

1. Planet orbits around Star A

2. Planet orbits around Star B

3. Planet orbits around the gravitational centre of Stars A and B

In cases 1 and 2, the upper limit for how many planets can have a stable orbit around each individual star will depend a great deal on the closest distance between stars A and B as they orbit each other. If the minimum distance is several hundred AU, then functionally it’s nearly the same as if the stars weren’t orbiting each other at all. If the minimum distance is within a few AU or less, then neither star is likely to have planets in stable orbits around them and only them.

In case 3, it depends on the maximum distance between the stars. If the stars are close to each other and have a relatively tight orbit with low eccentricity, then stable planetary orbits will be able to exist closer to the dual-star system, which in the end means more planets will be able to form and orbit around the two stars. If instead the stars have a highly elliptical orbit around each other, they may not have any planets orbiting around the dual system at all. 

As a nearby example, the Alpha Centauri system has three stars. A and B orbit around each other in an elliptical orbit. Their closest approach is distant enough for each star to have 3, maybe 4 planets each in stable orbits around each star, but no more than that. Any planets that orbit around the gravitational centre of the system would be many hundreds of AU distant and would have orbital periods of many thousands of years; these would all be uninhabitable frozen snowballs.

Proxima Centauri is extremely distant from the other two stars, so much so that their gravitational influence on it and its surroundings is negligible. So Proxima can have just as many planets around it as it could have if the other two stars weren’t even there.