How can we find the curvature of a star or planet?

[Tahmeed]

Suppose we are watching a star that is spehrical. But we will see its one face that is toward us(suppose its not rotating). We will see that face to be plane circular. Is there any way we can find the curvature? or tell if its spherical or disk shaped ?

 

[DrClaude]

Any sufficiently big celestial body will be spherical due to self-gravity: https://www.scientificamerican.com/article/why-are-planets-round/

 

[Bandersnatch]

You mean we can't use the existence of gravity to infer that it's spherical? (have you been arguing with flat-earthers?)

In that case, Doppler shift from rotation would give the easiest clue - note that pretty much the only case where it would not be rotating is for a tidally-locked moon.

And since in the case of a moon it's a close-by reflective solid body, you could use laser range finders (different return trip times for light signals aimed at different spots) to build a 3D map.
Observations of phases would work too (the varying curvature of terminator).

Another method for the star that would point to its spherical shape is to measure light reflected off of other planets at a range of times - a disc would not shine equally in all directions, so the amount of light reflected (after correcting for phases) would differ depending on the position of the planet w/r to the star.
Once you know it's spherical, the curvature from the observed disc gives the overall curvature.

 

[snorkack]

Any sufficiently big celestial body will be spherical due to self-gravity

No, it won't.
Centrifugal force can resist gravity no matter how big the body may be.

How can we tell a difference between a star which is not rotating or slowly rotating, and which therefore is spherical, and a star which is rotating rapidly and strongly flattened, but which we are observing along its axis and which therefore looks perfect circle?
Doppler shift will not help. The pole we are looking at is not moving, and the equator at the edge of the disc is moving rapidly but neither towards nor away from us, so no Doppler shift.

 

[Tahmeed]

No, it won't.
Centrifugal force can resist gravity no matter how big the body may be.

How can we tell a difference between a star which is not rotating or slowly rotating, and which therefore is spherical, and a star which is rotating rapidly and strongly flattened, but which we are observing along its axis and which therefore looks perfect circle?
Doppler shift will not help. The pole we are looking at is not moving, and the equator at the edge of the disc is moving rapidly but neither towards nor away from us, so no Doppler shift.

Exactly. Then what is the procedure? ?