First, presume I don’t know any advanced math because I don’t.
I’m familiar with the basic principle of staying in orbit; gravity pulls you toward the celestial body, horizontal velocity flings you away and you keep shooting beyond the body.
I’d like to know how to calculate the required velocity to stay in orbit and the size of the body.
Presume no air drag, presume that extremely dense materials are involved if need be, presume the celestial body is a perfect sphere.
If I were in a space ship that orbited an ultra dense celestial body with a gravity of 10m/s squared, how small would the body have to be for me to attain a circular orbit at an altitude of 1m above the surface at a velocity of 20m/s? How about 100m/s?
How about the other way around, if I know that the gravity of the body is 10m/s squared and that the diameter of the body is 1km, what velocity is required at an altitude of 1m to remain in circular orbit?
You can ignore the 1m altitude if that makes the calculation easier.
I know that no material is that dense, don’t bother pointing it out, it’s not the point. Don’t bother pointing out that orbiting that low is unsafe. I just want to know how to calculate the size of a planet for a given gravity and orbital velocity when a circular stable orbit has been achieved.
Well, the surprising answer is that you can’t.
The reason is that when you are outside a sphere, the gravitational force is only dependant upon your distance to the centre of the sphere and the sphere’s mass. There is no term for the sphere’s radius. It doesn’t matter.
So, so long as your orbit’s radius is larger than the radius of the planet, the planet can be any size you choose, and thus any density.
You need to solve two equations.
First, the centripetal acceleration of the orbiting spacecraft:
a[sub]c[/sub] = V[sup]2[/sup]/R
where:
a = spacecraft’s centripetal acceleration
V = spacecraft’s orbital velocity
R = radius of spacecraft’s circular orbit, measured to center of planet
Second, the gravitational acceleration at the altitude of the spacecraft’s orbit:
a[sub]g[/sub] = G * m/R[sup]2[/sup]
where:
a[sub]g[/sub] = acceleration due to gravity at the altitude of the spacecraft’s orbit
G = universal gravitational constant
m = mass of planet (equation assumes mass of spacecraft is much less than mass of planet)
R = radius of spacecraft’s orbit, measured to center of planet
This equation for gravitational acceleration is applicable anywhere outside of the planet’s surface (gravity fades to zero as you move from the surface of the planet toward its center).
For a stable circular orbit, set a[sub]c[/sub]=a[sub]g[/sub], and now you have the relationship between planetary mass, orbital radius, and orbital velocity; set two of those parameters, and solve for the third.
In your OP you posited specific accelerations, so you’re already halfway there: just solve the first equationdirectly for a V-to-R relationship. Choose a V, solve for R, now you know that your planet can be no larger in radius than that.