Question: Why does the following equation seem to predict stable orbits around the sun as well as for moons around planets without any involvement of balancing centripetal and gravitational forces.
The following is the equation that in my investigation seems to work for planetary motion just using geometric data.
C = 8*G^0.5 = 6.548E-5
Vs = surface velocity of rotating sphere (i.e. sun)
Rs = radius of rotating sphere (i.e. sun)
Vp = orbital velocity of body orbiting the sphere (i.e. planet)
Rp = distance of orbiting body from the center of the rotating sphere.
G = Newton’s gravitational constant
Answer: I am not sure how you derived this equation, but as it is not dimensionally consistent, it does not appear to be correct. Just checking the units of the left and right side of the equation, where Newton’s gravitational constant has units (using the cgs system) cm^3/(g*s^2):
cm^3/(g*s^2) * cm^2/s^2 * cm^2 = cm^2/s^2 * cm
cm^7/(g*s^4) = cm^3/s^2
As the units for the left and right side of the equation do not equate, your equation is not correct.