A larger planet at same density means it has more mass. More mass, more gravity. More gravity, the spaceship needs to be faster to go around it, cancelling out

A larger planet at same density means it has more mass. More mass, more gravity. More gravity, the spaceship needs to be faster to go around it, cancelling out the larger circumference. Volume and circumference do not grow to the same power of the radius, i haven't checked if that still accurately cancels each other away or if the astronauts statement is an approximation. Edit: Other poster linked mathematical proof https://en.wikipedia.org/wiki/Orbital_period#Effect_of_central_body's_density https://en.wikipedia.org/wiki/Orbital_period#Effect_of_centr...