In order for a planet to appear to orbit itself, it would need to rotate at a speed equal to or faster than the orbital speed required to maintain a stable orbit. This concept is known as synchronous rotation or tidal locking.
The required speed for a planet to achieve synchronous rotation depends on several factors, including the planet's mass and its distance from the object it is orbiting (such as a star or another planet). The relevant equation is derived from the balance between gravitational and centrifugal forces.
Assuming a planet is orbiting a central body with a mass much larger than itself, such as a star, the formula for the orbital speed required to achieve synchronous rotation is:
V = sqrt(G * M / R)
Where: V is the orbital speed required for synchronous rotation, G is the gravitational constant (approximately 6.67430 × 10^(-11) m^3 kg^(-1) s^(-2)), M is the mass of the central body (in this case, the star), R is the distance between the planet and the center of the central body.
To calculate the specific speed, you would need to know the mass of the central body and the distance of the planet from it. However, it's important to note that achieving such a speed is unlikely for planets in our solar system. For example, the Earth's rotational speed is much slower than the required speed for synchronous rotation.
It's worth mentioning that some celestial bodies, such as the Moon, have reached a state of synchronous rotation with respect to the Earth. This means that the Moon's rotational period is equal to its orbital period around the Earth, causing the same side of the Moon to always face Earth.
If you have specific data for a planet's mass and distance from its central body, I can help you calculate the required speed for synchronous rotation.