# A Hand-Waving Derivation of Planethood

by Hal Levison

The IAU recently defined a planet as any object large enough to clear its neighborhood of small bodies. I have been asked if there is a simple way to determine whether an object is a planet based on its mass and its location within the Solar System. In general, the answer is no, because it depends on the details of what its neighboring planets are like and the dynamical state of the small bodies.

However, being a good theorist, I will not let reality stop me. Indeed, it is possible to develop a back-of-the-envelope argument based on a system containing the Sun and a planet of mass Mp orbiting at a distance of ap. The pieces of this argument have already been developed for a different reason - the understanding of Oort cloud formation. Thus, much of what I say here can be found in Tremaine (1993. 'The distribution of comets around stars', ASP Conf. Ser. 36: Planets Around Pulsars 36, 335-344).

In this simple system, there are two ways in which a planet can remove a population of small bodies: 1) It can accrete all of them, or 2) It can gravitationally eject them from the Solar System. I will treat each of these separately.
1. Ejection: As a small body gravitationally interacts with a planet, it undergoes a random walk in orbital energy. A planet can eject a small body if it is able to change the body's orbital energy by an amount at least equal to its original value. Tremaine (1993) shows that this occurs if 2. Accretion: The probability that a small body will hit a planet each orbit is , where Rp is the radius of the planet, Fg is that classical gravitational focusing factor, and i is the typical inclination of a small body. The gravitational focusing factor is given by , where venc is the typical velocity of an enounter between the small objects and the planet and vesc is the planet's escape velocity. There is a transistion in behavior when venc ~ vesc. This transition is a function of the mass of the planet (through vesc) and its orbit (through venc). In particular, I can define a tranition mass (Mt) as: .
If Mp>Mt then gravitational focusing is important. Assuming that the small body has the same semi-major axis as the planet, we can show that for a planet to accrete all the nearby small bodies in the age of the Solar System (t*) it mass must be: Putting it all together, we find: The green dots are the 9 'classical' planets. The purple shows Ceres. Anything above the red line will accrete the small bodies in its neighborhood, while anything above the blue curve will eject them. The way to read this plot is that if an object is above either line it is a planet. Therefore, there are 8 planets.