n is also defined Expanding mean motion by expanding μ, where M is typically the mass of the primary body of the system and m is the mass of a smaller body. This is the complete gravitational definition of mean motion in a two-body system. Often in celestial mechanics, the primary body is much larger than any of the secondary bodies of the system, that is, . It is under these circumstances that m becomes unimportant and Kepler's 3rd law is approximately constant for all of the smaller bodies. Kepler's 2nd law of planetary motion states, a line joining a planet and the Sun sweeps out equal areas in equal times, or for a two-body orbit, where is the time rate of change of the area swept. Letting t = P, the orbital period, the area swept is the entire area of the ellipse, dA = ab, where a is the semi-major axis and b is the semi-minor axis of the ellipse. Hence, Multiplying this equation by 2, From the above definition, mean motion n = . Substituting, and mean motion is also which is itself constant as a, b, and are all constant in two-body motion. Mean motion and the constants of the motion Because of the nature of two-body motion in a conservative gravitational field, two aspects of the motion do not change: the angular momentum and the mechanical energy. The first constant, called specific angular momentum, can be defined as and substituting in the above equation, mean motion is also The second constant, called specific mechanical energy, can be defined, Rearranging and multiplying by , From above, the square of mean motion n2 = . Substituting and rearranging, mean motion can also be expressed, where the −2 shows that ξ must be defined as a negative number, as is customary in celestial mechanics and astrodynamics. Mean motion and the gravitational constants Two gravitational constants are commonly used in Solar System celestial mechanics: G, the Newtonian constant of gravitation and k, the Gaussian gravitational constant. From the above definitions, mean motion is By normalizing parts of this equation and making some assumptions, it can be simplified, revealing the relation between the mean motion and the constants. Setting the mass of the Sun to unity, M = 1. The masses of the planets are all much smaller, . Therefore, for any particular planet, and also taking the semi-major axis