The consistency of Newton's laws with an elliptical orbit. In cylindrical coordinates, assume some orbit r() with time t a free parameter. For a central force, L = mr²d/dt = constant. Hence L dt = mr² d, d/dt = (L/mr²)d/d, and specifically dr/dt = (L/mr²)dr/d = - (L/m)du/d where u = 1/r . Then the equation of motion m d²r/dt² - mr(d/dt) ² = -k/r² is transformed to d²u/d² + u = km/L². With y = u - km/L², y = constant x cos( - o), and u = 1/r = mk/L² [1+constant x (L²/mk)cos( - o)], the equation for an ellipse.

Kepler's third law (for the two-body problem). We start with the total energy of the system, E = 1/2 v² - GM/r, where M = m₁+m₂, =m₁ m₂,/M, and L = rv with v = v₁+v₂, the velocities evaluated in the center-of-mass system. From the geometry of the ellipse, with a and p standing for aphelion and perihelion, r_p=a(1-e), r_a =a(1+e), r_p/r_a = (1-e)/(1+e) = v_a/v_p. If E is now expressed once in terms of parameters at aphelion, once perihelion, then one obtains v_p² =(GM/a)(1+e)/(1-e), and L = v_pr_p =[GMa(1-e²)]^1/2, and finally E = -Gm₁m₂,/2a. Since the area swept out is dA/dt = 1/2 r²d/dt, we have also L = 2A/P = 2a² (1-e²)^1/2/P. The two forms of L yield P² =4²a³/GM. The energy equation becomes 1/2 v² = Gm₁m₂ (1/r - 1/2a).

Gravity within a spherical system.

1. The local gravity can be derived quite elegantly in terms of the differential equation div grad = -4 where is the gravitational potential and the mass density. Assuming spherical symmetry makes grad into a radial vector. Integration of both sides over a spherical volume out to radius r yields div grad dV = grad dA = 4r²grad = -4GdV, or grad = - GmM(r)/r².

2. The local gravity can be derived on a lower mathematical level by adding up all the gravitational forces (one needs only the components toward the center) acting on a particle of mass m at distance r from a thin ring of some mass dM that resides at an arbitrary distance r' < r from the center. (All parts of the ring are equidistant from the particle.) When one adds the effect of all the rings contributing to a shell at r', of mass dM(r'), the force becomes GmdM(r')/r² The gravity from that shell is the same as if its mass were at the center. Then it is easy to add the effect of all shells at r' < r, GmM(r)/r². If the limits of integration are handled carefully for shells at r' > r, indeed they turn out not to contribute.

The Virial Theorem. The proof for any system of self-gravitating particles is based on evaluating dQ/dt, with the definition Q = p_ir_i = 1/2 dI/dt, where the sum is over all particles and I is the moment of inertia. The differentiation of r_ileads to 2K. The differentiation of p_ileads to dp_i/dt r_i = F_ir_i = F_ijr_i = 1/2 (F_ij-F_ji)r_i = 1/2 F_ij (r _i-r_j) where is a sum over all pairs of stars i and j except i = j, F_ij = -F_ji has been used where the factor 1/2 appears, and the indices i and j have been interchanged in the last step. With the gravitational force between particles F _ij proportional to (r_i-r_j)/| r_i-r_j|³, the quantity to be summed turns into a scalar quantity which is the total gravitational energy. The end result is d²I/dt² = +2K. In equilibrium, = -2K.