The Eccentric Frame Decomposition of Central Force Fields Jared M. Maruskin,∗ Daniel J. Scheeres,† Fred C. Adams,‡ and Anthony M. Bloch§ September 15, 2021 Abstract The rosette-shaped motion of a particle in a central force field is known to be classically solvable by quadratures. We present a new approach of describing and characterizing such motion based on the eccentricity vector of the two body problem. In general, this vector is not an integral of motion. However, the orbital motion, when viewed from the nonuniformly rotating frame defined by the orientation of the eccentricity vector, can be solved analytically and will either be a closed periodic circulation or libration. The motion with respect to inertial space is then given by integrating the argument of periapsis with respect to time. Finally we will apply the decomposition to a modern central potential, the spherical Hernquist-Newton potential, which models dark matter halos of galaxies with central black holes. 1 Introduction arXiv:0712.3511v1 [astro-ph] 20 Dec 2007 1.1 Central Force Fields The motion of a particle in a central force field is known to be classically solvable by quadratures. Due to the spherical symmetry of the force field, ∗Ph.D. Candidate, Department of Mathematics, The University of Michigan †Associate Professor, Department of Aerospace Engineering, The University of Michi- gan ‡Professor, Department of Physics, The University of Michigan §Professor, Department of Mathematics, The University of Michigan 1 an angular momentum integral exists and the ensuing motion is confined to a single orbital plane so that, without loss of generality, we can assume the system to have 2 degrees of freedom. In polar coordinates, the Hamiltonian may be expressed as: 1 h2 H = r˙2 + − U(r), 2 r2 where h = r2θ˙ is the angular momentum and U(r) is the potential energy function. We will use the standard convention that dots refer to time deriva- tives, whereas primes refer to spatial derivatives. The corresponding Hamil- tonian system, h2 h r˙ = v v˙ = + U ′(r) θ˙ = h˙ =0, r r r3 r2 is integrable by quadratures: r(t) ±dr = t − t0 (1) 2 2 r(0) 2H +2U(r) − h /r Z p t h dt θ(t)= θ + . (2) 0 r(t)2 Zt0 The ensuing motion follows rosette-shaped paths (Arnold (1989), Whittaker (1988), etc.). 1.2 Osculating Orbital Elements One could, alternatively, proceed using Variation of Parameters and La- grange’s Planetary Equations (Brouwer & Clemence (1961), Roy 1988). In this case, one can write down differential equations of motion for the six os- culating classical orbital elements and then solve them by quadrature for all time. These equations are nonlinear and furthermore depend upon a choice of the “planetary” gravitational parameter µ. One assumes the potential is a perturbation of a Newtonian potential: µ U(r)= + R(r), r where R(r) is known as the disturbing function. For a general central force field where there is no nominal attracting body, such a choice is somewhat ar- bitrary. For instance, there is no primary “planet” when considering motion 2 in a galactic halo. A gravitational parameter for the unperturbed motion can nonetheless be artificially contrived, perhaps based on the total halo mass (if the motion evolves in the outskirts of the galaxy) or based on the mass of a central galactic bulge or black hole (if the motion evolves near the galac- tic core). Whatever the choice of gravitational parameter, a complete set of osculating orbital elements arises and the ensuing motion can be determined. 1.3 The Eccentric Frame We will define a gravitational parameter based on the central force field’s potential, with no reference to a main attracting body and perturbation theory. Following the classical analogy, we define µ(r) such that: µ(r) U(r)= . r We will show that this gives rise to a nonstatic eccentricity vector that rotates at a nonuniform rate. The eccentricity vector (Runge-Lenz vector) associ- ated with this spatially variable gravitational parameter function defines a preferred coordinate system which we call the eccentric frame. With respect to this frame, we will show that the motion follows a closed orbit. Depending on the value of energy, the particle will make closed circulations or librations in the eccentric frame. The eccentric frame decomposition gives rise to a set of orbital elements. We will discuss their physical implications and the key features of how they arise. In particular, one can have circular orbits in iner- tial space with nonzero osculating eccentricity. This feature is not unique to our method, it can arise from any choice of osculating orbital elements. The eccentric frame decomposition, however, illuminates the behavior and gives rise to a new standard description that better fits orbits of central force field potentials. 2 The Eccentric Frame Decomposition We first define the eccentric frame by means of specifying the nonstatic ec- centricity vector associated with the gravitational parameter function µ(r)= rU(r). We then show that the particle traces a closed orbit as viewed from this noninertial frame. Finally we compute the set of osculating orbital ele- ments that belong to this system. 3 2.1 Motion with respect to the Eccentric Frame Given a spherically symmetric potential energy field, we can recast the Hamil- tonian into the following form, reminiscent of its classical analogy: 1 h2 µ(r) E = v2 + − , (3) 2 r2 r where h = r2θ˙ is the magnitude of the angular momentum vector, H = r × r˙, (4) and (r,v,θ,h) are the symplectic coordinates, with v =r ˙. This gives rise to the following Hamiltonian equations of motion: h2 µ′(r) µ(r) r˙ = v v˙ = + − r3 r r2 h θ˙ = h˙ =0 r2 which can be recast in the following form h2 µ′(r) µ(r) ¨r = r¨ − e = − e , (5) r3 r r r2 r where r = rer. Consider now the eccentricity vector B = r˙ × H − µ(r)er. (6) Its evolution is governed by the following equations of motion: ′ B˙ = ¨r × H − µ (r)r ˙er − µ(r)θ˙eθ µ′(r) µ(r) = − r2θ˙e × Hˆ − µ′(r)r ˙e − µ(r)θ˙e r r2 r r θ µ′(r) µ(r) = − − r2θ˙e − µ′(r)r ˙e − µ(r)θ˙e r r2 θ r θ = −µ′(r)r˙. The vector B itself works out to be 2 B = r θ˙(r ˙er + rθ˙eθ) × Hˆ − µ(r)er 2 3 2 = −r r˙θ˙eθ + r θ˙ er − µ(r)er h2 = −hr˙e + − µ(r) e . θ r r 4 We thus find the magnitude of B is: B = 2h2E + µ2(r). (7) We define the argument of periapsis,p ω, to be the angle made between the inertial x-axis and the B-vector. The B-vector defines a rotating reference frame, which we call the eccentric frame. We define Bˆ to be a unit vector in the B direction. Hats will denote unit vectors. Let X and Y be the cartesian coordinates of the particle with respect to the eccentric frame and let x and y be the cartesian coordinates of the particle with respect to the inertial frame. The axes of the inertial frame are determined by the stationary unit vectors ˆi and ˆj. The polar angle of the particle measured with respect to the Bˆ direction is known as the true anomaly f. This notation is also used in Roy (1988). The polar angle of the particle in the inertial frame is related to the true anomaly by the following relation: θ = f + ω. (8) In some of the literature, the true anomaly f is denoted by ν; and the inertial polar angle (argument of latitude) θ is denoted by u. Decomposing the eccentricity vector B in the inertial frame, we see that h2 B = rh˙ sin θ + − µ(r) cos θ ˆi r h2 + −rh˙ cos θ + − µ(r) sin θ ˆj. r However, by definition, B = B(r)(cos ωˆi + sin ωˆj). Hence: cos ω 1 cos θ = A · sin ω B(r) sin θ 1 cos ω = A · F · (9) B(r) sin ω where we have defined (h2/r − µ(r))rh ˙ A = −rh˙ (h2/r − µ(r)) and cos f − sin f F = sin f cos f 5 and have further made use of the trigonometric identities cos θ = cos(f + ω) = cos f cos ω − sin f sin ω sin θ = sin(f + ω) = sin f cos ω + cos f sin ω. We recognize that the matrix premultiplying the vector hcos ω, sin ωi on the right hand side of (9) must be the identity matrix. Hence we have found an explicit expression relating the true anomaly and the radius: 1 h2 cos f = − µ(r) (10) B(r) r 1 sin f = rh.˙ (11) B(r) Thus we see that the particle traces out a closed path in the eccentric frame. By carefully considering (11), we see that periapsis is always achieved at f = 0, i.e. when r and B are parallel; and that apoapsis is achieved at f = π, i.e. when r and B are anti-parallel. If the angular momentum is positive, (11) tells us that r is increasing when the particle is in the upper half plane and is decreasing when the particle is in the lower half plane. The opposite is true for the case of a negative angular momentum. 2.2 The Osculating Eccentricity and Semi-Major Axis We would also like to point out that one can rearrange (10) into the following form: h2/µ(r) p(r) r = = , (12) 1+ B(r) cos f/µ(r) 1+ e(r) cos f completely analogous to its classical (µ(r) = const.) form.