Modified Kepler’s law, escape speed, and two-body problem in modified Newtonian dynamics-like theories

We derive a simple analytical expression for the two-body force in a subclass of modified Newtonian dynamics (MOND) theories and make testable predictions in the modification to the two-body orbital period, shape, precession rate, escape speed, etc. We demonstrate the applications of the modified Kepler’s law in the timing of satellite orbits around the Milky Way, and checking the feasibility of MOND in the orbit of the large Magellanic cloud, the M31 galaxy, and the merging bullet clusters. MOND appears to be consistent with satellite orbits although with a tight margin. Our results on two-bodies are also generalized to restricted three-body, many-body problems, rings, and shells.

Figure 1

An example of $N=6$ self-gravitating particles each with a mass of $m_i=M/(50N)$ initially on a ring of radius 16 kpc moving $280\mathrm{km}/\mathrm{s}$ tangentially around the Milky Way (the point at origin $M=5\times {10}^{10}$ solar masses); the arrows indicate the force towards the central mass; one can see the precession of the pericenter and the stretching of the ring for the past 1 Gyrs. The right panel shows a semilog plot of the Milky Way’s potential $2V(r)$ (in units of square km/s) for a tracer particle (solid line) and $N=6$ massive particles on a ring of mass $M/2$ (dashed line) as a function of its distance $r$, where we have labeled the location of the cut off radius, where $2V(r_{\mathrm{cut}})\sim 0$. This allows some stars to escape. In contrast, stars are unable to escape from a logarithmically divergent potential of an isolated MOND galaxy in an empty universe. All length units are in kpc.

Figure 2

The orbits computed for the LMC-MW pair with a LMC baryonic mass of $M/5$ (small circles) or $M/10$ (solid line) respectively, for the past 12 Gyrs; the LMC is presently moving to the left at $380\mathrm{km}/\mathrm{s}$, and at 45 kpc below the MW, which is held at the origin (shown by the dotted small circle around the origin). A generic difference with Newtonian Keplerian orbits is that the MONDian orbits depend on the mass ratio, and the direction of the pericenter or apocenter changes as the nearly elliptical orbit precesses.

Figure 3

Two possible orbits for the M31 (diamond symbols, lower right) and the MW (diamond symbols, upper left) for the past 12 Gyrs: the left panel shows a nearly radial orbit with the M31’s baryonic mass the same as that of the Milky Way ($M=5\times {10}^{10}$ solar masses) and the right panel shows a significantly nonradial orbit of the M31 with a mass of $2M$. The origin is set at the present position of the MW, while the M31 is presently 800 kpc away on the right. The arrows indicate the force towards the center of mass.

Figure 4

The orbit of the bullet subcluster with respect to the main cluster (the dot fixed at origin) with a mass ratio 1:2 (circles) and 1:3 (solid line) respectively, for 9 Gyrs earlier than its present age; the bullet is at 700 kpc to the right of the main cluster, and is moving to the right at $4000\mathrm{km}/\mathrm{s}$; both models assume the same combined mass, $M_1+M_2=6\times {10}^{14}$ solar masses.