We present a theoretical and experimental analysis of the elliptical-like orbits of a marble rolling on a warped spandex fabric. We arrive at an expression describing the angular separation between successive apocenters, or equivalently successive pericenters, in both the small and large slope regimes. We find that a minimal angular separation of ∼197° is predicted for orbits with small radial distances when the surface is void of a central mass. We then show that for small radii and large central masses, when the orbiting marble is deep within the well, the angular separation between successive apocenters transitions to values greater than 360°. We lastly compare these expressions to those describing elliptical-like orbits about a static, spherically symmetric massive object in the presence of a constant vacuum energy, as described by general relativity.
References
The theoretical and experimental work presented in this manuscript is based on the year-long senior research project of one of the authors (D.W.).
An embedding diagram is a 2D cylindrically symmetric surface that has precisely the same spatial curvature as a 2D spatial “slice” of a respective 4D spacetime.
The term apsides refers to the maximum and minimum radial distances of an orbiting body. The apocenter (pericenter) refers to the point on the elliptical orbit that is farthest from (closest to) the central mass (Ref. 11).
The exact solution to Newtonian gravitation is of the form , where r0 is known historically as half of the latus rectum. For small eccentricity, this expression can be expanded and equates to Eq. (5) to first-order in ε for ν = 1.
For an apocenter-pericenter-apocenter orbit, having radii rmax,1, rmin, and rmax,2, respectively, the average apocenter distance was first calculated through and then the eccentricity was calculated through . Likewise, for a pericenter-apocenter-pericenter orbit, having radii rmin,1, rmax, and rmin,2, respectively, the average pericenter distance was first calculated through and then the eccentricity was calculated through .
In the large slope regime, where an abundance of elliptical-like orbits are generated for a given run, we choose to neglect the angular separation between successive pericenters. This differs from the small slope regime where we measured the angular separation between successive apocenters and between successive pericenters.
