Embedding diagrams prove to be quite useful when learning general relativity as they offer a way of visualizing spacetime curvature through warped two dimensional (2D) surfaces. In this manuscript, we present a different 2D construct that also serves as a useful conceptual tool for gaining insight into gravitation: orbital dynamics—namely, the cylindrically symmetric surfaces that generate Newtonian and general relativistic orbits with small eccentricities. Although we first show that no such surface exists that can exactly reproduce the arbitrary bound orbits of Newtonian gravitation or of general relativity (or, more generally, of any spherically symmetric potential), surfaces do exist that closely approximate the resulting orbital motion for small eccentricities (exactly the regime that describes the motion of the solar system planets). These surfaces help to illustrate the similarities and differences between the two theories of gravitation (i.e., stationary elliptical orbits in Newtonian gravitation and precessing elliptical-like orbits in general relativity) and offer, in this age of 3D printing, an opportunity for students and instructors to experimentally explore the predictions made by each.

1.

Bertrand's theorem states that an orbiting body residing in a spherically symmetric potential will move in closed elliptical orbits only for a −1/r Newtonian gravitational potential or an r2 harmonic oscillator potential.

2.

The perihelion (aphelion) is the point of closest (farthest) approach of a planet in orbit around the Sun. These points march forward, or precess, in the azimuthal direction when additional orbiting bodies are included in the Newtonian treatment or when the theoretical framework of GR is employed. The pericenter (apocenter) refers to the point on the elliptical orbit that is closest to (farthest from) the central mass.

3.
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8.

This percent error is calculated from [|r(ϕ)rap(ϕ)|/r(ϕ)]*100%=(ε2cos2ϕ)*100%, where we used Eqs. (7) and (8) in calculating the second equality. This percent error is independent of r0, depending only on the eccentricity of the orbit and the angular position of the planet, and yields a maximum value when ϕ=0,π.

9.

Notice that the constant r0 is the characteristic radius of an elliptical-like orbit in both Newtonian gravitation and general relativity. Similarly, r¯0 in Eq. (15) is the characteristic radius of an elliptical-like orbit on the 2D surface.

10.

Notice that the numerical value of κ merely sets the scale for the 2D surface and, without loss of generality, can be normalized to take on values of κ ∈ (−1, 0, +1). This result is reminiscent of the curvature constant that arises in FRW cosmology11,12.

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James B.
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14.

The values for the mass of the Sun and the average Sun-planet distance, which were used to calculate the relativistic factor β, are found on the NASA Web site, <http://nssdc.gsfc.nasa.gov>. We note that we approximated the characteristic radius r0 as the average Sun-planet distance rave in calculating the β's for this table. For Mercury, where the discrepancy between rave and r0 is largest, the deviation between rave and r0 occurs in the second significant figure of β.

15.
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