In all the existing coordinate systems I am aware of, no orbits are strictly Keplerian. But perhaps in an Earth Centered coordinate system, one could say that the Earth's orbit is closed, since the Earth, by definition, is unmoving at [0,0,0].
Throughout history, CS (Coordinate Systems) have been a huge bone of contention. Einstein said:
Can we formulate physical laws so that they are valid for all
The struggle, so violent in the early days of science, between
the views of Ptolemy and Copernicus would then be quite meaningless.
Either CS could be used with equal justification. The two sentences,
'the sun is at rest and the Earth moves', or 'the sun moves and the
Earth is at rest', would simply mean two different conventions
concerning two different CS.
On a practical level, I think it is convenient to treat some orbits as Keplerian, since we can answer a lot of questions quickly and accurately off the 'back of the napkin' using Kepler's laws. As inspiration from Ptolemy, Kepler, and Einstein, I wondered if a coordinate system could be imagined in which every orbit is perfectly Keplerian and closed. The following is the result:
The Moon's orbit around the Earth is Keplerian and closed.
In order to track and quantify the motion of celestial objects, we define coordinate systems. Our choice of coordinate system is typically made to ease the calculation complexity for a particular problem. For example, Earth satellite calculations are often performed in an ECI (Earth Centered Inertial) coordinate system. That means the origin of the system is the center of the Earth, and the Earth rotates in place about the origin. This is a convenient system to study the motion of Earth bound satellites, since it is consistent with Kepler’s laws for Earth satellites, including the Moon. ECI:
In some cases, it's convenient to use an ECEF (Earth Centered Earth Fixed) coordinate system. This coordinate system fixes the rotation of the Earth, so the axes don't change with respect to the Earth’s surface. This is a convenient system for space launch since the coordinates of Earthed based sensors don’t change. ECEF:
We can define a coordinate system called ECMF (Earth Centered Moon Fixed). In this coordinate system, we set the x-axis to be coincident with the vector from the Earth to the Moon. As the Moon rotates around the Earth, the whole coordinate system moves with it. The z and y axes are offset by 90 degrees and lie in the plane orthogonal to the vector from Earth to the Moon.
In order to ‘fix the Moon’ in our ECMF coordinate system, we have to account for the variations in lunar distance due to the eccentricity of the orbit. If we switch from cartesian coordinates to polar coordinates, we see that we can set r, the distance from the Earth to the Moon to be equal to $k=500,000km$. ECMF (not to scale):
The coordinate transformation from ECEF to ECMF is dependent on the moon’s polar coordinates in ECEF at time $t$: [$\lambda$, $\phi$, $r$]. To translate a point $P = \alpha, \beta, d$ from ECEF to ECMF, $\alpha’=\alpha-\lambda$, $\beta’=\beta-\phi$, and $d’=d*k/r$. Note that the moon’s position [$\lambda$, $\phi$, $r$] in ECEF always becomes $[0,0,k]$ in ECMF.
The ECMF coordinate system has some really bad qualities. It non-uniformly stretches the rest of the universe based on time. Depending on the direction, light no longer travels in a straight line! Regular shapes in ECEF become irregular in ECMF. The z-axis becomes irregularly offset from the Earth's rotation axis within the range of the Moon's inclination from the equatorial plane. All kinds of bad stuff happens in ECMF.
From the ECMF coordinate system, we can make another coordinate system called a ECMFDR (Earth Centered, Moon Fixed Distance, Rotating) system. This coordinate system just rotates the ECMF system around the z’-axis so that a full revolution takes 1 year, or $p$. To translate a point P = $\alpha', \beta', d'$ from ECMF to ECMFDR, $\alpha’’=\alpha’$, $\beta’’=\beta’+2\pi(t-t_0)/p$, and $d’’=d’$. ECMFDR (not to scale):
In our usual ECEF system, the moon’s orbit is not quite Keplerian. Instead, it's perturbed by the irregular shape of the earth, it's perturbed by other gravitational bodies in the solar system, and it's slowly spiraling outward, away from the Earth. In the ECMFDR system, the moon is, by definition and construction, in a perfectly circular orbit which is both a Kepler orbit and a closed orbit.
If we can create this coordinate system for the Earth and the Moon, then we can generalize it to any pair of orbiting bodies.
Every orbit is Keplerian: Consider a body $b_2$ in orbit around a body $b_1$. For a particular time $t_0$, define a coordinate system with an origin at the center of mass of $b_1$, with the center of mass of $b_2$ at $[0,\sin(2\pi (t-t_0)/p),k]$, where $p$ is one year, and $k$ is one AU.
Then $b_2$’s orbit around $b_1$ is Keplerian because it complies with Kepler’s 3 laws,
$b_2$’s orbit traces out an ellipse (since it traces out a circle and a circle is an ellipse).
A segment from $b_1$ to $b_2$ sweeps out equal area in equal time (since distance between $b_1$ and $b_2$ is always 1 AU, $b_1$ is fixed, and $b_2$ maintains constant speed)
All orbits around $b_1$ have the same ratio of square of the SMA divided by the cube of the period, since all orbits have an SMA of 1 AU and a period of one year.
Every orbit is closed. Keplerian orbits are closed orbits because a Keplerian orbit traces out a closed shape (the ellipse). Since every orbit is a Keplerian orbit, every orbit is a closed orbit.