What is the rate of change of the Moon's eccentricity? So I know the Moon's current average eccentricity is ϵ≈0.039±0.006, but was this always the case? Was it ever increasing or decreasing, and if so is it known what the current rate of change is for it? Thank you!
 A: The Moon's orbit is rather complicated. It isn't really a Kepler ellipse, but it's convenient to describe orbits using osculating elements, which are the elements of an ideal ellipse which matches the true orbit at a given time. We can then use statistical techniques to derive so-called proper elements of the orbit, which can be used to approximate the body's position.
On Astronomy.SE I said:

The Moon's orbit around the Earth-Moon barycentre has an eccentricity of ~0.0549 and it's quite dynamic, with relatively short apsidal and nodal precession cycles, primarily due to perturbation by the Sun. Alternatively, the Moon's orbit around the Sun is strongly perturbed by the Earth. ;)

That answer has a graph showing the variation in the distance from the Earth to the Moon over the course of a year. You can see that the semi-major axis varies, mostly in response to the distance to the Sun. The other major perturbations to the Earth-Moon system are due to Jupiter and Venus.
Here's a daily plot of the Moon's eccentricity for 2020, using osculating elements courtesy of JPL Horizons. Each point is calculated for midnight UTC (really JPL's version of TDB).

Over a span of a few thousand years, the eccentricity pattern (and mean eccentricity) doesn't change much. Here's a plot for 4020.

The mean eccentricity value printed on those graph is calculated by integrating the Bézier curves that join the daily data points. The mean given in the 2020 graph should be fairly accurate because the curve starts and stops on days when the eccentricity was close to the mean. The 4020 graph doesn't do that, so you need to adjust its start & stop days if you want a more accurate mean.
You can make your own plots (from BC 9999 to AD 9999) using this Sage / Python script. You can save plots in SVG format for high quality printing.

Over the very long term, the eccentricity of the Moon's orbit is changing. Tidal forces tend to circularise orbits, but this can take a long time, and the process is further complicated due to the other perturbation forces influencing the Earth-Moon system. If the Earth and Moon were isolated, then eventually the orbit would circularise, but that's unlikely to happen in reality due to the periodic perturbations.
JPL doesn't use traditional orbital elements for its calculations. It produces its ephemerides by direct integration of the equations of motion (with relativistic corrections) of the major Solar system bodies (including 343 asteroids), and fine-tunes its model using both ground-based and space-based observations. Its lunar model is very good: it incorporates Lunar Laser Ranging data, and uses a sophisticated model of the Moon's interior.
