The 2 bodies in a 2-body system orbit around their common barycenter, which is located at the common primary focus of both their orbits. The empty foci don't seem to have a job in celestial mechanics.

I have heard the tidal bulge of a body in synchronous rotation points towards its "empty" focus (no reference). Do the empty foci have any other uses in celestial mechanics? Any "magical properties" like LaGrange Points or Wormhole portals?


2 Answers 2


Suppose that a moon $M$ orbits a much more massive planet $P$ and rotates at a uniform angular velocity synchronous with the orbital period. If we mark off the point on $M$ that faces $P$ at the periapsis and then follow the mark as $M$ orbits $P$, we find that the mark nearly faces the empty focus (this approximation is most accurate with low-eccentricity orbits). This may not reflect the actual location of the tidal bulge on $M$ (which can wobble back and forth under the gravity of $P$), but it may provide a reference point for gridding $M$ with latitude and longitude lines.

Our own Moon provides an interesting example of this effect because of the eccentricity of its orbit ($e=0.0549$), whichbis rather high for tidally locked moons but low enough for the approximation described above to show good accuracy. Sitting on the focus at Earth, we see the mark wobble through an angular range of $\pm2\sin^{-1}e$ as it points to the other focus. This is roughly $\pm6.3°$ for our Moon, so by this mechanism we can see, at one time or another, up to $192.6°$ of lunar longitude.

  • $\begingroup$ Cute! But I don't think it's exact. Here's a Sage / Python plot for eccentricity 3/5. i.stack.imgur.com/EMWOC.png And here's the script: $\endgroup$
    – PM 2Ring
    Feb 9, 2023 at 15:17
  • $\begingroup$ sagecell.sagemath.org/… $\endgroup$
    – PM 2Ring
    Feb 9, 2023 at 15:18
  • $\begingroup$ Not sure whatv"not exact" means. The vectors are pointing to $(-6,0)$ which would be the second focus. Also in the parabolic limit where the vector would have to point to infinity, the proposed orientation would ge exact. $\endgroup$ Feb 9, 2023 at 15:38
  • $\begingroup$ In my plot (which uses equal steps of eccentric anomaly), the red vectors go from the empty focus to the centre of the moon. The magenta dots rotate with the mean anomaly. If you look closely you can see that the magenta dots aren't exactly on the red lines, especially on the left side of the plot. It's more noticeable at higher eccentricity. $\endgroup$
    – PM 2Ring
    Feb 9, 2023 at 15:47
  • $\begingroup$ This script plots the difference (mean anomaly - empty focus angle) vs eccentric anomaly, all angles in degrees. sagecell.sagemath.org/… It's quite small for typical synchronous moon orbits. $\endgroup$
    – PM 2Ring
    Feb 10, 2023 at 7:52

The empty focus is useful for finding true anomaly from the current distance to the focus — geometrically.

Elliptic orbits have the powerful property that:

$$r + r_{empty} = 2a$$

Or in other words, you can use a piece of string attached to the two foci to draw an ellipse.

But this also means you can set up some equations from the right angle triangles formed to get $h$ the distance down to the apside line. From which you can calculate the true anomaly as $\sin^{-1}\left(\frac{h}{r}\right)$

empty focus

Your proposition that the tidal bulge of a a body in synchronous rotation points towards the empty focus is indeed true, though I don't have a source at hand either.

I remember it being possible to obtain this result from Kepler's third law. The special case of 90º rotation is more straightforward, with the red and blue swiping areas below being equal

third law

  • 1
    $\begingroup$ The answer claims that the empty focus is involved in calculating the true anomaly from the current distance to the focus. But there is no description or explanation so far of any procedure that achieves such a result! Please complete the answer by providing that information! $\endgroup$
    – terry-s
    Feb 5, 2023 at 23:38

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