# Why is the Earth-Moon distance not the same at each perigee/apogee?

I wonder why the Earth-Moon distance is not the same at each perigee/apogee. Isn't the Moon's orbit a fixed ellipse with Earth at one of the foci? If it is so, shouldn't the distance at perigee/apogee be a fixed value?

• Remember that the Moon's orbit is perturbed by the Sun (and other planets for that matter). It is very much an n-body problem. – Mick Jan 24 '18 at 3:52
• The moons orbit would be elliptical and AP/PE always the same if and only if only earth and moon existed in the whole universe and both were perfect point masses. But in reality, other planets, the sun etc disturb those orbits. – Polygnome Jan 24 '18 at 12:11
• – rmaddy Jan 24 '18 at 18:59

Isn't the Moon's orbit a fixed ellipse with Earth at one of the foci?

No, it's not. This isn't even true for the planets' orbits about the Sun. Each planet perturbs the orbits of the other planets, making Kepler's ellipses approximately correct rather than exact. The Moon's orbit is strongly perturbed by the Sun in a number of ways. The Moon's orbit deviates from being a fixed ellipse in a number of ways. One result of these solar perturbations (and to a much lesser extent, perturbations from Venus and Jupiter, and to an even lesser extent, from the other planets) is that the Moon's orbit precesses in a number of ways.

One such precession is the apsidal precession. The line from the Earth to the point at which the Moon reaches perigee does not point to a fixed position in space. It instead precesses with a period of about 8.85 years. This is what results in the so-called supermoons, which occur when the Moon's orbit is close to perigee when the Moon is full.

Another such precession is the nodal precession. The line of nodes (where the Moon crosses from above to below the ecliptic, and vice versa), also precesses, but with a period of about 18.6 years. We only get eclipses when the Moon is very close to a node at a syzygy (either a full Moon, resulting in a lunar eclipse, or a new Moon, resulting in a solar eclipse).

• Any refs to quantitative data on relative effect of the sun vs. each planet on these deviations? (Yeah, sorry, got lazy about looking them up myself) – Carl Witthoft Jan 24 '18 at 16:51
• Check out the book and software on CelesTrak by D. Vallado: celestrak.com/software/vallado-sw.asp For completion to @David Hammen's comment, in addittion to gravitational perturbation forces, there are additional forces which affect the trajectory of a planet, some conservative, some non-conservative. The modelling of such forces is one core subject in astrodynamic research. – Ela Jan 24 '18 at 17:30
• And the reason for the nodal precession? That Earth is an oblate spheroid? – Peter Mortensen Jan 24 '18 at 22:00
• @PeterMortensen -- Both the apsidal and nodal precessions of the Moon are caused almost entirely by the Sun. The Sun makes the Moon's aside and nodes precess by +40.67 and -19.55 degrees per year, respectively. The Earth's contribution? +6.4 and -6.0 arc seconds per year. – David Hammen Jan 24 '18 at 23:28

If the Moon and Earth were far away from any other gravitational bodies, then the orbit would be not only very consistent but also very close to circular as well. Orbits like the Earth-Moon, where the mutual tidal force is strong and the inner body's rotational energy is transferred to the smaller body's orbital energy, those orbits tend to circularize over time.

The mathematics behind 3-body gravitation is pretty intense, and above my pay-grade, but I can explain with a visual. The easiest way to picture this is with tidal forces.

We think of tidal forces as only affecting a solid body like waves on the Earth or the permanent tidal bulge on the moon, but all tidal forces are is a variation in the gravitational pull over different distances and because the Earth and Moon are bound to each other by gravity, that means the solar tidal force can be applied to the Earth-Moon system. The gravitational pull from the Sun is stronger on the side of the planet closer to the sun and weakest on the opposite side. This also happens relative to the Earth and Moon when one or the other is closer to the Sun. When the Earth/Moon orbit are in full moon or new moon, then the tidal force exerted by the sun is stronger on the closer body, weaker on the further body and the orbit effectively stretches in the direction of the arrows on the image above.

When the Earth-Moon orbit are in last quarter or first quarter, the tidal force exerted by the sun is in the perpendicular direction inwards, and the orbit effectively squashes.

Interestingly, the forces also have effects at the quarter points as well and everywhere in-between. When the Moon is at waning crescent or waxing gibbous the Sun exerts more force on the closer object and less force on the further away object not resulting in a change in shape so much, but the force effectively accelerates the objects with respect to each other making them move slightly faster. The opposite happens at waning gibbous and waxing crescent: the Sun is effectively slowing down the relative speed between the Earth and the Moon.

In summary, the Sun is constantly pulling or pushing the moon relative to the Earth, so there's a continuous stretching and squashing and accelerating and decelerating of the Moon's orbit around the Earth (or around the barycenter for you purists). You might think this could shake the Moon loose from the Earth, and it would, if the Moon was about 30%-50% further away than it is now. It's this tidal pulling and stretching that defines the vague border that's the stable region of the Hill sphere.

This solar tidal effect is cyclical, operating every time the Moon completes a full moon cycle, which is a synodic orbit of about 29.5 days. The Moon's "Kepler orbit" is a sidereal orbit of about 27.3 days.

What does this look like? The overall effect, (noted in the other answer), is an unusually high lunar apsidal precession of just 8.85 years, or just over 118 sidereal (or Kepler) orbits.

This means is that the Moon's apogee and perigee shift about 3 degrees for each lunar orbit. The Moon can't settle into a consistent orbit because of the solar gravitation acting upon it, and the tidal force on the Earth-Moon system is significant.

The Earth, for comparison, has an apsidal precession, mostly driven by Jupiter and Saturn, of about 112,000 years, or 112,000 orbits. That's about a thousand times less angular change per orbit. As a sidebar, the objects inside the orbit, Venus for example, don't have much effect on Earth's orbit. It's the outer planets that primarily drive apsidal precession. Neptune, for example, has no outer-planets to speak of, and if planet 9 is found, it would be too far away, so Neptune's orbit is nearly circular.

The Moon's successive apogee/perigee distances from the Earth do indeed undergo changes: these changes are almost cyclical, and they have a main period close to 205.89 days (nearly 7 synodic months). A main contributory factor to the changes in perigee distances is the periodic solar perturbation known as the evection. Then, in decreasing order of maximum size, a second contribution is due to the perturbation known as the variation.

The rest of this answer summarises explanations how the evection (along with the variation) affects the perigee distances: also offered is a numerical example of extreme lunar-perigee data from the Astronomical Almanac ('AA') for 2011: these data indicate how the combination of the two effects can account for nearly all of the observed range in lunar perigee distances. The natures and sizes of the two effects also indicate features by which the Moon's real orbit differs (considerably) from a simple Keplerian fixed ellipse.

The evection: Older textbooks used to discuss the way in which the evection gives rise to changes in apogee/perigee distances -- for example H Godfray (1859), Elementary Treatise on the Lunar Theory. Godfray's explanation proceeds by showing the practical equivalence between two forms in which the moon's longitude and radius vector &c. can be expressed:

(1) A first form is the usual modern form of trigonometrical series representation, which effectively supposes a constant (mean) eccentricity for the Moon's orbit. Among many other terms is a main separate term for the evection, in a form first given by Euler, with an argument nowadays often expressed as $(2D-l)$ , where $D$ is the Moon's mean elongation from the Sun and $l$ is the Moon's mean anomaly, i.e. Moon's mean elongation from the current position of its mean perigee.

(2) The second form is an older representation of the Moon's motions, which supposes a cyclically variable eccentricity, and thus also a cyclically variable perigee distance, greatest equation, &c.

Godfray's book gives the explanations fairly fully for the effects on longitude and equation of center (at p.66, art.70 along with preceding derivations), and then a much briefer summary of the analogous demonstration of the effects on radius vector (at pp.76-77, art.85). (In a little detail: what is shown is that the lowest-order elliptical term and the evection term can be trigonometrically combined and rearranged, to give as their equivalent an approximation to a variable ellipse, in which the eccentricity cyclically fluctuates and the angular orientation of the apogee/perigee cyclically librates as well as showing its well-known mean overall rate of rotation. A corresponding modern trigonometrical development shows essentially the same relation between the two forms for the longitude series, going as far as third-order -- S A Wepster (2010), at pp.100-104 in his historical and mathematical study of Tobias Mayer's 18th-century lunar theory and tables.)

Independently of this older type of explanation, details in appendix A below show, with reference to modern data, how the main term of the evection reinforces the main elliptic term when the Sun is in line with the Moon's line of apses, and opposes it when the Sun is at 90° to that line.

The variation: Next in size after the evection, the variation (in radius vector) brings the Moon closer to the Earth at new and full moon, and takes it farther away at the lunar quarters. This effect was shown to result from the solar perturbing force by Newton, in Principia Book 1 Prop.66 corollaries 2-5, and in Book 3 Props.26-29; results later refined by numerous authors, notably G W Hill, see especially his data for inverse radius vector e.g. at p.143 in (1895) Astron J 15, 137-143. (In Hill's paper, $\tau$ (tau) means the same as $D$ above.) The instantaneous amount of the variation depends on lunar phase, and so it also contributes to changes in perigee distance, because the mean period between perigees (~27.55 days) is about two days shorter than the mean period between new moons (~29.53 days), hence successive perigees occur at different phases in the lunation and are differently affected by the variation.

Numerical example: Appendix A below cites recently-refined modern values (Paris Observatory) for the amplitude of trigonometric terms affecting the Moon's radius vector. The main term of the evection is close to 3699 km in amplitude, and the main term of the variation is close to 2956 km. Ignoring many smaller periodic effects, one may expect from what has already been mentioned, that when a new or full moon occurs at perigee (implying also that the Sun is in the line of apses), the main evection and variation terms both act to reduce the perigee distance, by about the sum of the two amplitudes, i.e. by about 6655 km. When on the other hand a perigee occurs at one of the lunar quarters (implying also that the Sun is at 90° to the line of apses), both the two terms have the opposite effect, i.e. to increase the perigee distance by about 6655 km. Thus the main terms of the evection and variation, when they both reach maximum in the same sense, can be expected to make for a range of about 13310 km in lunar perigee distances.

This trigonometric-based expectation can be compared with data from almost any recent Astronomical Almanac ('AA'). (In recent years, the lunar distance data in AA come from a numerically-integrated ephemeris, version DE405 for years 2003-2014, see AA for 2011, page L4. The integrations were fitted to modern lunar laser-ranging data, independently of classical trigonometric analysis.) The AA for 2011 (at hand while writing this answer) tabulates lunar distances daily at 0h TT (using units of earth-equatorial-radius, 6378.14 km), and provides the following example-data (see esp. pages D1, D8, D14). (i) A smallest tabulated local-minimum moon-distance for the year occurred on March 20 (0h) at 55.912 earth-radii, close to a perigee at March 19 19h and a full moon at March 19 18h 10m; and (ii) a largest tabulated local-minimum moon-distance for the year occurred on July 8 (0h) at 57.951, close to a perigee at July 7 14h, and to a lunar first-quarter at July 8 6h 29m. At the dates for which the distances were tabulated, the phases and configurations were close but not exact, the moon was a very few degrees off exact perigee and also a little off exact syzygy or quadrature. Neglecting this inexactness, one may reckon, for the reasons mentioned above and shown in the Appendix, that on both dates the evection and variation act in the same sense and rather close to their maxima; both of them reduced the perigee distance at date (i), and both increased it at date (ii).

By difference between data (i) and (ii) from AA 2011, the range of the tabulated local-minimum (near-) perigee distances was 2.039 earth-radii, equivalent to about 13000 km. This differs less than 2.5% from the combined peak-to-peak range (13310 km) of the main terms of evection and variation. The calculation and comparison is of course rather rough, both by the inexactness of the configurations, and also because many smaller trigonometric terms are ignored. Nevertheless, it is close, and helps to indicate how the evection together with the variation can account for nearly all of the range in lunar perigee distances seen in a year.

Appendix:

Shown here are (A) how the effects mentioned above are also quantitatively inherent in the most recent analytical accounts of the lunar motions; and (B) how some (now historical) accounts have attempted to outline separately the gravitational causes of the evection -- a somewhat more awkward enterprise, involving approximations and engagement with older historical forms for expressing the motions.

A: Quantitative description of the varying lunar perigee distances is given here in terms of modern analytical expressions for the Moon's orbital longitude and radius vector. The following data are rounded from "ELP 2000-85 - A semi-analytical lunar ephemeris adequate for historical times", by Michelle Chapront-Touzé and Jean Chapront (1988) Astronomy & Astrophysics 190, 342-352, especially at page 351: this represents one of several versions of the authors' 'ELP' (Ephémérides Lunaires Parisiennes), see also this page at one of the Paris Observatory websites.

The three largest trigonometric terms describing the time-varying differences between the Moon's true and mean radius vector, and its true and mean orbital longitude, are known respectively as the largest of the elliptical terms, and the main terms of the evection and variation. They are close to --

(a) $-20905.355 \cos (l) -3699.111 \cos (2D-l) -2955.968 \cos (2D)$ (for the true radius vector (in km), relative to the mean distance of 385000.529 km), and

(b) $+22639.586" \sin (l) +4586.438" \sin (2D-l) +2369.914" \sin (2D)$ (for the difference true minus mean orbital longitude, in arc").

$D$ and $l$ have the meanings already mentioned.

The largest elliptical term (left-hand term in (a) and (b)) can be considered as the largest (lowest-order) term in a trigonometrical series with arguments in multiples of $l$ only. These sub-series can be excerpted from the long series of terms, in many arguments, given on page 351 of the cited 1988 paper, thus:

(c) $-20905.355 \cos (l) -569.925 \cos (2l) -23.210 \cos (3l) ...$ for radius vector, and

(d) $+22639.586" \sin (l) +769.026" \sin (2l) +36.124" \sin (3l) ...$ for orbital longitude.

These are approximately close to series for the equation of the center (in radius vector or orbital longitude) that could be developed for an exact Keplerian elliptical orbit with constant ('mean') eccentricity about 0.0549 (compare for example the forms given in Brouwer and Clemence (1961) Methods of Celestial Mechanics, pages 76-77, equations 73 and 75). Together, series (c) and (d) express approximately a mean ellipse that the Moon could follow in the absence of perturbations. Under this hypothetical condition, the lunar perigee distances for such a mean ellipse would of course always be the same, about 363502 km according to the three initial periodic terms excerpted here.

Then each second term in the three-term excerpts (a) and (b) above is the principal term responsible for the evection. To see the effect of the evection terms, the argument $(2D-l)$ can be considered as effectively $(l -(2l-2D))$ , which differs from $l$, the argument of the elliptical inequalities, by a slowly-changing quantity $(2l-2D)$.

The period of $l$ ('anomalistic month') is about 27.55 days, but the period of $(2l-2D)$ is about 205.89 days (it is the mean interval between passages of the Sun past the Moon's line of apses, of which one direction points to apogee, the other to perigee). (The mean interval between passages of the Sun past the Moon's mean apogee is double the foregoing, about 411.78 days, just under 14 mean synodic months.)

Two configuration-cases can usefully be pointed out: (i) When the quantity $(2l-2D)$ is zero (which happens once in each 7-month cycle, when the sun's position is conjoined/opposed to the direction of the moon's apogee/perigee) then it can be seen from the series-excerpts above that the quoted evection term in each series reinforces the effect of the main elliptic term. (ii) In the other case, at the opposite extreme, when $(2l-2D)$ is 180° (which happens when the sun's position is 90° from the direction of the moon's apogee/perigee) it can be seen that the evection term in each series opposes the main elliptic term.

The result is like a 'beat' effect between two oscillations. On account of this, the maximum excursions from the mean, both in radius vector and in orbital longitude, are not the same in each cycle of $l$ : the local maxima fluctuate in amount, with a period of ~205.89 days, just under 7 mean synodic months.

The expressions above thus show how the perigee distance of the moon varies, on account of the main evection term, over a range of about +/- 3699 km. The perigee distance is closer to the Earth in configuration-case (i), when the Sun conjoins/opposes the direction of the Moon's apogee/perigee; at this point the principal evection term(s) reinforce the elliptic terms), and the excursions in longitude are larger too. Then the perigee distance is larger in the second case, when the Sun is 90° away from the line of apses; at this point the evection term(s) and main elliptic terms are opposed, and here the excursions in longitude are smaller too.

In sum, the effects of the evection terms on perigee distance and on orbital longitude are approximately similar to effects that would arise from an increased orbital eccentricity in the first case, and from a reduced eccentricity in the second. The results are modified by the variation according to the phase of the lunation.

The (simpler) effect of the main term of the variation on radius vector has already been mentioned: the Moon is brought closer by about 2956km at new and full moon, and farther off by the same amount at the quarters. The exact perigee distances are also affected by other and generally smaller periodic terms.

(These effects, when considered together, also show how full moons at about the closest possible perigee distances, and hence with largest apparent diameter, tend to occur at intervals of about 14 synodic months: these are the effects sometimes called 'super moons' which cause peaks of media interest.)

B: Accounting gravitationally for these selected features of the Moon's perturbations is somewhat awkward. From the mid-18th century to the early 20th, analytical solution-techniques typically treated at least the main known perturbing forces on the Moon as a whole, to give approximate series solutions for the lunar motions. Such methods generate masses of trigonometric terms, and leave it practically impossible to see which (if any) particular parts of the perturbing forces are responsible for the evection effects. Neither do modern numerical techniques show any easily-separable parts of the perturbation effects.

There have been at least two attempts to show, mainly geometrically and qualitatively, how the effects of the evection can arise gravitationally. For this purpose the evection is taken to be represented by fluctuations in orbital eccentricity, an equivalence discussed above and in the Godfray reference already cited. The more recent of the two expositions was given by F R Moulton's (1914) Introduction to Celestial Mechanics (at chapter 9, esp. from p.321-360). The original exposition was given by Newton in Book 1 of the Principia, Proposition 66, especially corollary 9 (pp.243-5 in 1729 English translation from Latin). The explanations depend on examining the way in which the perturbing force alters the net power-law for the attraction of the Earth on the Moon, and does so differently in different parts of the Moon's orbit, making the inverse power a little more than 2 in some parts of the orbit and a little less in other parts. Beyond that it would take too much space to describe those explanations here, the originals are available in online archives.

It is also worth noting that (1) Absence of solar perturbing force would not render the moon's orbit circular or nearly so: the eccentricity is a free parameter corresponding to an arbitrary constant in the integration of the two-body problem: for example Bate, Mueller, White (1971) Fundamentals of Astrodynamics at pages 19-21 give a notably transparent demonstration of this.

(2) The solar force perturbing the Moon in its motion around the Earth is sometimes described as if represented by the absolute attraction of the Sun on the Moon: but it is really represented by the (vector) difference between the Sun's attraction on the Moon and the Sun's attraction on the Earth (Newton, Principia, Corollaries 1, 2 and 6 to the laws of motion and Book 3, Proposition 25).

(3) The rotation (precession) of the line of apses in itself does not change the perigee distances, it alters the angular places of the perigee and the times when the moon reaches perigee.

(4) The orbit of the Moon is quite far from a Keplerian ellipse or any ellipse, it combines features of a variational orbit (nearly elliptical but with the Earth near the center not at a focus) and an ellipse of varying eccentricity and fluctuating line of apses. Newton already in an unpublished paper expressed an approximate recognition that the real orbit of the Moon is not exactly an eccentric Keplerian ellipse, nor exactly a central ellipse due to the variation, but "an oval of another kind" (see D T Whiteside (ed.) (1973), The Mathematical papers of Isaac Newton, Volume VI: 1684-1691, Cambridge University Press, at page 533.

• Fascinating answer and wonderful supporting references! – uhoh Feb 18 '18 at 3:44