The Moon From the Dents Alone
This memo runs the Sun, the Earth and the Moon as three dents that add — nothing else, for six years — and reads the Moon’s motion off the result. Every classical term comes out: the evection at 1.275° against 1.274° observed, the Variation at 0.659° against 0.658°, the perigee circling in 8.86 years against 8.85.
Two inputs come from observation — the length of the month and the average stretch of the orbit. Everything else is output. The Sun’s whole hand on the Moon is in the dents.
What this memo does and does not do
It tests the dent method of 1.12 — The Actions of the Aether against the hardest ordinary problem in the sky: three bodies, the Moon’s motion, measured for centuries.
The summed dents are Newton’s arithmetic, so this confirms the method. It does not by itself choose between the Aether and curved space — section 6.
1. The Moon Does Not Keep Time
A single body round a single mass runs a fixed ellipse forever. The Moon does not, and the differences were seen long before anyone could explain them.
| When | Who | What was found |
|---|---|---|
| Antiquity | — | The Moon’s nearest point circles the Earth in about 9 years; its tilt line turns backwards in about 18.6. |
| c. 150 AD | Ptolemy, after Hipparchus | The evection: the Moon runs up to 1.27° ahead of or behind a plain ellipse, over about a month. |
| 1590 | Tycho Brahe | The Variation: faster than expected at new and full moon, slower at the quarters. |
| c. 1600 | Tycho, Kepler | The annual equation: the month runs slightly long in January, when the Earth is nearest the Sun. |
| 1687 | Newton | Shows in the Principia that the Sun causes the Variation. Gets about half of the perigee’s motion. |
| 1749 | Clairaut | Finds the other half: the approximations, not the law, were short. |
| 1919 | E. W. Brown | Tables of the Moon’s motion from Newton’s law, carried to hundreds of terms. |
2. Why The Sun Moves The Moon
The Sun’s gradient at the Moon is 2.2 times the Earth’s. The Moon still goes round the Earth, because the Earth and the Moon fall toward the Sun together — the same reason the space station crew feel nothing. The shared part moves the pair round the Sun and does nothing to the Moon’s orbit about the Earth.
What is left is the difference between the Sun’s gradient at the Moon and at the Earth. At new and full moon it points outward along the Sun line, 1.13 % of the Earth’s hold on the Moon. At the quarters it points inward at half that. The same difference raises the solar tide on the Earth.
Around the Earth, the Earth’s dent and the Sun’s difference combine into the dent the Moon actually sits in:
| Orbit | Distance | Sun’s difference as a share of the Earth’s hold |
|---|---|---|
| ½ × | 192,000 km | 0.14 % |
| The Moon | 384,000 km | 1.13 % |
| 2 × | 769,000 km | 9.0 % |
| 3 × | 1.15 million km | 30.5 % |
| 4 × | 1.54 million km | 72 % |
| 4.46 × | 1.71 million km | 100 % — the edge of the Earth’s hold |
The Sun’s share grows as the cube of the distance. The Moon sits well inside, where the dent is nearly round.
Outward at new and full moon, inward at the quarters, the same on both sides. Every effect in this memo follows from that pattern.
3. The Run
Set upThree masses, three dents
| Item | Value | Where from |
|---|---|---|
| Masses | Sun 1.98847 × 1030, Earth 5.9722 × 1024, Moon 7.342 × 1022 kg | standard figures |
| Force on each body | its mass × the gradient of the other two dents | 1.12, rules 1–6 |
| Earth’s orbit | 1 AU, eccentricity 0.0167 | observed |
| Moon’s tilt | 5.145° to the Earth’s orbit | observed |
| Input 1 | month set to 27.3217 days | observed — fixes the Moon’s distance |
| Input 2 | average stretch set so the main ellipse term is 6.289° | observed — fixes how elliptical |
| Integration | six years, 12-figure accuracy | — |
No relativity, no shadow term, no fitted coefficient. The results in section 4 were read off the run afterwards.
Two setup errors, found and corrected
A Moon started on an exactly circular orbit rang. With the Sun present, a perfect circle is not an orbit the Moon can hold, and it set off a false 1.3° wobble at the Moon’s own 27.6-day period. The fix: shoot for the starting speed that leaves no free wobble.
The stretch was first set at its peak, not its average. That gave a main ellipse term of 4.06° instead of 6.29°. The fix: set the start so the average matches.
4. What Came Out
| Effect | From the dents | Observed | Ratio |
|---|---|---|---|
| Perigee, full circle | 8.86 years | 8.85 years | 1.001 |
| Nodes, full circle backwards | 18.59 years | 18.61 years | 0.999 |
| Evection | 1.275° | 1.274° | 1.001 |
| Variation | 0.659° | 0.658° | 1.002 |
| Second ellipse term | 0.214° | 0.214° | 0.999 |
| Annual equation | −0.182° | −0.186° | 0.976 |
| Tilt term | −0.107° | −0.114° | 0.943 |
| Orbit’s stretch, range | 0.026 to 0.077 | 0.026 to 0.077 | — |
Left over after these terms: 0.089°, the sum of the many smaller terms not fitted.
Why the Moon runs fast and slow. Between new and full moon and the quarters, the Sun’s difference leans along the orbit:
The Variation, drawn. On a Moon started with no free wobble, its speed round the Earth through one month:
The evection, drawn. The stretch of the orbit, averaged over each month, across the six years:
5. The Two Small Misses
The annual equation comes out 2 % small and the tilt term 6 % small. Both trace to starting values set once and not refined: the Moon’s tilt was set at its starting value rather than its average, and the Earth started at the nearest point of its orbit. Neither points to missing physics; both close by tuning the start the same way the stretch was tuned.
6. What This Proves, And What It Does Not
ProvedThe dents carry the Sun’s whole hand on the Moon
Summing three dents, and nothing else, gives every classical irregularity of the Moon — the ones that took from Ptolemy to Clairaut to explain — to within a few parts in a thousand.
What it does not settle
The summed dents are Newton’s arithmetic, so this is the result Newton’s law gives. Relativity’s corrections to the Moon’s orbit are around a millionth the size of these terms. So the run confirms the method; it does not choose between the Aether and curved space.
That choice is made where the two give different numbers. The next place is Mercury: the same method with every planet gives its orbit 531 arcseconds a century of turning against 574 measured. The author proposes Aether flow for the remaining 43 — E.4 Mercury’s Perihelion and T.7.
7. The Numbers
- Sun’s gradient at the Earth: 5.930 × 10−3 m/s². Earth’s at the Moon: 2.698 × 10−3. The Sun’s is 2.2 times larger.
- Difference in the Sun’s gradient, Moon to Earth, at new and full moon: 3.05 × 10−5 m/s², 1.13 % of the Earth’s hold.
- Moon’s speed: 13.29°/day at new and full moon, 12.79°/day at the quarters.
- Evection 1.275°; Variation 0.659°; second ellipse term 0.214°.
- Perigee 8.86 years; nodes 18.59 years.
- Orbit’s stretch: 0.026 to 0.077, peaking about every 205 days.
Terms Used Here
| Word | What it means |
|---|---|
| Dent | The lowered pressure a mass makes in the Aether around it. |
| Gradient | How fast the depth of a dent changes with distance. Times a body’s mass, the force on it. |
| Eccentricity | How stretched an orbit is: 0 is a circle. |
| Perigee | The Moon’s nearest point to the Earth. |
| Nodes | Where the Moon’s tilted orbit crosses the plane of the Earth’s. |
| Evection | The swing in the Moon’s position caused by the Sun stretching and relaxing its orbit. |
| Variation | The Moon running fast at new and full moon and slow at the quarters. |
| Annual equation | The month running long when the Earth is nearest the Sun. |
Sources
- Ptolemy, Almagest (c. 150 AD); Tycho Brahe (1590s); I. Newton, Principia, Book III (1687); A. C. Clairaut (1749); E. W. Brown, Tables of the Motion of the Moon (1919). History carried as reported.
- Observed values of the lunar terms (evection 1.274°, Variation 0.658°, annual equation 0.186°, second ellipse term 0.214°, tilt term 0.114°, main ellipse term 6.289°) and the perigee and node periods. Standard lunar-theory values; evection amplitude and period checked 21 September 2026.
- The dent run: three bodies, forces from summed dents, integrated with an eighth-order method to twelve-figure accuracy. The script is kept with this memo and reproduces every figure here.
Where this sits in the series
The dent method run against the Moon’s measured motion.
- 1.12 — The Actions of the Aether — The rules.
- T.7 — Whirlpools All the Way Down, and Up — The flow of the Aether.
- E.4 — Mercury’s Perihelion — The next test.