J. Rogers, SE Ohio 

Tides Fall Out of the Landscape: Reading the Standard Tide Laws Off the Simulation

Draft. Companion to “Solar System as Moving Deformations” and “The Algebraic Decoding of Newton's Vector.”

Abstract

The simulation contains no tide. Its only rule is that each object reads the slope of the landscape at its own coordinates, and heights add. Two nearby points of one body read slightly different slopes, and the difference between the readings is the tide. This paper shows that the standard tide laws are that subtraction, repeated. The tidal tensor is the slope of the slope. The two bulges are the near side reading a steeper slope than the center and the far side a shallower one. The 1/d³ scaling and the Moon's factor of 2.2 over the Sun follow from differentiating the 1/r shape once more. The equilibrium tide is the sea surface settling level in the summed landscape. We measure each of these by calling the simulation's own landscape function, and every one matches the standard law with no tide term added.

The mapping in five steps

Step	The standard tide law	In the simulation (no tide term anywhere)
1. Tide is a difference	The tidal acceleration is the difference of gravitational acceleration between two points of the body.	Two points read the landscape. Subtract the readings.
2. The tidal tensor	T = (GM/d³)(3nnᵀ − I): stretching 2 along the line to the perturber, compression 1 across it.	The slope of the slope. For the 1/r shape, d(X/r²)/dr = −2X/r³, and the slope's direction turns by X/r³ across.
3. Two bulges	A bulge toward the perturber and one away from it.	The near side reads a steeper slope than the center, the far side a shallower one. Both stretch away from the center. No centrifugal force is used.
4. Size	Proportional to M/d³. The Moon's tide is about 2.2 times the Sun's.	The ratio of X/d³ for the two deformations.
5. Equilibrium tide	The sea surface is a level surface of the combined potential, with high minus low of 0.54 m (Moon) and 0.25 m (Sun).	The sea settles where the summed height is level. The heights alone give the numbers.

Each step is derived in section 3 and measured in section 4.

1. The standard tide laws

The standard account of tides rests on a few statements.

• A tide is a difference of gravity across a body. If g(x) is the gravitational acceleration at a point, the tidal acceleration at x + δ relative to the body's center at x is g(x + δ) − g(x).
• To first order it is the tidal tensor. For a perturber of mass M at distance d, with n the unit vector toward it, the tidal acceleration of an offset δ is (GM/d³)[3n(n·δ) − δ]. That is stretching of 2GMδ/d³ along the line and compression of GMδ/d³ across it.
• It falls as 1/d³ and is proportional to M. The Moon's tide on Earth is about 2.2 times the Sun's.
• There are two bulges. One faces the perturber and one faces away.
• The equilibrium tide is the shape the ocean takes if it settles to a level surface of the combined potential. For the Moon its high minus low is about 0.54 m, and for the Sun about 0.25 m.
• Roche and Hill limits mark where the tide across a body beats its own gravity.

The standard treatments introduce each of these as a separate result. They all come from one subtraction.

2. What the simulation contains, and does not

The simulation's landscape has one function, downhill. An object calls it with its own coordinates and gets back the sum, over every other deformation, of depth divided by distance squared, pointing toward the source. Heights add, so slopes add. Coordinates are measured from the Sun, so each object's change is reported against the Sun's own reading.

There is no tidal term, no tidal tensor and no list of bulges anywhere in the code. If tides appear, they appear because two readings at two coordinates differ.

3. Derivation: a tide is the slope of the slope

Take a deformation of depth X and shape 1/r. Its height at distance r is h = −X/r, and the slope a body reads is X/r², pointing toward the source.

Now take a body of small size δ whose center sits at distance d.

Along the line. The far side of the body is at d + δ and reads a slope X/(d + δ)². The change of the slope with distance is d(X/r²)/dr = −2X/r³. So the far side reads a slope weaker than the center's by 2Xδ/d³, and the near side reads one stronger by the same amount. Relative to the center, the near side is pulled toward the source and the far side is left behind. Both move away from the center, which is a stretch of 2Xδ/d³.

Across the line. A point offset sideways by δ reads the same size of slope but along a direction rotated by the angle δ/d. The part of that slope pointing back toward the line is (X/d²)(δ/d) = Xδ/d³. That is a compression toward the center, of half the stretch.

These three numbers, +2X/d³ and −X/d³ twice, are the tidal tensor (X/d³)(3nnᵀ − I). They add to zero because the 1/r shape is the shape a point source makes in three dimensions.

The frame. The center of the body is also reading the landscape, and it falls with the uniform part of that reading. Subtracting it is the same bookkeeping as measuring coordinates from the Sun: the uniform part shifts every part of the body alike, and only the difference between readings shows up in the body's shape.

Why two bulges without a centrifugal force. Textbook explanations of the far-side bulge often invoke a centrifugal force. In the landscape the far-side bulge needs nothing extra: the center reads a steeper slope than the far side, so the center falls toward the source faster, and the far side is left behind.

4. Measured from the simulation's own landscape function

We called the simulation's downhill function with the Moon, then the Sun, as the only deformation, and read the slope at points on Earth's surface and at its center. The tide is the reading at the point minus the reading at the center. Nothing else was computed.

Tidal acceleration at Earth's surface (m/s²).

Perturber	Near side	Far side	Side, toward the center
Moon, from the readings	+1.128 × 10⁻⁶	−1.073 × 10⁻⁶	5.497 × 10⁻⁷
Moon, standard (GMR/d³ and 2GMR/d³)	+1.100 × 10⁻⁶	−1.100 × 10⁻⁶	5.499 × 10⁻⁷
Sun, from the readings	+5.051 × 10⁻⁷	−5.051 × 10⁻⁷	2.526 × 10⁻⁷
Sun, standard	+5.051 × 10⁻⁷	−5.051 × 10⁻⁷	2.526 × 10⁻⁷

The stretch along the line is twice the compression across it. For the Moon, the near side comes out about 5% stronger than the far side. The first-order tidal formula has no such difference. It comes from the next term of the same subtraction, of relative size 3R/d, which is 5% for the Moon since R/d is 1.66%. It needed no extra term.

Scaling with distance. The near-side tide of the Moon at one, two and four times its distance falls by successive factors of 0.1234 and 0.1242, against 1/8 = 0.1250. That is 1/d³ with the same small correction.

The ring pattern. Reading 72 points around Earth's center in the plane of the Moon and splitting each tide into radial and tangential parts gives the standard pattern, a_r = (GMR/d³)(3cos²θ − 1) and a_t = −(3/2)(GMR/d³)sin 2θ, to within 1.69% and 2.32% of the peak. That is the size of R/d. The tide pulls outward at 44 of the 72 points, those within about 55 degrees of the line to the Moon, which is the two bulges.

The ratio of the two perturbers. The Moon's near-side tide is 2.233 times the Sun's from the readings. The first-order standard ratio is 2.177, and the difference is the same 5% effect.

The equilibrium tide from heights alone. The sea is not given any tide law. It slides until no slope remains along its surface, which means its surface is a level surface of the summed height. Writing each deformation's height as −X/r, removing the uniform slope that the center falls with, and dividing by Earth's own slope at its surface (9.82 m/s²) gives the displacement of the sea surface:

Perturber	Near-side bulge	Far-side bulge	Sides	High minus low	Standard high minus low
Moon	+0.363 m	+0.351 m	−0.178 m	0.541 m	0.535 m
Sun	+0.164 m	+0.164 m	−0.082 m	0.246 m	0.246 m

The Moon's equilibrium tide is 2.20 times the Sun's. The standard bulge heights are +0.357 m and +0.164 m. The same galactic-scale measurement is the “differs by” column in the simulation's readout, where the outside structures' slopes differ between the Sun and Neptune by about 1e-14 AU/yr² across 30 AU.

5. Roche and Hill limits are the same comparison

The Roche limit is where the tide across a body beats its own slope. In the landscape this is a comparison of two readings. A small body of depth X_m and radius r_b reads its own slope X_m/r_b² at its surface, and the tide from a larger deformation of depth X_M at distance d tries to pull a surface point away at 2X_M r_b/d³. Setting them equal gives

d = r_b (2X_M / X_m)^(1/3).

Because depth scales as density times radius cubed, this is d = R_M (2ρ_M/ρ_m)^(1/3), the standard rigid-body Roche limit. For the Earth and the Moon it is 9,483 km.

The Hill sphere is the same comparison for bodies orbiting a small deformation inside a bigger one's landscape. In an earlier run of the simulation, with no tide term, the Moon launched on a circular loop stayed bound at 0.005 AU from Earth and was lost at 0.0055 AU. Earth's Hill radius is 0.01 AU, so the Moon was lost at about 0.55 of it.

6. What the simulation does not have

• Extended bodies. The simulation's objects are points. The deformation of a real body, its elastic or viscous response, and the lag of its bulge are absent. The tidal friction that slows Earth's rotation, and tidal locking, are not in the simulation.
• A dynamical ocean. The equilibrium tide assumes the sea reaches a level surface. Real tides have basins, resonances and phase lags that this does not describe.
• New physics. A tide is the name for the difference between two readings. Nothing in this paper adds a force. It shows that the standard tide laws are consequences of reading the same landscape at two points.

7. Conclusion

The tidal tensor, the two bulges, the 1/d³ scaling, the Moon's factor of 2.2 over the Sun, the equilibrium tide heights and the Roche limit all appear when each object reads the slope at its own coordinates and the readings are compared. The simulation needed no tide formula, no centrifugal force and no tidal term, because a tide is the slope of the slope, and the landscape has that already.

Appendix: names used

• X: the depth of a deformation, G·m in the simulation's units.
• Reading: the slope that downhill returns at a coordinate, X/r² summed over the other deformations and pointing toward each source.
• Tide: the reading at a point minus the reading at the body's center. It is a name for that difference, not a quantity in the code.
• R and d: the body's radius and the distance to the perturber.

References

• Newton, I. (1687). Philosophiae Naturalis Principia Mathematica, Book III.
• Cartwright, D. E. (1999). Tides: A Scientific History. Cambridge University Press.
• Murray, C. D. and Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press.
• Heisler, J. and Tremaine, S. (1986). The influence of the galactic tidal field on the Oort comet cloud. Icarus, 65, 13–26.
• Companion: “Solar System as Moving Deformations.”
• Companion: “The Algebraic Decoding of Newton's Vector.”