Planetary systems

Orbits round a real star

Two point masses under the pinned 1/r orbit as Kepler said (chapter 5). A real star is not a point, and real planets are not alone, and in 4D both matter more than in ours.

Orbits round a real star precess

Without a shell theorem, a round star doesn't pull exactly like a point. Outside it, its pull carries a small correction, −1/r + ⟨r²⟩/8r³, where ⟨r²⟩ is the mean square of where its mass sits (0.32 R² for the 4D star solved above). That turns an orbit's closest point backwards by

Δϖ = −(3π/4) ⟨r²⟩ / p² per orbit

(p the orbit's semi-latus rectum, from Landau's formula for a 1/r³ perturbation). Checked by integrating orbits in the solved star's own pull, it agrees to 1.3% three star radii out, where the next, smaller correction still shows, and to 0.09% at twelve. For a star the Sun's size, an orbit at Earth's distance would turn back 3.4 arcseconds per orbit, and one at Mercury's 23. Mercury's own, from relativity, is about 0.1. It's slow, but an astronomer would find it, and it tells the routes apart: under route 4's nonlinear law the shell theorem holds and this precession is zero.

Planets in different planes barely notice each other

In 4D two orbits can lie in completely orthogonal planes, xy and zw, which meet only at the star. Then the two planets' positions are always at right angles, so their distance is √(r₁² + r₂²). For circular orbits that never changes, and neither can tug the other along its orbit. Run with Jupiter-mass planets for 3,000 orbits:

Equal radii is the strangest case. The two planets share a period, so their tugs on each other add up orbit after orbit, and each plane turns all the way round: in the run, planet 1's orbit swung completely into the plane planet 2 had started in, and back, in about 1,500 orbits. The two planes stayed at right angles throughout, and the planets never came closer than √2.

A 4D planetary system can pack its planets far tighter than ours, at nearly the same distance from the star, in different planes. Planes partly turned apart lie in between: at 30°, two planets 5% apart in radius came within 0.53 of each other and still survived 3,000 orbits.

The planes don't sit still. The planets' pulls turn their planes through 4D, by as much as 88% of planet 1's orbit swinging out of where it began. But the angle between the two planes holds, the 4D version of how our planets' tilts to one another keep while their nodes slide round:

Days with two rates

A planet spins too, and a 4D spin is a turning in a plane, or in two planes at once at two rates (chapter 4). A planet spinning both ways would have a sky that cycles on two clocks, its sunrises returning only when the two line up. Which way real planets spin depends on how they formed and how tides wear them, and that's open.

Checked here: Kepler's closed orbits in 4D, the precession formula against integration, and the two-planet runs above, 3,000 orbits each. The map below runs 100,000.

Which pairs last computed

3,000 orbits is short against a planetary system's life. So, on the compute runner: two planets on circular orbits, the outer's radius 1 to 1.6 times the inner's, their planes turned apart by every angle from 0 to 90°, at three masses, each pair run for 100,000 orbits of the inner one. In 4D there are two ways to turn one plane from another. One is the only tilt our planets can have: the planes share a line and tilt about it. The other is 4D's own: the planes turned by the same angle both ways, which at 90° makes them completely orthogonal.

Each cell is a pair of planets: across, the angle between their planes; down, a₂, the outer orbit's radius over the inner's. Brighter: the more either orbit was stretched (its largest eccentricity, dark for circular). ✕: a planet thrown out. Hover a cell for its numbers.

4D's own tilt keeps planets apart; ours throws them out. Turned equally both ways by 30° or more, Jupiter-mass planets stayed nearly circular (eccentricity at most 0.06) at every spacing, even sharing an orbit's size. Ten Jupiters need 60°, and completely orthogonal planes kept every pair at every mass (eccentricity at most 0.01). Tilted our way, about a shared line, the same planets fare worse than in one plane: at Jupiter's mass every pair tilted 45° or more was thrown out, at ten Jupiters every tilted pair was, and even at a tenth of Jupiter's the tilt is traded into stretched orbits, eccentricity up to 0.79.

In one plane the problem is exactly ours, since the pinned pull is Newton's within a plane, and the runs agree with our world's answer. Hill's criterion, in Gladman's form, guarantees two planets never meet beyond radii in the ratio 1.15, 1.36 and 1.97 for the three masses. No pair beyond it was lost; the losses stop at 1.1, 1.2 and 1.45, inside the guarantee, where (as in ours) a pair can still last a long time.

So a 4D planetary system packs its planets with their planes turned equally both ways, and can pack them far tighter than ours: tilted our way, they'd be thrown out.

Moons: one plane is safe computed

A moon circles its planet while both circle the star. The star's pull is a little stronger on the moon's near side than its far side, and over many orbits that uneven tug can stretch a moon's orbit until it's thrown off. In our world it does this to steeply tilted moons (Kozai's effect): a moon on a polar orbit doesn't last. In 4D there's one plane where the star can't tug at all. The star always lies in the planet's orbital plane, and that plane has exactly one completely orthogonal partner. A moon there is always at right angles to the star, so its distance to the star is √(R² + r²), the same all the way round its orbit: the star pulls it evenly and can't drag its plane.

Every 4D planet has one plane where its moons are safest, the one completely orthogonal to its own orbit, and moons there sit the quietest of all. A planet's moons make an orrery of their own, and its natural plane is the one its planet's orbit leaves free. The orrery puts these results together: a system whose planes come from the nebula below, spaced by the stability map, its moons in their safe planes, run live.

Tides: a locked planet keeps turning, and never sees a sunset computed

A star pulls a planet's near side harder than its far side and stretches it into a bulge. A planet that yields slowly carries the bulge a little way round with its spin before it settles, so the bulge points just off the star, and the star's pull on it brakes the spin. In our world that ends in tidal locking: the spin slows to the orbit's rate and the planet keeps one face to the star, as our Moon does to us.

Under the pinned law the tidal pull has the same form as ours in every dimension (it's the next term in 1/r, and nothing else). What changes in 4D is the spin. A 4D planet turns in two planes at two rates (chapter 4), and the star always sits in the plane of the planet's orbit. A turning in the plane completely orthogonal to the orbit moves nothing in the orbit's plane, so it can't carry the bulge off the star, and the torque on a bulge always lies in planes that hold the star's direction, so it can't touch that turning either. Tides lock the spin in the orbit's plane, damp the parts that tilt between the planes, and leave the completely orthogonal turning exactly as it was.

Rates in units of the orbit's own (1). The bulge model is the constant time-lag model our planets are studied with (Mignard; Hut), written for a turning in any number of planes. The second check knows nothing of it: five masses at a 4-simplex's corners, joined by springs with dampers, orbiting the star as point masses. It yields and lags because it's elastic and lossy, and it locks the same way, keeping its completely orthogonal angular momentum to 1.4 parts in 10,000 (its rate there moves 1.2% only because spinning stretched it).

So a locked 4D planet keeps turning, but its turning brings no night. How much light falls on a point of the surface goes with the angle to the star, and the star's direction lies in the orbit's plane, which the surviving turning leaves fixed point by point. On a locked planet still turning in its free plane, the star's height above any spot holds steady (to 0e+00 in the computation) while the planet goes round: a day side and a night side, as a locked planet of ours has, and over them the rest of the sky wheels by in the free plane. How soon a planet locks depends on how far it yields and how slowly, which belong to its interior and aren't known for 4D; the dependence on distance is ours, the sixth power.

Starlight on a planet follows

Light from a star thins as 1/r³ (chapter 1). A planet catches it over its shadow, which in 4D is a solid ball, and glows it away from its whole surface, a 3-sphere, as T⁵ (chapter 3). So a planet's temperature falls as r^−3/5 (ours as r^−1/2), and the share of its surface's worth of light it catches is 2/(3π) = 0.212 (ours 1/4).

For a star as hot at its surface as our Sun (5772 K), a black planet at Earth's distance in star radii (215) would sit at 278 K in ours and 169 K in 4D; to be as warm as ours it would orbit at 93 star radii, less than half as far. And a band of temperatures is narrower in 4D: a liquid that stays liquid over temperatures in the ratio of our water's (373 : 273 K) has orbits in the ratio 1.87 in ours and 1.68 in 4D. But in 4D more planets can share a band: they can orbit at nearly the same distance in planes turned apart (above). A 4D star's surface temperature isn't known yet: it depends on how light escapes a 4D star, which is open.

Does a nebula flatten? computed

Our planets lie nearly in one plane because the cloud they formed from flattened into a disk. The geometry chapter gave the reason: gas circling a star covers a sphere, a sphere can't be combed, so the streams cross, collide and settle into one plane. A 3-sphere can be combed, so a 4D cloud needn't flatten. Whether it does is a question for simulation.

A cloud's spin in 4D is a turning, and a turning can go in two planes at once at two rates (chapter 4). We let rotating clouds of gas fall onto a star and collide: tens of thousands of particles, pulled as 1/r², which is Newton in 3D and the pinned law in 4D, so only the dimension differs. Particles that come close while approaching lose part of their approach speed. That keeps momentum and spin exactly and only takes energy away, which is all a disk needs to form. The same code in 3D is the control.

Flatness is the cloud's two smallest extents over its two largest: 1 round, 0 a flat disk. Time in orbits of the cloud's edge over 2π.

A 4D nebula flattens to a disk only if it spins in one plane. Spinning in both, it thins and then stops, the thicker the closer its two rates: its flatness ends near the ratio of the rates to the power 0.67. At equal rates it doesn't flatten at all, and every orbit in it runs round one of Hopf's circles. The stop is real, not slowness: run nearly four times longer, the half-rate cloud ends at 0.66, against 0.62 at the shorter runs' end. And it holds up: with a second seed, collisions twice as hard, a start three times hotter and twice the particles (that run stopped two thirds of the way), the half-rate cloud ends between 0.61 and 0.64.

Which clouds are typical? If a cloud's spin is the sum of many random eddies, its two rates differ: the slower is typically a fifth of the faster (half of all clouds fall between 0.10 and 0.36), and equal rates are rare (0.1% within a tenth of equal). So the typical 4D nebula ends neither a disk nor a ball, but a thick lens, about 35% as thick as it is wide, and its planets would form in planes spread round one of Hopf's families rather than in one.

Each run's end, 500 of its particles, orbiting the star live. Turn it through w to see a 4D cloud's thickness.

The gas here has no pressure and no gravity of its own, and the time it takes depends on the collision model, which is why the shapes matter and the times don't. Half or more of the gas ends on the star. Next: how planets grow in a thick nebula.

Where the orbits' second alignment stops computed. After the shape has stopped, the orbits' planes went on lining up in the other half of their turning (the right half, where the cloud turns slower), which no flattening shows. Run to 500 orbits, that stops too: by 150 it has settled, moving under 0.007 after. It ends just below the cloud's own ratio of right-half to left-half turning, (1 − q)/(1 + q) for a second plane at q of the first's rate: 0.569 against 0.600 at q = 0.25, 0.308/0.319 against 0.333 at q = 0.5, 0.125 against 0.143 at q = 0.75. A cloud spinning in one plane only (q = 0) lines up both halves completely (0.998), as a flat disk must. So each orbit ends up with nearly the cloud's own mix of left and right turning, a little short of it, and the orbits' planes stay spread round Hopf's family by that much.

What's computed and what's open

Computed and checked: orbits' precession round a real star (against Landau's formula), two planets in one plane and in planes turned apart, the planes turning while their angle holds, moons in five planes, nebulae against our world's, the stability map (two planets for 100,000 orbits, over spacings, tilts of both kinds and masses), and tides on a spin in two planes (against our world's tidal model and a planet of springs). Follows: starlight on a planet. Open: a 4D star's brightness and colour; how planets grow in a thick nebula, and the spins they're born with; how soon tides lock them (it needs their interiors); and systems of many planets. The orrery puts what's computed together.