Where you can live around a hole and a star

Stand still on the pair

Take a body so light that it does not disturb the pair, and take the pair on a perfect circle. Now ride on a turntable that turns once per orbit of the pair. In that frame Vesper and Ember sit still, and the orbiting body feels their pull plus two fictitious forces: the centrifugal force and the Coriolis force.

Use units where the total mass is 1, the separation is 1 and the pair turns once per $2\pi$ time units. Put the heavier body (Vesper, mass $1-\mu$) at $x=-\mu$ and the lighter (Ember, mass $\mu$) at $x = 1-\mu$, so the centre of mass is at the origin. With $r_1$ and $r_2$ the distances to them, define

$$\Omega = \tfrac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$

The equations of motion in the turning frame are

$$\ddot x - 2\dot y = \frac{\partial \Omega}{\partial x}, \qquad \ddot y + 2\dot x = \frac{\partial \Omega}{\partial y}$$

Multiply the first by $\dot x$, the second by $\dot y$ and add. The Coriolis terms cancel, because a force at right angles to the velocity does no work, and what is left is a total derivative:

$$\frac{d}{dt}\left(\tfrac{1}{2}v^2\right) = \frac{d\Omega}{dt} \quad\Rightarrow\quad C = 2\Omega - v^2 \text{ is constant}$$

This is the Jacobi constant. Neither energy nor angular momentum is conserved on its own, because the pair keeps tugging, but this combination is, and it is the only conserved quantity the problem offers.

A wall you can't cross

Since $v^2 \ge 0$, a body can only be where $2\Omega \ge C$. The curves where $2\Omega = C$ are walls: a body with that value of $C$ stays on its side forever.

The shape of the walls depends on $C$. Large $C$ means small, closed pockets around each body and a separate region far outside. Lower $C$ and the pockets grow until they touch at a saddle point between Vesper and Ember, and the body may pass from one to the other. Lower it further and the walls open to the outside, first behind Ember, then behind Vesper.

Those saddles are the collinear Lagrange points, where $\nabla\Omega = 0$ on the line through the pair. For $\mu = 0.055$, solving numerically puts L1, the pass between the two, 1.1 AU from Vesper and 0.36 AU from Ember. L2 lies 0.43 AU beyond Ember and L3 1.5 AU beyond Vesper, on the side away from the star. Their Jacobi constants, in the same units, are

$$C_{L1} = 3.44\ ,\quad C_{L2} = 3.37\ ,\quad C_{L3} = 3.06$$

A body circling Vesper close in has $C$ well above $C_{L1}$: it cannot reach the pass, so it can never leave its pocket, whatever it does inside. This is called Hill stability, and it needs nothing but the value of $C$. As the orbit grows, its $C$ falls toward $C_{L1}$ and the pocket's neck opens at the saddle. Vesper's safe limit sits at 0.58 of the way to L1, and Ember's at 0.43 of the way: in both cases resonances end stable orbits well before the wall itself opens. For a pair on a circle, Mudryk and Wu showed that the Hill test and the resonance picture below mark nearly the same boundary.

The pocket around a small body has a simple size. When $\mu \ll 1$, L1 and L2 sit close to the Hill radius $a\,(\mu/3)^{1/3}$, which for Ember is 0.40 AU. Studies of moons around planets find prograde moons stable out to about 0.49 of that radius on circular orbits, the rule the Numbers section used as a check on Kindling.

The triangle points are unstable here

There are also two equilibrium points off the line, L4 and L5, each forming an equilateral triangle with the two bodies. At either one, $r_1 = r_2 = 1$ and $x^2 + y^2 = 1 - \mu + \mu^2$, so

$$C_{L4} = C_{L5} = 3 - \mu(1-\mu) = 2.95$$

They are equilibria for any mass ratio, but stable ones only when the masses are very unequal. Linearize the equations of motion about L4 and the small oscillations go as $e^{\lambda t}$ with

$$\lambda^4 + \lambda^2 + \tfrac{27}{4}\mu(1-\mu) = 0$$

Both roots for $\lambda^2$ are real and negative, so every small motion is a bounded wobble, only if the discriminant is positive: $1 - 27\mu(1-\mu) > 0$. This is Routh's criterion,

$$27\,\mu(1-\mu) < 1 \quad\Longleftrightarrow\quad \mu < 0.0385$$

For Vesper and Ember, $27\,\mu(1-\mu) = 1.4$. The star is a little too heavy: 5.5 percent of the pair's mass against a limit of 3.9 percent. The roots turn complex, one of them has a positive real part, and a nudge away from L4 grows exponentially, by a factor of e every 0.74 laps of the pair. A body parked there drifts off within a few laps. Around the Sun and Jupiter, where the planet's share of the mass is far smaller, the same points hold the Trojan asteroids.

Why far orbits are safe

Seen from far away, the pair pulls as if all its mass sat at the centre, plus a smaller, lumpy correction because the mass is really two blobs. With the origin at the centre of mass the next term vanishes, and the first correction scales as $\mu(1-\mu)(a/r)^2$ against the main pull. At the limit around both that ratio is 0.014 of the main pull. Further out it falls with the square of the distance.

Closer in, the lumpy pull is stronger, and it turns with the pair, so it can lock onto the body's own rhythm and stack its pushes: a resonance. Where the resonances sit close enough together to overlap, the orbit wanders chaotically from one to the next until the body leaves. Mudryk and Wu worked that picture out for orbits around one star of a binary. For orbits around both, Holman and Wiegert's simulations show the resonances directly, as the next fold tells. Murray and Dermott's textbook covers the machinery of the restricted problem.

What a "limit" means

Holman and Wiegert obtained these limits from numerical experiments. For each mass ratio and eccentricity they started test bodies on circular, flat, forward-going orbits at eight points round the orbit and watched them for 10,000 laps of the pair. Around one star the limit is the largest radius at which all eight survived; around both, the smallest. Their formulas fit those results to a few percent typically, and to 11 percent in the worst case for close orbits and 6 percent for far ones.

So every limit here is for flat, forward-going orbits; tilted and backward ones have their own. 10,000 laps of Vesper and Ember is 5,155 years, far longer than the Vigil has existed but short against the lifetimes it plans for. And this pair's mass ratio lies outside the range the fits were made from, so the far limit comes from integration. challenge

The edge is ragged

Around both bodies the boundary is not clean. Just beyond it, in most of Holman and Wiegert's runs, sat an island of instability: a narrow band where bodies died anyway, at the first resonance outside the unstable region where a body goes round once while the pair goes round $n$ times. For Vesper and Ember that is the 3:1 resonance, at 3.1 AU. The Ashfall starts beyond it, at 4 AU.

Lawrence Mudryk and Yanqin Wu showed in 2006 where the edge comes from for a planet circling one star of a binary. The companion's pull splits each resonance into close copies and shifts them until they overlap, and a body inside an overlap wanders chaotically, usually until it is thrown out. For eccentric pairs the boundary is jagged, and the widest stable orbit could be up to 20 percent smaller than the fits say. For a circular pair, they found, the Hill test from the previous fold gives nearly the same boundary. Vesper and Ember are on a circle by design; Gaia BH1, the nearest known hole with a star, is on an eccentric orbit, and a pair like that would have less room.

The light moves

Seen from Vesper, Ember goes round once every 188 days, so the direction to the star turns by 1.9 degrees a day. A radiator film catches starlight in proportion to the sine of its tilt from edge-on. It absorbs as much as one of its faces sheds at 30 K at a tilt of 0.46 milliradians near Vesper. On the star side of Zone D's outer edge, where Ember's light is 2.8 times stronger, the tolerance shrinks to 0.17 milliradians. The films also orbit Vesper, once every 49 days at the outer edge, so the pointing is a continuous control task. A black sphere, which sheds over four times the area it catches over, would need to sit 35 AU from the star to fall to 30 K, and a thin two-sided film held face-on, the worst case, 49 AU, far beyond the zones. design

Do planets live around pairs?

Yes. In the 2010s the Kepler space telescope caught several crossing in front of their stars. Kepler-16b, found by Laurance Doyle and colleagues in 2011, crosses the faces of both. It goes round its pair once every 229 days while the stars circle each other every 41, which puts it 3.1 separations out. Holman and Wiegert's formula, with Kepler-16's own mass ratio (0.22) and eccentricity, puts the limit at 2.9: the planet lives about 9 percent outside the edge. William Welsh, Jerome Orosz and colleagues added Kepler-34b and Kepler-35b in 2012 and estimated that at least about 1 percent of close double stars have giant planets orbiting in the plane of the pair.

Relativity and the companion

Near a heavy mass, an orbit's long axis turns slowly, as Mercury's does in the Sun's field: by $6\pi GM/(c^2 r)$ per lap for a near-circular orbit. At the limit around both that is 8.3 × 10⁻⁷ radians a lap, one full turn in about 10 million years. At Vesper's limit it is 3.4 × 10⁻⁶ radians a lap.

The companion turns the same orbits much faster. For a body circling a mass $m_1$ at radius $r$, with a companion $m_2$ at distance $a$, the companion's tidal pull averaged over both orbits advances the long axis by

$$\Delta\varpi \approx \frac{3\pi}{2}\,\frac{m_2}{m_1}\left(\frac{r}{a}\right)^3$$

per lap. At Vesper's limit, with Ember as the companion, that is 0.024 radians, about 7,116 times the relativistic advance. For Kindling, with Vesper as the companion, it is 0.012 radians, a full turn of the orbit every 14 years. Relativity shapes the details of an orbit near the horizon and hardly touches where the safe zones are.

The live scene does not apply these estimates. simulation

Still open

• The limits are for one body at a time. A crowded swarm is a many-body problem, and the Vigil keeps its machines apart using traffic control and thrusters as well as orbital separation. The swarm cannot disturb the pair, though: all its machinery weighs 4 × 10⁻¹⁸ of the pair's mass.
• The pair never merges (A pair too wide to hear). Ember's representative MIST track matches its canonical radius and luminosity at EEP 326; its actual age is unknown. Turnoff follows in about 30.3 billion years, then the sampled giant path engulfs Kindling at 32.8 billion years, ending useful work. The evolving lobe fills 29.9 million years later under isotropic-wind widening. Transfer and restart are unmodelled, so overflow does not extend computing simulation frontier.

Further reading

1. Holman and Wiegert (1999), Long-term stability of planets in binary systems, The Astronomical Journal 117, 621The simulations and the fitted formulas for the close and far limits, the islands of instability beyond the far limit, and the L4/L5 mass condition.
2. Mudryk and Wu (2006), Resonance overlap is responsible for ejecting planets in binary systems, The Astrophysical Journal 639, 423Why the edge around one star is where it is, why it is jagged, and why the Hill test works for a circular pair.
3. Domingos, Winter and Yokoyama (2006), Stable satellites around extrasolar giant planets, Monthly Notices of the Royal Astronomical Society 373, 1227How far out a moon can orbit a planet and survive: about half the Hill radius going forward, about the whole Hill radius going backward.
4. Murray and Dermott (1999), Solar System Dynamics, Cambridge University PressThe restricted three-body problem, the Jacobi constant and the Lagrange points, in a standard textbook.
5. Doyle et al. (2011), Kepler-16: a transiting circumbinary planet, Science 333, 1602A Saturn-sized planet seen crossing both stars of a pair, and the masses and orbits of all three bodies.
6. Welsh et al. (2012), Transiting circumbinary planets Kepler-34 b and Kepler-35 b, Nature 481, 475Two more such planets, and the estimate that at least about 1 percent of close pairs host coplanar giant planets.