Astronomy & Celestial Mechanics
the sky, computed
Concept

Restricted three-body problem — where it appears

Two masses on circular orbits about their common centre, and a third of negligible mass moving in the field they make. It is the simplest system that is not integrable, and its one conserved quantity — the Jacobi constant — is what bounds where the third body can go.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Zero-velocity curves, mass fraction 0.15. Level sets of the Jacobi constant in the frame that rotates with the two bodies. A particle with a given value of C is confined to the side of its own curve where the kinetic energy would be positive; as C falls the curves open, first at the inner point between the bodies, then behind the smaller one, and finally around the whole system.

The curve that says where a body cannot go

The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time.

gravitation · The three-body problem The surface a star stops at. The equipotential through L₁ — the Roche lobe — at mass fractions 0.50, 0.20, 0.05, in the frame that rotates with the pair. Each is a level set of exactly the same function the zero-velocity curves are level sets of, at exactly the critical value, so nothing here is a new construction: the Roche lobe is the last closed equipotential, and it is closed only because the two lobes touch at a single point. Material that reaches that point is no longer bound to the star it came from, and it leaves through an opening of zero area. The lobes are drawn in the orbital plane; in three dimensions each is a teardrop, and its volume-equivalent radius is what "the size of a Roche lobe" means. As the mass ratio becomes extreme the smaller star's lobe shrinks towards it, which is why a white dwarf accreting from a companion has a lobe smaller than the Sun.

The surface a star stops at

Around each star of a close pair there is a last closed equipotential, and the two touch at a single point. A star that swells to reach it hands its outer layers to its companion through an opening of zero area — and the transfer, once started, makes itself worse.

gravitation · Lagrange points The contours a planetary encounter cannot cross. Contours of the Tisserand parameter with respect to Jupiter in the plane of semi-major axis and eccentricity, drawn for a coplanar orbit. An encounter with Jupiter moves a comet along one of these curves and never across one, because T is what the encounter conserves. The heavy contours are at T = 3 and T = 2, and they are the boundaries the comet families are defined by: T > 3 means no encounter is possible at all, since v∞²/v_J² = 3 − T and a negative squared speed is not a trajectory; 2 < T < 3 is the Jupiter family; below 2 the approach speed exceeds Jupiter's own orbital speed and the orbits are the nearly isotropic ones. The shaded boundary is the crossing condition — an orbit whose pericentre is outside Jupiter's, or whose apocentre is inside it, never meets the planet whatever its T. The five comets are placed at their JPL elements and labelled with the T the literature quotes; all five with inclination sit off the coplanar contours by exactly the cos i in the definition, which is why 1P/Halley's is negative — a retrograde orbit meets Jupiter at nearly twice Jupiter's speed.

The number that survives the encounter

A comet that passes Jupiter comes away with every orbital element changed. One combination of them is not changed, and it is enough to recognise the comet afterwards, to sort the comet families, and to say where a spacecraft can and cannot go.

gravitation · Tisserand parameter An invariant that moves by 8.0e-3 once the planet's orbit is real. The Tisserand parameter of a comet on an orbit of semi-major axis 5 and eccentricity 0.8, followed through a close passage of Jupiter, integrated twice. The lower trace has Jupiter on a perfect circle, which is the problem the parameter is an exact constant of: it survives the encounter having moved by 1.0e-4, which is the integrator's own error and not a physical change, and the spike at the moment of closest approach is the osculating elements being briefly meaningless while the comet is inside Jupiter's sphere of influence rather than the constant failing. The upper trace is the same encounter with Jupiter on its real orbit, eccentricity 0.0489. The parameter comes out changed by 8.0e-3, 79 times as much, because the Jacobi constant exists only when the rotating frame is uniformly rotating and a planet on an ellipse does not provide one. That number is small and it is not negligible: comet families are separated by boundaries in this parameter placed to two decimal places, and a comet that drifts across one over several encounters has changed class without anything having happened to it that a single encounter could account for.

An invariant that is only almost one

The Tisserand parameter survives a close encounter with Jupiter exactly, and comet families are separated by boundaries in it drawn to two decimal places. The exactness holds for a Jupiter on a circle. Jupiter's eccentricity is 0.0489, and integrating the same encounter twice shows what that costs.

orbits · Tisserand parameter

Named alongside it

The objects these essays reach for when they reach for this one.

Jacobi constantTisserand parameterGravity assistLagrange pointsMass transferOsculating elementsRoche lobeAccretionAlgol paradoxBinary starsCircular restricted problemClose encounter

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