In 1983, aerospace engineer Kathleen Howell received her PhD from Stanford for her dissertation “Three-dimensional, periodic halo orbits in the restricted three-body problem”. A short paper she wrote with some of the results, published in 1984, is available here: “Three-Dimensional, Periodic, ‘Halo’ Orbits” by Kathleen Connor Howell, Celestial Mechanics, 32 (1984) 53-71.
But what exactly are these orbits that Howell discovered?
The restricted three-body problem concerns two massive bodies orbiting their common centre of mass, while a third body of negligible mass moves in their vicinity, subject to their gravity but too light to influence their motion at all. So it can apply to (relatively light) planets in a binary star system, or to satellites moving in the Earth-Moon or Earth-Sun system, in contexts when it’s reasonable to neglect all the other bodies in the system.
The orbits Howell discovered have the nice property that they are periodic when viewed in the rotating coordinate system locked to the two massive bodies as they move around in their own orbits. In that rotating frame, these “halo” orbits appear as fixed trajectories, with the combination of gravity and centrifugal and Coriolis forces bringing the third body back to exactly the same location, with the same velocity.
The image on the right shows the lighter of two massive bodies (with 0.04 of the total mass) as a black sphere, with the L1 Lagrange point (the point between the two massive bodies where all forces in the rotating frame balance) as a green sphere, and the L2 Lagrange point (the point beyond the lighter of the two bodies where all forces balance) as a red sphere. The heavier of the two massive bodies isn’t shown, as that would make the orbits themselves too small to see clearly.
Most of the halo orbits shown (those in grey) are unstable to small perturbations, but the coloured ones are stable to first order.
[The precise meaning of “stable to first order” for a periodic Hamiltonian system is quite subtle! Intuitively, it means the trajectory as a geometrical curve will remain at a bounded distance from the original trajectory, for any infinitesimal perturbation. For a nice account of the details, see: “Periodic Orbits in Gravitational Systems” by John D. Hadjidemetriou.]
There are lots of stable orbits in the L2 family that essentially collide with the light massive body (unless it’s tiny), so they’re not much use in practice. But there’s a small range of stable orbits in both the L1 and L2 families that don’t suffer from that problem.
The easiest way to find these orbits is to work in the rotating frame in which we want the trajectories to be periodic. We choose the x-axis to be a line containing the two massive bodies, and put the y-axis in the plane of their orbit. To try to find a halo orbit that passes through the xz-plane at a given value of x, we vary the value of z and of dy/dt, and numerically integrate particles that are launched from (x, 0, z) with a velocity of (0, dy/dt, 0) — that is, orthogonally to the xz-plane — and follow them until they “fall back” to the xz-plane under the influence of the forces at play in the rotating frame: the gravitational pull of the two massive bodies, and the centrifugal and Coriolis forces.
In general, when the test particle crosses the xz-plane again, it will not be orthogonal to it, but will have some non-zero values for dx/dt and dz/dt. But we have two degrees of freedom to work with in the initial conditions: z and dy/dt at the time of the launch. So, by varying these appropriately, we can hope to end up making dx/dt and dz/dt zero when the particle crosses the plane again. If we can do that, we will have a trajectory that is orthogonal to the xz-plane at two points. By the symmetry of the situation, that trajectory and its time-reversed mirror image in the xz-plane, taken together, will yield a periodic orbit!
In Howell’s paper, she explains how to integrate not just the trajectories, but also a 6 × 6 matrix that gives the partial derivatives of the current coordinates and velocities with respect to their initial values. This matrix lets us see how to vary the initial conditions to eliminate the unwanted components of the “landing velocity” in the xz-plane and make the second crossing orthogonal. It also contains the information that lets us know whether the orbit is stable or not.
In a sense, that gives the whole story: if we’re able to make the orbit cross the xz-plane orthogonally at two points while travelling in opposite directions, under the influence of all the forces acting in the rotating frame, then it must extend into a periodic orbit, in a fixed shape that rotates along with the frame.
But there is still something that seems a bit puzzling here. Normally, an orbit has a kind of “gyroscopic” property: it will tend to maintain its orientation in inertial space. In fact, it’s not hard to prove that if you suppose that a test particle moves at a constant speed around a circular orbit centred at a point on the x-axis and orthogonal to that axis, and allow the normal vector to this circle to turn within the xy-plane, it will act just like a gyroscope, and turn in such a way (in the rotating frame) as to cancel out the rotation completely, and maintain a fixed orientation in inertial space.
So, how does these halo orbits avoid doing that?
The image above shows one full halo orbit, viewed in inertial coordinates rather than a rotating frame.
The tight loop in the centre of the image is traced out when the test particle comes closest to the lighter of the two massive bodies, where it speeds up considerably, going backwards in the rotating frame faster than the rotating frame is carrying it forwards.
If you stare at this animation long enough, it becomes clear that it is the eccentricity of the halo orbit — which makes it highly asymmetric in the xy-plane — that allows the angular momentum to turn with the rotating frame, instead of remaining fixed in inertial space. The purple angular momentum vector doesn’t change direction much while the test particle is far from the massive body, but during the close approach it swings around, just enough to provide the right amount of overall rotation required.