Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

A boom held upright by a difference in gravity

A long spacecraft in orbit is pulled into line with the vertical for nothing — its near end feels slightly more gravity than its far end, and the difference is a torque. The torque restores and never dissipates, so the vehicle swings like a pendulum whose clock is the orbit. Whether it is held at all comes down to three inequalities between its moments of inertia, and one region that satisfies all three is destroyed by the damper every such spacecraft needs.

Assumes Attitude control and Tides.

16 min read 6 figures Small perturbations, large consequencesTwo bodies, and no more

Every spacecraft feels the gradient of gravity across its own body as a disturbance, a torque that fills its reaction wheels whenever it is not lined up with the local vertical. The same torque can be turned around. A spacecraft built long, and allowed to hang, is pulled into line with the vertical and held there with no wheels, no thrusters, no sensors and no power.

The Moon has done it for billions of years. It keeps one face towards the Earth because, once tides had slowed its rotation to match its orbit, its slightly elongated figure has been held pointing at the Earth by exactly this torque. Spacecraft have copied it since the 1960s. The Long Duration Exposure Facility, a twelve-sided cylinder about nine metres long, was released by the Space Shuttle in 1984 and collected in 1990, and in the whole of that time it held its long axis along the vertical with no attitude control system at all.

The method is free and it is not simple. A spacecraft held this way swings, and nothing in the torque that holds it removes the swing. Its three axes are held with very different stiffness, one of them hardly at all. And whether a given shape is held at all is decided by three inequalities between its moments of inertia, which have a surprising solution: an island of shapes that are held perfectly well until the spacecraft is given the damper it needs.

The difference that makes a torque

Gravity falls off with distance, so the end of a spacecraft nearer the Earth is pulled slightly harder than the far end. Seen from the spacecraft’s centre of mass, which follows the orbit, the near end is pulled down and the far end is left behind, pulled up — exactly the difference that raises two tides on the Earth rather than one. On a spacecraft whose long axis is tilted from the vertical, the two pulls are not in line, and they make a couple that turns the long axis back towards the vertical.

For a spacecraft pointing at the Earth, the axes are conventionally named for an aircraft flying along the orbit: roll about the direction of motion, pitch about the normal to the orbital plane, yaw about the vertical. A tilt in pitch, by an angle θ from the vertical, meets a restoring torque

τ=32 n2 (Iroll−Iyaw) sin⁡2θ,\tau = \tfrac{3}{2}\,n^2\,(I_{\rm roll} - I_{\rm yaw})\,\sin 2\theta ,

where nn is the orbital angular rate. The factor n2n^2 is μ/r3\mu/r^3 in disguise, the inverse cube that governs every tidal effect. The torque does not depend on the mass of the spacecraft, only on the difference of two moments of inertia, and a body with the two moments equal feels none.

Divide by the pitch moment and the equation of motion is a pendulum, θ¨=−32n2ksin⁡2θ\ddot\theta = -\tfrac{3}{2}n^2 k\sin 2\theta, with one shape factor

k=Iroll−IyawIpitch.k = \frac{I_{\rm roll} - I_{\rm yaw}}{I_{\rm pitch}} .

For a thin boom hanging along the vertical, the yaw moment is nearly zero and the roll and pitch moments are nearly equal, so kk is close to one. A stubbier body has a smaller kk.

A 20° pitch swing that nothing removes. The pitch angle of an Earth-pointing spacecraft released 20° from the local vertical, against orbits, for a thin boom, k = 1 and a stubbier body, k = 0.3, where k — the roll moment of inertia less the yaw moment, divided by the pitch moment — measures how elongated the body is along the vertical. The gravity gradient pulls the long axis back towards the vertical with a torque proportional to sin 2θ, and the body swings about it: with k = 1 once every 0.596 orbits, against 0.577 for a vanishingly small swing at √(3k) times the orbital rate; with k = 0.3 once every 1.087 orbits, against 1.054 for a vanishingly small swing at √(3k) times the orbital rate. With no damping the swing is still 20.0° in the last orbit drawn: the gradient restores, and nothing dissipates, so a gravity-gradient spacecraft needs something aboard to turn the swing into heat. No body swings faster than the thin boom: k cannot exceed one, because no moment of inertia is larger than the sum of the other two.
Fig. 1 Pitch angle from the local vertical against orbits for an Earth-pointing spacecraft released twenty degrees off, for a thin boom with k = 1 and a stubbier body with k = 0.3. Both swing back and forth about the vertical with no loss. The boom completes a swing every 0.60 orbits and the stubbier body every 1.09, each a little longer than the small-swing values of 0.58 and 1.05 because twenty degrees is not a small swing.

The swing’s period depends on the orbit and the shape and on nothing else. The size of the spacecraft does not enter, so a boom one metre long and a boom a hundred metres long, with the same proportions, swing together.

Because the torque goes as sin⁡2θ\sin 2\theta rather than sin⁡θ\sin\theta, the pendulum has two resting positions half a turn apart. A boom hanging down towards the Earth and a boom standing up away from it are equally stable. The gradient cannot tell up from down, and a gravity-gradient spacecraft that is captured upside down stays upside down until something turns it over.

A pendulum whose clock is the orbit

For small swings the pendulum’s frequency is 3k\sqrt{3k} times the orbital rate. The shape factor cannot exceed one, since no moment of inertia is larger than the sum of the other two, so no body swings faster than 3\sqrt 3 times its orbital rate. In a ninety-minute orbit that is a swing every fifty-two minutes at best. The gradient is a stiff enough spring to hold a boom and a slow one; it reacts on the timescale of an orbit, not of a second.

The Moon is a spacecraft with a tiny shape factor. The fractional difference between its moments about the axis pointing at the Earth and the axis along its orbit is about two parts in ten thousand, and with kk that small the same formula predicts a free swing in longitude about once every 2.9 years. That free swing is one of the physical librations measured by laser ranging. Mercury obeys the same pendulum equation on an eccentric orbit, where the torque’s sin⁡2θ\sin 2\theta is what allows it to lock at three turns for every two orbits, and the size of its rocking about that lock measures how much of its interior turns with the surface.

Large swings take longer, as they do for any pendulum.

A 60° pitch swing that nothing removes. The pitch angle of an Earth-pointing spacecraft released 60° from the local vertical, against orbits, for a thin boom, k = 1 and a stubbier body, k = 0.3, where k — the roll moment of inertia less the yaw moment, divided by the pitch moment — measures how elongated the body is along the vertical. The gravity gradient pulls the long axis back towards the vertical with a torque proportional to sin 2θ, and the body swings about it: with k = 1 once every 0.793 orbits, against 0.577 for a vanishingly small swing at √(3k) times the orbital rate and 0.793 for a pendulum released at 60°; with k = 0.3 once every 1.447 orbits, against 1.054 for a vanishingly small swing at √(3k) times the orbital rate and 1.447 for a pendulum released at 60°. With no damping the swing is still 60.0° in the last orbit drawn: the gradient restores, and nothing dissipates, so a gravity-gradient spacecraft needs something aboard to turn the swing into heat. No body swings faster than the thin boom: k cannot exceed one, because no moment of inertia is larger than the sum of the other two.
Fig. 2 The same two bodies released sixty degrees from the vertical. The boom now takes 0.79 orbits per swing and the stubbier body 1.45, both about a third longer than a small swing, and both exactly what the elliptic-integral period of a pendulum released at that angle predicts. A body released at more than ninety degrees would pass over the horizontal and settle into the upside-down position instead.

The lengthening matters for capture. A boom that is deployed while the spacecraft is still turning, or that is knocked well away from the vertical, spends most of each swing near its extremes, and the time to damp it out is set by those slow, wide swings rather than by the fast small-swing rate.

Nothing takes the swing out

The gradient’s torque depends only on the angle, not on how fast the angle is changing, so it stores and returns energy without losing any. A boom released at twenty degrees swings at twenty degrees forever.

How strong the holding torque is can be put in numbers. A boom ten metres long with a one-kilogram mass at its tip has a moment of inertia of about a hundred kilogram square metres about its roll and pitch axes and almost none about its length. At 500 kilometres the orbital rate is about 1.1 thousandths of a radian a second, and tilting such a boom ten degrees from the vertical produces a restoring torque of about sixty millionths of a newton metre. Sunlight pressing on a square metre of panel whose centre of pressure sits ten centimetres from the centre of mass makes about half a millionth. The gradient wins by two orders of magnitude, which is why a boom is held at all, and a steady disturbance of that size tilts the boom by less than a tenth of a degree. The same numbers show how slow the spring is: sixty millionths of a newton metre acting on a hundred kilogram square metres takes about four minutes to turn the boom through a single degree.

A gravity-gradient spacecraft therefore always carries something whose only job is to lose energy. Some used a small mass on a spring at the tip of the boom, moving through a viscous fluid or past magnets that induce eddy currents; others used rods of magnetically soft material that heat slightly as they turn through the Earth’s field. Each converts swing into heat.

A 20° pitch swing that a damper removes. The pitch angle of an Earth-pointing spacecraft released 20° from the local vertical, against orbits, for a thin boom, k = 1 and a stubbier body, k = 0.3, where k — the roll moment of inertia less the yaw moment, divided by the pitch moment — measures how elongated the body is along the vertical. The gravity gradient pulls the long axis back towards the vertical with a torque proportional to sin 2θ, and the body swings about it: with k = 1 once every 0.588 orbits, against 0.577 for a vanishingly small swing at √(3k) times the orbital rate; with k = 0.3 once every 1.073 orbits, against 1.054 for a vanishingly small swing at √(3k) times the orbital rate. With a damping of 0.05 of the orbital rate, the swing falls to 11.9° by the last orbit drawn. No body swings faster than the thin boom: k cannot exceed one, because no moment of inertia is larger than the sum of the other two.
Fig. 3 The same release at twenty degrees with a damper that removes energy in proportion to the pitch rate, at five per cent of the orbital rate. The swing falls from twenty degrees to about twelve in four orbits. The period is almost unchanged, because a damper this light alters the swing’s size and not its rhythm.

Four orbits to lose forty per cent of a swing is typical of what a light damper can do, and the result is a spacecraft that points to within a few degrees rather than a few arcseconds. That is good enough for an antenna with a broad beam, for exposure panels facing space, or for an instrument that does not care exactly where it looks. It is not good enough for a telescope, which is why pointed observatories pay for wheels and propellant instead.

The damper also connects this problem to the energy argument that decides which way a spinning body settles. A damper does not choose an attitude. It removes energy, and the spacecraft ends wherever energy has a minimum. For pitch, that is the vertical. For roll and yaw the answer is less obvious, and it is where the gradient’s weakest point and its most surprising region both appear.

Roll, yaw, and the axis the gradient cannot grip

The gradient’s torque on a body is 3n2 r^×(I r^)3n^2\,\hat r\times(\mathbf I\,\hat r), where r^\hat r is the direction to the Earth. A cross product with r^\hat r is always perpendicular to r^\hat r, so the gradient never exerts any torque about the vertical. A boom hanging along the vertical can be turned about its own length by anything at all, and the gradient does not resist.

Yaw is held all the same, and it is held by the orbit. An Earth-pointing spacecraft must turn once per orbit about its pitch axis to keep facing down, so it carries angular momentum about that axis. A yaw error tips that angular momentum, which shows up as a roll motion, and roll is restored by the gradient. Roll and yaw are therefore coupled into two joint oscillations, and yaw inherits its stiffness second-hand, through the coupling.

Three swings, and how weakly the gradient holds yaw. The natural frequencies of an Earth-pointing spacecraft's small oscillations, in units of its orbital rate, against k₁ at a fixed k₃ = 0.2. Pitch is decoupled and swings at √(3k) times the orbital rate. Roll and yaw are coupled by the orbital motion and swing together in two modes, whose frequencies are the roots of ω⁴ − (1 + 3k₁ + k₁k₃)ω² + 4k₁k₃ = 0 and whose product is fixed at 2√(k₁k₃). At k₁ = 1.00 the pitch swing is at 1.73, the faster roll–yaw mode at 2.00 and the slower at 0.45 times the orbital rate. The slow mode is mostly yaw, and it is slow because the gradient exerts no direct torque about the vertical at all: yaw is held only through its coupling to roll, which is why a gravity-gradient spacecraft points its boom at the Earth well and holds its heading about that boom poorly.
Fig. 4 The natural frequencies of an Earth-pointing spacecraft’s small swings, in units of the orbital rate, against the shape ratio k1k_1 — the pitch moment of inertia less the yaw moment, divided by the roll moment — at a fixed k3k_3 of 0.2, the pitch moment less the roll moment divided by the yaw moment. Pitch is independent of the other two. Roll and yaw swing together in two modes whose frequencies multiply to a fixed product. For a flat shape at the right-hand edge the pitch swing is at 1.73 times the orbital rate, the fast roll–yaw mode at 2.00 and the slow one at 0.45. The slow mode is almost pure yaw.

The slow mode is the weakness of the whole scheme. A yaw swing that takes more than two orbits to complete is held by a restoring stiffness a small fraction of what holds pitch, and any small disturbing torque about the vertical — sunlight on an asymmetric panel, a leak, the drag of the atmosphere on a boom that is not quite symmetric — pushes it a long way. Gravity-gradient spacecraft point their booms at the Earth well and hold their heading about the boom poorly. The usual remedy is a small wheel spinning at constant speed along the pitch axis, which adds angular momentum and so stiffens the coupling that holds yaw; it costs a little power, and it is still far cheaper than holding all three axes actively.

The three stiffnesses can be read off their origins. Written for small angles, each swing’s restoring term is the square of the orbital rate times a difference of two moments of inertia times a whole number, and the whole numbers are three for pitch, four for roll and one for yaw. Pitch gets its three entirely from the gradient. Roll gets the same three from the gradient and one more from the orbit’s rotation, because a body turning once an orbit about its pitch axis resists being tipped out of that rotation, as any spinning object does. Yaw gets nothing from the gradient and only that one gyroscopic unit. The coefficients three, four and one are the whole of the small-swing theory of a gravity-gradient spacecraft, and the smallness of the last is the reason heading is hard to hold. It also explains why the second island below exists at all: when the moments are arranged so that the gradient pushes roll and yaw away instead of back, the one gyroscopic unit is the only thing left holding them, and it can do so only as long as nothing drains energy from the motion.

Three inequalities

Putting the three swings together gives the conditions under which an Earth-pointing orientation is stable at all. Pitch is stable when the roll moment exceeds the yaw moment. Roll and yaw are stable together when two more conditions hold on the ratios k1k_1 and k3k_3, and the combination can be drawn as a map.

Where the gradient of gravity holds a spacecraft still. The plane of the two inertia ratios that decide whether a spacecraft pointing at the Earth is held there by the gravity gradient: k₁ — the pitch moment of inertia less the yaw moment, divided by the roll moment — across, and k₃ — the pitch moment less the roll moment, divided by the yaw moment — up, with roll along the velocity, pitch normal to the orbit and yaw towards the Earth. Shaded points satisfy all three conditions of the linear theory — pitch is stable when k₁ > k₃, and roll and yaw together when k₁k₃ > 0 and 1 + 3k₁ + k₁k₃ > 4√(k₁k₃). The large region at upper right, 12.4 per cent of the square, is the one in which the pitch moment is the largest and the yaw moment the smallest, the arrangement of a long boom hanging towards the Earth. The small region just left of the vertical axis and below the horizontal one, 2.0 per cent, is a second, narrow island of stability with the moments in a different order, found by DeBra and Delp in 1961. There the orientation is a maximum of the potential in roll and yaw rather than a minimum, held only by the gyroscopic coupling of the two, and a damper — the very thing the long-boom region needs — destroys it: with damping of 0.05 of the orbital rate, a swing of a hundredth of a radian at (−0.10, −0.21) grows to a full radian within 11 orbits, while the same swing on the long boom shrinks 435-fold in 40. A boom along the vertical, pitch moment largest sits at (0.97, 0.40) and is stable; the same boom with roll and pitch moments swapped sits at (0.93, −0.40) and is unstable: the same boom, with two nearly equal moments exchanged, crosses from one side of an axis to the other.
Fig. 5 The plane of the two shape ratios, with the shaded points meeting all three stability conditions of the linear theory. The large triangle at upper right, 12.4 per cent of the square, holds every shape whose pitch moment is the largest and yaw moment the smallest: the long boom. The narrow strip just below the origin and left of the vertical axis, 2.0 per cent, is a second island with the moments in a different order. A boom with moments 10, 10.2 and 0.5 about roll, pitch and yaw sits inside the large region; the same boom with the roll and pitch moments exchanged sits below the horizontal axis, outside it.

The long-boom region is the one every gravity-gradient spacecraft has used, and its rule is easy to say: the moment about the orbit normal must be the largest and the moment about the vertical the smallest. Deployed solar arrays are placed to add to the pitch moment for this reason.

The pair of marked points shows how little margin a nearly symmetric vehicle has. A boom whose roll and pitch moments differ by two per cent is stable one way round and unstable the other, because exchanging two nearly equal moments carries it across the line k3k_3 = 0. A spacecraft that looks like a boom is not automatically a gravity-gradient spacecraft; its moments have to be in the right order, and for a nearly axisymmetric body the order is decided by small things — where the batteries sit, which way the arrays unfold, how much propellant is left.

An island that a damper sinks

The narrow second island was described by D. B. DeBra and R. H. Delp in 1961, and it is stranger than it looks.

In the long-boom region the orientation is at the bottom of a valley: tip the spacecraft in any direction and the combined effect of the gradient and the orbit’s rotation raises its energy. In the second island, the pitch moment is the smallest of the three, and the orientation is at the top of a hill in both roll and yaw. Tipping it in roll or yaw lowers its energy. It is held only because the orbit couples the roll and yaw swings gyroscopically, so that the tendency to fall one way is turned, a quarter-cycle later, into a motion the other way. A spinning top is held upright on the same principle.

Gyroscopic stability of this kind has a known fragility, set out by William Thomson and Peter Guthrie Tait in the nineteenth century and proved in general by Nikolai Chetaev: an equilibrium that sits at an energy maximum and is held only by gyroscopic coupling is destroyed by any dissipation, however small.

One damper, two stable orientations, at 0.05 of the orbital rate. The largest roll–yaw swing in each orbit, on a logarithmic scale, for two Earth-pointing spacecraft that the linear theory calls stable: a long boom at (k₁, k₃) = (0.97, 0.40), and the point at (−0.10, −0.21) deepest inside the second, narrow island of stability. Both start with the same one-hundredth-radian swing. With nothing aboard to dissipate energy the second island's swing stays bounded, never more than 1.4 times its start. Add a damper removing 0.05 of the swing's rate each radian of orbit and the long boom's swing falls 435-fold in 40 orbits, while the second island's reaches a full radian after 11 — the end of any orientation worth the name, and of the linear theory that describes it, which is why the curve stops there. The long boom sits at the bottom of a valley of potential and friction lets it settle. The second island sits on a summit in roll and yaw, kept there only because the orbit couples the two swings into a gyroscopic dance, and anything that takes energy out of the dance lets it fall off.
Fig. 6 The largest roll–yaw swing in each orbit, on a logarithmic scale, for the long boom and for the point deepest inside the second island, both started with a swing of about a hundredth of a radian. With no damping the second island’s swing stays bounded. With a light damper, the long boom’s swing falls 435-fold in forty orbits, and the second island’s grows until it reaches a full radian after eleven — the end of the orientation, and of the small-swing equations that describe it.

The same damper does opposite things to the two regions. On the long boom it lets the spacecraft settle into its valley. In the second island it lets the spacecraft roll off its hill. A gravity-gradient spacecraft must carry a damper, since without one its pitch swing never dies, and so no gravity-gradient spacecraft can use the second island. The island is real, it is stable in the mathematics of a perfectly rigid body, and it is useless for any body that is not.

This is the result of the migration of a spinning spacecraft in a different setting. There, an energy sink walked a spin away from the orientation a designer chose and towards the one of least energy. Here, a damper walks a spacecraft out of the orientation the linear theory calls stable and towards one of least energy. In both cases the equilibrium that survives in a real vehicle is not the one that is stable, but the one that is lowest.

What the gradient gives, and what it takes

The accounting is short. The gradient holds a long body pointing at the planet for nothing, for as long as the body stays in orbit, and does so with a stiffness set by the orbital rate. It holds pitch best, roll less well and yaw hardly at all. It cannot tell up from down. It needs a damper, and the damper both makes the scheme work and rules out the only other set of shapes it could have used.

It also stops working where the torques that compete with it grow. Low orbits have more air: LDEF itself was left in orbit years longer than planned, and the drag that brought its orbit down nearly brought it into the atmosphere before a shuttle could reach it. High orbits have a weaker gradient, falling as the cube of the orbital radius, so that far from any planet sunlight and the spacecraft’s own leaks outweigh it. The scheme belongs to a band of altitudes, and the band is set by the same competition between torques that fall off with distance at different rates — the air exponentially, the gradient as an inverse cube, sunlight not at all — that sets the momentum budget of every other spacecraft.

Still open: what keeps the Moon swinging?

The Moon’s free libration in longitude — the swing of a gravity-gradient satellite with a very small shape factor, about once every 2.9 years — is measured by laser ranging to be small but not zero. Internal friction in the Moon, and friction between its mantle and its small core, should have damped such a swing out on timescales far shorter than the Moon’s age, just as a damper takes the swing out of a boom. Something must therefore have excited it recently, or must still be exciting it. Candidates include resonances with small periodic terms in the Moon’s orbit, impacts, and motions in the core, and none has been shown to supply the observed amplitude. It is the same shape of question as a wobble in the Earth that should have stopped, asked of a body a hundred times less massive and much harder to measure.

Where this goes next

An oscillation that does not matter, on a ramp that does. Stored angular momentum in a reaction wheel over 160 days at 550 kilometres, with a capacity of 25 newton metre seconds. The total environmental torque is 1.75e-4 newton metres, of which 35 per cent is taken to survive averaging over an orbit. The fast oscillation is the part that does not survive: it has the 95.6-minute orbital period, reaches 0.10 newton metre seconds, and returns to where it started every revolution, so it consumes capacity and nothing else. The ramp under it is the secular part, and its slope measured between two instants a whole number of orbits apart is 6.117e-5 newton metres, which is the secular torque and is how the figure checks itself. The wheel fills in 4.7 days and has to be emptied 33 times in the span drawn. Every attitude-controlled spacecraft in the collection lives on this sawtooth, and the vertical drops are the only part of it that costs anything: the store can be moved between wheels for nothing, and taken out of the vehicle only by pushing against something outside it. The spin that has to be put somewhere A spacecraft holding an attitude is not resisting a force. It is absorbing a slow, one-directional trickle of angular momentum from the gradient of gravity across its own body, from sunlight, from the last of the atmosphere — and every store it has for that trickle fills up. The sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats. The face that is not quite fixed The Moon keeps one face turned toward the Earth, and the sentence is exactly true only of a fictitious Moon on a circular orbit. The real one rocks by a few degrees each month, in two directions and for two unrelated reasons, and the rocking has shown 59 per cent of its surface to people who never left the ground. The path of a spin across the sphere of fixed momentum. A body with principal moments 1, 8, 8.6, started about its axis of least inertia with a 3° wobble and an internal energy sink strong enough that the whole motion fits in 80 turns and each circuit of the path can be seen. The disc is the near hemisphere of the sphere of fixed angular momentum, seen from a direction between all three body axes, with the near end of each axis marked. Each thin curve is a contour of kinetic energy on that sphere — a polhode, one of the paths the angular momentum can follow in the body with no dissipation — and the thick curve is the separatrix through the intermediate axis, which divides motions that circle the axis of least inertia from motions that circle the axis of greatest. The coloured path is what the integration did: solid on the near hemisphere, dashed where it passes behind. It starts at the open dot and ends at the filled one. A spin that left the axis it was given The first American satellite was spun about its long axis, like a rifle bullet, and soon after launch it was tumbling end over end. Nothing outside it had pushed. A body that cannot change its angular momentum but can lose energy has exactly one place to end up, and a long body spun about its length is as far from that place as a spin can be.

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Attitude controlGravity gradientGravity gradient torqueLibrationMoment of inertiaPrincipal axesTidal locking