Generator

The momentum-store generator

Four torques, and the altitude where the answer changes

Four torques, and the altitude where the answer changes. The four environmental torques acting on one spacecraft — 10 square metres of area, principal moments differing by 200 kilogram metres squared, a centre of pressure 10 centimetres from the centre of mass and a residual magnetic dipole of 1 ampere metre squared — plotted against altitude, both axes logarithmic. The aerodynamic torque falls by six orders of magnitude across the range because the density does; the gravity gradient falls only as the cube of the distance from the Earth's centre, which over this range is barely a factor of three; the magnetic torque falls as the same cube; and sunlight does not change at all, which the figure checks rather than states. The leader changes at 438 kilometres, from the atmosphere to the gravity gradient, and that crossing is measured off the drawn curves. Nothing here is a disturbance in the ordinary sense: each of these acts in the same direction for a large part of every orbit, so what matters is not the size of the torque but the part of its integral that survives an orbit — and the wheel that stores that integral is emptied on the schedule these curves set.

The four environmental torques acting on one spacecraft — 10 square metres of area, principal moments differing by 200 kilogram metres squared, a centre of pressure 10 centimetres from the centre of mass and a residual magnetic dipole of 1 ampere metre squared — plotted against altitude, both axes logarithmic. The aerodynamic torque falls by six orders of magnitude across the range because the density does; the gravity gradient falls only as the cube of the distance from the Earth's centre, which over this range is barely a factor of three; the magnetic torque falls as the same cube; and sunlight does not change at all, which the figure checks rather than states. The leader changes at 438 kilometres, from the atmosphere to the gravity gradient, and that crossing is measured off the drawn curves. Nothing here is a disturbance in the ordinary sense: each of these acts in the same direction for a large part of every orbit, so what matters is not the size of the torque but the part of its integral that survives an orbit — and the wheel that stores that integral is emptied on the schedule these curves set.

6 essays call momentum-store. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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.

Three motions, three decades apart, and that is the point. The three periods of a trapped 1-MeV electron's motion against the shell it is trapped on, on a logarithmic time axis. It gyrates about a field line in 0.22 milliseconds, bounces between its mirror points in 0.33 seconds, and drifts right round the planet in 16 minutes — ratios of 1498 and 3015 at L = 4. Each motion carries a conserved quantity: the magnetic moment, the longitudinal invariant, and the magnetic flux the drift shell encloses. The separation is what makes them conserved. An invariant survives anything that changes slowly compared with its own period, so a disturbance lasting minutes destroys the third and leaves the first two untouched — and a particle that keeps its magnetic moment while being moved inward to a stronger field must gain energy. That is not a loophole; it is how the belts are filled.

Three clocks and nothing to fall onto

A charged particle in a dipole field gyrates, bounces and drifts, on timescales a millisecond, a second and a quarter of an hour. The three periods are three decades apart, and that separation is not a curiosity — it is the reason each motion has a conserved quantity, and the reason a magnetic storm can accelerate particles rather than merely stir them.

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.

$A spin about the middle axis of a 1:2:3 body, turning over every 2.9 turns. A body with principal moments of inertia 1, 2, 3, spun about its intermediate axis with a hundredth of its angular momentum knocked onto the axis of least inertia, integrated with no dissipation and no external torque. The curves are the components of the angular momentum along the three body axes, as fractions of its fixed size. The intermediate component stays near one for 1.5 spin periods, then swings through zero to minus one — the body turns over, end for end — and keeps doing so every 2.9 periods, 14 times in the span drawn. Energy and angular momentum are both conserved throughout, the energy to better than one part in a billion; nothing is being lost and nothing drives the flips. A spin about the intermediate axis is an equilibrium like a pencil balanced on its point, and the smallest disturbance grows exponentially, here by a factor of e every 0.28 spin periods, until it carries the body to the opposite equilibrium and back.$ Spaceflight

A wingnut that turns over on its own

Spin a rigid body about the axis whose moment of inertia is neither the largest nor the smallest and it turns end over end, again and again, with nothing pushing it and nothing lost. The flip was noticed aboard a space station in 1985 and was already implicit in equations written in 1765. How long it waits is a logarithm, and no care in setting up the spin can make the logarithm infinite.

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.

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.

A tumble removed with a coil and a compass, at 500 km. The rotation rate of a small spacecraft — principal moments 0.0067, 0.041, 0.043 kg m², the proportions of a three-unit cubesat — tumbling at 8.8° a second after release, against orbits at 500 km, with nothing to control it but magnetic coils driven by the B-dot law: a dipole opposite to the rate of change of the field measured aboard, capped at 0.2 A m². The field is a dipole tilted 9.2° from the Earth's axis and turning with the Earth. In a polar, 97.4°, orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.58 orbits; in a 51.6° orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.32 orbits; in an equatorial orbit the rate settles at 2.44° a second over the last orbit drawn. The law needs no knowledge of the spacecraft's attitude: a tumbling body sees the Earth's field swing round in its own frame, and a dipole opposing that swing produces a torque that removes the part of the spin perpendicular to the field. The polar orbit does not reach zero. It settles at 0.94 of twice the orbital rate, 0.13° a second, and twice the orbital rate is how fast the field direction itself turns round a polar orbit: a body turning with the field sees little change to oppose. An equatorial orbit keeps the field pointing nearly the same way all the way round, so the spin about it is reached only through the dipole's tilt and the Earth's turning, and 28 per cent of the starting rate is still there at the end.

A tumble stopped by the field it tumbles through

A small satellite leaves its deployer tumbling, and the first thing most of them do is stop, using nothing but a magnetometer and three coils. The law they run needs no idea where the satellite is pointing. What it cannot do, at any instant, is touch the spin about the local field line — so how much tumble survives is decided by how much the field's direction changes along the orbit, and the stillness it reaches is defined by the field rather than by the stars.

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