Astronomy & Celestial Mechanics
the sky, computed
Concept

Moment of inertia — where it appears

The mass-weighted mean square distance of a body's material from an axis, which converts a rotation rate into an angular momentum. Expressed as a fraction of MR² it says how centrally condensed the body is: 0.4 for a uniform sphere and 0.07 for the Sun.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

A clock that runs down and says what it is

A pulsar hands over two measured numbers, a period and its derivative. Everything else usually quoted about it — an age, a magnetic field, a luminosity — is derived from those two through a model, and the one measurement that tests the model refutes it.

stars · Pulsars A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.

A wind that takes no mass and all the spin

The Sun loses about a ten-thousandth of itself to its own wind over its whole life. It loses most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor — the gas is forced to keep turning with the star until it is a dozen or more radii out from the surface it left.

stars · Magnetic braking 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.

spaceflight · Attitude control Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 61.1 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.

Ninety-nine per cent of the mass and none of the spin

The Sun holds 99.87 per cent of the solar system's mass and 0.6 per cent of its angular momentum. Jupiter holds a thousandth of the mass and three-fifths of the spin. That is not a curiosity of accounting — it is the record of the single operation that had to succeed before a star could form at all.

orbits · Angular momentum Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

A neutron star born turning too slowly

Collapse an iron core a few thousand kilometres across down to twelve, and conservation of angular momentum multiplies its rotation rate by about eighteen thousand. A model with no transport in it delivers a newborn pulsar at the break-up limit; the ones that are observed turn twenty times slower, which is a measurement of the core before it fell.

stars · Supernovae A circular orbit has one lock, and an eccentric one has several. The strength of each spin–orbit resonance against orbital eccentricity, as the Hansen coefficient H(p, e) that multiplies the restoring torque on a permanently non-spherical body. At zero eccentricity every curve but the synchronous one is exactly zero — the figure checks that rather than showing it — so a body on a circular orbit can lock only by turning once per orbit. Away from zero the others switch on: at Mercury's eccentricity of 0.2056 the 3:2 resonance has 73 per cent of the synchronous one's strength and more than twice the 2:1's. A planet spinning down through this family therefore meets the 3:2 before the 1:1 and has a real chance of being caught there, which is what happened — Mercury turns three times for every two orbits, a fact discovered by radar in 1965 after a century of assuming it was locked. The free libration of that locked state follows from the same coefficient and the measured 2.03e-4 for (B − A)/C: 12.1 years, against a measured period near twelve. What the figure cannot show is the capture probability itself, which depends on how the tide dissipates and ranges from a few per cent for a simple constant-lag tide to more than half once friction between a liquid core and the mantle is included.

A rotation locked to the orbit, but not one to one

Mercury turns exactly three times for every two circuits of the Sun. That was not what anybody expected, and it is not an accident — on a circular orbit a tidally despun body has exactly one place to lock, and on an eccentric one it has several — with the strength of each set by a coefficient that vanishes when the eccentricity does.

gravitation · Resonance A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 150 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.

A wobble that should have stopped

The Earth's rotation pole wanders across its own crust in a circle a few metres wide. A rigid Earth would do it in 305 days; it takes 433, and the difference is a measurement of the planet's elasticity. At the observed damping it should have died out within a human lifetime, and it has not.

sky · Precession A hundredfold expansion is a four-order slowdown. Equatorial rotation speed against radius for three stars leaving the main sequence at 2, 10, 100 km/s, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed along each track and checked at four radii rather than asserted, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 on the main sequence to 0.02 in a centrally condensed giant envelope. A star at 100 km/s crosses the 8 km/s line — the usual boundary for calling a giant a rapid rotator — at 45.6 solar radii, and everything larger is slower. That is why the observed giants are almost all under two kilometres a second, and why the one or two per cent that are not cannot be explained by anything the star did on its own: the angular momentum has to have been delivered, by a swallowed companion or a merger. The figure assumes no mass loss, which for the largest radii drawn is the weakest of its assumptions.

A surface that slowed because the star grew

A star leaving the main sequence expands by a factor of a hundred. Conserve its angular momentum and its equator slows by four orders of magnitude, which is why every red giant is a slow rotator. One or two per cent are not, and a giant turning at fifty kilometres a second has to have been given the angular momentum by something else.

stars · Stellar evolution A formula everyone uses, and the number no pulsar has. Above: the braking index measured for the 4 pulsars whose spin-down has been followed long enough to give a second derivative, against the value a magnetic dipole rotating in vacuum requires. That value is exactly 3, and it is what every catalogued field strength and every characteristic age assumes. Not one measurement reaches it: they run from 1.4 to 2.839, and all of them fall short in the same direction, which is the signature of a systematic rather than of noise. Below: what that costs. The age a spin-down history gives is the period divided by (n − 1) times its derivative, so the ratio to the quoted characteristic age is 2/(n − 1) — 5.00 for Vela. The numbers are not thereby useless: an exponent recovered from the data is exactly the kind of correction a measurement can absorb. What has gone is the claim that the field strength printed beside a pulsar is a measurement of a field. It is a measurement of a spin-down rate, read through a model the same pulsar refutes.

The exponent no pulsar has

Every field strength in the pulsar catalogue comes from one formula, which assumes the star is a magnetic dipole rotating in a vacuum and therefore that its spin-down obeys an exponent of exactly three. Where that exponent has been measured it is 2.51, 2.84, 1.4 — never three, and always short.

stars · Pulsars 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.

spaceflight · Attitude control 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.

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.

spaceflight · Attitude control 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.

spaceflight · Attitude control 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.

spaceflight · Attitude control The ceiling a measured response has to be read against. The fluid Love number of a layered body — the value it would have if its outer shell offered no resistance whatever — against the radius of its core, for cores 1×, 1.6×, 2.4×, 3.6× the density of the material outside. Every curve starts at 3/2, because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, since the tidal forcing is strongest out where there is then very little mass to move. A body 0.6 of the way core at 3.6× density has a fluid ceiling of 0.787. A measurement above the relevant curve is impossible for any interior of that layering; a measurement below it says only that something is resisting, and does not say what.

An ocean is detected and its depth is not

A tidal response thirty-six times too large for any solid body proves a moon has a liquid layer under its crust. It does not say how deep the liquid is, how thick the crust above it is, or what the core beneath it is made of — because the same one number is produced by a whole surface of interiors.

gravitation · Love numbers

Named alongside it

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

Angular momentumAttitude controlAngular momentum transportBraking indexCharacteristic ageMagnetic brakingNeutron starPrincipal axesSeparatrixSpecific angular momentumAsteroseismologyChandler wobble

All concepts