Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 34

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence. Essays 661–680 of 909.

Galaxies

Where the unknown stops being a number and becomes a profile — and most of it is not light.

The black holes that can be weighed are the heavy ones. 4000 synthetic galaxies spread evenly through a sphere of 120 megaparsecs, with black-hole masses scattered by 0.45 dex about an M–σ relation of slope 4.24 (faint points), and the 511 whose spheres of influence exceed 0.1 arcseconds (filled), which are the ones a dynamical measurement can resolve. At a given dispersion a heavier black hole has a larger sphere and is resolvable further away, in a larger volume, so the resolved sample is dominated by holes above the relation. Fitted to the resolved galaxies alone, the relation comes out with normalisation 8.74 and slope 2.95 (dashed), against 8.11 and 4.20 for all of them (solid) — 0.63 dex too high at 200 km/s, and more at low dispersions.

The black holes that can be weighed are the heavy ones

A black hole's mass is measured from the motions of stars or gas only inside its sphere of influence, the region where its gravity beats the galaxy's, and that region has to be resolved. At a given galaxy, a heavier hole rules a larger sphere and can be weighed further away, in a much larger volume — so the sample of weighed holes is drawn from the top of the distribution, and the relations built from it describe the heavy end of every galaxy rather than the typical one.

5 figures · Galactic nuclei Mass profiles that pinch together near the half-light radius. The enclosed mass of each of the 13 haloes that fit the same flat dispersion of 10 km/s, against radius in units of the stars' three-dimensional half-light radius, 391 parsecs. At a tenth of that radius they span a factor of 17, and at four times it a factor of 2.5; they pinch together near the half-light radius, closest at 1.47 of it, where they agree to 14 per cent. The dot is the estimate 3σ²r/G at the radius where the stars' density falls as r⁻³, 0.94 of the half-light radius; there the fitted masses lie between 0.65 and 1.03 times it. Each model trades a wrong guess about the orbits against a wrong guess about the mass, in opposite senses inside and outside.

The one radius where the orbits stop mattering

A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted.

5 figures · Velocity dispersion

Gravitation

Two bodies pulling on each other, and everything that goes wrong at three.

The tide's spectrum, with every frequency set by the sky. The equilibrium tide-raising potential broken into its largest constituents, each a sinusoid whose frequency is a sum of whole-number multiples of six astronomical rates — the Earth's turning relative to the Moon, the Moon's and the Sun's motions round the sky, and the slow turning of the lunar perigee and node — and whose height is its equilibrium amplitude relative to M₂. They fall in three bands: near two cycles a day (M₂ at 1.9323, S₂ at exactly 2, N₂, K₂), near one (K₁, O₁, P₁, Q₁), and near zero (fortnightly, monthly, semi-annual). Every frequency is known to many decimal places from astronomy; what a harbour does to each, its own amplitude and phase, is not, and is the only thing a tide table has to measure.

A tide table is astronomy with the harbour fitted

Nobody can compute the tide at a harbour from the Moon and the Sun, because the ocean's response depends on coastlines and depths nobody models well enough. Tides are predicted to centimetres years ahead anyway, because the frequencies are pure astronomy and only the response to each one has to be measured. The method needs a record long enough to tell neighbouring frequencies apart, and a correction for a cycle of 18.6 years.

5 figures · Tides Gravity per kilometre of topography, by wavelength. The free-air gravity anomaly per kilometre of topography — the admittance — on Earth, against the wavelength of the topography, for elastic plates 5, 25, 100 km thick, with crust of density 2800 kg/m³ over mantle of 3300 and compensation at 35 km. Short wavelengths are uncompensated and give the full attraction of a slab of crust, 117 mGal per km. Long wavelengths are compensated and give nearly nothing, because the mass of the mountain is cancelled by the root beneath it. Where the curve falls between the two is set by the plate's thickness, so a measured admittance spectrum is a measurement of how strong the lithosphere was when the load was applied.

A mountain's gravity says what holds it up

A mountain can stand because the rock beneath it is strong, or because it floats on a deep root of light crust like an iceberg. The two leave different gravity behind them, and the difference depends on the mountain's size — so the ratio of gravity to topography, measured wavelength by wavelength, is a measurement of how stiff a planet's outer shell was when the mountain was built.

6 figures · Moment of inertia A slow shock still heats the cluster. The energy a Plummer star cluster gains from a compressive tidal pulse, as a fraction of what the same impulse delivered instantaneously would give, against the pulse's duration times the orbital frequency at the half-mass radius. The points come from integrating 1200 stars through the pulse one by one. The lines are prescriptions applied to the same stars: an oscillator's exponential suppression and a power law, each applied star by star at the star's own orbital frequency, and the shortcut of one power-law factor for the whole cluster, evaluated at the half-mass radius. At the longest pulse drawn the cluster keeps 0.28 of the impulsive energy; the star-by-star rules give 0.24 and 0.36, bracketing it, and the half-mass shortcut gives 0.017, 17 times too little. The total is set by the outermost stars, which carry most of the impulsive energy and are always slower than the shock.

A slow shock heats the halo and robs the core

The textbook treatment of a slow tidal shock multiplies the instantaneous answer by a suppression factor that falls steeply as the shock lengthens. Integrated star by star through shocks of intermediate length, a cluster behaves differently. Its total heating falls slowly, because the outer stars are always slower than any shock and carry most of the energy, and its core does not merely escape the heating: on balance its stars lose energy, which no factor that multiplies a positive number can describe.

5 figures · Impulse approximation

Orbits

Kepler's three laws, and the family of curves a single force allows.

A fall into the Sun from a standing start. The distance of a body released at rest 1 AU from the Sun, as a fraction of its starting distance, against time as a fraction of the fall's total duration, 64.6 days: the exact radial orbit (solid) and a fall at the constant acceleration felt at the start (dashed). The exact fall begins exactly like the constant one and finishes far faster, because the pull grows as the inverse square of the shrinking distance: it reaches the Sun at 79 per cent of the time the constant pull would take. The curve is a cycloid, and the total time is a fifth of the circular orbital period at the starting distance — exactly 1/(4√2) of it — because a straight fall is half of an ellipse whose semi-major axis is half the starting distance.

A fall is half an orbit with no width

Stop a planet dead and it falls into the Sun, in a time that is not a quarter of its year or a half but exactly one over four times the square root of two. The fall is an ellipse squeezed to a line, it spends four-fifths of its duration in the outer half of its path, and in a uniform cloud every shell of it arrives at the centre at the same moment — which is where the free-fall time of star formation comes from.

6 figures · Conic sections One detection, and every density it could mean. The number density of interstellar objects implied by one detection of a given size in 8 years, against the distance from the Sun inside which a survey could have seen it, for objects arriving at 26.3 km/s, if the survey caught 100 per cent, 10 per cent, 1 per cent of those passing inside that distance. Each estimate is the count divided by the rate of passages per unit density, πq²v∞ with gravitational focusing, times the efficiency. If ʻOumuamua-sized bodies could be seen inside 0.5 AU at 10 per cent efficiency, there are about 0.05 per cubic AU; if inside 2 AU with every one caught, 7.9 × 10^−4. The inferred density is a statement about the survey as much as about the visitors.

A census of visitors is a census of the survey

Three bodies from other stars have passed through the inner solar system in eight years of looking. Turning that count into a number of such bodies per cubic astronomical unit needs the volume in which they could have been seen, and that volume is set by the Sun's gravity, by each body's size and by how many weeks a survey had to catch it — so the first number the census gives is a measurement of the survey.

5 figures · Hyperbolic orbits

The observed sky

Coordinates, seasons, phases and shadows — geometry seen from inside it.

The Sun bends light from every direction in the sky. The gravitational deflection of starlight by the Sun, as seen from the Earth, against the angle between the star and the Sun, as general relativity predicts it (solid) and as a Newtonian particle of light would be bent (dashed), exactly half. At the Sun's limb the deflection is 1.75 arcseconds, the value measured at the eclipse of 1919. It falls only as the cotangent of half the angle, so it is 4.07 milliarcseconds at right angles to the Sun and 9.8 at 45°, the closest a space astrometry mission looks; it reaches zero only directly opposite the Sun. Every star in a modern catalogue has been displaced by an amount far larger than the precision of its position, and the shading marks the angles such a mission actually observes.

Every star in the catalogue is bent by the Sun

The bending of starlight by the Sun is remembered as an eclipse measurement of 1.75 arcseconds at the solar limb. The same bending reaches every direction in the sky, falling only slowly away from the Sun, and at right angles to it is still four milliarcseconds — two hundred times the precision of a modern star position. Every catalogue corrects every star for it, and the correction has become the most precise test of how much space a mass curves.

5 figures · Aberration The edge of the Sun, computed. The brightness of the Sun at 500 nanometres across its limb, as a fraction of the brightness at the centre of the disc, against height above the level where the photosphere's radial optical depth is one, from the radiative transfer along each line of sight through a grey model atmosphere. Inside the disc the brightness falls gently with limb darkening; at the edge it falls to nothing over about two hundred kilometres. The steepest point of the fall — the inflection point, which is what a solar radius conventionally means — is 335 km above the radial τ = 1 level, because a ray grazing the limb passes through 270 times more atmosphere than a vertical one and becomes opaque that much higher up, at about 336 km. The brightness falls to one per cent 555 km above the τ = 1 level and to a thousandth at 694.

The Sun is three hundred kilometres larger seen edge-on

The radius of the Sun measured from images and eclipses and the radius measured from its oscillations disagree by about three hundred kilometres, and neither is wrong. A line of sight grazing the limb passes through hundreds of times more atmosphere than a vertical one and becomes opaque higher up — so the edge a telescope sees sits above the surface the waves feel, and every other definition of the edge, from the detector's threshold to the colour of the light, moves it again.

5 figures · Phases and eclipses

Spaceflight

Celestial mechanics used forwards — where to burn, and what it costs.

The plane change wants a gentle flyby. The lowest inclination to the Moon's orbit that one flyby can leave a spacecraft in, against the flyby's height above the Moon, for a spacecraft arriving from an orbit of perigee 6678 km inclined 51.6°: with no constraint on where the new orbit goes (dashed), and requiring the new perigee to lie between the atmosphere and 20,000 km, so that the spacecraft comes back to the Earth (solid). Unconstrained, any flyby closer than tens of thousands of kilometres can bring the inclination to nearly zero. But a close flyby turns the velocity so far that it also flings the spacecraft outward, and no aim point brings it back near the Earth; only from about 9,500 km up does a returning orbit become possible. Between that height and about 18,000 km the flyby can both bring the spacecraft back and leave its orbit nearly in the Moon's plane; further out still, the turn is too small to remove the inclination. The useful flyby is a gentle one, in a window.

A plane change borrowed from the Moon

Turning an orbit's plane by the engine costs a fixed fraction of the spacecraft's speed, which is why it is done as far from the Earth as possible. Farther still, at the Moon's distance, a spacecraft is moving so slowly that the Moon can turn its velocity through tens of degrees for nothing — but a flyby close enough to turn it hard also flings it outward, and the useful flyby is a gentle one, in a window of heights the arithmetic picks out.

5 figures · Plane change Two ways out of a transfer orbit. The eccentricity against the semi-major axis of a spacecraft spiralling out from a geostationary transfer orbit — perigee 300 km up, apogee at geostationary height — under a continuous thrust of 0.2 millimetres a second squared, integrated through hundreds of revolutions, for two steering laws. Thrust always along the velocity (red) raises the orbit efficiently but leaves its shape almost unchanged, reaching the geostationary semi-major axis after 64 days still with an eccentricity of 0.59 — an orbit swinging between 17,000 and 67,000 km. A law that points the thrust to reduce a weighted distance from the target in both elements (blue) circularises as it raises, and reaches a circular geostationary orbit in 129 days.

The thrust that rounds an orbit turns once a revolution

An electric engine pushes gently for months, and the obvious way to point it — along the velocity — raises an orbit as fast as possible and leaves its shape almost untouched. Starting from an elongated transfer orbit, that is the wrong job. Rounding the orbit off needs the thrust forward at apogee, backward at perigee and sideways in between, a full turn of the thrust vector every revolution, and the law that does it can be written down from the equations for how each element changes.

5 figures · Low-thrust transfer

Starlight

Distance, brightness and colour: what can be measured when nothing can be visited.

The Sun's motion, written across a group of stars. Proper motions of 400 stars all at 300 pc, against their angle from the solar apex, each with its own random velocity of 25 km/s in every direction. Filled dots are the component along the great circle away from the apex, υ; open dots are the component at right angles to it, τ. One star's υ is dominated by its own motion — the scatter is 17.6 mas/yr against a reflex of at most 13.7 — so no single star gives a distance. Together the υ components follow the Sun's reflex, 19.5 km/s times sin λ, and the fitted amplitude, 13.70 ± 1.06 mas/yr, gives a distance of 300 pc ± 8 per cent against the true 300 pc. The τ components scatter about zero at every angle, because the Sun's motion has no part across the great circle.

A baseline that grows four astronomical units a year

The Earth's orbit gives a parallax a baseline of two astronomical units and never more. The Sun's own motion through the nearby stars gives one that lengthens by four a year without limit — but the stars move too, by as much, so the long baseline yields a distance only for a group whose motions average away, and for a century it was the only way to reach past a few hundred parsecs.

6 figures · Parallax

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

Brighter by mass, fainter by temperature. Main sequences from 0.5 to 1.3 solar masses in luminosity against effective temperature, hotter to the left, for solar composition, [Fe/H] = −1.5, from homology applied to the solar-composition relation. The dots join stars of the same mass: lowering the metals moves each one up and to the left, so a 0.8 solar-mass star goes from 0.41 to 0.66 solar luminosities and from 5049 to 5879 K. The main sequence is steep, so a move of that size to the left carries a star past solar stars much brighter than itself: at a fixed temperature of 5800 K the metal-poor sequences lie 0.59 magnitudes below the solar one. A metal-poor star is brighter than a solar one of the same mass and fainter than a solar one of the same temperature — which is why such stars were called subdwarfs.

Brighter for its mass and fainter for its colour

Two stars of the same mass are almost the same star only if they are made of the same material. Take most of the metals out and the opacity falls, the radiation leaks out faster and the star settles brighter and much hotter — so far hotter that, at a given colour, it sits below the ordinary main sequence. The relation between mass and light has composition inside it, and every mass, age and distance read off it inherits that.

6 figures · The mass–luminosity relation Two critical mass ratios that move with where the mass goes. The logarithmic response of the donor's Roche lobe to its own loss of mass, ζ, against the mass ratio donor over accretor, for conservative transfer (dashed) and for all of the transferred mass lost from the accretor, from the donor and to a ring. The horizontal lines are two donors' own responses: −1/3 for a deep convective envelope, which swells as it is stripped, and +2 for a radiative one, which shrinks. Transfer is stable where the donor's line lies above the lobe's curve, so each crossing is a critical mass ratio — conservatively 0.63 and 1.72. Re-emission from the accretor moves them to 0.96 and 2.25, a wind from the donor to 1.24 and beyond the range drawn; and with everything fed to a ring the convective donor is unstable at every mass ratio drawn and the radiative one only below 0.19. Nothing about the donor has changed between the curves; only the question of where a gram that is not accreted leaves from.

Mass that leaves takes the orbit's spin with it

Most of the gas a star pours onto its companion never stays there. Where it leaves from decides how much angular momentum it carries, and that one bookkeeping entry moves the largest donor that can transfer stably by a factor of two — without anything about either star having changed.

6 figures · Mass transfer A period set by the donor's density, running in two directions. The orbital period at which a donor exactly fills its Roche lobe, against the donor's mass, onto a 0.7 M☉ white dwarf. A lobe-filling star's period is fixed by its mean density alone, so the period follows the donor's mass–radius relation. A main-sequence donor shrinks as it is stripped and grows denser, and the period falls: from 6.0 hours at 0.6 M☉ to 1.5 hours at the hydrogen-burning limit. A white-dwarf donor swells and grows less dense, and the period rises — 3.5 minutes at 0.2 M☉, 13 at 0.05, 47 at 0.01. The binaries with helium-rich spectra and periods between five minutes and an hour lie along the second curve and nowhere near the first, and their period is a mass gauge for a star that cannot be seen.

A donor that grows as it is eaten

A white dwarf that loses mass gets bigger, so a binary in which one white dwarf feeds another runs backwards — its period lengthens as the donor shrinks. Whether the pair survives first contact at all depends less on the donor than on whether the stream finds room to form a disc, and for most pairs it does not.

5 figures · Mass transfer Orbital energy spent against the envelope it has to lift. The orbital energy a 0.6 M☉ companion releases by spiralling in from the onset of a common envelope down to a final separation, against that separation (curve), and the binding energy of the 0.85 M☉ envelope of a 1.2 M☉ giant with a 0.35 M☉ core, divided by the efficiency, for αλ of 0.25, 0.5, 1 (horizontal lines). Energies are in units of GM☉²/R☉. The giant fills its lobe at a separation of 118 R☉, and the companion releases nothing net until the orbit has shrunk below 34 R☉, because the envelope's mass leaves with the envelope; from there it falls until the energy released pays for the envelope, so each crossing is a final orbit — 1.29, 2.48, 4.63 R☉. The released energy grows only as one over the separation, so halving the efficiency halves the final orbit: a factor of four in αλ is a factor of four in the outcome, and there is no measurement of αλ that does not run through outcomes like these.

The least known number in binary evolution

When a giant swallows its companion, the companion spirals inward until the orbital energy it has released is enough to throw the envelope off. The final orbit is set by one product of two unknowns, it moves in proportion to that product, and the merger time of the white dwarfs it leaves moves as its fourth power.

6 figures · Mass transfer

Gravitation

Two bodies pulling on each other, and everything that goes wrong at three.

How far a turbulent disc bends the field it drags. The tangent of the angle by which an accretion flow bends a vertical magnetic field at the disc's surface, against the magnetic Prandtl number Pm — the ratio of the turbulent viscosity that drives the inflow to the turbulent resistivity that lets the field slip — for discs of aspect ratio 0.03, 0.05, 0.1. In a steady state the inflow drags the field in at exactly the rate it diffuses back, and that balance sets tan i = (3/2) Pm H/R. The dashed line is 30° from the vertical, the inclination a field needs before gas is flung off along it. Turbulence whose own Prandtl number is of order one bends the field by about the disc's aspect ratio, a few degrees; reaching 30° needs Pm of 13, 8, 4. The same turbulence that carries the gas in lets the field out almost as fast, because both are the same eddies.

A field the turbulence will not let in

A disc that accretes through its own turbulence drags its magnetic field inward, and the same eddies let the field slip back out almost as fast. A disc of ordinary thickness bends its field by a few degrees instead of the thirty a wind needs, so a turbulent disc cannot gather the flux that would let it stop being turbulent.

6 figures · Magnetorotational instability A star fed in bursts by a disc that cannot pass its supply on. The accretion rate onto a 0.5 M☉ protostar over 100 thousand years, for a steady supply of 10⁻⁶ M☉ a year into a disc whose active layer can carry only 10⁻⁸ M☉ a year at 1 AU. The excess fills the dead zone until the column inside 1 AU reaches 4.4·10⁴ grams per square centimetre — 0.016 M☉, the column at which it becomes gravitationally unstable and its spiral arms heat it to ignition — and then drains on the hot disc's viscous time of 189 years, at 8.3·10⁻⁵ M☉ a year. The cycle repeats every 16,083 years, 6 times in the span drawn. The star spends 1.2 per cent of its time in outburst and gains 99 per cent of its mass there: a census at a random moment finds almost every protostar faint, while almost all of the mass arrives during events nobody happens to be watching.

A disc that stores its inflow and spends it in a century

Where only a thin ionised skin of a young disc can transport angular momentum, the gas arriving from the cloud cannot be passed on, and it piles up in the dead midplane until gravity stirs it and the whole column ignites. The star then gains most of its mass in bursts of a century every ten thousand years — which is why almost every protostar ever observed is fainter than its growth requires.

5 figures · Magnetorotational instability

Orbits

Kepler's three laws, and the family of curves a single force allows.

The plane a moon's orbit turns about, between equator and ecliptic. The inclination of the Laplace plane — the plane about which a satellite's orbit precesses — to the Earth's equator, against distance from the Earth in Earth radii, for an obliquity of 23.44°. Close in, the Earth's equatorial bulge dominates and the plane is the equator; far out, the Sun dominates and it is the ecliptic, 23.44° away. The dashed curve uses today's bulge, which makes the transition at 8.4 Earth radii. The solid curve lets the bulge grow with the faster spin the Earth had when the Moon was closer — a 5.3-hour day at 5 Earth radii, from the system's angular momentum alone — which moves the transition out to about 14. The Moon today, at 60.3 Earth radii, has a Laplace plane within 0.0022° of the ecliptic, so its 5.15° inclination is entirely free: nothing forces it.

A tilt the Moon should not have

A moon's orbit precesses about a plane that sits on its planet's equator close in and on the planet's own orbit far out. A Moon born in the Earth's equator and carried outward by the tides should have followed that plane all the way to the ecliptic, because the angle it keeps to it is an adiabatic invariant. It is inclined by five degrees instead, and slow recession cannot have put them there.

6 figures · Angular momentum Four numbers that carry a position and a velocity round an orbit. The Lagrange coefficients f, g, ḟ and ġ — g and ḟ scaled by the mean motion so that all four are pure numbers — against time as a fraction of the period, for an ellipse of eccentricity 0.6 and periapsis 0.5 AU, each computed from one universal-variable solve starting at periapsis. The new position is f times the old position plus g times the old velocity, and the new velocity ḟ times the old position plus ġ times the old velocity: the state at any time is a linear combination of the state at the start, with no semi-major axis, eccentricity or anomaly formed. f and ġ return to one and g and ḟ to zero after a full period, which is the orbit closing. At every one of the 401 instants the combination f ġ − ḟ g equals one to 6·10⁻¹⁵, which is the conservation of angular momentum written as the area of a parallelogram.

Four numbers that carry a state without an orbit

Once the universal variable is solved, a body's position and velocity at any later time are a linear combination of its position and velocity now, with four coefficients and no orbital elements formed at all. The four obey one identity exactly — the conservation of angular momentum, written as an area — and its failure measures the error a calculation has made without knowing the answer.

5 figures · Universal variables

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