Astronomy & Celestial Mechanics
the sky, computed
Depth

Series — page 6

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
Three bodies, integrated. Three equal masses integrated forward under mutual gravity: the figure-eight choreography — all three bodies on one closed curve. Every point is a step of the equations of motion, and the total energy is conserved to 2.5e-11 across the run.

The three-body problem

  1. 1 Three bodies, and what "no solution" actually means
  2. 2 The curve that says where a body cannot go
  3. 3 A third body kept at a safe distance
  4. 4 The lightest body leaves
  5. 5 A periodic orbit that the sky cannot keep
5 essays · gravitation Two lengths that cross at 1.1·10⁸ solar masses, above which nothing is seen. The radius at which a star of 1 solar radius and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 1.14·10⁸ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.

Tidal disruption

  1. 1 A flare that puts a ceiling on a mass
  2. 2 The line a star is swallowed at is not the horizon
  3. 3 The debris that returns fastest lights up last
  4. 4 The stars a black hole eats come from a narrow band
  5. 5 A flare rate a hole cannot afford for long
5 essays · galaxies A torque that stops when the patch lets go. Left, the mechanism: a protogalactic patch drawn as an ellipsoid of axis ratios 1:0.72:0.5, with the principal axes of the surrounding tidal field drawn across it at 30 degrees to its own. The torque is proportional to the difference of the patch's principal moments times the sine of twice that angle, so it is exactly zero when the two sets of axes agree — checked at both alignments — and largest at forty-five degrees. A spherical patch takes no torque whatever the field around it does, which is why the spin of every galaxy begins as a statement about its shape. Right, the angular momentum against time in units of the turnaround time: in linear theory the torque acts on a patch still expanding with the universe and the angular momentum grows as the first power of time, measured off the drawn curve as t^1.000. At turnaround the patch detaches from the expansion, its quadrupole shrinks, and the torque switches off — so a galaxy's spin is fixed before it has collapsed at all, by neighbours it will never interact with again. What the figure cannot show is the sign: the same mechanism gives no preferred direction, and the observed near-absence of alignment between neighbouring galaxies' spins is the check on that.

Tidal torque theory

  1. 1 Spin acquired before there was anything to spin
  2. 2 The shape a galaxy was given before it was lensed
  3. 3 A spin that remembers the field and not its sign
  4. 4 An alignment a disc shows only when it is tipped
  5. 5 A conversion measured without an alignment
5 essays · cosmology The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.

Tides

  1. 1 The tide is a difference, which is why there are two of them
  2. 2 The distance at which a moon stops holding together
  3. 3 A day five hours long
  4. 4 A tide table is astronomy with the harbour fitted
  5. 5 The tide that holds up the deep ocean
5 essays · gravitation Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.

The two-body problem

  1. 1 Neither body is still, and the wobble is how planets are found
  2. 2 Two bodies replaced by one that does not exist
  3. 3 A floor under the unseen companion
  4. 4 A black hole weighed by a wobble on the sky
  5. 5 A gap that counting can test and impostors can fill
5 essays · gravitation Leavitt's law, through the Milky Way calibrators. Mean absolute magnitude against the logarithm of the pulsation period, for classical Cepheids with independently known distances, and RR Lyrae itself for comparison. The line is M = −2.81 log P − 1.43: a tenfold longer period is 2.81 magnitudes brighter, a factor of 13 in luminosity. The vertical axis runs the astronomers' way, with brighter upward.

Variable stars

  1. 1 A star that tells its distance by how slowly it blinks
  2. 2 The valve that has to sit at the right depth
  3. 3 A distance that rests on one unmeasured number
  4. 4 A scatter that points the way dust does
  5. 5 A metallicity term too small to close a gap
5 essays · stars The energy budget of an orbit at e = 0.7. Kinetic, potential and total energy per unit mass against distance from the primary, in units where GM = 1. The total is a horizontal line — it depends only on the semi-major axis — and the vis-viva relation is that statement solved for the speed.

Vis-viva

  1. 1 One equation for the speed anywhere, and the eccentricity is not in it
  2. 2 The same burn is worth more when moving fast
  3. 3 Every fragment comes back through the same point
  4. 4 A kick speed read off the slope of a histogram
  5. 5 A collision moves the cloud's centre, and an explosion cannot
5 essays · orbits The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round.

Accretion

  1. 1 The disc that has to throw angular momentum away
  2. 2 A spectrum that is a stack of temperatures
  3. 3 The wake and the meal are one calculation
  4. 4 The period a star is pulled towards
4 essays · stars Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.

Angular momentum

  1. 1 Equal areas in equal times, which is angular momentum in disguise
  2. 2 Ninety-nine per cent of the mass and none of the spin
  3. 3 A tilt the Moon should not have
  4. 4 A tilt built from a thousand near misses
4 essays · orbits A 128.8-million-year-old collision, dated from the shape of a scatter plot. The Erigone family: 165 members drawn at their diameters and their proper semi-major axes, with inverse diameter up the page. The cloud is a V, and the V is a clock. Each member has been drifting in semi-major axis ever since the collision at a rate that goes as one over its diameter, with a sign set by which way it spins — prograde outward, retrograde inward — so after 130 million years the small members have moved far and the large ones have barely moved at all. Plotted against 1/D that envelope is a straight line through the family's centre, and its slope is the drift rate for a one-kilometre body multiplied by the elapsed time. Fitting the two edges of the points actually drawn here returns 128.85 million years against the 130 the members were generated from. The rounding at the bottom is not an artefact: it is the ejection velocity, some 15 metres per second, which every member got at the moment of the collision and which is the same for all sizes. The picture cannot show the interlopers — background asteroids that happen to lie inside the V and have nothing to do with the family — and it cannot show the members that have drifted into a resonance and left the belt entirely, which is the reason the oldest families have the softest edges.

Asteroid families

  1. 1 A collision dated by a scatter plot
  2. 2 A family whose size is a choice
  3. 3 An age read from angles that meet
  4. 4 A family checked by a colour it was not chosen by
4 essays · orbits A one-parameter model, and 9 times too many metal-poor stars. The metallicity distribution of long-lived stars near the Sun — the bars — against two models of how a galaxy enriches itself. The closed box is as simple as a model of a galaxy can be: gas turns into stars, stars make metals and return them, nothing enters and nothing leaves. It has exactly one parameter, the yield, and it predicts the whole curve. It gets the peak roughly right and the tail catastrophically wrong: 13.7 per cent of its stars fall below [Fe/H] = -1 against an observed 1.6 per cent, a factor of 9. This is the G-dwarf problem, and the reason it is an argument rather than a discrepancy is that the failure cannot be fixed by changing the yield: the yield sets where the peak is, and moving the peak to fix the tail moves it away from the data. What is wrong is a boundary condition. Let gas keep arriving — pristine, at roughly the rate it is being consumed — and the gas is never both abundant and metal-poor for long, so few stars form while it is. That is the second curve, with 3.2 per cent below -1, and it needed no new nucleosynthesis and no new parameter beyond the fact of accretion. The picture cannot show the thing that would settle it directly, which is the infall itself: the gas arriving on the disc now is a few solar masses a year spread over twenty kiloparsecs, and it has never been securely detected.

Chemical evolution

  1. 1 A histogram that says the box was not closed
  2. 2 The metals a galaxy keeps measure what it threw away
  3. 3 A gradient the old stars have walked away from
  4. 4 The iron clock has no single delay
4 essays · galaxies A period below which every orbit is round. Orbital eccentricity against period for binaries in four clusters of 0.125, 0.625, 6, 4 billion years, with the eccentricities drawn from one seeded distribution and then damped by exp(−age/τ), where τ rises as the 5.333 power of the period. Each cluster shows the same thing: below a boundary period nothing survives eccentric, above it the original distribution is untouched, and there is almost nothing in between because the timescale is so steep. The boundary is a clock. It moves as the three-sixteenths power of the age, which the figure checks against the drawn curves, and the calibration puts it at 6.5 days at 125 million years, 8.8 at 625 million and 12.5 at four billion — against measured cut-offs near 7.2, 8.5 and 12.5 days in the Pleiades, the Hyades and M67. The boundaries are also read back off the plotted points rather than trusted, and required to move outward with age. This is the cleanest measurement of tidal dissipation in ordinary stars that exists, and its cleanliness comes from the ages: a cluster's age is read off its main-sequence turn-off and owes nothing whatever to the tide being measured.

Circularisation

  1. 1 A cut-off period that is an age
  2. 2 A wall measures a ratio, and a ratio is a line
  3. 3 Two damping times, one crossing, and the slope that separates them
  4. 4 A wall that leans towards the body that damps
4 essays · orbits A cluster weighed three ways, and its stars weighed once. A cluster of galaxies with a velocity dispersion of 1000 km/s, gas at 8 keV and a strong-lensing Einstein radius of 25 arcseconds, each turned into a mass inside 1.5 Mpc by its own relation and nothing else: 7.0, 8.9 and 15.2 × 10¹⁴ M☉. The three assume, respectively, that the galaxies are in equilibrium, that the gas is, and nothing whatever — so their agreement to within a factor of 2.2 is not three restatements of one assumption. The lensing bar is the loosest of the three and is drawn that way deliberately: it measures the mass inside a cylinder of radius 109 kpc, 1.11×10¹⁴ M☉, and carrying that out to 1.5 Mpc as though it were a sphere overstates it. The stars are 2.9 per cent of it.

Clusters

  1. 1 A cluster weighed three ways
  2. 2 The baryons that are not in the galaxies
  3. 3 Red, gas-poor, and still spiral-shaped
  4. 4 A cooling flow that never arrives
4 essays · galaxies Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.

Conic sections

  1. 1 Every orbit one force allows, and the number that picks between them
  2. 2 The third thing that is conserved
  3. 3 A fall is half an orbit with no width
  4. 4 An orbit seen at an angle keeps everything but its focus
4 essays · orbits The energy at which the sky begins to point. Gyroradius against energy for a singly charged particle, in three field strengths, with the thickness of the galactic disc and the size of its halo marked. A cosmic ray is not an image of anything: its path is a helix about a field line, and by the time it arrives the direction it came from has been erased. The erasure is quantitative. At 3 µG a proton's gyroradius equals the disc's half-thickness at 4.2·10¹⁷ electronvolts and the halo's radius at 4.2·10¹⁹, so below the first the particle is stored and stirred for tens of millions of years, and only above the second does it travel in something like a straight line. The measured spectrum has a break — the knee — at about 3×10¹⁵ eV, which is within a factor of a few of the first of those; the sky only starts showing structure above 10¹⁹, which is the second. Two features of a spectrum measured on the ground, both located by one straight line on this plot.

Cosmic rays

  1. 1 The energy at which a sky begins to point
  2. 2 A horizon drawn by light at three kelvin
  3. 3 One slope read twice, and the reading that disagrees
  4. 4 A positron that cannot have come far
4 essays · galaxies Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.

Degeneracy

  1. 1 A star held up by a rule about counting
  2. 2 The mass a cold star cannot exceed
  3. 3 A radius that decides what matter can be
  4. 4 The pulse that shows the back of the star
4 essays · stars The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.

Distance ladder

  1. 1 Every distance is measured with the last one
  2. 2 Four distances to the same galaxy
  3. 3 A sample brighter than the population it came from
  4. 4 Two distances that must agree if light is conserved
4 essays · starlight A field strength read off the scatter of directions. The Davis–Chandrasekhar–Fermi estimate: field strength against the dispersion of polarisation angles across a cloud, at a density of 10⁴ molecules per cubic centimetre and a turbulent velocity of 0.9 kilometres a second. The polarisation directions themselves contain no field strength — an aligned grain reports which way the field points and nothing about how hard it is pulling. The strength is in the disorder. Turbulence of a given energy bends a stiff field less than a weak one, so the angular scatter is the ratio of the turbulent velocity to the Alfvén speed, and inverting it gives the field: 220 µG at a 9-degree dispersion. The relation is exactly inverse, so the estimate is most trustworthy where the field is most ordered and worst where it matters most. The factor of 0.5 in front is not theory — it is what simulations of a known field, analysed this way, turn out to need, and the honest statement is that this method is calibrated rather than derived.

Dust polarisation

  1. 1 A direction measured by something with no strength in it
  2. 2 A foreground drawn by the field it traces
  3. 3 The grains that glow and the grains that align
  4. 4 A falling polarisation and the angle that says why
4 essays · galaxies The wake, computed from the streamlines that make it. Left: 26 streamlines past a point mass, in the mass's own frame, each integrated from far upstream with the same speed and a different impact parameter, and mirrored about the axis. Nothing is drawn to converge — every track is the hyperbola its own impact parameter gives it, and they cross downstream because an attraction focuses. Right: the density that focusing produces at 3.2 focusing radii behind the mass, as (b/y)(db/dy) — the Jacobian of the map from starting radius to arrival radius — which peaks at 15.53 times the background at 0.03 radii off the axis. The overdensity is behind the mass, and that is the entire mechanism: the wake pulls backwards on the body that made it. What the figure cannot show is the steady state, because it has no time in it: a real wake is continuously replenished, and the drag is the sum over an infinite train of these encounters, which is where Chandrasekhar's logarithm comes from.

Dynamical friction

  1. 1 A drag with nothing to drag against
  2. 2 A drag that is strongest in the middle
  3. 3 A drag computed with a logarithm nobody can pin down
  4. 4 A drag that stops where every orbit keeps time
4 essays · gravitation The main sequence against its own ceiling. Luminosity against mass, in solar units, with the Eddington limit drawn on the same axes. The limit is exactly linear in mass. The main sequence is not — until it is: above about 55 solar masses the relation flattens to a straight proportionality as well, and the two lines then run parallel at a fixed ratio of 83 per cent. They never cross on this range, which is the more interesting outcome: the most massive stars do not approach their limit gradually, they are built at a fixed fraction of it. The Sun, by contrast, radiates 0.0026 per cent of its own ceiling.

Eddington limit

  1. 1 A brightness that would blow the star apart
  2. 2 A ceiling on the light and not on the mass
  3. 3 A limit written for free electrons
  4. 4 A nebula that cannot be beamed at
4 essays · stars

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