Astronomy & Celestial Mechanics
the sky, computed
Depth

Series — page 4

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.
A bump at a hundred megaparsecs. The two-point correlation function of galaxies, multiplied by the square of the separation so that the interesting part is not buried under the power law. The smooth dashed curve is the broad-band clustering — the ordinary fact that galaxies are near other galaxies, which carries no cosmological information and is treated as a nuisance term in the real analysis. The bump on top of it is the whole measurement, and its position here is not fitted: it is the comoving sound horizon at the drag epoch, integrated in this file from ∫c_s da/a²H with c_s = c/√(3(1+R)), which comes out at 146.9 Mpc — 99.0 in the h⁻¹ Mpc a survey works in. The excess says that a galaxy is very slightly more likely to have a companion at that separation than at 90 or 115, by about one part in two hundred, and the reason is that a pressure wave in the photon–baryon fluid ran outward from every overdensity for four hundred thousand years and stopped where it was when the photons let go. The points are drawn with the error a survey of a million galaxies achieves; a single pair of galaxies at 100 Mpc means nothing, which is why the measurement waited for the surveys.

Baryon acoustic oscillations

  1. 1 The same ruler measured twice, ten billion years apart
  2. 2 A ruler measured along and across
  3. 3 A blur undone with the field that made it
  4. 4 A ruler read in absorption
  5. 5 A wave pulled ahead by particles that outran it
5 essays · cosmology Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium.

Binary heating

  1. 1 A pair that heats what is trying to cool
  2. 2 A core that makes what the cluster asks for
  3. 3 The binaries a core keeps are spread evenly in energy
  4. 4 Ejected from a globular, merged in a nucleus
  5. 5 A population that cannot share the cluster's temperature
5 essays · gravitation Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses.

Binary stars

  1. 1 The only stars whose masses are known
  2. 2 Two orbits of one pair, and a distance falls out
  3. 3 Two radii, from a light curve alone
  4. 4 Three causes with three shapes in one curve
  5. 5 A precession slowed by stars lying on their sides
5 essays · stars From six gravitational radii to one. The radius of the innermost stable circular orbit against the dimensionless spin a = Jc/GM², in units of GM/c², for orbits prograde and retrograde with the hole's rotation. Both curves are the Bardeen–Press–Teukolsky expression and are checked at the three places it has exact values: 6 at zero spin, and 1 and 9 at the extremal limit. The separation is the observable consequence of frame dragging — space near the hole is itself circulating, so an orbit going the same way can stay closer before it becomes unstable, and one going the other way cannot come as close as a non-rotating hole allows. The prograde branch is required to fall and the retrograde branch to rise at every step drawn, which is a claim about the direction of the effect rather than about its size. The marked spin of 0.998 is not the extremal value but the equilibrium a hole fed by a thin disc actually reaches, because photons emitted by the disc are preferentially captured on retrograde orbits and spin the hole down again. Nothing here depends on what the hole is made of: two numbers fix the whole geometry, and this figure is the first of them holding still while the second moves.

Black hole spin

  1. 1 The second number a black hole has
  2. 2 A spin that is one length in disguise
  3. 3 A ring whose size is almost only a mass
  4. 4 A spin that is measured as an angular momentum
  5. 5 Precessions bought one hertz at a time
5 essays · gravitation The sky from latitude 52°. The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination 20 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint.

Celestial sphere

  1. 1 The sphere that is not there, and why it is still the right model
  2. 2 Where a star is depends on who is asking
  3. 3 A frame made of things that are not points
  4. 4 A frame fixed before its centre was found
  5. 5 A midplane that depends on how far it is averaged
5 essays · sky A separation of 10⁻¹⁰ becomes 0.0285. The distance between two copies of Burrau's problem, started 10⁻¹⁰ apart in one coordinate of one body and then stepped in lockstep — one loop, one step size, both states advanced by it — for 60 time units. The vertical axis is logarithmic, so the straight stretch is exponential growth, and its slope is the Lyapunov exponent: 0.3413 per time unit, fitted over the 604 samples lying between ten times the starting offset and a tenth of the system's own size. That is an e-folding every 2.93 time units, so the separation multiplies by ten every 6.75 and by 7.8·10⁸ across the whole run. It has not saturated within the drawn interval: the growth rate over the last tenth of the run is still 0.6002 per time unit, 176% of the fitted exponent, and the separation has reached only 0.90% of the system's own size. The exponent is what sets a prediction horizon, and this run is drawn short of it on purpose: the straight line is the measurement, and the flat part that follows is arithmetic about how far apart two bounded systems can get.

Chaos

  1. 1 A prediction with an expiry date
  2. 2 Where the chaos comes from
  3. 3 A barrier that leaks at a rate that can be computed
  4. 4 A chaotic orbit clings to the edge of order
  5. 5 A particle on a line that can spell any word
5 essays · gravitation Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.

Collisional cascade

  1. 1 Where the mass is and where the light is
  2. 2 The drag that sorts a disc by size
  3. 3 The size at which a body is easiest to break
  4. 4 A ceiling that forgets how much there was
  5. 5 Two clocks in the dust of one collision
5 essays · orbits The effective potential, for three angular momenta. The radial motion of an orbiting body is one-dimensional motion in an effective potential: the attraction −GM/r plus the centrifugal term L²/2r² that the angular momentum contributes. The barrier at small radius is what stops a body with any angular momentum at all from reaching the centre, and the bottom of each well is the circular orbit.

Effective potential

  1. 1 The wall that angular momentum builds
  2. 2 Only two force laws let an orbit come back
  3. 3 The well moves, and the body keeps its share of it
  4. 4 The size forgets how the mass was lost, and the shape does not
  5. 5 A kick that writes over the record of the pulses
5 essays · orbits The extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V.

Extinction

  1. 1 Dust makes everything look further away
  2. 2 A magnitude measured where nothing was measured
  3. 3 The dust is not lost light, it is moved light
  4. 4 A size read from the slope of a colour
  5. 5 A glow that weighs the grains reddening cannot see
5 essays · starlight GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst.

Gravitational waves

  1. 1 A distance with no ladder under it
  2. 2 A detector the size of the galaxy
  3. 3 One number where two masses were
  4. 4 Every pair arrives circular
  5. 5 A distance tangled with an angle
5 essays · gravitation Three horizons, and a light cone that bulges. Cosmic time upwards, comoving distance sideways, for the Planck 2018 cosmology. Galaxies sit still in these coordinates, so their worldlines are the vertical grey lines: comoving distance is defined to take the expansion out. The solid inner curve is the past light cone — the set of events whose light reaches here and now — and it reaches out to the particle horizon at 46.1 billion light years, which is the diagram's central number and the one that sounds impossible. The universe is 13.80 billion years old and light has travelled for 13.80 billion years, and yet the material that emitted the oldest light is now 46 billion light years away. Nothing has outrun light: the comoving distance covered is ∫c dt/a, and dividing by a scale factor that was small early on makes the integral three times ct. The dashed curve is the Hubble sphere, where the recession speed equals c, at 14.5 Gly today — and the light cone lies outside it for most of its length, which is exactly why a galaxy can be observed while receding faster than light. The outer dot-dash curve is the event horizon, at 16.7 Gly: a signal sent from here today never reaches anything beyond it.

Horizons

  1. 1 A horizon three times larger than the age allows
  2. 2 The galaxies that are already out of reach
  3. 3 The clock on which light travels in straight lines
  4. 4 Whether there is a horizon at all
  5. 5 Two horizons that differ only in who is inside
5 essays · cosmology The Hertzsprung–Russell diagram. Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.

The HR diagram

  1. 1 The diagram that sorted the stars, by plotting two things against each other
  2. 2 The main sequence is a place stars sit, not a track they travel
  3. 3 An age read off a bend
  4. 4 A lone star is dated only as it leaves
  5. 5 Four clocks and the stars none of them reads
5 essays · stars The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.

Impulse approximation

  1. 1 An answer obtained along a path that was not taken
  2. 2 The part of a kick that heats nothing
  3. 3 Stars lost to a kick that adds no energy
  4. 4 A slow shock heats the halo and robs the core
  5. 5 Pairs of stars too loosely bound to survive heavy dark matter
5 essays · gravitation Two densities in thermal balance at one pressure, 121 times apart. Thermal equilibrium for interstellar gas, drawn as pressure against density with both axes logarithmic. Every point on the curve is a temperature between 40 and 9000 K at which cooling exactly balances the 2·10⁻²⁶ erg per second per hydrogen nucleus that grain photoelectrons deliver. The curve is not monotonic: it rises to 5007 K cm⁻³, falls to 1597, and rises again, so a horizontal line anywhere between those two crosses it three times. At the 3000 K cm⁻³ of the local medium the three crossings are a warm phase at 0.47 cm⁻³ and 6354 K, a cold phase at 57 cm⁻³ and 52 K, and one in between drawn dashed because it cannot survive — a parcel there that is squeezed cools and keeps contracting, and one that expands heats and keeps expanding. The two survivors differ by a factor of 121 in density and by exactly the same factor in temperature — necessarily the same, because their product is the pressure both are held at — and yet they press on one another equally, which is why they can share the same volume of the disc indefinitely rather than mixing.

Interstellar medium

  1. 1 Two temperatures, and nothing in between
  2. 2 A disc held open by what cannot be photographed
  3. 3 A temperature that belongs to none of the gas
  4. 4 An atomic ceiling set by the dust
  5. 5 A molecule that forms too slowly to wait for
5 essays · galaxies The Balmer maximum is at 9,870 K, and it is a maximum in temperature. What two absorption lines actually count, each normalised to its own maximum. The first curve is the fraction of all hydrogen sitting in n = 2 — the only hydrogen a Balmer line can absorb — which is a Boltzmann factor climbing with temperature multiplied by the neutral fraction falling with it. The product peaks at 9,870 K, where 35.2% of the hydrogen is still neutral and only 8.73·10⁻⁶ of all of it is in n = 2 at all. Below the peak there is plenty of hydrogen and almost none of it excited; above it there is plenty excited and almost none of it neutral. The second curve is the fraction of calcium that is singly ionised, which is what the Ca II K line counts, and it peaks at 6,335 K — cooler, because calcium gives up its first electron at 6.113 eV. At 6,000 K the Balmer curve is at 0.1% of its own maximum while Ca II is near its peak, and calcium is 4.5·10⁵ times rarer than hydrogen in the same gas. A spectrum in which Ca II K is the strongest line is not a spectrum of a calcium star. Reading it as one is precisely the error that had the Sun made of iron until 1925.

Ionisation

  1. 1 Why hydrogen's lines are strongest where hydrogen is not
  2. 2 A luminosity class is a density measurement
  3. 3 A spectrum with no continuum left in it
  4. 4 The continuum is made by one atom in ten thousand
  5. 5 An ionised sphere that is a count of photons
5 essays · starlight Flight time against semi-major axis, for a fixed 135° sweep. Lambert's theorem drawn: the time to fly between two points 1 and 1.524 AU out and 135° apart, against the semi-major axis of the orbit that does it. Nothing else about the orbit enters — not its eccentricity, not where its periapsis is, not how it is oriented — which is the content of the theorem and the reason a two-point transfer is a one-dimensional search rather than a six-dimensional one. Two branches: the lower one is the ellipse whose arc stays short of apoapsis, falling towards the parabolic floor at 103.2 days as a grows without limit; the upper one is the ellipse of the same size whose arc runs through apoapsis, rising without limit. They meet at a = s/2 = 1.2161 AU, 244.2 days, which is the minimum-energy transfer and the slowest ellipse available — every faster one is bigger. Each branch is monotone, checked point by point across the drawn range, so a horizontal line cuts each at most once: for a given pair of points and a given time there is exactly one ellipse, and at 260 days it is a = 1.2189 AU on the upper branch. The freedom a mission designer has is not in this picture: it is the choice of the two points, which is what a porkchop plot sweeps.

Lambert's problem

  1. 1 Two places and a clock decide the path
  2. 2 One time of flight and five ways round
  3. 3 The variable that makes Lambert's problem easy
  4. 4 An answer from two bodies, flown through a flattened Earth
  5. 5 A correction that tilts the arc it meant to aim
5 essays · orbits The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in.

Limb darkening

  1. 1 The light that is missing from the edge
  2. 2 The depth is not the area
  3. 3 A sight line stops at two thirds
  4. 4 A limb brighter than the middle
  5. 5 A profile that averages a decade of depth
5 essays · starlight The curve of growth. The equivalent width of an absorption line against the number of absorbers along the sight line, both logarithmic, computed by integrating a Voigt profile with damping parameter 0.005. Three regimes: the width grows in proportion to the abundance while the line is weak, then almost not at all for two decades once the core saturates, then as the square root once the damping wings dominate. A line measured in the middle stretch carries almost no information about the abundance, and most strong lines in a stellar spectrum are there.

Line formation

  1. 1 How much of a line is not in its depth
  2. 2 The same width for three different reasons
  3. 3 A continuum that was never observed
  4. 4 The lines a synthesis keeps pay for the ones it lacks
  5. 5 A rotation found where the transform goes dark
5 essays · starlight The magnitude scale, plotted. The logarithm of the received light against magnitude from -27 to 32, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller. 9 landmarks are marked, from the Sun to the deepest exposures, spanning a factor of 1.2·10²³ in received light.

Magnitudes

  1. 1 A scale that runs backwards, multiplies, and works
  2. 2 A brightness is a distance only if something is known
  3. 3 A magnitude in a band the source never had
  4. 4 A redshift that is wrong by three
  5. 5 A mean that a few per cent of strays can drag
5 essays · starlight Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through.

Mass transfer

  1. 1 The flow that narrows its own channel
  2. 2 Three clocks and a runaway
  3. 3 Mass that leaves takes the orbit's spin with it
  4. 4 A donor that grows as it is eaten
  5. 5 The least known number in binary evolution
5 essays · stars

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