Astronomy & Celestial Mechanics
the sky, computed
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Series — page 8

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.
Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end.

The mass–luminosity relation

  1. 1 Mass decides everything, by a power of three and a half
  2. 2 An exponent that is a slope, not a law
  3. 3 Brighter for its mass and fainter for its colour
  4. 4 A star that has to make its own catalyst
4 essays · stars The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.

Molecular clouds

  1. 1 The cloud that cannot hold itself up
  2. 2 Support that cannot be squeezed away
  3. 3 The support and the seed are the same motions
  4. 4 A lifetime read from how gas and stars sit apart
4 essays · galaxies Two equilibria become four, and three worlds sit near the join. The Cassini equilibria of a spin axis, drawn against the ratio of its own precession rate to the rate at which its orbit plane turns, for an orbit inclination of 1.5 degrees. Each column of dots is the full set of obliquities at which the two precessions keep step at that ratio, found by root-finding rather than by tracing a remembered curve. Below α cos ε/|g| = 1.135 there are two such obliquities and above it there are four, and the figure checks both counts on either side of the join. The three marked bodies are placed by their own measured precession constants: the Earth with the Moon at 2.67, safely on the four-state side; the Earth without it at 0.86; and Mars at 1.06. Two of the three sit within a few tenths of the bifurcation, which is the whole reason their obliquities are not constants: near the join the equilibria are close together, the libration around them is wide, and a body pushed between neighbouring resonances wanders. The Moon's contribution to the Earth's precession constant is what moves the first mark away from that region, and the second mark is the same planet with that contribution removed. This is a two-frequency model of a many-frequency system, and the real chaos comes from the overlap of resonances it does not contain.

Obliquity

  1. 1 A tilt that is not a constant
  2. 2 A tilt held still by being too fast to resonate
  3. 3 A past that can only be given as odds
  4. 4 Layers that keep the rhythm and lose the dates
4 essays · sky Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc.

Orbit determination

  1. 1 Five directions and no distance among them
  2. 2 Three observations and no orbit at all
  3. 3 An error that is nearly all in one direction
  4. 4 An eccentricity that cannot be zero
4 essays · orbits Four averages of one distance, and the two of them that are the semi-major axis. The average distance of a body from its primary, against eccentricity and in units of the semi-major axis, computed four ways: averaged over time, over true anomaly, over eccentric anomaly, and as the harmonic mean in time. Every curve is a quadrature over the orbit — 2,048 panels uniform in eccentric anomaly, with Kepler's equation supplying the time weight — and not a closed form. Two of the four are exactly a at every eccentricity, which is why they are drawn as one line: the eccentric-anomaly average, because the mean of cos E over a turn is zero, and the harmonic mean in time, because the time weight cancels 1/r at every node before the sum begins. The other two are not: the time average is a(1 + e²/2), which rises to 1.4050 a at e = 0.9, and the true-anomaly average is a√(1−e²) — the semi-minor axis — which falls to 0.4359 a there. So a is the average distance in two senses out of four, and the ordering b ≤ a ≤ ⟨r⟩ₜ holds at every eccentricity with equality only on the circle. At Earth's e = 0.0167 the four agree to 0.014%, and at Mercury's e = 0.2056 the spread is 2.14%. The distinction is invisible for the planets and unavoidable for a comet, and it is the reason a quoted "mean distance" has to say which mean.

Orbital averages

  1. 1 The average depends on what is being averaged
  2. 2 An average that precession cannot move
  3. 3 A hotter summer that is also a shorter one
  4. 4 An ice sheet that skips a beat of the tilt
4 essays · orbits An orbit at i = 42°, Ω = 35°, ω = 55°. The three orientation elements. The orbit is generated in its own plane and rotated by the standard sequence, so the inclination, the node line and the argument of periapsis are the angles that produced the drawn curve rather than labels applied to it.

Orbital elements

  1. 1 Six numbers that fix an orbit for all time, and the sixth is the awkward one
  2. 2 Three rotations that put an orbit in space, and they do not commute
  3. 3 The elements that stop existing
  4. 4 A shower that is a reading of an orbit
4 essays · orbits A wave whose crests count out 35 grams per square centimetre of ring. Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 17.6 km and has shortened to 3.58 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 grams per square centimetre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring.

Planetary rings

  1. 1 A ring weighed by the wave crossing it
  2. 2 An edge is a balance, not a boundary
  3. 3 A resonance with the planet itself
  4. 4 A ring that is thinner from one side
4 essays · gravitation Lane–Emden solutions for n = 0, 1, 1.5, 3, 4.5, 5, and the one that has no surface. The dimensionless density θ against the dimensionless radius ξ, for polytropic indices 0, 1, 1.5, 3, 4.5, 5. Each curve is the whole structure of a star whose pressure is K times its density to the power 1 + 1/n: the equation of state and hydrostatic equilibrium leave one second-order differential equation, and this is its solution. Every curve starts at θ = 1 with zero slope, because the density is greatest at the centre and has no cusp there. What separates them is where they end. At n = 0 the density is uniform and the surface is at ξ₁ = 2.4495; by n = 3 it has moved out to 6.8968 and the central density is 54.2 times the mean. At n = 5 the curve reaches zero only at infinity — a configuration of infinite radius and, remarkably, finite mass — and every index above it has neither. The three curves that have closed forms, n = 0, 1 and 5, are drawn from the same numerical integration as the rest and agree with those forms to better than two parts in a million, which is what licenses reading the others off the picture. What the figure cannot show is the scale: ξ is radius divided by a length that depends on the central density and on K, so two stars of the same index and wildly different sizes have the same curve here.

Polytropes

  1. 1 An equation of state is already a star
  2. 2 A core too heavy for the gas around it
  3. 3 A star tipped over at a thousand times its horizon
  4. 4 A soft centre the rest of the core outvotes
4 essays · stars A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 51 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic.

Quasars

  1. 1 A size measured from a delay
  2. 2 The factor that multiplies every quasar mass
  3. 3 A delay that tells which way the gas is moving
  4. 4 A lag too short for the light
4 essays · galaxies The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye.

Resonance

  1. 1 Resonance clears a gap in one place and locks a moon in another
  2. 2 A chain that could not have been assembled in place
  3. 3 Capture is a direction, not a strength
  4. 4 A rotation locked to the orbit, but not one to one
4 essays · gravitation The 2.3-hour spin barrier, and the small bodies that are allowed through it. Rotation period against diameter for a synthetic asteroid population, both axes logarithmic, with the horizontal lines marking where a body held together by nothing but its own gravity would fly apart. That limit is P = √(3π/Gρ) and it contains only the density: 3.30 hours at 1 gram per cubic centimetre, 2.33 hours at 2 gram per cubic centimetre, 1.91 hours at 3 gram per cubic centimetre. Size does not appear in it, which is what makes the figure's shape informative rather than obvious — a barrier that depended on size would be drawn as a slope, and a horizontal line crossing five decades of diameter is a much stronger statement. The observed population respects it. Above about two hundred metres nothing rotates faster than the 2.3-hour line, and the crowding just below that line is real: bodies pile up against a limit they cannot cross. Below two hundred metres the wall stops applying and the fast rotators appear, some of them turning in minutes. Nothing about gravity changes at that size. What changes is that a body small enough is a single coherent rock with tensile strength, while a body large enough is a pile of fragments with almost none, and the barrier is a measurement of which is which. The picture is a synthetic population drawn at random from a fixed seed rather than a catalogue, so the individual points are not asteroids; what is real is the barrier, its value, and the fact that only the smallest bodies are found beyond it.

Rubble piles

  1. 1 A wall with no size in it
  2. 2 A spin barrier with a corner in it
  3. 3 A split that decides whether the piece can leave
  4. 4 A surface that slides before the body sheds
4 essays · orbits Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap.

Secular theory

  1. 1 No planet has an eccentricity of its own
  2. 2 The bound that holds only in the linear theory
  3. 3 A denominator that emptied the inner belt
  4. 4 A deficit the giants hold and Mercury could borrow
4 essays · orbits Seven sources over twelve decades of flux, and one threshold below the one that matters. The Sun's neutrino spectrum at the Earth, with the continua drawn per unit energy and the two monoenergetic lines as spikes at their own energies. The pp reaction supplies 91 per cent of all of them and its endpoint is at 0.4233 MeV. The vertical lines are experimental thresholds, and they are the figure's argument: only gallium sits below that endpoint. Chlorine, which produced the deficit and held it for twenty years, could not see a single pp neutrino — it counted ⁷Be and ⁸B, which are a rare branch of a rare branch, together under a per cent of the total — and the water detectors that followed were higher still. So the discrepancy that eventually turned out to be a property of the neutrino was measured, for two decades, using the least representative one per cent of the flux available. The total drawn here is 6.54·10¹⁰ cm⁻² s⁻¹, and it is checkable without any stellar model at all: every completed chain turns four protons into helium, releases 26.73 MeV of which 0.59 leaves as neutrinos, and emits two neutrinos — so the solar constant of 1361 W m⁻² fixes the number at 6.5·10¹⁰ cm⁻² s⁻¹, within 0.6 per cent of the sum of the model's own branches. The dominant flux is a consequence of the Sun shining and of nothing else.

Solar neutrinos

  1. 1 The only thing that leaves the centre
  2. 2 A flux that is a thermometer to a tenth of a per cent
  3. 3 A survival that depends on the energy
  4. 4 A rise a detector shrinks to a tilt
4 essays · stars The winding problem, drawn. A straight spoke of stars in the Milky Way from 2 to 15 kpc, and the same stars after 0.15, 0.3, 0.6 Gyr. Nothing has been done to them except let them orbit: the star at radius R has turned through Ω(R)t, and Ω falls as 1/R wherever the rotation curve is flat, so the inner end laps the outer one. By 0.6 Gyr the arm has a pitch angle of 3.4° at 8 kpc, which is tighter than any spiral galaxy that has ever been photographed, and the disc is thirteen gigayears old rather than 0.6.

Spiral structure

  1. 1 An arm that cannot be made of stars
  2. 2 Where a pattern is allowed to turn
  3. 3 An arm that is undone by the work it does
  4. 4 The bar leaves its gas where the orbits turn
4 essays · galaxies The star-formation law, and the clock inside it. 46 model discs on the Kennicutt–Schmidt plane, built from Σ_SFR = 2.5e-4 Σ_gas^1.4 with 0.28 dex of scatter, with the exponent then fitted back off the drawn points at 1.42. The diagonal dashed lines are constant depletion times: gas divided by the rate consuming it. Every disc on this plane consumes its gas within a few gigayears, and a galaxy that has been forming stars for ten of them therefore cannot have been working from the gas it started with. The slope steeper than one is what makes the depletion time shorter where the gas is richer, which is the opposite of the intuition that a full tank lasts longer.

Star formation

  1. 1 The gas runs out before the galaxy does
  2. 2 A birth rate measured from light nothing young emitted
  3. 3 A cloud that cannot become a star
  4. 4 A threshold with no free parameter in it
4 essays · galaxies A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.

Stellar streams

  1. 1 A stream is not the orbit it came from
  2. 2 A hole that says mass times time
  3. 3 A spectrum that counts what no single gap can
  4. 4 A stream that points the wrong way
4 essays · galaxies A wind that writes its own terminal speed on the blue edge, at 2017 km/s. A P Cygni profile, computed by integrating a spherical wind on a grid rather than drawn. The wind accelerates outward as one minus the inverse radius to the power 0.8, reaching 2000 kilometres a second, and its density falls as the inverse square of the radius over its own speed; the horizontal axis is the Doppler shift in units of that terminal speed, with blue to the left. The column directly in front of the stellar disc is moving toward the observer at every speed from nearly nothing near the surface up to the terminal value far out, so it takes light out of the beam across that whole range and the absorption trough runs from the rest wavelength to the blue edge at -1.01 of the terminal speed. That edge is the measurement: no model of the star, its distance or its mass enters it, only the geometry of a column seen end-on. The emission comes from everywhere else, where the wind moves across the line of sight and can only add photons; it is Infinity times stronger to the red than to the blue, because the part of the shell receding directly away is hidden behind the star and its blueshifted counterpart is not. Two features, one shell, and between them a velocity and a mass-loss rate.

Stellar winds

  1. 1 A speed read off an edge
  2. 2 The flare that arrives from somewhere else
  3. 3 A rate the wind has no choice about
  4. 4 A wind that may be too hot to see
4 essays · starlight A light curve that is a decay chain. Bolometric luminosity against days since explosion, for three Type Ia models synthesising 0.3 M☉, 0.6 M☉, 0.9 M☉ of ⁵⁶Ni, from Arnett's one-zone diffusion model. Nothing in the shape is fitted: the two timescales are the laboratory half-lives of ⁵⁶Ni and ⁵⁶Co, 8.8 and 111.3 days, and the rise is those decays seen through an envelope that takes 11.3, 15.5, 18.6 days to leak. Each peaks at 14, 18, 21 days, and at maximum the luminosity equals the instantaneous deposited power to within 0.6% — Arnett's rule, and the only reason a peak brightness can be read as a mass of nickel. After maximum the curves settle to nearly straight lines on this logarithmic axis, declining at 0.0360 magnitudes a day against ⁵⁶Co's own 0.0098. The difference is gamma-ray escape: the ejecta become transparent to the very photons that are supposed to be heating them, and the tail is therefore steeper than the isotope. The one thing the tail is not is a property of the star — it is a half-life, plus a column density that is falling as t⁻².

Supernovae

  1. 1 The candle that has to be standardised
  2. 2 Two explosions told apart by a missing line
  3. 3 The companion survives, and it is moving
  4. 4 A neutron star born turning too slowly
4 essays · stars What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -1.5 magnitudes wherever it is put.

Surface brightness

  1. 1 The brightness distance cannot touch
  2. 2 Where the sky is the measurement
  3. 3 A radius that is a measurement of the threshold
  4. 4 One profile for a galaxy that has two
4 essays · galaxies An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

The ellipse

  1. 1 The orbit is an ellipse, and the Sun is not in the middle of it
  2. 2 The position that has no formula, and is computed anyway
  3. 3 An orbit can look exactly like a circle and still not be one
  4. 4 The formula that exists, and is not used
4 essays · orbits

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