Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 20

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 381–400 of 514.

Galaxies

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

A single-epoch mass, good to a factor of 2.6. The error budget of a black-hole mass obtained from one spectrum, as probability densities in the logarithm of the ratio of the estimate to the truth. Four independent widths go in. The radius comes from the luminosity through a relation with 0.19 dex of scatter about it. The line width has to be measured on a profile that is blended with narrow lines and with iron, and it enters the mass squared, so 0.12 dex. The luminosity was not measured at the epoch the relation was calibrated for and the object has varied since, which is 0.053 dex. And the term this figure exists for is orientation: the virial factor is one number fitted to a sample, every individual object has its own according to how its broad-line region is inclined, and the spread of that is 0.35 dex — the largest of the four by some margin. In quadrature the total is 0.42 dex — a factor of 2.6 — and the drawn total's half-width is checked against that combination rather than assumed. The bracket at the foot is a different kind of error and is not in the quadrature: the absolute calibration of the virial factor, uncertain by about a factor of 3, which moves every quasar mass ever published by the same 0.24 dex and widens nothing.

The factor that multiplies every quasar mass

A reverberation lag and a line width give a length times a velocity squared, which is a mass multiplied by an unknown number of order one. That number is fixed by making a few dozen nearby active galaxies lie on a correlation measured in quiescent ones — so every black-hole mass at redshift two rests on a fit performed at redshift zero.

8 figures · Quasars A gap's depth is one number: 2GmT ÷ v b². The central density of the gap against impact parameter, 4 billion years after the encounter, for subhaloes of 10⁶, 10⁷, 10⁸ solar masses. Points are read off the drawn density profiles; the curves are 1/(1 + 2GmT/v b²), the stretch the map applies at the encounter point, and the two agree to 3.0 per cent wherever the histogram has enough stars left in the gap to measure a depth at all. Everything about the encounter enters the depth through that one combination. The consequence is the horizontal reading: a gap of a given depth is produced by every point along a locus on which the mass rises as the square of the impact parameter — half depth at 0.42, 1.33, 4.19 kiloparsecs for the three masses drawn, an exponent of 0.500 against the half the algebra requires. What breaks that particular degeneracy is the gap's width, which scales as the impact parameter itself while the depth does not: rescale s by b and the map is identical, so the profile is one shape stretched. Depth and width together give the impact parameter and the product mT. They do not give the mass.

A hole that says mass times time

A gap in a stellar stream is the strongest evidence available that dark subhaloes exist, and its depth depends on the perturber's mass, the impact parameter and the elapsed time only through one combination. Two of those three are unobservable, so a gap is a measurement of a product.

8 figures · Stellar streams

Gravitation

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

The spin a mismeasured radius reads as. The spin inferred from the disc's inner radius, against the fractional error in that radius, for four true spins. Both spectral methods reduce to one measurement — where the disc stops — and one conversion, the ISCO relation, so this figure is the error propagation both of them share. The behaviour is not uniform and it is the reverse of what the difficulty of the measurement suggests. A ten per cent error reads a hole of true spin 0.3 as anywhere from 0.16 to 0.44, while the same ten per cent moves a hole of true spin 0.99 only between 0.98 and 1.00. The reason is that the ISCO falls from six gravitational radii to one over the whole range of spin and does most of that falling in the last few per cent, so near the extremal limit a small change in radius is a large change in spin and the inversion is stiff. The published spins clustering near 0.9 and above are therefore the ones least sensitive to the systematics, and a published spin of 0.3 carries an error bar the method cannot really support. An overestimated radius always reads slow, so anything that stops the disc outside the last stable orbit biases every measurement the same way.

A spin that is one length in disguise

Both ways of measuring a black hole's spin from its light measure the same thing — where the accretion disc stops — and convert it through the same relation. That conversion is stiff at high spin and slack at low, so the published spins near one are the trustworthy ones and the published spins near a third are barely measurements.

8 figures · Black hole spin The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one.

The part of a kick that heats nothing

Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot.

8 figures · Impulse approximation

Orbits

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

June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is.

An average that precession cannot move

Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent.

8 figures · Orbital averages A family of 276, or of 582, depending on one number. Membership of a synthetic family against the cutoff velocity used to define it, for a family of 260 genuine fragments ejected at 15 metres a second sitting in a background of 417 unrelated bodies. The clustering is single linkage in the standard proper-element metric, started from one object and grown until nothing more is within the cutoff of anything already absorbed. Three curves: the number of objects claimed, scaled to its largest value; the fraction of the real family recovered; and the fraction of the claim that is background. The shape is the whole difficulty. At low cutoff the family is fragmented and only its core is found. There is then a plateau — near 58 metres a second here, giving 276 members of which 6 per cent are background — and that plateau is what every published family list is chosen at. Past it the count runs away, because single linkage absorbs an object and then searches from the object it has just absorbed: one chance interloper bridges the family to the belt and the algorithm returns half the main belt. Completeness and contamination rise together, so there is no cutoff at which both are good and the plateau is a compromise rather than a discovery.

A family whose size is a choice

Nothing observable distinguishes a collisional fragment from an asteroid that happens to be nearby. Membership is assigned by clustering at a cutoff velocity, and the same family has 276 members or 582 according to which cutoff is used — with completeness and contamination rising together, so no cutoff is good.

8 figures · Asteroid families Nine dates, and a gap from 0.8 to 3.1 billion years. The lunar crater chronology and the samples that fix it: crater density per square kilometre against age, the density axis logarithmic. Every point is a laboratory measurement on returned rock — six mare and highland units dated by crystallisation, four young craters dated by how long their ejecta has been exposed to cosmic rays. The curve is a fit of N(T) = A(e^(λT) − 1) + BT with the exponential rate swept over 6 to 7.9 per billion years and the two amplitudes refitted at each; the best fit lands at λ = 6.95, against the 6.93 the published chronology uses, which is a check on the fitting rather than an input to it. What the figure exists to show is where the anchors fall in time. Six of them lie between 3.15 and 3.92 billion years, four between 0.026 and 0.80, and there is nothing whatever between 0.8 and 3.1 — a gap covering half the age of the solar system. The consequence is drawn as the spread of the acceptable curves: a surface whose crater density says 3.5 billion years is dated to ±0.03 billion by this family, and a surface at 2 billion to ±0.13. The chronology is a measurement where the astronauts landed and an interpolation everywhere else.

Nine dates for every surface in the solar system

Every absolute age quoted for a planetary surface — Martian volcanism, Mercury's plains, the resurfacing of Europa — descends from radiometric dates on rocks returned from nine landing sites on one body. Six of them fall between 3.15 and 3.92 billion years, and there is nothing at all between 0.8 and 3.1.

8 figures · Surface chronology

The observed sky

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

Why the Moon holds the tilt still. Spin-axis precession rate against obliquity, with the solar system's own secular frequencies drawn across it. The rising-and-falling curves are α cos ε for three precession constants: the Earth as it is, with the Moon supplying about two thirds of the torque; and a Moonless Earth at two plausible rotation rates. The horizontal lines are the nodal eigenfrequencies of the planetary secular system — s₆, s₃, s₄, s₂, s₁, s₇, s₈ — which are the rates at which the Earth's own orbit plane wobbles. A crossing is a resonance: the axis is precessing about a plane that is itself turning at the same rate, the resonant angle stops circulating, and the obliquity librates instead of holding still. The present Earth precesses at 50.3 arcseconds a year, faster than every secular frequency in the list, and its nearest crossing is at 61.3°, which is 37.8 degrees from the obliquity drawn. Take the Moon away and α falls by a factor of three; the crossings come down with it, 3 of them land below 85°, and where several overlap the obliquity has no stable value at all. The Moon does not hold the axis by pulling on it. It holds it by making the precession too fast to resonate with anything.

A tilt held still by being too fast to resonate

The Moon does not hold the Earth's axis by pulling on it. It triples the rate at which the axis precesses, which lifts that rate clear of every frequency at which the Earth's own orbit plane wobbles — and a precession with nothing to resonate with cannot wander.

8 figures · Obliquity A factor of 100 in the wind, 8 degrees in the aurora. The latitude of the last closed field line against the solar wind's dynamic pressure, for a dipole of 0.31 gauss at the equator. The chain is short and every link is exact. Pressure balance puts the magnetopause at a distance going as the sixth root of the field pressure over the ram pressure; a dipole line reaching equatorial distance L returns to the surface at colatitude arcsin(1/√L); so the polar cap's edge is that, and the aurora sits just equatorward of it on the outermost closed lines. Across the 100-fold range of pressure drawn — which covers everything from a quiet wind to a severe storm — the magnetopause moves from 13.1 to 6.1 planetary radii and the oval from 74.0 to 66.1 degrees. The shaded band is where the oval is actually observed. The open magnetic flux is exactly the reciprocal of the standoff distance, so it rises by a factor of 2.15 over the same range — which is the quantity that actually matters for a storm, and it is the only one of the three that changes by much.

An aurora that is a sixth root inside an arcsine

The magnetopause distance fixes which field lines are open, the open ones map to a cap around each pole, and the cap's edge is where the aurora is. A hundredfold change in the solar wind moves that edge by eight degrees — and across planets whose fields differ by four orders of magnitude it moves by ten.

8 figures · Magnetosphere

Spaceflight

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

1,236 fragments anybody can see, and 63,397 that can kill. The cumulative fragment size distribution from the standard breakup model, for a catastrophic collision involving 1500 kilograms and for an explosion of the same object, both axes logarithmic. The exponents are −1.71 and −1.60, measured off the drawn curves; they are empirical, fitted to ground tests and to the observed clouds of real events, and they are steep. What follows is the reason a catalogue of tracked objects is not a catalogue of the hazard. Above ten centimetres — the size a ground radar can follow in low orbit — the collision makes about 1,236 pieces. Above one centimetre, which is the size that goes through a spacecraft at ten kilometres a second and cannot be shielded against, it makes 63,397. Above a millimetre, which shielding does stop but which erodes a surface, 3,251,385. The tracked population is under two per cent of the lethal one, and the difference is not a gap in the catalogue that better radars will close — objects of a centimetre at a thousand kilometres are beyond any sensor that has been proposed. An explosion makes fewer large pieces than a collision and a comparable number of small ones, because an explosion divides one object and a collision destroys two.

The fragments nobody can see and cannot shield against

A catastrophic collision in low orbit makes about a thousand pieces big enough to track and sixty thousand big enough to destroy a spacecraft. The catalogue is under two per cent of the hazard, the gap is not one better radars will close, and the fragments' lifetimes span a factor of twenty-five within a single event.

8 figures · Orbital debris Windows every 780 days, costing between 5 and 13. The cheapest departure energy available in each of 8 consecutive launch opportunities, each found by solving a grid of Lambert problems around the window and taking the minimum. Opportunities recur every 780 days — the synodic period of Earth and Mars, which is exact and which is why the interval between missions is always about twenty-six months. Their cost is not periodic on that interval. The cheapest here is 5.0 km²/s² and the dearest 12.9, a factor of 2.59, and the pattern repeats on a period of about fifteen years rather than on the synodic one. The cause is Mars's eccentricity of 0.093: a transfer that arrives near Mars's perihelion has less distance to cover and meets a faster-moving planet, and whether an opportunity does that depends on where Mars is in its own orbit — which drifts relative to the synodic cycle by a fixed amount each time and comes back into phase after seven windows. In launch mass the factor is larger than it looks: departure energy enters the rocket equation through an exponential, so a C₃ of 13 rather than 5 costs roughly 1.08 times the propellant at departure.

The window that comes back and the cost that does not

Launch opportunities to Mars recur every 780 days exactly, because that is the synodic period and a synodic period is arithmetic. What they cost does not repeat on that interval at all — the cheapest window is a factor of two and a half below the dearest, and the pattern comes back every fifteen years rather than every two.

8 figures · Launch windows

Starlight

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

A few per cent of diameter, hidden in the second lobe. Visibility against baseline in the natural variable πθB/λ, for stars with linear limb-darkening coefficients of 0, 0.3, 0.6, 0.9, each rescaled to the uniform disc that best fits its own first lobe. In the first lobe the four curves are within 0.72 per cent of one another; past the first null they differ by up to 4.2 per cent. That is the whole difficulty of measuring a stellar diameter. A limb-darkened star has a faint edge, so a uniform-disc fit returns a diameter 9.7 per cent too small at u = 0.9 and 2.5 per cent too small at u = 0.3 — and the information needed to tell which is in a region where the visibility is under five per cent and the calibration errors of a real interferometer are comparable to the signal. The correction from what is measured to what is wanted is taken from a model atmosphere, because the observation that would supply it is the hardest one there is. The fit here is done by least squares on the drawn curves rather than read from a conversion table, and the zero-coefficient case returns the uniform disc to 0.00 per cent, which is the fit checking itself.

A diameter that depends on a model atmosphere

An interferometer measures fringe visibilities and somebody fits a disc. A uniform disc and a limb-darkened one agree to within a per cent across the whole of the first lobe and differ by five in the second — where the visibility is under five per cent and the calibration errors are the same size.

8 figures · Angular diameter 742 kelvin between a dwarf and a supergiant, at the same ionisation. The fraction of calcium still neutral against temperature, for three surface gravities: a dwarf, a giant and a supergiant. The three curves are the same curve slid sideways. The Saha equation carries the electron pressure in its denominator, and the photospheric pressure follows the surface gravity as its square root — hydrostatic equilibrium gives a gas pressure of order g over the opacity, and the opacity in a cool star is set by the electrons themselves. So the pressure runs from 19 newtons a square metre in the dwarf down to 0.48 in the supergiant, a factor of 40, and the half-ionisation point moves from 4427 kelvin to 3685 — 742 kelvin apart. At fixed temperature the ionisation ratio goes as g^-0.50, exactly the square root the algebra requires. A star's luminosity class is a measurement of the density of its photosphere, read off which stage of an element the lines belong to.

A luminosity class is a density measurement

The Saha equation has an electron pressure in its denominator, and a supergiant's photosphere is forty times less dense than a dwarf's at the same temperature. So the same element sits in different ionisation stages in the two, the spectrum says which, and the second axis of stellar classification is a barometer.

8 figures · Ionisation

Stars

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

An exponent between 2.3 and 4, not a number. The local logarithmic slope of the mass–luminosity relation against mass — the exponent that "L ∝ M^3.5" stands in for — read off the same fit to eclipsing binaries that the relation itself is drawn from. It is a step function with two corners, and the steps are 2.3 below 0.43 M☉, 4 0.43 to 2 M☉, 3.5 2 to 55 M☉. The horizontal lines are what homology predicts: a star in radiative equilibrium has L ∝ μ⁴M³/κ̄, which gives an exponent of 3 when the opacity is electron scattering and a constant, 5.5 when it is Kramers and depends on density and temperature, and 1 where radiation pressure holds the star up and the luminosity is pinned to the Eddington value. The measured steps sit between those predictions rather than on them, which is what a real star does — no star is homologous, the low-mass ones are convective throughout and the heavy ones have convective cores, and the fitted exponents are what is left after all of that. The useful statement is not that the exponent is 3.5 but that it is between 2 and 5 everywhere and 4 in the middle, and that the reason it moves is a change of which opacity is carrying the energy out.

An exponent that is a slope, not a law

The phrase “L goes as M to the three and a half” stands in for a curve with three straight pieces and two corners. The exponent is 2.3 below half a solar mass, 4 in the middle and falls towards 1 at the top — and each change of exponent is a change in which opacity is carrying the energy out.

8 figures · The mass–luminosity relation A 1.1 solar-mass white dwarf held up for 2.8 extra billion years. The time a white dwarf takes to reach the crystallisation luminosity, and the two delays that follow, against mass. The lower band is bare Mestel cooling — thermal energy of the ions leaking out through an envelope whose opacity is Kramers'. On top of it sits the latent heat of crystallisation, 0.85 kT per ion released when the liquid interior freezes into a lattice; and on top of that the gravitational energy of ²²Ne settling through what is left, taken here as 0.6 kT per ion. Neither is fuel: both are energy the star already had, released late and radiated at the low luminosity it has by then, which is why so little of it buys so much time. The delay rises from 1.88 billion years at 0.5 solar masses to 2.81 at 1.1 — 58 per cent of the cooling already done. A white dwarf age computed from the bare law is too young, and it is too young by more the heavier the star is, which is exactly the direction that matters, because the massive white dwarfs are the ones used to date the oldest populations.

A clock that stops while its interior freezes

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling — until the interior crystallises. The latent heat of that phase change, and the settling of a heavy isotope through what is left, hold a massive white dwarf up for nearly three extra billion years.

8 figures · White dwarf cooling The same equation, run away at q = 0.8 and settled at q = 0.5. The donor's overfill of its own Roche lobe against time, integrated for 3 mass ratios at a donor adiabatic response of −0.33 — a star with a deep convective envelope, which expands as it loses mass. The overfill sets the transfer rate and the transfer rate changes the overfill, and the sign of that feedback is the stability criterion: where the lobe shrinks faster than the star does, ζ_L > ζ_ad, the overfill grows and the growth is exponential. At q = 0.8 the lobe responds at 0.03 and the overfill runs away; At q = 1.4 the lobe responds at 1.32 and the overfill runs away. At q = 0.5 it responds at -0.62 and the transfer throttles itself back. The critical ratio for this donor is 0.63 — below equal masses, which is the result the whole subject turns on: a giant transferring to a lighter companion is already unstable before the mass ratio has reversed, and what follows is not accretion but a common envelope. Nothing here is a fitted rate. The vertical scale is logarithmic and the runaway is a straight line on it, which is what an exponential is.

Three clocks and a runaway

Whether mass transfer between two stars is stable is a comparison of two logarithmic derivatives. How fast it runs is a separate question with three possible answers fourteen orders of magnitude apart — and the answer decides whether the companion accretes, is buried, or is swallowed.

8 figures · Mass transfer

Cosmology

One object, seen once, from inside — and every number in it the output of a model.

The deuterium burning rate against the expansion, at three baryon densities. The rate at which a deuteron is destroyed, divided by the expansion rate, against temperature. Temperature falls to the right, so the picture reads left to right as time, and the horizontal line at one is where the burning stops mattering: above it a deuteron is destroyed many times over before the universe doubles in size, below it the reaction has effectively ceased. The rate drawn is D(p,γ)³He at the NACRE parameterisation, multiplied by the free-proton density — three quarters of the baryons by number once ⁴He has taken the rest — and the expansion rate is the same 1.66√g* T²/mPl the freeze-out calculation raced the weak interactions against. The three curves differ in one number and one only: the baryons per photon, 3, 6, 12 in units of 10⁻¹⁰. They are therefore vertical translations of each other, exactly in proportion to η, because the reaction is two-body and the expansion is not. That is the entire mechanism by which an abundance measures a density. A denser universe crosses the line later — 21.4 keV at η₁₀ = 3, 16.0 keV at η₁₀ = 6, 12.3 keV at η₁₀ = 12 — and every extra second below the crossing is deuterium that does not survive. What the figure does not do is predict the abundance itself: the residue depends on the whole reaction network and on the ⁷Be and ³He channels that feed back into it, and the curve the abundance is read off is a fit to that network rather than to this.

The residue that failed to burn

Deuterium's abundance is not a measure of what the first three minutes made. It is a measure of what escaped being used — a two-body destruction rate losing a race to a one-body expansion, which is why its curve against the baryon density is steep and helium's is flat.

8 figures · Nucleosynthesis A factor of 2.9, and the density that would remove it. The predicted ⁷Li abundance against the baryon density, with the halo-star measurement as a horizontal band and the microwave background's density as a vertical one. At η₁₀ = 6.13 the network gives 4.70e-10 and the stars show 1.60e-10, a factor of 2.94 — 0.47 in the logarithm, which is the number every proposed resolution has to produce. The obvious cure is drawn as the second marker: ⁷Li rises as η², so the density that would reproduce the observation is η₁₀ = 3.58, a third below the measured one. That is where the cure fails, and it fails on a different element. Deuterium falls as η^(−1.6), so at 3.58 the predicted D/H is 5.946e-5 against the 2.527e-5 measured in quasar absorbers — 2.35 times too much, and 114 times deuterium's own error bar. Helium, meanwhile, moves to 0.2445 and stays inside its measurement, because its curve is flat. The three light elements do not fail together, and that is what rules out a single wrong parameter: whatever is wrong is wrong about mass seven specifically.

A factor of three, and the flatness that prices every cure

The oldest stars in the Galaxy show a third of the lithium the first three minutes should have left. Three kinds of resolution have been offered, and the datum that rules on all three is not how much lithium is missing but how uniformly it is missing.

8 figures · Nucleosynthesis A bound on the baryon density from an abundance nobody can extrapolate. Primordial deuterium against the baryon density, with the upper bounds that the solar system's own deuterium and helium-3 place on it. The inequality is D_p ≤ D_obs + ³He_obs/g₃, and it holds for any star-formation history whatever: a deuteron entering a star becomes ³He, so the pair can only be moved from one member to the other and then destroyed, never increased. With pre-solar values of D/H = 2.0e-5 and ³He/H = 1.5e-5, the bounds are 3.50e-5 at g₃ = 1, 5.00e-5 at g₃ = 0.5, 8.00e-5 at g₃ = 0.25 — and because deuterium falls with density, each upper bound on the abundance is a lower bound on η: η₁₀ > 4.98, η₁₀ > 3.99, η₁₀ > 2.97. The weakest of them, at a survival fraction of 0.25, still excludes 52 per cent of the density range below the answer. That was the state of the measurement for most of the 1980s, and the shape of it is the thing worth carrying: an abundance too poorly understood to be extrapolated at all still constrained the quantity, because its direction of change under processing was known even though its magnitude was not. The microwave background's 6.13 sits above every bound drawn, which is not a coincidence and is not evidence — a bound that excluded the answer would have been an error, and a bound that admits it is only a bound.

An abundance with no direction to correct in

Every primordial abundance is measured today and extrapolated backwards, and the extrapolation works because processing moves each species one way. Helium-3 is made by small stars and destroyed by large ones, so the sign of its correction is not merely uncertain — it is unknown.

8 figures · Nucleosynthesis A lumpy universe makes more of both, and one of them was already too much. The deuterium–lithium plane, with the curve a homogeneous universe traces as its baryon density varies and the points a two-zone universe reaches at a fixed mean density of η₁₀ = 6.13. The dense zone occupies 15 per cent of the volume, and the contrast between the zones runs from 1 — which is the homogeneous case — to 100. Every mixture lies up and to the right of its own homogeneous point, and that is not a modelling choice: an abundance is measured per baryon, so what a telescope averages is η times the abundance — and for both deuterium and lithium that product is a convex function of the density, whose average therefore exceeds its value at the average. The excess is large. At a contrast of 100 the mixture gives D/H = 1.13e-4 against 2.51e-5 smooth, and ⁷Li/H = 1.77e-8 against 4.70e-10, a factor of 37.7. The extra deuterium is exactly what the proposal was for: it lets the mean baryon density be raised while the observed D/H is still matched, which in the 1980s was the one way to make the baryons account for all the matter that dynamics required. The extra lithium is what it costs. The measured abundance was already a factor of 2.9 below the homogeneous prediction, and every step toward the lumpy universe that fixes the density makes that discrepancy worse — which is why the answer to "the baryons are lumpy" turned out to be that the extra matter is not baryons.

The universe that was lumpy at one second

If the baryons were unevenly spread when the network fired, each region ran its own nucleosynthesis and what is observed is an average. For a decade that was the one way to make ordinary matter account for all the matter — and the reason it fails is a theorem about convex curves.

8 figures · Nucleosynthesis

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