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Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

2 November 2026

18 essays on cosmology, exoplanets, galaxies, gravitation, orbits, the observed sky, spaceflight, starlight and stars

A clump that does not grow until the radiation is outweighed. The amplitude of a dark-matter overdensity inside the horizon against the scale factor in units of its value at matter–radiation equality, both logarithmic, from y = 10⁻⁴ to today near y = 3400. The solution of the growth equation, δ = 1 + 3y/2, stays flat until equality and then rises in proportion to the scale factor; matter domination alone would have grown it in proportion to the scale factor from the start. Over the whole range the true growth is 5.1e+3 against 3.4e+7 — radiation withheld a factor of 6666, which is the interval from y = 10⁻⁴ to about y = 0.67. Cosmology

A clump that waits for the radiation to thin

Dark matter feels no pressure from light, so nothing should have stopped it clumping in the radiation era. Almost nothing did stop it, and yet a clump inside the horizon at that time barely grows. The radiation does not push on it. It only makes the universe expand too fast for the clump's own gravity to keep up — and the length of that wait is the bend in the spectrum of every structure in the sky.

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The static universe is a saddle in the plane of size and speed. Every history of a closed universe of dust and an exactly balancing cosmological constant, as curves of constant energy in the plane of size against rate of change of size, in units of the static radius and 1/√(4πGρ). The two thick curves cross at the static point: they are the histories that approach it for an infinite time or leave it after an infinite time. Above them a universe expands from a big bang through forever or contracts from forever; to the left it rises from a big bang and falls back; to the right it contracts, bounces, and expands. No history other than the point itself stays near it. Cosmology

A balance that is a saddle

In 1917 Einstein added a constant to his equations to hold the universe still, and the constant does exactly that: there is one size at which the attraction of matter and the repulsion of the constant cancel. Eddington showed in 1930 that the balance is a pencil on its point. A thousandth out and it is gone in seven units of its own free-fall time. The instability is not a flaw of that one model. It is why the real universe could cross the same balance without stopping.

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A shift in the time and a change in the length, a quarter-cycle apart. The transit timing shift and duration change of a planet of 0.5 Jupiter masses in a 287-day orbit, carrying a moon of 10 Earth masses at 20 planetary radii, against the moon's orbital phase at the moment of transit, for two cycles. The timing swings by ±42 minutes and the duration by ±10.1 minutes. The moon's period is 12.9 days. Timing is largest when the moon is farthest along the track, and duration when the moon is moving fastest along it, so they are in quadrature; a perturbing planet would produce nothing of that kind in the duration. Exoplanets

A moon read from how late and how long

A moon pulls its planet round their common centre, and at each transit the planet is somewhere slightly different along that small orbit, moving slightly differently. So the transit arrives early or late, and it lasts longer or shorter. The two effects depend on the moon in different ways and they run a quarter of a cycle apart. Neither alone says anything about the moon. Their ratio gives its period, which a series of transits sampled once an orbit cannot otherwise reach.

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Water that is on the star and not on the planet. Excess transit depth, in parts per million, against wavelength, for a planet of depth 7000 ppm around a star of 2600 K. The upper curve is a bare rock with no atmosphere on a star whose disc is 10% covered by 2300 K spots that hold more water than the rest of the photosphere: it shows a broad slope and a band near 1.4 μm, 36 ppm from its shoulder. The lower curve is the signal of an Earth-like air column with water on an Earth-sized planet, about 93 ppm at the band. The band in the spot spectrum is an imposed illustration rather than a model atmosphere, but its sign is measured: it adds where a planet's own water would add. Exoplanets

Water that is on the star and not the planet

A transit depth is the planet's shadow divided by the brightness of the whole star, and the shadow falls on only one strip of it. If the strip is cleaner than the rest of the disc, the depth comes out too large, by an amount that changes with colour. The star's own spots hold water, so the error has water's fingerprint. On the smallest red dwarfs, whose rocky planets are the best targets there are, that false signal is larger than the real atmosphere being looked for.

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The middle of a cluster cools faster than the universe ages. The time the gas of a cool-core cluster, of central electron density 0.05 per cubic centimetre, would take to radiate its heat at constant pressure, against radius, both logarithmic. Inside 76 kiloparsecs it is less than 7.7 billion years, the time since a redshift of about one when the cluster's core is thought to have settled, and at 10 kiloparsecs it is 0.8 billion years. Beyond a few hundred kiloparsecs it is longer than the age of the universe, and the gas is simply hot. Galaxies

A cooling flow that never arrives

The gas at the centre of a rich cluster radiates its heat in under a billion years, so it should be sinking inward and turning into stars at hundreds of solar masses a year. For two decades that was the accepted picture and it was treated as a measurement. Then X-ray spectra looked for the gas on its way down and found that it stops at a third of the temperature it began with, while the galaxy in the middle forms stars a hundred times too slowly. Something pays the radiative bill, and bubbles of the right size are visible in the X-ray images.

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The same added mass, a different size. A galaxy's half-light radius against its mass, relative to its starting values and both on doubling axes, as it grows by repeated dry mergers with companions of mass ratio 1:1, 1:3, 1:10, 1:30, each companion's squared dispersion scaling as the square root of its mass. Equal mergers keep radius proportional to mass; small mergers approach the limit of radius as mass squared. To reach twice the mass, 1:1 gives ×2.0, 1:10 gives ×3.6. The straight dashed line is the mass-squared limit. Galaxies

A galaxy that grows faster by eating small

Two equal elliptical galaxies merge and the result has twice the mass and twice the size. A galaxy that takes in the same mass as thirty small companions ends up nearly four times the size. The difference is one line of the virial theorem, in which companions bring mass without much energy, and it may answer a puzzle thrown up by the first deep infrared surveys: the most massive galaxies at redshift two are a fifth of the size of anything like them today.

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One orbit, chosen to spell 1, 3, 1, 4, 1, 5. The height of the particle on the axis of an equal-mass binary of eccentricity 0.3, from launch at phase 210° and speed 1.3105, found by a seeded nested search for the sequence of excursions 1, 3, 1, 4, 1, 5, in whole binary periods. Between each pair of crossings the particle's excursion lasts the listed number of periods, to the nearest whole one. Moser's theorem is about every sequence of large enough integers at small enough eccentricity and is exact; this is one finite word at one finite eccentricity, found numerically to show the kind of thing the theorem says exists. Gravitation

A particle on a line that can spell any word

Two equal stars orbit each other, and a speck of dust moves up and down on the line through their centre, perpendicular to their orbit. It has one coordinate and cannot leave that line. In 1973 Jürgen Moser proved that for almost any list of whole numbers, as long as they are large enough, there is an orbit whose successive excursions last exactly that many periods of the stars. The list can be chosen at random, so the orbit can do anything a coin can.

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A valley of limits, and how much of it each experiment holds. The strength α a Yukawa term could have, against its range, with both axes logarithmic and everything above the curve excluded at about 95% confidence. The curve is a rounded envelope of published limits: laboratory torsion balances below a metre, tower and mine gravimetry to a few kilometres, the orbits of geodetic satellites near 10⁷ metres, lunar laser ranging near 4×10⁸ metres at a few parts in 10¹¹, and planetary ranging beyond. It does not rule out a term at ranges under about ten micrometres or above about 10¹³ metres; the limits are weakest where no pair of masses of the right size can be arranged. Gravitation

A force that can hide at any one distance

The inverse-square law is tested by comparing the strength of gravity at two distances, and every instrument that does so is sensitive over about one decade of range. A Yukawa force with a range outside that decade is invisible to it: much shorter and it never reaches, much longer and it only rescales the constant. So testing the law is not one experiment made more precise but a chain of unrelated ones, from torsion pendulums to the orbits of planets, and the chain has weak links exactly where no pair of masses of the right size can be arranged.

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Jupiter and Saturn trading a degree of longitude over nine centuries. Mean longitudes of Jupiter and Saturn in arcminutes, with their straight trends taken out and a 40-year smoothing applied, from a planar integration of the Sun, Jupiter and Saturn from their elements at the year 2000. Jupiter swings ±20′ and Saturn ±50′, in opposite senses, with a period of about 870 years. Comparing Ptolemy's positions with those of his own day, Kepler could not fit one mean motion to both, and by 1676 Halley reported Jupiter gaining and Saturn losing. Before 1785 that looked as if the Solar System were unstable; the curve shows that it is a very long period. Orbits

A term too small to keep that a denominator kept

Every planetary theory is an expansion, and the order of its terms is meant to say which can be dropped. In the 1770s the largest unexplained motion in the Solar System, Jupiter speeding up and Saturn slowing down by a degree over centuries, came from a term that the ordering called negligible. Five of Saturn's orbits nearly match two of Jupiter's, and that one near-match divides a tiny third-order coefficient twice by a tiny frequency. An integration of the three bodies shows the swap directly, and shows what Laplace saw: it reverses every nine centuries.

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The first-order perigee rate is right for a slower Sun, and half wrong for ours. The rate at which an orbit's perigee turns, in turns per orbit, against the ratio m of the Sun's motion to the orbit's, from separate integrations of a satellite with the Moon's eccentricity under a solar tide on a circular path. The first-order rate (3/4)m² gives 0.0042 at the Moon's m of 0.0748, a perigee period of 18 years. The integrations give 0.0085, 8.9 years; the Moon's perigee takes 8.85. At m = 0.01 the integration gives 8.3e-5 against a first-order 7.5e-5. So the disagreement is not a matter of neglected forces: it depends on how large m is. Orbits

A perigee that moved twice as fast as Newton

The Sun's tide on the Moon should turn the line of its perigee once in eighteen years. It takes less than nine. For sixty years that was the outstanding failure of the law of gravitation, and in 1747 Clairaut told the Paris Academy that the inverse-square law would have to be amended. Two years later he withdrew the remark. The law was right and the calculation was unfinished: the series for the perigee has a small parameter that is not small, and its second term is seventy per cent of its first.

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The Moon is late by anything from a quarter of an hour to well over an hour. Minutes between one moonrise and the next at latitude 52° N, against the Moon's longitude along its orbit, with the node at 0°. The daily delay is 10 minutes at longitude 354° and 84 at 222°. The mean of 50 minutes everyone quotes is true only as an average. A full Moon stands opposite the Sun, so it passes the shallow point near the September equinox, and the crescent passes it in spring. The observed sky

A full Moon that stops rising later

The Moon rises about fifty minutes later each day. That number is an average that hardly any night obeys. At fifty-two degrees north, in late September, the full Moon rises only ten or twenty minutes later for several nights running, so that a bright Moon stands in the evening sky for most of a week, just when harvest once needed it. Nothing about the Moon changes. It is the angle at which its path meets the horizon, and that angle goes through its lowest value at one point of the ecliptic, which the autumn's full Moon happens to occupy.

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Rays from a ring of sunsets, bent into the shadow. Sunlight grazing the Earth at heights of 1, 5, 15 km, bent toward the axis by the atmosphere by twice the refraction at that height, reaching the Moon's distance, with the shadow's geometry in the plane through the Sun: the distance axis is compressed by about sixty to one against the vertical. The low rays cross the axis well before the Moon and reach the far side of the shadow; the higher rays only bend enough to enter its edge. The geometric umbra at the Moon has a radius of 4584 km, the two dashed lines, so without air the Moon would disappear inside it. Light arrives from the whole ring round the Earth, which is every sunrise and sunset at that moment. The observed sky

A Moon lit by every sunset at once

With no air, the Moon would vanish inside the Earth's shadow. Instead it turns a dull copper, because the atmosphere bends sunlight round the planet, and what reaches the Moon has gone through a ring containing every sunrise and sunset on Earth at that moment. Each height in the air lights one place in the shadow, so the colour across the disc is a map of the atmosphere by altitude. It has sometimes been black, after the largest volcanoes, and that makes a century of eclipse records a measurement of the stratosphere older than any instrument sent up to sample it.

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An outward path and a return path that cross, seen from a frame turning with the Moon. A trajectory in the Earth–Moon restricted three-body problem, in the frame rotating with the Moon. It leaves a 185 km parking orbit at 10.95 km/s, passes 716 km behind the Moon 3.0 days later, and returns to a perigee of -9 km without an engine firing, 6.0 days after departure. The crossing of the two legs is why it is called a figure of eight. The Moon's pull turns the direction of travel, and in a stationary frame the Earth is a much weaker influence on the far side; neither fact alone makes the geometry close. The Earth and Moon are at true scale. Spaceflight

A way home that needs no engine

A spacecraft can leave the Earth so that the Moon's gravity turns it round and sends it home with no further burn. Drawn in the frame that turns with the Moon the path is a figure of eight. It was flown to the Moon by every early Apollo mission for one reason: if the engine fails, the crew comes back. That safety is exquisitely narrow. A tenth of a metre a second at departure moves the perilune by tens of kilometres and the return by thousands, and the perilune it allows is not one that most landing sites can use.

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Seven orbits in fifteen years, each turned from the last. The ellipse of an Aldrin cycler, semi-major axis 1.60 AU and eccentricity 0.390, in the Sun-centred frame, for 7 successive synodic periods, the line of apsides advanced by 48.7° each time. The inner circle is the Earth's orbit and the next is Mars's. Over about fifteen years the pattern has gone once round. The same spacecraft would need to be carried from each ellipse to the next by a flyby of the Earth, or a small burn. The drawing is the circular, coplanar case; the real one has to be recalculated for the actual orbit of Mars. Spaceflight

A timetable that orbits the Sun

Every trip to Mars leaves when the planets line up and accelerates everything it carries. Buzz Aldrin proposed in 1985 that there need be only one departure: a large ship on an orbit that returns to the Earth each synodic period, with crews riding small taxis out to catch it. The orbit almost exists without any propellant. It has a semi-major axis fixed by a single condition, an aphelion that passes Mars with nothing asked for, and one unavoidable bill: it must turn by about fifty degrees every twenty-six months, and the Earth can pay for all but seven of the eighty the turn requires.

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Almost everything is ionised, and then almost nothing is. The ionised fraction of hydrogen against distance from an O7 star in gas of 10 per cubic centimetre, from integrating photoionisation balance outward, in units of the counted radius of 14.62 parsecs. The fraction is near one until about the edge and then falls from 90% to 10% within 0.192 parsecs, 1.3% of the radius. That width is about 18 mean free paths of an ionising photon in neutral gas (0.011 pc). Nothing about the star sets it. A region with no gradual boundary is a surprising result of a smooth input, and it comes from the fact that a photon that gets as far as the neutral gas is absorbed at once. Starlight

An ionised sphere that is a count of photons

A hot star sits in a cloud of hydrogen and ionises it, out to a radius, and there the ionisation stops within a tiny fraction of the distance. That edge was predicted in 1939 from a single count: every recombination uses up one photon, so the star can hold ionised only as many atoms as its light can pay for. By the equation that describes a stellar atmosphere at the same temperature, the same gas would hold ten orders of magnitude fewer neutral atoms than it does. In a nebula the temperature is not what decides the ionisation. A budget is.

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Whether an excited ion radiates depends only on what reaches it first. The rate per second at which an oxygen ion in the state that gives the 5007 Å line loses its energy, by radiating and by collision, against electron density, both logarithmic. Radiating takes about 51 seconds whatever the surroundings. Collisions speed up with density and cross at 6.8e+5 per cubic centimetre. The air at the Earth's surface, 2.7×10¹⁹ molecules per cubic centimetre, has a collision rate about 4×10¹³ times the radiative one, so the ion loses its energy without radiating. A nebula at 10⁴ per cubic centimetre is sixty times below the crossing. The line has a tiny transition probability; the atom was never forbidden to make it, and it requires only the emptiness in which there is time. Starlight

An element that was only empty space

In 1864 William Huggins turned a spectroscope on a planetary nebula and saw a bright green line that matched nothing on Earth. It was given a name, nebulium, and it was treated as a new element for sixty years, until the periodic table had no room left for it. In 1927 Ira Bowen showed it was ordinary oxygen, emitting by a route that collisions close long before any laboratory vacuum could reach it. The line was never forbidden. It simply needs about a minute of being left alone, and nothing on Earth gets that.

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A binary that goes silent for part of its shrinking. The orbital period of a white dwarf of 0.8 M☉ and a donor from 0.5 M☉, in a one-zone evolution, against time. Mass transfer drives the period down fast while magnetic braking acts. At 3.17 hours the donor reaches 0.2 M☉ and loses its radiative core and with it the brake. Its radius, swollen by the fast transfer, relaxes and it falls inside its Roche lobe; transfer stops (dashed, no accretion, nothing to see). Gravitational waves alone close the orbit, until contact returns at 2.30 hours, 0.60 billion years later. The observed gap runs from 2.15 to 3.18 hours. The donor's thermal time at 0.2 M☉, set here to 3×10⁸ years, is chosen to place the upper edge, so that edge is fitted and the lower one is not. The period reaches a minimum of 68 minutes against an observed 80, a shortfall every model of this simplicity shares. Stars

A binary that goes silent for an hour of period

Binaries of a white dwarf feeding off a small star can be found at almost any orbital period down to about eighty minutes, except between about two and three hours, where they are strikingly rare. They have not stopped existing there. They have stopped shining. The moment the donor loses its radiative core, the wind's magnetic brake appears to switch off. The star, swollen by the transfer it was being driven through, shrinks back inside its own Roche lobe, and the feeding stops. Only gravitational waves remain to close the orbit, and it takes them about half a billion years.

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A reservoir so large that it does not notice. The fractional increase in the heat of the Sun's convective envelope if a fraction 0.001, 0.003, 0.01 of the surface is held under spots, against time. The envelope holds about 7×10⁴⁵ erg, 58 thousand years of the Sun's light. Spots covering 0.3% of the surface for an eleven-year cycle add 4.2e-7 of that, a temperature change of 0.4 parts per million at most, spread over the whole envelope. So the blocked heat simply remains in the star, and the surface flux shows the deficit. It levels off only after the envelope's thermal time, when the star has adjusted. Stars

A spot whose missing heat goes nowhere

A sunspot blocks most of the energy trying to come up beneath it, and a large group visibly dims the whole Sun by a thousandth. Energy is conserved, so the blocked flux ought to come out somewhere else, and for a century observers looked for the bright ring around every spot. There is none to speak of. The heat has gone into the rest of the convection zone, spread so fast and so widely that the surface cannot notice, and it will stay there for longer than human history. For the time being the Sun simply is fainter, without any contradiction.

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Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

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