Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 23

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 441–460 of 534.

Exoplanets

Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.

An ephemeris fitted to 120 days, 43 minutes wrong within a year against a band of ±5.1. Transit times of a 6 Earth-mass planet on a 10-day orbit, perturbed by a 14 Earth-mass planet at 15.24 days, integrated for 1460 days and compared with a straight-line ephemeris fitted only to the transits in the first 120 days — the shaded window. Inside the window the line fits to 2.1 minutes. Outside it the pair's 317-day super-period carries the transits away from the line, and within a year of the window closing the prediction is 42.8 minutes early of the observed transit, 347 days after the last fitted one. The narrow band is the formal three-sigma uncertainty of the same line for a timing precision of 0.5 minutes per transit, which at that date is ±5.1 minutes: the error is 8.4 times the band. A statistical uncertainty assumes the residuals are noise, and these are a signal, so the band describes a planet that does not exist.

A forecast that fails on a schedule

A transiting planet perturbed near a resonance keeps a clock that wanders, and a straight-line ephemeris fitted to part of the wander predicts the next transit with a confidence the wander does not deserve. The error is not noise and does not average down; it grows on the pair's super-period, it is many times the formal uncertainty within a year, and how soon it appears depends on which stretch of the wander happened to be observed. When a model that includes the known perturber still fails, the failure has a period, and the period is a planet.

6 figures · Transit-timing Io's eclipses, early at opposition and late at conjunction. The delay in the timing of Io's eclipses by Jupiter's shadow caused by the changing distance between the Earth and Jupiter, against days from an opposition, for circular orbits at 1 and 5.2026 AU. At opposition the two planets are closest and the eclipses arrive 8.3 minutes early against the average; 199 days later, near conjunction, they are furthest apart and arrive 8.3 minutes late. The whole swing, 16.6 minutes, is the time light takes to cross the diameter of the Earth's orbit, and the pattern repeats every 399 days, the synodic period. This is transit timing done on a moon in 1676, with a clock that ran on Io's 42.5-hour orbit and a residual that had nothing to do with Io.

A transit late by the width of an orbit

A transiting planet's clock can run fast and slow for a reason that has nothing to do with gravity acting on the planet. If its star is itself in orbit about a distant companion, each transit's light has further or less far to travel, and the timing wanders by the light-travel time across the star's orbit. It is the measurement that first showed light has a speed, made again on a different kind of clock.

6 figures · Transit-timing 5 perturbers that draw one 58-day timing signal. Every perturbing planet that gives a 3-day transiting planet the same timing signal — a sinusoid with a 57.8-day super-period and an amplitude of 1.27 minutes — placed wide of the nearest first-order commensurabilities inside and outside its orbit, with its mass found by integrating until the amplitude matched. outside, near 4:3 at 4.070 days needs 7.6 Earth masses and would move the star by K = 3.1 m/s; outside, near 3:2 at 4.620 days needs 12.0 Earth masses and would move the star by K = 4.8 m/s; outside, near 2:1 at 6.329 days needs 25.6 Earth masses and would move the star by K = 9.2 m/s; inside, near 3:2 at 1.966 days needs 6.9 Earth masses and would move the star by K = 3.7 m/s; inside, near 2:1 at 1.462 days needs 45.8 Earth masses and would move the star by K = 26.7 m/s. The period ratio is along the bottom on a logarithmic axis and the required mass up the side. A super-period fixes the distance from some resonance and not which resonance it is, and the amplitude then fixes a mass for each guess — so the timing alone returns a list rather than a planet. The velocity semi-amplitudes differ by a factor of 8.5 across the list, which is one of the two ways the list is shortened.

One timing curve and five planets that could draw it

A transiting planet whose times wander at a 58-day period, by just over a minute, is being pulled by something — but the period says only how far from some resonance the pull comes, not from which. Perturbers inside and outside the orbit, near four different commensurabilities, each with its own mass, reproduce the same curve to a fraction of a per cent. Timing alone returns a list, and even the detail that shortens it hides a coincidence of its own.

5 figures · Transit-timing A 10 Earth-mass Trojan started 10° from its point, seen in the planet's transits. A Jupiter-mass planet on a 4-day orbit about a 1 solar-mass star, with a 10 Earth-mass companion started 10 degrees beyond the leading Lagrange point of the same orbit, integrated for 160 days. Above, the angle between the companion and the planet as seen from the star: it swings about 60 degrees, between 51.1 and 70.3, with a period of 48.5 days measured off the curve, against 49.8 from the small-amplitude formula P/√(27μ/4). Below, the planet's own transit times minus a straight line: they swing by ±300.0 seconds at the same period, because the planet and the companion orbit their common centre of mass and the companion's libration moves that centre along the orbit. The companion shares the planet's period exactly, so it produces no timing signal at any orbital period of its own and would not transit on a schedule distinct from the planet's; the libration period is the only clock it has.

A companion on the same orbit, seen in the planet's clock

A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve.

6 figures · Transit-timing

The observed sky

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

Mars: a sundial 40 minutes ahead and 51 behind. The equation of time on Mars over one of its years, 668.6 sols long, against sols since the northern spring equinox, computed from Kepler's equation for an orbit of eccentricity 0.0934 and an axis tilted 25.19°, with perihelion at solar longitude 250.87°. The solid curve is the difference between true and mean solar time: it runs from −51.1 local minutes, 611 sols after the equinox, to +39.9, 385 sols after it — −52.5 to +41.0 in Earth minutes, since a local minute is a 1,440th of a sol. The dashed curves are its two parts. The eccentricity term, one cycle a year, swings by ±42.9 minutes; the obliquity term, two cycles, by ±11.4; the first is 3.74 times the second. Perihelion falls 485 sols after the equinox, marked, and the vertical lines are the equinoxes and solstices.

On Mars the orbit outweighs the tilt

The equation of time is the sum of two terms, one from the shape of the orbit and one from the tilt of the axis, and on the Earth they are nearly the same size. On Mars the orbit's term is almost four times the tilt's, a sundial runs from fifty-one minutes behind the clock to forty ahead, and the figure-of-eight the Sun traces in the Earth's sky becomes a teardrop. Nothing about the two terms is different; only their ratio is.

6 figures · Equation of time A Sun that runs backwards for 8.1 days. The rate at which the Sun moves across the sky of a planet with a 3:2 spin–orbit ratio and eccentricity 0.2056, in degrees of hour angle per Earth day, against days from perihelion, over one 87.97-day orbit. The rate is the spin rate minus the rate at which the Sun's direction turns because the planet moves along its orbit, and by Kepler's second law that orbital rate peaks at perihelion, at 1.551 times its mean. The spin is 1.5 times the mean orbital rate, so for 8.1 days around perihelion, from −4.0 to +4.0 days, the orbital rate wins, the rate is negative, and the Sun moves backwards across the sky by 1.11 degrees before resuming. At perihelion it is moving at 0.21 degrees a day in the wrong direction. Away from perihelion the Sun crosses the sky at up to 3.4 degrees a day, and a whole solar day, noon to noon, takes 175.9 days — two orbits.

A Sun that stops and runs backwards

On Mercury the equation of time is not a correction but a reversal. The planet turns three times for every two orbits, and near perihelion its orbital motion briefly outruns its spin, so the Sun halts, backs up by a degree over eight days and resumes. From one longitude that is three noons in a week; from another, a sunrise, a sunset and a second sunrise — and a slightly rounder orbit would have stopped it happening at all.

6 figures · Equation of time Solar noon by the clock, in four places on ordinary time. The clock time at which the Sun crosses the meridian, through a year, for London (0.13°W, clocks on UTC+0), Madrid (3.70°W, clocks on UTC+1), Vigo (8.72°W, clocks on UTC+1), Kashgar (75.99°E, clocks on UTC+8), with summer time where it is kept, from the end of March to the end of October. In London noon falls between 11:44 (4 November) and 13:07 (27 July); in Madrid noon falls between 12:58 (4 November) and 14:21 (27 July); in Vigo noon falls between 13:18 (4 November) and 14:41 (27 July); in Kashgar noon falls between 14:40 (4 November) and 15:10 (12 February). The smooth wave on each curve is the equation of time, the same ±16 minutes everywhere. The steps are summer time, a whole hour. And the vertical offset of each curve is where the place sits inside its time zone, which for a city in the far west of a wide zone is larger than both: noon near three in the afternoon, by the clock, is an ordinary consequence of one time zone spanning a large country.

The smallest term between a clock and the Sun

A clock and a sundial disagree for three reasons, and the one astronomy supplies — the equation of time, sixteen minutes at most — is usually the smallest of them. Where a place sits inside its time zone can put noon three hours after twelve, and summer time adds a whole hour on top. The equation of time is visible in ordinary life only where the other two happen to vanish.

5 figures · Equation of time An hour that was a twelfth of the daylight. The length of a daylight hour when the time from sunrise to sunset is divided into twelve, through a year, at Alexandria (31.2°N), Rome (41.9°N), London (51.5°N), Stockholm (59.3°N). At Alexandria the hour runs from 51 minutes at the winter solstice to 71 at the summer solstice; at Rome the hour runs from 46 minutes at the winter solstice to 76 at the summer solstice; at London the hour runs from 39 minutes at the winter solstice to 83 at the summer solstice; at Stockholm the hour runs from 30 minutes at the winter solstice to 93 at the summer solstice. The dashed line is sixty minutes, the length it has at both equinoxes everywhere. This is the hour of the ancient Mediterranean world and of medieval Europe until mechanical clocks: an hour defined by the Sun, which a sundial with suitably drawn lines reads exactly, and in which the equation of time does not exist, because nothing is being compared with a uniform clock. Sunrise, sunset and noon are each defined by the Sun, and the clock that would disagree with them had not been built.

An hour that stretched with the season

For most of recorded history an hour was a twelfth of the daylight — seventy-six minutes at a Roman midsummer and forty-six in midwinter — and a sundial read it exactly. In that system there was no equation of time, because nothing uniform was being compared with the Sun. The sixteen-minute correction became real only when the hour was made equal, and measurable only when clocks could keep time more steadily than the Sun by more than it.

6 figures · Equation of time

Spaceflight

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

The path of a spin across the sphere of fixed momentum. A body with principal moments 1, 8, 8.6, started about its axis of least inertia with a 3° wobble and an internal energy sink strong enough that the whole motion fits in 80 turns and each circuit of the path can be seen. The disc is the near hemisphere of the sphere of fixed angular momentum, seen from a direction between all three body axes, with the near end of each axis marked. Each thin curve is a contour of kinetic energy on that sphere — a polhode, one of the paths the angular momentum can follow in the body with no dissipation — and the thick curve is the separatrix through the intermediate axis, which divides motions that circle the axis of least inertia from motions that circle the axis of greatest. The coloured path is what the integration did: solid on the near hemisphere, dashed where it passes behind. It starts at the open dot and ends at the filled one.

A spin that left the axis it was given

The first American satellite was spun about its long axis, like a rifle bullet, and soon after launch it was tumbling end over end. Nothing outside it had pushed. A body that cannot change its angular momentum but can lose energy has exactly one place to end up, and a long body spun about its length is as far from that place as a spin can be.

6 figures · Attitude control A spin about the middle axis of a 1:2:3 body, turning over every 2.9 turns. A body with principal moments of inertia 1, 2, 3, spun about its intermediate axis with a hundredth of its angular momentum knocked onto the axis of least inertia, integrated with no dissipation and no external torque. The curves are the components of the angular momentum along the three body axes, as fractions of its fixed size. The intermediate component stays near one for 1.5 spin periods, then swings through zero to minus one — the body turns over, end for end — and keeps doing so every 2.9 periods, 14 times in the span drawn. Energy and angular momentum are both conserved throughout, the energy to better than one part in a billion; nothing is being lost and nothing drives the flips. A spin about the intermediate axis is an equilibrium like a pencil balanced on its point, and the smallest disturbance grows exponentially, here by a factor of e every 0.28 spin periods, until it carries the body to the opposite equilibrium and back.

A wingnut that turns over on its own

Spin a rigid body about the axis whose moment of inertia is neither the largest nor the smallest and it turns end over end, again and again, with nothing pushing it and nothing lost. The flip was noticed aboard a space station in 1985 and was already implicit in equations written in 1765. How long it waits is a logarithm, and no care in setting up the spin can make the logarithm infinite.

6 figures · Attitude control Where the gradient of gravity holds a spacecraft still. The plane of the two inertia ratios that decide whether a spacecraft pointing at the Earth is held there by the gravity gradient: k₁ — the pitch moment of inertia less the yaw moment, divided by the roll moment — across, and k₃ — the pitch moment less the roll moment, divided by the yaw moment — up, with roll along the velocity, pitch normal to the orbit and yaw towards the Earth. Shaded points satisfy all three conditions of the linear theory — pitch is stable when k₁ > k₃, and roll and yaw together when k₁k₃ > 0 and 1 + 3k₁ + k₁k₃ > 4√(k₁k₃). The large region at upper right, 12.4 per cent of the square, is the one in which the pitch moment is the largest and the yaw moment the smallest, the arrangement of a long boom hanging towards the Earth. The small region just left of the vertical axis and below the horizontal one, 2.0 per cent, is a second, narrow island of stability with the moments in a different order, found by DeBra and Delp in 1961. There the orientation is a maximum of the potential in roll and yaw rather than a minimum, held only by the gyroscopic coupling of the two, and a damper — the very thing the long-boom region needs — destroys it: with damping of 0.05 of the orbital rate, a swing of a hundredth of a radian at (−0.10, −0.21) grows to a full radian within 11 orbits, while the same swing on the long boom shrinks 435-fold in 40. A boom along the vertical, pitch moment largest sits at (0.97, 0.40) and is stable; the same boom with roll and pitch moments swapped sits at (0.93, −0.40) and is unstable: the same boom, with two nearly equal moments exchanged, crosses from one side of an axis to the other.

A boom held upright by a difference in gravity

A long spacecraft in orbit is pulled into line with the vertical for nothing — its near end feels slightly more gravity than its far end, and the difference is a torque. The torque restores and never dissipates, so the vehicle swings like a pendulum whose clock is the orbit. Whether it is held at all comes down to three inequalities between its moments of inertia, and one region that satisfies all three is destroyed by the damper every such spacecraft needs.

6 figures · Attitude control A tumble removed with a coil and a compass, at 500 km. The rotation rate of a small spacecraft — principal moments 0.0067, 0.041, 0.043 kg m², the proportions of a three-unit cubesat — tumbling at 8.8° a second after release, against orbits at 500 km, with nothing to control it but magnetic coils driven by the B-dot law: a dipole opposite to the rate of change of the field measured aboard, capped at 0.2 A m². The field is a dipole tilted 9.2° from the Earth's axis and turning with the Earth. In a polar, 97.4°, orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.58 orbits; in a 51.6° orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.32 orbits; in an equatorial orbit the rate settles at 2.44° a second over the last orbit drawn. The law needs no knowledge of the spacecraft's attitude: a tumbling body sees the Earth's field swing round in its own frame, and a dipole opposing that swing produces a torque that removes the part of the spin perpendicular to the field. The polar orbit does not reach zero. It settles at 0.94 of twice the orbital rate, 0.13° a second, and twice the orbital rate is how fast the field direction itself turns round a polar orbit: a body turning with the field sees little change to oppose. An equatorial orbit keeps the field pointing nearly the same way all the way round, so the spin about it is reached only through the dipole's tilt and the Earth's turning, and 28 per cent of the starting rate is still there at the end.

A tumble stopped by the field it tumbles through

A small satellite leaves its deployer tumbling, and the first thing most of them do is stop, using nothing but a magnetometer and three coils. The law they run needs no idea where the satellite is pointing. What it cannot do, at any instant, is touch the spin about the local field line — so how much tumble survives is decided by how much the field's direction changes along the orbit, and the stillness it reaches is defined by the field rather than by the stars.

7 figures · Attitude control

Galaxies

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

Five speeds from the same galaxies, and the relation each one gives. The exponent and the scatter of the baryonic Tully–Fisher relation fitted to the same 90 model galaxies with five different speeds, fitted as speed on mass. outer speed: exponent 3.98, scatter 0.080 dex in mass; peak speed: exponent 3.86, scatter 0.110 dex in mass; at 2.2 scale lengths: exponent 3.29, scatter 0.167 dex in mass; W50 ÷ 2: exponent 3.73, scatter 0.119 dex in mass; W20 ÷ 2: exponent 4.05, scatter 0.118 dex in mass. The galaxies were built with the relation in their outer halo speed, and the other four speeds each lose some of it: the ones read from the inner curve inherit how concentrated each galaxy's stars are, and the line widths add the turbulence of the gas, which matters most in the slowest galaxies.

The speed a line width stands in for

A galaxy does not have a rotation speed. It has a rotation curve, rising in the smallest galaxies and peaking early in the largest, and the Tully–Fisher relation is fitted to whichever single number is read off it. Build galaxies with the relation placed in their outer speed and read four other speeds from the same curves, and each gives a shallower or looser relation — which is why the choice of speed is a statement about where the relation lives.

7 figures · Tully–Fisher How the scatter depends on how much dispersion is added to rotation. The scatter of the baryonic Tully–Fisher relation, in dex of mass, when the same 80 turbulent model discs are measured with S = √(K Vᵣₒₜ² + σ²), against the weight K given to the rotation, for rotation measured at 1 and 2.2 disc scale lengths. Measured at 1 scale length the scatter is smallest, 0.076 dex, at K = 0.55, and is 0.076 dex at K = 0.5; measured at 2.2 scale lengths the scatter is smallest, 0.077 dex, at K = 0.23, and is 0.134 dex at K = 0.5. For an exponential disc with constant dispersion the pressure correction is exactly K = one over twice the radius in scale lengths, so the best weight depends on where the rotation is measured. The widely used K = 0.5 is the value that makes a pure rotator and a pure isothermal sphere of the same mass agree; it is also the pressure correction for rotation measured at one scale length, and not at any other.

A disc that turns slower than its mass requires

The star-forming discs of ten billion years ago were not the thin, cold, orderly discs of today. Their gas moved randomly at tens of kilometres a second, and that motion is a pressure that holds up part of each disc, so it turns more slowly than its mass alone would require. Read those rotation speeds as if rotation did all the work and the Tully–Fisher relation tilts and scatters as if galaxies had evolved, when what differs is how much of their weight is carried by disorder.

6 figures · Tully–Fisher

Starlight

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

Where the axes of rapid rotators point, in a sample chosen by brightness, at a darkening exponent of 0.19. The distribution of rotation-axis inclinations — 0° pole-on, 90° equator-on — for gravity-darkened stars whose axes point at random in space, surveyed down to a limit in apparent brightness, with the surface temperature following the local gravity to the power 0.19. The dashed curve is random orientation, which puts 13.4 per cent of stars within 30° of pole-on. At 0.9 of the critical rotation rate a star looks 1.093 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 14.4 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.09 times as common as random orientation would make them. At 0.98 of the critical rotation rate a star looks 1.188 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 15.3 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.19 times as common as random orientation would make them. Averaged over random orientations the apparent luminosity of each star equals its true luminosity to better than half a per cent, as it must; the tilt towards pole-on comes entirely from choosing stars by how bright they look.

The pole-on stars a brightness limit prefers

A rapidly rotating star looks brighter and hotter from its pole than from its equator, so a survey that picks stars by how bright they look should pick more of them pole-on. It does — and the surprise is how little. Averaged over random orientations, a gravity-darkened star's apparent luminosity is exactly its true one, because every photon goes somewhere; a brightness limit restores only a fraction of a per cent of bias, while the error in any single star is ten times larger and of either sign.

7 figures · Gravity darkening How fast the circulation a rotating star drives would stir it, against its lifetime. The classical Eddington–Sweet circulation time — the star's Kelvin–Helmholtz time divided by the ratio of centrifugal to gravitational acceleration at its surface — as a multiple of its main-sequence lifetime, against rotation rate as a fraction of critical, for stars of 2, 5, 15 solar masses, on logarithmic axes. Below the line at one, the circulation would turn the star over within its life. For 2 solar masses that happens above 0.101 of critical; for 5 solar masses that happens above 0.123 of critical; for 15 solar masses that happens above 0.115 of critical. The dot is the Sun, turning at 0.0084 of its critical rate, whose circulation would take 150 times its main-sequence life. By this estimate almost every star turning at more than a tenth of its critical rate should be stirred from core to surface; the measured surface compositions of such stars show that it is not, because the composition gradient left by core burning resists the circulation, and how strongly it resists is what the nitrogen at their surfaces measures.

The circulation that should have stirred every fast rotator

A rotating star in radiative equilibrium cannot be balanced in pressure and in heat at the same time, and the mismatch drives a slow circulation from pole to equator. The classical estimate of its speed says that any star turning at more than a tenth of its break-up rate should be stirred from core to surface within its life, bringing the nitrogen of hydrogen burning up with it. Some fast rotators show that nitrogen and some do not, and some slow rotators show it when they should not — including more than the angle of their axes can explain.

6 figures · Gravity darkening

Orbits

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

The two force laws whose orbits close. The apsidal angle — the angle swept from periapsis to the next apoapsis — against the exponent of the force law, for F ∝ r^p. The dashed curve is the near-circular limit π/√(3+p), which has a closed form; the solid curve is the same angle for an orbit of eccentricity 0.4, computed by quadrature of ∫(L/r²)dr/√(2(E−U)) between its two turning points, with U the effective potential. An orbit closes when the apsidal angle is a rational multiple of π, and an orbit closes at every eccentricity only where the two curves meet: p = −2 at exactly 180° and p = +1 at exactly 90°, which is Bertrand's theorem. The quadrature returns 180.0000° and 90.0000° at those two exponents and departs from the near-circular curve by 1.4° at p = 0. The angle diverges as p approaches −3, where the circular orbit stops being stable and there is no well left to oscillate in.

Only two force laws let an orbit come back

That a planet returns to the same point of its own path after one lap is not a fact about orbits. It is a fact about the exponent in the force, and out of the whole continuum of attractions only two — the inverse square, and a spring — bring every bound orbit back to where it started.

8 figures · Effective potential The same well after the primary has lost 45 per cent of its mass. Two effective potentials for one body: the solid curve before the primary loses mass and the faint one after, both at the same angular momentum, because a central force of any strength exerts no torque. The well shallows and its floor moves out from r = 1.00 to 1.82. The body's own level moves with it — from E = -0.420 to -0.127 — and the two horizontal lines are drawn where the radial action is conserved, which puts the turning points at 0.71–1.67 before and 1.30–3.03 after. The ratio between them is 2.3333 in both, so the orbit is the same shape at a larger size: everything about the body's path has scaled and its eccentricity of 0.4 has not moved. That is what a slow change leaves behind, and it is not what a sudden one leaves.

The well moves, and the body keeps its share of it

When the Sun becomes a white dwarf it will throw away half its mass, and every planet's orbit will swell by the same factor. Their eccentricities will not change at all — provided the loss is slow, and the only meaning "slow" has here is slow compared with one orbital period.

8 figures · Effective potential

Starlight

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

Where titanium oxide comes apart, and what the pressure does to it. The fraction of titanium oxide dissociated into Ti + O, against temperature, for a gas of equal parts at 10, 300, 6000 N/m² of total pressure. The equilibrium is Saha's with a free atom in place of the free electron and the pair's reduced mass in place of the electron's — the two atoms' partial pressures multiplied together, divided by the molecule's, equal an equilibrium constant Kₚ(T) — which for equal abundances gives a dissociated fraction of √(Kₚ/(Kₚ+P)) exactly. A molecule is one particle becoming two, so pressure suppresses the reaction exactly as it suppresses ionisation — the half-dissociation point moves from 3145 K at 10 N/m² to 4164 K at 6000. The bond energy is 6.87 eV against hydrogen's ionisation potential of 13.6, which is why this happens at a third of the temperature. Rotational and vibrational partition functions are included, classically and harmonically; the atoms keep their ground-term weights.

A spectrum with no continuum left in it

Below about four thousand kelvin a photosphere stops being a gas of atoms. Titanium oxide, water and carbon monoxide take over, the same equilibrium settles how much of each survives with a bond energy where an ionisation potential used to be — and the bands are so crowded that the level every line depth is measured against is nowhere on the plate.

9 figures · Ionisation Where a photosphere's free electrons come from. The share of the free electrons donated by metals rather than by hydrogen, against temperature, at a gas pressure of 12000 N/m² and at solar abundance, 1 dex below solar, 2 dex below solar. The electron pressure is not assumed here — it is solved for, as the value at which the ionisation of the gas supplies exactly the electrons the gas contains. Below about 5897 K at solar abundance every free electron in the gas comes from an element present at one part in ten thousand, because hydrogen's 13.6 eV keeps it neutral while magnesium's 7.6 does not. The handover is fast: solar crosses a half at 5897 K, 1 dex down crosses a half at 5088 K, 2 dex down crosses a half at 4467 K. At 5,772 K and this pressure the solved electron pressure is 1.78 N/m².

The continuum is made by one atom in ten thousand

The Sun's light leaves through an ion that exists only because a hydrogen atom will hold a second electron by three-quarters of an electronvolt. The electrons it holds come almost entirely from magnesium, silicon and iron — so the level against which every solar line depth is measured is rationed by elements present at one part in ten thousand.

7 figures · Ionisation

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