Astronomy & Celestial Mechanics
the sky, computed
Concept

Scale height — where it appears

The vertical distance over which an atmosphere's density falls by a factor of e, set by its temperature and the gravity holding it down. It is what makes an entry corridor narrow and an orbital lifetime steep, since a density falling by a factor of e every few kilometres changes everything over a small change of altitude.

Named by 17 essays across 5 fields — each of them below, with the objects they name alongside it.

7 clocks, and a body 233 kilometres across. A stellar occultation reduced. Each horizontal segment is one observer's chord: the star vanished, the star came back, and the interval multiplied by the shadow's 21.4 km/s across the ground is the length drawn. The longest, at an offset of -28 km, is 252.3 kilometres. The dashed ellipse is the limb fitted to the chords by least squares in the half-length squared, and its equivalent-area diameter is 233.9 km against the silhouette's true 232.9 — the residual is the shape the fitted ellipse cannot hold, not an error in any timing. The two open marks are observers inside the predicted path who saw nothing, and they are measurements: they bound the limb inside their own offsets, which is what fixes the extent when the positive chords all fall on one side. At 0.12 s per contact the length precision is 2.6 km, 1.10 per cent of the body — a size measured with a clock rather than with an angle, on an object no telescope resolves.

A shape measured by the edge of a shadow

When a small body passes in front of a star the observable is two times. Multiply the interval by the body's speed across the sky and it becomes a chord across a silhouette no telescope can resolve — and enough chords give a profile to a kilometre or two, for an object a few hundred across.

sky · Occultations Down by 280 km, and faster by 164 m/s. A circular orbit at 400 km with a ballistic coefficient of 100 kg/m², integrated down to 120 km through a tabulated atmosphere at solar minimum and solar maximum. At solar min it takes 1.2 years; at solar max it takes 147 days — a factor of 2.9 for the same satellite in the same orbit, decided by an eleven-year cycle nobody controls. The rising curves are the orbital speed on the right-hand scale, and they are the point: the drag force is opposite the motion and takes energy out, and the body goes faster, from 7673 to 7836 m/s. There is no contradiction in it. The specific energy is −μ/2a, so removing energy shrinks a, and the circular speed √(μ/a) rises when a falls; the kinetic energy gained is exactly half the potential energy lost, and the other half is what the air took. Every point on every curve was integrated from da/dt = −(ρ/β)√(μa), and the speed at each point is √(μ/a) at that point rather than a separate model.

An orbit that speeds up as it is slowed down

Drag takes energy out of a satellite and the satellite goes faster. There is no paradox in it, only a sign — and the same sign makes a re-entry date a space-weather forecast rather than an orbital computation, which is why Skylab was predicted for 1983 and came down in 1979.

spaceflight · Atmospheric drag A radius that depends on the colour it is measured in. Transit depth against wavelength for a planet of 1.38 Jupiter radii at 1400 K, whose atmosphere has a scale height of 538 km — computed from H = kT/µg, not assumed. One scale height of extra opacity adds 153 parts per million to a transit of 1.40 per cent, so the whole spectral signal is 862 ppm at its strongest: one part in 16 of the transit that carries it. The features are at real band centres — sodium at 0.589 µm, water at 1.4 µm, carbon dioxide at 4.3 µm — with the rise at the blue end the Rayleigh slope of scattering off the smallest particles.

A radius that depends on the colour it is measured in

Measure a transit in one colour and then another, and the planet is a different size. The difference is a few atmospheric scale heights, which is a few hundred parts per million of an already tiny signal.

exoplanets · Exoplanet atmospheres Pluto and Triton sit on the nitrogen line, and the Earth sits between helium and nitrogen. Escape speed against exospheric temperature, with a criterion line for each molecular species at v_esc = 6 v_th — the speed at which the Jeans loss time is comparable to the age of the solar system. Every line has slope one half, because the thermal speed goes as √T; a body above a line keeps that gas and a body below it does not. The horizontal axis is the temperature at the exobase, which for the Earth is near 1000 K rather than the 255 K of its equilibrium — the difference is extreme-ultraviolet heating, and using the wrong temperature puts the Earth above the hydrogen line, keeping an atmosphere it observably lost. Three readings are worth making. The Earth falls between helium and nitrogen and does exactly that: it loses helium as fast as radioactive decay supplies it, and keeps nitrogen for ever. Titan sits just above nitrogen and just above methane, which is why it has a thick nitrogen atmosphere and is slowly losing its methane. And Pluto and Triton sit on the nitrogen line, within 1 per cent — which is why both have atmospheres that are marginally bound and measurably escaping. Where it fails it fails in one direction only. Mercury, the Moon and the Galilean satellites all plot above lines for gases they do not have, because retention is necessary and not sufficient: a body also needs a source, and needs to survive the non-thermal losses this criterion says nothing about. Venus is the sharpest case — it sits above the hydrogen line and has still lost an ocean, because that hydrogen left by charge exchange with the solar wind rather than by moving fast enough.

The gas a planet cannot keep

Escape is a statement about the tail of a distribution rather than about its mean, so the threshold is not a speed but a dimensionless number near thirty — and past that number the loss rate falls by twelve orders of magnitude. Then, for a hot Jupiter, the whole picture fails and the atmosphere leaves as a wind.

exoplanets · Atmospheric escape The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in.

A planet ten times larger in one colour

A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving.

exoplanets · Atmospheric escape One mass and one radius, and every composition that gives them. A planet of 5 Earth masses and 1.6 Earth radii, and the compositions consistent with it. The horizontal axis is the fraction of the planet's mass in an iron core and the vertical axis the fraction in a water layer outside the rock; the heavy curve is every pair that reproduces the measurement exactly, and the band around it is what the 0.05 Earth-radius uncertainty allows. The answer is a curve, not a point, and that is not a failure of precision. Two numbers cannot determine three components: a planet can be made denser by adding iron or lighter by adding water, and along this locus the two changes cancel exactly. The ends of it are not small variations on one planet. At the left is a body with no iron at all and 0 per cent of its mass in water; at the right, one with an iron core like Mercury's and 23 per cent water. Those have different formation histories, different interiors, different everything, and the same mass and radius to the precision anybody can measure them. Breaking the degeneracy needs an observation that is neither a mass nor a radius. The usual one is a transmission spectrum, which measures the atmosphere's scale height and so its mean molecular weight — a hydrogen envelope and a steam envelope differ by a factor of nine in that, and the corresponding factor in the size of the spectral features. What the picture assumes is that the planet is differentiated into clean layers, which is the standard assumption and is false in detail: water dissolves into silicate melt at these pressures, and a mixed interior sits at neither end of this curve.

One density, and every planet that has it

A mass and a radius are two numbers, and a differentiated planet has at least three components. The set of compositions matching a measurement is therefore a curve rather than a point — and its two ends are a body with no iron and half its mass in water, and a body with a Mercury-like core.

exoplanets · Planet composition A layer several times thicker than its heat can explain. The vertical density profile of the neutral gas layer, drawn twice. The narrow curve is what thermal pressure alone supports: an 8-kilometre-a-second sound speed against the vertical gravity of the stellar disc gives a scale height of 47 parsecs. The broad curve adds the three pressures that are not heat — turbulent motion, the magnetic field and the cosmic rays — which together roughly triple the total and, because the scale height goes as the square of the effective dispersion, thicken the layer by a factor of 3.1 to 145 parsecs. The observed half-thickness of the H I layer is about 150. The narrow curve is a real prediction and it is wrong, and what is missing from it is precisely the two terms that emit nothing: a field measured by Zeeman splitting and Faraday rotation, and a particle population measured by what it does to a detector on a mountain. A galaxy's gas disc is inflated by things that cannot be photographed, and the thickness is how the inflation is measured.

A disc held open by what cannot be photographed

Solve hydrostatic equilibrium for the galaxy's gas layer using its temperature and the disc's gravity, and the answer is a layer several times thinner than the one that is there. What is missing from the calculation is turbulence, a magnetic field and a population of cosmic rays — three pressures of comparable size, two of which emit nothing.

galaxies · Interstellar medium Three planets that are the same spectrum. A model transmission spectrum, in scale heights of apparent radius, drawn three times: once as it is, once with the reference radius raised by 0.45 scale heights and the abundance reduced to compensate, and once with a cloud deck truncating the features. The three differ by 0.21 scale heights root-mean-square against features of 2.1, which is well inside the error bars of any real observation. The reason is structural rather than observational: a transmission spectrum measures a difference in apparent radius with wavelength and never an absolute radius, so the level is a free parameter, and shifting the level trades against the abundance almost exactly. Adding a cloud deck adds a third parameter that flattens features and trades against both. Three unknowns and one curve is why the quoted abundance uncertainties from transmission spectroscopy are so much larger than the photometric precision suggests.

A spectrum flattened by cloud, or by nothing

A transmission spectrum measures how a planet's apparent radius changes with wavelength, and never the radius itself. That missing level is a free parameter, it trades almost exactly against the abundance of whatever is absorbing, and a cloud deck adds a third unknown to a curve that constrains two.

exoplanets · Exoplanet atmospheres Where the missing kilometres a second go. The two losses along a gravity-turn ascent, against the vehicle's thrust-to-weight ratio at lift-off, from an integration of the trajectory rather than from a table. Every run spends the same 9400 m/s of ideal Δv and every one is flown as the same manoeuvre: one pitch kick, solved by bisection so that the vehicle is horizontal at burnout, and thereafter zero angle of attack so that gravity alone turns it — which makes the steering loss identically zero and the comparison a fair one. What differs is how much of the Δv survives as speed. The gravity loss is ∫g sin γ dt, the part of the thrust spent holding the vehicle up rather than accelerating it, and it falls as the thrust rises because a vehicle that leaves quickly spends less time doing it: 1263 m/s at T/W = 1.15 against 381 at 2.2. The drag loss is ∫(D/m) dt and it rises, because the same haste means reaching high speed lower down where the air is: 119 m/s against 1874. The sum is least at T/W ≈ 1.5, at 1198 m/s, and the minimum is shallow — which is why real vehicles cluster between 1.2 and 1.5 and none of them is there because of this curve. At the marked 1.5 the 9400 m/s of ideal Δv leaves the vehicle at 8202 m/s and 42 km, against a circular speed of 7884 m/s there, so the 1198 m/s of loss is the whole of the answer to why orbit costs about 9.4 km/s when orbital speed is under 7.9.

Orbit costs 7.8 and a launch buys 9.4

The gap between orbital speed and the velocity change a launcher spends is not overhead. It is three integrals along the ascent, only one of which can be reduced by flying better, and the two that can be traded move in opposite directions.

spaceflight · Rocket equation The zenith sky after sunset, as single scattering predicts it. The brightness of the zenith sky at 550 nm, in magnitudes per square arcsecond with brighter upward, against the Sun's depression below the horizon, computed by single scattering in a spherical atmosphere: US Standard Atmosphere densities, Rayleigh scattering, a 300 Dobson-unit ozone layer, and every photon scattered exactly once. The Earth's shadow climbs the zenith as R(sec d − 1) — 8.7 km at 3°, 35.1 at 6°, 142 at 12° — through air that thins by a factor of e every eight kilometres or so, so the brightness does not fall steadily: it falls 1.4 magnitudes a degree between 3° and 6°, and 2.7 a degree between 7° and 10°. The model gives 14.3 at the end of civil twilight and 27.0 at the end of nautical. The dashed line is the natural night sky, 21.9 magnitudes per square arcsecond. Single scattering reaches it at 9.25° of depression — 8.8 degrees before the 18° at which astronomical twilight is observed to end. The difference is not an error in the arithmetic; it is the light this model leaves out, scattered more than once.

The shadow that climbs the zenith

After sunset the sky overhead is lit only above the Earth's own shadow, and the shadow climbs as the square of the Sun's depression. Scattering computed once from that geometry predicts a sky that dims faster and faster and is as dark as night by nine degrees. The real sky takes eighteen, and the difference is light that has been scattered more than once.

sky · Twilight How long a planet's hydrogen lasts, against how much of it there is. The time a young Sun-like star's saturated X-ray and ultraviolet output would take to remove a planet's whole hydrogen envelope, at 100 times the Earth's insolation, against the envelope's share of the planet's mass, for cores of 3, 5, 8 Earth masses. It is computed with energy-limited escape and an interior fit for the envelope's thickness, and it is not monotonic. A heavy envelope takes long to remove because there is a lot of it. A very light one takes long because the planet is small and intercepts little light. In between the time peaks, and it peaks where the envelope has swollen the planet most for its mass: 3 Earth masses at an envelope of 1.9 per cent, 129 Myr, where the envelope is 1.31 times as thick as the core's radius; 5 Earth masses at an envelope of 2.8 per cent, 364 Myr, where the envelope is 1.30 times as thick as the core's radius; 8 Earth masses at an envelope of 4.0 per cent, 946 Myr, where the envelope is 1.30 times as thick as the core's radius. The peak is what makes a valley. A planet above it losing gas moves up the curve, its remaining envelope lasting longer and longer, and settles; a planet below it moves down, lasting less and less, and loses everything. The horizontal lines are the saturated phase, 100 Myr, and the 300 Myr of saturated-equivalent exposure the whole history delivers: a core whose peak lies below the second line cannot keep any envelope at all.

The envelope that doubles a planet lasts longest

The gap in the radii of small planets is not merely a place where planets are rare — it is nearly empty, and the reason is a peak. The time a young star needs to strip a planet's hydrogen is longest for the envelope that swells the planet to a little over twice its core's size. Anything thinner runs away to nothing, and anything thicker settles back towards the peak.

exoplanets · Radius valley How far the Sun has to sink before the sky darkens, on four worlds. The dimming of the zenith sky relative to sunset, in magnitudes, against the Sun's depression, for single scattering in an isothermal atmosphere with the stated scale height and vertical optical depth, scattering isotropically: the Earth (R 6371 km, H 8.5 km, τ 0.097); Mars (dust) (R 3389.5 km, H 11.1 km, τ 0.5); Titan (haze) (R 2574.7 km, H 50 km, τ 4); Pluto (haze) (R 1188 km, H 50 km, τ 0.02). These are representative values for the layer that does the scattering — air on the Earth, dust on Mars, haze on Titan and Pluto — not full models of those atmospheres. The angle over which a sky darkens is set by the height of the scatterers against the size of the planet, √(2H/R), because that is the depression at which the shadow over the zenith has risen one scale height. The Earth dims by ten magnitudes at 8.6° of depression, which at the equator of a world whose solar day is 24 hours takes the Sun 35 minutes. Mars (dust) dims by ten magnitudes at 13.4° of depression, which at the equator of a world whose solar day is 24.66 hours takes the Sun 55 minutes. Titan (haze) dims by ten magnitudes at 29.8° of depression, which at the equator of a world whose solar day is 382.7 hours takes the Sun 31.7 hours. Pluto (haze) dims by ten magnitudes at 43.9° of depression, which at the equator of a world whose solar day is 153.3 hours takes the Sun 18.7 hours.

The air's height against the planet sets the twilight

On the Earth the sky overhead fades over about nine degrees of the Sun's descent. That angle is not a property of air or of sunlight. It is the square root of twice the height of the scattering layer divided by the radius of the planet, and on a small world with a tall haze it grows to tens of degrees — so that twilight on Titan lasts more than a day.

sky · Twilight A shadow whose centre is brighter than no shadow at all. The flux an observer records against their distance from the centre of the shadow, for a body of half-light radius 1180 km with an isothermal atmosphere of scale height 55 km. Away from the centre the curve is the ordinary occultation light curve — the star fading as refraction spreads its light — and near it the two limbs' contributions both carry a geometric factor of the impact parameter over the shadow position, which grows without bound on the axis. The spherical atmosphere reaches 34.79 of the unocculted flux. The ray that arrives on the axis has impact parameter 1019 km, which is 161 km — 2.9 scale heights — below the half-light level, at a pressure 19 times higher. That is the only part of an occultation that reaches there. The peak is finite only because the star is not a point: the geometric factor is softened at 3 km, which is the star's own size projected to the shadow. The second curve is the same atmosphere flattened by 2.0 per cent, which spreads the focus over 20 km and drops the peak to 19.8. A real flattened body gives a caustic rather than a broad peak — several sharp spikes, spread in two dimensions rather than one — so the width here is right, the structure is not, and the height is an upper bound.

The brightest instant of an occultation is its middle

A spherical atmosphere is a lens with a focal length of astronomical units, and an observer standing at the exact centre of the shadow is standing at its focus. The star does not disappear there — it brightens, by more than it would have been unocculted, and the ray that arrives has come from far deeper than anything else in the event.

sky · Occultations Plasma beta from the corona's base outward. The ratio β of gas pressure to magnetic pressure against distance from the Sun's centre, one to ten solar radii on logarithmic scales, for a static isothermal corona in a dipole field. quiet Sun, 5 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.41 and crosses one again at 3.46 solar radii; quiet Sun, 10 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.1 and crosses one again at 5.00 solar radii; active region, 30 G at the base, with 3·10⁹ electrons per cubic centimetre at 2.5 million kelvin: β starts at 0.058 and crosses one again at 2.97 solar radii. The gas pressure falls steeply at first and then levels off, because gravity weakens with distance; the field's pressure, falling as the sixth power, never levels. Every corona therefore has a second place where the gas takes over, and it sits at two to four solar radii — where the closed loops of the corona end and the streamers' tips begin.

A corona held by its field and opened by its gas

A thousand kilometres above the Sun's surface the gas pressure has fallen so far that the magnetic field takes charge, and the corona's loops are the shape the field gives it. But the gas pressure of a hot atmosphere stops falling a few solar radii out, while the field's keeps falling as the sixth power of distance. So there is a second place where the gas takes over — and it is where the loops end, the streamers begin, and the solar wind is let out.

stars · Plasma beta The zenith sky after sunset, with light scattered twice. The brightness of the zenith sky at 550 nm in magnitudes per square arcsecond, brighter upward, against the Sun's depression: light scattered once (dashed), light scattered twice (dash-dot), and their sum (solid), in a spherical atmosphere with US Standard densities, Rayleigh scattering and 300 Dobson units of ozone. The dashed horizontal line is the natural night sky, 21.9 magnitudes per square arcsecond. Once-scattered light reaches it at 9.2°; the sum, dominated beyond about seven degrees by light scattered twice, reaches it at 13.2°, and it falls to a hundredth of the night sky — five magnitudes fainter, about the smallest excess a careful measurement of the sky can detect — at 17.5°, where once-scattered light alone would have reached that level at 11.9°. The second order falls more slowly than the first because it can reach the zenith by two grazing hops — sunlight to the lit air low towards the Sun, and that air to the shadowed air overhead — each of which needs only part of the depression a single hop would. Orders beyond the second are left out, so the solid curve is a lower bound on the sky.

The night is late because the light bounces twice

Light scattered once can reach the zenith only while the Earth's shadow over the observer is still low in the air, and by that reckoning the sky should reach the darkness of night about nine degrees after sunset. It does not until eighteen. The difference is light scattered twice — first by the sunlit air hundreds of kilometres towards the sunset, then again by the shadowed air overhead — and adding it to the calculation moves the end of twilight almost exactly to where observers have always put it.

sky · Twilight A disc weighed by how far its stars rise. The share of the circular speed at 2.2 disc scale lengths supplied by the disc, against the vertical velocity dispersion of its stars measured at 1.5 scale lengths, for a disc of scale length 3 kpc whose rotation curve reaches 200 km/s, with scale heights of 0.30, 0.41, 0.60 kpc. The share grows linearly with the dispersion, because the surface density goes as its square and the disc's speed as the root of the surface density. A maximal disc, supplying about 85 per cent of the speed (dashed), needs a dispersion of 44 km/s at the middle thickness. A measured 24 ± 3 km/s (shaded) gives 0.41–0.52: a sub-maximal disc, with the halo supplying most of the speed even at the disc's own peak.

A disc weighed by how far its stars climb

The rotation curve cannot say how much of a galaxy's speed belongs to its disc, because any split between disc and halo fits it equally well. The stars' motions perpendicular to the disc can, with no halo in the argument at all — and when that was done for thirty face-on spirals the discs came out light. But the vertical weighing needs the disc's thickness, which a face-on galaxy hides, and it needs the dispersion of the stars that hold the disc up, which the light measured may not belong to.

galaxies · Rotation curves The Sun's height, against the radius it is measured over. The Sun's height above the midplane as a survey would measure it, fitting the midplane to the stars within a projected radius R, against R, for a corrugation of 15 pc on a 2500-pc wavelength with the Sun 15 pc above the midplane beneath it. Nearby the answer is 15 pc whatever the phase; over several kiloparsecs it is the height above the mean plane: 30 pc with the Sun on a crest, 0 pc with it in a trough. Between them every intermediate value is a correct answer to a different question — a spread of 31 pc, comparable with the five to thirty parsecs the published measurements span.

A midplane that depends on how far it is averaged

The Sun sits a little above the middle of the Galaxy's disc — twenty parsecs, by the most careful recent counts, and anywhere from five to thirty by others. If the disc were flat that scatter would be error. It is not flat: the disc near the Sun ripples vertically by tens of parsecs on kiloparsec wavelengths, and on a rippled disc the Sun's height above 'the' midplane depends on how far around it the midplane is averaged. Every value in the scatter can be a correct answer to a different question.

sky · Celestial sphere

Named alongside it

The objects these essays reach for when they reach for this one.

Rayleigh scatteringEnergy-limited escapeMean molecular weightPhotoevaporationTwilightBallistic coefficientThe central flashEarth shadowThe half-light radiusHydrodynamic escapeHydrostatic equilibriumMagnetic pressure

All concepts