Astronomy & Celestial Mechanics
the sky, computed
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A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

Equation of time

  1. 1 The Sun is a bad clock, by up to sixteen minutes
  2. 2 Two fictitious suns in sequence
  3. 3 The earliest sunset is not the shortest day
  4. 4 On Mars the orbit outweighs the tilt
  5. 5 A Sun that stops and runs backwards
  6. +2 more
7 essays · sky 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.

Atmospheric drag

  1. 1 An orbit that speeds up as it is slowed down
  2. 2 A corridor a degree and a half wide
  3. 3 A density model wrong by a factor of two
  4. 4 A coefficient that belongs to the surface, not the satellite
  5. 5 A weather forecast made out of orbits
  6. +1 more
6 essays · spaceflight A magnification, and a spike inside it. The brightness of a background star as a foreground one passes in front of it. The smooth curve is exact: a point mass magnifies a point source by (u² + 2)/(u√(u² + 4)), which peaks at 6.7 for this track. The spike is the planet, of mass ratio 0.001, lensing one of the two images. Its duration is the Einstein time scaled by √q — about 23 hours against 30 days — so the whole planetary signal is a few hours in an event lasting a month, and it never repeats. The deviation is drawn in the approximation that the planet lenses the image in isolation; a real caustic crossing has structure this smooths over, and its true height is set by the source's size rather than by the geometry.

Microlensing

  1. 1 A star magnified by a planet nobody will see again
  2. 2 A light curve with a fold in it
  3. 3 An asymmetry that the Earth's own orbit puts in
  4. 4 A centroid that moves when the brightness does not
  5. 5 An event of a few hours and no host
  6. +1 more
6 essays · exoplanets Four abundances, one free parameter. The abundances big-bang nucleosynthesis predicts, against the one number it is free to choose: η₁₀, the ratio of baryons to photons in units of 10⁻¹⁰. Four curves spanning nine decades, from a helium mass fraction of about a quarter down to a lithium abundance of one atom in ten billion, and they are not four independent predictions — they all come out of the same reaction network run at the same density. The horizontal bands are what is measured in the sky, each at its published one sigma. The measurement that matters is deuterium, because its curve is the steep one: inverting the drawn curve at D/H = 2.527e-5 gives η₁₀ = 6.11, and the ends of the observed interval give 6.06 to 6.15. The vertical band is what the microwave background gives, 6.13 ± 0.04, from the height of the second acoustic peak relative to the first. Those two agree to 0.4 per cent, and they have nothing whatever in common: one is a nuclear-reaction network run in the first three minutes and read off a quasar absorption line, the other is a fluid oscillation at four hundred thousand years read off a sky map. Lithium is the exception and it is not a small one — the network predicts 4.70e-10 at the microwave background's density and the halo stars show 1.60e-10, a factor of 2.9 too much, which is unresolved.

Nucleosynthesis

  1. 1 Four abundances and one free parameter
  2. 2 A particle count taken from a dwarf galaxy
  3. 3 The residue that failed to burn
  4. 4 A factor of three, and the flatness that prices every cure
  5. 5 An abundance with no direction to correct in
  6. +1 more
6 essays · cosmology 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.

Occultations

  1. 1 A shape measured by the edge of a shadow
  2. 2 The edge of a shadow is a wave
  3. 3 A star that blinked before it should have
  4. 4 The brightest instant of an occultation is its middle
  5. 5 A population counted by shadows that never repeat
  6. +1 more
6 essays · sky How much of a rocket has to be propellant. Propellant fraction against the velocity change bought, for three real propellant combinations. The curves approach 1 and cannot reach it, so every extra kilometre per second costs a larger share of what remains — which is why staging exists.

Rocket equation

  1. 1 The exponential that decides what can be flown
  2. 2 The stage that has to be thrown away
  3. 3 The split that is not an equal split
  4. 4 Choosing a propellant is choosing a molecular weight
  5. 5 Orbit costs 7.8 and a launch buys 9.4
  6. +1 more
6 essays · spaceflight A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.

Transit timing

  1. 1 Planets found by a transit running late
  2. 2 A mass that is only a mass once the eccentricity is known
  3. 3 A forecast that fails on a schedule
  4. 4 A transit late by the width of an orbit
  5. 5 One timing curve and five planets that could draw it
  6. +1 more
6 essays · exoplanets Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.

Apparent motion

  1. 1 The loop a planet does not make
  2. 2 The solar system measured from inside one orbit
  3. 3 A right angle short by a seventh of a degree
  4. 4 A triangle of meetings that turns in eight centuries
  5. 5 Brightest as a crescent, and not as a disc
5 essays · sky An oscillation that does not matter, on a ramp that does. Stored angular momentum in a reaction wheel over 160 days at 550 kilometres, with a capacity of 25 newton metre seconds. The total environmental torque is 1.75e-4 newton metres, of which 35 per cent is taken to survive averaging over an orbit. The fast oscillation is the part that does not survive: it has the 95.6-minute orbital period, reaches 0.10 newton metre seconds, and returns to where it started every revolution, so it consumes capacity and nothing else. The ramp under it is the secular part, and its slope measured between two instants a whole number of orbits apart is 6.117e-5 newton metres, which is the secular torque and is how the figure checks itself. The wheel fills in 4.7 days and has to be emptied 33 times in the span drawn. Every attitude-controlled spacecraft in the collection lives on this sawtooth, and the vertical drops are the only part of it that costs anything: the store can be moved between wheels for nothing, and taken out of the vehicle only by pushing against something outside it.

Attitude control

  1. 1 The spin that has to be put somewhere
  2. 2 A spin that left the axis it was given
  3. 3 A wingnut that turns over on its own
  4. 4 A boom held upright by a difference in gravity
  5. 5 A tumble stopped by the field it tumbles through
5 essays · spaceflight The same recession law from two different galaxies. Twenty galaxies at fixed comoving positions after the whole picture has been multiplied by 1.34, with each galaxy's displacement drawn from where it was to where it is. Every arrow's tail sits at the old separation and its tip at the new one, so the arrow is exactly 0.34 times its tail's distance from the highlighted galaxy — proportional to separation for one reason and no other: a uniform scaling moves everything in proportion to its distance from whatever point the scaling is measured about. The right-hand panel measures from a different galaxy and gets the identical law with the identical constant. That is the content of a linear velocity–distance relation. It is the signature of an expansion with no centre, and the observation that every galaxy recedes is therefore not evidence that this one is the centre — it is evidence that none of them is.

Expansion

  1. 1 The shift that is not a Doppler shift
  2. 2 A redshift that changes while it is watched
  3. 3 An orbit feels the acceleration and never the rate
  4. 4 The age of the universe weighs the Local Group
  5. 5 The Sun's speed counted in quasars comes out twice too large
5 essays · cosmology GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst.

Gravitational waves

  1. 1 A distance with no ladder under it
  2. 2 A detector the size of the galaxy
  3. 3 One number where two masses were
  4. 4 Every pair arrives circular
  5. 5 A distance tangled with an angle
5 essays · gravitation three revolutions, on a turning Earth. The ground track of a circular orbit at 420 km and 51.64° inclination, over 3 revolutions, on an equirectangular graticule. The latitude is a sine wave bounded by ±51.64° exactly — sin φ = sin i sin u, so the inclination is the highest latitude the orbit ever passes over, and it is reached twice per revolution. Each successive pass is displaced west by (ω⊕ − Ω̇) × 92.90 min = 23.61°, of which 0.32° is the orbital plane's own regression and the rest is the planet turning underneath: the vehicle comes back to nearly the same place in inertial space and the place has moved. The period used is the nodal one, 92.899 min against the Keplerian 92.970: J₂ makes the two differ by 4.31 s, which is 0.018° of walk per revolution and 102° in a year — the difference between a repeat track and a track that used to repeat. The map is equirectangular and therefore wrong about area everywhere; what it is right about is longitude difference, which is the whole of what this figure measures.

Ground tracks

  1. 1 The line under a satellite
  2. 2 A swath is sized at the equator
  3. 3 The circle a station can see
  4. 4 A stationary satellite that draws a figure of eight
  5. 5 Five satellites and not four
5 essays · spaceflight The habitable zone, and the radius inside which a day never ends. The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8.

Habitable zone

  1. 1 One number sets the zone
  2. 2 The band moves and the orbit does not
  3. 3 A thermostat that only halves the error
  4. 4 A year too short to feel its own eccentricity
  5. 5 Steam before the zone existed
5 essays · exoplanets Three horizons, and a light cone that bulges. Cosmic time upwards, comoving distance sideways, for the Planck 2018 cosmology. Galaxies sit still in these coordinates, so their worldlines are the vertical grey lines: comoving distance is defined to take the expansion out. The solid inner curve is the past light cone — the set of events whose light reaches here and now — and it reaches out to the particle horizon at 46.1 billion light years, which is the diagram's central number and the one that sounds impossible. The universe is 13.80 billion years old and light has travelled for 13.80 billion years, and yet the material that emitted the oldest light is now 46 billion light years away. Nothing has outrun light: the comoving distance covered is ∫c dt/a, and dividing by a scale factor that was small early on makes the integral three times ct. The dashed curve is the Hubble sphere, where the recession speed equals c, at 14.5 Gly today — and the light cone lies outside it for most of its length, which is exactly why a galaxy can be observed while receding faster than light. The outer dot-dash curve is the event horizon, at 16.7 Gly: a signal sent from here today never reaches anything beyond it.

Horizons

  1. 1 A horizon three times larger than the age allows
  2. 2 The galaxies that are already out of reach
  3. 3 The clock on which light travels in straight lines
  4. 4 Whether there is a horizon at all
  5. 5 Two horizons that differ only in who is inside
5 essays · cosmology One length, leaving and returning. The comoving Hubble radius c/aH against the scale factor, both logarithmic. Everything to the right of the kink is exact and is the same cosmology as every other figure here: after inflation the comoving Hubble radius grows, as a in the radiation era and as √a in the matter era, then turns over once Λ takes hold. Everything to the left is a schematic — the energy scale of inflation is unmeasured, so neither the height of the plateau nor the 62 e-folds drawn is a number anybody has — but the shape is not negotiable: in any accelerated expansion H is nearly constant, so c/aH falls as 1/a. The consequence is the mechanism. The solid horizontal line is a fixed comoving length of 209 Mpc: it starts inside the Hubble radius, is carried outside during inflation, sits frozen while nothing can act across it, and re-enters at z = 1090, which is last scattering — it is the scale the first acoustic peak is made of. The dashed line above it is the comoving size of the whole observable universe, 14148 Mpc, and the figure's quiet second finding is that it is still outside the Hubble radius: the very largest angular scales in the microwave background have never re-entered, which is why they show the primordial spectrum with no acoustic processing on it at all. One crossing does two jobs: it makes the sky uniform, because everything now visible was once in causal contact, and it makes it not quite uniform, because a quantum fluctuation stretched beyond the horizon has nothing left that can smooth it out.

Inflation

  1. 1 Two coincidences with one mechanism
  2. 2 The tilt knows the slope and not the height
  3. 3 A ratio that is an energy and a distance
  4. 4 The epoch nobody saw moves the tilt
  5. 5 A test that can only fail one way
5 essays · cosmology A blackbody at 2.7255 kelvin, filling the sky. The Planck function at 2.7255 K in the units the measurement is reported in, with the peak marked where Wien's law in frequency puts it: x = hν/kT = 2.8214, so ν = 160.2 GHz and the intensity there is 384 MJy per steradian. The points are drawn at the twenty-one frequencies across the FIRAS band, displaced from the curve by a Gaussian of 50 parts per million of the peak, which is the root-mean-square deviation the instrument actually reported. At the scale of this plot that displacement is a fifth of a pixel and the points sit on the line — which is the entire finding. Nothing else in astronomy is a blackbody to a part in twenty thousand: a stellar spectrum is a blackbody with absorption lines cut into it and a continuum that is the wrong shape at both ends. A thermal spectrum this exact requires that the radiation was once in equilibrium with matter, which requires that the universe was once opaque, which requires that it was once hot and dense.

Microwave background

  1. 1 The most perfect blackbody ever measured
  2. 2 A standing wave frozen at one instant
  3. 3 The surface the background actually is
  4. 4 A blur that measures a depth
  5. 5 A dipole a hundred times the signal
5 essays · cosmology Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

Pulsars

  1. 1 A clock that runs down and says what it is
  2. 2 A clock read against a model of everything in between
  3. 3 A corner of the diagram that has to be earned
  4. 4 The part of a star that never slowed down
  5. 5 The exponent no pulsar has
5 essays · stars A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there.

Radius valley

  1. 1 A gap in a histogram that says how planets are built
  2. 2 The envelope that doubles a planet lasts longest
  3. 3 The stripping runs ahead of the starlight that drives it
  4. 4 A smaller star puts the valley lower
  5. 5 The young Sun's spin decides what the Earth kept
5 essays · exoplanets The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number.

Stellar evolution

  1. 1 The star that swells because its centre shrank
  2. 2 The explosion that never reaches the surface
  3. 3 The mass a star does not keep
  4. 4 A surface that slowed because the star grew
  5. 5 A convective boundary with no theory to fix it
5 essays · stars One cluster moved to z = 1.8: one signal unchanged, the other 57 times fainter. The same cluster of galaxies placed at a series of redshifts, with two ways of detecting it compared. The Sunyaev–Zel'dovich decrement is flat, because it is a fraction of the microwave background and the microwave background has the same surface brightness at every redshift a cluster can sit at — moving the cluster further away shrinks it on the sky but does not make its shadow shallower. The X-ray surface brightness of the identical object falls as (1+z)⁻⁴, the dimming every surface brightness suffers in an expanding universe, and by z = 1.8 it is 57 times below where it started. This is why the cluster surveys that reach the early universe are millimetre surveys: an X-ray telescope's cluster catalogue thins out with distance and a Sunyaev–Zel'dovich catalogue is limited only by how large the cluster looks and by how faint a fractional distortion the instrument can measure. What the effect cannot supply on its own is a distance — the signal that does not know how far away the cluster is also cannot say — so every one of these clusters still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it.

Sunyaev zeldovich

  1. 1 A shadow that does not get fainter with distance
  2. 2 A length in centimetres, measured against an angle
  3. 3 A velocity that has the colour of the sky
  4. 4 A null that moves with the temperature
  5. 5 Too few clusters, or a scale that reads light
5 essays · cosmology

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