Astronomy & Celestial Mechanics
the sky, computed
Depth

Series — page 2

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.
Mass against radius, and what lies between the curves. Planetary radius against mass on logarithmic axes, in Earth units, with composition curves computed from interior models rather than drawn through the points. The solid-planet curves are R ∝ M^(1/3.7) — flatter than a constant density because a heavier planet compresses itself — and the hydrogen curve turns over near three Jupiter masses, where degeneracy pressure takes over and adding mass makes the planet smaller. A mass alone or a radius alone places a planet on a line; only both together place it between two curves, and that is the whole argument for measuring a planet twice.

Planet composition

  1. 1 Two methods, and one density
  2. 2 One density, and every planet that has it
  3. 3 A radius no cold planet is allowed
  4. 4 A planet radius is a stellar radius
  5. 5 More iron than the star could give
  6. +2 more
7 essays · exoplanets The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close.

Precession

  1. 1 The pole star has a shelf life, and the sky has a slow hand
  2. 2 The wobble inside the wobble
  3. 3 A wobble that should have stopped
  4. 4 A core that wobbles at its own period
  5. 5 A rhythm in the rock that measures the Moon
  6. +2 more
7 essays · sky Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.

Primordial fields

  1. 1 The seed that cannot be remembered
  2. 2 A field that would have moved the ruler
  3. 3 A lower limit made of three lengths and a missing echo
  4. 4 A void heated from the inside
  5. 5 An echo heard only from the weakest fields
  6. +2 more
7 essays · cosmology The wobble of a star with a circular companion. The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.

Reflex velocity

  1. 1 The star moves, and the mass is a lower bound
  2. 2 A planet that was the star's own rotation
  3. 3 One eccentric planet, or two circular ones
  4. 4 An exposure time chosen by the star
  5. 5 A model of the star that can carry the planet
  6. +2 more
7 essays · exoplanets Catching a target 40° ahead. A phasing manoeuvre. Dropping into an orbit 6% lower shortens the period to 0.9553 of the target's, so the chaser gains 16.1° each lap and closes 40° in 3 revolutions. Speeding up would have lost ground instead.

Rendezvous

  1. 1 Catching up by slowing down, which cost Gemini 4 its fuel
  2. 2 A burn that moves the wrong way
  3. 3 The relative orbit that breathes
  4. 4 A camera that cannot see how far
  5. 5 A grapple point that moves faster than the orbit
  6. +2 more
7 essays · spaceflight 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
  6. +2 more
7 essays · cosmology Twilight at latitude 52°. Solar altitude through the second half of the day at 52°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 34 min, 40 min, 42 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. At this latitude the curve for June solstice never reaches −18° at all: astronomical twilight does not end, and there is no astronomically dark night from about 48.6° upward.

Twilight

  1. 1 Three definitions of night
  2. 2 The shadow that climbs the zenith
  3. 3 Ozone keeps the twilight zenith blue
  4. 4 The air's height against the planet sets the twilight
  5. 5 The night is late because the light bounces twice
  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 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.

Atmospheric escape

  1. 1 The gas a planet cannot keep
  2. 2 A planet ten times larger in one colour
  3. 2 The shield that is also a funnel
  4. 3 An escape that stops following the starlight
  5. 4 A helium line that weighs the star first
  6. +1 more
6 essays · exoplanets 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
  6. +1 more
6 essays · spaceflight The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection.

Dark matter

  1. 1 The mass that is not the light
  2. 2 Three mass models that fit the same curve
  3. 3 Stars that weigh and do not shine
  4. 4 One acceleration, and every galaxy's missing mass
  5. 5 A galaxy born without a halo
  6. +1 more
6 essays · galaxies The same universe, four times, as a fraction of itself. Each bar is the fractional contribution of the four components to the total density at one epoch, computed from the Planck 2018 parameters by scaling each component from today: radiation as (1+z)⁴, both kinds of matter as (1+z)³, and Λ as a constant. The familiar figure — five per cent baryons, twenty-six dark matter, sixty-nine dark energy — is the top bar and only the top bar. At recombination the same universe is three-quarters dark matter and Λ is one part in ten million; before matter–radiation equality it is mostly radiation. A pie chart of the contents of the universe is therefore a statement about a moment, and the moment is the one it happens to be drawn in.

Density parameters

  1. 1 A budget whose familiar part is five per cent
  2. 2 Half the ordinary matter was missing, and a millisecond found it
  3. 3 A mass measured by what it stopped from forming
  4. 4 A coincidence that is a factor of fourteen
  5. 5 A flatness that needs a second distance
  6. +1 more
6 essays · cosmology Contrast against separation, which is where direct imaging lives. Planet-to-star brightness ratio against apparent separation, both logarithmic, for a system 10 parsecs away. The reflected-light curves are A_g(R_p/a)² and fall as the inverse square of the orbit; the thermal curve is the ratio of two Planck functions at 10 µm and does not, which is why every imaged planet so far is young and hot rather than merely large. The vertical lines are diffraction limits λ/D — nothing inside a telescope's own line is reachable by it at any contrast at all. An Earth at ten parsecs sits at 6.3e-10, which is 4 orders of magnitude below the faintest planet yet imaged. The four imaged planets are plotted at their measured near-infrared contrasts rather than at 10 µm, because the near infrared is the band they were found in — which is itself part of the argument, since a young planet is hot enough to be bright where its star is not.

Direct imaging

  1. 1 Nine orders of magnitude, half an arcsecond apart
  2. 2 A star subtracted using the star
  3. 3 A reference built from other stars
  4. 4 A null filled in by the star itself
  5. 5 Dust that does not dim with distance
  6. +1 more
6 essays · exoplanets The same spectrum, at rest and at 900 km/s. A set of absorption lines at rest and shifted by a radial velocity of 900 km/s. The displacement is proportional to wavelength, so the reddest line here moves 2.06 nm and the bluest 1.18 nm — which is why the measured quantity is the ratio Δλ/λ and not a distance.

The Doppler effect

  1. 1 A shift in a line is a speedometer, and it works at any distance
  2. 2 The metre per second that is not the star
  3. 3 A velocity that is an average of lines that disagree
  4. 4 The photons on the sides of the lines
  5. 5 A spot a slow star hides inside its velocity
  6. +1 more
6 essays · starlight Circular and escape speed. Orbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.

Escape

  1. 1 The speed that does not come back, and the √2 that separates it
  2. 2 The ratio that is not √2
  3. 3 The speed that is left over
  4. 4 The cheapest circle to stop in
  5. 5 A thruster that must arrive slow
  6. +1 more
6 essays · spaceflight A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

Gravity assist

  1. 1 Stealing speed from a planet, which does not notice
  2. 2 The same planet, three times
  3. 3 The planet pays, and it shows
  4. 4 Braking on a moon before braking with the engine
  5. 5 A flyby that multiplies its own aiming error
  6. +1 more
6 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
  6. +1 more
6 essays · exoplanets 1I/ʻOumuamua: an orbit at e = 1.201, and the angle it turned through. The open branch of a conic at eccentricity 1.201 and periapsis 0.2559 AU, with the Sun at the occupied focus. Two numbers fix everything else: |a| = q/(e−1) = 1.273 AU, an impact parameter b = |a|√(e²−1) = 0.847 AU, an asymptote at ν∞ = arccos(−1/e) = 146.37° from periapsis, and a deflection of δ = 2ν∞ − 180° = 112.74° — which is the same statement as sin(δ/2) = 1/e. The asymptotes cross at |a|e = 1.529 AU from the Sun on the periapsis side, where a bound orbit's centre would be on the other. The speed left over at infinity is √(μ/|a|) = 26.40 km/s. Schematic in one respect: the drawing runs at about 119 px to the AU, so the Sun's own disc would be far smaller than its marker.

Hyperbolic orbits

  1. 1 The orbit that has no period
  2. 2 Aiming at a plane instead of at a planet
  3. 3 A spike only first-time visitors can be in
  4. 4 A census of visitors is a census of the survey
  5. 5 Jupiter keeps only the visitors already nearly bound
  6. +1 more
6 essays · orbits An eccentricity and an inclination trading, at 65° of mutual tilt. The secular equations integrated from a nearly circular orbit (e = 0.02) inclined at 65° to a distant perturber's plane, over three oscillations. Above, the eccentricity; below, the inclination, with the constant √(1−e²)cos i drawn as the flat line it is. The eccentricity climbs to 0.8380 and the inclination falls to 39.25° at the same instant, and neither is a coincidence: the product is fixed, so one can only rise as the other falls. That floor is the same for every starting tilt — at maximum eccentricity j = √(5/3)Θ, so cos i = √(3/5) and the inclination arrives at 39.23° whether the orbit began at 50° or at 89°. The closed form for a circular start is e_max = √(1 − (5/3)cos²i₀) = 0.8380, which contains nothing about the perturber — not its mass, not its distance. Those set the clock and not the amplitude, and the period here is 4.83 Kozai times. What the figure cannot show is what happens at the top of the cycle in a real system: at e = 0.838 the pericentre is 0.1620 of the semi-major axis, where tides, general relativity or a stellar surface all intervene, and the quadrupole picture ends.

Kozai–Lidov

  1. 1 An inclination that turns into an eccentricity
  2. 2 The precession that switches the cycle off
  3. 3 A cycle that builds what the wave needs
  4. 4 A ceiling the outer orbit imposes
  5. 5 An orbit that turns over
  6. +1 more
6 essays · orbits The cost of going to Mars, against when to leave and how long to take. Contours of departure energy C₃ over a grid of 64 × 56 solved Lambert problems: each point is a departure date, a flight time, and the unique single-revolution conic that connects the two planets between them. The cheapest transfer on this grid costs 5.1 km²/s², leaving in Mar 2001 with a flight time of 220 days, against 8.7 for the idealised Hohmann transfer between circular orbits of the same radii. A launch window is a region on this plane, not a moment, and its shape is what a launch period is negotiated against.

Launch windows

  1. 1 Two dates decide a mission
  2. 2 The window that comes back and the cost that does not
  3. 3 A wall across the porkchop
  4. 4 The wall sells inclination at the dearest price
  5. 5 A climb out of the ecliptic paid for in orbits
  6. +1 more
6 essays · spaceflight

All essays