Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 28

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 541–560 of 909.

Galaxies

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

A sky wrong by a thousandth, and the disc that ends or does not. The surface-brightness profile of an exponential disc with a central surface brightness of 21.65 magnitudes per square arcsecond, against radius in units of its scale length, drawn straight — a pure exponential is a straight line in magnitudes — and then as it would be measured after subtracting a sky of 21.5 magnitudes per square arcsecond wrong by +0.1%, −0.1%, +0.3%, −0.3%. A sky left in bends the profile up into a shallower outer part, the shape called an anti-truncation; a sky taken out too hard bends it down into a break and then drops it out of existence where the residual goes negative. The +0.1% error departs by 0.2 magnitudes at 5.2 scale lengths, 27.3 magnitudes per square arcsecond; The −0.1% error departs by 0.2 magnitudes at 5.0 scale lengths, 27.1 magnitudes per square arcsecond; The +0.3% error departs by 0.2 magnitudes at 4.1 scale lengths, 26.1 magnitudes per square arcsecond; The −0.3% error departs by 0.2 magnitudes at 3.9 scale lengths, 25.9 magnitudes per square arcsecond. Real discs are observed to break both ways, at 24 to 27 magnitudes; every one of those breaks lies in the range where a sky error of a few tenths of a per cent would make one.

Where the sky is the measurement

The outskirts of a galaxy at 29 or 30 magnitudes per square arcsecond are a thousand times fainter than the night sky they are seen through. Photon noise averages down with exposure; an error in the sky subtracted from under them does not. A sky wrong by a tenth of a per cent draws a break into a disc that has none, or a halo around a galaxy that has none, and the telescope's own scattered light draws another.

6 figures · Surface brightness The same disc measured smaller at every redshift. The isophotal radius, at a limit of 26.5 magnitudes per square arcsecond, of three exponential discs that do not change at all — central surface brightnesses 20.5, 21.65, 23, the same scale length throughout — against the redshift they are placed at. Surface brightness falls as (1+z)⁴, three magnitudes at z = 1 and six at z = 3, before anything about the galaxy evolves or any band shift is accounted for, so every isophote moves inward. The discs vanish from an isophotal catalogue entirely at z = 2.98, 2.05, 1.24. The dashed line is the half-light radius, 1.68 scale lengths, which is defined by the galaxy's own light and does not move. Galaxies at high redshift are genuinely smaller than their present-day counterparts; how much smaller depends on which of these two kinds of radius is quoted, and the isophotal kind would report a shrinkage for galaxies that have none.

A radius that is a measurement of the threshold

A galaxy has no edge. Its light fades exponentially for as far as anyone can measure, so every size ever quoted for one is the radius at which the fading crossed some line that somebody drew. Drawn at a fixed surface brightness, the line measures how bright the galaxy's centre is as much as how big the galaxy is — and at high redshift it measures the expansion of the universe as well.

6 figures · Surface brightness A cusp pulled deeper by the disc inside it. The dark-matter density of a halo of 10¹² solar masses and concentration 10 before and after an exponential disc of 5·10¹⁰ solar masses and scale length 3 kpc settles slowly into its centre, computed shell by shell from the adiabatic invariant r·M(r). Far outside the disc the two profiles coincide — 4.8% apart at 83 kpc — because contraction moves mass inward without changing how much there is. Inside, the dark matter is pulled in after the baryons: at half a disc scale length the density is 2.9 times higher, and at 1 kpc the logarithmic slope goes from −1.09 to −1.25. The cusp that dark-matter-only simulations predict is therefore the shallowest the inner halo of a disc galaxy can be before feedback acts, not the profile feedback starts from. The calculation assumes circular orbits and slow settling, and simulations that include gas find the true response weaker than this.

The halo a galaxy drags inward

The cusp that dark-matter simulations predict is a halo with no galaxy in it. When gas cools and settles into a disc at the centre, the dark matter is pulled in after it, and a quantity conserved on slowly changing orbits says by how much. In a large disc galaxy the pull triples the dark matter at the centre and multiplies tenfold the energy needed to dig a core; in a dwarf it barely registers.

5 figures · Halo profiles

Gravitation

Two bodies pulling on each other, and everything that goes wrong at three.

A band and a strip, and the constant where they cross. The plane of the two quantities a planetary ephemeris cannot separate: the fractional rate of change of the gravitational constant, vertically, and of the Sun's mass, horizontally, both in units of 10⁻¹⁴ per year. The orbits respond only to their sum, the rate of change of GM☉, so a measurement of that sum — drawn here as −9.0 ± 3.0 × 10⁻¹⁴ per year, the precision recent ephemerides reach, not any one published value — is a diagonal band. On its own it allows any Ġ/G at all, provided the Sun's mass is changing to compensate. The vertical strip is the Sun's mass loss computed from its luminosity and its wind, −9.9 to −8.8 × 10⁻¹⁴ per year. Where the two overlap, Ġ/G lies between −3.2 and 3.9 × 10⁻¹⁴ per year. The bound on a change in a fundamental constant is therefore set as much by how well the solar wind is measured as by how well the planets are.

A constant measured through a Sun that is getting lighter

Whether the gravitational constant changes with time is tested by watching the planets, and the planets report only the product of G with the Sun's mass. The Sun is losing mass — four million tonnes a second as light, a million and a half more as wind — at nearly a ten-trillionth of itself a year, and every bound on a changing G is a statement about the solar wind as much as about gravity.

5 figures · Gravitational constant

Orbits

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

A doubling rule, and the planet that broke it. Each planet's real semi-major axis against the value the Titius–Bode rule 0.4 + 0.3 × 2ⁿ gives it, both in astronomical units on logarithmic axes; points on the diagonal are fitted exactly. The rule gives the Earth 1.0 by construction and fits Venus, Mars, Jupiter and Saturn within a few per cent. It needed the value n = 3 left empty until Ceres was found in its slot in 1801, and it predicted Uranus, discovered in 1781, within 2%. Neptune, discovered in 1846 by a calculation that assumed the rule, sits 22% inside its predicted place, and Pluto occupies the place Neptune should have had. Mercury is given n = −∞, which is a free choice made to fit it. The labels give each body's departure from the rule.

A spacing law that random orbits obey

The planets' distances roughly double from one to the next, and a rule written down in 1766 predicted the asteroid belt and Uranus before either was found. It failed on Neptune, and the reason it worked at all turns out to be that almost any set of orbits forbidden to crowd each other looks geometric on a logarithmic scale — as a test against thousands of random systems shows.

6 figures · Harmonic law A hotter summer that is also a shorter one. Daily-mean sunlight at 65°N through one year beginning at the March equinox, for an orbit of eccentricity 0.0167 and obliquity 23.44°, drawn twice: with perihelion at the June solstice, when the northern summer is closest to the Sun, and with perihelion at the December solstice, half a precession cycle later. The first summer peaks 6.9% higher, and it is also shorter, because the Earth moves fastest near perihelion: days above 275 W m⁻² number 139 against 144. The energy delivered on those days — the shaded area above the line — is 5.055 against 4.989 GJ m⁻², a difference of 1.3%. The second law makes the intensity and the duration trade against each other almost exactly, so a summer measured by how much energy it delivers barely feels precession at all.

A hotter summer that is also a shorter one

When the Earth passes closest to the Sun in northern summer, the northern summer is brighter — and, by Kepler's second law, shorter by exactly the right amount to deliver almost the same energy. Measured by the energy it delivers above a melting threshold rather than by its brightest day, a summer barely feels the precession cycle at all and feels the tilt of the axis several times more strongly.

6 figures · Orbital averages The weakest bodies are a few hundred metres across. The energy per unit mass needed to shatter a body and disperse half of it, against the body's diameter, for basalt and water ice, from fits to impact simulations calibrated on laboratory fragmentation. Small bodies are held together by material strength, which falls with size because a larger body contains larger flaws; large bodies are held together by their own gravity, which rises with size because the fragments have to be thrown apart as well as broken. The two regimes meet in a minimum at 235 m for basalt and 204 m for water ice, where a body needs the least energy per kilogram to destroy of any size — 127 J/kg and 57 J/kg, the kinetic energy per kilogram of motion at only 16 and 11 metres a second. A collisional population is therefore not self-similar: it has a preferred size, and its size distribution records it.

The size at which a body is easiest to break

A pebble is held together by the strength of its rock and a moon by its own gravity, and somewhere between them is a size that has too little of either. Bodies a few hundred metres across are the weakest in the solar system — sixteen metres a second of impact energy per kilogram is enough to scatter one — and a population grinding itself down records that weakness as a knee in its size distribution.

5 figures · Collisional cascade

The observed sky

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

A square field imaged as a squashed one. A square grid of stars 1° across, centred 70° from the zenith, and how refraction moves each one relative to the field's centre after the displacement common to the whole field has been removed. The arrows are magnified 200 times. Refraction lifts every star towards the zenith, but it lifts the lower ones more because refraction grows towards the horizon, so the field is compressed vertically by 2377 parts per million; and every star moves along its own vertical circle, which converge at the zenith, so the field is compressed horizontally by 286 parts per million. The corners move by up to 4.4 arcseconds against the centre — a hundred times the precision of a modern relative astrometric measurement, and a distortion that changes as the field crosses the sky.

A square field imaged as a squashed one

Relative astrometry measures positions against neighbouring stars so that the atmosphere's refraction, common to them all, cancels. It does not cancel. Refraction changes across the field, so a square patch of sky seventy degrees from the zenith is imaged squashed by a few parts in a thousand — and stars of different colour are moved by different amounts, by more than the parallaxes being measured.

5 figures · Refraction Rays that follow the curve of the Earth. Rays of light leaving an observer five metres above a cold surface at elevations between −0.2° and +0.45°, traced through an atmosphere whose lowest 150 metres warm upwards by 0.15 K per metre, drawn as height above the ground against distance along it — so the Earth's curvature has been removed and a ray that follows the curve runs flat. Rays that start downwards or only slightly upwards are bent back by the inversion before they can leave it, glance off the surface, and are guided along the layer in shallow hops: 5 of the 8 traced remain inside it for the whole 300 km. Only the steeper rays escape upwards. Traced backwards, the trapped rays are lines of sight: an observer inside the duct sees objects hundreds of kilometres away that lie far below the geometric horizon, lifted into view.

An atmosphere that bends light round the Earth

A horizontal ray of light curves towards denser air, and in an ordinary atmosphere it curves at a fifth of the rate the ground falls away. Put eleven degrees of warmer air above every hundred metres of cold air over sea ice and the ray curves as fast as the Earth does. It follows the curve, and a Sun five degrees below the horizon rises two weeks early.

6 figures · Refraction Where every star appears to an observer moving fast. The apparent angle of a star from the direction of motion, against its angle in the rest frame of the stars, for observers moving at 0.3c, 0.6c, 0.9c. The diagonal is a stationary observer. Every star except the two on the line of motion is displaced forward, and the displacement grows with speed until, at 0.9c, a star at right angles to the motion appears 26° from the direction of travel and one directly behind still appears behind. A star at a true angle of 90° is seen at the angle whose cosine is β — 72.5°, 53.1°, 25.8°. At the Earth's orbital speed, ten thousandths of c, the curve is indistinguishable from the diagonal, and the whole of annual aberration is its first-order departure from it.

The sky crowded forward by the observer's speed

Annual aberration is the first-order term of a law that has no higher terms Bradley could have seen. Written exactly, it maps the whole sky onto itself, crowding every star towards the direction of motion — by half a milliarcsecond at the Earth's speed, which a modern astrometric catalogue must carry, and at half the speed of light by enough to put three-quarters of the stars in the forward half of the sky.

5 figures · Aberration How steady each kind of clock is, by how long it is averaged. The Allan deviation — the typical fractional difference in frequency between successive averages of length τ — against τ, for representative noise models of caesium beam, hydrogen maser, caesium fountain, optical lattice clock. Each falls as τ^−½ while random frequency noise dominates, then levels off at a floor set by slower, correlated noise, and a clock with a systematic drift eventually rises again. A caesium beam is still improving at the longest averaging drawn, reaching 10⁻¹⁴; a hydrogen maser reaches its best, 7.3·10⁻¹⁶, after about 27 hours; a caesium fountain is still improving at the longest averaging drawn, reaching 1.2·10⁻¹⁶; an optical lattice clock is still improving at the longest averaging drawn, reaching 10⁻¹⁸. No clock is best at every averaging time — the maser is steadiest over hours and wanders over weeks, the fountain is noisier over minutes and more accurate over months — and a time scale is built by using each where it is best. The curves are typical shapes, not the specification of any particular instrument.

A time scale that is finished after it is used

The world's reference time is not kept by any clock. It is an average of about four hundred and fifty, weighted by how steady each has been, published a month late, and then recomputed a year later against the handful of clocks that define the second — so the best available time for any moment is known only long after the moment has passed. The best clocks are now so good that averaging them requires knowing each one's height to a centimetre.

5 figures · Timescales

Spaceflight

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

The weak spot in the field. Contours of the strength of the Earth's magnetic field at the surface, in microtesla, over longitude and latitude, from an eccentric dipole: a dipole tilted to pass through the geomagnetic poles and displaced about 570 km from the Earth's centre towards the western Pacific. The field is strongest near the poles and weakest near the equator, as any dipole's is, but the displacement makes the side of the Earth furthest from the dipole's centre weaker still: the minimum, 22.9 µT, lies at 8° S, 40° W, over the South Atlantic. The measured anomaly sits further south and west, near 26° S and 55° W, and is slightly deeper, because the real field has higher harmonics that no dipole carries; the eccentric dipole gets the side of the planet and the size right and the exact place approximately. Trapped particles mirror where the field reaches a fixed strength, so wherever the field is weak they mirror closer to the ground — and over this region the inner belt reaches down into the upper atmosphere.

A belt emptied over one ocean

The Earth's magnetic field is not centred on the Earth. Its dipole sits five or six hundred kilometres towards the western Pacific, so the opposite side of the planet — the South Atlantic — has the weakest field at any altitude, and a trapped particle that mirrors safely above the atmosphere everywhere else dips into it there. The inner edge of the inner radiation belt is drawn by that one region, and it is moving.

6 figures · Radiation belts

Starlight

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

A source that erases its own polarisation. The polarised fraction of a uniform slab that both emits and Faraday-rotates, relative to its intrinsic value, against wavelength, for total Faraday depths of 10, 40, 150 rad m⁻². Light from the back of the slab is rotated by the whole depth and light from the front by none, so the polarisation vectors from different depths fan out and cancel. The fraction falls as |sin(Φλ²)/(Φλ²)| and vanishes altogether at λ² = π/Φ — 56.0 cm for 10, 28.0 cm for 40, 14.5 cm for 150 — then returns weakly and vanishes again. A source measured only at wavelengths beyond its first null looks unpolarised, and a source measured across the null shows a polarised fraction that depends on the wavelength in a way no foreground screen can produce. The shape of the fall is a measurement of how the rotating plasma is arranged along the line of sight.

A source that rotates its own light

The rotation-measure method assumes the plasma that turns the polarisation sits in front of the source. When it is mixed into the source instead, light from the far side is turned more than light from the near side, the contributions cancel, and the straight line in wavelength squared bends, halves its slope and then vanishes. The cure is to stop measuring in wavelength at all.

7 figures · Faraday rotation

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

The most pressure a core can supply, and what its envelope asks for. The pressure at the boundary of a star's isothermal core, on a logarithmic scale, against the fraction of the star's mass the core holds. The coloured curves are the most pressure a core of that mass can supply at its edge — the maximum of the isothermal sphere, 1.40 c⁸/(G³M²) for a core of mass M — for cores of mean molecular weight 0.6, 1.0, 4/3; the flat grey line is the pressure the envelope, of molecular weight 0.6, exerts on the core at the same temperature. The core's ceiling falls as the square of its mass, and it can hold the envelope up only while it lies above the line. The curves cross it at core fractions of 0.370, 0.133, 0.075: a core of the same molecular weight as its envelope could hold 37 per cent of the star, and a fully ionised helium core, μ = 4/3, under this envelope only 7.5 per cent. The limit exists because the core is heavier per particle than the gas it has to hold up.

A core too heavy for the gas around it

When a star's centre runs out of hydrogen, the helium left behind stops making energy, settles to one temperature, and becomes the one member of the polytrope family with no surface of its own. It can hold up the envelope around it only while it is less than about a tenth of the star. Past that there is no balance to be had, and the star crosses to the giant branch in a few per cent of its lifetime — which is why that part of the diagram is empty.

6 figures · Polytropes A survival probability with a step in the middle of the spectrum. The probability that an electron neutrino made in the core of the Sun is still an electron neutrino at the Earth, against its energy, from the resonant conversion it undergoes in the Sun's own electrons: mixing angle sin²θ₁₂ = 0.307, splitting Δm² = 7.4·10⁻⁵ eV², and an electron density at the point of production of 90 moles per cubic centimetre. At low energy the matter potential is small beside the vacuum term and the probability is the vacuum average, 0.54 at 0.3 MeV (the flat limit is 0.55). At high energy the matter term dominates, the neutrino is born almost entirely in the heavier mass state, and the probability falls to 0.32 at 10 MeV, close to sin²θ₁₂. The two terms are equal at 2.1 MeV, which falls in the gap between the ⁷Be line and the bulk of the ⁸B spectrum. The points are the measured survival probabilities for each source separately — pp, ⁷Be, pep and ⁸B from Borexino, and ⁸B from SNO — and every one of them lies within about two standard deviations of the curve: low-energy neutrinos survive about half the time, high-energy ones about a third.

A survival that depends on the energy

A low-energy neutrino from the Sun reaches the Earth still an electron neutrino about half the time, and a high-energy one about a third of the time. The difference is made by the Sun's own electrons, and the energy at which it switches over — near two million electronvolts — sits in the gap between the neutrinos that never meet a resonance and the ones that all do.

7 figures · Solar neutrinos A thousand times the supply, and eight times the light. The luminosity of an accretion disc, in units of its Eddington luminosity, against the rate at which gas is supplied to it, in units of the rate that would make a thin disc shine at exactly that limit — both on logarithmic axes. Below the limit the luminosity is proportional to the supply, the dashed line, with a tenth of the rest-mass energy radiated. Above it the disc cannot radiate the excess: the luminosity grows only as 1 + ln ṁ, so ten times the supply gives 3.3 times the limit, a hundred times gives 5.6, and a thousand times gives 7.9. The limit is a ceiling that bends rather than breaks — and whatever the supply is doing above it, the light does not say, because the same curve follows whether the surplus is blown away in a wind or carried into the hole.

A ceiling on the light and not on the mass

Pour gas onto a black hole a thousand times faster than the Eddington rate and it shines at only about eight times its limit. The rest of the energy is either blown away or swallowed, and those two fates give the same luminosity while differing by a factor of the supply in how fast the object grows — so the light a super-Eddington source emits cannot say which one it is.

6 figures · Eddington limit

Starlight

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

A push that depends on how fast the wind is already going. The force multiplier — the radiative push on a line-driven wind as a multiple of the push on its free electrons — against t, the electron-scattering optical depth across one thermal Doppler width, which is small where the wind accelerates steeply. The solid curve is an ensemble of spectral lines whose strengths follow a power law, the number of lines stronger than a given strength falling as that strength to the power 1 − α, with α = 0.6. If every line were optically thin the push would be the flat line at 2000, independent of the flow; if every line were thick it would fall as 1/t, set only by the velocity gradient. The ensemble lies between: over most of the range it falls as t to the power −0.59, because as t rises lines cross from thin to thick one after another. That single exponent is the whole of the theory — it fixes the terminal speed, the mass-loss rate and how both scale with the star — and it is a property of the atomic physics of the ions in the wind, not of the star.

A rate the wind has no choice about

A hot star's wind is driven by starlight caught in hundreds of thousands of spectral lines, and the way those lines are distributed in strength fixes everything else. The mass-loss rate turns out not to be a free parameter at all but the one value for which a smooth flow can leave the surface — and multiplying it by the wind's speed and the square root of the star's radius cancels the star's mass, leaving a relation with the luminosity alone.

5 figures · Stellar winds The power a source loses to its own photons, against how bright it is. The ratio of the power a synchrotron source loses by inverse-Compton scattering of its own photons to the power it radiates as synchrotron emission, against its brightness temperature, both on logarithmic axes. To first order the ratio goes as the fifth power of the brightness temperature over 10¹² K — so at 10¹¹ K it is 10⁻⁵, negligible, and at 10¹² K it is one. The scattered photons are scattered again, and each order multiplies by the same factor, so the second-order curve rises as the tenth power: at twice the limit the total loss is 1056 times the synchrotron power, and the electrons radiate away their energy in hours as X-rays and gamma rays. A steady source cannot sit above the line where the ratio is one. That is the inverse-Compton catastrophe, and it makes 10¹² K a ceiling on the brightness temperature of any incoherent synchrotron source at rest.

A temperature that is a speed

No steady synchrotron source can be brighter than about a trillion kelvin, because above that its electrons lose their energy scattering their own photons faster than they can radiate it. So when a radio source is measured brighter — as the cores of quasars routinely are — the excess is not a hotter source. It is a source coming towards the observer, and the brightness temperature becomes a measurement of how fast.

5 figures · Synchrotron radiation

Exoplanets

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

A loss rate that stops following the starlight. The rate at which a hot Jupiter of 1.38 Jupiter radii and 0.69 Jupiter masses loses its atmosphere, in grams per second, against the ionising flux it receives, in erg per square centimetre per second, both on logarithmic axes. The dashed line is the energy-limited rate with a heating efficiency of 0.3: linear in the flux, 1.04·10¹⁰ g s⁻¹ at 10³. The solid curve follows it at low flux and turns over near 10⁴, where the base of the wind becomes nearly fully ionised and each recombination radiates its energy as Lyman-α instead of heating the flow; above that the rate grows only as the square root of the flux. At 10⁶ erg cm⁻² s⁻¹, the flux such a planet receives from a young, active star, the energy-limited formula overestimates the loss by a factor of 11. A single efficiency cannot describe both ends of the curve: fitted at one flux, it is wrong at the other.

An escape that stops following the starlight

The standard estimate of how fast a planet loses its atmosphere assumes the loss grows in proportion to the ultraviolet light it absorbs. It does, up to a point. Beyond about ten thousand erg per square centimetre per second the escaping gas is fully ionised, its heat leaves as Lyman-α light instead of lifting anything, and the loss grows only as the square root of the flux — so the one efficiency every calculation needs is really a function of where the planet sits.

5 figures · Atmospheric escape A fringe that is dark on the star and bright on the planet. The fraction of a point source's light an interferometric nuller transmits, against its angle from the star along the baseline, in milliarcseconds, for two telescopes 10.3 m apart observing at 10 μm — the baseline that puts the first bright fringe exactly on a planet 1 AU from a star 10 pc away, 100 milliarcseconds out. The solid curve is the two-telescope null, sin²: it is zero on the star and rises as the square of the angle. The dashed curve is a quartic null, sin⁴, made by combining four apertures: it is darker near the star and narrower at the planet. The shaded strip is the star itself, 0.93 milliarcseconds across. It looks negligible at this scale, but the null is not perfect across it, and what leaks from that strip is the quantity the whole technique has to beat.

A null filled in by the star itself

Two telescopes combined out of phase cancel a star on the axis and pass a planet beside it, and the cancellation rises only as the square of the angle away from the axis. A star is not a point, so its own edges sit off the axis — and for the Sun seen with the baseline that finds the Earth, they leak through at a hundred and fifty times the Earth's brightness, at every distance, because the leak depends only on the size of the star compared with the planet's orbit.

5 figures · Direct imaging

Every sequence · Every named object · Search