Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 38

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 741–760 of 909.

Stars

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

In a mixed field the torque follows the dipole and ignores the rest. The torque of a field that is part dipole and part quadrupole, with the total surface strength fixed, as a fraction of a pure dipole's, against the fraction of the field strength in the dipole, for wind magnetisations of 300, 3,000, 30,000, using the broken power law fitted to simulated mixed-field winds. Above a dipole fraction of 0.39, 0.24, 0.15 respectively, the torque is exactly what the dipole component alone would give: the quadrupole has fallen below the dipole long before the wind reaches its Alfvén radius, and contributes nothing to the lever. Below it, the quadrupole sets a floor. A star whose surface is mostly quadrupole by strength can therefore be braked almost entirely by its small dipole — and a measurement of the total field says little about the torque unless it says how much of the field is dipolar.

A lever set by the dipole and not the field

A star's wind carries away its spin because the magnetic field holds the gas in step with the surface out to the Alfvén radius, and the torque goes as the square of that radius. How far out the field can hold depends less on how strong it is than on how it is arranged: a quadrupole of the same surface strength holds the Sun's wind to half the distance and brakes it at a fifth of the rate. In a mixed field only the dipole reaches far enough to matter — so the field that brakes a star is the part of it that is hardest to measure.

5 figures · Magnetic braking

Spaceflight

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

How far inside the door a day of compression reaches. The innermost drift shell at which phase-space density has halved after a day of compression, against where the compression put the last closed drift shell, for diffusion at Kp = 4, 5, 6, 7. With the door at L = 5 and Kp = 6 the loss reaches L = 3.8; at Kp = 4 it reaches only 4.3. The diagonal is the door itself: everything above the curves and below the diagonal was emptied without ever being opened. A strong storm both brings the door in and speeds the diffusion that drains the shells behind it, and the two together carry the loss a full Earth radius or more inside the shells that were opened.

A belt drained through its outer door

When a storm's solar wind squeezes the magnetopause inward, the outermost drift shells of the radiation belt are cut open and their electrons leave within minutes. The shells inside are untouched by the boundary and by the atmosphere alike, and they empty anyway: the opened region is a sink, and the same diffusion that fills the belt from outside carries electrons out to it. A day of compression drains the belt a full Earth radius inside the shells that were actually opened.

5 figures · Radiation belts A dropout that partly comes back. The flux of 1 MeV electrons at L = 4.5, relative to before the storm, through a storm whose ring current deepens to Dst = −150 nT over eight hours and decays with a one-day e-folding. Dashed: the adiabatic part alone, with phase-space density unchanged — the flux falls to 18 per cent at the storm's peak and returns completely. Solid: the same with a real loss of 60 per cent of the phase-space density during the main phase, as compression and outward diffusion would cause. The flux falls to 7 per cent and recovers only to 39. Seen in flux at fixed energy, the two causes look the same while the storm is on, and only the recovery — days later — tells how much of the dropout was electrons lost and how much was the field.

A dropout that is only the field

During a storm the ring current weakens the field the radiation belt sits in, and every electron, keeping its adiabatic invariants, loses energy. A detector counting electrons at a fixed energy and place sees the flux fall by half or more — and not one electron has gone anywhere. When the ring current decays, the electrons regain their energy and the flux comes back. Only in the coordinates the electrons themselves conserve can a storm's real losses be told from the field's sleight of hand, and the bigger the storm, the larger the sleight.

5 figures · Radiation belts

Starlight

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

The ratio of two distances, for a universe with dark energy and one with dust. The distance-duality ratio η — the luminosity distance divided by (1 + z)² times the angular-diameter distance — against redshift. In a universe with a cosmological constant, or any universe in which light travels on the geometry and is conserved, it is exactly one (solid). In a universe of matter alone, decelerating, with grey intergalactic dust dimming its supernovae just enough to make their distances match the accelerating one's, it rises with redshift: 1.19 at redshift 0.5, 1.30 at 1, because the dust makes supernovae look farther than the geometry puts them while leaving the acoustic-scale distances untouched. The shaded band is η = 1 ± 0.03z, about what supernova and acoustic-scale distances together allow. The dusty universe's η rises at the start with a slope of about 0.45 per unit redshift, more than ten times the band's, and leaves it at once.

Two distances that must agree if light is conserved

A supernova's distance comes from how faint it is; an acoustic ruler's distance comes from how small it looks. In any universe whose light follows the geometry and is neither created nor destroyed, the two are tied by an exact factor of (1 + z)², whatever the universe contains. Dust that dimmed distant supernovae without reddening them could have faked the accelerating expansion — and would have broken that tie by a fifth by redshift a half. The tie holds to a few per cent.

5 figures · Distance ladder Where the photons stop mattering and the star starts. Noise in a 13-hour window — the length of an Earth-like planet's transit across a Sun-like star — against the star's brightness, for a space telescope that reaches 17 parts per million in 6.5 hours on a twelfth-magnitude star. Photon noise falls by a factor of ten for every five magnitudes of brightness; the star's granulation, averaged over the window, stays at 5.1 parts per million whatever the brightness. The two are equal at magnitude 10.1; brighter than that, the star is the floor. The dashed line is an Earth's transit depth across the Sun, 84 parts per million: a single transit on a twelfth-magnitude star is detected at 6.4 standard deviations, and on the brightest stars at no better than 17, set by the star alone — and supergranulation, spots and faculae add more on these timescales.

A floor the star sets for itself

Above the atmosphere, with comparison stars and a big enough telescope, every noise in a measurement of a star's brightness can be beaten down except one: the star's own surface, a mosaic of convective cells each brighter or fainter than the rest. That flicker does not average away against anything, and it is larger for stars that are bigger and puffier than the Sun. For the brightest stars, an Earth's transit is limited by the star, not the photons — and the stars Kepler watched turned out twice as noisy as the Sun.

5 figures · Photon noise

The observed sky

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

Phase curves that agree where they can be measured. Modelled reflected light of a cloud-covered planet — geometric albedo times phase function, on a logarithmic axis — against phase angle, for clouds of asymmetry parameter 0.00 and 0.65, from scattering evenly to throwing most light forward. Each curve's single-scattering albedo — with the planet's radius known, as a transit measures it — has been chosen to match the first curve between 30° and 110° (shaded), the range the measurement is assumed to cover: the fitted albedos are 0.9900, 0.9998, and inside the window the matched curve differs from the first by at most 1%. Beyond it they part: at 165°, a thin crescent, the forward-scattering cloud is 6.0× as bright as the isotropic one. The model is an optically thick cloud with exact single scattering and a similarity-scaled multiple-scattering term.

Two clouds that shine alike until the crescent

A grey cloud that scatters light evenly and a whiter one that throws it forward return almost the same light at every phase angle from thirty degrees to a hundred and ten. They part only at the thin crescent, where a planet round another star is hardest to see — and the measurement that tells them apart in the middle of the orbit is the polarisation, a ratio with no albedo in it.

6 figures · Apparent motion One pulsar's stability, limited by the background every pulsar shares. The cubic-fit stability of a single millisecond pulsar, on logarithmic axes, against averaging time from 0.3 to 40 years. White timing noise of 100 ns every 14 days averages down as τ^(−3/2) (falling dashed line). The gravitational-wave background at the amplitude the timing arrays reported in 2023, A = 2.4·10⁻¹⁵, raises the curve as τ^(2/3) (rising dashed line). Their sum (solid) is lowest at 2.8 years, at 2.8·10⁻¹⁵; averaging longer than that makes this pulsar a worse clock, not a better one, because the background grows faster in the residuals than the white noise shrinks. At ten years the background alone is 5.5·10⁻¹⁵.

A clock whose best interval is three years

A pulsar time scale was expected to improve for as long as anyone kept timing it. The background the timing arrays found in 2023 says otherwise — a single pulsar is steadiest over about three years and worse over every longer interval, and how far an array of them can push that floor down depends less on how many pulsars it holds than on how evenly they cover the sky.

6 figures · Timescales

Cosmology

One object, seen once, from inside — and every number in it the output of a model.

One cluster weighed two ways, and the redshift where the two swap places. The fractional error on the mass of a single cluster of 3·10¹⁴ solar masses, on a logarithmic axis, against the cluster's redshift. Weighed by the shapes of the galaxies behind it (rising curves), the error is 50 per cent at redshift 0.3 and 750 per cent at 1.5 for a survey of today's depth, 10 galaxies per arcmin²; 26 per cent at redshift 0.3 and 187 per cent at 1.5 for a survey of the next depth, 30 galaxies per arcmin²: the galaxies far enough behind the cluster to be lensed by it run out. Weighed by the dipole it draws on the microwave background (nearly flat curves), the error is 461 per cent at 0.3 and 337 per cent at 1.5 for a map with 10 μK-arcmin of noise; 224 per cent at 0.3 and 153 per cent at 1.5 for a map with 3 μK-arcmin of noise — never good for one cluster, and almost the same at every distance, because the source is at redshift 1090. The curves cross at z = 1.22 (10 galaxies per arcmin², 10 μK-arcmin), z = 0.89 (10 galaxies per arcmin², 3 μK-arcmin), z = 1.77 (30 galaxies per arcmin², 10 μK-arcmin), z = 1.41 (30 galaxies per arcmin², 3 μK-arcmin). Below the crossing a galaxy survey weighs each cluster better; above it, only the background weighs it at all.

A mass scale checked near and carried far

A cluster can be weighed by the galaxies behind it or by the microwave background behind it, and the two share no systematic error. They do not share a redshift range either: the galaxies run out at about one, the background never does, and a comparison can only be made where both still work.

6 figures · Sunyaev zeldovich The number of lit dwarfs at a void's centre, against the date it was reionised. The number of haloes heavier than 3.2·10⁹ M☉/h that had collapsed from the matter of a 5 Mpc/h sphere by the redshift at which that sphere was reionised, on a logarithmic axis, against that redshift — for a sphere at the centre of a void with 20 per cent of the mean density today (solid) and an average sphere of the same size (dashed). If only those haloes could make stars, they are the sphere's dwarf galaxies. Reionised at 5.3: 4.3 in the void sphere, 195 in the average one; reionised at 6: 2.0 in the void sphere, 121 in the average one; reionised at 7.7: 0.22 in the void sphere, 29 in the average one. Each unit of delay in the date multiplies the void's count by 3.2 and the average sphere's by 2.1: the void's dwarfs sit further out in the tail of the distribution, and a tail is steep. A count of a few galaxies at a void centre is a statement about when that void's gas was heated, to within a fraction of a unit of redshift — provided the threshold is known.

A void dated by the dwarfs that lit in time

Reionisation did not heat the universe all at once, and the centres of the large voids were among the last places it reached. A dwarf halo there could make stars only if it had grown heavy enough before its own gas was heated — so the number that did is a clock for one void, and a clock that runs faster the emptier the void.

6 figures · Large-scale structure

Exoplanets

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

The heaviest atom a hydrogen wind can carry, against the strength of the wind. The crossover mass — the heaviest atom an escaping hydrogen flow drags out with it — against the hydrogen flux at the base of the flow, on a logarithmic axis, for the Earth's gravity and gas at 1000 K. Atoms lighter than the crossover leave with the hydrogen; heavier ones stay. The crossover rises in proportion to the flux: lifting neon takes 9.7·10¹² hydrogen atoms per square centimetre per second, argon 1.8·10¹³, krypton 4.3·10¹³ and xenon 6.6·10¹³. The Earth loses about 3·10⁸ today, which lifts nothing heavier than hydrogen. Driven by a young Sun at its saturated X-ray and ultraviolet output, an energy-limited flow would reach 2.2·10¹³ on Mars, a crossover of 114; 2.3·10¹³ on Venus, a crossover of 50; 10¹³ on the Earth, a crossover of 21 — the smaller planet's weaker gravity both raises the flux per unit area and lowers the force holding the heavy atoms back.

The heaviest atom a hydrogen wind can lift

A planet losing hydrogen fast enough to blow off as a wind drags other gases with it, but only those lighter than a crossover mass set by how strong the wind is. Below it every gas leaves in proportion to its distance from the limit and its isotopes are sorted by the same amount per mass unit; above it nothing goes. The Earth's xenon is sorted and its krypton is not, which no wind of neutral atoms can arrange.

6 figures · Radius valley Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find.

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

6 figures · Transits

Galaxies

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

Haloes that agree on the line of sight, and disagree across the sky. The dispersion of the tangential component of motion on the sky — across the radius vector, as a proper motion measures it — predicted by each of the 13 dark haloes that fit the same flat line-of-sight dispersion of 10 km/s, against projected radius in half-light radii, for stars in a Plummer profile. Coloured by the halo's inner slope; the dashed line is the line-of-sight value all of them reproduce. At the half-light radius the haloes' line-of-sight dispersions agree to 2.0 per cent, and their tangential dispersions range from 8.3 to 10.8 km/s, a factor of 1.29. The anisotropy each halo needed to match the line of sight is written into the motion across it, which no line-of-sight measurement sees.

Motions across the sky measure the orbits first

A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.

6 figures · Velocity dispersion A compact galaxy with a faint envelope, seen at redshift two. The surface-brightness profile of a model galaxy at redshift 2, in magnitudes per square arcsecond against radius in kiloparsecs on a logarithmic axis (solid): a compact de Vaucouleurs core of 1 kpc half-light radius and an exponential envelope of 10 kpc holding 30 per cent of the light, dimmed by 4.77 magnitudes by the expansion. The dotted curve is the core alone. Horizontal lines mark survey limits of 27 and 30 magnitudes per square arcsecond. A single Sérsic profile fitted to what is brighter than each limit (dashed) gives a half-light radius of 1.3 kpc and index 4.6 above 27, and a half-light radius of 1.5 kpc and index 5.1 above 30. The galaxy's true half-light radius, core and envelope together, is 2.1 kpc. The envelope is above the deeper limit out to 16 kpc, and the fit still does not follow it: a single profile weighted towards the bright centre cannot bend to two components, and more depth only nudges the compromise it settles on.

One profile for a galaxy that has two

The worry about the compact galaxies of ten billion years ago was that their growth since might be partly a faint envelope the sky hides. Modelled, the envelope does hide — but not mainly under the sky. A single profile fitted to a core with an envelope reports a compromise at any depth, too large for a tight envelope and too small for a wide one, and the growth it measures is an understatement.

6 figures · Surface brightness

Gravitation

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

How long a chaotic orbit stays away, with and without islands. The fraction of recurrence times to a small square in the chaotic sea of the standard map that exceed a given number of iterations, on logarithmic axes, counted along one orbit of 6·10⁷ iterations at K = 1.5 and K = 7 (772,165 recurrences, mean 78; 548,043 recurrences, mean 109). Dashed: exponentials with the same means. At K = 7, where the islands have all but vanished, the recurrences are exponential: the orbit forgets where it was in a few iterations. At K = 1.5 they part from the exponential after a few hundred iterations and fall instead as a power of time, with exponent 1.17 between a thousand and a hundred thousand iterations; the step at the left is the share of visits that simply stay in the square for a second iteration. A recurrence a thousand times longer than the mean is found once in 257,388 recurrences, where an exponential would give one in e to the thousand. Each mean is the area of the chaotic sea divided by the area of the square — 110 when the sea fills the torus, 78 at K = 1.5, where the islands take 29 per cent of it. The islands shorten the average absence and lengthen its tail.

A chaotic orbit clings to the edge of order

In a chaotic sea with no islands, an orbit forgets where it was within a few steps, and the time it stays away from any region falls off exponentially. Put islands of regular motion in the sea and the long absences multiply by factors of millions: the orbit reaches an island's edge, is caught in the hierarchy of smaller islands round it, and behaves as if it were regular for thousands of steps. The tail becomes a power law, and a rare few absences carry a large share of the orbit's whole history.

6 figures · Chaos Dark companions' masses with and without a gap, as an astrometric survey would see them. The distribution of the masses of dark companions in a sample found by their stars' astrometric orbits, per unit of the logarithm of mass, for two hypotheses: neutron stars near 1.35 solar masses and black holes near 8 in equal numbers with nothing between 2 and 5 (solid), or the same with the gap filled at the density the black holes have at 5 (dashed; 16 per cent of the sample in it). Each is drawn as measured with mass errors of 10 per cent (blue) and 30 per cent (orange). With 10 per cent errors the empty gap stays empty: 0.2 per cent of the sample lands between 2.5 and 4.5 solar masses, against 10 per cent if it is filled. With 30 per cent errors the edges spill in — 4.3 per cent against 12 per cent — and the gap becomes a shallow dip that a modest sample cannot distinguish from a filled one.

A gap that counting can test and impostors can fill

The black holes in X-ray binaries leave a gap between the heaviest neutron stars and about five solar masses, and nobody knows whether it is real or an artefact of how those masses are measured. Astrometric orbits give masses that need no light curve, and with errors of ten per cent a few dozen of them would settle it. The difficulty is not the counting. It is that pairs of white dwarfs and neutron stars, dark and close, can weigh exactly what a gap object would.

6 figures · The two-body problem

Orbits

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

How much a dying star stirs its companion's orbit, against their separation. The rms change in orbital eccentricity of a 0.8 M☉ companion when a 2 M☉ star becomes a 0.6 M☉ white dwarf, against the binary's separation before the loss, both logarithmic. Blue: the mass loss alone — twelve thermal-pulse episodes 500 years wide and a superwind 10,000 years wide, each adding a kick of random sign whose size falls off exponentially once it lasts longer than about a sixth of an orbit. Below 100 au the orbit is slow to every episode and the eccentricity is untouched. The pulses switch on near 229 au, and by 3,000 au the change is 0.26 and the superwind too has become sudden. Dashed: the change a velocity kick to the white dwarf of 0.05 or 0.2 km/s would make on its own, which grows as the square root of the separation. A 0.05 km/s kick outweighs the pulses out to 251 au; a 0.2 km/s kick outweighs the pulses at every separation drawn.

A kick that writes over the record of the pulses

A star that becomes a white dwarf loses half its mass in pulses, and a companion far enough away feels each pulse as a sudden jolt to its orbit's shape. The wide binaries of white dwarfs should therefore carry the pulses' record in their eccentricities. They would, except that the white dwarf itself is born moving at a few tenths of a kilometre a second, and a kick that small writes more eccentricity than all the pulses together.

6 figures · Effective potential The age a handful of craters allows. The probability of each age for a 100 square-kilometre surface on Mars, given a count of 0, 3, 10, 30, 100 craters 50 metres across or larger, per unit of the logarithm of age, with a uniform prior on age. No craters: younger than 45 Myr at 95 per cent, and nothing more; 3: 55 Myr, from 31 Myr to 88 Myr; 10: 158 Myr, from 116 Myr to 212 Myr; 30: 457 Myr, from 382 Myr to 547 Myr; 100: 1,502 Myr, from 1,359 Myr to 1,660 Myr. Each tenfold increase in the count narrows the range by about the square root of ten, from a factor of 2.8 with three craters to 1.22 with a hundred. The counting error is only part of the story: the chronology converting a crater density into an age is uncertain by a factor of about 1.6 on Mars before any crater is counted.

A young surface dated by a handful of holes

The surfaces that decide whether a planet is still geologically alive — the freshest lava on Mars, the smoothest ice on Europa — are the ones with fewest craters to count. Their ages are Poisson estimates from a few dozen holes, they cannot be younger than the area and crater size allow, and a fraction of a per cent of older ground in the counting area can double them. What survives best is not when a surface formed but which of two surfaces formed first.

6 figures · Surface chronology

The observed sky

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

Lucky frames chosen by the twinkle instead of the sharpness. The mean Strehl ratio of a target against its separation from the reference star, drawn as the high layer's shift in units of the aperture (32 arcseconds at the right for a 2.5-metre telescope and a layer at 8 km), for the best 2 per cent of 1600 frames chosen three ways: sharpest on the reference (orange), least scintillation of the reference's light (green), and best on both ranks together (blue); half the turbulence at the ground, both layers varying in strength frame to frame. Chosen by sharpness, the frames give 0.52 on axis and 0.34 at the edge. Chosen by scintillation, 0.22 and 0.22: the same everywhere, because the high layer — the only one whose blur differs between directions — has been selected directly, and lower everywhere, because the ground layer, which blurs every direction alike, has not been selected at all. Both ranks together give 0.44 and 0.34, a little more at the edge than sharpness alone at some cost at the centre. All frames average 0.10.

A twinkle that knows the high air and not the low

Lucky imaging keeps the frames in which a reference star is sharp, and that choice happens to favour moments when the high turbulence was quiet, which is why it reaches further across the field than it should. A star's twinkling reports the high layer directly. Choosing frames by the twinkle alone gives a field that is equally sharp everywhere and sharp nowhere, because the twinkle cannot see the air at the ground. Choosing by both buys a little more field for a little less centre, and only where the high air comes and goes.

6 figures · Seeing The summer sunshine at Mars's north pole over the last million years. The sunshine reaching Mars's north pole at the summer solstice, relative to its average, over the last 1,000 thousand years (blue), from a forcing built from the main periods of Mars's recent orbit: the tilt varying by about 3.5 degrees in 120,000 years and 1 degree in 99,000, the eccentricity by 0.03 in 95,000 years and 0.02 in 2.4 million, and the perihelion going round the seasons once in 51,000 years. The part due to the tilt alone is dashed. The sunshine swings by a factor of 2.06 between its extremes: the tilt sets a slow swell with a period of 120,000 years, and the precession, whose strength the eccentricity sets, cuts a faster ripple of 51,000 years into it, strongest when the orbit is most eccentric. These two periods, and their ratio of 2.35, are what a record of the sunshine should carry.

Layers that keep the rhythm and lose the dates

Mars's polar caps are stacks of ice and dust laid down while the planet's tilt and orbit changed the summer sunshine, and the question is whether they can be read as a record of that sunshine. They can — for the rhythm. Even when the ice accumulates faster under weak sunshine than under strong, the cycles of the tilt and the precession survive in the depth of the stack in their correct proportion. What does not survive is the date of any one layer.

6 figures · Obliquity

Spaceflight

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

The propellant a long departure burn wastes, under three ways of pointing it. The gravity loss of a departure from a 300-km orbit to a hyperbolic excess speed of 3 km/s with an exhaust speed of 4.41 km/s, against the stage's initial thrust-to-weight ratio, both logarithmic, for thrust held along the velocity (orange), held fixed in space at the best direction (gold) and turned at the best constant rate (blue). At a thrust-to-weight of 0.05: 1264 m/s, 3597 m/s and 1521 m/s. At 0.2: 166 m/s, 328 m/s and 154 m/s. At 0.8: 12 m/s, 20 m/s and 10 m/s. No one of the three is best everywhere. For burns of a few minutes to half an hour the constant-rate turn beats following the velocity by up to a tenth; for the longest burns, which sweep most of an orbit, following the velocity wins, because the direction that does most good turns faster as the vehicle gains speed and no constant rate keeps up. A fixed attitude, the simplest to fly, costs 88 per cent more than the best turn even for the strongest stage drawn, and several times more for weak ones. The true optimum, which no simple family contains, lies below all three and is computed numerically for each mission.

No single way to point a long burn

A departure burn that lasts tens of minutes wastes propellant because it cannot all happen at the lowest, fastest point of the orbit, and how much it wastes depends on where the engine points while it runs. Three simple rules — follow the velocity, hold one direction in space, turn at a steady rate — are each best somewhere and none everywhere. A steady turn saves up to a tenth for burns of minutes; following the velocity wins for burns that last most of an orbit; holding still costs twice as much as either.

6 figures · Orbital transfer

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