Astronomy & Celestial Mechanics
the sky, computed

The sky, computed.

Nothing in this subject can be picked up, weighed, or visited. Every number in it was inferred — from an angle, a brightness, or something that repeats — and the inference is nearly always geometric. This is a collection of essays about how that is done, one idea at a time, illustrated to the point where the argument becomes visible.

514 essays · 181 ladders · 885 named objects · what is new

An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.
Fig. 1 An orbit at eccentricity 0.6, with the primary at a focus rather than at the centre. Every length in the picture follows from that one number: the focal offset is 0.6 of the semi-major axis, and the second focus — marked, and empty — sits the same distance the other side. The dashed circle is the auxiliary circle the ellipse is a vertical compression of.

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one way into each of the 9 fields · all 514 essays

An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00. Orbits

The orbit is an ellipse, and the Sun is not in the middle of it

Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty.

9 figures Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller. Gravitation

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

9 figures The sky from latitude 52°. The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination 20 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint. The observed sky

The sphere that is not there, and why it is still the right model

The stars are at wildly different distances and the celestial sphere is a fiction. It is also the most useful fiction in observational astronomy, because for pointing at things, distance is exactly the information to throw away.

9 figures Parallax for a star at 1.3 parsecs. The same star observed from two ends of a baseline. The two sight lines are 1.54″ apart, so the parallax — half of that, the shift seen from a one-astronomical-unit baseline — is 0.769″ for a star 1.3 parsecs away. The definition of the parsec is the distance at which it would be exactly one. Starlight

The triangle that reaches the stars, and stops

Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it.

8 figures The Hertzsprung–Russell diagram. Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius. Stars

The diagram that sorted the stars, by plotting two things against each other

Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy.

10 figures A Hohmann transfer, 2.6 to 1 in radius. Two circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation. Spaceflight

The cheapest way between two orbits, and why it is so slow

Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait.

9 figures A fold 0.06 Einstein times wide, and two configurations that make the same one. A binary lens of mass ratio 0.003 at a projected separation of 1.5 Einstein radii. Top left: the source plane, with the caustic — the set of source positions at which the magnification is formally infinite — and the track of a background star across it. A single lens has no such curve; it magnifies smoothly and diverges only at one point. A second mass makes the lens mapping fold, and the fold has edges: crossing one, the number of images changes from 3 to 5, because a pair is created out of nothing on the critical curve. The small closed curve near the origin is the central caustic, always there; the larger one at 0.83 Einstein radii is the planetary caustic, and its distance from the origin is s − 1/s, which is where the planet's own image lies. Top right: the central caustic drawn twice, once for s = 1.5 and once for s = 0.667. They are 0.0184 and 0.0169 Einstein radii across and they lie on top of each other. That is not a coincidence of these numbers: to the order that a central-caustic anomaly is measured, a close binary and a wide one with the reciprocal separation produce the same perturbation, so an event with only a central anomaly returns two separations and no way to choose. Below: the light curve along the track. The smooth part is what a single lens of the same total mass would do; the spikes are the two crossings, 0.06 Einstein times apart, so a few hours inside an event lasting a month. The two curves differ in one thing only — the size of the source. A point source diverges at each fold and reaches 27; a source of angular radius 0.006 Einstein radii averages over its own disc and reaches 9 — an eighth of a source radius inside the fold the two are 8 and 4, with the divergence replaced by a rounded shoulder whose width is the source's own diameter. Everywhere else in this collection the finite size of a star is a nuisance that degrades a measurement. Here it is the ruler: the fold is a straight edge of known sharpness sweeping across a disc, so the shape of that shoulder gives the source's angular radius, and dividing it by the crossing time gives the angular Einstein radius — which is the one quantity a light curve otherwise cannot supply. Exoplanets

A light curve with a fold in it

A single lens magnifies smoothly. A second mass makes the lens mapping fold, and a fold has an edge — a curve across which two images appear out of nothing and the magnification formally diverges. Crossing it turns the finite size of the source star from a nuisance into a ruler.

9 figures The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not. Galaxies

A rotation curve that refuses to fall

Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure.

9 figures 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. Cosmology

The shift that is not a Doppler shift

Every galaxy beyond the Local Group has its lines shifted to the red, by an amount proportional to its distance. Read as a velocity that looks like a confession that everything is fleeing from here; read as a change of scale it says the opposite, because a uniform expansion produces exactly the same law measured from any galaxy in it.

8 figures

One idea, several depths

the deepest of the 181 ladders — an anchor idea, and the distinct arguments that stand against it

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. +3 more
7 essays · sky

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. +2 more
6 essays · spaceflight

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. +2 more
6 essays · exoplanets

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. +2 more
6 essays · cosmology

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. +2 more
6 essays · sky

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. +2 more
6 essays · spaceflight

Threads running through

themes, not chapters

Everything is a conic

One force law, four curves, and the discovery that a comet and a planet differ only in how fast they were going.

82 essays

Measuring the unreachable

Nothing in this subject can be picked up, weighed or visited. Every number in it was inferred from an angle or a brightness.

259 essays

Two bodies, and no more

The two-body problem is solved exactly and the three-body problem is not solved at all. Almost everything since has been about living with that.

117 essays

The sky as a clock

Periods that repeat to the second, used to build calendars, find longitude, and detect planets nobody can see.

118 essays

Light is the only messenger

One thin stream of photons carries distance, temperature, composition, motion and age — and everything visible is already old.

159 essays

Small perturbations, large consequences

The corrections that were supposed to be negligible: Neptune found in the residuals, Mercury's extra arcseconds, and orbits that are stable only for a while.

175 essays

Scales that defeat intuition

Distances, times and densities so far outside experience that the only reliable instrument is arithmetic.

197 essays