Astronomy & Celestial Mechanics
the sky, computed
Concept

Black hole — where it appears

A region whose escape speed exceeds the speed of light, detected only through its gravity and through the radiation of material falling towards it. In this collection it appears as a mass inferred from orbits or from a velocity dispersion rather than as an object with an interior.

Named by 8 essays across 6 fields — each of them below, with the objects they name alongside it.

Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.

The law that links period to size, and weighs everything

Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since.

orbits · Harmonic law The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.

The tide is a difference, which is why there are two of them

The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

gravitation · Tides The tide across a moon, against the moon's own gravity. The tidal acceleration across a satellite and the satellite's own surface gravity, both in units of that surface gravity, against distance from the primary in planet radii. The tide falls as the inverse cube and the self-gravity does not fall at all, so they cross once — at 2.23 radii for the density ratio drawn. Inside the crossing the tide wins and a body held together only by its own weight comes apart.

The distance at which a moon stops holding together

The tide across a body falls as the inverse cube; the body's own gravity does not fall at all. There is therefore exactly one crossing, and Saturn's rings end within a few per cent of it.

gravitation · Tides 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.

A shift in a line is a speedometer, and it works at any distance

A spectral line has a wavelength fixed by physics. Measuring where it actually arrives gives the source's speed toward or away — and the measurement does not degrade with distance.

starlight · The Doppler effect How long a star lasts, against its mass. Main-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.

Why the biggest stars die first, and take the galaxy with them

A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of it.

stars · Stellar lifetimes Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.

A star held up by a rule about counting

A white dwarf makes no energy and does not collapse. What holds it up is not heat or pressure in any ordinary sense — it is a quantum rule forbidding two electrons from occupying the same state.

stars · Degeneracy 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.

The speed that does not come back, and the √2 that separates it

Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving.

spaceflight · Escape Two signals that peak at different separations. The astrometric centroid shift and the photometric magnification against the source–lens separation in Einstein radii, on a common horizontal axis and their own vertical ones, for θ_E = 1 milliarcseconds. The shift is u/(u²+2) times θ_E: it vanishes at u = 0, because the two images are then symmetric about the lens and their centroid is the lens itself; it vanishes as u grows, because the minor image fades and the major one approaches the source; and it is largest in between, at u = √2 exactly, where it is θ_E/2√2 = 0.3536 mas. The magnification at that separation is only 1.15, which is a nineteen per cent brightening — a signal a survey would barely flag. The two observables are therefore complementary rather than redundant. The photometric event is short, bright and centred on the closest approach; the astrometric one is broad, largest on the wings, and lasts several times longer, falling only as 1/u once the source is well away. That slow decline is why astrometric microlensing needs years of monitoring and why it was a prediction for sixty years before it was a measurement.

A centroid that moves when the brightness does not

The two images a lens makes are never resolved, but their centre of light is displaced from where the source would be — by an amount that is largest at a separation where the magnification is only 1.34, long after the photometric event is over.

exoplanets · Microlensing

Named alongside it

The objects these essays reach for when they reach for this one.

EccentricityGravity gradientRedshiftRoche limitTidal forceTidal lockingWhite dwarfAngular einstein radiusAngular momentumAstrometric microlensingAstrometryAstronomical unit (AU)

All concepts