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 12 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 The companion's mass against the visible star's, for f = 6.08 M☉: never below 6.08, and 7.3 M☉ for a 0.7 M☉ star seen edge-on. The mass of the unseen companion against the mass of the visible star, both in solar masses, for a mass function of 6.08 M☉, at inclinations of 90°, 70°, 55°. Every curve starts at or above 6.08 when the visible star weighs nothing and rises as it gets heavier, because a heavier visible star moves less for the same companion. For a visible star of 0.7 solar masses — V404 Cygni's donor is a cool subgiant of about that mass — the companion is 7.3 M☉ at 90°, 8.6 M☉ at 70°, 12.4 M☉ at 55°. The shaded band is the range of neutron-star masses, up to about 2.3 M☉; no curve comes near it. A mass function above that limit is a proof that the companion is too heavy to be a neutron star, without knowing the inclination or the visible star's mass at all.

A floor under the unseen companion

A star that swings back and forth in velocity with nothing visible beside it is orbiting something. One velocity curve gives one number about the something — the mass function — and that number is not an estimate of its mass but a floor under it, reached only if the visible star weighed nothing and the orbit were seen exactly edge-on. When the floor stands higher than any neutron star can, the companion is a black hole, whatever the inclination turns out to be.

gravitation · The two-body problem The path of a star with a dark companion: a 2.66 mas orbit riding on its parallax and proper motion. The track on the sky of a star like the visible member of Gaia BH1 over 3 years, in milliarcseconds, east to the left: proper motion of about 27 mas a year, a yearly parallax loop of 2.09 mas, and on top of both the star's own orbit about the barycentre it shares with an unseen companion, a 185.6-day ellipse of semi-major axis 2.66 mas, eccentricity 0.45, inclined at 126.6°. The dashed line is the path the star would follow alone. The difference between the two, a loop more than 1.3 times the parallax's size, is the orbit — a semi-major axis of 1.27 AU at the parallax's distance, which with the visible star's mass gives the companion's.

A black hole weighed by a wobble on the sky

A velocity curve weighs an unseen companion only up to the tilt of its orbit. The same orbit followed on the sky, as a loop a few thousandths of an arcsecond across riding on a star's parallax, shows the tilt as the shape of the ellipse — and with the parallax turning the angle into astronomical units, Kepler's third law gives the companion's mass outright. That is how the nearest black holes were found: silent, unfed, and on orbits no X-ray survey could have reached.

gravitation · The two-body problem 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.

gravitation · The two-body problem Which populations can share a cluster's temperature. The share of a cluster's mass in a heavy population against the mass of its members relative to an average star of 0.5 solar masses, both on logarithmic scales, with Spitzer's limit: a heavy component can reach energy equipartition with the stars only below the line where its share of the mass times its members' mass ratio to the power three halves equals 0.16. Above it the heavy bodies sink, cannot slow down enough to match the stars' kinetic temperature, and decouple into a subsystem of their own. Stellar black holes, 10 solar masses and about 2% of the mass, sit at 1.8 — 11 times beyond the limit. Neutron stars sit at 0.05 and massive white dwarfs at 0.14, below it; binaries, at a tenth of the mass, sit at 0.28, just beyond it, which is part of why they gather toward the centre. Only the black holes are beyond it by an order of magnitude.

A population that cannot share the cluster's temperature

Encounters between stars push a cluster toward equal kinetic energies, so heavy bodies slow down and sink. Spitzer showed in 1969 that past a certain point they cannot slow down enough: the heavy population falls away from the rest and becomes a cluster of its own. Stellar black holes are past that point by a factor of ten, in every globular cluster that kept them — and the subsystem they form, not the stars' binaries, is what holds most clusters' cores open until the black holes have been spent.

gravitation · Binary heating

Named alongside it

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

AstrometryMass functionWhite dwarfBinary starsEccentricityGravity gradientNeutron starOrbital inclinationParallaxRadial-velocityRedshiftRoche limit

All concepts