Astronomy & Celestial Mechanics
the sky, computed
Concept

Dimensional analysis — where it appears

Reasoning about a quantity from the units it must be built out of, which fixes the form of a relation without solving anything. It is what turns a pulsation period into a mean density and a crossing time into a free-fall time, and it fixes the exponents while leaving a constant of order unity undetermined.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

G: 14 determinations in two families. Published determinations of G, each with its quoted one-sigma interval, sorted into two families — torsion balance, in one form or another, against beam balance, pendulum, atom interferometry. The shaded band behind each family is that family's inverse-variance weighted mean: 6.67435 ± 0.00004 across 11 of them, against 6.67343 ± 0.00009 across 3. The difference is 0.00092 ± 0.00010 10⁻¹¹ m³ kg⁻¹ s⁻², which is 9.3 standard deviations, computed here from the quoted errors alone. The arithmetic is the same one the Hubble figure uses and here it should be distrusted, because the scatter inside each family already exceeds what the intervals allow: eleven torsion-balance determinations spread over 500 parts per million with quoted intervals of 12 to 130 cannot all be right, whatever the difference between the families comes to. That is why the recommended value's uncertainty is expanded far beyond any single experiment's rather than being the weighted combination drawn here — the disagreement is between laboratories using the same method, not between methods.

Nothing in the sky is weighed in kilograms

The Sun's gravitational parameter is known to eleven significant figures. The Sun's mass is known to five. The two statements are about the same object and the difference between them is a constant measured in basements, which is the worst-determined fundamental constant in physics.

gravitation · Gravitational constant Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround.

One number that is an age, a size and a density

The slope of the velocity–distance relation has units of inverse time, so it can be read as an age, multiplied by c to give a length, or squared to give a density. All three readings are natural, all three are quoted, and not one of them is the quantity it appears to be.

cosmology · Hubble constant How far the Sun has to sink before the sky darkens, on four worlds. The dimming of the zenith sky relative to sunset, in magnitudes, against the Sun's depression, for single scattering in an isothermal atmosphere with the stated scale height and vertical optical depth, scattering isotropically: the Earth (R 6371 km, H 8.5 km, τ 0.097); Mars (dust) (R 3389.5 km, H 11.1 km, τ 0.5); Titan (haze) (R 2574.7 km, H 50 km, τ 4); Pluto (haze) (R 1188 km, H 50 km, τ 0.02). These are representative values for the layer that does the scattering — air on the Earth, dust on Mars, haze on Titan and Pluto — not full models of those atmospheres. The angle over which a sky darkens is set by the height of the scatterers against the size of the planet, √(2H/R), because that is the depression at which the shadow over the zenith has risen one scale height. The Earth dims by ten magnitudes at 8.6° of depression, which at the equator of a world whose solar day is 24 hours takes the Sun 35 minutes. Mars (dust) dims by ten magnitudes at 13.4° of depression, which at the equator of a world whose solar day is 24.66 hours takes the Sun 55 minutes. Titan (haze) dims by ten magnitudes at 29.8° of depression, which at the equator of a world whose solar day is 382.7 hours takes the Sun 31.7 hours. Pluto (haze) dims by ten magnitudes at 43.9° of depression, which at the equator of a world whose solar day is 153.3 hours takes the Sun 18.7 hours.

The air's height against the planet sets the twilight

On the Earth the sky overhead fades over about nine degrees of the Sun's descent. That angle is not a property of air or of sunlight. It is the square root of twice the height of the scattering layer divided by the radius of the planet, and on a small world with a tall haze it grows to tens of degrees — so that twilight on Titan lasts more than a day.

sky · Twilight

Named alongside it

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

Age of the universeAstronomical unit (AU)Critical densityDeceleration parameterDensity parameterEarth shadowGlobular cluster agesGravitational constantHubble constantHubble distanceHubble timeKepler's third law

All concepts