Illustrated Physics
physics, drawn
Concept

Sound speed — where it appears

The speed at which small pressure disturbances travel through a medium, about 343 m/s in air at 20 °C and 1,480 m/s in water. A flow faster than it cannot be signalled ahead by pressure, which is what makes shocks and supersonic flow different from subsonic flow.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Which wavelengths grow instead of oscillating. The dispersion relation of a self-gravitating isothermal gas, ω² = c²k² − 4πGρ, with each axis measured in the scale the gas sets for itself. Short wavelengths oscillate: pressure wins, and the disturbance is a sound wave. Long wavelengths do not: ω² is negative, so the disturbance grows exponentially instead of travelling, and the region collapses. The changeover is at λ_J = c√(π/Gρ), and it happens because pressure support acts on a sound-crossing time that grows with the region while gravity's collapse time does not depend on the size at all. Four densities spanning 6 decades are drawn and they lie exactly on top of one another, to 6e-16, because the criterion has no scale of its own: it is the same curve for a diffuse cloud and for a protostellar core, with different numbers written on the axes. In this gas at 10 K those numbers are 2.12 pc and 28.72 solar masses at 10² cm⁻³, 0.21 pc and 2.87 solar masses at 10⁴ cm⁻³, 4372 AU and 0.29 solar masses at 10⁶ cm⁻³, 437 AU and 0.03 solar masses at 10⁸ cm⁻³.

The disturbance that grows instead of travelling

A sound wave in a gas oscillates because pressure restores what the disturbance displaced. Add the gravity the gas exerts on itself and the restoring force acquires a competitor that does not weaken with size — so above one wavelength the sum changes sign, the frequency becomes imaginary, and the disturbance stops travelling and starts growing. It is the same wave equation with one term subtracted.

astrophysics · Self-gravity Every steady flow of gas onto a point mass. The Mach number of steady, spherical, isothermal flow round a point mass against distance from it, both on logarithmic axes, distance in units of the sonic radius, where the inflow reaches the speed of sound. Every curve is a contour of Bondi's conserved integral, which ties the Mach number to the distance. The faint curves are flows that never pass the speed of sound, or that exist only close in or only far out. Two curves cross at the saddle (1, 1): the solid one is accretion, starting from rest far away, reaching the speed of sound exactly at the sonic radius and falling supersonically within it; the dashed one is its time-reverse, a wind. Only the crossing flow joins a gas at rest to a hole that swallows it, and it fixes the rate: λ = exp(3/2)/4 = 1.120 in Ṁ = 4πλ(GM)²ρ∞/c³.

The gas a hole catches from far away

A particle drifting past a black hole is captured only if its aim is almost perfect; miss by a little and it swings round and leaves. Gas is different. Streams passing on opposite sides collide, lose their sideways motion as heat, and fall in, so a hole in warm gas draws from the whole sphere inside which its gravity beats the gas's thermal motion. Bondi worked out the flow in 1952: one steady solution passes smoothly from rest to free fall, it crosses the speed of sound at a single radius, and the rate it fixes grows as the square of the hole's mass — fast enough, left alone, to run away in a finite time.

astrophysics · Horizons

Named alongside it

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

AccretionAngular momentumBernoulli equationBlack holeBondi radiusDispersion relationEddington limitFree fall timeGravitational instabilityHydrostatic equilibriumJeans massMach number

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