Illustrated Physics
physics, drawn
Concept

Hydrostatic equilibrium — where it appears

The state of a still fluid in which pressure rises with depth exactly fast enough to carry the weight of everything above. The same condition governs a water column, a planetary atmosphere and a star's interior, and it fixes the pressure gradient without saying anything about the temperature.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

The isothermal atmosphere against the real one. Pressure as a fraction of its sea-level value, against altitude. The curves are the isothermal barometric formula at 220, 288, 400 kelvin, whose scale heights are 6.4, 8.4, 11.7 kilometres. The points are the measured standard atmosphere. At 20 kilometres the 288 kelvin model is 71 per cent out, because the air up there is not at 288 kelvin.

Why the air thins with height, and why that is the same law as the speeds

The pressure of the atmosphere falls exponentially with altitude, and the distribution of molecular speeds falls exponentially with energy. These are not two results that happen to look alike. They are one statement read on two axes.

thermodynamics · Kinetic theory Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

The pressure that only knows depth

A litre of water and a swimming pool press equally hard on a floor at the same depth. Pressure in a still fluid is a scalar with no direction of its own, it depends on how far down and on nothing else, and the shape of the container falls out of the arithmetic entirely.

fluids · Hydrostatics Force multiplied, distance paid. Two pistons on one body of fluid, of areas in the ratio 16 to 1. A force of 200 N on the small one holds 3.20 kN on the large one, and pushing the small piston 16 cm raises the large one by 10.0 mm. The two products are the same number: nothing is gained except the shape of the bargain.

Force multiplied, and nothing gained

A push on a small piston becomes a much larger push on a large one, in the ratio of their areas, with no machinery in between except the liquid. What the liquid will not do is give anything away — the distances shrink by the same factor the forces grow by, and the product is untouched.

fluids · Hydrostatics Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

The weight of the water that is not there

A submerged object is pushed up by the weight of the fluid it has displaced — not by something like it, not approximately, but exactly. The reason is that the pressures on its faces do not cancel, and the sum that survives has forgotten everything about the object except its shape.

fluids · Buoyancy Four tubes, four heights. Water in tubes of radius 0.2, 0.4, 0.8, 1.6 mm, with each meniscus drawn as the spherical cap a 20° contact angle forces and each height computed from it. The narrowest rises 70 mm and the widest 9 mm — in inverse proportion to the radius, with nothing about the glass or the volume of water entering it.

How high water will climb

Water rises up a narrow tube against gravity, and the narrower the tube the higher it goes. The height is set by a curved surface pulling on a circumference while gravity pulls on an area, and the two scale differently — which is the whole of it.

fluids · Capillarity The surface a spin decides. The free surface of a liquid in a dish of radius 0.5 m turning at 10, 20, 40 revolutions a minute. In the rotating frame the surface is a level set of gz − ½ω²r², so it is a paraboloid exactly and not to some approximation, and nothing about the liquid appears in its shape: the same curve is got with mercury, water or oil. The rim stands 14.0 mm, 55.9 mm, 223.6 mm above the centre at those rates. A parabola z = r²/4f has focal length f, so these surfaces are mirrors of focal length g/2ω² — 4.47 m at 10 rpm, 1.12 m at 20 rpm, 0.28 m at 40 rpm. That is checked here on the drawn curves rather than quoted: a vertical ray reflected off the surface at a quarter, a half, three quarters and the whole of the radius crosses the axis at the same height to 0.00%, which is what a mirror with no spherical aberration means. Doubling the spin quarters the focal length, and there is no other adjustment: the dish can only ever look straight up.

The surface a spin decides

Spin a dish of liquid and its surface settles into a paraboloid — exactly, with nothing about the liquid in the shape. A parabola of that form has a focal length of g over twice the spin rate squared, so a bucket of mercury turning at twenty revolutions a minute is a telescope mirror figured by a clock instead of by grinding.

fluids · Hydrostatics Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another.

The layer a parcel cannot leave

Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.

fluids · Stratification Cold matter, and the mass above which nothing holds it up. The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.452 solar masses, against 1.456 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it.

The mass no cold matter can hold up

A white dwarf gets smaller as it gets heavier, which no ordinary object does. Follow that curve upward and the radius reaches zero at 1.46 solar masses — because once the electrons are relativistic the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball does not depend on its radius at all.

astrophysics · Self-gravity 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 The distance that does not know how big the moon is. How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all.

The distance that forgets the moon

A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities.

astrophysics · Self-gravity The gravity that grows on the way down. The acceleration due to gravity inside the Earth against distance from the centre, from Gauss's law applied to the Preliminary Reference Earth Model's density: g(r) = G M(r)/r², counting only the mass inside each radius. The model reproduces the Earth's mass to 0.02 per cent, a surface gravity of 9.82 m/s² and a moment-of-inertia factor of 0.3308, none of which it was fitted to. For a uniform Earth, drawn dashed, gravity would fall in a straight line to zero at the centre. The real one does the opposite for the first 2891 km: it rises through the whole mantle to 10.69 m/s² at the core–mantle boundary, 8.8 per cent above its surface value, and only then falls to zero through the core. Going down through the mantle removes very little mass and brings the dense core much closer, and the second effect wins.

The pull that grows on the way down

Inside a uniform ball, gravity falls in a straight line from the surface to nothing at the centre, and that is the answer usually given for the Earth. It is wrong for almost three thousand kilometres. Going down through the mantle, gravity rises, reaching nearly nine per cent above its surface value where the core begins — because Gauss's law counts only the mass inside, and the local form of the law says gravity grows inward wherever the rock is lighter than two thirds of the average beneath it.

electromagnetism · Gauss's law

Named alongside it

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

DensityFree surfacePressureSelf-gravityApparent weightBuoyancyGauge pressureTemperatureAdiabatic processArchimedes principleThe Boltzmann factorBulk modulus

All concepts