Illustrated Physics
physics, drawn
Concept

Virial theorem — where it appears

The relation, for a bound system in a steady state, between the average kinetic energy and the average of force times position. For an inverse-square attraction twice the kinetic energy plus the potential energy is zero, so a star's total energy is minus its kinetic energy.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does.

The ball of gas that heats up as it cools

Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

astrophysics · Self-gravity What a coordinate is worth, by the shape of its energy. The mean energy stored in one coordinate, in units of kT, against the power with which that coordinate enters the energy. The curve is 1/n and the points are the same average obtained by integrating the Boltzmann weight numerically, agreeing to 3.3e-7 per cent at worst. an energy going as |x|^1 holds 1.0000 kT; an energy going as |x|^1.5 holds 0.6667 kT; an energy going as |x|^2 holds 0.5000 kT; an energy going as |x|^3 holds 0.3333 kT; an energy going as |x|^4 holds 0.2500 kT; an energy going as |x|^6 holds 0.1667 kT. The familiar half a kT is the case n = 2 and nothing more general than that: a coordinate whose energy is linear in it — the momentum of an ultrarelativistic particle, or a field with no restoring force but a constant tension — carries a whole kT, and one confined by a very steep wall carries almost nothing. Equipartition is a theorem about quadratic terms, and calling it a theorem about degrees of freedom is the substitution that makes it fail.

The share that is not half a kT

Equipartition is quoted as half a kT for every degree of freedom, and it is nothing of the kind. It is half a kT for every *quadratic* term. A coordinate whose energy is linear in it carries a whole kT, and a gas hot enough that its particles' energy is pc rather than p²/2m therefore holds twice what the counting says — which drops its ratio of specific heats to four thirds and puts a star on the edge of being able to hold itself up.

thermodynamics · Equipartition Twice the kinetic energy, and what it equals. Twice the time-averaged kinetic energy of a bound orbit, divided by its time-averaged potential energy, against the power with which that potential depends on separation. Each point is a measurement: an eccentric orbit integrated for more than a hundred radial periods, with the two averages accumulated along it, and the radius checked to vary by at least a fifth so that the orbit is not trivially circular. The line is the exponent itself, and the points miss it by at most 6.6e-4. Two cases carry everything. At n = 2, a harmonic well, the two energies are equal — which is the ordinary equipartition statement, half a kT to the kinetic term and half a kT to the potential one. At n = −1, which is gravity and the Coulomb force, twice the kinetic energy equals minus the potential energy, so the total energy of a bound system is minus its kinetic energy. Nothing about temperature entered, and nothing about equilibrium: the relation holds for one orbit averaged over time as well as for a crowd averaged over members.

Weighing what cannot be put on a scale

Summed over every coordinate of a bound system, equipartition stops being a statement about temperature and becomes a relation between two averages: twice the kinetic energy equals n times the potential energy for a potential going as the nth power. For gravity that fixes a bound system's total energy from how fast its parts move — so a Doppler shift and an angular size return a mass, and for the Coma cluster the mass they return is fifty times the mass that shines.

thermodynamics · Equipartition The paths in space: orbits of one period, and a throw straight up. The same free falls drawn in space around the Earth, which is the filled disc. All start at the marked point 2 Earth radii from the centre. The circle is the circular orbit. The ellipses, of eccentricity 0.2 and 0.4, have the same period, so they come back to the start at the same moment. The straight line is the thrown clock's path: straight up to 4.46 Earth radii and back down the same line, arriving as the orbits complete one revolution. The Earth's rotation is ignored and it is treated as a point mass for the paths that pass close to it.

The orbit that ages less than a throw

A clock in orbit and a clock thrown straight up leave the same point at the same moment and meet there again one period later. Both fall freely the whole way, so both follow paths of stationary proper time — and the thrown clock comes back 4.1 microseconds older. Even a clock held still by a rocket, which is not falling at all, beats the orbit. Free fall picks out a path that is stationary, not one that is longest.

relativity · Time dilation Four plasma columns that need exactly the same current. Left: four pressure profiles across a 1 cm column — flat-topped, parabolic, peaked and hollow — each holding the same 10¹⁹ ions per metre at the same 10 keV, so the same total ∫2πr p dr. Right: the current each needs to hold itself together, from solving the force balance J × B = ∇p with Ampère's law, radius by radius. All four come out at 0.800, 0.800, 0.800, 0.800 MA: Bennett's 0.800 MA. The shape of the column decides where the current flows and how strong the field is inside it, and has no effect at all on how much current there must be.

The current that squeezes what carries it

A current flowing along a column of hot gas makes a magnetic field that wraps round the column, and the field pushes the current inward. If the column is a plasma, free to move, it is squeezed. Ampère's law and a balance of forces then fix how much current a column of given temperature needs to hold itself together — and the answer does not depend on how wide the column is or how its plasma is arranged inside. Ten thousand electronvolts in ten million million million ions per metre needs 0.8 million amperes, whatever the shape. Above about 1.4 million amperes the same arithmetic says a hydrogen column radiates faster than its current can heat it, and collapses.

electromagnetism · Ampere law Three orbits that are one orbit, rescaled. Three orbits round the same mass, each an ellipse of eccentricity 0.6, with sizes in the ratio 1 : 2 : 3, integrated from the inverse-square force. The dots mark twelfths of each orbit's period, at the same fractions of the way round. The shapes are exact enlargements of one another and so are the motions, with time stretched as size to the power 3/2: the measured periods are 1.000, 2.828, 5.196 times the smallest, against 1.000, 2.828, 5.196 from the rule. That is Kepler's third law, obtained without solving for any orbit.

The third law that falls out of a change of scale

Kepler spent a decade of tables to find that the squares of the planets' periods go as the cubes of their distances. The law can be had in three lines, without solving for a single orbit. Gravity's potential energy halves when distances double; stretch every length of an orbit by some factor and every time by that factor to the power three-halves, and the action changes only by a constant, so the stretched orbit is an orbit too. The same move gives the swing of a spring that keeps time at any amplitude, the virial theorem that makes a star heat up as it loses energy, and the spacing of quantum levels.

mechanics · Least action The shift a member galaxy shows against the centre. The predicted gravitational shift of a cluster member's light relative to the central galaxy's, in km/s, against the member's projected distance from the centre, averaged along the line of sight with the members assumed to follow the mass out to twice R200, for clusters of 1 × 10¹⁴ M☉, 3 × 10¹⁴ M☉, 1 × 10¹⁵ M☉. Every member is blueshifted relative to the centre, because its light climbs out of a shallower part of the well. For the heaviest the shift is −13.2 km/s at half a megaparsec and −21.6 at two; for the lightest, whose members end at 1.9 Mpc, −4.4 and −5.7 at one and a half. Projection blunts the profile: members seen near the centre include many that lie far in front or behind it.

The light that climbs out of a cluster of galaxies

A cluster of galaxies is a well a few megaparsecs wide, and its galaxies sit at different depths in it. Light from the one at the bottom climbs furthest, so every other member should look slightly blue beside it — by ten kilometres a second, against galaxies that rush about at a thousand. No single cluster can show it. Thousands stacked together can, but only once a second effect of exactly the same size, the slowing of the galaxies' own moving clocks, has been taken out — and it cannot be separated by choosing heavier clusters, because both are set by the same depth.

astrophysics · Gravitational redshift

Named alongside it

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

EquipartitionDegrees of freedomGravitational collapseHeat capacityKinetic theoryPhase spaceAdiabatic indexAmperes lawBohr sommerfeld quantisationThe Boltzmann factorBremsstrahlungConjugate point

All concepts