Astronomy & Celestial Mechanics
the sky, computed
Concept

Power law — where it appears

A relation in which one quantity is a fixed power of another, appearing as a straight line on logarithmic axes. It has no characteristic scale, so a power-law spectrum carries no temperature and its slope is the whole of its content.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Three slopes: 2.5, −0.75 and −1.25, and only the middle one is about the electrons. A radio source's spectrum across five decades of frequency, computed for an electron population with index p = 2.5 in a uniform field. Nothing in this curve is a temperature, because a power law has no scale and therefore nothing a thermometer could read. The straight section between the two bends has slope −0.75, measured here off the drawn curve, and the electron index follows from it and from nothing else: α = (p − 1)/2, so a flux ratio between two frequencies is a measurement of the energy distribution of particles in a place no detector will ever visit. The two bends are the other two measurements. Below 80 MHz the source is opaque to its own radiation and rises as ν^2.5 — a slope fixed at 5/2 by the geometry alone, whatever the electrons are doing — and the frequency at which that happens gives an angular size for a source nothing has resolved, because the turnover is where the brightness temperature meets the electrons' own. Above 12 GHz the spectrum steepens by 0.50: the electrons that radiate at high frequency lose their energy fastest, so the top of the distribution has already emptied, and the frequency of the break is a clock. What no part of this curve gives is the magnetic field. Only the field and the particle density together enter the emission, and every field strength ever quoted for a radio source comes from assuming the two share the energy equally.

A spectrum with no temperature in it

Every spectrum in this collection so far has been a thermometer. A radio lobe's is a power law, and a power law has no scale — so there is nothing for a thermometer to read, and what the shape carries instead is the energy distribution of the particles that made it.

starlight · Synchrotron radiation One population, three integrals, two ends. The same mass function weighted three ways, each normalised so that the area under it is one, against mass on a logarithmic axis — so a share of the page is a share of the total. The light is a zero-age population's, every star still on the main sequence, which is the only age at which the top of the range is present at all. The steep curve on the left is the number of stars, which peaks at 0.08 M☉ and falls away because the function is steeper than m⁻¹; the middle one is the mass they carry; the one on the right is the light they emit, computed from this file's own main-sequence relation and peaking at 55.0 M☉ — 682 times further up the axis. The population that is counted and the population that is seen are two different populations. Above 8 M☉ there are 0.63% of the stars, 21% of the mass and 99% of the light. The slopes are α = 1.3 above 0.08 M☉, α = 2.3 above 0.5 M☉, and the low-mass end is where the honesty runs out: it has never been measured in another galaxy, and every mass inferred from a luminosity assumes it.

Most stars are small, and most of the light is not

The mass function is steep and the mass–luminosity relation is steeper, so counting stars and measuring their output are integrals of the same function dominated by opposite ends of it. Every mass inferred from a brightness passes through that mismatch.

stars · Initial mass function Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.

Where the mass is and where the light is

A population that grinds itself up settles into a size distribution with a fixed slope, and that slope puts almost all the mass in the largest bodies and almost all the cross-section in the smallest. So a debris disc's brightness measures a population whose mass it says nothing about, and the two are connected only by the exponent.

orbits · Collisional cascade The weakest bodies are a few hundred metres across. The energy per unit mass needed to shatter a body and disperse half of it, against the body's diameter, for basalt and water ice, from fits to impact simulations calibrated on laboratory fragmentation. Small bodies are held together by material strength, which falls with size because a larger body contains larger flaws; large bodies are held together by their own gravity, which rises with size because the fragments have to be thrown apart as well as broken. The two regimes meet in a minimum at 235 m for basalt and 204 m for water ice, where a body needs the least energy per kilogram to destroy of any size — 127 J/kg and 57 J/kg, the kinetic energy per kilogram of motion at only 16 and 11 metres a second. A collisional population is therefore not self-similar: it has a preferred size, and its size distribution records it.

The size at which a body is easiest to break

A pebble is held together by the strength of its rock and a moon by its own gravity, and somewhere between them is a size that has too little of either. Bodies a few hundred metres across are the weakest in the solar system — sixteen metres a second of impact energy per kilogram is enough to scatter one — and a population grinding itself down records that weakness as a knee in its size distribution.

orbits · Collisional cascade

Named alongside it

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

Collisional cascadeDebris discSize distributionAsteroid beltAsteroid familyBrightness temperatureBrown dwarfCooling breakCross-sectionDisruption thresholdDohnanyi slopeDust

All concepts