Concept

Cross-section — where it appears

The effective target area a process presents, defined so that the rate is the cross-section times the flux. For gravitating bodies it exceeds the geometric area by one plus the square of the ratio of escape speed to relative speed, which is why slow encounters are common.

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

The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.

An answer obtained along a path that was not taken

Integrate the force of a passing mass along the straight line the body would have followed if the encounter had not happened, and out comes an exact deflection with no free constant in it. The approximation is circular, it is wrong in a known direction by a known amount, and it is the reason stellar dynamics has closed forms at all.

gravitation · Impulse approximation

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

Prograde and retrograde, the same encounter. The same encounter twice: a companion of 1 times the primary's mass passing at 1.5 disc radii, with the disc spinning the same way the companion orbits and then the other way. Nothing else differs. The prograde disc grows a bridge towards the companion and a tail away from it; the retrograde one is barely disturbed. The reason is a resonance rather than a force: in the prograde case the outer particles keep pace with the companion for a substantial part of the encounter and are pulled the whole time, and in the retrograde case they sweep past it and the impulses cancel.

Whether it merges is a ratio of two times

Two galaxies passing each other either merge or do not, and what decides it is not how close they come. It is whether the encounter lasts long enough for the internal motions to respond — a slow, prograde passage transfers orbital energy into stellar orbits and the pair is bound; a fast one leaves both galaxies heated and still moving.

galaxies · Galaxy interactions

A flux that measures a temperature to 0.12 per cent. Neutrino flux against central temperature, in units of the standard model's, for the three main solar channels. The exponents are not arbitrary: each reflects how far up the Gamow peak the reaction has to reach, so the channel with the largest Coulomb barrier is the steepest. The boron-8 flux goes as roughly the twenty-fourth power, which means a measurement good to 3 per cent constrains the Sun's central temperature to 0.12 per cent — better than any other technique by an order of magnitude. The same steepness is why the flux is useless as a check on anything else: a stellar model whose central temperature is uncertain at the half-per-cent level predicts this flux to within a factor, and the disagreement between two model families is far larger than the measurement.

A flux that is a thermometer to a tenth of a per cent

The boron-8 neutrino flux from the Sun's core rises as roughly the twenty-fourth power of the central temperature. That makes it the sharpest thermometer in astrophysics and simultaneously the most fragile prediction — a model uncertain in its central temperature by half a per cent predicts the flux to within a factor.

stars · Solar neutrinos

A factor of three, and the flatness that prices every cure

The oldest stars in the Galaxy show a third of the lithium the first three minutes should have left. Three kinds of resolution have been offered, and the datum that rules on all three is not how much lithium is missing but how uniformly it is missing.

cosmology · Nucleosynthesis

Named alongside it

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

Adiabatic invariant Dynamical friction Impulse approximation Abundance problem Asteroid family Baryon-to-photon ratio Big bang nucleosynthesis Central temperature Close encounter Collisional cascade Convective envelope Coulomb logarithm

All concepts