Astronomy & Celestial Mechanics
the sky, computed
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Collisional cascade — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · orbits
  2. Below 0.46 microns a grain is not in orbit at all. The ratio of the radiation force to the gravitational force on a dust grain, against the grain's radius, for three densities. Every line has slope exactly −1 because gravity acts on the mass and radiation on the cross-section, and the ratio of a volume to an area is a length. Two horizontal lines matter and they are different statements. At β = 1 the star does not attract the grain at all. At β = 1/2 a grain released at rest from a circular orbit is already unbound, because it keeps the speed appropriate to the full stellar mass while feeling only half of it — and since dust is made by breaking up larger bodies that were on circular orbits, the lower line is the one that applies. For rock at 2500 kilograms a cubic metre that is 0.46 microns; for ice it is 1.15, and for iron 0.15. A collisional cascade that grinds material finer runs into this floor and stops, and the material that would have been finer leaves the system on a hyperbola.

    The drag that sorts a disc by size

    Starlight does three different things to a dust grain depending on how big it is — blows it out of the system, drags it inward over millennia, or ignores it entirely. Which one happens is decided by a single length, and whether it happens at all is decided by how crowded the disc is.

    part 2 · orbits

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