Astronomy & Celestial Mechanics
the sky, computed
Concept

Random walk — where it appears

Accumulation of independent steps of random sign, so that the total grows as the square root of their number. An unbiased numerical error grows this way; a biased one grows linearly in the number of steps and quadratically in time.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has.

The part of a star that boils

Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite.

stars · Energy transport A step size below which a smaller step is worse. The error left in a long integration against the step size, for methods of three different orders, with both contributions drawn. The falling lines are truncation error, whose slope on these axes is exactly the order of the method. The rising line is round-off, identical for all three because it is a property of the arithmetic and not of the algorithm: every operation loses a few bits, the losses are independent, and they accumulate as the square root of the number of steps — which is why its slope is −1/2 and why it rises as the step shrinks. Each method's total has a minimum, at a step of 1.0e-6, 5.0e-6, 9.7e-4 for orders 1, 2, 4. Below that minimum every halving of the step costs time and makes the answer worse. That is the practical reason a solar-system integration is not run at an arbitrarily fine step, and it is a reason with nothing to do with computer time.

An error that grows like a random walk

A long integration accumulates two errors with opposite habits. One falls when the step is made smaller and grows in proportion to the time; the other grows when the step is made smaller and accumulates as a square root. Which of the two dominates decides whether a billion-year integration means anything.

gravitation · Numerical integration

Named alongside it

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

Adiabatic gradientBrouwers lawCNO cycleCompensated summationConvectionEnergy driftError propagationFloating-pointGranulationHelioseismologyMixing length theoryOpacity

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