Calculations

A calculation recorded here is one whose working is visible. The purpose is that it can be checked, by someone else or by yourself later, which a bare answer cannot be.

Why show the working

Every number on this site was computed rather than recalled, and several errors were caught that way. A squared term computed wrongly, a unit conversion dropped, a constant misremembered: these produce answers that look entirely plausible, and the only defence is a visible derivation and an independent check.

The specific failure mode is that a wrong number rarely announces itself. An orbital rate 12% low, a radiator area off by a factor of two, a delay wrong by a factor of three: none of these looks wrong in a table.

Structure

Rules that catch errors

Carry units through every step

A dimensional inconsistency is the cheapest error to catch and it catches a large fraction of algebra mistakes.

State the source of every constant

Whether mu is 398600.4418 or 398600.5 rarely matters; whether it is the right body always does.

Do not round intermediate values

Round only at the end. Rounding partway through a chain accumulates error that is hard to attribute later.

Report significant figures honestly

A result computed from a three-figure input is not known to seven figures, regardless of what the calculator shows.

Check a limiting case

Set a parameter to zero, or to infinity, and confirm the expression reduces to something known.

The sanity check

The most valuable field, and the one most often omitted.

An independent estimate need not be accurate. It needs to be independent, so that it fails differently from the calculation it checks. An order-of-magnitude estimate that agrees within a factor of two rules out the errors that matter most, which are factors of ten and sign errors.

The known-case check is the strongest. Computing the nodal regression rate for a 400 km orbit at 51.6 degrees and getting −5.0 degrees per day confirms both the formula and its implementation, because that value is independently documented. Getting −4.4 would indicate an error somewhere, and locating it is then a bounded problem.

Presentation

Calculations belong in prose and mathematical notation, not in code listings. The style guide prohibits code in content, and the reasons apply with particular force here: a derivation stated in mathematics is checkable by anyone, survives a change of language, and shows the reasoning rather than only the operations.

Where an implementation exists, link to the repository and record what it was validated against. See Code.

Worked example structure

A short illustration of the form, using a result recorded elsewhere on this site.

Question. What radiator area is required to reject one kilowatt at room temperature?

Givens. Stefan-Boltzmann constant $\sigma = 5.670 \times 10^{-8}$ W m⁻² K⁻⁴; emissivity $\varepsilon = 0.85$ for a typical radiator finish; radiator temperature $T = 300$ K; sink temperature taken as 0 K for deep space.

Assumptions. The radiator sees only deep space, with no Earth infrared or solar load. This is optimistic and gives a lower bound on the area.

Derivation.

Sanity check. The scaling is $T^4$, so at 350 K the area should fall by $(350/300)^4 = 1.85$, giving 1.4 m². That matches the value tabulated in Thermal control, computed independently.

Sensitivity. The result scales as $T^{-4}$, so temperature dominates entirely. A 10% error in assumed radiator temperature is a 40% error in area. Emissivity enters linearly and is known to perhaps 5%.

References

Calculations cite the sources of their constants. This page is a convention.