Uncertainties and error propagation

A random (statistical) error scatters repeated measurements around the true value, and averaging $N$ of them shrinks it by $\sqrt N$. A systematic error shifts every measurement the same way, and averaging does nothing to it.

The 2023 LMC problem shows the difference clearly. Six distances scatter by $s = 0.24$ kpc, giving a statistical error of $0.24/\sqrt6 = 0.09$ kpc. But every distance depends on the same surface-brightness calibration, whose 0.8% bias adds a systematic error of 0.40 kpc that no amount of averaging removes. Quote both: $49.45\pm0.09_{\rm stat}\pm0.40_{\rm sys}$ kpc.

Write uncertainties as absolute ($\Delta x$, with units) or relative ($\Delta x/x$, dimensionless). Sums and differences combine absolute errors; products and powers combine relative errors.

For independent errors, combine in quadrature:

• Sum or difference, $q = x\pm y$: $\Delta q = \sqrt{\Delta x^2 + \Delta y^2}$.
• Product or quotient, $q = xy$ or $x/y$: $\dfrac{\Delta q}{|q|} = \sqrt{\left(\dfrac{\Delta x}{x}\right)^2 + \left(\dfrac{\Delta y}{y}\right)^2}$.
• Power, $q = x^n$: $\dfrac{\Delta q}{|q|} = |n|\,\dfrac{\Delta x}{|x|}$.
• Any function: $\Delta q = \sqrt{\sum_i\left(\dfrac{\partial q}{\partial x_i}\Delta x_i\right)^2}$.

The linear sum $\Delta q = \sum|\partial q/\partial x_i|\,\Delta x_i$ gives a worst-case bound. It is larger (roughly 1.4 to 1.7 times for two or three comparable terms) and is acceptable when a question asks for "error margins", but say which one you used.

Worked example (2016 D1). $P_B = \dfrac{P_t}{P_0}\dfrac{2\pi c}{a_t}$, with relative errors $0.02/0.67 = 3.0\%$, $0.02/7588 = 0.0003\%$ and $0.06/1.71 = 3.5\%$. In quadrature, $4.6\%$, so $P_B = 1.126\pm0.052$ d. The radius $r\propto P_t^2/a_t$ doubles the first term: $\sqrt{(2\times3.0)^2 + 3.5^2} = 6.9\%$.

Notice how the largest term dominates. Squaring makes small contributions negligible, so find the biggest relative error first and you already know the answer to within a few percent.

Astronomy is full of logarithms, and their error rules are worth memorising:

Since $m = -2.5\log_{10}F + C$,

so 0.01 mag is about a 1% flux error. Going the other way, the distance modulus $\mu = 5\log_{10}d - 5$ gives

An error of 0.1 mag in a distance modulus is a 4.6% distance error.

Parallax. $d = 1/\pi$, so $\Delta d/d = \Delta\pi/\pi$ and $\Delta M = \dfrac{5}{\ln10}\dfrac{\Delta\pi}{\pi}$. In 2017 D1, RT Aur's $\pi = 2.40\pm0.19$ mas gives $\Delta M = 2.17\times0.079 = 0.17$ mag, the same in every band because it comes from the distance alone.

Some formulas amplify errors enormously. Look out for differences of nearly equal numbers and for sines and tangents of tiny angles.

In the 2016 Moon problem the Sun's distance is $d_{\rm Moon}\sin\phi_M/\sin\phi_S$ with $\phi_S = 0.09^\circ$. An error of only $0.05^\circ$ in the elongation changes $\phi_S$ by more than 50%, and $d_{\rm Sun}$ with it. This is why Aristarchus' method failed in antiquity.

If a result depends on such a difference, say so, and estimate the uncertainty honestly instead of quoting the spuriously precise number your calculator shows.