Prerequisites
Differentials, variances, covariance, logarithms, and inverse-square flux.
For isotropic emission without attenuation, L = 4 pi d^2 F. Linearizing gives delta L/L approximately 2 delta d/d + delta F/F. Squaring and averaging introduces a covariance term; independence is an assumption, not a default fact.
Worked example
Independent small fractional errors of 3% in distance and 8% in flux give approximately sqrt((2*0.03)^2 + 0.08^2) = 10% fractional standard deviation in luminosity.
Model and Assumptions
An invented nearby-source model uses isotropic luminosity L = 4 pi d^2 F. Fractional standard deviations are 0.05 for d and 0.10 for F. Their correlation is rho. First-order relative variance is 4(0.05)^2 + (0.10)^2 + 4 rho(0.05)(0.10).
Further inquiry
Derive the variance formula from the differential, including the covariance factor. Explore rho from -1 to 1. Explain why a zero first-order variance under perfect anticorrelation would not guarantee an exact nonlinear luminosity.
Review criteria
- Carry the cross term with its correct factor.
- Distinguish first-order cancellation from exact cancellation.
- State the physical assumptions required by the flux relation.