Distance & Parallax

Part 5: The Inverse-Square Law — Two Derivations

The second pillar of distance measurement: how light spreads from a star to Earth.

Geometric Derivation: Concentric Spheres

A star of luminosity $L$ (total power radiated in all directions) sits at the center of a sphere. At distance $d$ the energy is spread over area $A = 4\pi d^2$, so the flux (power per unit area) is

$F(d) = \frac{L}{4\pi d^2}.$

The same total energy $L$ flows through every spherical shell; at larger radius the area grows as $d^2$, so the flux drops as $1/d^2$. Double the distance, quarter the flux; ten times farther, one-hundredth. The $4\pi$ is the full solid angle of a sphere.

Dimensional Analysis Derivation

Assume $F \propto L^a d^b$ and match dimensions in $[M\,L\,T]$: flux $[M\,L^0\,T^{-3}]$, luminosity $[M\,L^2\,T^{-3}]$, distance $[L]$.

$[M\,L^0\,T^{-3}] = [M^a\,L^{2a+b}\,T^{-3a}] \;\Rightarrow\; a = 1,\quad 2a + b = 0 \Rightarrow b = -2.$

So $F \propto L/d^2$; the constant $1/(4\pi)$ comes from geometry, which dimensional analysis can’t supply. Two independent routes give the same $1/d^2$ scaling — that’s confidence.

The Flux-Luminosity-Distance Equation