Q.Describe the measurement of Earth's shadow (umbra) radius during a total lunar eclipse.

Step 1. Setting up the geometry. During a total lunar eclipse, the Moon passes through Earth's umbra (the fully dark inner shadow) surrounded by the lighter penumbra. As the Moon moves through and exits the umbra, its illuminated crescent shape directly traces the curved edge of Earth's shadow.

Step 2. Photographing the eclipse. By photographing the Moon at the moment it exits the umbra (using a digital camera, as done for the 31 January 2018 eclipse visible from Tamil Nadu), the apparent radius of the umbra shadow disc at the Moon's distance, $R_s$, and the Moon's own apparent radius, $R_m$, can both be measured directly off the photograph (e.g. $R_s\approx13.2$ cm and $R_m\approx5.15$ cm as measured on the print/image).

Step 3. Computing the ratio. From these apparent (image) measurements, the ratio works out to $\dfrac{R_s}{R_m}\approx2.56$.

Step 4. Converting to a true radius. Since the Moon's true radius is already known independently ($R_m=1737$ km), and the ratio $R_s/R_m$ is the same whether measured on the image or in real distances (both are just scaled versions of the same geometry), the umbra's true radius is

$R_s=\frac{R_s}{R_m}\times R_m\approx2.56\times1737\approx4446\ \text{km}.$ …

Measure apparent radii on an eclipse photograph, compute their ratio, then scal …

• Using the apparent (image) radius of the umbra directly as if it were already the true radius in kilometres, without scaling by the ratio. …