Chapter 25, Problem 48P

To determine

The angular resolution limit set by diffraction for the 100-inch mirror Mt Wilson telescope.

Answer to Problem 48P

Solution:

The angular resolution limit of the 100-inch mirror Mt Wilson telescope is found to be $\left(1.54\times {10}^{-5}\right)°$.

Explanation of Solution

The image of a point object spreads out in the form of a disk due to diffraction effects. Two objects when viewed through a telescope appear to be just resolved, if the central maximum of one of the diffraction pattern of one object just falls on the first minimum of the other. This criterion of resolution is called Rayleigh’s criterion for resolution.

Given:

The diameter of the mirror $D=254\text{ cm}=2.54\text{ m}$

Wavelength of the light $\lambda =560\text{ nm}=5.60\times {10}^{-7}\text{m}$

Formula:

Angular resolution limit due to diffraction is given by,

$\theta =\left(\frac{1.22\lambda }{D}\text{rad}\right)$

Calculation:

Substitute the given values of diameter and wavelength in the expression and calculate the angular resolution.

$\begin{array}{c}\theta =\left(\frac{1.22\lambda }{D}\text{rad}\right)\\ =\frac{\left(1.22\right)\left(5.60\times {10}^{-7}\text{m}\right)}{\left(2.54\text{ m}\right)}\\ =2.689\times {10}^{-7}\text{rad}\end{array}$

Express the angular resolution in degrees.

$\begin{array}{c}\theta =\left(2.689\times {10}^{-7}\text{rad}\right)\left(\frac{360°}{2\pi \text{ rad}}\right)\\ =\left(1.54\times {10}^{-5}\right)°\end{array}$