Chapter 25, Problem 50P

Two stars 18 light-years away are barely resolved by a 66-cm (mirror diameter) telescope. How far apart are the stars? Assume λ= 550 nm and that the resolution is limited by diffraction.

To determine

The distance between two stars which are barely resolved by the telescope

Answer to Problem 50P

Solution:

The stars are $1.73\times {10}^{11}\text{m}$ apart from each other.

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.

The angular resolution is given by,

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

If the stars which are apart by a distance xare at a distance s from the Earth then,

$x=s\theta$

Given:

The diameter of the mirror $D=66\text{ cm}=0.66\text{ m}$

Wavelength of the light $\lambda =550\text{ nm}=5.50\times {10}^{-7}\text{m}$

The distance of the stars from Earth $s=18\text{ ly}$

Formula:

Angular resolution limit due to diffraction is given by,

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

The distance between the stars is given by,

$x=s\theta$

Calculation:

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

$\begin{array}{c}\theta =\left(\frac{1.22\lambda }{D}\text{rad}\right)\\ =\frac{\left(1.22\right)\left(5.50\times {10}^{-7}\text{m}\right)}{\left(0.66\text{ m}\right)}\\ =1.017\times {10}^{-6}\text{rad}\end{array}$

Calculate the separation of the stars using the expression,

$\begin{array}{c}x=s\theta \\ =\left(18\text{ ly}\right)\left(1.017\times {10}^{-6}\text{rad}\right)\\ =1.8306\times {10}^{-5}\text{ ly}\end{array}$

Express the distance x in m.

$\begin{array}{c}x=1.8306\times {10}^{-5}\text{ ly}\times \frac{9.46\times {10}^{15}\text{m}}{1\text{ ly}}\\ =1.73\times {10}^{11}\text{m}\end{array}$