Question

A series of optical telescopes produced an image that has a resolution of about 0.00350 arc second.

What is the smallest diameter telescope that could theoretically resolve these features using light with a wavelength of 1.70 μm? (Note: 1arcsec=1/3600∘)
Express your answer to three significant figures and include appropriate units.

Answer 1

The smallest diameter is
`D =122 \ m`

Step-by-step explanation:

From the question we are told that

The resolution of the telescope is
`\theta = 0.00350 \ arc \ second`

The wavelength is
`\lambda = 1.70 \mu m = 1.70 *10^(-6) \ m`

From the question we are told that

`1 arc \ sec = (1)/(3600^o)`

So
`0.00350 \ arc \ second = x`

Therefore

`x = 0.00350 * (1)/(3600 )`

`x = ( 9.722*10^(-7) )^o`

Now
`1^o = (\pi)/(180)`

So
`(9.722*10^(-7))^o = \theta`

=>
`\theta = (9.722*10^(-7)) * (\pi)/(180)`

`\theta = 1.69*10^(-8) rad`

The smallest diameter is mathematically represented as

`D = (1.22 \lambda )/(\theta )`

substituting values

`D = \frac{1.22 * 1.7 *10^(-6)} {1.69 *10^(-8) }`

`D =122 \ m`