Question

If an optical telescope focusing light of wavelength 550 nm had a perfectly ground mirror, what would have to be the minimum diameter of its mirror so that it could resolve a Jupiter-size planet around our nearest star, Alpha Centauri, which is about 4.30 light years from earth? (1 light year = 9.46×1015 m )

Answer 1

Diameter is 486.23 m

Solution:

Wavelength of the light,
`\lambda =  50\ nm`

Distance from the earth, d =
`4.30\ ly`

1 ly =
`9.46* 10^(15)\ m`

Since, the size of the planet is that of the Jupiter, therefore,

Diameter of the planet, D =
`1.38* 10^(8)\ m`

Now,

To calculate the minimum diameter:

The angular resolution is given by:

`\theta = (Diameter\ of\ the\ object,\ D)/(Distance\ of\ the\ object,\ d)`

Distance from the earth, d =
`4.30* 9.46* 10^(15)\ m = 4.067* 10^(16)\ m`

Now, putting the appropriate value in the above formula:

`\theta = (1.38* 10^(8))/(4.067* 10^(16)) = 3.39* 10^(- 9)\`

Also, we know:

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

`\theta = \frac{1.22* 550* 10^(- 9)}[1.38* 10^(- 9)} = 486.23\ m`