Quick check on telescope optics calculations please

[skydivephil]

Homework Statement

A Cassegrain telescope has a 500mm diameter , a focal length of 5000mm and a central obstruction of 80mm diameter.
Calculate:
1) the magnification with a 25mm eye piece
2) the light gathering power compared to aye of 5mm in diameter
3) the angular field in the sky if the apparent angular field is 50 degrees
4) the angular and linear resolution for 550 nm light in arc seconds and micro metersMost of this is easy but i just wanted to check I have to the conversion from angular resolution to linear resolution correct.

The Attempt at a Solution

1) mag= focal length scope/ eye piece length = 5000/25= 200
2) light gathering = (apeture of scope/apeture of eye)^2 =(500/5)^2 = 10,000
we have to subtract the central obstruction which is
(80/5)^2 = 256
so the answer is 10,000- 256 = 9744
3) angular field = apparent field /mag = 50/200 = .25
4) theta = 2.5 * 10^5 * lambda/diameter
= 2.5.10^5 * 550*10^-9*500810^-3
=2.5*10^5*1.1*10^-6
= 2.75 seconds of arc
To convert this into linear resolution in micro meters
apeture in um /206265
=500*10^3/206265 = 2.42
hence 2.42*2.75 = 6.66 um.

 

[anathema]

Your calculations for parts 1-3 appear to be correct. However, for part 4, the conversion from angular resolution to linear resolution is not quite accurate. The formula for angular resolution is θ = λ/d, where λ is the wavelength of light and d is the diameter of the telescope's aperture. To convert this into linear resolution, you need to multiply by a factor of 206265 (the number of arcseconds in a radian) and then divide by the focal length of the telescope. So the correct calculation would be:

θ = λ/d = 550*10^-9/500*10^-3 = 1.1*10^-6
Linear resolution = θ * 206265 / focal length = 1.1*10^-6 * 206265 / 500 = 0.453 μm

Alternatively, you could also use the formula for linear resolution, which is d = 1.22 * λ/f, where f is the focal length of the telescope. Plugging in the values, we get:

Linear resolution = 1.22 * 550*10^-9 / 5000*10^-3 = 0.133 μm

Both methods should give you the same result. So the final answer for part 4 would be:

Angular resolution = 2.75 seconds of arc
Linear resolution = 0.133 μm or 133 nm