Question

Two distant stars are separated by an angle of 35 arcseconds. If you have a refracting telescope whose objective lens focal length is \(3.5 \mathrm{~m},\) what focal length eyepiece do you need in order to observe the stars as though they were separated by 35 arcminutes?

Short answer

Answer: The required eyepiece focal length is approximately 5.83 cm.

Step-by-step solution

Convert angle from arcseconds to arcminutes

Given separation of stars = 35 arcseconds We need to convert it to arcminutes. 1 arcminute = 60 arcseconds Therefore, 35 arcseconds = \(\frac{35}{60}\) arcminutes.

Calculate the magnification needed

Initially, stars are separated by \(\frac{35}{60}\) arcminutes, and we want them to be separated by 35 arcminutes. Hence, the magnification required (\(M\)) can be calculated as: \(M = \frac{Final\ Separation}{Initial\ Separation}\) \(M = \frac{35}{\frac{35}{60}}\)

Simplify the magnification

Now, let's simplify the magnification: \(M = \frac{35 \times 60}{35}\) The 35s cancel out: \(M = 60\)

Calculate the eyepiece focal length

Using the magnification formula \(M = \frac{F_{objective}}{F_{eyepiece}}\) and the magnification value we found in Step 3, we can calculate the eyepiece focal length (\(F_{eyepiece}\)). Rearranging the formula to solve for \(F_{eyepiece}\), we get \(F_{eyepiece} = \frac{F_{objective}}{M}\) Plug in the given objective focal length 3.5 m and the magnification \(M=60\): \(F_{eyepiece} = \frac{3.5}{60}\)

Simplify the eyepiece focal length

Finally, let's compute the eyepiece focal length: \(F_{eyepiece} = \frac{3.5}{60}\) \(F_{eyepiece} = 0.0583 \mathrm{~m}\), or approximately 5.83 cm. So, in order to observe the two stars as though they were separated by 35 arcminutes, you will need an eyepiece with a focal length of approximately 5.83 cm.