Focal Length problem for astronomy class

[Finnegan]

Homework Statement

A telescope has a focal ratio of f/7.5. You wish to use it with a spectrometer that requires an f/10 beam at its input. Compute the focal length of a 50mm diameter lens that, when inserted in the beam 150mm in front of the unmodified focal plane, produces the required beam.

Homework Equations

Focal ratio = focal length/diameter of entrance aperture. The notation f/10 means that the focal ratio is 10, not that the diameter is 10.

The Attempt at a Solution

I tried drawing some diagrams, which obviously can't be reproduced with ease here. I think the main problem might be that I'm not familiar enough with telescopes to know how one uses a spectrometer or an extra lens--must they go outside the telescope, thereby making the focal length longer, or can one or both be placed inside the telescope?

I also tried making up variables--saying, for instance, that the telescope's original focal length is 750 and its entrance diameter 100, which fits for f/7.5. But then I was stymied by the above fact: I don't know how a spectrometer works. I think if someone can give me information about the mechanics of the problem I should be able to make a better attempt.

Thanks.

 

[chemisttree]

This isn't a question about the spectrometer. The only reason it was mentioned is to provide a rationale for the f10 requirement.

I will give you a hint: you need a lens with a negative focal length. These lenses are sometimes called barlows or Smythe lenses.

 

[Finnegan]

Okay. I understand that the question doesn't focus on the spectrometer, but I'm still pretty sure that I need to understand how the setup would work in order to make any progress. If I could just get a text diagram of how the device would look, like so:

telescope lens--extra lens--focal plane--spectrometer

Or

extra lens--telescope lens--focal plane--spectrometer

As of right now, I don't know how the variables (focal length and lens diameter) have to vary, so I can't make any inferences.

 

[chemisttree]

when inserted in the beam 150mm in front of the unmodified focal plane, produces the required beam.

Does this sound like it is possible for the 50 mm lens to be placed in front of the objective? 150 mm in front of the unmodified focal plane.

 

[DeeNos]

Hello,

I would approach this problem by first understanding the basic principles of a telescope and a spectrometer. A telescope is an optical instrument that uses lenses or mirrors to gather and focus light from distant objects. The focal ratio, or f-number, is a measure of the light-gathering ability of the telescope and is equal to the focal length divided by the diameter of the entrance aperture.

A spectrometer, on the other hand, is a device used to analyze the properties of light, such as its intensity and wavelength. It works by passing light through a narrow slit and then through a series of lenses or prisms to separate the different wavelengths of light. The f-number of a spectrometer is related to the size of the slit and the focal length of the lenses used.

Now, let's look at the given problem. We have a telescope with a focal ratio of f/7.5 and we want to use it with a spectrometer that requires an f/10 beam at its input. This means that the spectrometer needs a narrower beam of light compared to the telescope. To achieve this, we can insert a lens in the beam before it enters the spectrometer.

We are given that the diameter of the lens is 50mm and it is placed 150mm in front of the unmodified focal plane. This means that the light from the telescope will pass through the lens before entering the spectrometer.

To find the focal length of this lens, we can use the formula for focal ratio, f = focal length/diameter of entrance aperture. Rearranging this, we get focal length = f x diameter of entrance aperture. Substituting the values, we get f = (10/7.5) x 50 = 66.67mm.

Therefore, the required focal length of the lens is 66.67mm. This means that when placed 150mm in front of the unmodified focal plane, the lens will produce an f/10 beam, which is suitable for the spectrometer.

I hope this helps in understanding and solving the problem. It is important to have a good understanding of the basic principles involved in order to approach and solve scientific problems effectively.