Telescope optics question

I know what an f-ratio is: the ratio of a telescope’s focal length to its aperture. What I don’t understand is why a low f-ratio leads to a “faster” telescope and shorter exposure times for photography?

For a fixed aperture, and thus fixed amount of light entering the tube, how does the focal length affect the “brightness” of the image (yes, I know it’s not literally the brightness but you know what I mean)? So, I ask, for a fixed aperture how does varying the focal length affect the “brightness”? What’s the mechanism? For a refractor, wouldn’t a shorter focal length lead to a bigger image and thus make it duller (same amount of light spread over a bigger area)?

Thanks!

focal length divided by lens diameter.

Focal length being equal bigger lens diameter collects more light.

Lens diameter being equal the lens with a shorter focal distance concentrates light more

I tried a quick search, but did not find a complete answer to your question. But here’s a relevant tidbit from
http://www.skypub.com/tips/telescopes/choosing.html

For photography, normally the exposure time depends on how much light falls on the film per unit area of film. The diameter of the mirror determines the total amount of light gathered by the telescope/camera. The focal length determines how large an area this total light is projected onto - i.e. the magnification.

Say you have two 10cm diameter telescopes, one has f=100cm (F/10) and the other has f=50cm (F/5). (f is focal length) If you point both at the moon, they both gather the same amount of moonlight. But the F/10 telescope projects a 9mm diamter image of the moon onto the film. The F/5 telescope projects a 4.5mm diamter image of the moon onto the film. So the F/5 telescope provides more light per square millimeter of film - 4 times as much, in fact.

However, for astronomy the focal ratio doesn’t matter too much. For one thing, stars are point sources, and even if you magnify it a lot, it’s still a point source. It doesn’t get any dimmer. Also, if you use an eyepiece and do visual observations, you can change the magnification by changing the eyepiece. Actually, even for photographic systems, research observatories are usually equipped with lens systems to adjust the focal ratio for various instruments (just like the tele-extender you can buy for your camera). Fast telescopes have one big advantage though - size. The longer the focal length, the longer and heavier the telescope, which results in higher cost of the telescope itself as well as the support system and dome. Faster mirrors are expensive to make and test, but the difference is more than made up by the savings resulting from reduced physical size.

Isn’t it the other way around?