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It ALL comes down to flux; the amount of photons per unit area of sky. Both telescopes have the same aperture so they both capture the same amount of light. The difference between them is their f/ratios squared.
10^2/4^2 = 6.25
So the F/4 is 6.25x faster than the F/10
To show this, let’s use an ASI1600 with 3.8 micron pixels.
With the F/4 and a FL of 520mm, you have an image scale of 1.51”/pixel.
With the F/10 and 1300mm FL, an image scale of 0.6”/pixel.
Let’s say with a 130mm telescope you capture 100e- per square arcsec of sky per second. That’s the sky flux.
With the F/4 it’s (1.51^2)*100=228 electrons/s per pixel
With the F/10 it’s (0.6^2)*100=36 e-/s/pixel
So you’d have to capture 6.33 (rounding errors) times longer with the F/10 to capture the same amount of electrons.
Another example would be to have a 325mm aperture with 1300mm focal length. It’s a 12.8” F/4 so it captures 6.25x more light than the 130mm. It will also be imaging at 0.6”/pixel but the flux increases 6.25x.
So, (0.6^2)*625=225e-/s/pixel
Remember, 130mm gives a flux of 100e-/arcsec so a 325mm gives 625e-/arcsec.
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