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7.4.6.1 Expected surface brightness limit ¶

The Surface brightness limit was defined in Surface brightness limit of image. As we saw there, it is ultimately a theoretical extrapolation of one pixel’s noise level. Therefore, we can use a reference image’s surface brightness limit (\(SB_r\)) to derive an expected surface brightness limit of another image (on another telescope with another exposure time, but with the same filter; let’s call it \(SB_i\)). Assuming no correlated noise (which is only valid on a single exposure!), the \(\sigma_p\) described above is purely due to the Poisson noise of the background signal (for example Zodiacal light, light pollution or etc). Therefore, taking \(S\) to be the exposed surface of the telescope’s primary mirror (or lens) and \(t\) to be the exposure time, the signal of the background will increase with \(S\times t\) and therefore \(\sigma_p\propto \sqrt{St}\). Plugging this into two instances of the equation above, allows us to derive \(SB_i\) from \(SB_r\)

$$SB_i = SB_r + 2.5\log_{10}\left( {n_i \over n_r} \sqrt{{S_i \over S_r}{t_i \over t_r}{A_i \over A_r}} \right)$$

Since almost all mirrors are circular, we can simplify the relation above by replacing the exposed surface with the exposed radius (accounting for the non-exposed area in most reflective telescopes due to the secondary mirror) as shown in the equation below. Note that we didn’t simply say “radius” (and the equation does not have \(r\)), but “exposed radius” (\(r_e\) in the equation). This is a very important factor to have in mind. For example in the Vera C. Rubin Observatory (that will be conducting the LSST survey) the primary mirror has a diameter of 8.4 meters, however, the inner circular area of radius 5 meters is not used (due to the second and third mirrors)²²⁹. Therefore, the useful surface of the Vera C. Rubin telescope is \(\pi(8.4^2-5.0^2)=\pi45.56=\pi6.75^2\), giving it an exposed radius of \(r_e=6.75m\) (for an easy implementation of this equation in Gnuastro, see the sblim-diff operator of Unit conversion operators).

$$SB_i = SB_r + 2.5\log_{10}\left( {r_{ei} \over r_{er}}{n_i \over n_r} \sqrt{{t_i \over t_r}{A_i \over A_r}} \right)$$