D1 - surface brightness

for the case of a galaxy, one is dealing with an extended source, ie. an object that occupies a finite angular size on the sky
so, rather than distributing stellar flux over a circle, distribute it over a square area of the object

considering the angle subtended by a square patch of a galaxy, with side $D$ , on the sky
if $d >> D : α = D / d$
the intensity, or energy per unit area:

I = \frac{F}{α^{2}} = (\frac{L_{t o t}}{4 π d^{2}}) {(\frac{d}{D})}^{2} = \frac{L_{t o t}}{4 π D^{2}}

definitions	symbol	units
linear	$I (Σ)$	$L_{⊙} p c^{- 2}$
log	$μ$	magnitudes arcseconds $^{- 2}$

to convert from linear ( $I$ ) to a logarithmic ( $μ$ ) scale, starting from the definition of magnitude:

\begin{aligned} μ & = - 2. 5 \log_{10} (\frac{F}{α^{2}}) + c \\ = - 2.5 \log_{10} (F) + c + 5 \log_{10} (α) \\ = - 2.5 \log_{10} (\frac{L_{⊙}}{4 π d^{2}}) + c + 5 \log_{10} (\frac{D}{d}) \end{aligned}

μ = M_{⊙} + 5 \log_{10} (\frac{1}{10})

1 L_{⊙} {pc}^{- 2} = 25.7 {mag arc}^{- 2}

typically, galactic nuclei $μ \approx 18$ mag arc $^{- 2}$ , and overall $μ \approx 22$ mag arc $^{- 2}$
cf. night sky $μ \approx 24 - 25$ mag arc $^{- 2}$
therefore, galaxies are hard to detect against the brightness of the night sky
particularly true for low surface brightness galaxies (LSBGs)