Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 1 / c

The magnitude zero point is defined by the reference response $I_0$. With zero magnitude assigned to that reference, the apparent magnitude isThe two responses must use the same photometric passband and instrumental weighting. If the reference has assigned magnitude $m_0$ instead, add $m_0$ to the right-hand side.

The traditional infrared photometric bands extending the optical photometric system into this range are approximatelyTheir photometric passbands lie mainly in atmospheric windows between strong molecular absorption bands. This makes ground-based observations practical. The exact centers depend on the system: $L^{\prime }$ is near $3.8\mu \mathrm{m}$, and shortened $K_s$ and $M^{\prime }$ bands reduce unwanted background or absorption. At the longest wavelengths, atmospheric and instrumental thermal radiation make sky brightness especially important. A modern $Y$ band near $1.0\mu \mathrm{m}$ is an additional choice, rather than one of the traditional $JHKLM$ extensions.

For identical luminosities in a uniform Euclidean distribution, the inverse-square law gives flux $F\propto r^{-2}$, while the number of sources within distance $r$ is proportional to $r^3$. Therefore ten times as many objects require $r_{10}/r_1=10^{1/3}$ and $F_{10}/F_1=10^{-2/3}$. Applying the definition of apparent magnitude givesThis is the Euclidean magnitude-limited source count relation. It assumes no extinction, spatial variation of density, or cosmological change in the flux-distance relation.