IAO 2008, Теория, Група α — Задача 2. Absolutely black cat

• 66.5 mag

  The printed sketch gives $M_{abc}=66^m.5$.

2αβ. Absolutely black cat. The statistically average cat, living in Grignano (Trieste, Italy) weighs about 4 kg and, if considered to be an absolutely black body, radiates basically in the IR part of the spectrum. For simplicity we shall compare the cat to the Sun, which absolute stellar magnitude is equal to $M_S=+4^m.8$. Let us consider that the surface area of a usual 4-kilogram cat is approximately equal to $s=15\,\mathrm{dm}^2=1.5\cdot10^{-1}\,\mathrm m^2$, that is essentially less than the surface area of the Sun, $S=4\pi R^2=\pi D^2$, where $D=1392000$ km is the diameter of the Sun, $S=6.1\cdot10^{18}\,\mathrm m^2$.

Also the surface temperature of the cat $t=310$ K is essentially less than the temperature of the solar photosphere $T=5780$ K.

As the general power of radiation is proportional to $S\cdot T^4$, we may obtain, that the ration of the power of radiation of the cat to the power of radiation of the Sun is

That is, the cat radiates approximately $5\cdot10^{24}$ times less energy than the Sun. In stellar magnitudes this difference is

Thus, the absolute stellar magnitude of the cat is equal to

Of course, it is estimation only.