IAO 2003, Теория, Група B — Задача 2

• $1 \times 10^{-8}$ K

  Order-of-magnitude estimate, not more than of order 10⁻⁸ K.

Large telescopes began to appear only in the beginning of the 20th century. The working surface of one 5-m telescope is equal to the working surfaces of a hundred 50-cm telescopes. There were built only a dozen large telescopes by the end of the 20th century. As an estimate let us assume that the total area of all professional telescopes is 1000 m², and that they have observed for a period of 30 years, 365 nights a year, 8 hours a night. This is about 300 000 000 seconds. As usual astronomers do not observe bright objects, especially when using large telescopes, so we take an average magnitude of $1^m$ to be an upper limit for our estimate. (N.B. It is important for the solution to understand that only light visible by the eye of a researcher, photographic film or CCDs may be used for obtaining knowledge about the structure of the universe.) For flux from Sun, with magnitude $-26.8^m$, the Total Solar Irradiation Constant is equal to 1.37 kW/m², so the equivalent Total Irradiation Constant for $1^m$ is equal to $10^{-27.8/2.5}\cdot1.37\ \mathrm{kW/m^2} \approx 10^{-8}\ \mathrm{W/m^2}$. So the total energy accumulated by 1000 m² during 300 000 000 seconds from a star of $1^m$ is of order 3000 J, enough only to heat a glass of tea by a few degrees. (In fact, though this need not be taken into account, only about 10% of this energy may be used, since the efficiency of both photographic film and the today's CCDs is still low, a few percents. So more correctly: of order enough to heat a glass of tea for a tenth part of degree.)

The mass of the water in the swimming pool $50 \times 20 \times 2$ m is 2 000 000 kg or $10^7$ glasses. So the estimate of how many degrees it would be possible to the raise temperature of the water in that swimming pool is: not more than of order $10^{-8}$ K.

Proposed list of parameters and assumptions used in solution:
The total square of all professional telescopes is 1000 m².
Period of observing by them is about 300 000 000 seconds.
As usual astronomers observe not bright objects, $1^m$ to be upper limit for estimation.
For an object of $-26.8^m$ the Total Solar Irradiation Constant may be used.
Not more about 10% of energy may be used due to the low efficiency of CCDs, etc.