IAO 2014, Теория, Група β — Задача 5. Climate

Solution

In order to get an inner orbit of the star that would provide a seasonal cycle with a periodicity of 1 year, it is necessary that its close position with the Earth would take place once a year. That is, the synodic period of its revolution should be one year. It is not difficult to find that the sidereal period of revolution of the celestial body must be equal to

and the radius of its orbit, from the III Kepler’s law:

Thus, the new celestial body will once a year become close to the Earth at a distance of $R_0-R=0.37$ au, providing summer on the planet, and once a year move away to a distance of $R_0+R=1.63$ au, leaving the Earth in winter.

The ratio of energies received by the Earth from the new star in summer and winter will be

The ratio of these energies to the energy $\ddot E_0$, received from the new star at a distance of 1 au is

At the same time, the Earth will always receive additional energy $E_1$ from the cooler Sun, that is, $E_S=E_1+\ddot E_S$ in the summer, and $E_W=E_1+\ddot E_W$. To keep the heat balance, all the incoming energy should be radiated.

The ratio of the energy radiated by the Earth in summer and winter will be

Neglecting the effects of delays of the seasons compared to the maxima and minima of the energy income, let us assume that $\mathcal E_S=E_S$, $\mathcal E_W=E_W$. Then we can write:

The sum of $E_1$ and the average $\ddot E$ for the period must be the same as the Earth receives nowadays, $E_0$. To estimate the average value of $\ddot E$, let us take $\ddot E_0$, and so

5.1. Thus, the Sun in new conditions will emit 4.6% less energy, and so its temperature will be lower and equal to

5.2. To understand what spectral type will correspond to the star, we calculate by how many magnitudes it is fainter than the Sun:

From the HR diagram, we find that it is about the spectral type K3.

Source note: the winter ratio is printed with (R₀−R)² in its denominator although the preceding winter distance is R₀+R and the numerical 0.38 corresponds to the latter. The printed formula is retained. Calligraphic ℰ here represents the separate radiated-energy symbol used in the key.

Official grading criteria

• Understanding nature of the effect and conclusion synodic period is 1 year – 1 pt.
• Calculation sidereal period = 0.5 years – 1 pt.
• Radius of the orbit from the III Kepler's law – 1 pt.
• Energy balances — 2 pt.
• 5.1. Temperature of Sun — 1 pt.
• 5.2. ΔM=−(5/2)lg(0.0464)≈3.3 and from HR diagram spectral type K 3 — 2 pt.

Jury note: evaluate the solution in essence rather than formal presence of sentences/formulae; award points when subsequent steps de facto use the required positions.