IAO 2013, Теория, Група β — Задача 3. Observations from Gliese 581 g

3.1. Apparent magnitude

The magnitude of the Sun, visible from any remote location is

where $m_0$ is the apparent magnitude of the Sun visible from Earth, and p is the annual parallax of the location.

The magnitude of the Sun as seen from the system of Gliese 581 is

(And it is not the only way of solution. The value can be found in many other ways, e.g., by formula $m_{\odot}=M-5^{\mathrm m}+5^{\mathrm m}\lg p$).

Source note: the double minus in the intermediate expression is written here as $-0^{\mathrm m}.17-(-3^{\mathrm m}.98)$ with parentheses for readability; the alternative formula above is retained as printed, including its plus sign before $5\lg p$.

3.2. Approximate constellation

While observing from the planet Gliese 581 g, our Sun will be at a point opposite to where the system of Gliese 581 is seen from Earth. Because Gliese 581 is located in the zodiacal constellation of Libra, the sun will be approximately in the opposite zodiacal constellation, that is, in the constellation of Aries. (That solution is correct enough.)

Note: Using a sky-chart one can solve the problem more accurately. The point opposite to the location of the system Gliese 581 at the Earth’s sky, has the coordinates $03^{\mathrm h}\ 19^{\mathrm m}\ 27^{\mathrm s}$ and $07^\circ43^{\prime}20^{\prime\prime}$. This point is located in the constellation of Taurus (Tau), very close to the borders of the constellations of Cetus (Cet) and Aries (Ari). As the students were not provided by a sky-chart, and question was “what is the approximate constellation”, the answer “approximately in the constellation, opposite to the constellation of Libra, that is in the constellation of Aries” is correct.

3.3. Angular diameter

To keep the heat balance a Gliese 581 g should receive the same amount of energy per unit area as is received at Earth.

(As in the previous solution G indices refer to the system of Gliese 581, and indices $\odot$ and E — to the Sun and the Earth).

From the given tables one may find that the spectral class of the star Gliese 581 is M3V. According to the Hertzsprung–Russell diagram the temperature of the star, corresponding to this spectral class, is about $T_G=3300\ \mathrm K$. It is $T_{\odot}/T_G=5780/3300\approx1.75$ times less than the temperature of the Sun.

According to the Stefan–Boltzmann law the power emitted from the star is $P=\sigma\cdot ST^4$, and in just the same way we may discourse about the apparent angular area from which the radiation is coming.

Thus, since the apparent angular area of a star is proportional to the square of its apparent angular diameter, the following equations should be correct:

More than three times larger than the Sun!

Official basic criteria

3.1. Totally 3.5 pt, including:

• Formulae for magnitude of the Sun (any way solution) — 2 pt.
• Correct calculations — 1.5 pt.

3.2. Totally 2 pt, including:

• Understanding that Gliese 581 is located in the zodiacal constellation of Libra — 0.5 pt.
• Sun will be at opposite point — 1 pt.
• Answer: constellation of Aries — 0.5 pt.

3.3. Totally 2.5 pt, including:

• Heat balance, the same amount of energy per unit area — 0.5 pt.
• Finding spectral class and temperature (using HR diagram) — 0.5 pt.
• Using Stefan–Boltzmann law — 0.5 pt.
• Final formula — 0.5 pt.
• Final calculations and result — 0.5 pt.