IOAA Jr. 2022, Теория, тест с избираем отговор — Задача 9. The Moon, seen on the way to the Moon

Archived Grid 9 variant: equal brightness

The Moon, whose linear diameter is $d=3436\ \mathrm{km}$, at Earth in the distance $r_0=348000\ \mathrm{km}$, has the apparent magnitude $m_0=-12.7$. The apparent magnitude of the Sun, seen from Earth, is $m_s=-26.84$.

The distance, $r$, from the Moon, where a cosmonaut is, on the way of his spaceship to the Moon, the brightness of the Moon should be the same as the brightness of the Sun seen from Earth, is:

a) $r\approx675.25\ \mathrm{km}$; b) $r\approx568.88\ \mathrm{km}$; c) $r\approx763.23\ \mathrm{km}$; d) $r\approx837.44\ \mathrm{km}$.

Solution supplied for the archived Grid 9 variant

The approach of the cosmonaut to the Moon must be such that it determines a variation of the apparent magnitude of the Moon:

so that the apparent magnitude of the Moon, viewed from this new distance, is:

Knowing that the luminous intensity, $I$, of a star, in particular of the Moon, is:

that is, the energy flow emitted by the star, the Moon, in the unit of solid angle, then:

According to Pogson's formula, if $E_0$ and respectively $E$ are the brightnesses of the Moon, corresponding to the initial $(r_0)$ and respectively the final distances $(r)$, shown in the drawing in figure 1, it results:

where $I$ is the luminous intensity of the Moon;

this is the distance from which the Moon should be viewed, so that its brightness is the same as the brightness of the Sun, seen from Earth.