Some planet orbiting a distant star in a circular orbit has a satellite, and its orbit is also circular. During eclipses of a star

Some planet orbiting a distant star in a circular orbit has a satellite, and its orbit is also circular. During eclipses of a star by a satellite, when viewed from a planet, the apparent sizes of the star and the satellite coincide (like the Moon and the Sun on Earth), and when the satellite enters the planet’s shadow, its angular dimensions coincide with the angular dimensions of the shadow. Find the ratio of the geometric dimensions of the planet and the satellite, considering them significantly less than their mutual distance, and the distance itself – significantly less than the distance to the star

The satellite’s shadow is a cone with an opening angle a equal to the angular diameter of the star and the satellite when viewed from the center of the planet (the dimensions of the planet are considered to be significantly smaller than the distance to the satellite). Denoting the radius of the satellite through r, and its distance from the planet through L, we obtain the expression connecting them: r = L a / 2. The distance to the star is much greater than the radius of the satellite’s orbit, the shadow of the planet itself is a cone with the same opening angle. From the condition of equality of the sizes of the shadow of the planet and the satellite, we obtain R – r = L a / 2. Hence R = 2r, the planet is twice as large as its satellite in radius.