IOAA 2012, Отборен тур · английски — Задача 1

Part I

Imagine a binary stellar system, consisting of two main sequence stars, with one star at least 30 times more massive than the other. Now imagine a planet located at one of the Lagrange points of the system, L4 or L5. Imagine the orbital plane of the system is aligned to our line of sight, i. e., it is perpendicular to the plane of the sky (“edge-on”), so you can detect it using its light curve.

1) Make a schematic drawing of this system.

2) Create a hypothetical light curve for this system and use it to understand the mechanics of the system. Do not take into account any instrumental effect.

3) Assuming the light curve from item (2) is accurate, calculate the values of the relevant physical parameters for each of the three bodies: mass, radius, orbital period, mean orbital radius, luminosity, effective temperature.

Part II

Now let’s examine the planet from Part I, called Troll by its inhabitants, more closely. Troll has a rotation axis inclined by 30° with respect to the normal of the orbital plane, rotation period of five terrestrial days and mass of 0.5 terrestrial masses.

4) For an observer located at the Trollian equator, draw a sketch of the sky in the moment of the day when the incident light on the surface of the planet reaches its maximum intensity.

5) At which latitude during the Trollian year, is there at least one Trollian sidereal day that there is no night?

Part III

Suppose now that Troll has a moon that can’t be detected from Earth and whose radius is ¼ of Troll’s radius. The moon’s orbital plane is inclined 15° with respect of the orbital plane of the planet. The orbital period of the moon is about 30 Trollian sidereal days.

6) For an observer at a Trollian latitude φ, on the day that the light incidence is largest, calculate the time when the moon’s “Trollian ecliptic latitude” is largest.

7) Under the same conditions and in the same day of item (6), calculate the duration of the moon setting, that is, the time since the moon touches the Troll horizon until it disappears completely below it.

8) How many total (T) and how many partial (P) eclipses can be seen during a Trollian year?

Parte IV

Assume that the binary system is at 277 pc from Earth, located at equatorial coordinates $\alpha = 3^{h}$ and $\delta = -15^{\circ}$, as seen from Earth.

9) Is it possible for a Trollian observer to see the Sun being occulted by any other body of its system? Say YES if it possible and NO otherwise.

10) What is the “Trollian ecliptic latitude” of the Sun, to a Trollian observer?

11) In which conditions can the Trollian observer see a transit of the Jupiter in front of the Sun? Make a drawing to explain your answer.

12) Estimate the apparent magnitudes of the following stars, as seen from Troll sky: Sun, Vega, Sirius, Rigel, Aldebaran.