Russian Open School Astronomical Olympiad by Correspondence 2008 // List of Problems

Observer is situated in the definite point on the Earth’s surface. One definite moment he noticed that each point of ecliptic had met the mysterious property: the angular distance between this point and North Celestial Pole had been equal to the zenith distance of the same ecliptic point. Disregarding the refraction, please find the latitude of the observation point
The artificial satellite of the Earth has the mass equal to 100 kg and moves along the elliptical orbit with perigee altitude equal to 200 km and apogee altitude equal to 10000 km. Being close to perigee, the satellite is decelerated by the Earth’s atmosphere. Please estimate the time, during which the satellite’s orbit will become circular. The decelerating force of the atmosphere can be considered to be constant with the value 0.01 Newton, the path length of the satellite through the atmosphere each revolution is equal to the radius of the Earth.
The magnitude of total umbral lunar eclipse is equal to 1.865. Please find the duration of totality. The expansion of the umbra caused by atmosphere can be disregarded.
The grazing occultation of the star by the Moon is observed in the zenith at the Earth’s equator. The Moon is exactly in the first quarter. Please find the maximum possible angular distance between the star being occulted and the closest “horn” of the Moon (the crossing point of limb and terminator) in the grazing moment. The orbit of the Moon can be considered to be circular.
The minor planet moves around the Sun in the ecliptic plane, never coming inside the orbit of the Earth. The conditions of its observations exactly repeat in 2 years, and its visible magnitude changes on 8^m with the same period. Please find the minimum possible value of the eccentricity of the asteroid’s orbit. The asteroid is the smooth spherical uniform ball with constant surface albedo. Orbit of the Earth can be considered to be circular.
The observer on the Earth had measured the angular distance between the stars X and Y, both situated on the ecliptic, and obtained 30° with exactness 0.1″. The star X is situated westwards from the star Y, so it has less ecliptical longitude. During the observation moment both stars were situated westwards from the Sun, the ecliptical longitude difference of the Sun and star X was equal to 100°. Please find the angular distance between stars X and Y after three months. How will this distance change if we observe from the Sun? The parallax values of the stars X and Y are equal to 0.5″ and 0.2″, respectively. Please disregard the eccentricity of the Earth’s orbit, self motions of the stars and all atmospheric effects.
In March 1997 we saw the bright comet Hale-Bopp with magnitude –1.5^m. Being observed from Earth, the brightest inner part of the comet’s tail had the length about 10° and width about 1°. Imagine that the same time the spaceship with astronauts arrived to the comet and landed on its core at the side opposite relatively the Sun. Will the astronauts see the stars in the sky when they come to the surface of the core?
The star has the surface temperature 15000 K and the radius equal to 10 radii of the Sun. During the last 100 years this star produces the uniform stellar wind blowing with the velocity 20 km/s. This substance created the shell of gas and dust around the star with optical depth equal to 0.2. Please calculate the radii of inner and outer visible edges of the shell, find the dependence of the dust density in the shell on the distance from the star. Please find the mass of the shell and the mass loss rate of the star. The dust particles have the radius equal to 1 µm, density equal to 3 g/cm³ and fusion temperature equal to 1500 K. Consider that the mass of the gas is 200 times larger than the mass of the dust, but light absorption is caused only by dust.
The gamma-ray bursts sometimes happen in the distant galaxies. These are the short (about several seconds) bursts of gamma-ray emission with average energy of the photon equal to 1 MeV. To be registered on the Earth, the flux of such photons must be not less than 50 phot/(cm²·s). The luminosity of the burst is equal to 10⁴⁹ ergs per second, this energy is released inside two opposite cones with angle at the top equal to 10°. The gamma-ray bursts are registered on the Earth once a week. What is the frequency of gamma-ray bursts in one definite galaxy? How much times more or less bursts we would see, if the cones of their emission were two times narrower?
We know that the temperature of Cosmic Microwave Background in the direction with Galactic coordinates l = 264° and b = 48° is maximal, being by ΔT = 3.35 mK more than average value. Please find the velocity of our Galaxy as a whole relatively the Cosmic Microwave Background.