1. Having observed the sunrise every day in the same location, the astronomer noticed that the azimuth of the sunrise point changes in the range of 90^o during the year. Please find the latitude of the observation place. The refraction and solar disk size can be neglected.

2. The traveller came to the equator on the vernal equinox day. At the sunset moment he starts to climb on the northern slope of the mountain inclined to the horizon by the angle 10^o. He does it to see the centre of the solar disk on the horizon exactly and continuously. During what time will he succeed in doing this if he can move with the velocity up to 5 m/s? The relief surrounding the mountain and the refraction can be neglected.

3. Astronomers know the Metonic cycle of eclipses alongside with Saros for a long time. The Metonic cycle contains 254 sidereal months being very close to 19 tropical years. Owing to this, the cycle can be used to predict not only the eclipses, but also the lunar occultations of stars. Every Metonic cycle the consequence of occultations is almost the same. In 19 years after the occultation of the star Alcyone (h Tauri, the brightest star of Pleiades cluster) the similar one can occur. How many Metonic cycles in a consequence can contain the similar Alcyone occultation? The durations of sidereal and draconic months are equal to 27.321662 and 27.212221 days, respectively. The ecliptic latitude of Alcyone is equal to +4^o03'.

4. Large telescope of future generation is used for the visual observations of artificial minor planet - ideal mirror metal ball with diameter equal to the telescope lens diameter. The ball moves around the Sun by the circular orbit with radius equal to 3 a.u. Please find the minimal value of lens diameter. The sky background and influence of atmosphere can be neglected.

5. Two stars have the same physical parameters. They are observed close to each other in the sky, but their distances are different. Both stars and the observer are situated inside the uniform cloud of interstellar dust. The photometric measurements of these stars in B band gave the results 11^m and 17^m, in V band the results were 10^m and 15^m. What is the ratio of distances to these stars? Assume that the extinction property of interstellar dust is proportional to the wavelength in the degree of (-1.3).

6. The Cepheid variable star with period equal to 50 days is visible by the naked eye. Having observed it with the telescope, astronomers detected two-layers reflecting nebula around this star. The nebula scatters the Cepheid emission. Layers� angular radii are equal to 10'' and 21''. The brightness of both layers changes with the same period equal to 50 days, reaching the maximum in 30 and 18 days after the Cepheid maximum for inner and outer layer, respectively. Please find the distance to the Cepheid.

7. The Planck unit system is often used in astrophysics and cosmology. In this system the gravitation constant G, light velocity c and Planck constant h are equal to unity and have no dimension. Using this system, we can express any physical value in units of any other physical value. Please, use the Planck unit system to express the astronomical unit (the distance between Earth and Sun) in seconds, kilograms and Joules. Do these numbers have the physical sense?

The problems about comets evolution in the Solar System

8. Approximately once in 5 years people on the Earth can observe bright comets, whose nuclei have the radius about one kilometer. The orbits of such comets are close to parabolic. Assuming that these nuclei are uniformly filling the volume of spherical Oort cloud with radius 10000 a.u., estimate the number of large comet nuclei and mass of Oort cloud.

9. The comet with parabolic orbit comes to perihelion, approaching close to Jupiter. After the gravitational interaction with giant planet the comet comes to the new heliocentric orbit with the period equal to the half of Jupiter�s orbital period. Please find the angle of comet�s turn in the gravitational field of Jupiter. Consider the orbit of Jupiter as circular, the orbit planes of comet and Jupiter are the same.

10. At May, 16^th, 2006, Earth passed by the fragments of comet Schwassmann-Wachmann 3. Being observed from the Earth, the cluster of fragments had the string-type form with angular length equal to 40^o, the distances to the edges of this string were equal to 0.055 and 0.105 a.u. Assuming that the comet Schwassmann-Wachmann 3 had broken up by the momentary isotropic explosion near the perihelion in October, 1995, estimate the fragments scattering velocity during the explosion. In what time the meteor shower created by this comet will be annually observed from the Earth? The comet perihelion distance is equal to 0.939 a.u., the orbit eccentricity is equal to 0.693.

Problems written by Eugene N. Fadeev, Nicolay I. Perov, Oleg S. Ugolnikov