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It is known that the Moon when it is full has the apparent magnitude of approximately -12 mag and the Sun during the day has the apparent magnitude of -27 mag. Try to figure out what is the apparent magnitude of the Moon directly before a solar eclipse, if you know that the albedo of the Earth is approximately 0.36 and the albedo of the Moon 0.12. Presume that light after reflection disperes the same way on the surface of both the Moon and Earth.

How fast does the boundary between regions with and without sunlight move on the surface of the Moon? Is it possible to run away from dark when you are at the equator?

A favorite topic in astrology is the relation between cosmic catastrophes and planetary conjunctions. Imagine that, precisely at noon, all the large planets and the Sun lied on a ray which originated at the center of the Earth. What was the percantege change of your weight compared to a situation with only the Earth present in the solar system?

• Assume the validity of the Newton model derived in the text. For $E=0$ solve the case that the Universe is expanding and the energy of vacuum is constant. What is the future of the Universe according to this model?

• Since the Universe is full of stars, the light from each of them should reach the Earth sooner or later. However as you know the nights are pretty dark. Explain this paradox and support your answer with quantitative arguments.

• In the text a simple derivation of the existence of dark matter in galaxy clusters was presented. Can you figure out another way to prove the existence of dark matter in galaxy clusters? Suggestion is enough, you do not have to work out any calculations.

• Active galaxies appear as a point sources, same as stars. Try to find some ways to distinguish between star and AGN. The more ways, the better.

Radio observations of quasar 3C&nbsp273 showed that there's a blob of radio emission moving away from the nucleus with angular velocity μ = 0.0008 year^{ − 1}. Assuming the radio knot is moving perpendicular to the line of sight, and using the distance $d=440/hMpc$, where $h$ is Hubble constant, compute apparent velocity $v_{zd}$.

• Derive for which value of angle $φ$ is $β_{T}$ maximal?
• Let's assume that the supermasive black hole in the galactic centre is 30 % efficient. How much energy will the hole produce when swalloving an Earth size

• Spiral galaxies can be described using logarithmic spiral$r(φ)=r(0)\exp(φ\tanΦ)$, where $r$ and $φ$ are polar coordinates and Φ is the opening angle, which is an angle between the normal to the vector and the spiral tangent (opening angle increases in the negative direction, in general we use radians and the angle can exceed 2π). Assume Φ = 10$°$. Derive the relation for the ratio of the distances for two neighborings coils of one spiral arm from the galactic centre. How would the ratio change if there were four arms, egually distributed. Write the distance for the neighboring arms for r(0)=8kpc.

• Consider infinite universe with uniform stellar density and no extinction. Write the relation for the integral and differential star counts. What will happen if the apparent magnitude increases a lot?

• *Bonus:** What is the probability that two stars in the galaxy will be projected into one spot? Consider lonely stars, not binaries.

• Some stars are considered circumpolar. Does it mean that they can be seen the whole year? What stars are visible throughout the whole year in Czech Republic? What coordinate describes circumpolar stars? What is the situation in Czech Republic, at the pole and at the equator? We recommend that you download the program <a href=http://www.stellarium.org/>Stellarium</a> (GNU GPL license) where you can enter your location and look at these different cases.

• Compare the absolute magnitudes of Alpha Lyrae (7.79 pc far, apparent magnitude 0.01 mag) and Betelgeuze (Alpha Ori, approximately 200 pc far, apparent magnitude 0.42 mag). How would we see these stars if they exchanged their distances from the Earth? Discuss visibilities.

• Transformations and some more transformations. Find the transformation between galactic and equatorial coordinates or the second kind. Do not worry if the resulting equations do not look exactly like those found in literature.

• Janap likes to get lost once in a while. It is not always desired but it happens anyway. This time however she brought a theodolite – a magic box that can measure how high above a horizon a star is. She found out that the star Arcturus was located at 23.20 gon at 18:46:30 and the star Capella at 13.60 gon at 19:18:30. The scale of theodolite was in grad (gon), where 100 gon = 90°. What was her location?

How would the power of sunlight hitting the Earth in aphelion change if we were to accelerate the Earth in the direction of motion in such a way to extend a year by a week? Estimate the temperature of the Earth in aphelion and perihelion if its heat capacity is almost zero. For simplicity assume that the original trajectory of the Earth was circular (not the case after the velocity boost).