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Sometimes the Moon can look like a little ship. What are the conditions (time and place) for this to happen? If you get stuck with this problem you can consult a software like Celestia.

A solid sphere of density ρ is placed between two layers of immiscible liquids (see picture). Densities of the upper and lower liquids are ρ_{1} and ρ_{2} respectively. Assume that ρ_{1} ρ < ρ_{2}. Calculate the fraction of the sphere surrounded by the upper resp. lower liquid.

The year 2009 is international year of astronomy and remembers 400 years from discovering telescope. Lets go back 400 years to the times, when the telescope was already invented, but the classical physics was just in its beginnings. At this time there were two explanation of world: Copernicus's heliocentric and Ptolemaios's geocentric. Design an experiment, which will be able to decide which of above models is correct. Document, what results you can expect and how you can interpret it. You do not have to make the observation itself. Explain, why in geocentric model the Sun and Earth are connected by a line.

Rama travels between the stars in such way, that one half of time is constantly accelerating and second half of time is slowing down. Currently the Rama is on parabolic trajectory around the Sun with peak on Earth orbit. It gets energy from Sun light. Its surface absorbs 80 % of incident energy. Will it get enough energy to get to Sirius, which is in distance of 12 light years in less than 24 years?

• Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

• The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

• Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

Find the properties of material, which you need to make a rope for a lift from geostationary orbit to the Earth surface. Is such material available on Earth?

Make a measurement of the height of Sun above horizon at noon time and time from sunrise (middle of Sun disk) till its sunset. Then you can calculated theoretical duration of day and compare with reality and comment on differences.

Suggest several methods for estimating the mass of the Sun. Explain in detail each of them and estimate mass of the Sun.

The magnitude of binary star is changing with 4-day period with this sequence:

side minimum: m = 3.5

maximum: m = 3.3

main minimum: m = 4.2

maximum: m = 3.3.

The larger part of binary star has higher temperature then its escort. Assuming, that the Earth is in the orbital plane of binary star, calculate magnitudes of each component and ratio of its linear dimensions. Relation between magnitude $m$ and irradiation $E$ is given by

$m=-2,5\log(E⁄E_{0})$,

where $E_{0}$ is some defined value.

We have calculated spectral class of two stars forming binary star from reduced star spectra (from present spectral lines which does not change its position in time). From spectral class we have estimated its masses as 2 and 3 times the mass of the Sun. From the observation with telescope of focal length 3 m we know that the stars really orbit in constant angular distance of 5 angular minutes. One orbit takes 50 years.

Are you able to calculate distance of this binary star from the Sun? If yes, state which information you need and the result round accordingly. Comment on the precision of the result and how the uncertainty of input information (mainly the masses) impacts on accuracy.