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## astrophysics

### 1. Series 24. Year - 1. floating sphere

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.

Z ruských bylin vyčetl Marek

### 1. Series 22. Year - P. Copernicus versus Ptolemaios

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.

Významný důkaz chtěl připomenout Pavel Brom.

### 5. Series 21. Year - 4. Sun can

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?

### 5. Series 21. Year - S. sequence, hot orifice and white dwarf

• 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.

Zadali autoři seriálu Marek Pechal a Lukáš Stříteský.

### 3. Series 21. Year - 2. lift to the skies

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?

### 3. Series 21. Year - E. experimenting with Sun

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.

Experimentální úlohu navrhl Pavel Brom.

### 1. Series 21. Year - 3. weighting the Sun

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

K zahřátí mozků do nového ročníku FYKOSu zadal Pavel Brom.

### 6. Series 20. Year - 4. binary star

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.

Nepoužitá úloha z archivu.

### 3. Series 20. Year - 3. the distance of binary star

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.

Při astronomickém pozorování vymyslel Pavel Brom.

### 2. Series 20. Year - 4. how far is the Sun?

Let's return back to 18th century, when the value of Newton's gravitation constant was not known yet. And the distance between Sun and the Earth was unknown as well. During this times Edmond Halley (astronomer who discovered that the comet in the sky in 1682 is the same as the one in 1456, 1531 and 1607) suggested to find distance Earth-Sun by measuring time of transit of Venus over the Sun circle. Unfortunately the Venus cross the Sun irregularly, in pair each eight years and then there is century-long break. During Halley life there was no crossing.

However, the idea was not forgotten and with the next crossing was successfully resuscitated and in 1761 a next scientific experiment was prepared. Scientists went to many parts of world, including Siberia, China, South Africa and Indonesia. It was the first international scientific experiment.

After all people have returned, it was concluded that no conclusion can be made. Ironically it was due to too many contradicting observation. Successful was captain James Cook in 1769 on one summit in Tahiti. After his return the astronomers were able to calculate the average distance between the Earth and the Sun as approximately 150 millions kilometers.

It is up to you to calculate distance Earth-Sun using only data known at the time: Earth diameter and orbit time of Venus and Earth.

Úloha napadla Honzu Prachaře při čtení knížky Stručná historie téměř všeho.