EST-CZE possible problems

- 1.
Calculate what direction on a compass should one choose if she/he would like to travel from Tartu (\(58^{\circ}23′\) N \(26^{\circ}43′\)E ) to Bhubaneswar (\(20.27^{\circ}\) N \(85.84^{\circ}\) E)

- 2.
The distance between Tartu and Prague is around 1200 km. Assuming a ideally spherical model of Earth how tall should a tower be on top of which one could see the famous Prague landmark the Charles bridge from Tartu and also assuming a completely transparent atmosphere how big would the telescope have to be in order to make it visible in optical light?

- 3.
Astronomers observe a star in the Milky way which is located 150 pc from Earth. Spectroscopic measurements in the oxygen rest wavelenght (500.7 nm) reveal that the lines are redshifted to 502.2 nm. The observed star is known to have a previously observed proper motion of \(\mu\) of 1.5 arc seconds per year. Calculate the 3 dimensional velocity and parallax of the star.

- 4.
A red giant star has a radius 500 times greater than the Sun and its tempeature is 0.5 of the Sun. The Red giant is 500 000 further from the Earth as the Sun. Calculate the Red Giants apparent magnitude.

- 5.
The Sun is orbiting the galactic centre with a velocity of 230 km/s and has a orbital period aproximately \(2.3\times 10^{8}\) years. Estimate Milky way mass based on this information.

- 6.
Assume that agalaxy residues in a dark matter halo with a density profile of:

\begin{equation} \label{eq:NFW}\rho(r)=\frac{\rho_{\mathrm{0}}}{\frac{r}{R_{S}}(1+\frac{r}{R_{S}})},\\ \end{equation}where \(\rho_{0}\) is a constant that responds to dark matter density in galactic centre and \(R_{S}\) is the scale radius of the dark matter, which also is a measured constant. Based on this express the velocity function v(r) of the stars in the galaxy (stellar mass can be ignored and no radial velocities assumed).