# Astrophysical warm-up

### Problem 1: classical Doppler effect

Obtain the non-relativistic approximation (to the first order in $V/c$) for the Doppler effect.

### Problem 2: relativistic Doppler effect

Obtain the relativistic formula for the Doppler effect.

### Problem 3: Doppler effects for sound and light

What are the differences between the Doppler effect for light and the "analogous" effect for sound?

### Problem 4: Doppler effect and time dilation

Using the Doppler effect, how could we demonstrate that time is running differently for observers which move relative to each other?

### Problem 5: Solar luminosity

Every second about $1400 J$ of solar energy falls onto one square meter of the Earth. Estimate the absolute luminosity of the Sun.

### Problem 6: solar mass loss

Assuming that the constant luminosity stage for the Sun is of order of $10^{10}$ years, find the portion of solar mass lost due to radiation.

### Problem 7: Cepheids in LMC

Why was the connection between the luminosity of variable stars (Cepheids) and the period of their brightness' variation discovered from observation of stars in the Large Magellanic Cloud (LMC) rather than in our Galaxy?

### Problem 8: frequency of supernova's registration

A supernova in its maximum brightness reaches the absolute stellar magnitude of $M=-21$. How often will the supernova outbursts be registered if observation is carried out on the whole sky up to the limiting magnitude $m=14$? Assume that in a typical galaxy a supernova bursts on average once per 100 years, and galaxies are distributed with spatial density of one galaxy per $10 Mpc^3$.

### Problem 9: star formation rate

Calculate the average rate of star formation in our Galaxy.

### Problem 10: H resonant lines

Determine the wavelengths of hydrogen resonant lines.

### Problem 11: the $\alpha$-Lyman line

The main method for investigation of interstellar neutral hydrogen are observations in ultraviolet band. The strongest interstellar absoption line is $\alpha$--Lyman hydrogen line ($\lambda=121.6\mbox{nm}$). This line corresponds to transition of electron from state with quantum number $n=1$ to the state with $n=2$. At the same time, the Balmer series characterized by electron transition from excited $n=2$ state are not observed. Why does this happen?

### Problem 12: tunneling through the Coulomb barrier

Hydrogen burning (hydrogen to helium transformation) in stars is realized in the so-called $p$--$p$ cycle, which starts from the reaction of deuterium formation $$p+p\to d+e^+ +\nu.$$ To support such a reaction the colliding protons have to overcome Coulomb barrier (in order to enter the region where nuclear forces act: $r_{nf}\approx10^{-13}\mbox{ cm}$.) It requires energy as high as $$E_{c}=e^2/r_{nf}\approx1.2\mbox{ MeV.}$$ Typical solar temperature is only $T_\odot=10^7 \mbox{ K}\approx0.9\mbox{ keV}$. The Coulomb barrier can be overcome due to the quantum tunneling effect (classical probability to overcome the barrier for the protons in the tail of Maxwell distribution is too low). Estimate the probability of tunneling through the Coulomb barrier for protons on the Sun.

### Problem 13: elderly Sun

At present hydrogen in the Sun burns (transforms into helium) at temperature $1.5\cdot10^7\ K$, but much higher temperature will be required to synthesize carbon from helium (when hydrogen is exhausted) due to higher Coulomb barrier. What physical mechanism could provide the increase of the Sun's temperature at the later stages of its evolution?

### Problem 14: accretion and the Eddington limit

Accretion is the process of gravitational capture of matter and its subsequent precipitation on a cosmic body, i.e. a star. In such a process the kinetic energy of the falling mass $m$ transforms with some efficiency into radiation energy, which leads to additional contribution to the brightness of the accreting system. Determine the limiting brightness due to accretion (the Eddington limit).

### Problem 15: masses of the stars

Masses of stars vary in the limits $0.08{{M}_{\odot }}<{{M}_{star}}<100{{M}_{\odot }}.$ How could this be explained?

### Problem 16: double stars

How many quantities completely determine the orbit of a component of a double star?