# Astrophysical warm-up

## Contents

- 1 Problem 1: classical Doppler effect
- 2 Problem 2: relativistic Doppler effect
- 3 Problem 3: Doppler effects for sound and light
- 4 Problem 4: Doppler effect and time dilation
- 5 Problem 5: Solar luminosity
- 6 Problem 6: solar mass loss
- 7 Problem 7: Cepheids in LMC
- 8 Problem 8: frequency of supernova's registration
- 9 Problem 9: star formation rate
- 10 Problem 10: H resonant lines
- 11 Problem 11: the $\alpha$-Lyman line
- 12 Problem 12: tunneling through the Coulomb barrier
- 13 Problem 13: elderly Sun
- 14 Problem 14: accretion and the Eddington limit
- 15 Problem 15: masses of the stars
- 16 Problem 16: double stars

### Problem 1: classical Doppler effect

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

\[{\nu }'=\nu \left( 1+\frac{V}{c}\cos \vartheta \right)\]

### Problem 2: relativistic Doppler effect

Obtain the relativistic formula for the Doppler effect.

\[{\nu }'=\nu \frac{\sqrt{1-\frac{{{V}^{2}}}{{{c}^{2}}}}}{\left( 1-\frac{V}{c}\cos \vartheta \right)}\]

### 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.

$L_ \odot = 4\pi R^2 \cdot 1400 \mbox{J/s}^2c=3.94 \cdot 10^{24} \mbox{J/c}$

### 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.

$\Delta m = {{L_ \odot t} \over {c^2 }} = 1.4 \cdot 10^{27} \mbox{kg}=0.07\%M_{\odot}$.

### 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?

All stars of the Large Magellanic Cloud cloud are located at approximately equal distances from us and thus, observing the corellation between apparent luminosity of cepheids and their periods, astronomers figured out the connection between the period and apparent luminosity of this class of stars. In constrast, the determination of this relation while observing stars in our Galaxy would be more complicated, since it involves the measurement of the distance to every star to calculate its real luminosity. Moreover, additional complication arises from the difference of light absorbtion in different directions within the Milky Way (cepheids are young stars and thus could be found near the equator of the Galaxy), while the absorbtion of light from stars in the LMC in the interstellar medium of our Galaxy is approximately the same.

All stars of the LMC are located at approximately equal distances from us and thus, observing the corellation between apparent luminosity of cepheids and their periods, astronomers figured out the connection between the period and apparent luminosity of this class of stars. In constrast, the determination of this relation while observing stars in our Galaxy would be more complicated, since it involves the measurement of the distance to every star to calculate its real luminosity. Moreover, additional complication arises from the difference of light absorbtion in different directions within the Milky Way (cepheids are young stars and thus could be found near the equator of the Galaxy), while the absorbtion of light from stars in the LMC in the interstellar medium of our Galaxy is approximately the same.

### 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 $10 Mpc^3$.

Let's determine the distance $r$ up to which supernova outbursts would be registered. Absolute $M$ and apparent $m$ magnitudes are related through distance $r$ (in pc) as: $$M=m+5-5\lg r.$$ Thus, $r=100\mbox{Mpc}$. The volume of this region is $4\pi r^3 / 3 = 4 \cdot 10^6 \mbox{Mpc}^3 $ and it contains $4 \cdot 10^5$ galaxies. Since one outburst takes place once in 100 years in one galaxy, the number of outbursts during a year is 4000 or approximately 11 new outbursts every night. However, absorbtion of light in interstellar medium can significantly reduce these numbers.

### Problem 9: star formation rate

Calculate the average rate of star formation in our Galaxy.

The mass of the Milky Way is around $10^{11}M_{\odot}$. Since the age of our Galaxy is approximately $10^{10}$ years, stars formation proceeded with the average rate $10M_{\odot}$ per year. This estimate is an upper limit for the present time, since the rate of stars formation would have been much higher at earlier epochs.

### Problem 10: H resonant lines

Determine the wavelengths of hydrogen resonant lines.

Spectral lines of hydrogen are well described by Rydberg formula $$\frac{1}{\lambda} = R \left(\frac{1}{{n'}^2} - \frac{1}{n^2}\right),$$
where $n'$ and $n$ are principal quantum numbers and $n'=n+1, n+2\ldots$. Spectral lines which originate from transitions to main energy level are called resonant.

For example, let's consider the case $n'=1, n=2$:
\[\lambda =\frac{8\pi \hbar }{3R}\simeq 1216 \text{Å}\,;\quad R\simeq 13.6\,eV\]
This line is resonant and belongs to the Lyman series.

### 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?

Almost all hydrogen atoms in interstellar space are in ground state, therefore the Lyman line is the main absorption line. The lines of Balmer series appear in stellar atmosphere, where substantial fraction of hydrogen atoms is in the first excited state.

### 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.

The tunneling rate is $$ P=\exp \left[ -\frac{2}{\hbar }\int_{0}^{{{r}_{0}}}{dr}\sqrt{2m(V(r)-E)} \right], $$ where $m={{m}_{p}}/2$ is the reduced mass of two protons and ${{r}_{0}}$ is the turning point of a classical particle with energy $E$, so that $V(r)=E{{r}_{0}}/r$. Lower limit is set to $0$, since it's real value is much smaller than ${{r}_{0}}$. Calculating the integral we obtain $$ P=\exp \left( -\frac{\pi }{\hbar }{{r}_{0}}\sqrt{2mE} \right)=\exp \left( -\frac{\pi }{\hbar }{{e}^{2}}\sqrt{\frac{2m}{E}} \right)=\exp \left( -\frac{B}{\sqrt{E}} \right), $$ where $${{B}^{2}}=2{{\pi }^{2}}{{c}^{2}}{{\alpha }^{2}}m\approx 450 \;\mbox{KeV}.$$

### 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?

Negative heat capacity of the Sun will ensure the temperature necessary for reaction of helium generation at appropriate time.

### 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).

When a star of mass $m$ falls from infinity on the massive object (e.g., star) of mass $M$ and radius $R$, it's kinetic energy $K$ transforms into radiation with some rate $\varepsilon$. The contribution of this process to luminosity is $$ L = \varepsilon {dK\over dt} = \varepsilon {GM\over R} {dm\over dt}. $$ Generated photons would interact with falling particles. Effective accretion would terminate when pressure of radiation exceeds the gravitational forces. Let $n(r)$ be the density of photons crossing the sphere of radius $r$ with center at accretion object. This quantity can be expressed through luminosity as $$ n(r) = \left( {L/4\pi r^2 } \right)/\hbar \omega, $$ where $\omega$ is some mean frequency. The rate of collisions between photons and electrons in ionized gas is $n(r)\sigma _T $, where $\sigma _T $ is the Thomson cross section. In every collision the proton bounded with electron would receive momentum $\hbar \omega /c$. Therefore radiation pressure on proton would be $$ F_e \simeq n\sigma _T {\hbar \omega \over c} = \left( {{L \over {4\pi r^2 }}} \right)\left( {{1 \over {\hbar \omega }}} \right)\sigma _T \left( {{{\hbar \omega } \over c}} \right) = {{L\sigma _T } \over {4\pi cr^2 }}. $$ This force would compensate the gravitational force $F_g = GMm_p /r^2 $, that pulls the proton to the object of accretion, after reaching the luminosity $L_{cr} $, $$ L_{cr} = {{4\pi GMm_p c} \over {\sigma _T }}. $$

### 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?

Stars could be classified by their physical state as normal, which consist of non--degenerate matter (ideal gas) and inside which the thermonuclear fission is going on, and degenerate (white dwarfs, neutron stars), in which equilibrium is maintained due to pressure of degenerate fermions (specifically, electrons in white dwarfs and neutrons in neutron stars). Black holes form the distinctive class, since they are not stars in the usual sense. White dwarfs, neutron stars and black holes are all designated by the common title *compact remnants* since they are products of the evolution of regular stars. The total number of stars and their remnants in our Galaxy is estimated to be $\sim 2\cdot 10^{11}$.

Let's start with regular stars. Regular stars are not different from galaxies in their diversity, but the main characteristics of star, which determine its structure and evolution, are its initial mass $M$ and chemical composition (the ratio of helium and heavier elements to hydrogen). Masses of stars vary in the limits $\sim 0.08\; M_\odot $ to $\sim 100 \;M_\odot$. The lower limit is caused by the forbiddance of thermonuclear fission at lower masses, while the upper limit is related to the determining role of the radiation pressure in massive stars. At higher masses the luminosity of a star exceeds the Eddington limit $L_{Edd}\sim 10^{38}(M/ M_\odot)~\mbox{erg/s}$, so that radiation stripes the outer shell. Consequently, there are no stable stars at higher masses (see previous problem).

### Problem 16: double stars

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

Let's switch to the center of mass frame. The angular momentum of the system, which comprises three scalar quantities, determines the plane of the orbit. Two more quantites determine the direction of any of the axes of ellipsis in the plane of the orbit. Finally, two quantities determine ellipsis itself. So, the overall orbit of a component of double star is set by seven quantities.