Difference between revisions of "Astrophysical warm-up"

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

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

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

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

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.

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.

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.

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.

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.

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 $${{B}^{2}}=2{{\pi }^{2}}{{c}^{2}}{{\alpha }^{2}}m\approx 450 \;\mbox{KeV}.$$

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

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} $, $$ L_{cr} = {{4\pi GMm_p c} \over {\sigma _T }}. $$

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).

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.