Quantities large and small

Problem 1: parsec

From what distance will the length of one astronomical unit have the visible size of one angular second?

solution

1 pc = $3.18 \cdot 10^{16} $m.

Problem 2: Milky Way as viewed from Andromeda

What is the angular dimension of our Galaxy for an observer in the Andromeda galaxy, if the distance to it is about $700 kpc$? Compare it with the angular size of the Sun viewed from the Earth.

solution

$\alpha _G = arctg\left( {\frac{{30\;\mbox{ kpc}}}{{700\;\mbox{ kpc}}}} \right) = 2^ \circ 28' = 5\alpha _ \odot $

Problem 3: static stars?

A glance on the night sky makes the impression of invariability of the Universe. Why do the stars seem to us practically static?

solution

Average human lifespan is approximately 70 years. It is easy to demonstrate that during this period a star's shift is too small to be visible by the naked eye. Assuming that the average tangential velocity of a star is $100$ km/s and distance is 100 ly (distance to nearest star, $\alpha$ Centauri, is 4 ly), the star would shift by $\alpha = 48''$ in 70 years. Let's recall for comparison that Moon's angular size is $30'$.

Problem 4: distance and time to Andromeda galaxy

A supernova outburst in the Andromeda galaxy has been observed on Earth. Estimate the time since the star's explosion.

solution

$2.28 \cdot 10^6$ years.

Problem 5: receding galaxies

A galaxy at distance $R$ from us at the moment of observation recedes with velocity $V$. At what distance was it situated at the moment of emission of the observed light?

solution

Assuming that the distance to the galaxy is $R_e$ at the moment of emission, the light will reach us in $t=R_e/c$. During this period of time the galaxy will cover the distance $\Delta R = Vt = VR_e /c$. The distance to the galaxy at the moment of observation is $R_e (1 + V/c) = R$ and, thus, $R_e = {R}/(1 + V/c)$.

Problem 6: cosmic history on a logarithmic clock

Suppose that we have concentrated the whole cosmic history (14 billion years) in one day. Display the main events in the history of the Universe using the logarithmic time scale. Start from the Planck's time to avoid singularities.

solution

See figure:

• 

Problem 7: age of the Universe from isotopes' abundances

According to the Big Bang model the initial ratio of the uranium isotopes' abundances was $U^{235}/U^{238}\approx 1.65$, while the presently observed one is $U^{235}/U^{238}\approx 0.0072$. Taking into account that the half--value periods of the isotopes are equal to $t_{1/2}(U^{235})=1.03\cdot 10^9 $ years and $t_{1/2}(U^{238})=6.67\cdot 10^9 $ years, determine the age of the Universe.

solution

As is well known, $N_{} = N_0 e^{ - \lambda t}$, where $\tau = 1/\lambda$ is half--lifetime. Thus, the number $\left[ {U^{235} } \right]$ of $U^{235}$ nuclei and the number $\left[ {U^{238} } \right]$ of $U^{238}$ nuclei are $$ \left[ {U^{235} } \right] = \left[ {U^{235} } \right]_0 e^{ - T/\tau (U^{235} )}, $$ $$ \left[ {U^{238} } \right] = \left[ {U^{238} } \right]_0 e^{ - T/\tau (U^{238} )}, $$ where $T$ is the age of the Universe. Then, $$ {{\left[ {U^{235} } \right]} \over {\left[ {U^{238} } \right]}} = {{\left[ {U^{235} } \right]_0 } \over {\left[ {U^{238} } \right]_0 }}e^{ - T\left( {{1 \over {\tau (U^{235} )}} - {1 \over {\tau (U^{238} )}}} \right)}; $$ $$ \ln {{\left[ {U^{235} } \right]} \over {\left[ {U^{238} } \right]}} = \ln {{\left[ {U^{235} } \right]_0 } \over {\left[ {U^{238} } \right]_0 }} - T\left( {{1 \over {\tau (U^{235} )}} - {1 \over {\tau (U^{238} )}}} \right); $$ $$\displaystyle T = {{\ln {{\left[ {U^{235} } \right]_0 } \over {\left[ {U^{238} } \right]_0 }} - \ln {{\left[ {U^{235} } \right]} \over {\left[ {U^{238} } \right]}}} \over {\left( {{1 \over {\tau (U^{235} )}} - {1 \over {\tau (U^{238} )}}} \right)}}, $$ $$ T = {{\ln \left( {{{\left[ {U^{235} } \right]_0 } \over {\left[ {U^{238} } \right]_0 }}{{\left[ {U^{238} } \right]} \over {\left[ {U^{235} } \right]}}} \right)} \over {\ln 2\left( {{1 \over {t_{1/2} (U^{235} )}} - {1 \over {t_{1/2} (U^{238} )}}} \right)}}, $$ and, finally, $$ T = 9.6 \cdot 10^9 \mbox{years}. $$

Problem 8: mass of Milky Way

Estimate the mass $M_G$ of Milky Way and the number of stars in it, if the Sun is an average star of mass $M_\odot$, situated almost at the edge of our Galaxy and it orbits its center with the period $T_\odot=250$ millions years at the distance $R_G=30$ thousands light years.

solution

Given the period of Sun's revolution around the Galaxy and distance to its center, the mass of matter inside its orbit could be calculated as $$M_G \approx {{4\pi ^2 R_G^3 } \over {GT_ \odot ^2 }} \approx 2\cdot10^{44}\mbox{g} \approx 10^{11} M_ \odot.$$

Problem 9: average matter density in the Universe

Estimate the density of luminous matter in the Universe assuming that the Milky Way containing $\sim10^{11}$ stars of solar type is a typical galaxy, and average intergalactic distance is of order of $L=1 Mpc$.

solution

$\rho = {{NM_ \odot } \over {L^3 }} = {{10^{11} \times 2 \cdot 10^{33} } \over {\left( {3 \cdot 10^{24} } \right)^3 }} = 0.7 \cdot 10^{ - 29}\mbox{g/cm}^3.$

Problem 10: infinite Universe

Assume that the space is infinite and on average uniformly filled with matter. Estimate the distance from our observable part of the Universe to the part of the Universe with identical distribution of galaxies and the same Earth.

solution

Problem 11: two large numbers

Show that in the hydrogen atom the ratio of electrical forces to gravitational ones is close to the ratio of the size of the Universe to the size of an electron (this fact was first noted by P.Dirac).

solution

Problem 12: Bohr radius

Express the Bohr radius through the fine structure constant and Compton wavelength.

solution

$\displaystyle a_0 = {{\hbar ^2 } \over {m_e e^2 }} = {{\lambda _c } \over \alpha }$

Problem 13: electron's mean free path

Estimate the mean free path of a hydrogen atom in the intergalactic space.

solution

The mean free path is $l \sim 1/n\sigma $, where $\sigma$ is the cross--section of hydrogen atom: $\sigma \sim 10^{ - 15} \mbox{cm}^2$. Let's assume that average density of matter in interstellar space is equal to critical density: $\rho _{cr} \sim 10^{ - 29} \mbox{g/cm}^3$, then $n = {{\rho _{cr} } / {m_p }}$, and, thus, $l \sim 10^{18} \mbox{cm}\sim 1 \mbox{pc}$.

Problem 14: a cosmic race

• • Protons accelerated at LHC ($E=7 TeV$) and photons are participants of a cosmic Earth--Sun race. How much will the protons lose in time and distance?

solution

$\Delta l \approx 1.5 \times 10^3 \mbox{m},~\Delta t \approx 5 \times 10^{-6} c$.

Problem 15: on telescopes of the XX century

Estimate the total amount of energy collected by optical telescopes during the past XX century and compare it with the energy needed to turn over a page of a book.

solution

Consider the page with dimensions $20 \times 15 \mbox{ cm}$ and mass $m=2\mbox{ g}$. To turn it, one needs to spend the energy $$E_p = mgh = 10^{ - 3} \mbox{ J},$$ where $h$ is the distance to the center of mass of the page.
Let's estimate the upper limit of energy collected by optical telescopes in the XX century. It is known that a star with zero apparent magnitude gives on average $\sim 10^6 $ photons (estimate this) (?in unit time?). Let's assume also that we observe the star with magnitude 6 on a 5-meter telescope every night for 8 hours during the entire XX century. The energy collected by the telescope is $$E \approx 10^{ - 4} \mbox{ J}.$$ Thus, even this overestimated value is by order of magnitude smaller than the energy needed to turn a page of this book.

Problem 16: on relativity of weight

• Estimate your own weight on the surface of white dwarf, neutron star, black hole.

solution

Problem 17: density scales

Densities of astrophysical objects vary in a wide range. Estimate the ratio of a neutron star's density to the average density of Milky Way.

solution

Problem 18: the coolest cosmological process

What cosmological process releases the maximum amount of energy simultaneosly since the Big Bang?

solution

Problem 19: gravity in Standard Model

Demonstrate, that for any Standard Model particle quantum gravity effects are completely negligible at the particle level.

solution

Quantum gravity effects are negligible when the Compton wavelength $\lambda_c = h/mc$ of a particle is much larger than its Schwarzschild radius $r_s=2mG/c^2$. The role of quantum gravity is determined by the parameter $$ {\lambda_c\over r_s}\sim {m_{Pl}^2\over m^2}. $$ For electron, for example, this parameter is $$ {\lambda_c\over r_s}\sim {m_{Pl}^2\over m^2}\sim 10^{45}, $$ so quantum gravity effects are completely negligible. The same is true for all other Standard Model particles.