# 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?

### 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.

### 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?

### 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.

### 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?

### 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.

### 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.

### 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.

### 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 $L=1 Mpc$.

### 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.

### 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).

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

### Problem 13: electron's mean free path

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

### 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?

### 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.

### Problem 16: on relativity of weight

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

### 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.

### Problem 18: the coolest cosmological process

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

### Problem 19: gravity in Standard Model

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