Difference between revisions of "Quantities large and small"

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[[Category:Cosmo warm-up|2]]
 
[[Category:Cosmo warm-up|2]]
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<div id="razm2"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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From what distance will one astronomical unit length have visible size of one angular second?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm3"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm6"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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A glance on the night sky makes the impression of invariability of the Universe. Why do the stars seem to us practically static?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm4"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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A supernova outburst in the Andromeda galaxy has been observed on Earth. Estimate the time since the star's explosion.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm5"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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?
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm26"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm30"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm28"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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Estimate the mass $M_G$ of Milky Way and the number of stars in it, if the Sun is an avarage 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.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm29"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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 $L=1 Mpc$.
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm41"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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.
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm34"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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).
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
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</div></div>
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<div id="razm37"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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Express the Bohr radius through the fine structure constant and Compton wavelength.
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
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</div></div>
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<div id="razm38"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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Estimate the free path of a hydrogen atom in the intergalactic space.
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm39"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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** 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?
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>
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<div id="razm49"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
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</div></div>
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 +
<div id="razm25n1"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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* Estimate your own weight on the surface of white dwarf, neutron star, black hole.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
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</div></div>
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<div id="razm25n2"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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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.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
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 +
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<div id="razm52"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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What cosmological process releases the maximum amount of energy simultaneosly since the Big Bang?
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
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</div></div>
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<div id="razm52n1"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1 ===
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Demonstrate, that for any Standard Model particle quantum gravity effects are completely negligible at the particle level.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div></div>

Revision as of 14:05, 20 August 2012



Problem 1

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


Problem 1

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 1

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


Problem 1

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


Problem 1

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 1

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 1

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 1

Estimate the mass $M_G$ of Milky Way and the number of stars in it, if the Sun is an avarage 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 1

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 1

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 1

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).


Problem 1

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


Problem 1

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


Problem 1

    • 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 1

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 1

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


Problem 1

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 1

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


Problem 1

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

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