Quantities large and small - Revision history

Igor: /* Problem 1: parsec */

2012-10-11T13:14:32Z

‎Problem 1: parsec

Igor at 13:13, 11 October 2012

2012-10-11T13:13:28Z

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Igor: /* Problem 2 */

2012-10-11T13:00:08Z

‎Problem 2

Cosmo All: /* Problem 15 */

2012-08-20T14:33:37Z

‎Problem 15

Cosmo All: /* Problem 14 */

2012-08-20T14:29:31Z

‎Problem 14

Cosmo All: /* Problem 13 */

2012-08-20T14:29:11Z

‎Problem 13

Cosmo All: /* Problem 9 */

2012-08-20T14:28:16Z

‎Problem 9

Cosmo All: /* Problem 8 */

2012-08-20T14:27:52Z

‎Problem 8

Cosmo All: /* Problem 7 */

2012-08-20T14:27:09Z

‎Problem 7

Cosmo All: /* Problem 3 */

2012-08-20T14:26:10Z

‎Problem 3

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 === Problem 1: parsec ===
-From what distance will one astronomical unit length have visible size of one angular second?
+From what distance will the length of one astronomical unit have the visible size of one angular second?
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    <div class="NavHead">solution</div>
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 === 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.

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">$\alpha _\Gamma   = arctg\left( {\frac{{30\;\mbox{ kpc}}}{{700\;\mbox{ kpc}}}} \right) = 2^ \circ  28' = 5\alpha _ \odot  $</p>
+     <p style="text-align: left;">
    </div>
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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Consider the page with dimensions $20 \times 15 \mbox{\it cm}$ and mass $m=2\mbox{\it g}$. To turn it, one needs to spend the energy
+     <p style="text-align: left;">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
-$$E_p  = mgh = 10^{ - 3} \mbox{\it J},$$
+$$E_p  = mgh = 10^{ - 3} \mbox{ J},$$
 where $h$ is the distance to the center of mass of the page.
 <br/>
-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) \underline{(?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
+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) <u>(?in unit time?)</u>. 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
-$$E \approx 10^{ - 4} \mbox{\it J}.$$
+$$E \approx 10^{ - 4} \mbox{ J}.$$
 Thus, even this overestimated value is by order of magnitude smaller than the energy needed to turn a page of this book.</p>
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 === Problem 16 ===
 * Estimate your own weight on the surface of white dwarf, neutron star, black hole.

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">$\Delta l \approx 1.5 \times 10^3 \mbox{\it m},~\Delta t \approx 5 \times 10^{-6} c$.</p>
+     <p style="text-align: left;">$\Delta l \approx 1.5 \times 10^3 \mbox{m},~\Delta t \approx 5 \times 10^{-6} c$.</p>
    </div>
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 === Problem 15 ===
 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.

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">The mean free path is $l \sim 1/n\sigma $, where $\sigma$ is the cross--section of hydrogen atom: $\sigma  \sim 10^{ - 15} \mbox{\it cm}^2$. Let's assume that average density of matter in interstellar space is equal to critical density: $\rho _{cr}  \sim 10^{ - 29} \mbox{\it g/cm}^3$, then $n = {{\rho _{cr} } / {m_p }}$, and, thus, $l \sim 10^{18} \mbox{\it cm}\sim 1 \mbox{\it pc}$.</p>
+     <p style="text-align: left;">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, $l \sim 10^{18} \mbox{cm}\sim 1 \mbox{pc}$.</p>
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 === Problem 14 ===
 ** 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="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">$\rho  = {{NM_ \odot  } \over {L^3 }} = {{10^{11}  \times 2 \cdot 10^{33} } \over {\left( {3 \cdot 10^{24} } \right)^3 }} = 0.7 \cdot 10^{ - 29}\mbox{\it g/cm}^3.$</p>
+     <p style="text-align: left;">$\rho  = {{NM_ \odot  } \over {L^3 }} = {{10^{11}  \times 2 \cdot 10^{33} } \over {\left( {3 \cdot 10^{24} } \right)^3 }} = 0.7 \cdot 10^{ - 29}\mbox{g/cm}^3.$</p>
    </div>
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 === Problem 10 ===
 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 style="width:100%;" class="NavContent">
      <p style="text-align: left;">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
-$$M_G \approx {{4\pi ^2 R_G^3 } \over {GT_ \odot ^2 }} \approx 2\cdot10^{44}\mbox{\it ã} \approx 10^{11} M_ \odot.$$</p>
+$$M_G \approx {{4\pi ^2 R_G^3 } \over {GT_ \odot ^2 }} \approx 2\cdot10^{44}\mbox{g} \approx 10^{11} M_ \odot.$$</p>
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 === Problem 9 ===
 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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 and, finally,
 $$
-T = 9.6 \cdot 10^9 \mbox{\it years}.
+T = 9.6 \cdot 10^9 \mbox{years}.
 $$</p>
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 === Problem 8 ===
 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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 === Problem 4 ===
 A supernova outburst in the Andromeda galaxy has been observed on Earth. Estimate the time since the star's explosion.