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$\int$ Ze wzgledu na ekspansje Wszechswiata, czyli ciagla zmiane odleglosci pomiedzy dwoma wybranymi galaktykami, lub gromadami galaktyk istnieje w kosmologii potrzeba uzywania roznie zdefiniowanych odleglosci. Sa to:

odleglosc swiatla (ang. light travel distance):

$\int cdt = 0$

odleglosc wlasna (ang. proper distance), czyli odleglosc jaka przebywa swiatlo od chwile wyemitowania go przez rozwazany obiekt az do chwili jego obecnej obserwacji:

$\int^{t_0}_t \frac{cdt}{a(t)} = \int^{t_0}_t \frac{cdt}{\frac{1}{1+z(t)}}$ ,

gdzie

$a(t) = \frac{1}{1+z(t)}$ to czynnik skali.

tzw. odleglosc ruchu wlasnego (ang. proper motion distance), czyli odleglosc w jakiej obecnie znajduje sie obserwowany obiekt:

$d_{pm} = R_c sin (\frac{d_{proper}}{R_c})$ - w przypadku dodatniej krzywizny,

$d_{pm} = R_c sinh (\frac{d_{proper}}{R_c})$ - w przypadku ujemnej krzywizny,

gdzie $R_c = \frac{c}{H_0} (|\Omega_m + \Omega_{Lambda} - 1|)^{-\frac{1}{2}$ jest promieniem krzywizny.

-- GrzegorzNowak - 11 Mar 2004

Error during latex2img:

ERROR: problems during latex
INPUT:
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\begin{math}\displaystyle d_{pm} = R_c sin (\frac{d_{proper}}{R_c})\end{math}
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\begin{math}\displaystyle R_c = \frac{c}{H_0} (|\Omega_m + \Omega_{Lambda} - 1|)^{-\frac{1}{2}\end{math}
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\begin{math}\displaystyle \int cdt = 0\end{math}
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\begin{math}\displaystyle d_{pm} = R_c sinh (\frac{d_{proper}}{R_c})\end{math}
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\begin{math}\displaystyle \int^{t_0}_t \frac{cdt}{a(t)} = \int^{t_0}_t \frac{cdt}{\frac{1}{1+z(t)}}\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle a(t) = \frac{1}{1+z(t)}\end{math}
}
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STDERR:
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 restricted \write18 enabled.
entering extended mode
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LaTeX2e <2018-12-01>
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Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
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For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
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(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def))
(/usr/share/texlive/texmf-dist/tex/latex/base/latexsym.sty)
No file cUr5Zi9fZy.aux.
(/usr/share/texlive/texmf-dist/tex/latex/base/ulasy.fd) [1]
! Missing } inserted.
 
                }
l.22 ...ega_{Lambda} - 1|)^{-\frac{1}{2}\end{math}
                                                  
[2] [3] [4] [5] [6] (./cUr5Zi9fZy.aux) )
(see the transcript file for additional information)
Output written on cUr5Zi9fZy.dvi (6 pages, 2468 bytes).
Transcript written on cUr5Zi9fZy.log.

Topic revision: r2 - 12 Mar 2004, GrzegorzNowak

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