ProblemOdleglosciWKosmologii < Main

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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.

Error during latex2img:

ERROR: problems during latex
INPUT:
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\setlength{\mathindent}{0cm}
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\usepackage{latexsym}
\begin{document}
\pagestyle{empty}
\pagecolor{white}
{
\color{black}
\begin{math}\displaystyle R_c = \frac{c}{H_0} (|\Omega_m + \Omega_{Lambda} - 1|)^{-\frac{1}{2}\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle d_{pm} = R_c sinh (\frac{d_{proper}}{R_c})\end{math}
}
\clearpage
\end{document}
STDERR:
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=latex)
 restricted \write18 enabled.
entering extended mode
(/tmp/1_YKrikXQ_/4eNaUwvk8K
LaTeX2e <2022-11-01> patch level 1
L3 programming layer <2023-01-16>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2022/07/02 v1.4n Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/fleqn.clo)
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
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))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/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/graphics/mathcolor.ltx))
(/usr/share/texlive/texmf-dist/tex/latex/base/latexsym.sty)
(/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-dvips.def)
No file 4eNaUwvk8K.aux.
(/usr/share/texlive/texmf-dist/tex/latex/base/ulasy.fd)
! Missing } inserted.
 
                }
l.17 ...ega_{Lambda} - 1|)^{-\frac{1}{2}\end{math}
                                                  
[1] [2] (./4eNaUwvk8K.aux) )
(see the transcript file for additional information)
Output written on 4eNaUwvk8K.dvi (2 pages, 1180 bytes).
Transcript written on 4eNaUwvk8K.log.

Topic revision: r3 - 30 Jan 2004, GrzegorzNowak

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