\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:
\documentclass[fleqn,12pt]{article}
\usepackage{amsmath}
\usepackage[normal]{xcolor}
\setlength{\mathindent}{0cm}
\definecolor{teal}{rgb}{0,0.5,0.5}
\definecolor{navy}{rgb}{0,0,0.5}
\definecolor{aqua}{rgb}{0,1,1}
\definecolor{lime}{rgb}{0,1,0}
\definecolor{maroon}{rgb}{0.5,0,0}
\definecolor{silver}{gray}{0.75}
\usepackage{latexsym}
\begin{document}
\pagestyle{empty}
\pagecolor{white}
{
\color{black}
\begin{math}\displaystyle d_{pm} = R_c sinh (\frac{d_{proper}}{R_c})\end{math}
}
\clearpage
{
\color{black}
\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 R_c = \frac{c}{H_0} (|\Omega_m + \Omega_{Lambda} - 1|)^{-\frac{1}{2}\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle \int cdt = 0\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle d_{pm} = R_c sin (\frac{d_{proper}}{R_c})\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle a(t) = \frac{1}{1+z(t)}\end{math}
}
\clearpage
\end{document}
STDERR:
This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex)
 restricted \write18 enabled.
entering extended mode
(/tmp/29P3jx5__J/27Dim8UDOq
LaTeX2e <2017/01/01> patch level 3
Babel <3.9r> and hyphenation patterns for 30 language(s) loaded.
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h 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/base/latexsym.sty)
No file 27Dim8UDOq.aux.
(/usr/share/texlive/texmf-dist/tex/latex/base/ulasy.fd) [1] [2]
! Missing } inserted.
 
                }
l.27 ...ega_{Lambda} - 1|)^{-\frac{1}{2}\end{math}
                                                  
[3] [4] [5] [6] (./27Dim8UDOq.aux) )
(see the transcript file for additional information)
Output written on 27Dim8UDOq.dvi (6 pages, 2468 bytes).
Transcript written on 27Dim8UDOq.log.
Topic revision: r2 - 12 Mar 2004, GrzegorzNowak
 
This site is powered by FoswikiCopyright © CC-BY-SA by the contributing authors. All material on this collaboration platform is copyrighted under CC-BY-SA by the contributing authors unless otherwise noted.
Ideas, requests, problems regarding Foswiki? Send feedback