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):
d_{lt} = \int^{t_0}_t cdt = c(t_0-t)
  • 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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\begin{math}\displaystyle d_{pm} = R_c sinh (\frac{d_{proper}}{R_c})\end{math}
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{
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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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STDERR:
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=latex)
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entering extended mode
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LaTeX2e <2022-11-01> patch level 1
L3 programming layer <2023-01-16>
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Document Class: article 2022/07/02 v1.4n Standard LaTeX document class
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(/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
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(/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 UdeQr5pgD2.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] (./UdeQr5pgD2.aux) )
(see the transcript file for additional information)
Output written on UdeQr5pgD2.dvi (2 pages, 1180 bytes).
Transcript written on UdeQr5pgD2.log.
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Topic revision: r2 - 29 Jan 2004, GrzegorzNowak
 
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