% 6_vzdalenosti/README.tex
% Miroslav Broz (miroslav.broz@email.cz), Jan 2nd 2007

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\input ../povetron.tex
\input ../kurzmac.tex

\def\sect*#1\par{%
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\def\=>{$\Rightarrow$}
\def\->{$\rightarrow$}

\def\prnl{p�.\,n.\,l.}
\def\name#1{{\sc #1}}

\mathcode`,="002C  % �esk� desetinn� ��rka v matematice  

\def\app{\ifmmode\approx\else$\approx$\fi\,}
\def\myitem#1{\item{--}{\bf #1}}

\def\hvezd{\ifmmode *\else $*$\fi}
\def\slunce{\ifmmode \odot\else $\odot$\fi}
\def\zeme{\ifmmode \oplus\else $\oplus$\fi}

\parindent=0pt
\parskip=4pt
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\sect* 6.~kapitola o~m��en� vzd�lenost�

\itm definice metru v~SI, nepraktick� p�ikl�d�n� m���tka;

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\myitem{zatm�n� a z�kryty nebesk�ch t�les:}

\app 5.~st.~\prnl~--- \name{P�thagor�s ze Samu} (\app 582--\app 507~\prnl) z tvaru st�nu
                      p�i ��ste�n�m zatm�n� \mesic\ usoudil na kulat� tvar \zeme\
                      a ur�il, �e \mesic\ je 2,5~kr�t men�� ne� {\em ku�elov� st�n\/} \zeme\
                      ve vzd�lenosti \mesic\ \=> \mesic\ je 3,5~kr�t men�� ne� \zeme,

                      �hlov� velikost \mesic\ $30'$ \=> vzd�lenost 30~pr�m�r� \zeme

OBR ku�elov�ho st�nu \zeme\ a~�hlov�ho pr�m�ru \mesic

\app 290~\prnl~--- \name{Aristarchus ze Samu} (\app 310--\app 230~\prnl) ve spisu
                   \uv{O~velikostech a~vzd�lenostech Slunce a M�s�ce}
                   u�il geometrickou metodu pro v�po�et vzd�lenosti \slunce\ a~\mesic\ od~\zeme.

Pohledem na kotou�ek \mesic\ zjistil okam�ik 1.~�tvrti
\=> �hel $\zeme\mesic\slunce = 90^\circ$, zm��il �hel $\mesic{\zeme}\slunce = 87^\circ$
\=> pom�r vzd�lenost� $\oplus$--\mesic/$\oplus$--$\odot$ = 1/20.
%
�hlov� velikost \slunce\ $30'$ \=> 5~kr�t v�t�� ne�~\zeme\
({\em ale\/} spr�vn� hodnota �hlu je $89^\circ\,51'$ a pom�r 1/400).

Vendelinus (1630) tou� metodou, ale s pomoc� dalekohledu, z�skal paralaxu \slunce\ $15''$ (dvakr�t men�� ne� ve skute�nosti)

% telurium

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\myitem{obvod $\oplus$ z~d�lky vr�en�ho st�nu:}

\app 235 \prnl~--- \name{Eratosthenes z K�r�ny} (276--194~\prnl) ve spisu \uv{O m��en� Zem�}
               odvodil obvod \zeme\ s~dobrou p�esnost� z vr�en�ho st�nu.
               M��en� �hlu prov�d�l pomoc� {\em skaf�\/} p�i slunovratu v Alexandrii
               (a zjistil odchylku poledn�ho st�nu od svislice $7,2^\circ$),
               p�i�em� v�d�l, �e v Sy�n� (dne�n�m Asu�nu) b�v� Slunce v nadhlavn�ku.

               \=> absolutn� vzd�lenosti a velikosti \mesic\ a~\slunce.

OBR skaf�

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\myitem{3.~Kepler�v z�kon} (Kepler 1619) \=> pom�ry {\em v�ech\/} vzd�lenost� ve slune�n� soustav�;
%
synodick� \-> siderick� periody:

$${360^\circ\over 1\,{\rm yr}} - {360^\circ\over P_{\rm sid}} = {360^\circ\over P_{\rm syn}}$$

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\myitem{trigonometrick� paralaxa} (denn�, ro�n�):

\app 340~\prnl~--- \name{Aristotel�s ze Stageiry} (384--322~\prnl) nepozoroval ��dnou
                   paralaxu \hvezd\ \=> a) \hvezd\ jsou p��li� daleko, b) \zeme\ je nehybn�;
                   tvrd�, �e b)~je pravd�podobn�j��.

meteory (lze si snadno o�ima vyzkou�et, �e jsou ve v��k�ch \app 100\,km nad zem�)

%�LOHA na pr�se��k dvou rovin?

komety (lze se p�esv�d�it, �e jsou mimo atmosf�ru)

za��tek 17.~st.~--- \name{Johannes Kepler} (1571--1630) pou�il Tychonova vizu�ln� pozorov�n�
            Marsu k odhadu jeho paralaxy, co� se mu nepoda�ilo. Proto�e ale znal p�esnost
            pozorov�n�, usoudil, �e Slunce mus� b�t nejm�n� 3~kr�t d�l ne� tvrdil Aristarchus.


ro�n� paralaxa \hvezd\ 61~Cygni $0,3''$ (Bessel 1838), Vega (Struve 1838), $\alpha$~Centauri (Henderson 1839)

temn� pr�vodce S�ria (Bessel 1844)

denn� paralaxa Marsu na z�kladn� Cayenne, Francouzsk� Gyana -- Pa��� (Richer \& Cassini 1672)
\=> paralaxa Slunce $\pi_\slunce = 9,5''$ s chybou 30\,\%, prvn� dobr� m��en�

{\em p�echody Merkuru a Venu�e\/} p�es disk~\slunce\ (m��en� {\em rozd�l�\/} dob trv�n� p�echod� z r�zn�ch observato��):
observa�n� kampa� na dva p�echody Venu�e 1761 a 1769, kter� se opakuj� a� po 120~letech \=> $\pi_\slunce = 8,55"$ a� $8,88''$;
p�epo�et Enckeho (1835) $8,57''\pm 0,04''$

Galle (1872) navrhnul pozorovat asteroidy kv�li jejich bodov�m obraz�m,
m��en� Iris, Victoria a Sappho v letech 1888 a 1889 poskytlo hodnotu $\pi_\slunce = 8,802"$.

paralaxy bl�zkozemn�ch asteroid�: (433)~Eros byl objeven v roce 1898
a m��en p�i opozic�ch v letech 1900 a pak 1930--31, kdy se pozorovac� kampan�
��astnilo 40 observato�� \=> $\pi_\slunce = 8,790"\pm 0,001"$ (Spencer Jones 1942)

astrometrick� dru�ice Hipparcos ($10^{-3}\,{\rm arcsec}$, 1\,000\,pc) a~GAIA ($10^{-6}\,{\rm arcsec}$);

OBR  p�echod� Venu�e, Hipparcos, GAIA, p�esnost astrometrie

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\myitem{dynamick� poruchy\/} pohybu \mesic\ kolem \zeme\ vlivem \slunce\ (Hanson 1857, 1863) \=> $\pi_\slunce = 8,92''$

poruchy Marsu a Venu�e \zeme\ p�soben� \zeme\ (Leverrier, \app 1860) \=> $8,95''$
(tj. hodnota odli�n� od paralaktick�ch metod)

poruchy Erosu \=> $M_\zeme$ \=> $\pi_\slunce = 8,799''$ (Noteboom 1921);
s~vyu�it�m perturbac� od v�ech planet $8,79402"\pm 0,00012"$ (Lieske 1966)

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\myitem{aberace sv�tla:}

Foucault a Fizeau (1850) m��il rychlost sv�tla laboratorn� a aberaci \hvezd\ v~dalekohledu \=> ob�n� rychlost \zeme\ \=> $\pi_\slunce = 8,8''$

Michelson \& Newcomb (\app 1890) obdr�eli $8,80''\pm 0,01''$

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\myitem{\uv{rychlost kr�t �as}:} vysok� relativn� p�esnost m��en� �asu $\sim\!10^{-13}$

laser a koutov� odra�e� (t�i zrcadla/st�ny hranolu v z�kladn�ch rovin�ch)

laserov� skvrna na \mesic\ o~pr�m�ru 7\,km

OBR  koutov�ho odra�e�e na M�s�ci a pozemsk�ch stanic McDonald (USA), Calern (OCA, Francie)

radar, radarov� rce ($\propto r^{-4}$):

$$P_r = {P_t G_t A_r \sigma F^4 \over (4\pi)^2 r_t^2 r_r^2} \rightarrow {P_t G_t A_r \sigma \over (4\pi)^2} r^4\,,$$

kde
$P_r$~je p�ij�man� v�kon,
$P_t$~vys�lac� v�kon,
$G_t$~zisk ant�ny,
$A_r$~efektivn� plocha c�le,
$F$~faktor ���en�,
$r_t$~vzd�lenost vys�la�e od c�le,
$r_r$~vzd�lenost p�ij�ma�e od c�le.

odrazy od Venu�e (Victor \& Stevens 1961, Campbell et al. 2001), Merkuru, planetek

% uk�zka kusu zna�ky nebo sv�t�c�ch kalhot?

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\myitem{pohybov� \hvezd kupy} (Hy�dy), vertex (Boss 1908, Perryman et al. 1998);

$$\alpha_C = 96,6^\circ,\; \delta_C = +5,8^\circ$$

$$[\mu]_{''/{\rm yr}} = {[v_t]_{\rm km/s} \over 4.74047 [d]_{\rm pc}} = \sqrt{(\mu_\alpha \cos\delta)^2 + \mu_\delta^2}$$

$$v_t = v_r \tg\theta$$

tangenci�ln� rychlost $v_t$ nemohu m��it p��mo v km/s, ale vypo��t�m j� z radi�ln� $v_r$
(u kter� to lze ze spektra), proto�e zn�m �hel $\theta$ mezi hv�zdokupou a �b�n�kem!

$$\cos\theta = \sin\delta\sin\delta_C + \cos\delta\cos\delta_C\cos(\alpha-\alpha_C)$$

(kos�nov� v�ta ve sf�rick�m troj�heln�ku)

OBR Hy�d a vertexu na obloze

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\myitem{dynamick� paralaxa dvoj\hvezd:}

$${a^3\over P^2} = {G\over 4\pi^2} (M_1+M_2)$$

hmotnosti $M$ odhadnu ze spekter

$$\pi'' = \alpha'' (M_1+M_2)^{-{1\over3}} P^{-{2\over3}}$$

slab� z�vislost na $M$

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\myitem{luminozitn� vzd�lenost} $F = {L\over 4\pi r^2}$

nap�. hv�zdy stejn�ho spektr�ln�ho typu maj� stejn� $M$ a $L$
(s rozli�en�m obr� a trpasl�k� podle tlakov�ho roz���en� spektr�ln�ch �ar),
maxim�ln� jasnost nov v galaxii nebo charakteristick� pr�b�h sv�teln� k�ivky

nap�. novy v M\,31 maj� $m_{\rm max} = 16\hbox{ a� }17\,{\rm mag}$,
podle m��en� v~MW je $M = -6\hbox{ a� }-7\,{\rm mag}$
\=> {\em modul vzd�lenosti\/} $m-M = 24\,{\rm mag}$

Pogsonova rovnice $$m_2 - m_1 = -2,5 \log {E_2\over E_1}$$

$$m-M = -2,5 \log {{L/r^2}\over L/(10\,{\rm pc})^2} = -5 \log {10\,{\rm pc}\over r} = -5 (1-\log [r]_{\rm pc})$$

\=> $\log [r]_{\rm pc} = 0,2 (m-M) + 1 = 5,8$; $r = 600\,{\rm kpc}$

Je zdroj {\em izotropn�\/}? Zdroje z�blesk� gama (GRB) z�ejm� ne!

OBR  HR diagram, spektr�ln� typy, animace GRB

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\myitem{cefeidy, vztah perioda--sv�tivost:}

$M_V = a + b \log [P]_{\rm dny}$, $a = -1,7\,{\rm mag}$, $b = -2,54\,{\rm mag}$ pro typ I (klasick� cefeidy)

periodick� zm�ny opacity nitra \hvezd\ kv�li p�echod�m mezi jednou a dvakr�t ionizovan�m h�liem:

\table{%
\offinterlineskip
\halign{
\strut #\hfil & #\hfil & #\hfil\cr
%
\hfil HeII & $\leftrightarrow$ & \hfil HeIII \cr
%
velk� opacita,                     & & mal� opacita, \cr
absorpce z��en�,                   & & z��en� unik�, \cr
ionizace HeII $\rightarrow$ HeIII, & & rekombinace HeIII $\rightarrow$ HeII, \cr
cefeida sv�t� m�lo,                & & cefeida sv�t� hodn�, \cr
roste $T$ a $p$                    & & kles� $T$ a $p$ \cr
}}

Vrstva, kde prob�haj� p�echody h�lia, je {\em pod povrchem\/},
po ionizaci a poklesu opacity tlak {\em proch�zej�c�ho\/} z��en� rozepne vrstvy {\em naho�e\/}.

zm�na polom�ru cefeid o~$\sim\!10\,\%$ (typ~I) nebo $\sim\!50\,\%$ (typ~II)

Leavittov� (1912) --- vztah $P$--$L$ pro cefeidy v Magellanov� oblaku (v�echny jsou stejn� daleko),
ale nebyly tehdy je�t� zn�my podobn� \hvezd\ RR~Lyr, W~Vir

OBR  sv�teln� k�ivky cefeid

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\myitem{supernovy typu Ia:}

akrece na b�l�ho trpasl�ka (WD), p�ekro�en� Chandrasekharovy
meze $1,\!44\,M_\odot$, \uv{standardn� sv��ky};

SN~Ia nemaj� ve spektru ��ry vod�ku ani h�lia

$M_V = -19,30 + 5 \log (H_0/60)$

disperze jen 0,3\,mag, $B-V \doteq 0\,{\rm mag}$

OBR  spektrum SN Ia a II

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\myitem{Tullyho--Fisher�v vztah} (Tully \& Fisher 1977):

���ka ��ry $\Delta f$ na 21\,cm \=> rota�n� rychlost $\Delta V$ -- sv�tivost pro spir�ln� galaxie

$\mu_0 = 3,5\,{\rm mag} + 6,25\log \Delta V + m_{\rm pg} \pm 0,3\,{mag}$

%$M_{\rm bol} =  -9,95 \log_{10} \Delta V + 3,15 (typ Sa)$
%$M_{\rm bol} = -10,2  \log_{10} \Delta V + 2,71 (Sb)$
%$M_{\rm bol} = -11,0  \log_{10} \Delta V + 3,31 (Sc)$

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\myitem{rozd�len� poloh objekt� na obloze} a jejich korelace se zn�m�mi strukturami

OBR  rovina ekliptiky, galaxie, z�blesky $\gamma$ v~izotropn�m vesm�ru

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\myitem{Hubble�v z�kon} $v = H d$ (Slipher 1918, Hubble 1925, 1928),

Hubbleova konstanta $H \doteq 70\,{\rm km}\sks{\rm s}^{-1}\sks{\rm Mpc}^{-1} \doteq 2,5\,{\rm cm}/{\rm rok}/\hbox{vzd�lenost M�s�ce}$,

rud� posuv $z = {\Delta\lambda\over\lambda} \doteq {v\over c}\hbox{ pro }v\ll c$,
relativisticky $(z+1)^2 = {c+v\over c-v}$;

OBR sch�ma rozp�n�n� (kdybychom byli jinde, je rozp�n�n� stejn�), Mt Wilson

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%Eros: Historical background.
%http://near.jhuapl.edu/eros/history/parallax.html

%The distance to the Hyades
%http://www.pas.rochester.edu/~dmw/ast142/Lab_manual/Lab4.pdf

%Michael Fowler
%Measuring the Solar System
%http://galileo.phys.virginia.edu/classes/109N/lectures/gkastr1.html

%ur�ov�n� st���:
%\itm vlastn� pohyb \hvezd\ v asociac�ch --- rozb�haj� se z 1 bodu, z $\mu_\alpha$, $\mu_\delta$ \=> st��� \hvezd kupy

% Lunar Laser Ranging:
%http://www.lpi.usra.edu/expmoon/Apollo11/A11_Experiments_LRRR.html
%http://en.wikipedia.org/wiki/Lunar_laser_ranging_experiment
%http://science.nasa.gov/headlines/y2004/21jul_llr.htm
%http://physics.ucsd.edu/%7Etmurphy/apollo/lrrr.html
%http://www.csr.utexas.edu/mlrs/

%http://cfa-www.harvard.edu/oir/Research/supernova/RecentSN.html

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\end



