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Z w w6/ [ w60 \ w63 ] w ww64 ^ w6> _ ww66 `  w67 a w68 b w w69 c w w6: d w6C e x ww6@ f  w~6H g w wy6I h ~ ww6N i { wz6O j { wz6T k { ww6U l w wv6V m ww6l n w ww6e o ww6m p w ww6z q wy6} r ww6~ s x w~6 t w w|6 u ~ w6 v { ww6 w ww6 x w ww6 y w w6 z  w6 {  ww6 |  w6 }  w6 ~ ww6  w~6  w|6 w6  ww6 w|6 ww6 w ww6 ww6 x w|6 w w6 ww6 w~6 w{6 w w{6 ww6 w w6 w6 w ww6 w6 w6  w6 z w6 z ww6 w ww6  w6 z wx6  w6  ww6 z w~5Q6Ndq5{wPF6Vdq6wLH6zdq7x~8C6Hm3R q897 wHm3R H RH R FootnoteHr@ q98:7 Hr@ HzHz  Single LineH'q:9<7 ;;Footnote  5_;:     HD q<:=7 6HD HH  Double LineH q=<@76>?w Double Line 5c>?=5e?>=66 H q@=B7NAA Single Linew5hA@HZqB@C7q TableFootnoteEGxR qCB7 9EGxR EPwEPw  TableFootnoteod5pDRRHH5xE5 HH Fe   HH5zFN5 HHlEE DHH5{G6 HH @>?HDoe   HH5}HJ6HHlGG HUV 5~I6 HUV AAJUU`   HUV 5JLH6HUV lII H$ 5K6 FtnH$ ELUU` tn oH$ 5LJ6H$ lKK HUV 5M5 HUV NUUe !  HUV 5NPF5HUV lMM H$ 5O5 H$ P UUe "  H$ 5PN5H$ lOO HH5QD HHAR  ` #  Hubbles Law ;` $ The expansion of the universe X` %  u ' *Distant galaxies recede from us at higher @ ' $velocities than do nearer galaxies. ` ( o  ) 5/The increased recession velocity appears to be l@ ) "proportional to distance from us. ` *  #` + !  The universe is expanding! @` ,  ]` - !Implications for ` . F  Distance measurement ` /   The age of the universe ߀` 0 5  The Big Bang  ` 1   The dark matter problem HH5RD"HHlUQQ d5SOUU5HH5TS AHHt`Z`U ` & Redshift h;` 2 u eX 4 %@Redshift  z  is a change in observed wavelength  l  igu H 4 '6relative to the emitted wavelength  l 0 Z ` 5 Careful of notation: a` 6 ci,  Red  shift implies  z  > 0 or$` 7 f-  Blue  shift implies  z  < 0 +A` 8 iv e^` <   ` 9 Redshift and Doppler Shift s f` : . F > ea/One way to get a redshift is for the source to f t@ > 5move at velocity  v  relative to the observer. ` ; at  h = $If [ then \ and therefore] HH5USUHHlRXTT d5VXXHH5WV `HHzi RedshieiX ` 3 4 Hubbles Law ;` ? ge X A h 4For observable (i.e. nearby) galaxies, the redshift teu H A  3is proportional to the distance to the galaxy.^ : a` B ci0H  is called the Hubble Constant. We find ` J ʥ "H  50-100  km/(sec Mpc) ` C  8; h D +Nonrelativistic (nearby) galaxies: _ d D` E  ` F !Relativistic (distant) galaxies: r G s,The situation is more complicated. We will ci@ G iv*return to this when we discuss cosmology. ` I  f! h H d 'For a flat universe, we find ` UHH5XVHHlUcWW V݀ 5YZyj`A݀ 4'&equiv[(*q"Green"q*)char[(*q"Green"q*)z],over[(*q"Green"q*)plus[(*q"Green"q*)char[(*q"Green"q*)lambda],minus[(*q"Green"q*)indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)lambda],num[(*q"Green"q*)0.00000000,"0"]]]],indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)lambda],num[(*q"Green"q*)0.00000000,"0"]]]]NX5Z\SHblYYTZJN5[\M) JNJN&'E'oll[(*q"Green"q*)fract[(*q"Green"q*)char[(*q"Green"q*)v],char[(*q"Green"q*)c]],num[(*q"Green"q*)1.00000000,"1"]]gal]OLN5\Z^S m[[T[cio65]^ssso6o6V7'equal[char[lambda],times[indexes[0,1,char[lambda],num[0.00000000,"0"]],id[(*i1i*)plus[num[1.00000000,"1"],fract[char[v],char[c]]]]]]NOo6 5^\`S ]]T\y5_`eq;]P)F6e.[('list[(*q"Green"q*)equal[(*q"Green"q*)char[(*q"Green"q*)z],over[(*q"Green"q*)char[(*q"Green"q*)v],char[(*q"Green"q*)c]]],string[(*q"Green"q*)"Nonrelativistic motion"]]NjR5`^Sm*q__T]]]d5alccHH5ba HH}N  nnc  @ )vBrightness, Distance, and C @ @ 00 Hubbles Law f` K 5 ` M mHow do we test Hubbles Law? ` N =  Measure  z  and  d  for galaxies and plot! ˀ` P l[(How do we measure  d ? Hmmm.... ],` Q ]] [ R 005Simple test:  Consider a set of galaxies all with " H R 3the  same absolute magnitude  M . a X` S <  Plot apparent magnitude  m  versus  log z u` U *),It should be a straight line with slope 5 *qHH5caeq*HHlXlbb 5de_<;Q6e$5'equal[(*q"Blue"q*)char[(*q"Blue"q*)z],times[(*q"Blue"q*)over[(*q"Blue"q*)char[(*q"Blue"q*)H],char[(*q"Blue"q*)c]],char[(*q"Blue"q*)d]]]NR5egV ddW^Dian5fgwP!Ho'%equal[char[v],times[char[H],char[d]]]NN85geiV r ffW_5him.. d9- R'equal[times[char[c],char[z],over[plus[num[1.00000000,"1"],fract[char[z],num[2.00000000,"2"]]],power[id[plus[num[1.00000000,"1"],char[z]]],num[2.00000000,"2"]]]],times[char[H],char[d]]]NqQ5igVU*)hhW`li wd6jllHH6kj HH[5<;l ` L eqWhat Does It Mean? (*q;` O (* lX` T e".Hubbles Law implies that space is expanding. u` V   X 5/Any observer in any galaxy would also conclude Di@ X that Hubbles Law is correct. ` Y 0(There is nothing special about the Milky Way.) ch` Z  ` [ %History: This was a Big Controversy! #` \ /Before 1929, people believed in Steady State d@` ]  ]` ^ l[&This created the field of Cosmology ` ` ch! The universe had a beginning d` a 00  The Big Bang .00` _ es Is there any other evidence? Q` f # Will expansion continue forever? li` b  6` c  S` d 3We will return to this later in the course  ... HH6ljHHlcqkk Wh6mnl 2vt 'atop[(*j4j*)list[equal[plus[char[m],minus[char[M]]],plus[times[num[5.00000000,"5"],log[char[d]]],minus[num[5.00000000,"5"]]]],string["(distance in pc)"]],equal[char[m],plus[times[num[5.00000000,"5"],log[over[char[d],times[num[10.00000000,"10"],string[" pc"]]]]],char[M]]],equal[char[m],plus[times[num[5.00000000,"5"],log[id[over[times[char[c],char[z]],cdot[char[H],times[num[10.00000000,"10"],string[" pc"]]]]]]],char[M]]],equal[char[(*q"Red"q*)m],plus[cross[num[5.00000000,"5"],log[(*q"Red"q*)char[(*q"Red"q*)z]]],id[(*i1i*)plus[char[M],times[num[5.00000000,"5"],log[id[over[char[c],cdot[char[H],times[num[10.00000000,"10"],string[" pc"]]]]]]]]]]]]Nr6nacmmba cd6(ono qq ..HH6)po HHhcqvq ` W The Age of the Universe ;` e When did the Big Bang happen? X` g  iu` i [t+Galaxies move apart at velocity  v . num h j ],<  The separation between any two is  d  whereb 00` k ch,and  t  is the age of the universe. ]]` l l[ r h m [5+Hubbles Law says that c so we haved [z]k o es2In fact, the universe expanded faster when it was  H o d"more compressed, so e 0` n ed ) ( p ]]Write f with the ], H p 00&observation that g. Thereforeh C` q 00.Consistent with the age of globular clusters? `` r Maybe yes. Maybe no. HH6+qo(HHlltpp d6JrttHH6Ksr HHhe` e t B ` h Measuring H 0 ;` s xi(Independent measures of large distances X` t  u v we2Need to establish bright Standard Candles, i.e. @ v  4objects whose  intrinsic luminosity  is known r` w [5-  Bootstrap techniques to get distance ]` x es n` y e &Possible examples of standard candles ` z pr# ScI galaxies  (Example: M101)  )#` { ]]/ Giant elliptical galaxies  (Example: M87) erv@` | Th. Large HII regions  (Example: 30 Doradus) ]` } la Supernove z` ~ Ma++  Type I:  M=M WD  explosions q`  $+  Type II: Becoming more useful `  Tully-Fisher relationship ` 'Rotation velocity    luminosity `  Gravitational lenses  `  (` &Calibrate these with other techniques E` (Example: Cepheids in M81, M100,  ... urb` ce ` 6Combine candles    Are the results consistent? HH6MtrbctHHlqss ` w6QuvianӶ FX y'fequal[(*q"Green"q*)char[(*q"Green"q*)d],times[(*q"Green"q*)char[(*q"Green"q*)v],char[(*q"Green"q*)t]]]N56Rvxoe uupbP6Xwx DPP)a'pequal[(*q"Magenta"q*)char[(*q"Magenta"q*)v],times[(*q"Magenta"q*)char[(*q"Magenta"q*)H],char[(*q"Magenta"q*)d]]] u5SR6Yxvzorelwwpc 6]yz \:j'|equal[(*q"Magenta"q*)char[(*q"Magenta"q*)t],fract[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],char[(*q"Magenta"q*)H]]], Nn06^zx|oyypdle O6i{|-<ӶOq'equiv[(*q"Magenta"q*)lessthan[(*q"Magenta"q*)char[(*q"Magenta"q*)t],fract[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],indexes[(*q"Magenta"q*)0,1,char[(*q"Magenta"q*)H],times[(*q"Magenta"q*)char[(*q"Magenta"q*)T],char[(*q"Magenta"q*)o],char[(*q"Magenta"q*)d],char[(*q"Magenta"q*)a],char[(*q"Magenta"q*)y]]]]],indexes[(*q"Magenta"q*)0,1,fract[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],char[(*q"Magenta"q*)H]],num[(*q"Magenta"q*)0.00000000,"0"]]]"O 6j|z~o"ge {{pe"qd]d6o}~rdd 'Requal[char[H],cross[char[h],times[num[100.00000000,"100"],string[" km/secMpc"]]]]d6p~|otfr}}pf"Mens@6s, s@s@: 'Wleq[num[0.50000000,"0.5"],indexes[0,1,char[h],num[0.00000000,"0"]],num[1.00000000,"1"]]Ӷu@ 6t~o[q" pggea"6w")nl "$(*'approx[char[t],times[sn[id[times[num[10.00000000,"10"],string[" to "],num[20.00000000,"20"]]],num[9.00000000,"9"]],string[" years"]]]MN/G6xoes[ph*qagd6cr["MenHH6 6jHHjge"qd] ` u Values for H 0 ;` 1The current bottom line:  Still controversial [X` ], iu` ]]%Sandage-Tammann:  Ladder approach ` }2They find  H 0  = 50 10  km/secMpc ` /  No problem with age of globular clusters [nu` ], e` m[(Aaronson-Huchra-Mould:  Tully-Fisher ` 6t1They find  H 0  = 90 5  km/secMpc e#` 4  Big  problem with age of globular clusters @` (* ]` imOngoing research: z` tr/ Recalibration with Cepheids in M100,  ... 0,"` s"-   H 0  = 73 5  km/secMpc e` p( HST observations of Type I supernovae ` r[)   H 0  80  km/secMpc `  Type II supernovae  ` 1   H 0  = 73 6 7  km/secMpc (` & Re-examination of Type I supernovae E` om.   H 0  = 50 11  km/secMpc b`  HH6HHlt  d5 pLefth d6nuRightd7 ReferencendDhedSd dVMdagobdjludodr: dio a }-s f a Body. f b    Bulleted\t. f c CellBody. f d  CellHeading. f e V MFootnote. f fT Heading1Body. f gT  Heading2Body. f hT   HeadingRunInBody. f i l yIndented. f j g Numbered.\t. f kE  Numbered1.\tNumbered. f l f TableFootnote. f mT   TableTitleT:Table : .  f nP TitleBody. f o T   TableTitleT:Table : . f p   CellHeading. f q  CellBody. f r   CellFooting. f s  Body. @ t  lHeader. @ u  blaFooter. f v $ Body. f w Body. f x  Body. f y Body. Hf z Body. f {  Body. f |  Body. f ~ $ Body. $f  Body. f  Body. f  Body. f  Body. f  Body. f Body. $f Body. $f  Body. Hf Body. f Body. f   Body. f   Body. $f  Body. $f  Body. f  Body. Hf  Body.  )  Emphasis  EquationVariables )   ڝ  ڝ tu ڝ $)   [  )  w tu tu )  w )  ڝ ڝ w w w  ڝ  ڝ ڝ  ڝ ڝ ڝ tu w ڝ w tu w  ڝ )  uo&  $)  \S  tu tu )    Thin Medium Double Thick@  Very Thin     oH p q rH p q rH p q rH p q rH p q rFormat A   oH p q rH p q rH p q rH p q rH p q rFormat B U e V UComment d BlackT!WhiteddAReddd Greendd  Blued Cyand Magentad Yellow  Times-Roman Times-Bold Times-ItalicHelvetica-BoldSymbolSymbolTimes-BoldItalic ZapfDingbats ZapfDingbatsTimesy HelveticaSymbol ZapfDingbats RegularRegular BoldRegularItalicuz/2 u܏C dGj xr0k4Tjg\wYF%p1 HRX쮈v꠼+7tK/ls3`3C_=+"1tV?m^Rzjq