AaBd HH $ @d HHHHff@  d Footnote TableFootnote**.\t.\t/ - :;,.!?4 ` ( `TOCHeading1Heading2  EquationVariables7  5 V6 6 x6' z6- |6E 6M 6  ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year> +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm>\) -"<$monthnum>/<$daynum>/<$shortyear> .<$monthname> <$daynum>, <$year> /"<$monthnum>/<$daynum>/<$shortyear> 0 <$fullfilename> 1 <$filename> 2 <$paratext[Title]> 3 <$paratext[Heading1]> 4 <$curpagenum> 5 <$marker1> 6 <$marker2> 7 (Continued)d 8+ (Sheet <$tblsheetnum> of <$tblsheetcount>)loo 9Heading & Page <$paratext> on page<$pagenum> :Pagepage<$pagenum>( ;See Heading & Page%See <$paratext> on page<$pagenum>. < Table All7Table<$paranumonly>, <$paratext>, on page<$pagenum> =Table Number & Page'Table<$paranumonly> on page<$pagenum>   FFA x HHAE JJ LL NN PP RA$  in5y  soe>5|  s > 5  ui> 5 t)"5 ! u/um5 " ta5 # n$d5 $ v se5 % so>/5 & v s$ea5 ' sfi5 ( ~ s 5 ) v s 5 * sele5 + ~ ste5 , ~ s]5 - ~ sg5 . sa5 / ~ sa5 0 v sod)5 1 s+t 5 2 su<$5 3 wnoo5 4 znge5 5 vx p5 6 u5 7 veen5 8 z zHe5 9  ze<5 :  v e5 ; v5 < ve5 =  vl$p5 > vo<5 ? vTa5 @ v&'5 A v vay>5 B v vp>5 C v 5 D v vF5 E v v 6 F v6 G v v6 H vJ6 I v6 J v6 K v6 L v v6 M v6 N vR6 O v$ i6 P v vs6< Q vs6 R u6 S vt6 T vu6 U vt6 V 6 W xv6 X v v6# Y v vv6$ Z v v6% [ v v~6& \ v vv6* ] v v6+ ^ v v~6= _ ~60 ` v v~6> a v v6? b v v~6I c v vv6N d { v6O e { v6P f { {6Q g {6R h {6Z j {6[ k 6y s {z6 u {6 v { {6 w { {6 x { {6 y { {6 z { {6 { {6 | {6 } y {v6 ~ { {v6  { {6 { {v6 { {v6 { {6 { {v6 {7 {7 { {7 { {7 {v7 { {7 {7 {7 {v7 { {7 { {7 {7 {7 { {7 {7 {5|6dq5vPF6&dq6v~LH60dq7v~8C6NHm3R q897 {Hm3R H RH R FootnoteHr@ q98:7 y Hr@ HzHz  Single LineH'q:9<7 {;;Footnote  5_;:v     HD q<:=7 6HD HH  Double LineH q=<@77 >?v Double Line {5c>?=5e?>=77 {H q@=B7AA Single Line5hA@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 N UUe !  HUV 5NPF5HUV lMM H$ 5O5 H$ P UUe "  H$ 5PN5H$ lOO HH5QD HH>AR  ` # Distances and Magnitudes VC ` 3 of Stars f` $  ` % *First step in understanding the structure ` 4 and evolution of stars. ` & ˪ ` ' %Distance measurements using parallax ` ( Note: Studio exercise ` )  1` * #Magnitude: A measure of Luminosity N` +  Relative magnitude k` ,  Absolute magnitude ` -  Magnitude and wavelength ` . "  Color Index and temperature ` /  Bolometric magnitude ` 0  ` 1 Example: Color and magnitude ` 6 Ԫof star clusters 6` 2 5A hint of things to come. HH5RDHHlUQQ d5SUUHH5TS VHHvta   VVU  ` 5 inDistance and Parallax ;` 7 ev(The motion of a nearby star against a X h 8 $background of fixed stars   ` 9  o` : e HAstronomical Unit  (a.u.)  =  1.496    10 11  m no` =  +` < agBParsec  =  Distance d for  p =  1 second of arc ` ? an:=  1 a.u. / sin(1) = 3.086    10 16  m ` @ =  3.26 light years m4` A  Q ; *This has been measured only for less than n@ ; ت10,000 stars! HH5UStcoHHlRqTT H5VSQWWnT 5O5WXV O  j$bM5XWYV[i j$bMj4dZj4dZ H   W5YX[V  W f f IbN5Z_`VagrNj bM5[Y\VX]i j bMjdZjdZ H  JbM5\[]Vij oJbM[ [ H  jnbM5]\^V[^i 1jnbMj j H  jbM5^]_V]_i 1 jbMj j H  bM5_^ZV^i ;bM  H  $5`ZaV$6<%X5a`bV 6<%X6LdZ6LdZ SunH35bacV H3HdZHdZ Earth$`65cbdV$`6j$$`$-5dceVMj$-H $$$Wl5edfV$WlIbW$WWl5fegVWl5W Wfg45gfhV Hfg4 1 a.u.N 5hgiV HN ^dZ^dZ d[j =5ihjV\j j =X_j =5jikVj =\i6q5kjlVZ^6q6Z$5lkmVa6$HZ$\ -5mlnV \ -ll pǛbM5nmV ǛbM p  parallax angled5oqqHH5po HH05eq ` > IbStellar Magnitudes W;` B  gX` C #A left over way of measuring the u` R 5brightness of a star. ` D  ` E u.&Your eye has a logarithmic response ` F *  magnitude is a logarithmic scale. ` G \ j` H ԕAncient definition: # I 2First magnitude  is the brightest star you can @@ I see with the naked eye. ]` J  z` K 5%Sixth magnitude  is the faintest. ` L  ` M Modern definition: ` N % Divide up according to a standard. M O ) One unit of magnitude corresponds to a  @ O alspecific ratio in brightness. (` P  HH5qoHHlUtpp d6 rttStlaHH6 sr HH}fPov ux|t5 ` T Law of Stefan and Boltzmann ;` U hHow Bright is a Star? X` V F u` W is1It depends on  Temperature  and  Radius ` X An n h Y Integrated intensity   dfPfP` Z teswhere  s = 2p 5 k 4 /15 c 2 h 3  = 5.6710 -8   W / m 2 K 4 agn7fP` [ ai sTfP` \ L qfP h ] M2Total luminosity  (for a star of radius R)  fP` ^  afP h ` 1Detected Flux  (for star at a distance r)  oXfP` Q O lufP` _ n -  Later: Connect flux with magnitude HH6trHHlqwss dd6utwwHH6vu HHptw ` S te.Apparent Magnitude  (lower case  m ) 9` a  V f` h b @Two stars (1 and 2) with fluxes F 1  and F 2  :   h c An where  J` d gr+and L  = luminosity (emitted power) swhg` e 5 d  = distance to star ` f   8 f` g m;Example:  Consider m 1 -m 2 =1, so star 2 Lf`@ g ]*appears to be brighter than star 1, and f`` j  f` h /1) If the stars are at the same distance, then sta̰fP f`@ h 1L 1 / L 2  = 10 -0.4  = 0.4 CA3 k /2) If the stars have the same luminosity, then d`fP f`@ k 0d 1 / d 2  = 10 0.2  = 1.6 HH6wutHHltvv NH6!xzryys Ui^a/76"yxe MUi^a/7Ui^a/7$'equal[times[char[I],id[char[T]]],int[(*i2i*)times[indexes[0,1,char[R],char[lambda]],id[char[T]],diff[char[lambda]]],num[0.00000000,"0"],char[infty]],times[char[sigma],indexes[1,0,char[T],num[4.00000000,"4"]]]]NfPH6(zx|r{{s f` gY,^-<6){z Y,^-<Y,^-<$ap'equal[char[L],times[id[times[num[4.00000000,"4"],indexes[1,0,times[char[pi],char[R]],num[2.00000000,"2"]]]],char[I]],times[num[4.00000000,"4"],char[pi],indexes[1,0,char[R],num[2.00000000,"2"]],char[sigma],indexes[1,0,char[T],num[4.00000000,"4"]]]]e sNCfPH6.|zrd}}s 10jY?6/}|jY?jY?$'cequal[char[F],over[char[L],times[num[4.00000000,"4"],char[pi],power[char[r],num[2.00000000,"2"]]]]]6C~wdNet2f`'equal[plus[indexes[0,1,char[m],num[1.00000000,"1"]],minus[indexes[0,1,char[m],num[2.00000000,"2"]]]],times[num[2.50000000,"2.5"],log[over[indexes[0,1,char[F],num[2.00000000,"2"]],indexes[0,1,char[F],num[1.00000000,"1"]]]]]]fPNf`Vf`6Du~~v ,^6K,^dyl9El0eq'iequal[char[F],over[char[L],indexes[1,0,times[num[4.00000000,"4"],char[pi],char[d]],num[2.00000000,"2"]]]]IN_3@6Lue1, fv 2",cd60"46.HH6 HH?   ` s $Magnitudes at Different Wavelengths [n-` u ,cMore on filters J` v 2" ]g w 0Instruments generally do not measure light over @ w f`the whole spectrum. nd` y um 0 x s[.Instead, they are sensitive to separate wave H x .5length bands:  ,1,` z 00 " { ha2  Magnitude usually measured for flux emitted  { 2over a specific band, for example B (blue) or 7@ { V (visible) HH6qHHlw H+6IX 6c6&^6&^V8„;]}y ]`YKla)(-& 6 dest a6 e a--  Efficiency.$^6e.$^f`thW;U}yNi^KHIlF'(.$46 pat4EE  for observing,LV6 a LV]]  light of adAM6 fodAMuu certainV |m[6 |m[  wavelength"ѝbLbM6 "ѝbLbM" " Wavelength  l2$6c2$62$2l?6l?ll69b]ƞ`6 69b]ƞ`69bD69bD l 0Vl?6Vl?lVlf&.6 icyf&.v v Dld6;(HH6 HHa  r observ  ` | Color Index ; } &The difference between the B (blue) X } *and V (visible) magnitudes of a star is u } ܛ,called the color index. It indicates the @ } temperature T of the star. bL h ~   `  bL M`  a` @T  = 7090 / [ (B- V ) + 0.71]   Perfect Blackbody 6<`  Y` >T  = 8540 / [ (B- V ) + 0.865]  Realistic star HH69bHHl 6H6 .?6.?Dl???6?H?l]wy6li| lw{єv[6k1K]wyw6";vTO1 eHnC6u) HC< anH~ߛ* /iI5Rnf!a WB|\jt,Al=X6 t s+dZ+dZ+dZ log (Intensity)bL;bLbM6 ;bLbM;; Wavelength  l B; 6 ; ;; )6-6-H0&SS6 H0&SSH@ H@ 20,000KAS6 AS 3000Kl6lll-6-w--66 -- 6l%ٟ6 A=ۥ6 %ٟ6 A=ۥ%ٟBKC%ٟBKC Hot Star: B-V < 0MqO|ۥ6 MqO|ۥM~M~  Cool Star:VMAۥ6 VMAۥVV  B-V > 06-l?6l?SSld6SHH6 HH ` Examples: Star clusters ;` A hint of things to come. X`  u 1Consider clusters of stars, all at pretty much @ ˣthe same distance from us. -` ,  Magnitudes give relative luminosities `  ` ۥ&For each star, plot magnitude (V) and ` ٟcolor index (B-V). V #` ۥWhy? @` !Why not! (Youll see why later.) ]`  z` Try two different clusters: A` (1) The Pleiades (M45)Zeilik Fig.13-9 6` *2) Globular cluster M3Zeilik Fig.13-10 `   1Later: Well see what these plots are telling us  @ about these stars. HH6HHl  d5Leftd6: Righttd7A ReferencesdDdS rsdomudr de du dveosdd a |7itf a Body. f b l lBulleted\ti. 0 f c CellBody. f d  CellHeading. f e o uFootnote. 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  6.Body. f z Body. f { 6.Body. HHf ~ Body. f  Body. llf Body. 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