Aa HH $ @d HHHHff@  d Footnote TableFootnote**.\t.\t/ - :;,.!? . `$`TOCHeading1Heading2 EquationVariables6dAQB5AV5Ch5Dj65Er6JFw6XGy ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year> +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm>HD -"<$monthnum>/<$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) 8+ (Sheet <$tblsheetnum> of <$tblsheetcount>) 9Heading & Page <$paratext> on page<$pagenum> :Pagepage<$pagenum>! ;See Heading & Page%See <$paratext> on page<$pagenum>.ein < Table All7Table<$paranumonly>, <$paratext>, on page<$pagenum> =Table Number & Page'Table<$paranumonly> on page<$pagenum>i  FFAV HHA5r JJ LL NN PP RuA$  m hK5y  soe>5|  s > 5  ui> 5 t"5 ! u/um5 " ta5 # x xn$d5 $ x se5 % so>/5 & x$ea5 ' wfi5 ( x 5 ) x 5 * xele5 + xte5 , x]5 - v xg5 . x xa5 / xa5 0 xod)5 1 {+t 5 2 x xu<$5 3 x xn5 4 z xnge5 5 x xx p5 6 | xu5 7 x xeen5 8 } xHe5 9 } xe<5 : } x e5 ; x xe5 < x xe5 = x xl$p5 > x xo<5 ? x xTa5 @ v x&'5 A { xay>5 B xp>i5 C x5 D xF5 E w5 F x5 G xr6 H xJ6 I x6 J 6 K y 6 L x6 M x6 N xR6 O $ h6 P s6 Q s6 R u6 S t6 T ~ u6 U t6" V 6# W x60 X 61 Y 66 Z x6; [  x6< \ x x6= ] x6> ^ x x6? _ x6@ ` xv6A a xx6D b 6E c x6F d x6K e xx6T f xx6c g z5zE5dq5xePFHe5dq6xeLHe5dq7x8C'5Hm3R q897 Hm3R H RH R FootnoteHr@ q98:7  Hr@ HzHz  Single LineH'q:9<7 ;;Footnote  5_;:s     HD q<:=7 6"HD HH  Double LineH q=<@76;>? Double Line] 5c>?=5e?>=6F6K H q@=B7AA Single Linee5hA@HZqB@C7q TableFootnoteEGxR qCB7 9EGxR EPwEPw  TableFootnoteod5pDRRHH5xE5 HH  Fe   HH5zFN5 HHlEE ZHH5{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 HHAR UTUT ` ' Hertzsprung-Russell Diagrams 9` $  V` , Ia.k.a.  HR Diagrams s` .  F` % Valuable tool for studying ` 1  stellar structure and evolution o` #  ` & + Plotting and studying stellar properties ` /  !` ( 5# Discovered by accident in 1910 >` ) )Plots of brightness vs spectral class [` 0  x` * Now we understand the physics UV` + %Plots of brightness vs temperature HH5RDHHlUQQ Pd5SUUHH5TS HHmVDVVU ` - Layout of an HR Diagram ;` 2  TX h 3 HeA nHH5USHHlRfTT H 5VS ud}WcTAstc h5WXV #c h#=C$! 5XWYV rer{Nƨ! ${_{_ Something like brightnessd b)fg-PLr5YXZV Pl9iơLr9z*9z* %e.g.  Apparent or Visual MagnitudeDfg{O5ZY[V VT!T2T2 or  log(luminosity)H G5$5[Z\V G5$ G9X G9X Something like temperatureĝbM5\[]V UĝbM  'e.g.  Spectral Class, B - V,bTfg5]\^V VbTfgbTbT or  log(temperature)raRff5^]_V 3HeRffcc z3ff5_^`V z3ffzDzD ifa$5`_aV ifa$ Each star you considerˑiQ$5a`bV ˑiQ$ is a point on this plot.ZQH5bacVPZZZQHZ~Q5cbVAar~Qd5dffloHH5ed \HHk Sohjfte UU` 4 bMHR Diagrams and Stellar Radii 8` 5  U 6  9Recall the luminosity  L  of a spherical blackbody r H 6 with radius  R : C ` 7 te r 8 ffA  An HR diagram plots  L  and  T , so we can use @ 8 ff%it to determine the value of  R. .` 9  K` : i(We have seen this before.) h` < fa ` = /What are lines of constant R on an HR diagram? a h ; D iF` >  c` ? n /where  K = log (4ps)= constant. HH5fdHHlUmee 2 L45gh52 L42 L4$'equal[(*q"Blue"q*)char[(*q"Blue"q*)L],times[(*q"Blue"q*)num[(*q"Blue"q*)4.00000000,"4"],char[(*q"Blue"q*)pi],power[(*q"Blue"q*)char[(*q"Blue"q*)R],num[(*q"Blue"q*)2.00000000,"2"]],char[(*q"Blue"q*)sigma],power[(*q"Blue"q*)char[(*q"Blue"q*)T],num[(*q"Blue"q*)4.00000000,"4"]]]]alNH5hjdu ggeCҐ`5ijHdiҐ`Ґ`D c'atop[(*j4j*)equal[log[(*q"Red"q*)char[(*q"Red"q*)L]],plus[char[K],times[num[2.00000000,"2"],log[char[(*q"Green"q*)R]]],times[num[4.00000000,"4"],log[(*q"Red"q*)char[(*q"Red"q*)T]]]]],times[char[o],char[r]],equal[string[(*q"Red"q*)"magnitude"],plus[char[K],times[num[2.00000000,"2"],log[char[(*q"Green"q*)R]]],times[num[4.00000000,"4"],id[(*q"Red"q*)string[(*q"Red"q*)"spectral class"]]]]]]I5jhdiieDd5k5mm4HH5lk qluHH|"qy0000,"4"wyme" ` Z haWorked Example: um;` [ 002Find the radius of a yellow (G0) supergiant star. X` \ ], [u` ] 00'Refer to figures and tables in Zeilik. j` b $Use reasonable, approximate values. ` ^  ` _ ҐF Fig. 13-11   -7 M V -4  so take M V   -5.5 ` ` )L9 Fig. 13-6 (and Table A4-3)    T   6000 K n` a [4B M bol =M V +BC with (Table A4-3) BC   - 0.06 D` c ng  M bol  = -5.56 [chl h d 00* Determine luminosity  L  fromF ` e "]Iwith M bol (SUN)  = +4.72   L =1.310 4  L SUN ` f ( Use L =4p R 2 s T 4 , h g 5Dand T  T SUN   (this problem only!)  to getG HH5mkHHlfpll d6nppiuofHH6on \HH ]nd tablepj ` @ reWhat Can We Learn from uesC ` E ^ HR Diagrams? f` A g. -` B Lots of great stuff! ` C  ` D g.  How stars live ` L 00 ` F a  How stars are born ` M A )1` I 6   How stars die N` N - 6k` G d  Details of star clusters L ` H How old is the cluster? ` J 72#+ How far away is the cluster? N HH6 pn HHlmuoo  u63qruu$'equal[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)Delta],char[(*q"Red"q*)m]],plus[(*q"Red"q*)times[(*q"Red"q*)num[(*q"Red"q*)5.00000000,"5"],log[(*q"Red"q*)char[(*q"Red"q*)d]]],minus[(*q"Red"q*)num[(*q"Red"q*)5.00000000,"5"]]]]NiH64rsqqtEn omd67suu BHH68ts CHHig.rrru  K Distances to Star Clusters C @ K using HR Diagrams f` O  ` T (See also Hoff Exercise #28.) ` U    V 0Assume all stars in the cluster are at the same @ V e distance from the observer. ` W   P 0The main sequence is a standard in terms of u1@ P 63)Absolute Magnitude M as a function of T. N` Q (More on this later.) k` R *) r S ,c+Plot an HR Diagram as Apparent Magnitude m Red S )51as a function of T. By how much do we shift the m S m[1vertical axis to line it up with the standard? @ S Call this shift  D m . n ` X   h Y <  Determine the distance  d  (in parsec) fromE 68HH6:usg.HHlptt ހ 6Hvw4_bހ $'equal[(*q"Black"q*)plus[(*q"Black"q*)times[(*q"Black"q*)indexes[(*q"Black"q*)0,1,char[(*q"Black"q*)M],times[(*q"Black"q*)char[(*q"Black"q*)b],char[(*q"Black"q*)o],char[(*q"Black"q*)l]]],id[(*q"Black"q*)times[(*q"Black"q*)char[(*q"Black"q*)S],char[(*q"Black"q*)U],char[(*q"Black"q*)N]]]],minus[(*q"Black"q*)indexes[(*q"Black"q*)0,1,char[(*q"Black"q*)M],times[(*q"Black"q*)char[(*q"Black"q*)b],char[(*q"Black"q*)o],char[(*q"Black"q*)l]]]]],times[(*q"Black"q*)num[(*q"Black"q*)2.50000000,"2.5"],log[(*q"Black"q*)id[(*q"Black"q*)fract[(*q"Black"q*)char[(*q"Black"q*)L],indexes[(*q"Black"q*)0,1,char[(*q"Black"q*)L],times[(*q"Black"q*)char[(*q"Black"q*)S],char[(*q"Black"q*)U],char[(*q"Black"q*)N]]]]]]]] NH6IwykYvvlFan 6Vxy*e s7&?$'equal[(*q"Red"q*)char[(*q"Red"q*)R],times[(*q"Red"q*)num[(*q"Red"q*)113.00000000,"113"],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)R],times[(*q"Red"q*)char[(*q"Red"q*)S],char[(*q"Red"q*)U],char[(*q"Red"q*)N]]]],cross[(*q"Red"q*)num[(*q"Red"q*)7.90000000,"7.9"],times[(*q"Red"q*)power[(*q"Red"q*)num[(*q"Red"q*)10.00000000,"10"],num[(*q"Red"q*)7.00000000,"7"]],string[(*q"Red"q*)" km"]]],times[(*q"Red"q*)num[(*q"Red"q*)0.52000000,"0.52"],string[(*q"Red"q*)" AU"]]]q"N|H6Wywk"acxxlGch ckd5]]Left(*d6*)Right0d7(* ReferenceadDck,idSarladd*)*qdkk"chdnds a 9f a Body. f b *q) cBulleted\to. 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