Aad HH $ @d HHHHff@  d Footnote TableFootnote**.\t.\t/ - :;,.!?% ` `TOCHeading1Heading2  EquationVariables65E; 685l56S57U58W59\5:^6x<6=6> ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year>,Z +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm>\F -"<$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>. < Table All7Table<$paranumonly>, <$paratext>, on page<$pagenum> =Table Number & Pagei'Table<$paranumonly> on page<$pagenum> l FFAW HHAx JJ LL NN PP RA$  e5y  soe>5|  s > 5  ui> 5 tF"5 ! uum5 " ta5 # y wn$d5 $ v we5 % w wo>/5 & x w$ea5 ' w wfi5 ( w w 5 ) w w 5 * y wele5 + w wte5 , z w]5 - | wg5 . w wa5 / } wa5 0 w wod)5 1 w w+t 5 2 w wu<$5 3 w wn5 4 w wnge5 5 y wx p5 6 u5 7  seen5 8  He5 9  e<5 :   e5 ; 5 <  e5 = l$p5 > o<5 ? Ta5 @ &'5 A ay>5 B p>5 C l5 D F5 E 5 F  5 G 5 H J5 I 5 J 5 K 5 L 5 M  5 N R5 O $ 5 P s5 Q s6 R u6 S t6 T u65 U t66 V  y6 W v6 X w6[ Y  x6 Z w6\ [ w6] \ w6` ] y6a ^ w6b _ { z6g ` { |6l a w6m b  }6n c  w6o d  w6p e w6y f  w6z g w6 h y6 i 6 j 6 k 6 l  6 m 6 n 6 o 6 p  6 q 6 r 6 s  6 t 6 u  6 v 6 w  6 x 6 y  6 z 6 { 6 | 6 } 6 ~ 6  6 6 5X5dq5sPF6dq6tLH6 dq7x8C6\Hm3R q897 yHm3R H RH R FootnoteHr@ q98:7 m Hr@ HzHz  Single LineH'q:9<7z ;;Footnote  5_;:     HD q<:=7 6HD HH  Double LineH q=<@76>? Double Linew 5c>?=5e?>=66 H q@=B7AA Single Lines5hA@HZqB@C7q TableFootnoteEGxR qCB7 9EGxR EPwEPw  TableFootnoteod5pDRRHH5xE5 HH zFe   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 HHZA WSWR  ` $ .Absolute Magnitude  (upper case  M ) 9` %  V & .Absolute magnitude is what apparent magnitude s@ & *would be  if the star were 10 pc away. ` ' UU  h ( +For a specific star, we therefore have6 k` )   h * K+If we measure distance in parsec, then7 ` +    h , >Or, if we measure parallax  p  in arc seconds, then8 HH5RDHHl[QQ C@5SUD !TTQ6r85TSr8r8@\'equal[(*l-720896ll-720896ll-720896lr-65536r*)plus[char[m],minus[char[M]]],times[num[2.50000000,"2.5"],log[id[(*i1i*)over[over[char[L],times[num[4.00000000,"4"],char[pi],power[id[times[num[10.00000000,"10"],char[p],char[c]]],num[2.00000000,"2"]]]],over[char[L],times[num[4.00000000,"4"],char[pi],power[char[d],num[2.00000000,"2"]]]]]]]],times[num[5.00000000,"5"],log[id[(*i1i*)over[char[d],times[num[10.00000000,"10"],char[p],char[c]]]]]]]NH5USWD(VVQ7thef 95VU 9 9$rs'equal[(*q"Green"q*)char[(*q"Green"q*)M],plus[(*q"Green"q*)char[(*q"Green"q*)m],num[(*q"Green"q*)5.00000000,"5"],minus[(*q"Green"q*)times[(*q"Green"q*)num[(*q"Green"q*)5.00000000,"5"],log[(*q"Green"q*)char[(*q"Green"q*)d]]]]]]NZH5WUDXXQ85XW$'equal[(*q"Magenta"q*)char[(*q"Magenta"q*)M],plus[(*q"Magenta"q*)char[(*q"Magenta"q*)m],num[(*q"Magenta"q*)5.00000000,"5"],times[(*q"Magenta"q*)num[(*q"Magenta"q*)5.00000000,"5"],log[(*q"Magenta"q*)char[(*q"Magenta"q*)pi]]]]]]]d5Ym[[[ow[cHH5ZY [00HH1i[10.0000\^[r[ ` - Bolometric Magnitude ;` .  VX / ef4If you  can  determine the magnitude over the u / 94entire  wavelength spectrum, then you can figure n"@ / en#out the true luminosity of a star. n"q` 0 mi [ h 1 [(-We refer magnitudes to that for the sun:9 g_` 2 [( G| f` 3 :where we know that M bol (Sun)  = +4.72  from f`@ 3 other measurements. f`` 4  f` f` 5 >We can estimate M bol ( H ) from the visual magni H 5 en/tude using a bolometric correction (BC):: .00] # (*.Zeilik Appendix 4 gives values for bolometric z@ # )c corrections. "HH5[YHHlRbZZ Nv5\^Y]]Z90.5]\-i}R-SI(='qequal[plus[function[indexes[0,1,char[M],times[char[b],char[o],char[l]]],times[char[S],char[u],char[n]]],minus[function[indexes[0,1,char[M],times[char[b],char[o],char[l]]],char[(*f"ZapfDingbats"fV"Regular"V*)H]]]],times[num[2.50000000,"2.5"],log[id[over[indexes[0,1,char[L],char[(*f"ZapfDingbats"fV"Regular"V*)H]],indexes[0,1,char[L],times[char[S],char[u],char[n]]]]]]]] NHH5^\Y__Z:ts 5_^WeAo}#$$ t'*equal[(*q"Green"q*)indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)M],times[(*q"Green"q*)char[(*q"Green"q*)b],char[(*q"Green"q*)o],char[(*q"Green"q*)l]]],plus[(*q"Green"q*)indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)M],char[(*q"Green"q*)V]],times[(*q"Green"q*)char[(*q"Green"q*)B],char[(*q"Green"q*)C]]]]d5`bb0.HH5a` -HHqequal[bin ` 6 arBinary Star Systems ti;` : ], rX ; on#Important source of information on ar[u@ ; (*the properties of stars. *` 7 .5 0` 8 ov#Q:  Arent these special cases? ing` 9 )H4A:  No!  About half of all stars are binary. ]]` <  ` = Newtonian Mechanics Z#` >   Keplers Laws @` ? We  Measure stellar ]` @ t Masses [z` A es Radii "q` B en )` C "q.How do they depend on luminosity (magnitude)? ` D ,p [HH5b`("GHHl[eaa d5ceeHH5dc HHe ` E Kinds of Binary Star Systems r;` F em iX L ],0Note that a star system can belong to more than rmu@ L one of these categories. s` M  ` G Visual Binary Stars re N ca-The stars can be resolved into two images in b@ N ar(a telescope.  This may not be easy! ` H Ne n#` I Spectroscopic Binary Stars rs@ O -Study spectral lines and and identify binary @]@ O nature by the Doppler shift. z` J ) ` K HoEclipsing Binary Stars osi P /Angle of inclination very shallow, so one star @ P passes in front of the other. ` Q a HH5ecHHlbhdd dd6fhhHH6gf is HHmp rXllhNo ` R beMass Determination uC ` W onFrom the Orbit Parameters f` X  ` S ViUnique solution with ` T Th Visual binaries ` U s " Near enough to measure distance  h V ot5 y` Y H e f`` [ I_  M 1 / M 2   =  a 2 / a 1   Definition of C.of M. ina̰fP f`` \ OO Force  =  GM 1 M 2 / a 2 Newton's Law of Gravity Ec`fP f`` ] rsj Acceleration  = M 1 V 1 2 / a 1  = M 2 V 2 2 / a 2   ` ^  DffP f`` _ W+   (M 1 + M 2 ) P 2 = a 3    where P=period ep` ` Keplers Third Law HH6hfHHlekgg Hd6ikkHH6ji HH*it ` X k  ` Z wiMass-Luminosity Relation ;` d  X` a arDetermined from Binary Stars u` b  y h c HApproximate form: < ` f M 3` g More accurate form: P f` h i .o- If M   0.43 M Sun  then,= f`` j  f` f` h k n'- If M    0.43 M Sun  then,> eHH6ki HHlhjj H*67lf m}g5P$ ZZ6:mnlp$ ZZ$$$6;nmol6$$$$$6<onpl$$$H6$6=porlk6$H66l6Bqstlitll~Ql$6?rpslLinQl$;QlQl$6Asrqlytal$` bl;uZ6Ctqul;uZ;uuu~6DutvlMu~Sun uuu6Evuwl Mu unuuQu6Fwvxl QuluQu6Gxwyl67 Q6HyxzlnQ$QZq6Izy{l Zqjkjk M 1_q6J{z|l _q_"k_"k M 2Vfg6K|{~l Vfgff ?+$6Z}l ?+$OdZOdZ Center of MassqllqA6M~|l qllqAql|kql|k a 1%llqA6O~l %llqA%l|k%l|k a 2X~ 6Q}l X~ XdZXdZ ad6q MHH6r HHfP ` e 6GSomething New: C ` h How long does a star live? f` l  z` m Fuel proportional to mass M ` n + Burn Rate proportional to luminosity L ` o M" L  M 3  (roughly) ` p  ` q (+  Lifetime= Fuel / Burn Rate fPfP` r   1/M 2 7fP` s  \fP t 6ZLow mass stars live much fP @ t dZlonger than high mass stars! lfP` u 6M fP` v More later! fP` w  HH6t HHlk 6v2|Bg$2'approx[over[char[L],indexes[0,1,char[L],times[char[S],char[u],char[n]]]],power[id[over[char[M],indexes[0,1,char[M],times[char[S],char[u],char[n]]]]],num[3.00000000,"3"]]]Ng6wiw:j<Holo6 z||~B$l 'equal[over[char[L],indexes[0,1,char[L],times[char[S],char[u],char[n]]]],power[id[over[char[M],indexes[0,1,char[M],times[char[S],char[u],char[n]]]]],num[4.00000000,"4.0"]]]Nf`H6ifPj=/M6t6ZfP|^B$ t'equal[over[char[L],indexes[0,1,char[L],times[char[S],char[u],char[n]]]],times[num[0.23000000,"0.23"],power[id[over[char[M],indexes[0,1,char[M],times[char[S],char[u],char[n]]]]],num[2.30000000,"2.3"]]]]N*H6i6vj>2|d6poxch[LHH6 [erHHhau],char[00 ` x A Brief Recap ;` y  X z Ho+We have seen how to measure various properu z 0ties of stars, using telescopes and the physics L]@ z L]of binary systems. ],c` { [o [ | 1,,We have seen some interesting relationships 4.@ | !between these various properties ` } +  Brightness depends strongly on color 6#` ~ t*  Brightness depends strongly on mass @`  er&  Radius depends strongly on mass ]` ]] iz ,".Now the job is to try to understand why these @ r[relationships happen. 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