Aaÿ´_dÑÒÒÒÓÓÔÛHH $ @d HHHHÌÌÌÌÌÌff@€€€ÿö ø ø øÿÿÿ øÿÿÿÿdý ÿFootnote TableFootnote* à* à.\t.\t/ - Ð Ñ:;,.É!?`9`TOCHeading1Heading2 EquationVariables6ó5E<5Ö5Y6Q6m6 7f6¶8‰6s9u6v:6;…6À=Œ ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year>4zå +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm>Ä•Î -"<$monthnum>/<$daynum>/<$shortyear> .<$monthname> <$daynum>, <$year>ÒÓ /"<$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 <$tblsheetcount>)loo 9Heading & Page Ò<$paratext>Ó on page<$pagenum> :Pagepage<$pagenum>9 ;See Heading & Page%See Ò<$paratext>Ó on page<$pagenum>. < Table All7Table<$paranumonly>, Ò<$paratext>,Ó on page<$pagenum> =Table Number & Page'Table<$paranumonly> on page<$pagenum>‘ˆEFFA f‚HHAvƒJJ„LL…NN†PP‡RˆA$Xc5y soe>5| s > 5 ui> 5‚ tÎ"5…! u/um5ˆ" ta5‹# }n$d5$ x seÓ5Ž% ~ so>/5& ~ s$ea5' ~ sfi5”( € s5•) € s5–* } xele5—+ ~ yte5˜, ~ y]5›- € {g5œ. y ya5/ y {a5ž0 y yod)5£1 y+t 5¤2 † yu<$53 ynoo5®4 y ynge5¶5 ‡ yx p5°6 yu5·7 ‡ yeen5¸8 ‡ yHe5¹9 ‡ yeÒ<5º: y e5»; y5¼< ˆ ye5½= y yl$p5¾> y yo<5¿? yTa5À@ y&'5ÁA yay>5ÂB ˆ yp>5ÃC … yE5ÌD y yF5ÑE y y 5ÒF y y5åG y y5æH yJ5çI y y5èJ z y5ïK z z5õL z z5öM „ z6N ‰ zR6O z z$X6 P zs6Q zs6R „ zu6ÞS „ zt6T z wu6U š zt6ãV š š}6òW šx6X š z~6Y ž z~6Z z~6 [ z€6#\ z€6&] ‰ z}6>^ z z~6?_ z~6@` z z€6Aa „ zy6Bb z zy6Cc ‹ zy6Dd z z6Ee z z†6Kf ™ z6og y6ph ” ƒ‡6qi ‚ ‚6rj Š ‚‡6yk ‚ Š‡6~l z z‡6ªm — ˜6¦n | z6§o — zˆ6‰p ˜ zy6Ÿq — zy6«r ˜ ˜6Œs z6‘t z z6’u z zˆ6—v ™ z…6´w z zy6¹x z zy6µy zy6ºz “ zy6»{ z z6¼| z zy6Á} z zz6È~ z zz6É › zz5•X5ödqÁ5™z$PFÑ6dqÂ6™„tLHÒ6dqÃ7™x8CÓ6H´m3Rë™qÄ8š97 €H´m3Rë™H RH R ÍFootnoteHrý@™qÅ9š8:7@ zHrý@™Hz·¯Hz·¯ ÍSingle LineH'´ÿðqÆ:›9<7E z;;Footnote 5_;œ: ™ H¨ýD™qÈ<:=76¦H¨ýD™H°·¯H°·¯ ÍDouble LineH¹ÿðÔqÉ=›<@76Œ>?Double Lineu zÔ5c>ž?=yÔÔÔ5e?ž>=6¼Ô6Á zÔH…ÿúÔ qÌ@›=B7öAASingle Line$Ô5hAž@ÔÔHZ´ÿðqÎB›@C7qÃ TableFootnoteE¸ÏGxÅRªù™qÏC›B79E¸ÏGxÅRªù™E¸ÏPwE¸ÏPw Í TableFootnoteod5pD™RRÔ·¯HHÔˆ5xE›5ðHHÔˆEFe ÈHHÔˆ5zF›N5 HHÔˆlÿðEEDHHÔˆ5{G›6ÿðHHÔˆ@>?HDoe ÈHHÔˆ5}H›J6HHÔˆlÿðGG‚HíUVÔ 5~I›6HíUVÔ ÿÿ›AAJªªªªUU` ÉHíUVÔ 5€J›LH6HíUVÔ lÿðIIƒH$Ô 5K›6FtnH$Ô ÿÿE¸ÏL¸ÏªªªªUU` ÉtnoH$Ô 5ƒL›J6H$Ô lÿðKK„HíUVÔ 5„M›5HíUVÔ ÿÿNªªUUe! É›HíUVÔ 5†N›PF5HíUVÔ lÿðMM…H$Ô 5‡O›5H$Ô ÿÿPªªUUe" É›H$Ô 5‰P›N5H$Ô lÿðOO†HHÔˆ5ŠQ›DHHÔˆLAªªªªR`# Ï›Spectral Classification UVC`* Ï of Stars n`$ Ì ‘`% Ð,¥ Based on what stellar spectra Òlook likeÓ ÿÿ®`+ Ð Ë`& Ð,¥ Defined before the physics was understood è`, Ð› J`' Ð¥ Recall the physics so far: "`. ÎK ? ( Ñ-(1) General ÒshapeÓ (intensity as a function \( Ñ&of wavelength) similar to a blackbody y@( Ñ! spectrum. –`- Ñ5† ›³ ) Ñ+(2) Atomic Òabsorption linesÓ superimposed ÿðÐ@) Ñon top of the general shape. í`/ Î `0 Î ' 1 Ó3Today: ÔDefine spectral classes and relate them $D@1 Ô›=to the Ötemperature Ô and Öluminosity Ô of a star. HHÔˆ5ŒR›DŠ›HHÔˆlÿðUQQ‡d5¥S™UUÕicioHHÔˆ5¦T›SHHÔˆ!t stellaUk `2 ÏSpectral Classifications C`3 Ê pThe general idea of`4 Î, ›ƒ`6 Ô'Which absorption lines are fa `5 Ô. Present? ½`7 Ô(1 Missing? sÚ`8 Ôs Strongest? \÷`9 Ôof Weakest? )`: Ôkb 1`; ×!(Þ ÔThe Harvard Classification Scheme N`< ÖAtÒTemperatureÓ k`= Îed ðˆ`> Î n¥`D Îl pÂ`? Ô/$How broad are the absorption lines? ß`@ Ôy: Ôü`A ×la Þ ÔThe Morgan-Keenan Classes `B Ö Ö ÒLuminosityÓ HHÔˆ5¨U›SHHÔˆlÿðRXTT‡ld5ÍV™XXÖHHÔˆ5ÎW›VHHÔˆ@ÌÀ ˆ!YYX C ßHarvard Classification and `2C@C ßssTemperature f`E ÎTh eƒ F Ó1Question: Ô Is the temperature high enough so i HF Ô:that the following is possible for the gas atoms? Î5 e`G Îs t‚`H Ó+Answer: Ô Yes, if ×D ÔE ×» ÔkT Ÿ`I Î! ¼f``J ÐasHNote: Î ÕD ÎE Õ>> ÎkT ÕÞ Î Level E Ò2 Î not excited edÝf`f``K Î>C ÕD ÎE Õ<< ÎkT ÕÞ Î Excite levels higher than E Ò2 he þÌÀ`L Î ÌÀ M Ñ Ô1Therefore, absorption lines Òcome and goÓ as the e8ÌÀ@M ÑBtemperature is increased. HHÔˆ5ÐX›VHHÔˆlÿðUdWW‡HðÔ¨5ÕY›VZbW5Ñ¼uc5ØZ \YÑ¼ucÑ¼u4¼uþ¼u$c5Û[¡\]YCþ¼u$cssicaþ¼u‡þ¼u$Ñ¼uc5Ú\ Z[YEÑ¼ucFÑ¼u4¼u=¼u‡qC‹Žï5Ü]£[^Y=¼u‡qC‹Žï=¼u—Õk=¼u—Õk ÎE Ò1=¼uC‹Žï5Ý^£]_YGs =¼uC‹Žï=¼u"dZ=¼u"dZ ÎE Ò2üRâùGœ5à_£^`YI!üRâùGœüZüZ ÒÕYhŠHh—vŽï5á`£_bYÎnoYhŠHh—vŽïYhŠXdZYhŠXdZ ÕD ÎE=E Ò2 Õ- ÎE Ò1 d6a™dd×M ÔI›¦¹áBbM6b¡`Y tI›¦¹áBbMZZ ÙThermal Energy ÚÞHHÔˆ6c›aHHÔˆy‡ffd`N ÏMagnitude and A0 Stars ;`O Î X hP Ô*Recall our definition of ÒmagnitudeÓ:7 à Q Ô5Defines the êdifference Ô between one magnitude iý@Q Ô6and another, but what defines êzero magnitude Ô? `R Ñ¼u 7 S ÑC‹/The answer is historical, and may be different ŽïT@S Ñ)for different textbooks and astronomers: q`T Î ‹Ž U ÐdZ5¥ Old answer: ÎThe star Vega ( Õa Î-Lyra) has I«@U ÎGœ(apparent visual magnitude V Õº Î 0 È`V Î Îå X ÐŽï1¥ A Modern answer: ÎFind A0 stars (like Vega) @X ÎHwith U Ð- ÎB Õ= ÎB Ð- ÎV Õ=0 Îand correct for distance bM`W 1`Y áBRecall Zeilik Table 11-1: G`Z ¥ U = m( l»350 nm) ]`[ ¥ B = m( l»430 nm) s`\ ¥ V = m( l»550 nm) HHÔˆ6d›aOHHÔˆlÿðXicc‡Ý€ 6eŸfiern*8¬Rë«SI(Ý€ ä7'Óequal[(*q"Blue"q*)plus[(*q"Blue"q*)indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)m],num[(*q"Blue"q*)1.00000000,"1"]],minus[(*q"Blue"q*)indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)m],num[(*q"Blue"q*)2.00000000,"2"]]]],times[(*q"Blue"q*)num[(*q"Blue"q*)2.50000000,"2.5"],log[(*q"Blue"q*)id[(*q"Blue"q*)over[(*q"Blue"q*)indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)F],num[(*q"Blue"q*)2.00000000,"2"]],indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)F],num[(*q"Blue"q*)1.00000000,"1"]]]]]]]ikeN¨Èk6f›aÎ eec7 Îndd6'g™ iiØlleiHHÔˆ6(h›g HHÔˆˆ mB = mmi `] ÏGravitation in Stars 5;`^ Î X _ Ó0Question: Ô A star is a big ball of gas. Why u@_ ÔcdoesnÕt it just Òblow awayÓ? ’`` Îi r¯`a ó€'Answer: Ñ Gravity holds it together! lueÌ`b Îq* dé c â,10Question: ã If gravity holds it together, why "]@c ã*) doesnÕt the star just collapse? lu#`d Îue )@ e Ð,t4Answer: ÎThe internal pressure of the gas pushes [(]e ÎBl1itself outward and counteracts gravity. This is [z He Î[(.called åÒhydrostatic equilibriumÓ Î:6 HHÔˆ6*i›g000HHÔˆlÿðdlhh‡ d6Lj™llÙ HHÔˆ6Mk›jHHÔˆpˆul`g ÏHydrostatic Equilibrium ]P€`h inSpherically Symmetric Star |`i Î u§€ j Î i7Consider a small volume ÜdV=Adr Îinside the star itÓj Î [at radius Ür Î. The Ôdensity Î at radius Ür Î is ×r ê(r) Ü Îand þ€@j Î&the Ôpressure Î is êP(r) Î. * hk Îer 9: HHÔˆ6Ol›jlseHHÔˆlÿðiƒkk‡ThHÊÔ6Pm›gvn}h6rdnd«?~~6Vn¤ome-?üü«?~~Z«6‡†ÿy6Yo¤nqm$6 þò«6‡†ÿyZá6I¹™™6_p¡tsmá6I¹™™á dZá dZ ÙForceól6[qªorm6Móló‡l½cH?6\r¯qtm½cH?½¢cÆ¢]™™6`s¡pvmnSpÆ¢]™™Æ²dZÆ²dZ Ù of gravity-6-6^t°rpmvum-6-inse D-ZNzÈö6tu›j Ô]~~k9r ud›¦S¼h™™6bv¯swmjud›¦S¼h™™uuuu ÔOutward*u|›¦QOÜ™™6cw¯vxmu|›¦QOÜ™™uu ÔPressureM7›¦Cº\™™6dx°wymhM7›¦Cº\™™MHMH áInwardMO›¦QOÜ™™6ey°xzmMO›¦QOÜ™™M`M` áPressureMg›¦n©ø™™6fz°y{m6YMg›¦n©ø™™MxMx á(a little lessM›¦L§î™™6g{°z|msM›¦L§î™™MM áthan theM—›¦S¼h™™6h|°{}mM—›¦S¼h™™M¨M¨ áOutwardlM¯›¦YMÏ™™6i}°|mM¯›¦YMÏ™™MÀMÀ á Pressure)6u~ŸuÿÀ%‘ÿëÙèG´Þíüäu'áatop[(*j4j*)equal[string["Outward Force from Pressure"],string["Inward Force from Gravity"]],equal[cross[minus[id[(*i1iq"Green"q*)plus[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)P],id[(*q"Green"q*)plus[(*q"Green"q*)char[(*q"Green"q*)r],times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)r]]]]],minus[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)P],id[(*q"Green"q*)char[(*q"Green"q*)r]]]]]]],char[A]],times[char[G],over[times[char[M],id[char[r]],id[(*i1iq"Cyan"q*)times[(*q"Cyan"q*)char[(*q"Cyan"q*)d],char[(*q"Cyan"q*)m]]]],indexes[1,0,char[r],num[2.00000000,"2"]]]]],equal[minus[cross[id[(*i1iq"Green"q*)times[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)P]],times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)r]]],char[(*q"Green"q*)d],char[(*q"Green"q*)r]]],char[A]]],times[char[G],over[times[char[M],id[char[r]],id[(*i1iq"Cyan"q*)times[(*q"Cyan"q*)char[(*q"Cyan"q*)rho],id[(*q"Cyan"q*)char[(*q"Cyan"q*)r]],char[(*q"Cyan"q*)A],char[(*q"Cyan"q*)d],char[(*q"Cyan"q*)r]]]],indexes[1,0,char[r],num[2.00000000,"2"]]]]],equal[(*q"Red"q*)minus[(*q"Red"q*)over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)d],char[(*q"Red"q*)P]],times[(*q"Red"q*)char[(*q"Red"q*)d],char[(*q"Red"q*)r]]]],times[(*q"Red"q*)char[(*q"Red"q*)G],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)M],id[(*q"Red"q*)char[(*q"Red"q*)r]],char[(*q"Red"q*)rho],id[(*q"Red"q*)char[(*q"Red"q*)r]]],indexes[(*q"Red"q*)1,0,char[(*q"Red"q*)r],num[(*q"Red"q*)2.00000000,"2"]]]]]])NpÈH6w›uj("C€€k:har]6x€Ÿi(*(¢Ìô‹®¼3jä$Gr'¾equal[(*q"Blue"q*)string[(*q"Blue"q*)"Mass inside r"],times[(*q"Blue"q*)char[(*q"Blue"q*)M],id[(*q"Blue"q*)char[(*q"Blue"q*)r]]],int[(*i2iq"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)4.00000000,"4"],char[(*q"Blue"q*)pi],indexes[(*q"Blue"q*)1,0,char[(*q"Blue"q*)r],num[(*q"Blue"q*)2.00000000,"2"]],char[(*q"Blue"q*)rho],id[(*q"Blue"q*)char[(*q"Blue"q*)r]],diff[(*q"Blue"q*)char[(*q"Blue"q*)r]]],num[(*q"Blue"q*)0.00000000,"0"],char[(*q"Blue"q*)r]]]d6z™")cƒƒÚ"R"qHHÔˆ6{‚›h[(HHÔˆTÌ°(*)G],over……ƒme`f ßq*Application: Central Pressure ;`l Î"q hX n Îha4Rough approximation for a star of Ùradius R Î q*u@n Î)2and Ùmass M Î. Take ’`o Î›2¥ Integration in one giant step: Ùdr Ú= ÙR ¯f``q Î/¥ Zero pressure at r=R: ÙdP Ú=- ÙP c Ðf``m ÎGr!¥ Density is the same everywhere BóÌ°fP`p Õe *Þ Úr Ù(r)=M/(4 Úp ÙR ÿ3 Ù/3) Ì°`r Î(* l-Ì° hs Ô2iCentral pressure is then; [õÌ°`t Î00 "Ì° u ýq*8This is very high! Ù Î Later we will see that this lu/Ì°u Î"]3large pressure gives rise to the very high central q*)LÌ°@u Îq*.temperature which generates the power source. HHÔˆ6}ƒ›HHÔˆlÿðlˆ‚‚‡H6Ž„Ÿ…(J‚2àüaÀÔäW'8equal[(*q"Blue"q*)indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)P],char[(*q"Blue"q*)c]],times[(*q"Blue"q*)cross[(*q"Blue"q*)char[(*q"Blue"q*)R],char[(*q"Blue"q*)G]],over[(*q"Blue"q*)times[(*q"Blue"q*)char[(*q"Blue"q*)M],over[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)3.00000000,"3"],char[(*q"Blue"q*)M]],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*)3.00000000,"3"]]]]],power[(*q"Blue"q*)char[(*q"Blue"q*)R],num[(*q"Blue"q*)2.00000000,"2"]]]],over[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)3.00000000,"3"],char[(*q"Blue"q*)G],power[(*q"Blue"q*)char[(*q"Blue"q*)M],num[(*q"Blue"q*)2.00000000,"2"]]],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*)4.00000000,"4"]]]]]N}Ì°È«6…›0„„‚;d6˜†™ŽŸˆˆÛàüHHÔˆ6™‡›†[q"HHÔˆofPue)P],char‰Œˆ]]`v ß*qApplication: Line Broadening a;`w Îov (X y Ô*q9What is the Òacceleration due to gravityÓ ( êg Ô) on eu Hy Ô*)the surface of a star?8 "q*Ú`x Îe" n÷ z Î00=Consider stars with ÒnormalÓ masses (M Õ» ÎM ÒSUN Î) "z Î"q3but with very different radii R. Large stars have "q*1@z Î*)"much smaller values of Üg Î. N`{ Înu *k h| Î00-Therefore, the surface pressure gradient= rÐ } Îm[0is much smaller for large stars. This leads to m[í} Î00-narrow absorption lines. Also remember that cfPfP@} ã,n7large stars are bright Î since L Õµ ÎR ï2 Î. 6-fP`~ Î JfP Ú3Þ ÙLarge (and therefore bright) stars have very ŽgfP@ Ùnarrow absorption lines.HHÔˆ6›ˆ›†HHÔˆlÿðƒ‡‡‡NÅÈH6·‰›Œ† oaŠŠ‡8ov (6¸ŠŸ‰tn ¶iH¬Ù[-p=þä$'ºequal[(*q"Blue"q*)char[(*q"Blue"q*)g],over[(*q"Blue"q*)times[(*q"Blue"q*)char[(*q"Blue"q*)G],char[(*q"Blue"q*)M]],power[(*q"Blue"q*)char[(*q"Blue"q*)R],num[(*q"Blue"q*)2.00000000,"2"]]]]6¾‹ŸŒz†Í¥ºd¶=þä$'¤equal[over[times[char[d],char[P]],times[char[d],char[r]]],times[char[G],over[times[char[M],char[rho]],power[char[R],num[2.00000000,"2"]]]],times[char[g],char[rho]]]N»ÈH6¿Œ›‰†mbe‹‹‡=,nÑáÜardÑ5Left2dÒ6RightdÓ7(a ReferencebdÔÕDfPdÕÔÖSn dÖÕ×V›d×ÖØadØ×ÙgdÙØÚj6·ŒdÚÙÛŠdÛÚ† a ¡ v6æf™ aýBody. æf™ býz ¶Bulleted¥\t. æf™ cýCellBody. æf™ dýCellHeading. æf™ e ýV ›Footnote. 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Å Ë(€ Åý Emphasis€ Æý EquationVariables)ÕÞ Çý úÚÜ Èý úÚÜ É$úÚÜ Êý$)ÕÞ Ìý ž[ Íý úÚÜ Îý$)ÕÞ Ïÿ )ÕÞ Ðý úÚÜ Ñ úÚÜ Òý)ÕÞ Ó úÚÜ Ô wÇ Õýtuž ÖÿwÇ × úÚÜ ÙÿwÇ Úÿ)ÕÞ Ûý tuž ÜýúÚÜ Ý úÚÜ Þ $)ÕÞ ß úÚÜ àÿúÚÜ á )ÕÞ â úÚÜ ã uo«& åÿ tuž ê úÚÜ ì úÚÜ ïý)ÕÞ ó )ÕÞ ýÿ úÚÜ ÿÿúÚÜ ÿtuž úÚÜ ýwÇ ý™¹¢™šý›ý€œýž€ýžýŸýÜ ýZ¡ÿZ£ýZÜ€¤ýZªÿZ¯ ZÞ° Z ’€ ýThin ŽýMedium€ ýDouble ýThick@ ‘ý Very Thin ñ ñ ò oýýýH p q rH p q rH p q rH p q rH p q rFormat A ó Ž Ž oýýýH p q rH p q rH p q rH p q rH p q rFormat BUeVUÿCommentý d ýBlackT!þWhiteddAÿReddd Greendd ýBlued Cyand ÜMagentad Yellow¹ÉºTimes-Roman Times-BoldTimes-ItalicHelvetica-BoldSymbolSymbolTimes-BoldItalicTimesc HelveticaSymbol RegularRegular BoldRegularItalic‘FÂB:O%}†T´7]ýLðò~ŒâÉn1´¦:W‰FÜoÚŠ/ÄŠ`É£l<ŽÏ}ðªW"n3ÛÊÀ†;¥×ûñtÖlj™TåZÅU´3ûñ^z±Ñð•hX%"Ê~ìébµà–ÑøÒq%Z‘úâQµ³eRqes9‹LÁfg»ßtÏã2Ÿd!Ð