Aad HH $ @d HHHHff@  d Footnote TableFootnote**.\t.\t/ - :;,.!?#& ` 0 ` TOCHeading1Heading2  0 EquationVariables8  7 9 = ;% ?' A, C. E0 GEC I|OY K@t UH8n C ) 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 <$marker2> 7 (Continued)ot 8+ (Sheet <$tblsheetnum> of <$tblsheetcount>);! 9Heading & Page <$paratext> on page<$pagenum> :Pagepage<$pagenum>2 ;See Heading & Page%See <$paratext> on page<$pagenum>. < Table All7Table<$paranumonly>, <$paratext>, on page<$pagenum> =Table Number & Page'Table<$paranumonly> on page<$pagenum> ? WW YY [[AC ]] __ aaA| qA$  5  woe>5  v > 5|  ti> 5 u"5 ! v/um5 " ua7 # n$d5y $ te5 % xo>/5 & y$ea5 ' yfi5 ( y 5 ) z 5 * zele5 + zte5 , z]5 - {g5 . {a5 / {a5 0 |od)5 1 }+t 5 2 ~u<$5 3 n!6 4 nge6 5 x p6 6 u6 7 een6 8 He6 9 e<6 :  e6 ; 6 < e6 = l$p6 > o<6" ? Ta6 @ &'6 A ay>6 B p>5 C z?5 D W5 E z5 F Y5 G 5 H C5 I ]5 J 5 K _61 L 5 M |5 N q5 O $ 60 P w5 Q v5 R t5 S u5 T v6# U u5 V 5 W t5 X x6= Y y6M Z y6N [ y6V \ z6_ ] z6` ^ z6a _ z6s ` {6x a {6O b {6 c |6 d }6 e ~6 f 6 g 6 h 6 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 z6 w 6 x z6 y 6 z 6 { 6 | 6 } 6 ~ 6  6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 6 7 7 7 7 7 7 7L 7! 7" 7# 7; 7> 7& 7' 7. 7/ 70 71 72 77 78 7 7M 7O 7 8 s 8 s8 8 8 89 8: 8; 8* 8+ 84 8< 8= 8C 8N 8_ 8` 8a 8X 8b 8c 8h 8 8 57dq5W[7dq6]a7LNN579r]88s 7;Tx587TxTx' atop[(*j4j*)equal[(*q"Magenta"q*)string[(*q"Magenta"q*)"Outward Force (P) "],string[(*q"Magenta"q*)"Inward Force (G) "]],equal[(*x1507328xr327680r*)cross[id[(*i1iq"Green"q*)plus[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)P],id[(*q"Green"q*)char[(*q"Green"q*)r]]],minus[(*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]]]]]]]],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"]]]]]])tiNal597rqya::s r[,n!CB1D5:9[i1!CB1D!CB1DHee'equal[(*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]]]N`]6;;?{*ha<<| ed*) jc<6:<;r*q jc< jc<2d"'eequal[over[num[1.00000000,"1"],char[mu]],times[over[indexes[0,1,char[m],char[H]],char[rho]],char[n]]])N}̰6 =ues[0>>v  )rMr-TY6 >=Mr-TYMr-TYV'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"]]]]],cNN6g?;A{ @@| >JvD23>6f@? IRD5[>JvD23>*'^equal[(*q"Blue"q*)char[(*q"Blue"q*)n],plus[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)2.00000000,"2"],indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)n],char[(*q"Blue"q*)H]]],times[(*q"Blue"q*)num[(*q"Blue"q*)3.00000000,"3"],indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)n],times[(*q"Blue"q*)char[(*q"Blue"q*)H],char[(*q"Blue"q*)e]]]],times[(*q"Blue"q*)id[(*q"Blue"q*)plus[(*q"Blue"q*)num[(*q"Blue"q*)1.00000000,"1"],over[(*q"Blue"q*)char[(*q"Blue"q*)A],num[(*q"Blue"q*)2.00000000,"2"]]]],indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)n],times[(*q"Blue"q*)char[(*q"Blue"q*)m],char[(*q"Blue"q*)e],char[(*q"Blue"q*)t]]]]]]N.Z6uA?{r*q BB| *)00Ý x6e6vBAe*)Ý x6eÝ x6e-[('approx[(*q"Red"q*)over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],char[(*q"Red"q*)mu]],plus[(*q"Red"q*)times[(*q"Red"q*)num[(*q"Red"q*)2.00000000,"2"],char[(*q"Red"q*)X]],times[(*q"Red"q*)over[(*q"Red"q*)num[(*q"Red"q*)3.00000000,"3"],num[(*q"Red"q*)4.00000000,"4"]],char[(*q"Red"q*)Y]],times[(*q"Red"q*)over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],num[(*q"Red"q*)2.00000000,"2"]],char[(*q"Red"q*)Z]]]]q*)NO6CE~000DD )0,cev52A6DC*Blev52Aev52A+Bl'equal[(*q"Red"q*)indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)P],char[(*q"Red"q*)c]],times[(*q"Red"q*)char[(*q"Red"q*)P],id[(*q"Red"q*)num[(*q"Red"q*)0.00000000,"0"]]],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)rho],char[(*q"Red"q*)k],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)T],char[(*q"Red"q*)c]]],times[(*q"Red"q*)char[(*q"Red"q*)mu],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)m],char[(*q"Red"q*)H]]]]]xN?L6ECG~x*qFF m[q" 96FE,us 9 9)*)'(equal[(*q"Red"q*)indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)P],char[(*q"Red"q*)c]],times[(*q"Red"q*)cross[(*q"Red"q*)char[(*q"Red"q*)R],char[(*q"Red"q*)G]],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)M],char[(*q"Red"q*)rho]],power[(*q"Red"q*)char[(*q"Red"q*)R],num[(*q"Red"q*)2.00000000,"2"]]]]]ENQ6GE~0,cHH k ()<6HG2k ()<k ()<1[('"equal[(*q"Red"q*)indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)T],char[(*q"Red"q*)c]],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)G],char[(*q"Red"q*)M],char[(*q"Red"q*)mu],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)m],char[(*q"Red"q*)H]]],times[(*q"Red"q*)char[(*q"Red"q*)k],char[(*q"Red"q*)R]]]]NL6IqedJJ ]]x5?e6JIx*q5?e5?e1'equal[(*q"Red"q*)times[(*q"Red"q*)string[(*q"Red"q*)"Energy Release "],char[(*q"Red"q*)U]],approx[(*q"Red"q*)int[(*i1iq"Red"q*)times[(*q"Red"q*)over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)G],char[(*q"Red"q*)M]],char[(*q"Red"q*)r]],diff[(*q"Red"q*)char[(*q"Red"q*)M]]],num[(*q"Red"q*)0.00000000,"0"],char[(*q"Red"q*)M]],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)G],indexes[(*q"Red"q*)1,0,char[(*q"Red"q*)M],num[(*q"Red"q*)2.00000000,"2"]]],char[(*q"Red"q*)R]]]]Ha6K)LT [(- --6LMKMNh[(- --6@6MLNKLN R"q6@6Vʫ6Vʫ 1 H- -P6NMOK[q"- -PLM --6ONPKPQ),c --@6POQKOQ q@VʫVʫ 1 H -P6QPRK5 -POP zë`6RQSK s*q zë`00   510 -15  mU]];& H6SRTK("R;& Hd"qti;& & c& H6TSK("Rc& Hq"R"q& c& HD7YU("R tis[H$ V5 eq*H$  ],0,"2"]]]Wq*UUe   H$ 5WY5H$ l VV  HUa X5 LNHUa   H-YUUe q" HUa 5YW[5PHUa l XX OQHHZ5 ʫHH [e $  HH5z[Y5HHl ZZ H$ \6 qtiH$   K] UU` !  H$ 5]_6("RH$ l \\ HUa ^6 HUa  ]]_ UU` "  YHUa 5_]a6 HUa l ^^ HH`6 HH a e   [HH5}a_6 HHl `` dqbcnHm3R qcdb Hm3R H RH R FootnoteHr@ qdceb ZHr@ HzHz  Single Line qH'qedgbff Footnote  5_fe  "R   HD qgehb HD HH  Double LineUaH qhgkbij Double Line5cijh 5ejih H qkhmbll Single Line [5hlk HZqmknbq TableFootnoteEGxR qnmb dEGxR EPwEPw  TableFootnoteod5poqqHHpo HHb  q  UU`  Stellar Structure and Evolution "R8` %  U` & q, What is the physics of the Main Sequence? r` ' le How are stars born? ` (  How do they grow old? ` ) in ` *  c` + .Today:  The Center of a Main Sequence Star ` , 5e  ` - +  Hydrostatic equilibrium and pressure =` . (  Equation of state and temperature Z` / [  Nuclear reactions HH5qoHHl tpp d5rxttHHsr TaHH979t ` 0 Hydrostatic Equilibrium P` 1 Spherically Symmetric Star |` 2   3 Ϊ7Consider a small volume  dV=Adr  inside the star  3 [at radius  r . The  density  at radius  r  is  r (r)   and re  H 3 .the  pressure  is  P(r) .   HH5trcHHl qwss ar d5u wwm d HHvu aonHHq̰ur  ==won ` 4 5Application: Central Pressure ;` 5  X 6 t4Rough approximation for a star of  radius R  u@ 6 and  mass M . Take ` 7 2 Integration in one giant step:  dr = R  f`` 8 / Zero pressure at r=R:  dP =- P c rosf`` 9 ! Density is the same everywhere y̰fP` : *  r (r)=M/(4 p R 3 /3) ̰` ; um -̰ h < e Central pressure is then  ̰` = $Example:    Consider the Sun. r̰` > r)M  = 1.9910 30  kg 3/̰` ? suR  = 6.9610 8   m HL̰` @ 51  P c  = 2.710 14  N/m 2 ḭ` A + = 2.710 9  atmospheres  (!) HH5wu d HHl tzvv d5x=zz 4HHyx HHmation fz  ` B The Equation of State ;` C  X` D at.Relates properties of some state of matter. u` E ro e` F dP/For normal stars, use the  Ideal Gas Law: ty ` G he% PV=n m RT (chemistry) )` H 3or PV=NkT (physics) ` I e  e` J th)In stars, it is most convenient to write #` K un  P=nkT @` L 1. 1]` M ̰-where P=P(r)  is pressure at radius r mz` N @+ T=T(r)  is temperature at radius r N/` O F n=n(r)= r (r)/ m (r)m H  is  number density ` P  ` Q 'with  r (r)= mass density vv` R 2and  m (r)= mean molecular weight  ` S  (` T 6 Stellar model    Specify some  m (r) E` U f' Atoms are almost completely ionized! b` V St: m(0) [2 X +(3/4) Y +(1/2)Z] -1 1/2 ` W st$ Radiation pressure not included eHH5zxsthHHl w}yy  d6{r }}HH|{ a, HH]to  un ;A}  h X 1Estimating the Mean Molecular Weigh t   a` Y .Define mass fractions in terms of m H : ` Z TX  = ( n H V ) m H / M  =  n H m H /r ` [ ^Y  = ( n He V )(4 m H ) / M  =  4n He m H /r ` \ mo_Z  = ( n met V )(A m H ) / M =  An met m H /r so` ] *where A is the atomic number for metals ` ^  8 _ 0)1If all we have are hydrogen, helium, and metals, U H _ st2and  the electrons that go with them, then   ` ,where we assume that an element with atomic  H ` }:number  A  has  A/2  electrons. Therefore  U` a "where we are taking A >> 1. HH6}{toHHl z|| d6P~ ig YHH~  HH}  V )CG  ` b  Application: C ` c Y Central Temperature Vf` d   h e 49Use the equation of state at the center of the star:  n h f A;As before, use hydrostatic equilibrium for P c :  soY h g 4Combine to find central temperature T c :  ` h "Example:   Consider the Sun ` i m*All numerical values in Zeilik Appendix 7 ` j th e` k (T c   = 1.1510 7   K en;` l kT c  1000  eV X` m   eu` n e.Important clue for the power source in stars! HH6S~HHl } zd6HH HH|IIG ` o The Power Source of Stars ;` p Y  eX` q VUse the Sun as an example: u` r thBt SUN    t EARTH    4.510 9   years ` s se d` t um1What has kept it burning steadily for so long? n` u te r` v Gravitational Contraction? mpl` w r  # x i2Maybe the sun falls in on itself and the energy @ H x e!released comes out as light.  ` y   ` z l,For the Sun find U   3.810 41  J m` {  n` | ue-Energy output L  = 3.910 26   W ` }  :` ~ V  Max t SUN   =  U/L  = 10 15  sec  = 310 7  years W`   t` 5Gravitational contraction is  not  the answer! HH6HHl  e od6 qame:HH  HHt10KK ` kNuclear Reactions ;` n X` r7Generally take the form A+B    C+D +  Energy u`  ` e 8This  energy  is the stars source of luminosity. x` es t` NNuclear notation:  A (Z) e.g.  1 H,  3 He,  12 C, ... `  {` +Two kinds of reactions important in stars: #` < Hadronice.g.  1 H+ 2 H  3 He+ g @` 1K Weake.g.  1 H+ 1 H  2 H+ e + + n e ]` nt tz t3High temperatures  are necessary so that nuclei  H +can touch and undergo reactions:   e o`  2` PU Coulomb   =  q 2 /4pe 0 d  = 310 5   eV  O` '=  k  ( 310 9  K ) l` )Rates are very sensitive to temperature! HH6ay HHl  ` d6g  8noHH HH:  H,   1 ` Hydrogen Burning ;` s  rX` i.Primary energy source in Main Sequence stars. u`   ` %Conversion of hydrogen into helium ` Ha1 H+ 1 H+ 1 H+ 1 H    4 He+2e + +2 n e + energy e` ce$(This does not happen in one step!) ` an n` )Use masses to calculate energy released: #` loE  = (D m ) c 2 0@` D m =  mass defect ]` 3= 4 m(Hydrogen atom)  -  m(Helium atom) re z` te= 4.710 -29   kg `  a` 1 E = 4.310 -12  J  for each reaction ` 6 ` !Is this enough to power the sun?  ` ܚ*Assume 10% of all mass is usable hydrogen (` G  Total available energy U = ( 0.1  M/4m H )  E rniE` W  Max t SUN   =  U/L  = 310 17  sec  = 10 10  years ` b`  ` of"This looks like the right answer! HH61HHl  d7o san nHH tenHHU m UU 0 ` mThe Proton-Proton Chains ; h .One of the two ways to burn hydrogen.   HH7- HHl   = d7PHH 9HHmele hydro  ` laThe CNO Cycle ;` # %The other main way to burn hydrogen. X` UN u h =PStill  4 p  4 He, but uses  12 C  as a catalyst.  HH7S HHl  _7U_7q_;_I^r ,7]U I^r ,33 :p + p      2 H+e + + n e67^Unro6MM ,7_U yogMM ,11 5p +e - + p    2 H+ n e I^"7`U I^" ,p + 2 H     3 He+ g67aU6 67bU 6 CNI^ '7cU nayI^ '&& 13 He+ 3 He     4 He +p+pI^47dU calI^4&& /3 He+ 4 He     7 Be+ gl67eU6?|I^/X ,7fU ?|I^/X ,?? 37 Be+e -      7 Li+ n e|I^~7gU +e|I^~ ,7 Be+ p      8 B+ g-n67hU y-n6 ,$67iUp$6 -I^ 7jU -I^ -- 37 Li+ p      4 He+ 4 Hegn67kUn667lU6 CNI^a ,7mU nayI^a , 68 B      8 Be+e + + n e67nUl6& I^7oU  I^"" 18 Be      4 He + 4 He7U;8$7qU;8$7g;8;\cEQ7rU7cEQ  cEcrruu67sUuu6uuE-7tUjE-rqrqEQ7uUHQ;; 67U 6;_-7U^a;_-;_h_hqh7U h  hhh7Uh I^hh!Vc&$7U !Vc&$!Vs9X!Vs9X PPIHe$O7U $O$$ d7#/$7U 7r#/${{ PPII!#9V$7U !#9V$2{2{ PPIIIHH7 HH7tE  ` Helium Burning 7u;`  HX` .Can you continue after the hydrogen runs out? u`  ` 5Yes! The  first  thing you do is burn helium. `  ` 'However,  4 He is tightly bound.  ` /  It doesnt stick well to other nuclei. `  #` ا=Example:  4 He  +   4 He   8 Be @` V3That is,  8 Be decays soon after it is made! ]`  z` 5The solution: A lucky accident in  12 C allows ` 44 He  +  8 Be    12 C  + g ` "to proceed with high probability. ` 7 ` :  If T 10 8  K , Helium is consumed through  ` J4 He  +  4 He  +   4 He    12 C  + g (` Triple Alpha Process BurE`  b` ,Other elements burn at higher temperatures. ru` Later ... HH7thHHl  H 8ood.Y  \RK`8p \RK`\Rb \Rb -12 C + p     13 N +ge]F`8q h i]F`] ] .13 N    13 C + e + +nl.58r l.5ll 4\R>`8s + \R>`\RU \RU -13 C + p     14 N +g gM`8t 8 gM` g g -14 N + p     15 O +gcn.58u cn.5cc s@`8v @`W W .15 O    15 N + e + +n `8w h `  715 N + p     12 C + 4 He=68x\R=68p68yR 6 7뛦8 ]7뛦77  i068z 06+68{6l 268| 26 68}R 6+ R78U `R7cc pepө?8~ ? g)_q-?8.5_q-?sq_MZ-?8`MZ-?15MZzQZ-?8QZ-? hQ~ZZq-?8 Zq-?  ZqH?8RH?H)H+`=n$8 +`=n$"dZ"dZ  Proton in866)`=n$8 )`=n$)9X)9X  Proton in866Qi`=n$8 Qi`=n$QQ  Proton in866Z[#F8[#FQi`=n$8 Qi`=n$QQ  Proton inc8c66cZc[#F8c[#F8 66ZSisX $8 SisX $SS  Helium outd5LeftdZd6Rightdb Referencedo$dr du9XPrdxd{d~Q=nddQd8d d d  a Xf a Body. f b  Bulleted\t. f c CellBody. f d  CellHeading. f e  Footnote. f fT Heading1Body. f gT  Heading2Body. f hT   HeadingRunInBody. f i l yIndented. f j g Numbered.\t. f kE  Numbered1.\tNumbered. f l f TableFootnote. f mT   TableTitleT:Table : .  f nP TitleBody. f o T   TableTitleT:Table : . f p   CellHeading. f q  CellBody. f r   CellFooting. f s bl Body. . f t  Body. @ u    l Footer. @ v   d  Header. f w " Body. f x Body. $f y  Body. f z Body. f { Body. f | $ Body. f }  Body. f ~ Body. f  Body. f $  H Body. f H Body. $$f H Body. dy$Hf H Body. . $lf H Body. f  H Body. f H Body. f  Body. f  Body. f $ H Body. f  Body. f  Body. $f  H  Body. f  H  Body. f H  Body. f H  Body. $f  fH  Body. 6f  H  Body. f H  Body. f  H  Body. $f   Body. f   Body. f  Body. f  f Body. f $  H f Body. f $  H f Body. f  Hf  Body. f H  Body. f  H  Body. f $ H  Body. f  H  Body. f  H  Body. f  H  Body. f  H  Body. $f  Body. f  Body. Hf   Body. Hf  f Body. f $  Body. f   Body. f   Body. f   Body. f  Body. f 6 Body. Hf  Body. f   Body. f   Body. f   Body. f   Body. f  Body. f   Body. f  Body. f  Body. f   Body. f   Body. f   Body. f   Body. % 7  Emphasis  EquationVariables )   ڝ )  )  ڝ ڝ w w w ڝ  [   ڝ ")  tu )  ڝ $)  tu tu ڝ w $)  ڝ ڝ )  w ڝ )  tu uo&  tu tu w ڝ w )  ڝ uo&  w ڝ ڝ ڝ ڝ w w  ڝ  ڝ  ڝ  ڝ  ڝ )  ڝ ڝ    Z Z Z Z Z Z Z Z  Z Z Z Z  Z  Z  Z  Z  Z  Z  Z  Z  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-ItalicSymbolSymbolTimes-BoldItalicHelvetica-BoldTimesc HelveticaSymbol RegularRegular BoldRegularItalicpc8+q6xe P|O|Ys79&whevH9XCG!^ f,T<~J.@Ф{cxZ+nϑװZI&(ZJ,fS`&c8o = 7$9)uD{$qŊ2r8 6v7vO"kqaKB|bǰsGi܌||7@S.j