Aa3d  qHH $ @d HHHHff@  d Footnote TableFootnote**.\t.\t/ - :;,.!?  `;! `YeowTOCHeading1Heading2  EquationVariablesiItcme7 6_ymb6xRe6w6~ 66777 7&7P7 ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year> +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm> Aa -"<$monthnum>/<$daynum>/<$shortyear> .<$monthname> <$daynum>, <$year> /"<$monthnum>/<$daynum>/<$shortyear> 0 <$fullfilename>ff 1 <$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>n ;See Heading & Page%See <$paratext> on page<$pagenum>. < Table All7Table<$paranumonly>, <$paratext>, on page<$pagenum>b =Table Number & Page'Table<$paranumonly> on page<$pagenum>e ~ FFA HHA& JJ LL NN PP RA$  5y  soe>5|  s > 5  u> 5 ta"5 ! u/um5 " ta5 # xn$d5 $ w se5 % } so>/5 & v s$ea5 ' { sfi5 ( ~ vf 5 ) v 5 * v vele5 + vte5 , v]5 - vg5 . v va5 / va5 0 vod)5 1 v+t 5 2 v vu<$5 3 vn- 5 4 v vnge5 5 } vx p5 6 ~ vu5 7 { veen5 8 } wHe5 9 ve<5 : v e5 ; ~ v5 < ve5 = ~ ~l$p5 > ~o<5 ? ~ ~Ta5 @ v&'5 A ~ ~ay>5 B ~p>e5 C ~~5 D ~ ~F5 E ~ ~5 F y ~5 G ~ ~5 H ~J5 I v v5 J v5 K v v5 L v5 M v v5 N vR5 O v$ 5 P vs5 Q v vs5 R vu5 S vt5 T vu5 U vt5 V ~ vx5 W ~ ~w5 X ~}5 Y ~v5 Z ~ ~{5 [ ~~5 \ ~5 ] ~v5 ^ v v5 _ { v5 ` | v5 a v vv5 b v v6 c v6 d v v6 e vv6 f v6 g vv6 h v}6 i ~6 j x v{6 k v v}6 l { v6 m } v6 n } v~6 o } v6 p v v~6 q ~ v6 r ~ ~~7 s ~6 t ~ ~~7 u 7 v v v7 w } v~7 x v v~7 y { vy7 z v~7 { v v7 | vv7! } v v7" ~ vv7'  v v7( vv7) v7* ~ v7+ { v70 vv71 v72 ~ v73 ~ v7I ~ v75 y v~7= ~ v~7> ~ v7Q ~7@ y v~7J v7V v v7W v7X vv7f ~ v{7Z v|7[ ~ vv7\ vv7g v7o ~ ~v7p ~7 ~7 z 7 ~ z7 ~5`6dq5{PF6dq6v~LH6dq7~~8C7Hm3R q897 v~Hm3R H RH R FootnoteHr@ q98:7 ! vHr@ HzHz  Single LineH'q:9<7* ~;;Footnote  5_;:     HD q<:=7 7>HD HH  Double LineH q=<@77W>?v Double Line ~5c>?=v5e?>=77 zH 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 NUUe !  HUV 5NPF5HUV lMM H$ 5O5 H$ P UUe "  H$ 5PN5H$ lOO HH5QD HHAR  ` # White Dwarf Stars ;` $  X` % Slow Death of a typical star u` &  ` ( 5 End state of stars with M  <  a few M SUN ` ) ժ)More massive    Supernova (later) ` *  ` + ( Follows  Helium Burning  and ` , 5# Planetary Nebula  stages #` - 6  Hot, dense,  inert  carbon core left over @` . UV ]` / F Very, very dense! UVz` 0 %  Quantum mechanics is important ` 1 5!  These are strange objects! ` 2  ` 3  Cool-down time is very long HH5RDNHHlUQQ d5SUUHH5TS HHjf Stars U $ ` ' Example: Sirius B ;` 4  X` 5 $White dwarf binary companion to the wu` 8 fe&brightest star in the sky (Sirius A). ` 6 up o` 9 ) Radius from Luminosity and Temperature l` : ur*  R   710 -3   R SUN ` ;  t` < $ Mass from motion of binary system #` @ o  M  = 1.05  M SUN @` = ry e]` > /The mass of the sun in the size of the earth t z` ? 5 ` A ng b` B O Density  r =  M /(4p R 3 /3)= 410 9   kg/m 3 ` C = 410 6  r WATER ` D   ` E /  The distance between the carbon atoms is (` F bd  ( m C /r) 1/3  (12  m H /r) 1/3  = 1.710 -2    E` G  $b` H 7Tight squeeze! Typical atomic spacing    few . HH5UStthHHlRXTT ). d5V XX TpeHH5WV 10HHjN X  ` 7 sElectrons at High Density ;` I 05 X` J  Degenerate Fermi Gas ` >u` K t s` M thPauli Exclusion Principle ` N (  Electrons cannot scatter randomly y ` O (4  They are confined ` Q 3 ` P -What is the momentum of a confined electron? D#` R 0  Use  Heisenberg Uncertainty Principle: om@` S $D p D x   h /2p ]` T m-where  D x    atomic spacing d z` U 7 D p    typical electron momentum p e pac` V ˁ. ` W !Using values from Sirius B, find ` X ,p e  = 6.210 -23   kgm/sec ` Y _E e  = 2.110 -15   J  = 13  keV  <<  m e c 2   (!!) H ` Z N  (` [ ,Note: E e  = k ( 15210 6 K) 7E` \ %  Actual temperature irrelevant! b` ] *Degenerate   Zero Temperature HH5XVoPrHHlU[WW rand5Yeco[[3 HH5ZY oa HHj X  Use  _x[er  UU` L The Equation of State for a Gas 8` ^  U` ` e &Relation between Pressure and Density r` a   h b /Consider N particles in a box with side L: ` VY h c 0Pressure is average force from N particles: ` d e ` e Example: The Ideal Gas ` f # p=mv  Nonrelativistic motion " h g # Definition of temperature E X` i  b X` h    P=nkT 2HH5[YtuHHlX^ZZ  d5\^^PrHH5]\ [HH e~^[  ` _ The Equation of State C ` j for a Degenerate Fermi Gas f` k  x` m Momentum determined by Eq` n r Pauli Exclusion Principle and ` o e !Heisenberg Uncertainty Principle s` p a ` q b< Nonrelativistic electrons    v = p e /m e ` t sua Heisenberg     p e  = (h/2 p ) / d = (h /2p) n e 1/3 T1 h s @since  n e   =  one per  d 3  "` u De i h r re1So, finally, our  approximation  gives HH5^\HHl[|]] H6_wY`uZ^$?HH6`a_$?HHH$$$6a`c_$$$[$?HHH6beg_ jHHera FHl$$6 cad_ontl$$ Eql?lc$$6 dce_ olc$$Unctalc$c$$6edb_b $c$$lecon$HcQ6fik_ isQ  pQlHcH6gbh_ 1/HcHh sHccHH6hgi_HH` uDeHcHH6ihf_nHHcBV6jkl_ BVBVdZBVdZ L$6kfj_$^?$$6ljn_$$$$?H'6 mno_op?H'l6nlm_lHllOo.P%6!omp_mpOo.P%lOo.PuP?H66"poq_?H6moZVfg 6qpr_ bZVfg ZfZf v6r qs_ k-- 6s rt_ h$$ Force on side of box from+ |6t su_ + |<< particle collisions with wallk P66e6u t_ 2P6*6ek P66e/5o] 'equal[char[F],over[times[char[Delta],char[p]],times[char[Delta],char[t]]],over[char[p],fract[char[L],char[(*q"Green"q*)v]]],over[times[char[p],char[(*q"Green"q*)v]],char[L]]]|x96v w%6|ǔ9|x9$' equal[(*q"Red"q*)char[(*q"Red"q*)P],times[(*q"Red"q*)char[(*q"Red"q*)N],over[(*q"Red"q*)char[(*q"Red"q*)F],char[(*q"Red"q*)A]]],times[(*q"Red"q*)char[(*q"Red"q*)N],over[(*q"Red"q*)fract[(*q"Red"q*)id[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)p],char[(*q"Red"q*)v]]],char[(*q"Red"q*)L]],power[(*q"Red"q*)char[(*q"Red"q*)L],num[(*q"Red"q*)2.00000000,"2"]]]],times[(*q"Red"q*)over[(*q"Red"q*)char[(*q"Red"q*)N],char[(*q"Red"q*)V]],char[(*q"Red"q*)p],char[(*q"Red"q*)v]],times[(*q"Red"q*)char[(*q"Red"q*)n],char[(*q"Red"q*)p],char[(*q"Red"q*)v]]]NH6w_xYvvZpayMf+ X6xwY"yyZ 26yxeqd) X3 ],'_equal[(*q"Blue"q*)times[(*q"Blue"q*)over[(*q"Blue"q*)num[(*q"Blue"q*)1.00000000,"1"],num[(*q"Blue"q*)2.00000000,"2"]],char[(*q"Blue"q*)m],power[(*q"Blue"q*)char[(*q"Blue"q*)v],num[(*q"Blue"q*)2.00000000,"2"]]],times[(*q"Blue"q*)over[(*q"Blue"q*)num[(*q"Blue"q*)1.00000000,"1"],num[(*q"Blue"q*)2.00000000,"2"]],char[(*q"Blue"q*)k],char[(*q"Blue"q*)T]]]es[d6zqfr||q*imHH6{z eq*HHdL] (*q"Red"|Re !x!x ` l me'Implications of  P   r 5/3 "q*Dx` v *) ,ax` w ,c*Add mass to a white dwarf and it shrinks! ~x` x q" "x"h z "q* The Equation of State says that  x"h { + Hydrostatic Equlibrium says that  6r2"h |  Density is given by  e` }  e"h ~ )  Put this all together and find HH6|zl"qHHl^{{ ue"6}~e(* 2jA8"q'equal[times[string["Therefore "],char[P]],times[indexes[0,1,char[n],char[e]],indexes[0,1,char[p],char[e]],over[indexes[0,1,char[p],char[e]],indexes[0,1,char[m],char[e]]]],times[power[id[over[char[h],times[num[2.00000000,"2"],char[pi]]]],num[2.00000000,"2"]],power[indexes[0,1,char[n],char[e]],over[num[5.00000000,"5"],num[3.00000000,"3"]]],over[num[1.00000000,"1"],indexes[0,1,char[m],char[e]]]]]Nc6~\v*)}}]Adma6p <-ua'equal[(*q"Green"q*)indexes[(*q"Green"q*)0,1,times[(*q"Green"q*)string[(*q"Green"q*)"However "],char[(*q"Green"q*)n]],char[(*q"Green"q*)e]],times[(*q"Green"q*)num[(*q"Green"q*)6.00000000,"6"],indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)n],char[(*q"Green"q*)C]]],times[(*q"Green"q*)num[(*q"Green"q*)6.00000000,"6"],over[(*q"Green"q*)char[(*q"Green"q*)rho],indexes[(*q"Green"q*)0,1,times[(*q"Green"q*)num[(*q"Green"q*)12.00000000,"12"],char[(*q"Green"q*)m]],char[(*q"Green"q*)H]]]],over[(*q"Green"q*)char[(*q"Green"q*)rho],indexes[(*q"Green"q*)0,1,times[(*q"Green"q*)num[(*q"Green"q*)2.00000000,"2"],char[(*q"Green"q*)m]],char[(*q"Green"q*)H]]]]NQ6~\m.0]s[1,6u3.O0w(B9nd'equal[(*q"Red"q*)char[(*q"Red"q*)P],times[(*q"Red"q*)id[(*i1iq"Red"q*)times[(*q"Red"q*)power[(*q"Red"q*)id[(*q"Red"q*)over[(*q"Red"q*)char[(*q"Red"q*)h],times[(*q"Red"q*)num[(*q"Red"q*)2.00000000,"2"],char[(*q"Red"q*)pi]]]],num[(*q"Red"q*)2.00000000,"2"]],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)m],char[(*q"Red"q*)e]]],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],power[(*q"Red"q*)id[(*q"Red"q*)times[(*q"Red"q*)num[(*q"Red"q*)2.00000000,"2"],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)m],char[(*q"Red"q*)H]]]],over[(*q"Red"q*)num[(*q"Red"q*)5.00000000,"5"],num[(*q"Red"q*)3.00000000,"3"]]]]]],power[(*q"Red"q*)char[(*q"Red"q*)rho],over[(*q"Red"q*)num[(*q"Red"q*)5.00000000,"5"],num[(*q"Red"q*)3.00000000,"3"]]]]]No`6\cr[]Grn"d7 HH7  HHmnd'd" ` y s[Cooling a White Dwarf Star ime;`  [( RX` *) Homework problem (*qu` *q Doug Robinsons question 000` d" p` q*6Hot, dense ball of carbon atoms (N particles)  ... ` *q...  at zero temperature `  r q"3Structure (mass, radius, ...) stay the same as the )ti#@ [()star radiates away its thermal energy E: *@` q*E  =  NkT *)]` "q uz` 00-Energy radiated away at a rate L=dE/dt where w` [($L  = 4p R 2 s T 4 "R` 5" u"h 00  To find T(t) solve H 2Answer:  The white dwarf stays hot enough to e@ 6see (at least in the infrared) for billions of years! HH7HHl| d"7tDw`ίLb4$'equal[(*q"Green"q*)char[(*q"Green"q*)P],cross[(*q"Green"q*)string[(*q"Green"q*)"some constant "],power[(*q"Green"q*)char[(*q"Green"q*)rho],over[(*q"Green"q*)num[(*q"Green"q*)5.00000000,"5"],num[(*q"Green"q*)3.00000000,"3"]]]]]NxH7z {iu .7[(WE;20x *'~equal[char[P],cross[string[" another constant "],over[power[char[M],num[2.00000000,"2"]],power[char[R],num[4.00000000,"4"]]]]]NPxTx7z s{ 5"7)ol\/ P~9-2Th'equal[(*q"Cyan"q*)char[(*q"Cyan"q*)rho],cross[(*q"Cyan"q*)string[(*q"Cyan"q*)"different constant"],over[(*q"Cyan"q*)char[(*q"Cyan"q*)M],power[(*q"Cyan"q*)char[(*q"Cyan"q*)R],num[(*q"Cyan"q*)3.00000000,"3"]]]]]N2Q27z{7$*ha{fM3f91eg['equal[(*q"Red"q*)char[(*q"Red"q*)R],cross[(*q"Red"q*)string[(*q"Red"q*)"constant"],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],power[(*q"Red"q*)char[(*q"Red"q*)M],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],num[(*q"Red"q*)3.00000000,"3"]]]]]]NMe^ 7%z2 4@{ *d7,tt 2.00HH7- ]]]HH7 ` The Chandrasekhar Limit ol;`  X` Th/Suppose you keep adding mass to a white dwarf. ss[u` [()(This can happen. Well see how later.) y` n")  It shrinks and becomes more dense. ,` 00 0` 2!Can this go on forever?  NO! ` {  0Remember:  We assumed that the motion of the 3f#@ Gelectrons was nonrelativstic ( v e  = p e /m e ). rin@` ta ]] um@  At some point we must have  v e  = c  and the erz"H Equation of State becomes  3"3If you try to solve for R(M),  the radius drops 2@ 1out of the equation and all you get is a mass!!! #`  @ 9Find maximum mass  =  1.44M SUN . Speed of ]@ =electrons is  c  and if you add more mass then  ... z` r '...   something drastic happens! HH7/ itHHl e h7NsinӇ X6e) 'equal[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)N],char[(*q"Red"q*)k],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)d],char[(*q"Red"q*)T]],times[(*q"Red"q*)char[(*q"Red"q*)d],char[(*q"Red"q*)t]]]],times[(*q"Red"q*)num[(*q"Red"q*)4.00000000,"4"],char[(*q"Red"q*)pi],power[(*q"Red"q*)char[(*q"Red"q*)R],num[(*q"Red"q*)2.00000000,"2"]],char[(*q"Red"q*)sigma],power[(*q"Red"q*)char[(*q"Red"q*)T],num[(*q"Red"q*)4.00000000,"4"]]]]N!Z7O di  7ms!G|L9x4, 'equal[(*q"Black"q*)char[(*q"Black"q*)P],cross[(*q"Black"q*)string[(*q"Black"q*)"yet another constant"],power[(*q"Black"q*)char[(*q"Black"q*)rho],over[(*q"Red"q*)num[(*q"Red"q*)4.00000000,"4"],num[(*q"Red"q*)3.00000000,"3"]]]]]NR7 d57NLeftd6eRightd7l[ ReferencemdD*)r[dS"qs[dV)d[(dYd"r[d\"q],dzed00d*)wedq* a @d"f a Body. f b q"* aBulleted\t[. acf c CellBody. f d  CellHeading. f e V dFootnote. 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 Body. $f z Body. f { $ Body. f |  Body. f }  Body. f ~ Body. $f  .Body. Hf Body. f  Body. f  l.Body. f  Body. Hf  Body. Hf  Body. f Body. f  Body. f Body. f  Body. f Body. f .Body. f Body. Hf Body. f  Body. f " Body. f Body. f   Body. Hf Body. $f  .Body. f  Body. $f  Body. f  Body. f  H.Body. f  H.Body. f  Body. f  Body. $f Body. f  Body. $f  Body. f  Body. $f Body. f Body. % 9  Emphasis  EquationVariables )   ڝ  ڝ w ڝ $)   [  tu tu tu w w )  )  ڝ w w ڝ ڝ uo&  ڝ ڝ ڝ w ")  ڝ )  tu $ڝ  ڝ ڝ ڝ w w )  $w $w )   ڝ uo&   ڝ )  uo&  tu  ڝ tu )   ڝ  ڝ  ڝ  tu )   tu  tu  )    Z  Z Z  Z   Z& Z  Z   Thin Medium Double Thick@  Very Thin     oH p q rH p q rH p q rH p q rH p q rFormat A   oH p q rH p q rH p q rH p q rH p q rFormat B U e V UComment d BlackT!WhiteddAReddd Greendd  Blued Cyand Magentad Yellow  Times-Roman Times-Bold Times-ItalicSymbolSymbolHelvetica-BoldTimes-BoldItalicTimesc HelveticaSymbol RegularRegular BoldRegularItalic 9#3f3] VRd|NТWPb&llRioH5~C5+~(OЃn#)mbMH"tvMDD3YB3?t,[T d#j\G= 3q= *7:$o#~