ࡱ> ac`q` bjbjqPqP .N::%tttp"p"p"8"|$#D:tt#x$"%%6%s,s,s,@:B:B:B:B:B:B:$5<h>Hf:ts,*s,s,s,f:%6%{:111s,~"%(t6%@:1s,@:11V|8rt96%h# `Cp"-ft9 @::0:9x>W/>>99>t :4s,s,1s,s,s,s,s,f:f:1js,s,s,:s,s,s,s,d I. SMALL STAR (MMS < 8 M() DEATH DEGENERACY ENDS CORE CONTRACTION with TCORE < NEXT FUSION LEVEL OUTER ENVELOPE GREATLY EXTENDED, WEAKLY HELD SOMEHOW EJECTED (radiation pressure? acoustic waves?) APPEARANCE: JUMP from Red Giant to White Dwarf STAGE       PLANETARY NEBULA PROPERTIES: ( SPHERICAL SHELL (ring appearance due to line-of-sight) MASS ( 0.1 M( (per ejection?) v ( 10-30 km/sec (much slower than wind) PLANETARY NEBULA GONE in ( 105 years (too thin to see) ( CORE ALONE SEEN CORE COOLS & FADES ( WHITE DWARF (and beyond) Takes many B-yrs to cool completely Best guess: Coolest WDs in universe ( 4000 K now WHITE DWARFS So compact that es are forced into degeneracy, i.e., Pauli Exclusion Principle restricts available states. EXCLUSION PRINCIPLE: (p (x (  EQ \f(h,2()  ( (p (  EQ \f(h,2((x)  As (x ( 0, (p must increase, i.e., es must be in higher p states so not identical in same (x volume. Means es have KE >> thermal. From Virial Theorem: 2 KESYS + PESYS = 0 PESYS =  EQ - \f(3,5) \F(GM2,R)  (assumes ( = constant) KESYS = Ne (mev2) (ignores KE from slower non-degenerate nuclei) = Ne (me) EQ \b(\f(p,me))2 = Ne  EQ \f(p2,2me) (non-relativistic) = Ne ( EQ \r(,m02c4 + p2c2) ) ( Ne pc (highly relativistic) Ne =  EQ ( \f(M,mH) where ( = # es per nucleon So, for the non-relativistic case (larger white dwarfs): 2 ( EQ ( \f(M,mH))  EQ \f(p2,2me)  EQ - \f(3,5) \F(GM2,R) = 0 From Exclusion Principle: Say p ( (p and (x =  EQ \b(\f(V,N))\s(1/3) =  EQ \b(\f(\f(4,3)(R3, ( \b(\f(M,mH))))\s(1/3) or p (  EQ \f(h,2((x) =  EQ \f(h,2() \b(\f(3(M,4(mHR3))\s(\s(1/3))  Putting this into the Virial Theorem: 2 ( EQ ( \f(M,mH))  EQ \f(\b(\f(h,2() \b(\f(3(M,4(mHR3))\s(\s(1/3)))2,2me)  EQ - \f(3,5) \F(GM2,R) = 0 so  EQ \f(3.47 h2, ( G me) \b(\f((,4(mH))5/3  EQ \f(M5/3,R2) =  EQ \F(M2,R)    or ` K = 1.574 x 1016 m/kg1/3 for ( = Interestingly, this says white dwarfs get SMALLER when they are more massive! Even MORE interesting is what happens as they get extremely compact. Then more and more es are forced into higher and higher p states since so many of them are trying to be in the same small volume. In the limit that the mass is so great and compactness so extreme that the ps become so great that we need to use the relativistic expression for the KE, i.e., 2 ( EQ ( \f(M,mH)) ( EQ pc)  EQ - \f(3,5) \F(GM2,R) = 2 ( EQ ( \f(M,mH)) ( EQ \f(h,2() \b(\f(3(M,4(mHR3))\s(\s(1/3)) c)  EQ - \f(3,5) \F(GM2,R) = 0 This yields: 2 ( EQ \f((,mH))(EQ \f(5,3G)) ( EQ \f(h,2() \b(\f(3(,4(mH))\s(\s(1/3)) c)  EQ \b(\F(M4/3,R)) =  EQ \F(M2,R)   or This is only ONE mass value! It means that there is a limiting mass as es become ultra-relativistic, i.e., NO WHITE DWARF CAN EXCEED THIS MASS and maintain equilibrium (as Virial Theorem requires). WHITE DWARF PARAMETERS: MWD ( 1.44 M( (more careful calcs & for ( = ) Typical RWD ( few x 103 miles (planetary size) DENSITY ( 20 tons/in3 !! g ( 106 gEARTH !! Comments: a) High g ( smooth surface (max. mt. = 2 feet!) b) Person weight 150 Million pounds, squashed into a paint layer c) Drop mosquito on surface, speed on impact ( KE of 20 tons nitroglycerine !! d) Cant get sample for Smithsonian remove from high g and expands (explodes) back to normal state  WHITE DWARFS DO EXIST First seen in 1862 (Sirius B) much confusion then TODAY: A FEW THOUSAND KNOWN Selection Effect, due to faintness Of 50 nearest stars, 5 are WDs, none are RGs TWD = 100,000 K 5000 K (cooler would be too faint to see easily) LWD ( SMALL RWD BINARIES ( MWD ( 0.3 1.2 M( (i.e., < 1.44 M( ) Good news for theory, as mass limit seems valid Note importance of mass loss during late stages An 0.3 M( star wouldnt reach WD stage in universe age (15 B-yrs) Importance/necessity for mass loss at end of life revealed by: Examples of stars born together, i.e., of same age, but smaller mass is more evolved Sirius A (2.3 M() is on MS while Sirius B (1.0 M() is a white dwarf. These masses cant be original as the smaller mass star is more evolved here! 40 Eridani A (1.0 M( MS star) and 40 Eridani B (0.5 M( white dwarf). Again, smaller mass star is more evolved. Star cluster NGC 1818 has 7.5 M( MS stars but also contains white dwarf. Evolution theory requires that the WD must have been > 7.5 M( when on the MS yet must be < 1.4 M( now. CONCLUSION: IF M.S. MASS ( 8 M( (B5 to M), MASS LOSS ( MFINAL ( 1.4 M( SOURCES of mass loss: Planetary Nebula ejection Strong RG winds POSSIBLE LAST GASPS: WDs with RG COMPANIONS Mass lost from RG can impact WD companion (Chap. 12.1, 12.2) H from RG falls on low mass WD H builds up, ignites H ( He fusion ON SURFACE Very bright (for a few months) = NOVA Observational support: All known novae are from systems with RG + WD Some novae repeat (in 20-50 yrs) i.e., not destructive event Substantial RG wind on high mass WD Mass from RG increases MWD near 1.4 M( Gravity overwhelms degeneracy, star contracts, heats ( fusion of C,O & whole star explodes ( SUPERNOVA (Type I) (Well discuss SN more later, with large star death)     PAGE  PAGE 5 M2/3 =  EQ \f(10 hc(, 6(mHG)   EQ \b(\f(3(,4(mH))\s(\s(1/3))  or M = 1.38 M( (for ( = ) K = M1/3 R or R =  EQ \f(K,M1/3)  K = OBSERVATIONS RG RG+ RG++ Planetary Nebula  Z^rt        ! $ % & * E G H J K    " : ; ¾|| jh#CJ h#CJ H* jh#CJ H* j~h#CJ  jh#CJ h#CJjh#CJ UmHnHuh#h#5CJ \ h#CJ H* jh#>*CJ H*h#>*CJ H* h#>*CJ h#CJ h#CJ OJQJ^J0',Ost     ! % ' ( ) G H  hx h-  # % U  3 4  3 4 s t Q R  h; < G Y _ c 3 4 K L N O Q R S T ^ _ a b c d i j l m n o y z { ~  z jh#CJ h#CJOJQJ jh#CJ OJQJ jph#CJ jh#CJ U jh#CJ  jDh#CJ h#CJh#CJOJQJ jh#CJ OJQJh#6CJ OJQJ]h#CJ OJQJ h#>*CJ h#CJ 0   # & + . 4 7 9 < ? @ A T U Y Z e f r t w y | ˱hb(CJ OJQJ hb(CJ H* jrhb(CJ h#CJ H*jh#CJ U hb(CJ h#CJ H* h#CJ jDh#CJ h#6CJ ]h#CJ OJQJ h#CJ ?     # $ % & ' + , - . 0 1 3 4 6 7 W X [ \ ` a h i j k s t  )*+,459:L jDh#CJ h#CJ h#CJ h#>*CJ  jhhb(CJ  jh#CJ h#CJ H* h#CJ H*jh#CJ U hb(CJ H* h#CJ D  gkHIUV$a$`gdR$a$$ ha$ h$ ha$LOPQTUbefghiklvw~     !ٱ٫hD/h#CJaJ hb(CJ H* hb(CJ  jDh#CJ  jh#CJ h#CJ h#CJ H* jhhb(CJ  jph#CJ h#CJjh#CJ U h#CJ h#CJ H*ABEFGHIJ]^bcghklmz{|}~|u hzaCJ H*hzah#CJ  jphzaCJ hzahzaCJ hzaCJ H* hzaCJ jh#CJUmHnHu hD/CJ jh#CJ U h#CJ H* h#CJ h#CJ H*hb(hb(CJ H* hb(CJ  jph#CJ  jhhb(CJ h#CJ +HI#(]ÿÿÃxtthzahD/h#CJaJhb(hRh#CJaJ jhhRhRCJaJhRCJaJhRhRCJH*aJhRhRCJaJhRh#jh#CJUmHnHu h#CJ H* h#CJH*jh#CJ U h#CJ hzahzaCJ H*, !"#$%-03478KLPQghop廵囕 jhhzaCJ hzaCJ jhzaCJ U h#CJhb(hb(CJ H* hb(CJ  jph#CJ h#CJ H* h#CJ H* jhhb(CJ jh#CJ U h#CJ hD/h#CJaJh# h#>*7prstuwxTwz h#CJ h#>*CJ hD/>*CJ hD/ h#5\h#h#6>*CJ ]jh#CJUmHnHu h#CJ H* h#CJ jhhzaCJ  jph#CJ jh#CJ U h#CJ jhzaCJ UhzahzaCJ H* hzaCJ /.I^_9m^gdza^ hp  h h x h$ ha$  89DEMNQRTY`uv򯢯sh#6CJ OJQJ]h#6>*CJ ] hzaCJ h#jh#UmHnHu jh#CJ OJQJh#CJ OJQJ h#CJ H* hD/CJ H* jh#CJ  jmh#CJ  jh#CJ H* hD/CJ  jh#CJ h#CJ h#CJ H*-Dz{:lKO  IJ`gd:{$ & F 8^a$gd:{$a$gd:{gd:{^!HKlp AXxy Z[{|Nڳڥڟzzzz jh:{CJ H*OJQJh:{6CJ OJQJ]h:{CJ OJQJ h:{CJ  jh#CJ H*OJQJh#>*CJ OJQJ jh#CJ H* j@h#CJ  jh#CJ h#CJ OJQJ h#CJ H*h#6CJ ] h#CJ hD/CJ -NOno}   %'679>?!<>?TӾӱӪӞӔӾӱӎӈ~ztkӞh:{6CJ ] h:{CJh:{h:{5>*CJ \ h:{CJ h:{CJh:{5CJ H*\ jh:{5CJ \ h:{CJ \ jh:{5CJ H*\ jh:{5CJ \h:{5CJ\h:{5CJ \ jh:{CJ H*OJQJh:{CJ OJQJh:{>*CJ OJQJ*Jz%U*T 8^gd:{ & F ^`gd:{^gd:{^gd:{ & F ^`gd:{gd:{^gd:{ $ a$gd:{gd:{ ^`gd:{Ty}&CEPQ    ztztpzh# h#0Jjh#0JUh%jh%Uh:{5>*CJ OJQJ\^Jh:{5CJ \h:{6>*CJ ] jh:{CJ h:{5CJ OJQJ\ jh:{CJ H* h:{CJ H*h:{6CJ ] h:{>*CJ h:{CJh:{5>*CJ \ h:{CJ ,T   *+,-. $ !a$h]h&`#$ hgd:{h^hgd:{ 8^gd:{ &'()*,-./256BCFGHILMNOZ[]^_`hknortԽ԰ԤԚԽԽ԰ԤԚԑԽ~qchD/hD/CJ OJQJ\ jhD/5CJ H*\hD/hD/CJ \hD/5CJ \h#5CJ\h#5CJ H*\ jph#5CJ \ jhhzahza5CJ jh#5CJ U\h#5CJ H*\h#5CJ \h%h#h:{0JmHnHujh#0JU h#0J&̹̆y h:{CJ h#5\hR5CJ \ h#CJ hb(hb(5CJ H*\jhb(5CJ U\hb(hb(CJ \hb(5CJ \h#5CJ H*\h#5CJ \h# hD/h#hD/hD/CJ OJQJ\! jhhD/hD/CJ OJQJ\  hgd:{$a$,1h/ =!8"#$8% L@L Normal$CJOJQJ^J_HaJmH sH tH D@D Heading 1$ h@&>*CJ F@F Heading 2$ h@& 5CJ \@@@ Heading 3$ h@&CJ F@F Heading 4$$ h@&a$CJ DA@D Default Paragraph FontViV  Table Normal :V 44 la (k(No List 2B@2 Body TextCJ4@4 Header  !.)@. 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