% dr. dm d 2 r dt 2 = F g + F P,t + F P,b !"#$%&'()+(,-%&'./'0(/1(2%#'0( !"#$%&'()+(,-%&'./'0(/1(2%#'0( !"#$%&'()*+(,-%&'.

Size: px

Start display at page:

Download "% dr. dm d 2 r dt 2 = F g + F P,t + F P,b !"#$%&'()*+(,-%&'./'0(/1(2%#'0( !"#$%&'()*+(,-%&'./'0(/1(2%#'0( !"#$%&'()*+(,-%&'."

Edmund Black
5 years ago
Views:

1 /-.3&'(#(456.-3'.4#6(7/689&(/(:#(.-(%&( #%9/$&'&(/(#(%#'(;.%(8'#4&(#'&#<(<(#%(#( '#3.8('/9(4&-%&'(/(%#'<('<(6&-:%(3'<#-3(;.%( 3&-.%5<(+( dm d 2 r dt 2 F g + F P,t + F P,b >-65(/'4&(#4?-:(/-(456.-3&'(#'&(:'#7.%5(#-3( $'&8'&(@;.4(.(#6;#5(-/'9#6(%/(%&(8'#4&A( F P,t ( F P,b + df P ) P F A df P A F G G M rdm dm d 2 r dt G M rdm A 2 E/'4&(#%(%&(%/$(&F8#6(%&(/'4&(#%(%&(D/G/9( #-3(#(4/''&4?/-(%&'9(%#%(#44/8-%(/'(%&( '&384?/-(.-(%&('#3.#6(3.'&4?/-H( I'#7.%#?/-#6(/'4&( #$%&$'()*%+,-./$2324*25( dm Adr Adr d 2 r dt GA M dr r A 2 *( d 2 r dt G M r 2 dr dr G M r g J53'/%#?4(&F8.6.D'.89(K( #44&6&'#?/-(.(L&'/( MF-(/(53'/%#?4(&F8.6.D'.89(K(.-(/'3&'( /'(#(%#'(%/(D&(%#?4<(%&'&(98%(&C.%(#( $'&8'&(:'#3.&-%(%#%(D#6#-4&(:'#7.%5H( B'&8'&(.-4'&#&(;.%(3&4'&#.-:('#3.8H( $+.5-,$,'$7*++*$-,$,'$/#,*$%&$-$+,-*$%&$%#$ 9 :;< $%#$> :;< $)#+2,($?@?A$BC$5 DE F$ dr P s P c R s P c R s P c G M s# s R s 2.7 $ 4 Nm 2 P c R % c GM P c r #(r) % dr P s R s N/8:(&?9#%&(/(4&-%'#6( $'&8'&( O/'&('.:/'/8(4#6486#?/-(3&9#-3( 8-4?/-#6(/'9(/'(O ' (#-3(@'A<(-/%( :&-&'#665(#7#.6#D6&H( /9$8%&'(9/3&6.-:(5.&63(#(4&-%'#6( $'&8'&(/+(( PHQR(C()* )S (T9 KP (U(PHQ(C()* )) (#%9((

2 #$%&$5-++$/%#+*G-.%#( 2$&'.4#665(599&%'.4#6(&66(/(9#(3O<(;.%(%.4V-&(3'<(#%( 3.%#-4&('('/9(%&(4&-%&'(/(%&(%#'+( dm r ( 4 dr) H*++*$#$%&$+,-,( O#4'/4/$.4($'&8'&(4/9&('/9(%&(&F8#?/-(/(%#%&(/( %&(9#%&'.#6(@.3&#6(:#(6#;AH( PV NkT dm r dr 4 r2 O#(4/-&'7#?/-( &F8#?/-( W.4%#%&(/;(%&(.-%&'./'(9#(/(#(%#'( 98%(4#-:&(;.%(3.%#-4&('/9(%&( 4&-%&'H( V N k T X/689&( T89D&'(/($#'?46&( Y/6%L9#--Z(4/-%#-%( [&9$&'#%8'&( H*++*$#$%&$+,-,( H*++*$#$%&$+,-,( /-.3&'(:#(4/-%#.-&3(.-(#(456.-3&'(/( 6&-:%(C(;.%($#'?46&(7&6/4.?&(7 C ( f ma dp dt ft p 2p xˆ i t 2 x v x f 2p x t p xv x x &(U(E/'4&($&'($#'?46&(,9$86&('/9( 4/66../-( [.9&(D&%;&&-(4/66../-( f p xv x x p x v x v 2 v x 2 + v y 2 + v z 2 v x 2 v y 2 v z 2 v 2 p x v x 3 pv O/9&-%89($'/$/'?/-#6(%/( 7&6/4.%5( 3 2$&&3(.(%&(F8#3'#%8'&(89( /(7&6/4.?&(.-(&#4(3.'&4?/-( T/($'&&''&3(7&6/4.%5( 3.'&4?/-( [&($'/384%(/(9/9&-%89(#-3( 7&6/4.%5(.-(%&(C(3.'&4?/-(.()\Q( %&(%/%#6(7&4%/'($'/384%(

3 H*++*$#$%&$+,-,( f ( p) p v 3 x N # N p dp H*++*$#$%&$+,-,( W.7.3&(D/%(.3&(D5(#'&#+( F A 3 # N p Ax v pdp df( p) f (p)n p dp 3 N p x pvdp n p dp N p V dp T89D&'(/($#'?46&(;.%( 9/9&-%89($($&'(8-.%(7/689&( [/%#6(/'4&(&C&'%&3(D5( $#'?46&(4/66../-+( F 3 # N p x pvdp B'&8'&(.-%&:'#6( P # n p pvdp 3 #6486#%&($'&8'&<(:.7&-(%&(3.%'.D8?/-( 8-4?/-<(- $ 3$( I-+$7*++*$-#)$9-#$9%3/3-*$ J2C',( pdp mv 2 dv P n v mv 2 dv 3 28D?%8?-:(.-%/(%&($'&8'&(.-%&:'#6( I-+$H*++*$-#)$9-#$9%3/3-*$ J2C',( P g nkt kt µ m H I#($'&8'&(.H%H/H(9&#-( 9/6&486#'(;&.:%( n v dv n p dp $ m ' n v dv n& ) % 2#kT ( n ndv 3 / 2 e *mv 2 / 2kT 4# 2 dv B#'?46&(-89D&'(3.%'.D8?/-( O#C;&66KY/6%L9#--(7&6/4.%5( 3.%'.D8?/-( 2.-4&+( n m µ m m H O&#-(9/6&486#'(;&.:%(

4 9-#$9%3/3-*$J2C',( K%*$-$/%573,3($#,*-3$C-+8( O&#-(9/6&486#'(;&.:%(3&$&-3(/-(4/9$/.?/-(/( &F8#?/-AH(( E'&&(&6&4%'/-(98%(D&(.-4683&3(.-(%&(#7&'#:&(9#( $&'($#'?46&H( m n N m N m n µ n N m N N A N A m m H µ m /m H #$%&'()$*+#$#,'$-(..$ /&(2$/&$'(,$3(&4'$ K%*$-$/%573,3($#,*-3$C-+8( K%*$-$/%573,3($#,*-3$C-+8( µ i N A N ( + z ) )]L^(#44/8-%(/'(%&(-846&8($68(%&(-89D&'(/( '&&(&6&4%'/-(%#%('&86%('/9(4/9$6&%&65(./-.L.-:(#-(#%/9(/(%5$&(^H( µ n m H N N m number particles from mass particle from :(/7&'($#'?46&(;.%( 3._&'&-%(9#&( mass particle from total mass of gas µ n m H N N m,-7&'%(%&(&c$'&./-(/'(#7&'#:&($#'?46&(9#( #-3('&;'.%&((.-(%&'9(/(%&(9#('#4?/-H( N N A m H A m H X X L M $.(%&(9#('#4?/-(/(#%/9( /(%5$&(M( %/%#6(-89D&'(/($#'?46&( ((((%/%#6(9#(/(:#( µ n X A µ n X + 4 Y + A n Z ;&.:%&3(#7&'#:&(/(#66(&6&9&-%(.-(:#( &#7.&'(%#-(53'/:&-(#-3(&6.89H(

5 K%*$-#$2%#2N)$C-+8( µ i + z A X 2X + 3 µ i 4 Y + + z A i Z E/'(&6&9&-%(&#7.&'(%#%(J(#-3(J&+( [/%#6(-89D&'(/($#'?46&(.-4683&3+(/-(J( 4/-%'.D8%&()(&6&4%'/-($68(-846&8<(/-&(J&( 4/-%'.D8%&(P(&6&4%'/-($68(-846&8( + z A i 2 >G2O8( dr G M r g P n p pvdp 3 P g nkt kt µ m H dm r dr 4 r2 µ m m H µ n X + 4 Y + A n Z 2X + 3 µ i 4 Y + + z Z + z A i A i 2 O&#-(9/6&486#'(;&.:%(@-&8%'#6( :#A(( B'&8'&(,-%&:'#6( O#(4/-&'7#?/-(&F-( O&#-(9/6&486#'(;&.:%(@/'( 4/9$6&%&65(./-.L&3(:#A( O&#-(9/6&486#'(;&.:%( I#($'&8'&(&F-( MF-(/(53'/%#?4(&F8.6.D'.89( PG*-C$#*C($7*$7-*./38( K*52DQ2*-/$G+$R%+D2#+,2#( nkt mn v v 2 dv 3 n n v v 2 dv 3kT m v 2 3kT 2 2 mv kt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

6 >-)2-.%#$H*++*( P rad 3 u 3 at 4 >-)2-.%#$H*++*( T/;<(4&4V(%/(&&(;&%&'(.%(;#('&#/-#D6&(%/(-&:6&4%('#3.#?/-( $'&8'&(.-(%&(4/'&(;.%(/8'(3&'.7&3(4/'&(%&9$&'#%8'&+(( P tot kt µm H + 3 at 4 [/%#6($'&8'&(38&(%/(:#( #-3('#3.#?/-(( P tot kt µm H + 3 at 4 #6486#%&(%&(%&9$&'#%8'&(.-(%&(4/'&(/(%&(28-H(,:-/'&('#3.#?/-($'&8'&+( P rad.53 3 Nm #2.65% P gas T c P cµm H k.44 # 7 K 2%#-3#'3(/6#'(9/3&6+( [ 4/'& U)H`aC)* a b( sun 4kg m 3 µ ioninzed.62 P central 2.34 # 6 Nm 2 29#66(4/-%'.D8?/-('/9('#3.#?/-($'&8'&<(/( %#%(#89$?/-(;#('&#/-#D6&H( :,33-*$#*C($+%*/+( :,33-*$#*C($:%*/+( I'#7.%#?/-#6(B/%&-?#6(M-&':5+( U G Mm r E'/9(%&(X.'.#6(%&/'&9(@MF-(PHRa(.-(5/8'(D//VA<(#6(/(%&(4#-:&(.-(:'#7.%#?/-#6(BHMH(.(#7#.6#D6&(%/(D&('#3.#%&3(#;#5(K(%&(/%&'(#6( 8$$6.&(%&'9#6(&-&':5(%#%(&#%(%&(%#'H((( [&(:'#7.%#?/-#6(/'4&(/-(#($#'?46&(/(9#()5 2 (/8%.3&(#( $&'.4#665(599&%'.4(9#<(9(.+((( df g,i G M rdm i du g,i G M rdm i r G M r 4 # dr r R U grav 4G $ M r # rdr # ~ # M 4 3 R3 M R ~ 4 3 r3 # U grav ~ G# 2 R 5 ~ 3 GM 2 5 R E ~ 3 GM 2 R [/%#6(:'#7.%#?/-#6(BHMH( 89&(4/-%#-%(3&-.%5(U(9&#-(3&-.%5( $$'/C.9#%&(%&(9#(.H%H/H(9&#-(3&-.%5( [&-<(&?9#%&(%&(:'#7.%#?/-#6(BHM(( E'/9(%&(X.'.#6(%&/'&9<(%&(%/%#6( 9&4#-.4#6(&-&':5(.(#6(%&( :'#7.%#?/-#6(BHMH(

7 (cc(n A<( /;(984(&-&':5(;/863(#7&(D&&-(6.D&'#%&3('/9(:'#7.%#?/-#6( 4/66#$&d(( E grav (E f E i ) # E f ~ 3 t KH E grav L sun ~ 7 yr 2 GM sun R sun ~.$ 4 J 89.-:(-&#'65(4/-%#-%(689.-/.%5<(/;(6/-:(4/863(%&( 8-(&9.%(&-&':5(#%(%.('#%&d( b&67.-kj&69/6%l(?9&4#6&(k(d#&3(/-('#3./#4?7&(./%/$&(3#?-:<(%&(9//-(#-3(9&%&/'.%&(#'&(/63&'( %#-(%.<(%&'&/'&(:'#7.%#?/-#6(4/-%'#4?/-(4#--/%( #44/8-%(/'(&-&':5(/8%$8%(/(%&(28-H( :,33-*$#*C($:%*/+( &9.4#6('&#4?/-d( [5$.4#6(#9/8-%(/(&-&':5($&'(#%/9(-/%(6.V&65(%/(D&(9/'&(%#-()K)*(&XH(( I.7&-(%&(-89D&'(/(#%/9($'&&-%(.-(#(%#'<(%&(#9/8-%(/(4&9.4#6( &-&':5(.(#6/(#'(%//(6/;(%/(#44/8-%(/'(/6#'(689.-/.%5(/7&'(#('&#/-#D6&( $&'./3(/(?9&H(( M sun 2 3 kg M hydrogen.7 27 kg ~ 57 ev.6 8 J L sun W t E tot L sun T $#'?46& (@89&(8-(.()**e( 53'/:&-A( M-&':5($&'($#'?46&( f89.-/.%5(/(%&(8-( T/%(6/-:(&-/8:(%/(D&( 6*6 * 24 * 365 s/ yr 573 yr 4/-.%&-%(;.%(./%/$.4( 3#?-:(/(9//-(#-3( 9&%&/'.%&( </3-*$.5+/-3+( T846&#'('&#4?/-($'/384&(&-&':5(@O&X($&'( 'C-A( </3-*$.5+/-3( 89&(%&(28-(;#(/'.:.-#665()**e(53'/:&-(#-3(/-65(%&(.--&'( )*e(/(%&(28-z(9#(d&4/9&(/%(&-/8:(%/(8&(53'/:&-(%/( &6.89H(J/;(984(-846&#'(&-&':5(.(#7#.6#D6&(.-(%&(28-d( E nuclear..7 M sun c J 2/9&(9#(.(4/-7&'%&3(.-%/(&-&':5(@MU94 P AH(E/'(&#4( '&#4?/-+( t nuclear E nuclear L sun ~ yr T846&#'(8./-($'/7.3&(#-(&-&':5(/8'4&( %#%(.(4/-.%&-%(;.%(689.-/.%5(/(%&( 8-(#-3(#:&(/(9//-(#-3(9&%&/'.%&H( 9(U(*H*PgSha@9&#-(9/6&486#'(;&.:%A( 9(U(*Hae(

8 S-#,5$9/'-#2/-3$T##32#C( N&4#66(%#%(%&(/-65(7.#D6&(/8'4&(/(&-&':5(%/($'/7.3&( /D&'7&3(%&66#'(689.-/.%5(/'($'&89&3(6.&?9&(/( %&(28-(.(-846&#'(8./-H(( 89&(%#%(&-&':5('&F8.'&3(%/(/7&'4/9&(/86/9D( $/%&-?#6(.(:.7&-(D5(%&'9#6(&-&':5+( 2 mv kt classical Z Z 2 e 2 4 r T classical Z Z 2 e2 6 kr # K J/;&7&'<(4#6486#%&3(%&9$&'#%8'&(/(/6#'(4/'&(.(/-65()* a b( S-#,5$9/'-#2/-3$T##32#C( M7&-(;.%(%&(OY(7&6(3.%'.D8?/-<(?66(-/%(&-/8:(&-&':5(%/(#7&( #(.:-.i4#-%(8./-('#%&h((j/;&7&'(o(%8--&6.-:(#66/;($#'?46&(%/( 9#V&(46/&'(#$$'/#4H(( M?9#%&(%&($#'?46&(;#7&6&-:%(#(%&(3&Y'/:6.&(;#7&6&-:%+(( 2 µ v 2 m p2 Z Z 2 e 2 2µ m 4 # 4 # h Z Z 2 e 2 2µ m T quantum Z 2 Z 2 2 e 4 µ m h 2 k $7 K ( h /#)2 2µ m 2/67&(/'(%&(3&Y'/:6.&(;#7&6&-:%(#-3( 8D?%8%&(/'('(.-(%&($'&7./8(&F8#?/-( h p </3-*$>6#$*-,+$-#)$,'$I-5%O$H-B( </3-*$/*%++$+/.%#( O/'&(4#'&86(4#6486#?/-+('C-('#%&($&'(&-&':5(.-%&'7#6(.-(%&'9( /(%&(-89D&'(3&-.%5(/($#'?46&(;.%(&-&':5(.-($&4.i&3( '#-:&(4/9D.-&3(;.%(%&(%8--&6.-:($'/D#D.6.%5+( '/K&4?/-(.(%&(-89D&'(/('&#4?/-($&'(%#':&%(-846&8($&'( 8-.%(?9&(3.7.3&3(D5(%&(k8C(/(.-4.3&-%($#'?46&H( KE E µ mv 2 2 ( E ) number of rxn /nucleus/time number incident particles/ area / time n E de 2n 2 / 2 kt ( ) 3 / 2 E/ 2 e E / kt de f&%(6$3&-/%&(#(%#':&%($#'?46&(#-3(2(3&-/%&(#-(.-4.3&-%($#'?46&h([&( -89D&'(/('C-<()< <(.(%&(-89D&'(/($#'?46&(%#%(4#-(%'.V&(C(.-(?9&(.-%&'7#6(3%(;.%(#(7&6/4.%5+( v E ( ) 2E /µ m

9 </3-*$/*%++$+/.%#( dn E ( E )v( E )n i,e dedt n i,e de n i n n EdE, n i n i,e de rxns per nucleus dt # [&(-89D&'(/(.-4.3&-%($#'?46&(.(%&( -89D&'(4/-%#.-&3(.-(#(456.-3&'(/(7/689&( UV$GUV$),F(( dn dt ( E )v ( E ) n i n n EdE T89D&'(/(.-4.3&-%($#'?46&(;.%(%&( #$$'/$'.#%&(7&6/4.%5(@.H&H<(&-&':5A(.(#( '#4?/-(/(%&(%/%#6(-89D&'(/($#'?46&H( #5*-,%*( # )(4/9&('/9(%&($'&7./8(OY(3.%'.D8?/-H(T'$ #54*$%&$*6#+$7*$#/3+$7*$.5$2#,*G-3$O2,'$ #*C($4,O#$$-#)$$W$)F( </3-*$/*%++$+/.%#( r i,x E # n x n i ( E )v ( E ) n E n de ' ( ) $% & 2 $% ) h p ( E ) $e -2% 2 U c / E ( 2 *, $ + E U c E Z Z 2 e 2 /4%. r Z Z 2 e 2 µ m v 2 /2 2%. hv ( ) $e -be /2 b / %µ / 2 m Z Z 2 e 2 E 2 / 2. h E/'(# 6 (%#':&%<(%&(%/%#6(-89D&'(/('C-($&'(8-.%( 7/689&(.(/8-3(D5(.-%&:'#?-:(/7&'(#66($/.D6&( &-&':.&( [&(8-4?/-#6(/'9(/(%&(4'/K&4?/-(.(4/9$6.4#%&3H((#(.9$6.i4#?/-<(%.-V(/(.%(#(%&($5.4#6(4'/K&4?/-<(4#6&3( D5(%&(3&(Y'/:6.&(;#7&6&-:%( [&(%8--&6.-:($'/D#D.6.%5(.(&C$/-&-?#6<(:/.-:(%/(L&'/( #(%&(/86/9D($/%&-?#6(:/&(%/(.-i-.%5( /86/9D($/%&-?#6(&-&':5(( D(3&$&-3(/-(9#<(&6&4%'.4(4#':&( </3-*$/*%++$+/.%#( /9D.-.-:(%&(3&$&-3&-4.&(/'(%&(4'/K&4?/-+( r i,x 2 $ # & kt % ( ) S( E ) E 3 / 2 E n i n x µ m ' /2 ebe * ) S E ( ) / 2 ( )e (be (/2 e (E / kt de E'/9(%&(%8--&6.-:( $'/D#D.6.%5( E'/9(%&(OY(%&'9( I-5%O$H-B( [&($'/384%(/(%&(%;/(&C$/-&?#6(#4%/'($'/384&(#( %'/-:65($&#V&3(48'7&+( T846&#'('C-($'/D#D.6.%5(3&$&-3(/-(%&(V.-&?4( &-&':5(/(%&(4/66../-H([&(I#9/;($&#V( #'.&('/9(%&(OY(.:(&-&':5(%#.6(#-3(%&( /86/9D(D#''.&'(%&'9H( [.($6/%('&$'&&-%(4/66../-(/(%;/($'/%/-(#%( %&(4&-%'#6(%&9$&'#%8'&(/(%&(28-H(,(:UV(.(6/;65(7#'5.-:<(.%(4#-(D&(%'&#%&3(#(#( 4/-%#-%(#-3($866&3(/8%(/(%&(.-%&:'#6H((

10 9%)2X/-.%#+$,%$</3-*$*6#$*-,+8(,(%&'&(.(#('&/-#-4&(D&%;&&-(%&( &-&':5(/(%&(.-4/9.-:($#'?46&(#-3(( %&(/'D.%#6(&-&':5(6&7&6(/(%&( $&#V.-:(#%($&4.i4(&-&':.&H(( H%O*$3-O$&%*$*6#$*-,+( NC-('#%&(4#-(D&(;'.G&-(#($/;&'(6#;(4&-%&'&3(#%(#( $#'?486#'(%&9$&'#%8'&H(( r i,x r X i X x #' T $,%(.(#6/($/.D6&(%/(6/;&'(%&(/86/9D($/%&-?#6(D#''.&'(D5( 3/,*%#$+/*#2#C<(;.4(/448'(;.%(%&'&(#'&(9#-5('&&( &6&4%'/-(@./-.L&3(#%/9(#%(.:(4&-%'#6(%&66#'(%&9$&'#%8'&A( X i X x #' 2 $ O#('#4?/-(/(.-4.3&-%(#-3( %#':&%($#'?46&( B/;&'(6#;(&C$#-./-(/('C-('#%&(&F-( Z(lP(/'(%;/KD/35(4/66../-<((4#-('#-:&( '/9()(%/(#D/8%(R*H( H%O*$3-O$&%*$*6#$*-,+( /9D.-&(%&('C-('#%&(;.%(%&(&-&':5($&'('C-(%/(:&%(%&( #9/8-%(/(&-&':5($&'(&4/-3($&'(V:+( Y52#%+2,($C*-)2#,( [/%#6(689.-/.%5(.(%&(89(/(#66(&-&':5(@-846&#'<( 4&9.4#6<(:'#7.%#?/-#6A(:&-&'#%&3(D5(%&66#'(9#%&'.#6H( # i,x & % ( r i,x $ ' ' i,x X i X x ) T * [ W kg + ] ) )'+ M-&':5($&'(V:(?9&(%&('C-('#%&( [&(89(/(&-&':5('&6&#&3($&'( &4/-3($&'(V:(.(%&(%/%#6(-846&#'( &-&':5(:&-&'#?/-('#%&( dl dm nuclear + grav dl r dr 4r2 # [/%#6(&-&':5($&'(V:($&'(&4/-3(D5(#66( -846&#'('C-Z(#-3(D5(:'#7.%5( f89.-/.%5(38&(%/(&-&':5(:&-&'#%&3(;.%.-( %#'(.-%&'./'<(/8%(%/(%&('#3.8<('H(

11 </3%+(#,'+2+$-#)$/%#+*G-.%#$3-O+( T846&/5-%&.('&#4?/-(4/-&'7&(4#':&<(-846&/-( -&8%'.-/<(-FBF-F$32C',$,'2#C+A( R*-#/'2#C$>-.%+( ' pp ~,pp X 2 4 f pp # pp C pp T 6 ~.8 $ %2 W m 3 kg %2 E.'%(%&$(.(%&(6/;&%(K(.-7/67&(%&(3&4#5(/(#( $'/%/-(.-%/(#(-&8%'/-( e + e + 2# M6&4%'/-($68(#($/.%'/-(5.&63(%;/(@.:( &-&':5A($/%/-( #$%&'()#*+*,%-.#$/** -89D&'(/(9#G&'(6&$%/-(K(-89D&'(/(#-?9#G&'(6&$%/-( 5+,$4$/%#+,-#,(.()#/( A Z X L(.(%&(&6&9&-%<(Z(.(%&(-89D&'(/($'/%/-<(P(.(%&(-89D&'(/(-846&/-(@$'/%/-(](-&8%'/-A( M#4(%&$(/(%&($K$( 4#.-(#(#(3._&'&-%( 'C-('#%&(D&4#8&( 3._&'&-%(/86/9D( D#''.&'(#-3(4'/K &4?/-(#'&(.-7/67&3H( [&&('C-('#%&(#'&(/'( %&(4&-%'#6( %&9$&'#%8'&(/(%&( 28-H( [<\$[(/3( [<\$[(/3( 4 2He+ 4 2 He 8 4 Be 8 4 Be+ 4 2 He 2 6 C + # 3$ ~ #',3$ % 2 Y 3 f 3$ T 8 4 E#9/8(m%'.$6&(#6$#($'/4&n(4/-7&'%( &6.89(.-%/(4#'D/-H(( T/%&(%&(3'#9#?4(&-.?7.%5(%/( %&9$&'#%8'&(.-(%&($/;&'(6#;('C-('#%&H( E/'(#(-846&8(;.%((-846&/-<( :(p( $'/%/-<(%&(D.-3.-:(&-&':5($&'(-846&/-+( M D U94 P Uqp9 $ ]@KpA9 - (K9-846&8 r4 P ( 2 6C+ 4 2 He 6 8 O + 6 8O+ 4 2 He 2 Ne + E/'9#?/-(/(&#7.&'( &6&9&-%(%'/8:(%&( #33.?/-(/(#6$#( $#'?46&o<T<><T&<O:<( 2.<(E&( 2.9$65(%&(3._&'&-4&(D&%;&&-(%&(9#(/( Kp+(T89D&'(/(-&8%'/-( '&&($'/%/-(#-3(-&8%'/-(#-3(%&(9#(/( %&(D/8-3(-846&8H( %(%&(#%/9.4(9#(/(E&<(%&(D.-3.-:(&-&':5(.((((

12 #*C($T*-#+7%*,( dr G M r g dm r dr 4 r2 dl r dr 4r2 # M-&':5(%'#-$/'%+( )A PA QA 5&%(/'([@'AH(( N#3.#?/-H(N#-3/9(;#6V(D5($/%/-<(#D/'D&3<(4#G&'&3<('&K&9.G&3H( /-7&4?/-H(J/%(D/85#-%(9#(&6&9&-%(4#''5(&-&':5( /-384?/-H(['#-$/'%(D5(4/66../-(@:&-&'#665(-/%(.9$/'%#-%(.-(%#'AH(( >-)2-.G$T57*-,*$I*-)2#,( rad dr # c F rad 4 3 at 3 dr dr 3 # 4ac T F 3 rad F rad L 4$ dr 3 # L 4ac T 3 4$ N#3.#?7&(k8C('/9(%&(2%&#-K Y/6%L9#--('&6#?/-( N#3.#?/-($'&8'&(:'#3.&-%( N#3.#?7&(%&9$&'#%8'&(:'#3.&-%+(#(&.%&'( %&(689.-/.%5(/'(/$#4.%5(.-4'&#&<(%&( %&9$&'#%8'&(:'#3.&-%(D&4/9&(%&&$&'<H( 2%&&$&'(:'#3.&-%($'/9/%&(4/-7&4?/-H( H*++*$:/-3$]2C',8$/'-*-/,*2+./$3#C,'$+/-3$&%*$/%#G/.%#( K2*+,$Y-O$%&$T'*5%)(#-52/+( H P dr B'&8'&(4#6&(&.:%( du dq dw #-:&(.-(.-%&'-#6(&-&':5(U(&#%(#33&3(@3A( 9.-8(;/'V(3/-&(@3tAH( dr P H P lnp r H P U 3 k % $ ' T 3 2# µm H & 2 nrt,-%&'-#6(&-&':5(.(%&(#7&'#:&(%&'9#6(&-&':5( $&'(9#(&6&9&-%H(;$.(#(m%#%&n(8-4?/-(K(/-65( 3&$&-3(/-(48''&-%(4/-3.?/-(/(%&(:#<(-/%( /-(.%/'5H(,%(.(#(8-4?/-(/(4/9$/.?/-<(<( #-3(%&9$&'#%8'&( P P o e r / H P dr g J B (.(%&(3.%#-4&(;&'&(%&( $'&8'&(3'/$(D5()\B( 28D?%8%&(%&('#3.#?7&($'&8'&(:'#3.&-%(.-%/( &C$'&./-(/'(%&($'&8'&(4#6&(&.:%( nr k #(.(%&(-89D&'(/(9/6&((( µm H NUgHQ)RRaPs(9/6& K) (b K) (U(8-.7&'#6(:#(4/-%#-%( H P P g ~ R Sun

13 :7/2X/$]-,( :7/2X/$]-,$[^( C P Q P C V Q V #-:&(.-(@3A(.(88#665(&C$'&&3(.H%H/H(%&( $&4.i4(&#%<([ H $U+7/2X/$'-,$-,$/%#+,-#,$ 7*++*V$%*$[^$U+7/2X/$'-,$-,$/%#+,-#,$ G%35V( dw F $ # & dr PA $ # & dr PdV m% m % du dq ' PdV I#(&C&'%(#(/'4&<(EUB(/-(%&( &-3(/(#(456.-3&'<(#-3(3/&( ;/'V(9/7.-:(%&($.%/-(#( 3.%#-4&(3'+(tUE3'( %(4/-%#-%(7/689&+( dv ( du du V dq V du )Q )T V C V du C V,-3&$&-3&-%(/(%&( %&'9/35-#9.4($'/4&( E/'(#(9/-#%/9.4(:#+( du 3nR $ # & 2 % C V 3 2 nr :7/2X/$]-,$[ H ( du Q T P P V T P PV nrt PdV + V RTdn + nr P)2-4-./8$)S_A( du dq PdV PdV + V nr du C V du C V PdV C V %(4/-%#-%($'&8'&+( PdV nr P dv nr du C V C V Q T P nr C V C P nr C P C V + nr #6;#5o( C V 3 2 nr C P 5 2 nr C P C V 5 3 # PdV + V nr & % ( PdV $ ' C V PdV + V 2 3 PdV 5 PdV V 3 ) dv V P PV K u.-:(%&(.3&#6(:#(6#;+( ( ) P K'T

14 P)2-4-./$+%#)$+7)8( P)2-4-./$,57*-,*$C*-)2#,8( v sound B [&($&&3(/(/8-3(.('&6#%&3(%/(%&( 4/9$'&.D.6.%5(/(#(:#( P g kt µm H W._&'&-?#?-:(%.(/'9(/(%&(.3&#6(:#(6#;( B V #P #V [&(D86V(9/3868(/;(/;(984(%&(7/689&(/(%&( :#(4#-:&(;&-(.%(.(4/9$'&&3H( dr P dµ µ dr + P d dr + P T dr V # $ P K% W&i-.?/-(/'(X/689&+()\$&4.i4(3&-.%5( v sound $P 3.#D#?4(/8-3($&&3( dr % P d dr W._&'&-?#?-:(%&(#3.#D#?4(:#(6#;( P)2-4-./$,57*-,*$C*-)2#,8( P)2-4-./$,57*-,*$C*-)2#,8( dr P d dr + P T dr P d dr P d dr + P T dr $ # ' & ) T dr adiabatic % ( P dr dr #G M r P g kt µm H 89&(4/-%#-%(( P)2-4-./$,57*-,*$C*-)2#,8$ /%#G/.%#$ # & % ( µm H GM r dr adiabatic $ ' k g C P [&(4/-7&4?7&(4&66(98%('&9#.-(D8/5#-%(#%(&7&'5(6&7&6(.(.%(.(%/(4/-?-8&('..-:H(,(%&(3&-.%5(/(%&(D8DD6&(.(6&( %#-(%&(3&-.%5(/(%&(8''/8-3.-:(:#(.-(%&(%#'<(%&-( %&(D8DD6&(;.66('.&H(( /-3.?/-(/'(4/-7&4?/-+( > dr star dr adiabatic bubble N#3.#?7&(%'#-$/'%(;.66(#6;#5(D&($'&&-%v((/-7&4?/-(4#-( %8'-(/-(/'(/_(3&$&-3.-:(/-($5.4#6(3&$%(.-%/(%&(%#'<( &7/68?/-#'5(%#:&(&%4H( f#':&(%&9$&'#%8'&(:'#3.&-%(#'&(4#8&3(d5(.:(/$#4.%5h(( J.:65(/$#F8&(6#5&'(4/-%'.4%(%&(k/;(/(&#%('/9(D&6/;(K( %&(%#'(4#-(&C$#-3<(/'(8&(4/-7&4?/-(#(#(m#&%5(7#67&Hn(

(6#':&(@T>(4546&A( 9262#C$3#C,'$,'%*($%&$+7*-)2-4-./$/%#G/.%#8( 9262#C$3#C,'$,'%*($%&$+7*-)2-4-./$/%#G/.%#8( E8-3#9&-%#6(4'.%&'.#(/'(4/-7&4?/-+(3&-.%5(/(%&('..-:(D8DD6&(.( 6&(%#-(3&-.%5(/(%&(8''/8-3.

15 :,33-*$/%#G/.%#$ [%#)2.%#$&%*$/%#G/.%#8( E/'(#(9/-#%/9.4(:#<(%.(4/-3.?/-(/'(%&( %&66#'(#%9/$&'&(4#-(D&('&K;'.G&-+( d lnp dlnt < # 2.5 t&-()$6-h$`$)$6-t(c(ph`<(%&(%#'(.(%#d6&(#:#.-%( 4/-7&4?/-H(,-(:&-&'#6<(4/-7&4?/-(/448'(;&-+(@)A(%&66#'(/$#4.%5(.( 6#':&<(@PA(./-.L#?/-(4#8&(#(6#':&($&4.i4(&#%(#-3(6/;( #3.#D#?4(%&9$&'#%8'&(:'#3.&-%(@QA(%&(%&9$&'#%8'&( 3&$&-3&-4&(/(-846&#'(&-&':5('#%&(.(6#':&(@T>(4546&A( 9262#C$3#C,'$,'%*($%&$+7*-)2-4-./$/%#G/.%#8( 9262#C$3#C,'$,'%*($%&$+7*-)2-4-./$/%#G/.%#8( E8-3#9&-%#6(4'.%&'.#(/'(4/-7&4?/-+(3&-.%5(/(%&('..-:(D8DD6&(.( 6&(%#-(3&-.%5(/(%&(8''/8-3.-:(:#H( > dr star dr adiabatic bubble f&%(%&(d8dd6&(d&(#3.#d#?4<(#-3(%&(%&9$&'#%8'&(:'#3.&-%(/(%&( 8''/8-3.-:(D&(%&(%&9$&'#%8'&(:'#3.&-%(/(%&(%#'+( # T % $ dr ad dr star & # ( dr & % ( dr ' $ dr ' 2.-4&(%&9$&'#%8'&(:'#3.&-%(#'&(#6;#5(-&:#?7&+( l )H P 3(.(%&(9.C.-:(6&-:%<(] H (.(%&($'&8'&(4#6&( &.:%H(([&('#?/(/(%&&(%;/(6&-:%<(<( #(7#68&(D&%;&&-(*H`(#-3(QH( > dr bubble dr star q C P T ( )* MC4&(&#%(k/;($&'(8-.%( 7/689&('/9(%&(D8DD6&(%/(%&( 8''/8-3.-:(:#(

16 :,33-*$9%)3$R23)2#C( dr G M r g ]()*%+,-./$2324*258$7*++*$G+$C*-G2,($ :,33-*$9%)3$R23)2#C( dr G M r g ]()*%+,-./$2324*258$7*++*$G+$C*-G2,($ dm r dr 4 r2 9-++$/%#+*G-.%#$ dm r dr 4 r2 9-++$/%#+*G-.%#$ dl r dr 4r2 # Y52#%+2,($C*-)2#,$ dl r dr 4r2 # Y52#%+2,($C*-)2#,$ dr 3 # L 4ac T 3 4$ T57*-,*$C*-)2#,$U*-)2-.GV$ dr 3 # L 4ac T 3 4$ T57*-,*$C*-)2#,$U*-)2-.GV$ # & % ( µm H dr adiabatic $ ' k GM r g C P T57*-,*$C*-)2#,$U/%#G/.%#V( # & % ( µm H dr adiabatic $ ' k GM r g C P T57*-,*$C*-)2#,$U/%#G/.%#V(,(%&(%#'(.(%#?4(&-&':5(4/9&('/9(-846&#'('&#4?/-( :,33-*$9%)3$R23)2#C( (9/3&6(#%9/$&'&(4/-.%(/(#(%#D6&(/(-89D&'(:.7.-:(%&(/8'4&( 8-4?/-(@/'(%&(%&9$&'#%8'&A<(#-3($'&8'&(#(#(8-4?/-(/(/$?4#6(3&$%H(( W&$&-3.-:(/-(%&(#$$6.4#?/-<(#(9/3&6(9.:%(#6/($&4.5+$$ w$q#+2,($ w$i%5,*2/-3$)7,'$ w$p4+%*7.%#$/%a/2#,$ w$3/,*%#$7*++*$ w$9-#$2#,#+2,($ w$^3%/2,($ w$9-c#./$7*%7*.+$$ E'/9(%&(9/3&6<(%&(8'#4&(k8C(4#-(D&(4#6486#%&3+(.H&H<(%&(6.:%(&4#$.-:( %&(8'#4&($&'(8-.%(#'&#H(,-(/'3&'(%/(4#6486#%&(%&(689.-/.%5<(-&&3(%&(.-3&$&-3&-%65(3&%&'9.-&3(%&66#'('#3.8H( :,33-*$9%)3$R23)2#C(,-(#33.?/-(%/(%&(i7&(D#.4(%&66#'(&F8#?/-(@;.4(5/8Z66( -&&3(%/(9&9/'.L&A<(.-/'9#?/-(.(-&&3&3(#D/8%(%&( 4/9$/.?/-(/(%&(%#'(.-(&#4(6#5&'H$ P P(,T,composition) (,T,composition) # #(,T,composition) $$'/C.9#%&(;.%(.3&#6(:#(#-3( '#3.#?/-($'&8'&( #6486#%&(&C$6.4.%65(D5(.-%&'$/6#?-:(#( :'.3(/(T( u&($/;&'(6#;(/'(%&($$(4#.-(/'( T>(4546&(K(&#75(38%5<( 4/9$8%#?/-#6('&#4?/-(-&%;/'V( /9$/.?/-(4#-:&(;.%(?9&(#(&#7.&'(&6&9&-%(/'9H(NC-Z(%#V&($6#4&(.-(%&(4/'&( /(%&(%#'(;&'&($'&8'&(.(.:&%H( >D^&4%(;.%(9#(x(*H*aP(O 2uT (3/(-/%(#7&(.:(&-/8:(%&9$&'#%8'&\$'&8'&(/'($$( 4#.-H(2%#'(;.%(9#(:'&#%&'(%#-(h*(O 2uT (#'&(8-%#D6&(%/(%&'9#6(/4.66#?/-H((

17 :,33-*$9%)3$R23)2#C( 6/(8&(D/8-3#'5(4/-3.?/-(%/(D8.63(%&66#'(9/3&6+$ M r L r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i'%(&%(/( D/8-3#'5(4/-3.?/-H( :,33-*$9%)3$R23)2#C8$P#-3(./-3$P77*%-/'( f#-&km93&-(&f8#?/-(:.7&(#-(#-#65?4#6(/68?/-(/'(%&66#'(%'84%8'&h( J5$/%&?4#6(%&66#'(9/3&6(.-(;.4(%&($'&8'&(3&$&-3(/-(3&-.%5(.-(%&( /'9(H$_$b (V-/;-(#($/65%'/$&H( d % $ ' (G dm r dr # dr & dr d % $ ' (G( 4) ) dr # dr & % $ ' (4)G # dr & d d* % dr r2 g $ ' 4)G # dr & * g U g m B/./-Z(&F8#?/-(/'($&'.4#665( 599&%'.4#6(:'#7.%#?/-#6($/%&-?#6( &-&':5( :,33-*$9%)3$R23)2#C8$P#-3(./-3$P77*%-/'( K d $ % & #2 d dr r2 dr ' ( ) #4*G + n +. - K d $, n / dr r2 (#n )/ n d ' % & dr ( ) #4*G ( ) c [ D n ( r) ] n r $ + ( n +) K #n c & - %, 4*G $ + n & n + - %, ( ) / n ( ) K c.' ) /( ( N ) / n 4*G d $ dd n ' dr % & r2 dr ( ) #D n n.' ) /( / 2 :4+.,,8$$H$_$b Y,8$$$_U?W#V$`$#$$ 67*++$$2#$,*5+$%&$+/-32#C$&-/,%*$Q # $-#)$,'$/#,*-3$)#+2,($ / N&$'&&-%(%&(3.%#-4&(%&'9(D5((

18 :,33-*$9%)3$R23)2#C8$ P#-3(./-3$P77*%-/'( [&-<(;.%($ r n # I&%(%&(.-#9/8(f#-&KM93&-(&F8#?/-+( d 2 d dd 2 n % n $ ' (D n # d & ))2#C,%#$Y52#%+2,($Y252,( P t rad dr kt µm H + 3 at 4 c F L rad c 4# [&($'&8'&(.-(#(%#'(.(-/'9#665(3/9.-#%&3(D5(%&(:#( $'&8'&(%&'9H(Y8%<(/'(%&(.:&%(9#(@.:&%( %&9$&'#%8'&A(%#'<('&:./-(/(6/;(3&-.%5(4#-(D&4/9&( 3/9.-#%&3(D5('#3.#?/-($'&8'&H$ N#3.#?/-($'&8'&(:'#3.&-%(,9$/&(D/8-3#'5(4/-3.?/-(@4/-%#-%(/(.-%&:'#?/-A(K(%&-<( /68?/-(/(%.(WHMH(.-(%&'9(/((/'(#($&4.i4($/65%'/$.4(.-3&C<(-<( :.7&(#(3&-.%5($'/i6&(#(#(8-4?/-(/('#3.8H(( J/;&7&'<(%&(8&86-&(/(%.(#$$'/#4(.(6.9.%&3(K(/-65(#-#65?4#6( /68?/-(/'(%'&&($/65%'/$&+(-U*<)<`H((( O/-/%/9.4(.3&#6(:#+(-U)H`( W&:&-&'#%&(%#'+(-U)H`( M33.-:%/-(%#-3#'3(9/3&6+(-UQ( 5-3+#'&.$&6777$ dr $G M L Ed 4#Gc M E/'(%&(4/-3.?/-(/(53'/%#?4(&F8.6.D'.89<(&%(%&( &C$&'./-(/'(%&($'&8'&(:'#3.&-%(&F8#6(%/(&#4(/%&'(#-3( /67&(/'(%&(899:2;#2$<+-:2.:#H( ))2#C,%#$Y52#%+2,($Y252,( 9-++DY52#%+2,($*3-.%#( E/'(%#'(/-(%&(8$$&'(9#.-(&F8&-4&(;.%([l(`*<***b(@/(./-.L&3(K(;#%(.(%&($'.9#'5(/$#4.%5(/8'4&dA+( L Ed ~.5 3 M M sun L Ed M ~ L sun M sun [&(9#K689.-/.%5('&6#?/-(4#-(D&( 4#6486#%&3(/'(%#'H((( t.6&(%&(689.-/.%5('#-:&($#-('/9( )* KR (f 2uT (%/()* S( f 2uT <(%&(9#('#-:&(.( /-65(#(#4%/'(/(l)***H(( J.:(9#(%#'(#'(/8%$#4&(%&(&-&':5( $'/384?/-(/(6/;(9#(%#'<(#-3(%8( #7&(984(/'%&'(6.&?9&H( L Ed ~ L sun E/'(#(h*(O 2uT (%#'<(%&(M33.-:%/-(689.-/.%5(&C4&&3( %&(/D&'7&3(689.-/.%5(D5(#(#4%/'(/(Qo((%&(%#'(.(.-( 53'/%#?4(&F8.6.D'.89<(D8%(-/%(&C4&.7&65(/v(

to Highbury via Massey University, Constellation Station, Smales Farm Station, Akoranga Station and Northcote

to Highbury via Massey University, Constellation Station, Smales Farm Station, Akoranga Station and Northcote b v Nc, g, F, C Uv Hgb p (p 4030) F g (p 4063) F (p 3353) L D (p 3848) 6.00 6.0 6.2 6.20 6.30 6.45 6.50 6.30 6.40 6.42 6.50 7.00 7.5 7.20 7.00 7.0 7.2 7.20 7.30 7.45 7.50 7.30 7.40 7.42 7.50 8.00 8.5 8.20