Astérosismologie des étoiles de type solaire. Test de composition chimique et de structure interne. Applications aux étoiles centrales de système planétaire.

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HAL Id: tel-00269053

https://tel.archives-ouvertes.fr/tel-00269053

Submitted on 2 Apr 2008

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Astérosismologie des étoiles de type solaire. Test de composition chimique et de structure interne.

Applications aux étoiles centrales de système planétaire.

Marion Laymand

To cite this version:

Marion Laymand. Astérosismologie des étoiles de type solaire. Test de composition chimique et de structure interne. Applications aux étoiles centrales de système planétaire.. Astrophysique [astro-ph].

Université Paul Sabatier - Toulouse III, 2008. Français. �tel-00269053�

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(11)

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ρ

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~v

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ρ d~v

dt = − ∇ ~ p + ρ ~ f

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(13)

2

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dt + p dV dt

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V = 1/ρ

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dρ

dt )

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dt = Γ 1 p ρ

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(14)

2

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ρ 0

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26

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ξ r (r, θ, φ, t) = √

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p ⁰ (r, θ, φ, t) = √

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^N ² ^o^2!9 ^Y

...

(15)

2

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ξ r

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I8%/L/F%&*-@J5-_,:$'bw%@J1E:$,1H<:$%&*48:$,.+5<=0%5jP%@85j/*-:,)6

k h =

√ l(l+1) R

:; 8@G5-P%/"e$:$,125TEF&:$*-54o

"!#$8:$%@%&8@=I:,1+;R,<:$'bw%@>125-*-W,P12,:`8;<1KA`;<*=0&@8,.=0%

548H=*$5-H:$,R

| m | ≤ l

^o

9;0,1

l = 0

].5-i:&8*45-5-+/*4:$,@:$,.i%/1*-5-Dq+@/8*4W$,,Ci548,./%@G1<G5OQFL/:$*45-o+9,<

5-b&;*g/]0:,RFL/;1<*-25-H:$&*-5-5-+/*4:$,q,:$,f|}%/1*-5->N

l

=%&,1R,Q*4'7=?:$%n/9AC;5-5-9!+548;%/YLo EFG-<F%!'X"'J# J".)/ /,'X"' ' #','%&'X F%&

,*-,./%@:f1;<*4D548D&:5-;<&*-:$,1K8@/H:$%@'7J1,l5-DF8AC;0+&*-:$,GN

2 o2 4$YL]N

2 o2 Y]vN

2 o2 :CY]

N 2 o2!6 YLo\80%@'7:,*-AC;b&=<F8%&*4AC;88EF854*-'(*-,,.b548b*4,8:,,C;81F8AC;0+&*-:$,k1L|

!`*-,,,CK548K'(=54*4/;<18

ξ ˜ _r (r)

^]

p ˜ ⁰ (r)

]o4ogo09,<H5jb&;<*4/+]:$,M,:+/>5-K'(=5-*g/;1_&,K5-

&*-5-1<qLD5-^AC;0,./*g/F8^t95TEF8AC;*-54*-w%@q&,D54l

^

]=0%:$'('7:`1*4&Fo8^FAC;0+/*4:$,DQF8%&*g!$,C

54:$%&P6

dξ _r

dr = − ( 2 r + 1

Γ 1 p dp

dr )ξ _r + 1 ρc ² ( S _l ²

ω ² − 1)p ⁰ + l(l + 1)

ω ² r ² φ ⁰

^N ² ^o^2!= ^Y

dp ⁰

dr = ρ(ω ² − N ² )ξ _r + 1 Γ 1 p

dp

dr p ⁰ − ρ dφ ⁰ dr

N 2 o54

^ Y

1 r ²

d

dr (r ² dΦ ⁰

dr ) = 4πG( p ⁰ c ² + ρξ r

g N ² ) + l(l + 1)

r ² φ ⁰

^N ² ^o54 ² ^Y

ρ ⁰ = ρ

Γ ₁ p p ⁰ + ρξ r ( 1 Γ ₁ p

dp dr − 1

ρ dρ

dr )

^N ² ^o5434$Y

P;fm %@F8AC;8,<8828%/L/F%&*-n/*4A`;<82==%/*4&@8,.>10,<98>F8AC;0+&*-:$,k65j%&FA`;<8,

:$;n/*-AC;28%/L/F%&*-n/*4A`;<>:$;R%@F8AC;,8B1B'bw

S l

]01F0,<*->=0%26

S _l ² = l(l + 1)c ²

r ² = k _h ² c ²

^N ² ^o54 ^Y

5-k%&F8AC;,8S1 B_% <,Cn|}h&%*-'%5(%

N ²

⁶

N ² = g( 1 Γ 1 p

dp dr − 1

ρ dρ

dr )

^N ² ^o54?:CY

(16)

\26

,),:+/A`;<P5-q:fL3(8*-,./J18G@e`@&#8'(P1EF8AC;0/*-:,)&:,C_%&F85-]C1:$,

ω ²

^8n_%@F85T]

+Q@_tB1*4%&KAC;

ω

8n_%@F885v:$;(*-'7W$*4,0*4%&_=<;%8o`8i=I%@&;%&w0/*-:,l&:$,.q1:$,B6`'7:%@/*48]

:$; %&:$*4&/+,C&81M :$, Lmf=?:$,8,.&*-8545- N:$,y 5-:%&b1c5- 8:$,.!$&*-:$,vY]D:$;:$@8*-545j,.&8

N ==0%@*4&*-:$,1Q:$,<18/YLo

"$!"$# !& ;%% "$ % & !

!&

(' ' - 7.7.A 3! 1 -0/2143 5 3 ?41

8FA`;+/*4:$,bW$F,F8%&5-b1:$&*-5-5-+/*4:$,B8:$,<@/*g/;,.;, @e`@&#8'(M1E:$%&1%@ : AC;E*-5

@J1<*43(8*45-K1EF/;<1*-%q'G_8'(P"!89;,<9==<%&:`9ne`'7=<&:/*4AC;o ,)1*4'7*4,`;<5OQ:%&1%@G1;

nefn/#'7KW$%&8t>5TQ==%@:"m<*4')/*-:,k1Gr_:!7K5-*4,W>AC;*08:$,<&*-n/t>,F8W5-*-W8%^548D=?8%@&;%&w+/*4:$,

1;R=?:/,C&*-5W$%/"!`*4/+/*4:$,,5

Φ ⁰

^o

8KF8AC;0/*-:,K1!`*-,,,CP54:$%&>6

dξ r

dr = − ( 2 r − 1

Γ 1

H _p ⁻ ¹ )ξ _r + 1 ρc ² ( S _l ²

ω ² − 1)p ⁰

^N ² ^o54 ⁶ ^Y

dp ⁰

dr = ρ(ω ² − N ² )ξ _r − 1 Γ 1

H _p ⁻ ¹ p ⁰

^N ² ^o54 ⁸ ^Y

:3R

H _p ⁻¹ = − d ln p/dr

9,<_5-"+_1E:$&*-545j+&*-:$,1Q:%&1%@P%/+1*j50F5-L!$F]f5-i:$,</*4:$,i=%&:=%&q!+%&*48,.qwI8;f|

:$;=c=5-;<J%/=<*-1'7,CJA`;<>5-8HAC;0+,C&*4&F8Ktb5TEF8AC;*45-*4w%&+o<D,c=%&'7*4#8%@>==%@:"m<*4')/*-:,]C5-

nefn/#'721L!f*48,.S6

dξ r

dr = 1 ρc ² ( S _l ²

ω ² − 1)p ⁰

^N ² ^o54 ⁹ ^Y

dp ⁰

dr = ρ(ω ² − N ² )ξ r

N 2 o54

< Y

r_nefn/#'71QFA`;+/*4:$,=?8;<QF8%@*-%&7&:$;<b5jc:$%&'(71Q;,<F8AC;0+&*-:$, 1*[IF8%@8,./*485-54

1;R&8:$,<1:$%@1%&6

d ² ξ _r dr ² = ω ²

c ² (1 − N ² ω ² )( S _l

ω ² − 1)ξ r

N 2 o54 = Y

df:$*gH8,<8:$%@76

d ² ξ r

dr ² = − K (r)ξ r

N 2 o3^ Y

:3R

K(r) = ω ² c ² ( N ²

ω ² − 1)( S l

ω ² − 1)

(17)

,1<*-@&*-,W;91;fmR"26

df*

K(r) < 0

^]

ξ r

;W$'(8,./iLmf=?:$,,C&*-5-5-'7,.\:;S1*4'7*4,C;DLmf=?:$,8,.&*-8545-'7,C8o9,

B826

| N | < | ω | < S l

:$;

S l < | ω | < | N |

df*

K (r) > 0

^]

ξ _r

@;,2:$,L/*-:,M:$&*-5-5-,./91<B%ov9,KB826

| ω | > | N |

| ω | > S l

60'(:f1<8H=

| ω | < | N |

^L

| ω | < S l

60'7:`18HW

(' ' 3 5H7

J%&F8AC;,8i1D8'7:`18@~W$F,F8%&5-'7,./%&#W$%/+,1D=0%%&==?:$%@~tJ5-_%&FA`;<8,

1 B_% ,.@|Th %*4%5%)6

ω N

^o ,R=?8;<1:$,<BF8%@*-%&6

K(r) ∼ 1

c ² (ω ² − S _l ² )

^N ² ^o^X2 ^Y

S1<e`,0+'7*4A`;<G1G8J'7:`1q@J=<%&*-,<8*-=5-'7,Cl1<F/%&'(*-,FG=%_5-2!+%@*j+&*-:$,15j

!`*4&8@&1;{&:$,{8,

r

682'7:`182@(:$'(=I:$%n/,C21<:$,8:$'('(b18B:$,1<8B8:;@&*-AC;8 1:'7*4,F8G=0+%P5-G:$%@891k=<%&8@&*4:$,o)%@85-+/*4:$,N

2 o

X2

YH8@9&*-'(*-5-*-%@>t)5-7%&5j+&*-:$,R1

1*4&=?8%@&*-:,1Q;<,K:$,1K=5-,H:$;n/*-AC;SN

K(r)

8n_+&&*4'7*45-Fq;"%@%&FK1H5-98:$'(=I:/,./

%&1*j+5-1<;S,:'bw%&D1Q:$,<1 YoG,B'7:`1D=28n\5-H&;=?8%@=I:$@*4&*-:$,>1l=54;&*48;%@:$,1&:$,:%&8

=5-,8o

%@F8AC;8,<8H1 B_% ,.n|}h %+*-%+5(%K^5-%@F8AC;8,<8J1J~+'bwk@:$,./%&8F8^1,5-0W;%&

N 2 o2 Y_8,c:$,&*-:$,M1<;%& e:$,R125TEF/:*-5-N 10,<54>"K1;df:$548*45*-*Yo

, !`; =<%&F8F81'7'(8,.BAC;(5j=;54/+&*-:$, 18B'7:`18>@S"%&&F8%@*-&F=0%2;,

!+548;%^&85-54KA`;<

| ω | > S l

o`r_:$'('7K5TE*-545-;n/%@Jw*-,75-90W$;%@bN 2 o2 YL].5-l'7:`18l=7&:$,.q1:$,

=*4F8W$F10,~;,l%&FW$*-:,B1<q5OQFL/:$*45-^@*4/;<F8l8,./%@DV;,B=I:$*4,.1q%@&:$;%&,<8'(8,.@lL5-J&;%n8

1B5TEF&:$*-54N =%K%&FWLmf*-:$,0YLov8G=I:$*4,./K12%&L/:$;%@,8'(8,.G18G:$,1G:$%&%@8@=I:$,<18,.>+;fm

F8%&:1<

K(r)

^o

2%/"e$:$,R1;R=I:*-,.K12%&L/:$;%@,8'(8,.8]

r _t

]0&285-;5-2t=0%n/*4%H125jb%&5j+&*-:$, 6

ω = S l =

p l(l + 1)c

r t ⇒ r t =

p l(l + 1)c ω

N 2 o 4$Y

8:$'(=I:/,./2%&1*j+5-P1<;,:'bw%&21E:$,1<B@F8W$5-2tRN1 =<%&#8

2 o

X2 YP6

k _r ² = ω ² − S _l ² c ²

N 2 o

$

Y

d<7!+548;%G1*-'(*-,C;S"!$b5j(=%&::,18;<%b6I5-(&%/a@/:*-%&S1b5OQ:$,<1k71:$,k&8,1,8

tB&9:$;%&w?8%>N0W$;<%&

2 o4$Yo<k=<%&::$,<18;%q')m<*4')+5-HAC;E85-54P=?8;<J+&&8*4,1%&G8n

r t

of[D5-;

(18)

\29

2 o2

# # ! # ! ! #

!" # # ! '%$ ' # !& ! ' # # ! ( *),+ #-

. # !%/

S _l

^!

l =

^021#3 ^#546& ⁷ ^', ^! ^!

!

098:8

µ

^; ^# ^! ^! ^< ^! ^098:8:8

µ

^; ^# ³ ^#

# !&

o

5-S!+5-;%J1

l

^8@H +*-w54]ft;<,9%&FA`;<8,S1:,,F8+]=54;H5jS!+548;%_1

r t

8@K=?&*4&]<=<5-;

8k'(:`18B=I;<!$,.+@/8*4,1%@71k%@F8W$*4:$,B=%&:`8S1; 8,./%@S,:$;<S=?8%@'7L&/,.b1

@:$,1%=<%&::$,<1F8'(8,.5TE*-,./F%&*48;%J1<B5OQFL/:$*45-+o

(' '= 3 5

%&F8AC;,871=;54/+&*-:$, 18B'7:`18>@B*-,fF8%&*48;%@tM5j%&FA`;<8,71GB_% ,.@|

h&%*-'%5(%<of[\:;%K18:f1<%&8K%&1*j+;fmMW$%&,1]0:$,R

ω ² S _l ²

^L ^@ =I;<8EF8%&*-%@6

K(r) ∼ l(l + 1) r ² ( N ²

ω ² − 1)

^N ² ^o ^:CY

71<e`,0+'7*4A`;<S1B8P'(:f1<88@G1:$'(*-,<F8B=0%

N ²

]vE8n>t(1*-%@S=0+%5-(W%/"!`*4/F+o?l,

@&*4'7*45j,. @7N%Yt>;,,:'bw%&_1Q:,1].:,(:w</*48,.D5j9'G_8'(J%@85j/*-:,kAC;K=I:;%D18D:,18

12W$%&"!f*g/F+o0G,R'7:`19W(8n;,2&;=?8%@=I:&*4&*-:$,c1E:$,1<8K12W$%/"!`*4&Fo

==?85-:,HAC;25jb%&FA`;<8,S1 B_% ,.@|Th %*4%5%b8@1FL0,*4B=%>6

N ² = g( 1 Γ 1 p

dp dr − 1

ρ

dρ

dr )

(19)

2 o54

## ! # # # ! / 9 !6 #! 7) ' # !

! # !& # !:( )

l = 30

¹

ν = 3

^; ^# ^'^!# ⁾

l = 100

¹

ν = 3

^; ^,& ^# ^' ^#) ^'^! ^&# ^# ^# ^#

# !& # ''

1 3 8:8 4

df*

N ² ≥ 0

^]I5-:$%@

d ln(p)/d ln(ρ) ≥ 1/Γ ₁

^od`*:$, *4')+W$*-,2;,<bw;545-B1b'7+/*4#8%&BAC;*

'(:$,./c!$%&M5j{&;%n 81 5OQFL/:$*45-]D5OQ [L)1< 5j{W$%/"!`*4&F@c54:$%&(=54;)W$%&,1A`;< 5j

=?:$;@&F8P*-,1;<*4/G=0%q5- W;*-1G8,.!`*-%@:$,,0,.BN5jB=I:$;<&&F91ExP%@*-'(#81"YG6`5jkw;<5-5-%&L!`*-8,.

t/=?:$&*g/*4:$,*-,<*4/*-5-+]C18K:$@8*-545j+&*-:$,<_;f/:$;%H1<B5-k=?:$&*g/*-:,M1QFAC;*-54*-w%@>==0+%/*4&&,."o

UV5eM(1:,2=%&:$=W.+&*-:$,1Q:$,<1o

df*

N ² ≤ 0

]+5-:$%@G:$, 75j7@*4&;0+/*4:$,*4,.!$8%@&+o)w;545-B,b%&L!`*-8,.>=>t)&=I:$@*4&*-:$,

*4,*4&*j54(65-b'(:$;<!8'(8,.28nk+'7=54*40FS2;,<(:$,1:$,.!$/*4:$,==0%&-8o8>:,18

12W$%&"!f*g/F>AC;*@B=<%&:$=0+W$8,.Gt5TE*-,./F%&*-;%J1B8L&&B:$,2&:,CFL!+,8@88,.&88o

G:$,1*g/*-:,

N ² ≤ 0

F8AC;*4!+;f^;k8%&*g/#%&i1Hdf7_%&&*45-1=?:$;%~5TQ==0+%&*4&*-:$,>1Q;,<

:$,2128:$,.!8&*-:$,o

9,<J5-G"J1Q:$@8*45-5j/*-:,N

N ² ≥ 0

Y]f5-S%@85j/*-:, N 2 o :CYi,:;J*-,<1*-AC;AC;2 :$%&%@8n|

=?:$,1Rt5jb%&FAC;8,S'7mf*-'754PAC;2=I;<!$,.>+@/8*4,1%@>5-8H:,1812W$%& !`*g/Fo

r_8'(:f1K&:$,.G1:$,28;`m;&@*=*-FW$F8K10,<G;,<B:$,B1B5TEF/:*-5-bNL0W$;<%&)N

2 o2 Y&Yo

8S=I:*-,./B1)%@&:$;%&,<8'(8,.b:$%&%@8&=?:$,1<8,.t5-R:$,1*g/*4:$,

N = ω

o9,<b5-"+k1

FL/:$*45-8l1Ve`=IJ&:$5-*-%@8].*-5vLmf*-n/G18;fm7=?:$*-,.&q1G%&L/:$;%@,8'(8,.P6`;<,=%&:`G1;8,./%&

LK;,t5jbw&2125j:$,28:$,.!8&*4!o

PG:$,<18KW(,Q:,CK=K8,:$%&BF&FB1<F/&F88G10,K548HF&:$*-548H1>Ve`=?B@:$5j+*-%&+o

(' ' 3 5

DQ:%&1%@B%&1*-5,&@:f*-F2t0AC;S'7:`19@G:$*4&*^6

n > 0

=?:$;%H5-H'7:`1=o

n < 0

=?:$;%H5-H'7:`1W<o

n = 0

=?:$;%H5-H'7:`1KJN\=?:$;%J:$,1'7,.;fmYo

Astérosismologie des étoiles de type solaire. Test de composition chimique et de structure interne. Applications aux étoiles centrales de système planétaire.

HAL Id: tel-00269053

https://tel.archives-ouvertes.fr/tel-00269053

Submitted on 2 Apr 2008

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Astérosismologie des étoiles de type solaire. Test de composition chimique et de structure interne.

Applications aux étoiles centrales de système planétaire.

Marion Laymand

To cite this version:

Marion Laymand. Astérosismologie des étoiles de type solaire. Test de composition chimique et de structure interne. Applications aux étoiles centrales de système planétaire.. Astrophysique [astro-ph].

Université Paul Sabatier - Toulouse III, 2008. Français. �tel-00269053�

ξ r

β

α

− 1

µ

ι

ι

ι

ι

ι

∂ρ

∂t + div(ρ~v) = 0

ρ

~v

ρ d~v

dt = − ∇ ~ p + ρ ~ f

f ~

~g

~g

~g = − ∇ ~ Φ

Φ

∇ 2 Φ = 4πGρ

dq dt = dE

dt + p dV dt

dq/dt

V = 1/ρ

dq

dt = 1

ρ(Γ 3 − 1) ( dp

dt − Γ 1 p ρ

dρ

dt )

Γ 1

Γ 3

Γ 1 = ( ∂lnp

∂lnρ ) ad

Γ 3 = ( ∂lnT

∂lnρ ) ad

p

ρ

T

Γ

p = k b ρT µm u

k b

µ

m u

dq

dt = 0 = ⇒ dp

dt = Γ 1 p ρ

dρ dt

p(r, t) = p 0 (r) + p 0 (t)

δp(~r, t) = p(~ r 0 + δ~r) − p(~ r 0 ) = p 0 ( r ~ 0 ) + δ~r.~ ∇ p 0

ξ r

ξ h

δ~r

∇ ~ p 0 = ρ 0 g ~ 0 = − ρ 0 ∇ ~ Φ 0

∇ 2 Φ 0 = 4πGρ 0

dq

dt = 0

ρ 0 = − 1 r 2

∂

∂r (ρ 0 r 2 ξ r ) − ρ 0 ∇ ~ h · ξ ~ h

ρ 0

∂ 2 ξ r

∂t 2 = − ∂p 0

∂r − ρ 0 g 0 − ρ 0

∂Φ 0

∂r

∇ ² Φ = 4πGρ

dt − Γ ₁ p ρ

p(r, t) = p 0 (r) + p ⁰ (t)

δp(~r, t) = p(~ r 0 + δ~r) − p(~ r 0 ) = p ⁰ ( r ~ 0 ) + δ~r.~ ∇ p 0

∇ ² Φ 0 = 4πGρ 0

ρ ⁰ = − 1 r ²

∂r (ρ 0 r ² ξ r ) − ρ 0 ∇ ~ ^h · ξ ~ h

∂ ² ξ r

∂t ² = − ∂p ⁰

∂r − ρ ⁰ g 0 − ρ 0

∂Φ ⁰

1 r ²

∂r (r ² ∂Φ ⁰

∂r ) + ∇ ² h Φ ⁰ = 4πGρ ⁰

4π ξ ˜ r (r)Y _l ^m (θ, φ) exp( − iωt)

p ⁰ (r, θ, φ, t) = √

4π p ˜ ⁰ (r)Y _l ^m (θ, φ) exp( − iωt)

p ˜ ⁰ (r)

Y _l ^m

Y _l ^m (θ, φ) = ( − 1) ^m c lm P _l ^m (cos θ) exp(imφ)

P _l ^m

ξ ˜ _r (r)

p ˜ ⁰ (r)

dξ _r

dr )ξ _r + 1 ρc ² ( S _l ²

ω ² − 1)p ⁰ + l(l + 1)

ω ² r ² φ ⁰

dp ⁰

dr = ρ(ω ² − N ² )ξ _r + 1 Γ 1 p

dr p ⁰ − ρ dφ ⁰ dr

1 r ²

dr (r ² dΦ ⁰

dr ) = 4πG( p ⁰ c ² + ρξ r

g N ² ) + l(l + 1)

r ² φ ⁰

ρ ⁰ = ρ

Γ ₁ p p ⁰ + ρξ r ( 1 Γ ₁ p

S _l ² = l(l + 1)c ²

r ² = k _h ² c ²

N ²

N ² = g( 1 Γ 1 p

ω ²

Φ ⁰

H _p ⁻ ¹ )ξ _r + 1 ρc ² ( S _l ²

ω ² − 1)p ⁰

dp ⁰

dr = ρ(ω ² − N ² )ξ _r − 1 Γ 1

H _p ⁻ ¹ p ⁰