The self consistent field method for constructing stellar equilibria: When & why is convergence attained?

Transcription

1 The self consistent field method for constructing stellar equilibria: When & why is convergence attained? Charalampos Markakis (FSU Jena) Richard H Price (UT Brownsville) John L Friedman (UW Milwaukee) 1

2 Applications Single, non rotating star Single, rotating star Binary system 2

3 Articles where methods are used 3 R. James, Astrophys. J. 140, 552 (1964) R. Stoeckly, Astrophys. J. 142, 208 (1965) Y. Eriguchi and E. Müller, Astron. Astrophys. 146, 260 (1985) Y. Eriguchi and E. Müller, Astron. Astrophys. 147, 161 (1985) J. P. Ostriker and J. W. K. Mark, Astrophys. J. 151, 1075 (1968) I. Hachisu, Astrophys. J. Suppl. 61, 479 (1986) E. Butterworth and J. Ipser, Astrophys. J. 204, 200 (1976) M. Ansorg, A. Kleinwächter, and R. Meinel, Astr. Astrophys. 381, L49 (2002) S. Bonazzola and S. Schneider, Astrophys. J. 191, 195 (1974) S. Bonazzola, E. Gourgoulhon, M. Salgado, and J. Marck, Astron. Astrophys. 278, 421 (1993) H. Komatsu, Y. Eriguchi, and I. Hachisu, Mon. Not. R. Astron. Soc. 237, 355 (1989) H. Komatsu, Y. Eriguchi, and I. Hachisu, Mon. Not. R. Astron. Soc. 239, 153 (1989) N. Stergioulas, Living Reviews in Relativity (2003) U. M. Schaudt, Ann. Henri Poincaré 1 (2000) U. M. Schaudt & H. Pfister, Phys. Rev. Lett. 77, 16 (1996)

4 Equations of Stellar Structure Poisson 3 r ( r ) F () r = -Gò d r r - r Euler +F- = 1 2 ( h v ) 0 2 EOS h : = ò 0 r dp r 4

5 Equations of Stellar Structure Poisson 3 r ( r ) F () r = -Gò d r r - r Euler 1 2 h +F- v = k 2 EOS r = rh [] 5

6 Iterative procedure (SCF method) Poisson 3 r ( r ) F () r = -Gò d r r - r Euler 1 2 h +F- v = k 2 EOS r = rh [] 6

7 Variants of the SCF method ρ (ν) Fix κ, Ω Fix κ, R p /R e Fix ρ c, Ω Fix ρ c, R p /R e ρ (ν+1) 7

8 Iteration with fixed k r(r) r(r) Initial guess (n = 1 polytrope) 2nd iteration R r r r(r) r(r) 1st iteration 3rd iteration 8 r r

9 Iteration with fixed central density r(r) 10 r(r) Initial guess (n = 1 polytrope) 8 6 2nd iteration R r 2 r r(r) 10 r(r) st iteration 6 4 3rd iteration (n=3/2 polytrope) r r

10 Iteration Stability SCF method has been used for > 40 years Little is known about why it works when one quantity is fixed and not another Consider behavior of iterative scheme near an exact solution 10

11 Iteration Stability Exact solution 3 r ( r ) F () r = -Gò d r r - r Linear perturbation 3 dr( r ) df () r = -Gò d r r - r 1 2 h +F- v = k 2 dh + df = dk 11 r = rh [] Iteration operator dr dr = dh dh old new 3 r[ h ( r)] h () r = k + G d r ò r - r

12 Iteration Stability Exact solution 3 r ( r ) F () r = -Gò d r r - r Linear perturbation 3 dr( r ) df () r = -Gò d r r - r 1 2 h +F- v = k 2 dh + df = dk 12 r = rh [] Linear iteration operator dr dr = dh dh old new 3 dr dh ( r ) dh () r dk G d r = + ò ( r ) dh r -r

13 Iteration Stability Linear iteration operator old new 3 dr dh ( r) old dh () r dk G d r = + ò ( r ) º L( dh ) dh r - r Fixed k iteration: dk = 0 Fixed h(0) iteration: d h(0) = 0 13

14 Iteration Stability Linear iteration operator new 3 old old dh () r = ò d rk (, r r ) dh ( r ) º L( dh ) Fixed k iteration: d Krr (, r ) = G ( r ) 1 dh r -r 14 æ ö Fixed h(0) iteration: dr Krr (, ) G ( r) 1 1 = dh - r r ç - r çè ø

15 Iteration Stability Linear iteration operator old new 3 dr dh ( r) old dh () r dk G d r = + ò ( r ) º L( dh ) dh r - r Eigenvalue problem L( dh) = ldh Necessary condition for convergence: l < 1 15

16 Iteration with fixed k 16 Linear iteration operator old new 3 dr old dh = G d r dh ò º L( dh ) dh r - r Discretization dh = å L dh new old i ij j j dhr ( ) 1 dhr ( ) 2 dhr ( ) 3 dhr ( ) N

17 Demo 1 Fixed κ linear iteration 17

18 Iteration with fixed k Spectrum of perturbations about an n = 1 polytropic background æ öæ4 öæ ö ç ç L = v v v v (1) (2) (3) ( N ) ç 2 5 v v v v (1) (2) (3) ( N ) ç ç ç ç 4 ç ç ( 2 è øç 2N - 1) çè è ø ø -1 18

19 Iteration with fixed k dh() r l 2 = 4/3 2 unstable mode (l 1 = 4) r 19 pl r R = = pl r / R -1/2 sin( / ) 2 2 () ; l 4,4/3,4/5, -1/2 dhr

20 Iteration with fixed k 20 All eigenvalues < 1, except l 1.

21 Iteration with fixed central density æ ö new 3 dr 1 1 old old dh = Gò d r dh L( dh ) dh - º r r ç - r çè ø Discretization dh = å L dh new old i ij j j 21

22 Demo 2 Fixed ρ c linear iteration 22

23 Iteration with fixed central density A matrix is nilpotent iff it s eigenvalues are all 0 Nilpotency explains convergence $ only one eigenvector, with l=0 Eigenbasis not complete Generalized eigenvectors Jordan decomposition 23

24 Iteration with fixed central density Jordan form of L J æ0 1 ö 0 1 = 0 1 ç 0 çè ø 24

25 Iteration with fixed central density Jordan form of L J 2 æ0 0 1 ö 0 0 = ç 0 çè ø 25

26 Iteration with fixed central density Jordan form of L J N -1 æ0 0 1ö 0 0 = 0 ç 0 çè ø 26

27 Iteration with fixed central density Jordan form of L J N æ0 0 0ö 0 0 = 0 ç 0 çè ø 27

28 Demo 3 Fixed ρ 0 linear iteration explanation 3 dr( r ) df () r = -Gò d r r - r dh + df = dk dr = dr dh dh 28

29 Iteration with fixed central density Degenerate subspace spanned by generalized eigenvectors ( ) k L- li v = 0, k = 1, 2, 3, () k Equivalently, ( L- li) v = 0 ( L- I) v = v, = 2, 3, (1) l k ( k+ 1) ( k) 29

30 Iteration with fixed central density Jordan s theorem: Every square matrix M is similar to a Jordan matrix J: æ ö J1 æl 1 ö i J 2 l i ; J = J3 J = i 1 ç l ç çè i ø çè Jr ø # blocks = # eigenvectors 30

31 Iteration with fixed k Diagonalizable matrix æl ö 1 l 2 ç l 3 L = ç l çè N ø All eigenvalues distinct complete eigenbasis 31

32 Iteration with fixed central density Degenerate subspace spanned by generalized eigenvectors k ( L- li) v = 0, k = 1, 2, 3, () k Equivalently, ( L- li) v = 0 ( L- I) v = v, = 2, 3, (1) l k ( k+ 1) ( k) 32

33 Iteration with fixed central density Jordan decomposition in the continuum? ( L- li) k v = 0, k = 1, 2, 3, () k ( L- li) v = 0 ( L- I) v k = v k (1) l ( + 1) ( ) 33

34 Fixed central density continuum Unable to construct countable basis using thin shells Continuum operator L is not nilpotent k Lv = 0 not true for finite () k It is, however, quasi nilpotent Still able to construct generalized basis using ( L- li) v = 0 ( L- I) v k = v k (1) l ( + 1) ( ) k 34

35 Fixed central density continuum Jordan basis v = 1, v = Lv, v = Lv,, v = Lv p p (0) (-1) (0) (-2) (- 1) ( ) ( + 1) Example: n=1 polytrope 2 4 2p p v = 1, v = - r, v = r,, v = (-1) r (0) (-1) (-2) ( p) 3! 5! (2p + 1)! 35

36 Fixed central density continuum Jordan basis v = 1, v = Lv, v = Lv,, v = Lv p p (0) (-1) (0) (-2) (- 1) ( ) ( + 1) 36 L æ0 1 ö 0 1 = 0 1 ç 0 çè ø

37 Demo 4 Fixed ρ f linear iteration 37

38 38