Hamilton-Jacobi irrotational hydrodynamics for binary neutron star inspiral

Transcription

1 Hmilon-Jcoi irroionl hydrodynmics for inry neuron sr inspirl Chrlmpos Mrkkis Mhemicl Sciences, Universiy of Souhmpon work in progress in collorion wih: John Friedmn, Msru Shi, Nils Andersson, Nicls Moldenhuer, Dvid Hildich, Kourou Kyuoku, Bernd Brüegmnn

2 Inroducion Grviionl wves from neuron-sr nd lck-hole inries crry vlule informion on heir physicl properies nd proe physics inccessile o he lorory. Alhough developmen of lck-hole grviionl wve emples in he ps decde hs een revoluionry, he corresponding work for doule neuron-sr sysems hs lgged. Recen progress y groups in Frnkfur (Whisky), Kyoo (SACRA), Jen (BAM), Clech-Cornell-CITA-AEI (SpEC) ec. The Vlenci scheme hs een workhorse for hydro in numericl reliviy 1 ( ru ) = ( - gru ) = 0 -g 1 g T = ( -gt )-G T = 0 g -g

3 Inroducion Grviionl wves from neuron-sr nd lck-hole inries crry vlule informion on heir physicl properies nd proe physics inccessile o he lorory. Alhough developmen of lck-hole grviionl wve emples in he ps decde hs een revoluionry, he corresponding work for doule neuron-sr sysems hs lgged. Recen progress y groups in Frnkfur (Whisky), Kyoo (SACRA), Jen (BAM), Clech-Cornell-CITA-AEI (SpEC) ec. The Vlenci scheme hs een workhorse for hydro in numericl reliviy, u considering lernive hydrodynmic schemes cn led o furher progress Hmilonin mehods hve een used in ll res of physics u hve seen lile use in hydrodynmics

4 Inroducion Consrucing Hmilonin requires vriionl principle Crer nd Lichnerowicz hve descried roropic fluid moion vi clssicl vriionl principles s conformlly geodesic d dx dx r dp h - g d = 0 h = 1 + d d 0 r

5 Inroducion Consrucing Hmilonin requires vriionl principle Crer nd Lichnerowicz hve descried roropic fluid moion vi clssicl vriionl principles s conformlly geodesic d dx dx r dp h - g d = 0 h = 1 + d d 0 r Moreover, Kelvin s circulion heorem d hu dx 0 d c = implies h iniilly irroionl flows remin irroionl. S 0 S

6 Inroducion Consrucing Hmilonin requires vriionl principle Crer nd Lichnerowicz hve descried roropic fluid moion vi clssicl vriionl principles s conformlly geodesic d dx dx r dp h - g d = 0 h = 1 + d d 0 r Moreover, Kelvin s circulion heorem d hu dx 0 d c = implies h iniilly irroionl flows remin irroionl. S 0 S Applied o numericl reliviy, hese conceps led o novel Hmilonin or Hmilon-Jcoi schemes for evolving relivisic fluid flows, pplicle o inry neuron sr inspirl.

7 Crer-Lichnerowicz vriionl principles for roropic flows Crer s Lgrngin: Cnonicl momenum: Crer s superhmilonin: h h = g u u - =-h (on shell) 2 2 dx p = = hu ; u = u d 1 h = pu - = g pp + = 0 2h 2

8 Crer-Lichnerowicz vriionl principles for roropic flows Crer s Lgrngin: Cnonicl momenum: Crer s superhmilonin: h h = g u u - =-h 2 2 dx p = = hu ; u = u d 1 h = pu - = g pp + = 2h 2 0 Euler equion in Crer-Lichnerowicz form: dp - = p - = 0 (Euler-Lgrnge) u d x dp d + = u ( p - p ) + = 0 (Hmilon) x

9 Consrined Hmilonin pproch dx dx d h - g d = d h 1-g n n d = 0 d d -1 n = ( v + ) fluid velociy mesured y norml oservers v = dx / d fluid velociy mesured in locl coordines L n p = = h = hu v 2 1- n cnonicl momenum of fluid elemen

10 Consrined Hmilonin pproch dx dx d h - g d = d h 1-g n n d = 0 d d -1 n = ( v + ) fluid velociy mesured y norml oservers v = dx / d fluid velociy mesured in locl coordines L n p = = h = hu v 2 1- n cnonicl momenum of fluid elemen Consrined Hmilonin: H = p v - L =-p + h + g p p =-hu 2 Euler-L grnge equion: ( + ) p = u L Hmilon equion: p + v ( p - p ) =- H

11 Conservion of circulion Euler-Lgrnge equion: ( + ) p = L ( + ) w = 0 u u S V oriciy 2-form: w =p - p S 0 d d Kelvin's heor em: pdx dx x ( ) dx x 0 d = w d = + w d = u d 0 The mos ineresing feure of Kelvin's heorem is h, since is derivion did no depend on he meric, i is exc in ime-dependen spceimes, wih grviionl wves crrying energy nd ngulr momenum wy from sysem. In priculr, oscilling srs nd rdiing inries, if modeled s roropic fluids wih no viscosiy or dissipion oher hn grviionl rdiion excly conserve circulion Corollry: flows iniilly irroionl remin irroionl.

12 Irroionl hydrodynmics Irroionl flow: p - p = 0 p = S Hmilon equion: p + v ( p - p ) + H = 0 Hmilon-Jcoi equion: S + H = 0 Exmple: In he dus limi on Minkowsky ckground, one oins relivisic Burgers equion: ( / 1 ) ( 1 ) 0 S 1 ( S) u - u u 2 = = 0 Oined noncovrinly y LeFloch, Mkhlofnd nd Okumusur, SINUM 50, 2136 (2012) y lgeric mnipulion of he Euler equion in Minkowski nd Schwrzschild chrs. The fc h hese re Hmilon equions nd cn e oined covrinly for rirry spceimes is unnoiced. Soluions o HJ equion re NOT unique. Neverheless, 'viscosiy' soluions o HJ equion re unique.

13 Irroionl hydrodynmics Irroionl flow: p - p = 0 p = S Hmilon equion: p + v ( p - p ) + H = 0 Hmilon-Jcoi equion: S + H = 0 For roropic fluids, he ove equion is coupled o he coninuiy equion, resuling in sysem æ k r ö ær u ö + k k = p ç i çd H è ø è i ø 0 where r : = - gru = gru, g = de( g ) ij Chrcerisics : l = 0 k 1,2 k k 2 2 1/2 2 2 kk 2 k 2 1/2 k = (1 - c ) { (1 -c ) c (1 - ) [(1 - c ) -(1 -c )( ) ] } - 3,4 s s s s s l n n n n g n Complee eigensis The sysem is srongly hyperolic (for finie c s )

14 Nole feures: Conclusions æ k r ö ær u ö + k k = p ç i çd H è ø è i ø 0 Unlike Vlenci, recovery of primiives from conservives requires no mosphere: u i is recovered vi dividing p i =hu i y specific enhlpy h which is 1 he surfce (no division y zero) Like Vlenci, srong hyperoliciy is los when c s = 0: eigensis no complee, sysem ecomes wekly hyperolic insiliy on surfce Insed of rificil mosphere, cn use crus EOS wih smll u nonzero c s ner surfce: sound speed in relisic NS crus (ouer 1 km) c s ~ 0.05

15 Sound speed profile of TOV sr 2 c S c S ~0.05 R (km)

16 Nole feures: Conclusions æ k r ö ær u ö + k k = p ç i çd H è ø è i ø 0 Unlike Vlenci, recovery of primiives from conservives does requires no mosphere: u i is recovered vi dividing p i =hu i y specific enhlpy h which is 1 he surfce (no division y zero) Like Vlenci, srong hyperoliciy is los when c s = 0: eigensis no complee, sysem ecomes wekly hyperolic insiliy on surfce Insed of rificil mosphere, cn use crus EOS wih smll u nonzero c s ner surfce: sound speed in relisic NS crus (ouer 1 km) c s ~ Then, exrpoling he EOS o he exerior (h<1) llows one o evolve smooh fields nd oin poinwise convergence on he surfce, which is uninle wih n rificil mosphere. Scheme my e comined wih symplecic inegrion or consrin dmping mehods h preserve symplecic srucure nd circulion SPH schemes sed on he Lgrngin or Hmilonin formulion possile Exension eyond irroionl flows lso possile Reference C. Mrkkis, rxiv: