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Astrosismologie relativiste et ondes gravitationnelles :ce que pourrait nous dire la danse des etoiles a neutrons
Loıc Villain
DFA, Universidad de AlicanteE-03690 Alicante, Espana
etLUTH, Observatoire Paris-Meudon
F-92195 Meudon, [email protected]
base sur des collaborations avecSilvano Bonazzola (LUTH), Pawe l Haensel (CAMK, Warszawa)
Annecy, Seminaire LAPTH, Avril 2006
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 1 / 61
Outline
1 General relativity and gravitational waves
2 Neutron stars
3 Instabilities and oscillations of compact objects
4 Inertial mode instability and spectral methods
5 Summary, conclusions and perspectives
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 2 / 61
General relativity and gravitational waves
1
Gravitational waves in general relativity
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 3 / 61
General relativity and gravitational waves
1
Gravitational waves in general relativityA : brief history and main features
By. A. NAGAR
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 4 / 61
General relativity and gravitational waves
Prediction of gravitational waves (GWs)
First mention
1906, Poincare : any theory of gravitation with Lorentz invariance should lead tosome “ondes gravifiques” (but no valid theory found)
Oscillations in general relativity
1916, Einstein : “gravitationalwaves” exist in general relativity(GR) : progressive wave
1918, Einstein : “quadrupoleformula” (→ GW emissivity) forslow motions and weak gravitationalfield found in a given system ofcoordinates → coordinate wavesor physical waves ? ? ?
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 5 / 61
General relativity and gravitational waves
Theoretical proof
GWs and characteristic surface
Finzi (1949) : covariante proof of the deplacement at c of characteristicsurfaces associated to Einstein equations → quite unknown work (privatecommunication by S. Bonazzola)
GWs and energy
Pirani (1956) : “What would happen to my detector if a GW would gothrough my lab ?” → energy deposite → physical detection possible
Bondi, Schoen, Yau, Witten (and others) (1962-1982) : “What wouldhappen to an objects emitting GWs ?” → mass loss (≡ energy) until apositive value is reached → physical emission possible
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 6 / 61
General relativity and gravitational waves
GWs observation
Observational proof
Hulse & Taylor (1982) : discoveryand precise observation of a binarypulsar PSR B1913+16 → Nobelprize in 1993
First try of direct detection
Weber (1965) : first bar, no actualdetection
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 7 / 61
General relativity and gravitational waves
Hulse & Taylor : indirect observation
indirect experimental verification ofGWs existence but also very precisetest of GR : theoretical prediction
versus observational data
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 8 / 61
General relativity and gravitational waves
1
Gravitational waves in general relativityB : multipoles and emission
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 9 / 61
General relativity and gravitational waves
GWs emission and mass multipoles
Electromagnetism
at least time variations of electric dipole to emit electromagnetic waves
breaking of spherical symmetry (∼ massless spin 1 boson)
Gravitation
at least time variations of mass quadrupole to emit GWs in GR
∼ breaking of spherical and axial symmetries (∼ massless spin 2 boson)
Notes
axial symmetry with time variation of radial distribution also leads to GWs
in others relativistic theories of gravitation : scalar waves also
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 10 / 61
General relativity and gravitational waves
Current multipoles
Quadrupolar flux (l = m = 2) leading toa current quadrupole and GWs
Electromagnetism
other possibility : magnetic dipole
∼ breaking of electric flux sphericalsymmetry
Gravitation
in GR “current multipoles” lead toGWs
basic example : spherical mass wtihconstant density but non-symmetricinternal motions
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 11 / 61
General relativity and gravitational waves
Criteria for relevant emissivity
Emissivity
dE
dt∼ GN
c5s2ω6M2R4
s breaking of symmetry, M mass, R typical radius, ω typical frequency
problem : GN /c5 ∼ 3 × 10−53 I.S. → very weak emission
no GWs in lab (current quadrupole : another 1/c factor)
Astrophysical emissivity
solution : rewrite the formula with Schwarzschild radius, Rs = 2MGN /c2,typical velocity v, and ω ∼ v/R
dE
dt∼ c5
GNs2
(Rs
R
)2 (v
c
)6
with astrophysical units the 10−53 factor has become a 10+53 factor...
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 12 / 61
General relativity and gravitational waves
Astrophysical GWs
Typical amplitude in astrophysical units
h ∼ 2× 10−19
(M
M
)1/2 (1 Mpc
d
) (1 kHz
f
) (1 ms
τ
)1/2
ε1/2 ,
f frequency, d distance to the source, M solar mass, Mpc megaparsec(∼ 3× 1022 m) : Local group of galaxies, τ duration of the emission, εefficacity (= ratio between emitted and mass energies)
N.B. : detection of h and not h2 → 1/d signal but weak...
relevant sources
relativistic (v ∼ c) compact (high M/R) object with coherent internalmotions (to avoid destructive interferences)
compact binaries (black holes, neutron stars, strange stars) + isolatedcompact object : oscillating, rotating (without axial symmetry) and/oraccreting
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 13 / 61
General relativity and gravitational waves
1
Gravitational waves in general relativityC : direct detection and gravitational astronomy
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 14 / 61
General relativity and gravitational waves
Result of a GW
Recipies to detect a GW
several test masses (6= electromagnetism)
continuous (bar) or discrete (mirrors of an interferometer) mass distribution
narrow (bar) or wide (interferometers) band
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 15 / 61
General relativity and gravitational waves
GW detection in 2006
Detectors
Bars (and sphere) : numerous, more sophisticated than Weber’s ; looking forcorrelations ; drawbacks : narrow band...
Interferometers : various taking data (LIGO, TAMA, GEO), VIRGO in“commissionning”, LISA Pathfinder for 2007 ( ?) ; drawbacks : moreexpansive, more complex...
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 16 / 61
General relativity and gravitational waves
Historical parenthesis : Meudon Observatory’s detector
From left to right : Jean Thierry-Mieg, Georges Herpe, Silvano Bonazzola andMichel Chevreton (around 1965).
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 17 / 61
General relativity and gravitational waves
Worldwide network of bars and interferometers
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 18 / 61
General relativity and gravitational waves
European network
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 19 / 61
General relativity and gravitational waves
VIRGO sensibility curve
Noise
at low frequency : mainly sismic noise
at high frequency : quantum fluctuations of laser beam, thermal noise ofmirrors
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 20 / 61
General relativity and gravitational waves
All sensibility curves
Detectors
at low frequency : LISA (no sismic noise in space)
at high frequency : ground detectors (here LIGO)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 21 / 61
Neutron stars
2
Compact astrophysical objects (material ones)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 22 / 61
Neutron stars
2
Compact astrophysical objects (material ones)A : (pre-)history
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 23 / 61
Neutron stars
Prehistorical footsteps
Theoretical birth
1932, Chadwick : discovery of neutron
1932, Landau (following a legend) : idea of neutron stars on the same dayof the announcement ; → reality : “prediction” before the discovery(Haensel, et al.)
1934, Baade & Zwicky : very cautious proposal
With all reserve we advance the view that supernovæ represent thetransitions from ordinary stars into neutrons stars, which in their final stagesconsist of extremely closely packed neutrons
1939, Tolman, Oppenheimer et Volkoff : first structure calculation of aneutron star modelized as a relativistic Fermi gas of neutrons → modernmodels more complex : degeneracy pressure ∼ 1
3 of the resistance (stronginteraction crucial)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 24 / 61
Neutron stars
Discovery
History
1967, Bell & Hewish : firstobservation of a pulsar (lighthousemodel Gold and Pacini)
1996, Walter et al. : first opticalobservation of an isolated neutron star
2002, Cottam et al. : observation ofgravitational redshift of a NS’satmosphere spectrum
2004, Lyne et al. : first double pulsar(PSR J0737-3039A and PSRJ0737-3039B)
2006 : more than 1500 known pulsars
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 25 / 61
Neutron stars
A parallel history : strange stars
X-ray observation of 3C58 byChandra
History
1971, Bodmer : true fundamental stateof matter = deconfined quark plasmawith possible hypercharge
1984, Witten : same idea + proposalof “strange stars”
1986, Haensel et al., Alcock et al. :first numerical models (MIT bag model)
2002, NASA : announcement ofrevolutional discoveries : 2 isolatedneutron stars said to be strange stars →nothing more than a perfect illustrationof what a scientist should not do...
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 26 / 61
Neutron stars
2
Compact astrophysical objects (material ones)B : birth and main features
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 27 / 61
Neutron stars
Birth of compact objects
Gravitational collapse of massive stars iron cores
iron core mass > Chandrasekhar mass ∼ 1.5 Solar mass
collapse (in a few ms) → density increases → electronic captures +photo-dissociations
fission of nuclei, free neutrons appear
bounce (at saturation density) and possible ejection of outer layers (fallingslower : 1 s)
initial mass of the star > 45M : no supernova, black hole formation
central remnant = warm soup (T ∼ 6.1011 K) of neutrons, protons, electronsand neutrinos
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 28 / 61
Neutron stars
Gravitational waves from the collapse
Main mechanisms
during collapse, fast variation of mass quadrupole (Q) → GWs
strong convective motions in the remnant → GWs
bounce also gives fast variation of Q and GWs
if formation of BH : Q increases and then decreases
excited BH : loss of “gravitational hair” to give a Kerr BH (damping ofquasi-normal modes in a few ms)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 29 / 61
Neutron stars
Characteristics at birth
After collapse
(Proto)Neutron Star (PEN) with mass 1.2 to 2 M (∼ iron core mass)
T ∼ 6.1011 K : npeν matter opaque to ν but fast cooling
possible fall back of surrounding matter → possible black hole and GWs(typically : Mi > 20M → BH)
N.B. : true story depends on metallicity, angular momentum, existence of acompanion, magnetic field, etc. (stellar wind in very massive stars :Mi ∼ 60M can lead to a NS)
Birth of a neutron star (8− 10 < Mi/M < 20)
cooling through diffusion/emission of ν → ν-transparency in t < 1 minute(T < 3.1011 K)
anisotropic emission of ν → GWs
npe plasma becomes a degenerated Fermi liquide (TF ∼ 1011 K) of n and p(with ultrarelativistic e gas) at supranuclear density (ρ ∼ 10−14 g.cm−3)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 30 / 61
Neutron stars
More exotical features
Physics in very dense matter
strong interaction → possible formation of nucleon Cooper paires → BCSsuperfluidity (for some densities since effective interaction) : TF ∼ 1011 K,Ts ∼ 1010 K
at very high densities → exotical particles and states (superfluid hyperons,deconfined quarks, etc.)
→ during collapse, possible phase-transition leading to GWs
Collapse and compact object
radius ∼ 102 km to 10 km→ compactness parameter (M/R)= 0.1 after 10 s→ relativistic object (Sun : ∼ 10−6 ; Schw. black hole : R = 2 M)
magnetic flux conservation → very high magnetic fields : 108−9 G to 1013 G
(almost) conservation of angular momentum during collapse → very highangular velocity (period P ∼ 1 ms) → electromagnetic radiation (lighthousemodel) and possible instabilities leading to GWs
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 31 / 61
Neutron stars
Structure of an old compact star (Source F. Weber)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 32 / 61
Neutron stars
Density profile
Typical energy density profiles of a neutron and strange stars [Glendenning (1997)]
(rough) approximation : constant density (better for strange stars and moremassive ones)
N.B. : surface density of strange star 6= 0 (self-bound object due to QCD)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 33 / 61
Instabilities and oscillations of compact objects
3
Instabilities and oscillations of rotating compactobjects
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 34 / 61
Instabilities and oscillations of compact objects
3
Instabilities and oscillationsA : link between modes and instabilities
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 35 / 61
Instabilities and oscillations of compact objects
Modes of neutron stars
Modes of isotropic stars
a spherical star without“anisotropic physics” (stress inthe rigid crust, magnetic field,etc.) → all modes with the sameazimuthal number m have thesame frequency : wlm ≡ wl
Pattern speed (“apparentvelocity”) :positive m →σ+ = w
m = dφdt > 0 → prograde
negative m →σ− = w
m = dφdt < 0 →
retrograde
no axial part for the velocity field[parity (−1)l for polar, (−1)l+1
for axial]
NSs main modes for various equationsof state (from Kokkotas et al.)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 36 / 61
Instabilities and oscillations of compact objects
Modes of rotating stars
Rotating star
degeneracy breaking (splitting wlm 6= wl(m−1)) + modification offrequencies
wi = wr −m Ω +Clm(Ω2)
axial modes enter into the game
progrades and retrogrades modes are not affected in the same way : “sign” ofthe pattern speed may change
↔ trace of an instability
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 37 / 61
Instabilities and oscillations of compact objects
Non-axisymmetric instabilities of rotating stars
Main classes
Dynamical : hydrodynamical timescale (here millisecond)
Secular : related to some “dissipative process” → longer timescale [ex. :viscosity (τ > 103 s), angular momentum loss, etc.]
Dissipative processes triggers of secular instabilities in NSs
viscosity : conservation of angular momentum, dissipation of vorticity[physical basis : n-n diffusion (low T) or beta reactions n ↔ p + e + ν(high T)]
gravitational radiation : dissipation of angular momentum, conservation ofvorticity
Summary of the principle
viscosity : evolution from (E0, L, C0) → (E < E0, L, C < C0)
GWs : evolution from (E0, L0, C) → (E < E0, L < L0, C)→ both made possible by spontaneous symmetry breaking but competition
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 38 / 61
Instabilities and oscillations of compact objects
Secular instabilities
Some (almost) detail on the secular instabilities
viscosity driven instability tries to make the star non-axisymmetric and rigidlyrotating → GW emission
GW driven instability tries to make the star keep a given non-axisymmetricfixed shape → inner fluxes → differential rotation → damping due to viscosity
possibility to define some “canonical energies” (in the inertial frame or in therotating frame) such that for a given eigenmode, a negative value of theenergy means that the mode is unstable
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 39 / 61
Instabilities and oscillations of compact objects
CFS criterion
Chandrasekhar-Friedman-Schutz (1970-1978)
in a rotating fluid, modes that are progrades in the rotating frame butretrogrades in the inertial frame are driven to instability by any mechanismthat can evacuate angular momentum away from the fluid
Principle ∼ such a mode gives a negative contribution to the total angularmomentum of the star, but, through GWs (for instance), carries away apositive amount → total angular momentum of the star decreases → GWsemission makes the angular momentum of the mode more and more negative→ instability...
verified when wr wi < 0 ↔ wr (wr ± m Ω) < 0 with azimuthal number mand star angular velocity Ω
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 40 / 61
Instabilities and oscillations of compact objects
Direct consequences of the CFS criterion
Generic instability of rotating relativistic fluids
any rotating relativistic perfect fluid is unstable → for every mode, there is aminimal angular velocity of the star above which an instability appears : Ωl,m
for increasing values of m :
- (Ωl,m ) → (instability )
- (viscous timescale ) → damping of the instability easier
- [growing time of the instability (related to multipoles) ] → (instability )
relevance in actual relativistic stars (neutron stars) obtained through awindow of instability for each mode in the plane (Temperature, AngularVelocity)
but in actual stars, various phenomena make the game more tricky :compressibility, equation of states, magnetic field, differential rotation etc.
→ need to proceed to realistic and detailed numerical studies
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 41 / 61
Instabilities and oscillations of compact objects
f mode window of instability (l = m = 2)
f mode window of instability (l = m = 2) : angular velocity of the star in Keplervelocity units (= maximal velocity for which the star is not losing matter at the
equator). N.B. : minimal value > 90% ΩK .
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 42 / 61
Instabilities and oscillations of compact objects
3
Instabilities and oscillationsC : inertial modes instability
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 43 / 61
Instabilities and oscillations of compact objects
Inertial modes inertiels (in Newtonian perfect fluids)
Main features
inertial modes ↔ generated byCoriolis force : only in rotatingfluids
bidimensional : Rossby modes(cf. atmospheric physics)
mainly fluxes → very smalldensity perturbations → almostno mass quadrupole...
CFS instability
but in 1998, Andersson and Friedman & Morsink : r modes verify CFScriterion ∀Ω → large window of instability
explanation : wr = 2 m Ωl(l+1) → wi = −Ω m (l+2)(l−1)
l+1 → wi wr < 0
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 44 / 61
Instabilities and oscillations of compact objects
R-mode window of instability (l = m = 2)
R-mode window of instability (l = m = 2) : much larger than for all othermodes...
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 45 / 61
Instabilities and oscillations of compact objects
Possible implications of the instability
Baby NSs
explanation of a limit for young NSs period ? (Andersson et al., Lindblom etal.)
a way to identify strange stars among compact astrophysical objects ?(Andersson et al.)
Oldest NSs
NSs in low mass X-ray binaries
continuous transfert of angularmomentum (and matter) byaccretion → the reason why LowMass X-ray Binaries have verysimilar frequencies ? (Bildsten,Wagoner)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 46 / 61
Instabilities and oscillations of compact objects
Relevance of the r-mode instability ?
Open questions
influence of frame-dragging ? (∼ Lense-Thirring) → ∼ differential rotationΩ = Ω(r, θ)
exact characteristic of the modes in fast rotating relativistic stars ?
growing of the instability in young and hot NSs (PNSs) ?
influence of the crust ?of a magnetic fields ?
coupling to other modes ?
exotical composition :→ viscosity ? (hyperons superfluidity ?)→ fast cooling → time spent in the instability window ?
r-modes in superfluid stars ?
non-linear saturation amplitude ?
in binary systems, influence of accretion ?
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 47 / 61
Inertial mode instability and spectral methods
4
Inertial mode instability and spectral methodscf. Villain, Bonazzola & Haensel (2005)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 48 / 61
Inertial mode instability and spectral methods
Villain, Bonazzola & Haensel 2005
Goals
improve the study of inertial modes (typical time millisecond) in GR (mostof studies done with Newtonian gravitation and post-newtonian corrections)
investigate the influence of realistic equations of state (not only polytropsP ∼ nγ or barotrops P = P (n) as in most of past studies)
take into account stratification (inhomogeneous composition) and its effectthrough the existence of various microscopical timescales
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 49 / 61
Inertial mode instability and spectral methods
Methodology
Spectral numerical simulation
time evolution (linear here) through resolving of relativistic Euler equations(coming from ∇T = 0 for a perfect fluid)
tridimensional spectral code based on spherical coordinates → highnumerical stability → long simulations → very precise spectra
initial version of the code (analytical tests in simplest Newtonian andpost-Newtonian situations) Villain & Bonazzola (2002) upgraded to take intoaccount stratification
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 50 / 61
Inertial mode instability and spectral methods
Numerical and technical parenthesis
Usual solving of partial differential equations
functions known on a discrete lattive, derivative calculated by finitedifferences
Main advantage : fast implementation
Main drawback : no way to use spherical coordinates (singular operators) →badly defined surface of stars
Spectral solving
generalisation of Fourier transforms (cf. spherical harmonics for Poissonequation) → functions known by their coefficients in a reciprocal space,derivatives calculated by linear algebra ↔ (semi-analytic method)
Main advantage : very high precision (no numerical viscosity, less numericalpoints needed), spectral basis chosen to fit with the physical situationgeometry, etc.
Main drawback : implementation takes more time (elementary operations toimplement one by one), need to change everything for new topology, etc.
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 51 / 61
Inertial mode instability and spectral methods
Spectral methods in general relativity
Numerical relativity and relativistic astrophysics at Meudon Observatory (LUTH)
since the early 90ies DARC later LUTH : Silvano Bonazzola and Jean-AlainMarck, later E. Gourgoulhon, J. Novak, P. Grandclement, ...
relativistic astrophysic : compact objects and gravitational waves sources
up to recently : stationnary situations (in dynamical situations no easytemporal spectral basis)
hydrodynamics : finite differences for time coordinate
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 52 / 61
Inertial mode instability and spectral methods
Hypothesis
Main features and timescales
npe degenerated matter (T TF ) without superfluidity
perfect fluids (viscous time >> 1 ms) in comotion (strong interactioneffect)
electric neutrality (plasma frequency ∼ 10−20 s)
breaking of beta equilibrium (n ↔ p + e) due to millisecond oscillations(relaxation time through weak interaction : between some hours and somemonths)
influence of the modes on the background metric neglected : Cowlingapproximation, GW signal from multipoles
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 53 / 61
Inertial mode instability and spectral methods
Chosen problem
Questions
no relaxation → frozen composition for any lump of matter (nulleLagrangian perturbation of proton fraction x)
equations of state = effective barotrops at equilibrium Peq (nb, xeq(nb))(baryonic number density and proton fraction) but non-barotropic EOSs fordynamical situation
coupling between composition g modes and r-modes (l = m = 2)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 54 / 61
Inertial mode instability and spectral methods
Illustration of numerical results
Power spectra of density (m = 2) perturbation time evolution for various angularvelocity. Splitting of the two components m = ± 2 can be seen for g-modes ;
r-mode increases with Ω
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 55 / 61
Inertial mode instability and spectral methods
Energy fluxes (without stratification)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 56 / 61
Inertial mode instability and spectral methods
Energy fluxes (with stratification)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 57 / 61
Inertial mode instability and spectral methods
Summary, conclusions and perspectives
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 58 / 61
Summary, conclusions and perspectives
NSs oscillations and high frequencies GWs ?
NSs are natural laboratories to study nuclear matter at very high density andgravitation in the strong field regim, but current observations giveinformations only on global quantities (M,R,M/R) or on what happens atthe surface
to better understand the outside (magnetosphere, bursts, etc.) knowledge ofinner structure needed : modes are witnesses of what happens inside
moreover NSs are relativistic objects → GWs are direct witnesses of the innerstructure
during gravitational collapses, many mechanisms can emit GWs, among themoscillations
but not all modes emit GWs in a relevant way : instabilities are probablymore interesting
CFS criterion is a very useful tool to identify instabilities, but microphysicsmakes the game more tricky
r-modes are very promising, but probably other instabilities are still to bediscovered...
plenty of reasons to study the dynamics of compact objects and to improveGW detectors in the high frequency band
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 59 / 61
Summary, conclusions and perspectives
Works in progress and perspectives
coupling to Maxwell equations → MHD → study of soft gamma repeaters(cf. 17/12/2004 magnetar)
non-linear study → saturation amplitude of the instability and consequentlyof the expected gravitational wave signal
proto-neutron stars : warm fast rotating objects [collaboration with Alicante,cf. Villain et al. (2004)]
prediction of the gravitational signal by simulations with less approximations(collaboration with Meudon, Bonazzola, Gourgoulhon, Grandclement, Novak)
Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 60 / 61
Summary, conclusions and perspectives
References
Villain, Bonazzola & Haensel, Inertial modes in stratified rotating neutronstars : An evolutionary description, Phys.Rev. D71 (2005) 083001
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