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Binary Neutron Stars in General Relativity
John Friedman & Koji Uryu
I. Formalism and Analytic Results
University of Wisconsin-Milwaukee
Center for Gravitation and Cosmology
I. EINSTEIN EULER SYSTEM
II. HELICAL SYMMETRY
III. STATIONARY AND QUASI-STATIONARY
EQUILIBRIA FOR ROTATING STARS
IV. 1st LAW OF THERMODYNAMICS FOR
BINARY SYSTEMS
V. TURNING-POINT CRITERION AND
LOCATION OF INNERMOST STABLE
CIRCULAR ORBIT (ISCO)
I. EINSTEIN-EULER SYSTEM:
PERFECT FLUID SPACETIMES
A perfect fluid is defined by its local isotropy:
At each point of the matter there is a timelike
direction ua for which
Tab is invariant under rotations in the 3-space
orthogonal to ua ; that is, a comoving observer
sees no anisotropic stresses.
Denoting by qab = gab+ ub the projection ? ua ,
one can decompose Tab into a scalar Tab ua ub
a spatial vector qag Tag and a symmetric
tensor qag qbd Tgd .
Because any nonzero spatial vector picks out a
direction, qag Tag = 0 for a perfect fluid.
Similarly, the 3-dimensional tensor
is rotationally invariant only if it is proportional to
the 3-metric qab:
Then
For equilibria, these are the main corrections.
For dynamical evolutions -- oscillations,
instabilities, collapse, and binary inspiral, one
must worry about the microphysics governing,
for example viscosity, heat flow, magnetic
fields, superfluid modes (2-fluid flow), and
turbulence.
Binary NS inspiral is modeled by a perfect-fluid
spacetime, a spacetime M,g whose whose metric
satisfies
with Tab a perfect-fluid energy-momentum tensor.
The Bianchi identities imply
and this equation, together with an equation of state,
determines the motion of the fluid.
3+1 DECOMPOSITION FOR FLUID MOTION
The projection of
along ua expresses
conservation of energy, while the projection orthogonal
to ua is the relativistic Euler equation:
Interpretation of the equation:
The fractional change in fluid volume V in time dt is
implying
Then
means
the energy of a fluid element of volume V decreases
by the work p dV in proper time dt .
The projection of
perpendicular to ua is the
RELATIVISTIC EULER EQUATION :
Newtonian limit:
Barotropic flows: enthalpy and injection energy
A fluid with a one-parameter EOS is called
barotropic. Neutron star matter is accurately
described by a one-parameter EOS because it is
approximately isentropic: Neutron stars rapidly cool
far below the Fermi energy (1013K » mp), effectively
to zero temperature and entropy.
(There is, however, a composition gradient in neutron
stars, with the density of protons and electrons
ordinarily increasing outward, and this dominates a
departure from a barotropic equation of state in stellar
oscillations).
1-PARAMETER EOS
Then the Euler eqn
becomes
As you have seen in Nick Stergioulas’ first talk,
introducing h allows one to find a first integral of
the equation of hydrostatic equilibrium for a
uniformly rotating star. We’ll derive the equation
in a broader context including stationary binaries.
COMPACT BINARIES:
QUASISTATIONARY EQUILIBRIA
In the Newtonian limit, because a binary system does
not radiate, it is stationary in a rotating frame.
Because radiation appears only in the 2 1/2 postNewtonian order -to order (v/c)5£ Newtonian theory,
one computes radiation for most of the inspiral from a
stationary post-Newtonian orbit.
Time translations in a rotating frame are generated by
a helical Killing vector ka.
In the Newtonian limit and in the curved spacetime of
a rotating star, ka has the form
where ta and fa are timelike and rotational Killing
vectors. For a stationary binary system in GR, one
can choose t and f coordinates for which ka has this
form with tat and faf. .
(One can define a helical KV by its helical structure in
spacetime: There is a unique period T for which each point P
is timelike
separated from the
corresponding point
parameter distance T later
along the orbit.)
ka is timelike near the fluid
ka is spacelike outside the
light cylinder at v W = 1
Although ka is spacelike outside a large cylinder, one
can, as usual, introduce a 3+1 split associated with a
spacelike hypersurface S. Evolution along ka can again
be expressed in terms a lapse and shift,
na is the future pointing unit normal to S
ba a vector on S
In a chart t, xi , for which S is a t = constant surface,
the metric is
Equation of hydrostatic equilibrium
We found that the relativistic Euler equation is
When the spacetime has a Killing vector ka with ua along ka ,
we have
Proof of
:
The equation of hydrostatic equilibrium now takes the
form
with first integral
where
is a constant
( is the injection energy per unit baryon mass needed to
bring baryons at infinity to the same internal state as that in the
star, lower them, give them the speed of the baryons in a fluid
element, open a space to put them, and inject them into the
star).
For irrotational flow, a good approximation at late stages of
inspiral, u is not along a Killing vector, and the proof fails. It is
nonetheless still possible to recover an equation of hydrostatic
equilibrium.
Quasiequilibrium models are based on helically
symmetric spacetimes in which a set of field equations
are solved for the independent metric potentials and the
fluid density. From, e.g., the angular velocity and
multipole moments of a model, one can compute the
energy radiated and construct a quasiequilibrium
sequence.
Until recently, these sequences (and initial data for NS
binaries) were restricted to spatially conformally flat
metrics, the IWM (Isenberg-Wilson-Mathews)
approximation.
ISENBERG-WILSON-MATHEWS ANSATZ:
SPATIALLY CONFORMALLY FLAT METRIC
fab flat.
Five metric potentials y, a, b i are found from
five components of the Einstein equation:
1 Hamiltonian constraint
3 components of the Momentum constraint
Spatial trace of Einstein eq: gab(Gab-8p Tab)
IWM solutions have 5, not 6, metric functions and
satisfy only 5 of the 6 independent components of the
Einstein equation. An IWM spacetime agrees with an
exact solution only to 1st post-Newtonian order.
Initial data then has some spurious radiation and
cannot accurately enforce the W (r) relation.
Orbits from the data can be elliptical. One
improvement is obtained by adding the asymptotic
equality between Komar and ADM mass.
To do better, we need the remaining metric degree of
freedom.
Shibata and Uryu have recently solved the full set of
equations, in a waveless approximation in which time
derivatives of the extrinsic curvature are artificially
dropped.
Two advantanges of helically symmetric solutions
to the full set of Einstein equations:
 Energy in gravitational radiation is controlled
(smaller than that of the outgoing solution, for
practical grids).
 By satisfying a full set of Einstein-Euler equations,
one enforces a circular orbit.
Because data obtained by solving the initial value
equations alone or from an (spatially conformally
flat) approximation satisfy a truncated set of field
equations, they yield elliptical orbits.
An exact helically symmmetric solution is
not asymptotically flat,
because the energy radiated at all past times is present
on a spacelike hypersurface.
At a distance of a few wavelengths (larger than the
present grid size) the energy is dominated by the mass
of the binary system, and the solution appears to be
asymptotically flat.
Only at distances larger than about 104 M is the
energy in the radiation field comparable to the mass
of the binary system.
Work in progress by
Price, Beetle, Bromley;
Shinichirou Yoshida, Uryu, JF
seeks to solve the full equations without truncation
for a helically symmetric spacetime.
In a helically symmetric spacetime, the constraints
remain elliptic, but the dynamical equations have a
mixed elliptic-hyperbolic character:
elliptic where ka is spacelike and hyperbolic where ka
is timelike:
Flat space wave equation with helical symmetry:
In general,
Problem is not intrinsically elliptic, BUT after
spherical harmonic decomposition,
have coupled system of elliptic equations
(Helmholtz eqs), each of form
Uryu has a full code (a modification of the waveless
code). No convergence yet for the full problem, but
toy problems with nonlinear wave equation and two
orbiting point scalar charges as source converge:
Remarkably, iteration converges for l of order unity.
In example below (due to Shin Yoshida), l1.
Curt- This figure was too large – I’ll send it
separately.
For a change that locally preserves vorticity, baryon number
and entropy,
dM=WdJ
TURNING POINT STABILITY AND
LOCATION OF THE ISCO
The first law allows one to apply a turning point theorem (Sorkin 1981)
to sequence of binary equilibria.
The theorem shows that on one side of
a turning point in M at fixed J
or in J and fixed baryon mass M0,
the sequence is unstable.
The side on which M is smaller is more tightly bound - the stable side.
The other side is unstable. (The argument for the instability does not
imply that one can reach the lower energy state by a dynamical
evolution; the angular momentum distribution might need to be
redistributed by viscosity to allow transition to the lower energy state.
But the fact that there is a lower energy state with the same vorticity
suggests that we may have identified the point of dynamical
instability.)
Note:
The theorem gives a sufficient condition for secular instability.
It does not imply that the low-mass side is stable – there could be
other nearby configurations, not on the sequence, that have lower
mass.
Lai, Shapiro, Shibata, Uryu, Meudon , . . .
Baumgarte, Cook, Scheel, Shapiro, Teukolsky:
In the figure thin solid
lines of constant J are
plotted on a graph of rest
mass vs central density. A
red curve marks the onset
of orbital instability, the
set of models with
maximum rest mass M0
(and maximum mass M
along a sequence of
models with constant rest
mass M0. Blue segment is
sequence of corotating
stars, ISCO at intersection.
At much larger central
density, the stars are
widely separated, and
another set of massmaxima along the constant
J curves (e.g., point B in
the inset) marks the the
onset of instability to
collapse of the individual
stars. The turning point
method identifies both
instabilities.
Gravitational mass vs frequency at the ISCO
(Oechslin, Uryu, Poghosyan, Thielemann 2003)
for a hadronic EOS and an EOS with a quark core