Download Effects of r-modes induced differential rotation on long-term

Effects of
r-mode induced differential rotation on
the long-term evolution and
gravitational wave radiation of
neutron stars (preliminary results)
Yun-Wei Yu （俞云伟）
IOA, Huazhong Normal University
Outline
 Differential rotation induced by r-modes
 A phenomenological model for r-mode evolution
 Long-term evolution of isolated and accreting NSs
 Gravitational radiation from NSs
 Outlook
1、Differential rotation induced by r-modes
 R-modes in a perfect fluid star with arbitrary rotation
arise due to the action of the Coriolis force with positive
feedback (Andersson 1998; Friedman & Morsink 1998),
succumbing to gravitational radiation driven CFS instability
(Chandrasekhar 1970; Friedman & Schutz 1978).
Rossby waves on the earth
The r-modes of rotating barotropic Newtonian stars are
solutions of the perturbed fluid equations having (Eulerian)
velocity perturbations (Lindblom et al. 1998, PRD, 80, 4843)
where is the unperturbed angular velocity and
dimensionless amplitude of the perturbation.
is the
In spherical coordinates, the three components are
Which is obtained by solving the linear fluid equations
at the first order of the r-mode amplitude
 Second-order r-modes
At the second order, the perturbed Euler, continuity,
and Poisson equations in an inertial frame are
given, respectively, by (Sa 2004)
A and N are two constants determined by the initial
condition. Sa & Tome (2005) suggested
N = 2l -1 and redefined A by introducing a new free
parameter K as
This second-order solution gives a differential
rotation, producing large scale drifts of fluid
elements along stellar latitudes.
Rezzolla et al. (2000, 2001);
Sa (2004); Sa & Tome (2005)
2、 A phenomenological model for
r-mode evolution
 The physical angular momentum and energy of the l=2 r-
mode can be calculated up to the second order in alpha
as (Sa & Tome 2005)
For K = -2, J(2) vanishes and the expressions of Jr and Er
return to their canonical forms (Owen et al. 1998).
Namely, the case of K = -2 corresponds to the
non-differential (uniform) rotation case
 Both the physical angular momentum and energy of r-
modes are increased by gravitational radiation back
reaction and decreased by viscous damping, which yields
 The total angular momentum of the star contains two
components
where I is the moment of inertial of the star.
 For isolated magnetized (1010-1012G) NSs
 For accreting NSs in LMXBs with a weak magnetic field
 For isolated magnetized NSs
The
problem
to fixwith
thea five
 Forkey
accreting
NSs inisLMXBs
weak
magnetic field
timescales
As a preliminary work, we adopt the simplest EOS of
,
physicists
Owen et al. (1998)
strongly dependent on the EOS
and the stellar temperature
 Thermal evolution
also strongly dependent on the
composition of the stellar matter
3、 Long-term
evolution of
isolated NSs
1 prolongs duration
2 stops spin-down
3 enhances the
heating effect
3、 Long-term
evolution of
accreting NSs
4、Gravitational radiation from NSs
Using the obtained r-mode amplitude and angular
velocity, we can estimate the amplitude of the emitted
gravitational waves as follows
The frequency-domain gravitational wave amplitude
Outlook
 It is necessary to adopt some more realistic
EOSs (Hyperon? Pion? Quarks?
Superfluid? CFL?...) of the stellar materials
to give the composition and structure of NSs.
 The interactions between the r-modes and
the magnetic field in the presence of the
differential rotation. On one hand, the
differential rotation can distort the magnetic
field and increases its energy. On the other
hand, the distortion of the magnetic field
would prevent the r-mode oscillation.
Thank you for your attention !