VOLUME 80, NUMBER 22 PHYSICAL REVIEW LETTERS 1JUNE 1998
Gravitational Radiation Instability in Hot Young Neutron Stars
Lee Lindblom and Benjamin J. Owen
Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 91125
Sharon M. Morsink
Physics Department, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201
(Received 2 March 1998)
We show that gravitational radiation drives an instability in hot young rapidly rotating neutron
stars. This instability occurs primarily in the l 2 r-mode and will carry away most of the angular
momentum of a rapidly rotating star by gravitational radiation. On the time scale needed to cool a
young neutron star to about T 10
9
K (about one year) this instability can reduce the rotation rate
of a rapidly rotating star to about 0.076V
K
, where V
K
is the Keplerian angular velocity where mass
shedding occurs. In older colder neutron stars this instability is suppressed by viscous effects, allowing
older stars to be spun up by accretion to larger angular velocities. [S0031-9007(98)06212-7]
PACS numbers: 04.40.Dg, 04.30.Db, 97.60.Jd
Recently Andersson [1] discovered (and Friedman and
Morsink [2] confirmed more generally) that gravitational
radiation tends to drive the r-modes of all rotating stars
unstable. In this paper we examine the time scales asso-
ciated with this instability in some detail. We show that
gravitational radiation couples to these modes primarily
through the current multipoles, rather than the usual mass
multipoles. We also evaluate the effects of internal fluid
dissipation which tends to suppress this instability. We
find that gravitational radiation is stronger than viscosity
in these modes and so this instability severely limits the
rotation rates of hot young neutron stars. We show that
such stars can spin down by the emission of gravitational
radiation to about 7.6% of their maximum rotation rates
on the time scale (about one year) needed to cool these
stars to 10
9
K.
The r-modes of rotating barotropic Newtonian stars
are solutions of the perturbed fluid equations having
(Eulerian) velocity perturbations
d
$
y aRV
µ
r
R
∂
l
$
Y
B
ll
e
ivt
, (1)
where R and V are the radius and angular velocity of the
unperturbed star, a is an arbitrary constant, and
$
Y
B
lm
is the
magnetic-type vector spherical harmonic defined by
$
Y
B
lm
flsl 1 1dg
21y2
r
$
=3sr
$
=Y
lm
d . (2)
Papaloizou and Pringle [3] first showed that the Euler
equation for r-modes determines the frequencies as
v 2
sl 2 1dsl 1 2d
l 1 1
V. (3)
Further use of the Euler equation (as first noted by
Provost, Berthomieu, and Rocca [4]) in the barotropic
case (a good approximation for neutron stars) determines
that only the l mr-modes exist, and that d
$
y must have
the radial dependence given in Eq. (1). These expressions
for the velocity perturbation and frequency are only the
lowest order terms in expansions for these quantities in
powers of V. The exact expressions contain additional
terms of order V
3
.
The lowest order expressions for the (Eulerian) density
perturbation dr can also be deduced from the perturbed
fluid equations (Ipser and Lindblom [5]):
dr
r
aR
2
V
2
dr
dp
3
"
2l
2l 1 1
s
l
l 1 1
µ
r
R
∂
l11
1dCsrd
#
Y
l11l
e
ivt
,
(4)
where dCsrd is proportional to the gravitational potential
dF and satisfies
d
2
dC
dr
2
1
2
r
ddC
dr
1
"
4pGr
dr
dp
2
sl 1 1dsl 1 2d
r
2
#
dC 2
8pGl
2l 1 1
s
l
l 1 1
r
dr
dp
µ
r
R
∂
l11
. (5)
Equation (4) is the complete expression for dr to order
V
2
. The next order terms are proportional to V
4
.
Our interest here is to study the evolution of these
modes due to the dissipative influences of viscosity and
gravitational radiation. For this purpose it is useful to
consider the effects of radiation on the evolution of the
energy of the mode (as measured in the corotating frame
of the equilibrium star)
˜
E:
˜
E
1
2
Z
"
rd
$
y?d
$
y
p
1
µ
dp
r
2dF
∂
dr
p
#
d
3
x.
(6)
This energy evolves on the secular time scale of the
dissipative processes. The general expression for the time
0031-9007y98y80(22)y4843(4)$15.00 © 1998 The American Physical Society 4843
VOLUME 80, NUMBER 22 PHYSICAL REVIEW LETTERS 1JUNE 1998
derivative of
˜
E for a mode with time dependence e
ivt
and
azimuthal angular dependence e
imw
is
d
˜
E
dt
2
Z
s2hds
ab
ds
p
ab
1 zdsds
p
dd
3
x
2vsv1mVd
X
l$2
N
l
v
2l
sjdD
lm
j
2
1 jdJ
lm
j
2
d .
(7)
The thermodynamic functions h and z that appear in
Eq. (7) are the shear and bulk viscosities of the fluid. The
viscous forces are driven by the shear ds
ab
and expansion
ds of the perturbation, defined by the usual expressions
ds
ab
1
2
s=
a
dy
b
1=
b
dy
a
2
2
3
d
ab
=
c
dy
c
d, (8)
ds =
a
dy
a
. (9)
Gravitational radiation couples to the evolution of the
mode through the mass dD
lm
and current dJ
lm
multipole
moments of the perturbed fluid,
dD
lm
Z
drr
l
Y
p
lm
d
3
x , (10)
dJ
lm
2
c
s
l
l 1 1
Z
r
l
srd
$
y1dr
$
yd?
$
Y
Bp
lm
d
3
x ,
(11)
with coupling constant
N
l
4pG
c
2l11
sl 1 1dsl 1 2d
lsl 2 1dfs2l 1 1d!!g
2
. (12)
The terms in the expression for d
˜
Eydt due to viscosity
and the gravitational radiation generated by the mass
multipoles are well known [6]. The terms involving the
current multipole moments have been deduced from the
general expressions given by Thorne [7].
We can now use Eq. (7) to evaluate the stability of the
r-modes. Viscosity always tends to decrease the energy
˜
E, while gravitational radiation may either increase or de-
crease it. The sum that appears in Eq. (7) is positive defi-
nite; thus the effect of gravitational radiation is determined
by the sign of vsv1mVd. For r-modes this quantity is
negative definite:
vsv1lVd2
2sl21dsl 1 2d
sl 1 1d
2
V
2
, 0. (13)
Therefore gravitational radiation tends to increase the en-
ergy of these modes. For small angular velocities the en-
ergy
˜
E is positive definite: The positive term jd
$
yj
2
in
Eq. (6) (proportional to V
2
) dominates the indefinite term
sdpyr2dFddr
p
(proportional to V
4
). Thus, gravi-
tational radiation tends to make every r-mode unstable
in slowly rotating stars. This confirms the discovery of
Andersson [1] and the more general arguments of Fried-
man and Morsink [2]. To determine whether these modes
are actually stable or unstable in rotating neutron stars,
therefore, we must evaluate the magnitudes of all the dis-
sipative terms in Eq. (7) and determine which dominates.
Here we estimate the relative importance of these
dissipative effects in the small angular velocity limit using
the lowest order expressions for the r-mode d
$
y and dr
given in Eqs. (1) and (4). The lowest order expression for
the energy of the mode
˜
E is
˜
E
1
2
a
2
V
2
R
22l12
Z
R
0
rr
2l12
dr . (14)
The lowest order contribution to the gravitational radi-
ation terms in the energy dissipation comes entirely from
the current multipole moment dJ
ll
. This term can be eval-
uated to lowest order in V using Eqs. (1) and (11):
dJ
ll
2aV
cR
l21
s
l
l 1 1
Z
R
0
rr
2l12
dr . (15)
The other contributions from gravitational radiation to
the dissipation rate are all higher order in V. The
mass multipole moment contributions are higher order
because (a) the density perturbation dr from Eq. (4) is
proportional to V
2
while the velocity perturbation d
$
y is
proportional to V; and (b) the density perturbation dr
generates gravitational radiation at order 2l 1 4 in v
while d
$
y generates radiation at order 2l 1 2.
The contribution of gravitational radiation to the imag-
inary part of the frequency of the mode 1yt
GR
can be
computed as follows:
1
t
GR
2
1
2
˜
E
√
d
˜
E
dt
!
GR
. (16)
Using Eqs. (14)–(16) we obtain an explicit expression for
the gravitational radiation time scale associated with the
r-modes:
1
t
GR
2
32pGV
2l12
c
2l13
3
sl 2 1d
2l
fs2l 1 1d!!g
2
µ
l 1 2
l 1 1
∂
2l12
Z
R
0
rr
2l12
dr .
(17)
The time derivative of the energy due to viscous dissipa-
tion is driven by the shear ds
ab
and the expansion ds
of the velocity perturbation. The shear can be evaluated
using Eqs. (1) and (8) and its integral over the constant
r two-spheres performed in a straightforward calculation.
Using the formulas for the viscous dissipation rate Eq. (7)
and the energy Eq. (14), we obtain the contribution of
shear viscosity to the imaginary part of the frequency of
the mode,
1
t
V
sl 2 1ds2l 1 1d
Z
R
0
hr
2l
dr
√
Z
R
0
rr
2l12
dr
!
21
.
(18)
The expansion ds, which drives the bulk viscosity
dissipation in the fluid, can be reexpressed in terms of
the density perturbation. The perturbed mass conservation
law gives the relationship ds 2isv1mVdDryr,
where Dr is the Lagrangian perturbation in the density.
The perturbation analysis used here is not of sufficiently
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VOLUME 80, NUMBER 22 PHYSICAL REVIEW LETTERS 1JUNE 1998
high order (in V) to evaluate the lowest order contribution
to Dr. However, we are able to evaluate the Eulerian
perturbation dr as given in Eq. (4). We expect that the
integral of jdryrj
2
over the interior of the star will be
similar to (i.e., within about a factor of 2) the integral of
jDryrj
2
. Thus, we estimate the magnitude of the bulk
viscosity contribution to the dissipation by
1
t
B
ø
sv1mVd
2
2
˜
E
Z
z
drdr
p
r
2
d
3
x . (19)
Using Eqs. (4) and (14) for dryr and
˜
E, Eq. (19)
becomes an explicit formula for the contribution to the
imaginary part of the frequency due to bulk viscosity.
To evaluate the dissipative time scales associated with
the r-modes using the formulas in Eqs. (17)–(19), we
need models for the structures of neutron stars as well
as expressions for the viscosities of neutron star matter.
We have evaluated these time scales for 1.4M
Ø
neutron
star models based on several realistic equations of state
[8]. We use the standard formulas for the shear and bulk
viscosities of hot neutron star matter [9]
h 347r
9y4
T
22
, (20)
z 6.0 3 10
259
r
2
sv1mVd
22
T
6
, (21)
where all quantities are given in cgs units. The time scales
for the more realistic equations of state are comparable
to those based on a simple polytropic model p kr
2
with k chosen so that the radius of a 1.4M
Ø
star is
12.53 km. The dissipation time scales for this polytropic
model (which can be evaluated analytically) are ˜t
GR
23.26 s, ˜t
V
2.52 3 10
8
s, and ˜t
B
6.99 3 10
8
s for
the fiducial values of the angular velocity V
p
pG ¯r
and temperature T 10
9
K in the l 2 r-mode. The
gravitational radiation time scales increase by about 1
order of magnitude for each incremental increase in l,
while the viscous time scales decrease by about 20%.
The evolution of an r-mode due to the dissipative
effects of viscosity and gravitational radiation reaction is
determined by the imaginary part of the frequency of the
mode,
1
tsVd
1
˜t
GR
√
V
2
pG ¯r
!
l11
1
1
˜t
V
√
10
9
K
T
!
2
1
1
˜t
B
√
T
10
9
K
!
6
√
V
2
pG ¯r
!
. (22)
Equation (22) is displayed in a form that makes explicit
the angular velocity and temperature dependences of the
various terms. Dissipative effects cause the mode to
decay exponentially as e
2tyt
(i.e., the mode is stable) as
long as t.0. From Eqs. (17)–(19) we see that ˜t
V
. 0
and ˜t
B
. 0 while ˜t
GR
, 0. Thus gravitational radiation
drives these modes towards instability while viscosity tries
to stabilize them. For small V the gravitational radiation
contribution to the imaginary part of the frequency is
very small since it is proportional to V
2l12
. Thus, for
sufficiently small angular velocities, viscosity dominates
and the mode is stable. For sufficiently large V, however,
gravitational radiation will dominate and drive the mode
unstable. It is convenient to define a critical angular
velocity V
c
, where the sign of the imaginary part of the
frequency changes from positive to negative: 1ytsV
c
d
0. If the angular velocity of the star exceeds V
c
then
gravitational radiation reaction dominates viscosity and the
mode is unstable.
For a given temperature and mode l the equation for the
critical angular velocity, 0 1ytsV
c
d, is a polynomial of
order l 1 1 in V
2
c
, and thus each mode has its own criti-
cal angular velocity. However, only the smallest of these
(always the l 2 r-mode here) represents the critical an-
gular velocity of the star. Figure 1 depicts the critical
angular velocity for a range of temperatures relevant for
neutron stars. The solid curve in Fig. 1 represents the crit-
ical angular velocity for the polytropic model discussed
above. Figure 2 depicts the critical angular velocities for
1.4M
Ø
neutron star models computed from a variety of re-
alistic equations of state [8]. Figure 2 illustrates that the
minimum critical angular velocity (in units of
p
pG ¯r)
is extremely insensitive to the equation of state. The
minima of these curves occur at T ø 2 3 10
9
K, with
V
c
ø 0.043
p
pG ¯r. The maximum angular velocity for
any star occurs when the material at the surface effectively
orbits the star. This “Keplerian” angular velocity V
K
is
very nearly
2
3
p
pG ¯r for any equation of state. Thus the
minimum critical angular velocity due to instability of the
r-modes is about 0.065V
K
for any equation of state [10].
To determine how rapidly a young neutron star is
allowed to spin after cooling, we must compare the
rate it cools with the rate it loses angular momentum
10
5
10
7
10
9
10
11
0.0
0.2
0.4
0.6
0.8
1.0
Temperature (K)
Ω
c
/(π
G
ρ)
1/2
FIG. 1. Critical angular velocities for a 1.4M
Ø
polytropic
neutron star with (solid line) and without (dashed line) bulk
viscosity. Also the evolution of a rapidly rotating neutron
star (dash-dotted line) as the star cools and emits gravitational
radiation.
4845
VOLUME 80, NUMBER 22 PHYSICAL REVIEW LETTERS 1JUNE 1998
0.041
0.042
0.043
0.044
0.045
Temperature (K)
10
9
2x10
9
3x10
9
Ω
c
/(π
G
ρ)
1/2
FIG. 2. Critical angular velocities of realistic 1.4M
Ø
neutron
star models.
by gravitational radiation. We approximate the cooling
with a simple model based on the emission of neutrinos
through the modified URCA process [11]. We compute
the time evolution of the angular velocity of the star by
setting dJydt Jyt, where J is the angular momentum
of the star and t is the time scale given in Eq. (22).
The result is a simple first order differential equation for
Vstd which we solve for initial angular velocity V V
K
and initial temperature 10
11
K. The solution is shown as
the dash-dotted line in Fig. 1. The gravitational radiation
time scale is so short that the star radiates away its angular
momentum almost as quickly as it cools. The angular
velocity of the star decreases from V
K
to 0.076V
K
in a
period of about one year [12]. Thus, we conclude that
young neutron stars will be spun down by the emission of
gravitational radiation within their first year to a rotation
period of about 13P
min
, where P
min
2p yV
K
. The
Crab pulsar with present rotation period 33 ms and initial
period 19 ms (based on the measured braking index)
rotates more slowly than this limit if P
min
, 1.5 ms.
Our analysis is based on the assumption that a young
hot neutron star may be modeled as an ordinary fluid.
Once the star cools below the superfluid transition tem-
perature (about 10
9
K) the analysis presented here must
be modified [13]. We expect the r-mode instability to
be completely suppressed (with V
c
V
K
) when the star
becomes a superfluid [14]. This makes it possible for old
recycled pulsars to be spun up to large angular veloci-
ties by accretion if they are not reheated above 10
9
Kin
the process. If nonperfect fluid effects enter above 10
9
K,
however, the spin-down process may be terminated at a
higher angular velocity than the 0.076V
K
figure com-
puted here. The detection of a young fast pulsar [15]
would provide evidence for such effects at temperatures
higher than 10
9
K. Magnetic fields could also damp these
modes; however preliminary estimates based on standard
magnetosphere-mode coupling models [16] suggest that
such damping is too weak to suppress the relatively low
frequency r-mode instability.
We thank N. Andersson, J. Friedman, J. Ipser, S. Phin-
ney, B. Schutz, and K. Thorne for helpful discussions.
This research was supported by NSF Grants No. AST-
9417371, No. PHY-9507740, No. PHY-9796079, NASA
Grant No. NAG5-4093, the NSF graduate program, and
NSERC of Canada.
[1] N. Andersson, gr-gc/9706075, Astrophys. J. (to be pub-
lished).
[2] J.L. Friedman and S. M. Morsink, gr-gc/9706073, Astro-
phys. J. (to be published).
[3] J. Papaloizou and J.E. Pringle, Mon. Not. R. Astron. Soc.
182, 423 (1978).
[4] J. Provost, G. Berthomieu, and A. Rocca, Astron. Astro-
phys. 94, 126 (1981).
[5] J. Ipser and L. Lindblom (unpublished).
[6] J. Ipser and L. Lindblom, Astrophys. J. 373, 213 (1991).
[7] K.S. Thorne, Rev. Mod. Phys. 52, 299 (1980).
[8] See references in S. Bonazzola, J. Frieben, and E.
Gourgoulhon, Astrophys. J. 460, 379 (1996).
[9] C. Cutler and L. Lindblom, Astrophys. J. 314, 234 (1987);
R.F. Sawyer, Phys. Rev. D 39, 3804 (1989).
[10] Note that the minimum V
c
is rather small, and thus the
small V expansions used here are expected to be quite
good. Also, the approximation used to evaluate t
B
in
Eq. (19) is not expected to have a large effect on this
minimum value; e.g., if ˜t
B
were off by a factor of 10, then
the high T portion of the curve in Fig. 1 would be moved
along the T axis by a factor of about 1.5 øs10d
1y6
. Since
the shear viscosity influence on the curve (the dashed line
in Fig. 1) is so flat in this temperature range, the minimum
value of V
c
would not be significantly changed.
[11] S.L. Shapiro and S. Teukolsky, Black Holes, White
Dwarfs, and Neutron Stars (Wiley, New York, 1983). Vis-
cosity also converts rotational energy into heat. However,
in the high temperature range this energy is converted pri-
marily by bulk viscosity into neutrinos which are quickly
radiated away. We do not expect this rotational reheating
to significantly impede the cooling process.
[12] Our assumption that the spin-down time scale is t of
Eq. (22) is based on the fact that the r-mode will grow
(within a few minutes) to a point where the mode contains
a substantial fraction of the total angular momentum of the
star before being saturated by nonlinear effects. Equation
(22) is a low V expansion so the early time values of t are
also somewhat uncertain. Fortunately, the final angular
velocity state of the star is quite insensitive to the details
of the cooling and spin-down.
[13] The formation of a solid crust below r ø 2 3
10
14
gycm
3
also affects these modes; however, this den-
sity appears to be too low to affect the r-mode instability.
[14] L. Lindblom and G. Mendell, Astrophys. J. 444, 805
(1995).
[15] F.E. Marshall et al., astro-ph/9803214.
[16] O. Blaes et al., Astrophys. J. 343, 839 (1989).
4846