THE ASTROPHYSICAL JOURNAL, 2003
Preprint typeset using LATEX style emulateapj v. 11/12/01

ONE-ARMED SPIRAL INSTABILITY IN DIFFERENTIALLY ROTATING STARS
Motoyuki Saijo 1 , Thomas W. Baumgarte 2,3 , and Stuart L. Shapiro 3,4,5
1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2 Department of Physics and Astronomy, Bowdoin College,

8800 College Station Brunswick, ME 04011-8488, USA
3 Department of Physics , University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA
4 Department of Astronomy , University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
5 NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received 2003 January 22ABSTRACT
We investigate the dynamical instability of the one-armed spiral m = 1 mode in differentially rotatingstars by means of 3 + 1 hydrodynamical simulations in Newtonian gravitation. We find that both a soft
equation of state and a high degree of differential rotation in the equilibrium star are necessary to excite
a dynamical m = 1 mode as the dominant instability at small values of the ratio of rotational kinetic
to potential energy, T /|W |. We find that this spiral mode propagates outward from its point of origin
near the maximum density at the center to the surface over several central orbital periods. An unstablem = 1 mode triggers a secondary m = 2 bar mode of smaller amplitude, and the bar mode can excite
gravitational waves. As the spiral mode propagates to the surface it weakens, simultaneously damping
the emitted gravitational wave signal. This behavior is in contrast to waves triggered by a dynamicalm = 2 bar instability, which persist for many rotation periods and decay only after a radiation-reaction
damping timescale.
Subject headings: Gravitation — hydrodynamics — instabilities — stars: neutron — stars: rotation

We have studied the conditions under which Newtonian, differentially rotating stars are dynamically unstable to an m = 1 one-armed spiral instability, and foundthat both soft equations of state and a high degree of differential rotation are necessary to trigger the instability.
For sufficiently soft equations of state and sufficiently
high degrees of differential rotation we found that stars
are dynamically unstable even at the small values of theratio of rotational to potential energy T /|W | ∼ 0.14 considered in this paper.
While we find that a toroidal structure alone is not
sufficient for the m = 1 instability, all the models thatare unstable do have a toroidal structure, suggesting that
this may be a necessary condition. The growing m = 1
mode redistributes the matter in the unstable star anddestroys the toroidal structure after a few central rotation periods.
Quasi-periodic gravitational waves emitted by stars
with m = 1 instabilities have smaller amplitudes thanthose emitted by stars unstable to the m = 2 bar mode.
For m = 1 modes, the gravitational radiation is emitted
not by the primary mode itself, but by the m = 2 secondary harmonic which is simultaneously excited, albeit
at a lower amplitude. Unlike the case for bar-unstable
stars, the gravitational wave signal does not persist of
many periods, but instead is damped fairly rapidly inmost of the cases we have examined.
We have plotted typical wave forms for stars unstable
to m = 2 bar modes and for stars unstable to one-armed
spiral m = 1 modes. Characteristic wave frequencies
fGW are seen to be the order of central angular velocity∼ Ωc , and are considerably higher than the equatorial
angular velocity at the surface Ωeq ∼ (M/R3 )1/2 due toappreciable differential rotation. For supermassive stars

1.5
1II (a)II (b)II (c)II (d)-1.5 -1 -0.5 0 0.5 1 1.5
x/ R-1.5 -1 -0.5 0 0.5 1 1.5
x/ R

y/ R

0.5
0-0.5
-1-1.5
1.5
1y/ R

0.5
0-0.5
-1-1.5

Fig. 1.— Final density contours in the equatorial plane forModels II in Saijo, Baumgarte, & Shapiro (2003)1 . The stiffness of the equation of state decreases from model (a) to (d).
(0)
Snapshots are plotted at (t/Pc , ρmax /ρmax , d) = (a) (36.8, 1.24,0.220), (b) (23.8, 2.63, 0.267) (c) (17.3, 6.91, 0.333), and (d)(18.6, 9.09, 0.333). The contour lines denote densities ρ/ρmax =10−(16−i)d (i = 1, · · · , 15).

(M & 105 M ) the amplitudes and frequencies of thesewaves fall well within the detectable range of LISA.
A more detail discussion is presented in Saijo, Baumgarte, & Shapiro (2003)1 .

1 Saijo, M., Baumgarte, T. W., & Shapiro, S. L. 2003, ApJ, submitted (astro-ph/0302436)

1