### Abstract

We study the dynamical stability against bar-mode deformation of rapidly and differentially rotating stars in the first post-Newtonian approximation of general relativity. We vary the compaction of the star M/R (where M is the gravitational mass and R the equatorial circumferential radius) between 0.01 and 0.05 to isolate the influence of relativistic gravitation on the instability. For compactions in this moderate range, the critical value of β ≡ T/W for the onset of the dynamical instability (where T is the rotational kinetic energy and W the gravitational binding energy) slightly decreases from ∼0.26 to ∼0.25 with increasing compaction for our choice of the differential rotational law. Combined with our earlier findings based on simulations in full general relativity for stars with higher compaction, we conclude that relativistic gravitation enhances the dynamical bar-mode instability, i.e., the onset of instability occurs for smaller values of β in relativistic gravity than in Newtonian gravity. We also find that once a triaxial structure forms after the bar-mode perturbation saturates in dynamically unstable stars, the triaxial shape is maintained, at least for several rotational periods. To check the reliability of our numerical integrations, we verify that the general relativistic Kelvin-Helmholtz circulation is well conserved, in addition to rest-mass energy, total mass energy, and linear and angular momentum. Conservation of circulation indicates that our code is not seriously affected by numerical viscosity. We determine the amplitude and frequency of the quasi-periodic gravitational waves emitted during the bar formation process using the quadrupole formula.

Original language | English |
---|---|

Pages (from-to) | 919-931 |

Number of pages | 13 |

Journal | Astrophysical Journal |

Volume | 548 |

Issue number | 2 PART 1 |

DOIs | |

Publication status | Published - 2001 Feb 20 |

Externally published | Yes |

### Fingerprint

### Keywords

- Gravitation
- Hydrodynamics
- Instabilities
- Relativity
- Stars: neutron
- Stars: rotation

### ASJC Scopus subject areas

- Space and Planetary Science

### Cite this

*Astrophysical Journal*,

*548*(2 PART 1), 919-931. https://doi.org/10.1086/319016

**Dynamical bar instability in rotating stars : Effect of general relativity.** / Saijo, Motoyuki; Shibata, Masaru; Baumgarte, Thomas W.; Shapiro, Stuart L.

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 548, no. 2 PART 1, pp. 919-931. https://doi.org/10.1086/319016

}

TY - JOUR

T1 - Dynamical bar instability in rotating stars

T2 - Effect of general relativity

AU - Saijo, Motoyuki

AU - Shibata, Masaru

AU - Baumgarte, Thomas W.

AU - Shapiro, Stuart L.

PY - 2001/2/20

Y1 - 2001/2/20

N2 - We study the dynamical stability against bar-mode deformation of rapidly and differentially rotating stars in the first post-Newtonian approximation of general relativity. We vary the compaction of the star M/R (where M is the gravitational mass and R the equatorial circumferential radius) between 0.01 and 0.05 to isolate the influence of relativistic gravitation on the instability. For compactions in this moderate range, the critical value of β ≡ T/W for the onset of the dynamical instability (where T is the rotational kinetic energy and W the gravitational binding energy) slightly decreases from ∼0.26 to ∼0.25 with increasing compaction for our choice of the differential rotational law. Combined with our earlier findings based on simulations in full general relativity for stars with higher compaction, we conclude that relativistic gravitation enhances the dynamical bar-mode instability, i.e., the onset of instability occurs for smaller values of β in relativistic gravity than in Newtonian gravity. We also find that once a triaxial structure forms after the bar-mode perturbation saturates in dynamically unstable stars, the triaxial shape is maintained, at least for several rotational periods. To check the reliability of our numerical integrations, we verify that the general relativistic Kelvin-Helmholtz circulation is well conserved, in addition to rest-mass energy, total mass energy, and linear and angular momentum. Conservation of circulation indicates that our code is not seriously affected by numerical viscosity. We determine the amplitude and frequency of the quasi-periodic gravitational waves emitted during the bar formation process using the quadrupole formula.

AB - We study the dynamical stability against bar-mode deformation of rapidly and differentially rotating stars in the first post-Newtonian approximation of general relativity. We vary the compaction of the star M/R (where M is the gravitational mass and R the equatorial circumferential radius) between 0.01 and 0.05 to isolate the influence of relativistic gravitation on the instability. For compactions in this moderate range, the critical value of β ≡ T/W for the onset of the dynamical instability (where T is the rotational kinetic energy and W the gravitational binding energy) slightly decreases from ∼0.26 to ∼0.25 with increasing compaction for our choice of the differential rotational law. Combined with our earlier findings based on simulations in full general relativity for stars with higher compaction, we conclude that relativistic gravitation enhances the dynamical bar-mode instability, i.e., the onset of instability occurs for smaller values of β in relativistic gravity than in Newtonian gravity. We also find that once a triaxial structure forms after the bar-mode perturbation saturates in dynamically unstable stars, the triaxial shape is maintained, at least for several rotational periods. To check the reliability of our numerical integrations, we verify that the general relativistic Kelvin-Helmholtz circulation is well conserved, in addition to rest-mass energy, total mass energy, and linear and angular momentum. Conservation of circulation indicates that our code is not seriously affected by numerical viscosity. We determine the amplitude and frequency of the quasi-periodic gravitational waves emitted during the bar formation process using the quadrupole formula.

KW - Gravitation

KW - Hydrodynamics

KW - Instabilities

KW - Relativity

KW - Stars: neutron

KW - Stars: rotation

UR - http://www.scopus.com/inward/record.url?scp=0035916505&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035916505&partnerID=8YFLogxK

U2 - 10.1086/319016

DO - 10.1086/319016

M3 - Article

AN - SCOPUS:0035916505

VL - 548

SP - 919

EP - 931

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 2 PART 1

ER -