1. Warps in Triaxial Haloes Sungsoo S. Kim (Kyung Hee Univ.) Myoung Won Jeon (Kyung Hee Univ.) Hong Bae Ann (Pusan Natl. Univ.)

2. Early Studies on the Warp of Our Galaxy Kahn & Woltjer (1959) –Intergalactic gas flow past the disc Idlis (1959) –Distortion by impact with the Magellanic Clouds Elwert & Hablick (1965), Avner & King (1967) –Tidal distortion due to the Magellanic Clouds Lynden-Bell (1965) –Free mode of oscillation (free precession) of a disc

3. Discrete Mode of Vertical Oscillation (Bending) 21 out of 133 isolated galaxies are warped like an integral sign (Reshetnikov & Combes 1998). Apparently the bending mode is the best model to describe the integral-sign warps in isolated galaxies. Necessary conditions to explain the warps with the bending modes: –The mode must be discrete. –The mode must be sufficiently different from a rigid tilt. –Nodal points must not significantly wind up.

4. History of the Bending Mode Isolated self-gravitating discs –Lynden-Bell (1965) : Free precession –Hunter (1969a,b), Hunter & Toomre (1969) : Discrete bending modes are available only when the disk has an unrealistically sharp edge. density radius

5. History of the Bending Mode Discs in a flattened halo –Toomre (1983), Sparke (1984), Sparke & Casertano (1988) : No need for sharp edge. –Dekel & Shlosman (1983), Toomre (1983), Dubinsky & Kuijken (1995), Ideta et al. (2000) : Discs initially misaligned to the halo plane. Ideta et al. found that only prolate haloes can sustain the warps. –Nelson & Toomre (1995), Dubinsky & Kuijken (1995), Binney et al. (1998) : Dynamical friction between a disc and a halo may cause the warp to disperse within the Hubble time. But Shen & Sellwood (2006) find that the dynamical friction may not be important. (cont.)

6. History of the Bending Mode Reorientation of a massive galactic halo –Ostriker & Binney (1989), Quinn & Binney (1992), Jiang & Binney (1999), Shen & Sellwood (2006) : Cosmic infall –Revez & Pfenninger (2001), Lopez-Corredoira et al. (2002) : Accretion of the intergalactic medium (cont.)

7. The Present Work Many theoretical and numerical studies suggest that galaxies are situated in (flattened) triaxial dark matter haloes. –We like to see if the results of Ideta et al. (2000) are applicable for triaxial haloes. We have performed a series of numerical simulations for self- gravitating, initially tilted discs in a fixed triaxial halo. –We will later compare these results with those from live halo simulations.

8. The Galaxy Model Disks –Exponential density profile –Initially 30 o tilted relative to the z-axis of the halo Haloes –Triaxial version of the Hernquist model (1990) –M h /M d = 17

9. The Simulation GADGET-2 (parallel tree n-body) –Modified for addition of external potential –3-d interpolation of pre-calculated grid of forces. 200,000 particles for the disk Evolution up to t = 8 Gyr.

10. Triaxiality Parameters abc Oblate 10 7.5 Prolate 7.5 10 Oblate-like 108.57.5 Prolate-like 7.58.510 Bar-like 106.58.5

11. Oblate Halo

12. “Type I” warp in the beginning  Decays later. The inclinations of the inner and outer disks are significantly different. But, the outer disk keeps precessing more slowly than the inner disk. Outer Inner Outer Inner

13. Prolate Halo

14. “Type II” warp in the beginning  Survives for a long time. The inclinations of the inner and outer disks are significantly different. And, the inner and outer disks have similar precessing frequencies. Outer Inner

15. Oblate vs. Prolate ( Two-annulus analysis; Sparke 1984 & Ideta et al. 2000) Oblate –  –  is negative.  Retrograde precession –|  –  | decreases as r .  Trailing nodes  Outer disc inclination increases (Type I).  Larger inclination gives a slower precession.  Outer disc trails even further. Prolate –  –  is positive.  Prograde precession –|  –  | decreases as r .  Trailing nodes  Outer disc inclination decreases (Type II).  Smaller inclination gives a faster precession.  Outer disc catches up the inner disk. dominant & negative dominant & positive

16. Oblate-like Halo

17. “Type I” warp in the beginning  Unlike the oblate halo case, the warp survives for a long time. The inclinations of the inner and outer disks are noticeably different. And, the inner and outer disks have very similar precessing frequencies. Inner Outer

18. Oblate vs. Oblate-like Oblate –  –  is negative.  Retrograde precession –|  –  | decreases as r .  Trailing nodes  Outer disc inclination increases (Type I).  Larger inclination gives a slower precession.  Outer disc trails even further. Oblate-like –  –  is negative.  Retrograde precession –|  –  | decreases as r .  Trailing nodes  Outer disc inclination increases (Type I).  Triaxial contribution makes the precession rates of inner and outer discs similar. dominant & negative periodic (m=2) dominant & negative

19. Prolate-like Halo

20. “Type II” warp in the beginning  Unlike the prolate halo case, the warp appears and disappears repeatedly. The inclinations of the inner and outer disks oscillate at a different phases. And, the inner and outer disks have similar precessing frequencies. Outer Inner

21. Prolate vs. Prolate-like Prolate –  –  is positive.  Prograde precession –|  –  | decreases as r .  Trailing nodes  Outer disc inclination decreases (Type II).  Smaller inclination gives a faster precession.  Outer disc catches up the inner disk. Prolate –  –  is positive.  Prograde precession –|  –  | decreases as r .  Trailing nodes  Outer disc inclination decreases (Type II).  Triaxial contribution makes the precession rates of inner and outer discs similar. dominant & positive dominant & positive periodic (m=2)

22. Bar-like Halo

23. The disk quickly aligns its angular momentum vector with the minor axis, whose normal plane has the most flattened configuration. This is consistent with Steiman-Cameron & Durisen (1984), who found that the precession around the intermediate axis can never occur. As in the oblate-like halo, a warp develops about the minor axis and sustain for an extended period of time. Outer Inner

24. Conclusions We confirm that –Warps in an oblate halo does not sustain for an extended period. –Warps in a prolate halo sustains for an extended period. We find that –Warps in an oblate-like halo can be developed and maintained.  More consistent with the dark matter halo simulations. –Warps in a prolate-like halo can be developed but disappears repeatedly. –Disks in a bar-like halo align their angular momentum with the minor axis of the halo, and develop warps. We will –Perform the same simulations with live haloes and see the significance of the dynamical friction.