1. Grand Rotation Curveｓ　& Galactic Roche Lobes in the Galaxy and M31 Yoshiaki Sofue (Meisei Uni.) at Meisei University

2. Contents 1. Rotation Curve 2. Mass models 3. Grand RC 4
Contents 1. Rotation Curve 2. Mass models 3. Grand RC 4. Local DM Density 5. Roche Lobes Galactic Territory

3. 1. Rotation Curve

4. 銀河中心

5. inside/outside Solar Circle
Rotation velocities
inside/outside Solar Circle

6. Define: Vrot =Vvirial≒ √2 VGC
Rotation curve at R>30 kpc assuming random motion of satellite galaxies
Define:
Vrot =Vvirial≒ √2 VGC

7. Solar Circular Velocity V0=200 km/s @ R0=8kpc

8. 2. Mass Model

9. RC Fitting by (1) Bulge: de Vaucouleurs Law (2) Disk: Exponential
(3) Dark Halo: NFW

10. de V ∝ exp(-r1/4) Expo. Plummer Kepler

11. (1) Bulge de Vaucouleurs: exp(-r1/4)

12. Mass models: Bulge: de Vaucouleurs (r1/4)

13. (2) Exponential Disk +Arm/Ring
(If Δ=0, V(R)=written by modified Bessel functions)

14. Disk: Exponential

15. (3) Dark Halo Density Profile (Navaro-Frenk-White 1994)
NFW Profile
(Navaro-Frenk-White 1994)

16. Volume Densities for
Bulge, Disk, and Halo（Isoth, NFW, Burkert)

17. Sofue, Honma, Omodaka 2009

18. Total Masses
Dark Halo
Expo. Disk
de Vaucl. Bulge

19. 3. Grand Rotation Curve

20. Dark Halo Density Profiles
(X=R/h)
NFW density ∝ 1/X(1+X) 2
(Navaro-Frenk-White 1994)
Burkert ∝ 1/(1+X)(1+X2)
Isothermal ∝ 1/(1+X2) =>X-2

22. Local Group of Galaxies

23. Grand Rotation Curve
Sattelite’s velocity
=>Virial　Mass
V(R)=√２ Vr

24. Solar Circular Velocity V0=200 km/s @ R0=8kpc

25. Grand RC

26. MW ⇔ M31
border
de V. Bulge
Milky Way GRC
Expo. Disk
NFW Halo

27. 4. Galaxy Parameters & Locad DM Density

28. Dispersion
Scale radius fit
Mass fit

30. The Local Dark Matter Density @ Sun
ρ☉= 0.235±0.030
GeV cm-3

31. M ~ RV2/G => ρ0 ~ M/R3 ~ (V/R)2 ~ Ω2
V0/R0=Ω0 (km s-1 kpc-1) ρ0 (GeV cm-3)
Sofue (2012)
200/8.0 ＝ ±0.030
( 248/8.2 ＝ ± )
Salucci, et al. (2011)≒Honma et al (2012)
248/8.2 ＝ ± 0.11
( 200/8.0 ＝ ± 0.07 )
Weber and de Boer (2010)
0.2 ~ 0.4

32. 6. Galactic Roche Lobe - Territory problem –

33. Grand RC of M31
√2 Vr
Vrot

34. GRC of M31
M31⇔ MW
de V. Bulge
Expo. Disk
NFW Halo
M31

35. Grand RC of M31 ≒ Galaxy MW M31 de V. Bulge NFW Halo NFW Halo
Expo. Disk
NFW Halo
M31⇔ MW
de V. Bulge
Expo. Disk
NFW Halo
M31

36. M31 ≒ Galaxy>> M33, dwarfs
MW+M31≒Binary

37. 2D Matsuda et al. 2000

38. Binary (2 point masses) Roche Lobes

39. Galactic Roche Lobe Φ=ΦG+ΦM31+ΦΩ Rigid rot coordi
Galactic Roche Lobe Φ=ΦG+ΦM31+ΦΩ Rigid rot coordi. = -GM1/r1 - GM2/r2 - (1/2)Ω2 r2

40. NFW Roche Lobe Mh(R) = 4πρ0h3{ln(1+X)−X/(1+X)} X=R/h } :

41. Galactic DM Roche Lobe
MW-M31 Binary

43. Local Group of Galaxies

44. Galaxy + M31 ＝＞What in future？

45. Galaxy – M31

46. Summary 1. Rotation Curve 2. Mass models 3. Grand RC (Log RC) 4
Summary 1. Rotation Curve 2. Mass models 3. Grand RC (Log RC) 4. Local DM Density & G-parameters 5. Roche Lobe Galactic Territory MW vs M31

47. ７．Grand RC (Log RC) for the Galactic Center

48. Bulge is in between de Vaucouleurs and Exponential

49. de V ∝ exp(-r1/4) Expo. Plummer Kepler

50. MW ⇔ M31
border
de V. Bulge
Milky Way GRC
Expo. Disk
NFW Halo