] Mass of CND [M SUN

The Galactic Centre

Determination of the Mass Distribution in the Galactic Centre from Stellar Motions

Jaroslava Schovancov ´a, Ladislav ˇSubr

Astronomical Institute, Charles University in Prague Argelander Institute for Astronomy, University of Bonn

Where is the Galactic Centre?

Genzel et al. (2003)

◮ dynamical centre of Galaxy

◮ R₀ = (7.62 _± 0.32) kpc

Eisenhauer et al. (2005)

◮ Celestial position: Sgr

α= 17^h45^m40^s, δ= -29^◦00’ 28" (J2000.0)

Reid & Brunthaler (2004)

◮ harbours

⊲ super-massive black hole

⊲ stellar clusters

young and old stars

⊲ ISM

. – p.1

The Central Body: Sgr A*

◮ Detection in radio: Balick & Brown (1974)

◮ Detection in NIR: Becklin & Neugebauer (1975)

◮ Compact radio source

◮ Rejected Candidates: would have lower luminosity and density than observed

⊲ Stellar cluster of neutron stars and white dwarfs

⊲ Fermion ball

⊲ Boson star

◮ Super-massive black hole

M• = (3.61 _± 0.32) _× 10⁶M⊙ Eisenhauer et al. (2005)

Stars and Gas in the GC

Genzel et al. (2003)

◮ Length scaling:

1 " =_b 0.037 pc

◮ Young stars in central 1 "

◮ Stellar disks of young stars inside 12 "

◮ Circum-nuclear ring of molecular gas, radius 45 "

◮ Spherical cluster of old stars in central 100 "

. – p.3

Cluster of Old Stars

Genzel et al. (2003)

◮ Old, metal-rich stars, 1-10 Gyr

◮ Broken power-law cusp:

ρ(r) ∝ r^−α, R_br = (6±1) ”

α =

(1.19 ± 0.05 r ≤ R_br 1.75 ± 0.05 r > R_br

Schödel et al. (2007)

◮ Mass _∼ 1 M• inside 2 pc

The Circum-nuclear Disk (CND)

Christopher et al. (2005)

◮ Molecular ring:

HCN and HCO⁺, . . .

◮ Well defined radius 1.6 pc

◮ Uncertain total mass:

M_CND ≈ 10⁴ M⊙

Genzel et al. (1985) M_CND ≈ 10⁶ M⊙

Christopher et al. (2005)

◮ Considered as a gas source for star formation in the GC

. – p.5

The CND Mass

100 1000 10000 100000 1e+06 1e+07

1980 1985 1990 1995 2000 2005 2010 Mass of CND [M SUN]

Year

CND mass estimates over past three decades

Neutral gas Ionized gas Neutral and ionized Not specified

Planar Structures in the GC

Paumard et al. (2006)

◮ Two coherent disks of massive O- & B-type stars _≃ 0.1 pc;

Genzel et al. (2003), Ghez et al. (2005)

◮ Well defined inner (0.04 pc) and outer (0.5 pc) radii

◮ Geometrically thick:

h/R ∼ 0.13

. – p.7

Stellar Disks in the Galactic Centre

◮ Young stars: (6 _± 2) Myr _⇒ recent star formation Paumard et al. (2006)

◮ Similar disks detected in the centre of M31 Bender et al. (2005)

◮ Flat mass function, mass _∼ 10⁴ M⊙

◮ Significant eccentricities for some of stellar orbits

◮ Clockwise disk (CWS): e_rms ∈ [0.2; 0.3]

Paumard et al. (2006), Beloborodov et al. (2006)

◮ Counter-clockwise disk (CCWS): e_rms ∈ [0.6; 0.7]

◮ Hot topic: origin?

The Observed Angular Momentum

-1.5 -1 -0.5 0 0.5 1 1.5

0 2 4 6 8 10 12

j = J z/J z,max

p [arcsec]

Genzel et al. (2003), Paumard et al. (2006) j = J_z

J_{z, max} = xv_y − yv_x

q(x² + y²)(v_x² + v_y²) _{. – p.9}

Cosine Pattern of the Disk

◮ Normal vector to the disk

~n = (sin i cos Ω, − sin i sin Ω, − cos i)

◮ Velocity vector of the k-th star

~v_k = k~v_kk(sin θ_k cos φ_k, sin θ_k sin φ_k, cos θ_k)

~n · ~v_k = 0 ⇒ cotg θ_k = tg i cos(Ω + φ_k)

Paumard et al. (2006) ^{. – p.10}

Determination of the Mass Distribution in the Galactic Centre from Stellar Motions

. – p.11

Model of the GC

◮ System dominated by the SMBH central potential

◮ Two “perturbations”:

⊲ spherical stellar cluster

Φ_SPHE(r) = 4πGρ₀r₀^α

(α − 2)(α − 3)r^2−α

⊲ axi-symmetrical CND

Φ_CND(r) = −2Gλ

ra_CND

R kK(k),

k² = f(a_CND, z_CND, R, Z)

◮ CWS disk considered as a set of test particles

Thesis Aims

◮ Limit the mass of the CND

◮ Confine the spatial structure of the CWS disk How?

⊲ Deformation of the CWS disk

⊲ Dependence of the deformation on the parameters of the perturbing potentials

⊲ Compare simulation results with observations

. – p.13

Useful Tools and Techniques

◮ Kozai mechanism, Kozai (1962), Lidov (1962):

⊲ Evolution of a hierarchical triple system; motion of an asteroid under influence of Sun and Jupiter

⊲ Secular evolution of the orbital elements e, i and ω

⊲ Hamiltonian perturbation theory & averaging

technique to get rid of fast-changing variable, the mean anomaly

⊲ Integrals of motion: a, c = √

1 − e² cosi, Φ^¯_perturb

⊲ Convenient tool for study of motion of a test

particle in the potential dominated by the central mass and perturbed by an axi-symmetrical

potential and a spherical potential

The Φ ¯ perturb Isocontours

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

c=0.0

e sinω

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

c=0.2

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

c=0.4

e sinω

e cosω

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1 c=0.8 e cosω

M_CND/M• = 0.01, a_CND/a∗ = 2, c = 0.2 _{. – p.15}

Secular Evolution of Orbits

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0 5 10 15 20 25

<e>(t)

Time [10³ P]

60 70 80 90 100 110 120

0 5 10 15 20 25

<ω>(t) [o ]

Time [10³ P]

0 5 10 15 20 25

45 50 55 60 65 70 75 80

<i>(t) [o ]

Time [10³ P]

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

e sinω

e cosω

c=0.2

Composite Perturbation

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

M_CND/M• = 0.01, M_CND/M_SPHE = 0.5, a_CND/a∗ = 2, c = 0.2

. – p.17

The GC Model

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

M_CND/M• = 0.33, M_CND/M_SPHE = 0.33, a_CND/a∗ = 4.5, c = 0.1

The “Quadrupole Equations”

de

dτ = +15 8 e

q

1 − e² sin²(i) sin(2ω) di

dτ = −15 8

e²

p1 − e² cos(i) sin(i) sin(2ω) dω

dτ = +3 4

p 1

1 − e²

2(1 − e²) + 5 sin²(ω)

e² − sin²(i)

dΩ

dτ = −3 4

cos(i)

p1 − e² [1 + 4e² − 5e² cos²(ω)]

. – p.19

Evolution of the Orbital Elements

◮ Disk deformation depends more on Ω than on e, i, ω

◮ Quadrupole equations DO NOT describe system with a heavy spherical perturbation!

⇒ alternative timescale estimate necessary P_Ω =?

◮ P_Ω = f(M_CND; M_SPHE, α_SPHE; a∗,0, e∗,0, i∗,0, ω∗,0)

Exploring the P _Ω Dependences

Dependence on

◮ M_CND: P_Ω ∝ M_CND⁻¹

◮ a∗,0: P_Ω ∝ a⁻∗ ^3/2

◮ e∗,0: P_Ω ∝ q

1−e_∗,0

1+e_∗,0

◮ i∗,0: P_Ω ∝ |cos i∗,0|⁻¹

◮ M_SPHE, α_SPHE: no dependence has been found for mass range M_SPHE/M• ∈ [0.5; 4] and profiles

α_SPHE ∈ [1.0; 2.0]

. – p.21

The P _Ω Estimate

P_Ω Myr

=

a a_CND

⁻3/2

M_CND M•

⁻1

1

| cos i₀|

r1 − e0

1 + e₀ fn

Isocontours of P _Ω

0.01 0.1 1 10

0.001 0.01 0.1 1

M CND [M BH]

cos i₀

Formula result Numerical result M_CND, Christopher et al., 2005

. – p.23

FIG: jak vypada jz(p) pro ruzne konfig

– kumulovane cetnosti

FIG: The Modelled Angular Momentum

TODO: obrazky pro par bodu podel P Omega=108 Myr, par bodu pro delsi a par bodu pro kratsi periodu.

– snapshot disku – odpovidajici j_z(p)

– Aitoffova projekce ~j tehoz?

. – p.25

Conclusions

TODO: – zminit nutnost vyssich excentricit nez

"pozorovanych"

– mass of CND is ...

– pocatecni rozevreni

Thank you for your attention!

. – p.27