Документ взят из кэша поисковой машины. Адрес оригинального документа : http://nuclphys.sinp.msu.ru/conf/epp10/Dokuchaev.pdf Дата изменения: Sat Sep 7 17:14:19 2013 Дата индексирования: Fri Feb 28 02:47:33 2014 Кодировка: Поисковые слова: universe

Spin and mass of the nearest sup ermassive black hole

Vyacheslav I. Dokuchaev
Institute for Nuclear Research, Russian Academy of Sciences Moscow, Russia

16th Lomonosov Conference

MSU, 2013


Rotating (a 1) black hole are welcomed !
There are no Schwarzschild black holes in the Universe

`Rotation' angular velo city of the Kerr-Newman metric: (d - dt )2

=

2Mr - e 2 a, (r 2 + a2 )2 - a2 sin2

= r 2 - 2r + a2 + e

2

Relativistic jets, sho cks, ultra-high-energy cosmic rays generation: Penrose mechanism Extraction of rotaion energy from black hole ergosphere Unipolar induction Electric potential of rotating black hole Blandford-Znajek mechanism Electromagnetic extraction of rotation energy from black hole


Very Large Telescop e Array (interferometer)
Chile, ESO 2635 m, VLTA = 4 в Ь8.2 m + 4 в Ь1.8 m


Sgr A : Quaisi-p erio dic oscillations (QPO)
QPO observations in the near Infra-Red of the Galactic center by VLTA

arXiv:astro-ph/0310821


X-ray Multi-Mirror Mission

Newton

NASA-ESA 1999, 3.8 tons в10 m в16 m, 0.2 - 12 keV


Sgr A : Quasi-p erio dic oscillations (QPO)
QPO observations in the X-rays of the Galactic center by Newton telescope

arXiv:astro-ph/0401589


Test particle motion in the Kerr-Newman metric
Parameters

M

black hole mass
GM c
2

a J= 0a1 e µ

black hole angular moment black hole spin parameter

black hole electric charge particle mass

particle electric charge

Integrals of motion

E L Q

total particle energy particle azimuthal angular momentum Carter constant

At Q = 0 the motion is in the equatorial plane At a = 0 the total particle angular momentum J = Q + L2


Equations of motion of test particles
in the Boyer-Lindquist coordinates
dr dL 2 dL d 2 d dL dt 2 dL µ,

B. Carter 1968

2

= ± Vr = ± V = L si n

-2

+ a(
2

-1

= a(L - aE sin ) + (r 2 + a2 )
particle proper time

P - E)

-1

P

L=

Effective radial potential

Vr = P 2 - [µ2 r 2 + (L - aE )2 + Q ]
Effective latitude potential (nutation)

V = Q - cos2 [a2 (µ2 - E 2 ) + L2 sin P = E (r + a ) + er - aL,
2 2 2

-2

]
±

= r + a2 cos2

2

= r 2 - 2r + a2 + e 2 Horizons = 0, r = r r+ = 1 + 1 - a2 - e 2 external horizon (event horizon) 2 - e2 r- = 1 - 1 - a internal horizon (Cauchy horizon)


Infall to the rotating black hole
numerically calculated and viewed from the black hole north pole

Planet infall: parabolic orbit E = 1, zero angular momentum L = 0

Photon infall: zero impact parameter b = L/E = 0


Infall to the rotating black hole
Planet infall: L = -3 (negative!), E = 1

Photon infall: b = L/E = -3 (negative!)


Test particle infall trajectories to the black hole
G = c = 1, Q = 0.3M 2 µ2 , E E /M = 0.85, L L/M = 1.7, r (0) r (0)/M = 4.4

a=0 a = 0.998 Angular velo city h and rotation perio d Th of the horizon: h = (r+ ) = a 2 = T h 2(1 + 1 - a2 )


Planet infall trajectory to the rotating black hole
a = 0.998, Q = 1, E = 0.85, L = 1.7, r (0) = 4.4

h = (r+ ) =

2 a = T h 2(1 + 1 - a2 )


Photon infall trajectory to the rotating black hole
a = 0.998, Q = 2, b = 2, r+ = 1.063

h = (r+ ) =

a 2 = T h 2(1 + 1 - a2 )


`Synchrotron' fo cusing of outgoing radiation at r

r+
h

Observed frequency modulation with the horizon rotation frequency

Light curve of the source at the circular orbit
C. W. Mizner, Phys. Rev. Lett. 28, 994­997 (1972) J. M. Bardeen, W. H. Press and S. A. Teukiolsky Astrophys. J 178 347 (1972) C. T. Cunningham J. M. Bardeem, Astrophys. J. 173, L137 (1972) A. G. Polnarev, Astrophysics, 8, 273 (1972)


Quasi-p erio dic orbit, n > 7 turns (years)
a = 0.9982, Q = 2, E = 0.92, L = 1.9, rp = 1.74, ra = 9.48,
max

= 3 6 .2




The same orbit, viewed from the north p ole
a = 0.9982, Q = 2, E = 0.92, L = 1.9, rp = 1.74, ra = 9.48,
max

= 3 6 .2




Latitude oscillation of the hot plasma clump
over the opaque accretion disk

a = 0.9982, Q = 2, E = 0.92, L = 1.9, r+ = 1.06, rp = 1.74, ra = 9.48, max = ±36.2


Latitude oscillation of the hot plasma clump
in the thin opaque accretion disk with a frequency

a = 0.65, Q = 0.1, E = 0.91, L = 2.715, r+ = 1.76, rp = 3.86, ra = 5.01, max = ±6.6
Hot sp ots on the surface of the accretion disk R. A. Suynyaev 1972


Latitude oscillation frequency of the hot sp ot clump



L 2xE - aL (-z- , k ) + K (k ) = ( z+ )1/2 2 a K (k ) =
/2 dx 0 (1-k 2 sin2 x )1/
2

-1

, (n, k ) =
2

/2 dx 0 (1+n sin2 x )(1-k 2 sin2 x )1/

2

z± = cos2 ± = (2 )-1 [ + ± = (Q + xx = L2
3

)/a , = 1 - E

2

( + )2 - 4Q ], k 2 = z- /z , (Q = 0) =
1 x
3/2

+

D. C. Wilkins, Phys. Rev. D, 5, 814 (1972)

(3Q -Qx +x 2 )+a2 Q 2 -a(x 2 +3Q ) x 5 -a2 [x 2 +Q (x +3)]

+a

Thin accretion disk (Q 0)
Q 0

VD, arXiv:1306.2033

=

2 = T

x 2 - 4ax 1/2 + 3a x (x 3/2 + a)
ms

2

Minimal radius of the stable circular orbits x = x x
ms

= 3 + Z2 -

(3 - Z1 )(3 + Z1 + 2Z2 ),



ms

= (x = x

ms

)

Z1 = 1 + (1 - a2 )1/3 [(1 + a)1/3 + (1 - a)1/3 ],

Z2 =

3a2 + Z

2 1


Quasi-p erio dic oscillations of hot clumps
in the accretion disk

Frequencies and h are independent on the accretion mo del Frequencies and h depend only on the black hole parameters (mass M and spin a) Mo dulation of accretion radiation with frequencies and h (two spikes in the power spectrum) Mo dulation of accretion radiation with frequency is wide spreaded (absence of the corresponding spike in the power
spectrum)


Sgr A : observed quasi-p erio dic oscillations
2000 T
h

Sgr A

1500 T QPO2 T sec


1000

QPO1 T 500
ms

0

0.0

0.2

0.4 a

0.6

0.8

1.0


Sup ermassive black hole Sgr A: spin and mass
1 -error region
4.6

4.5 Sgr A 4.4 a 0.65 0.05 M 4.2 0.2 106 M 4.3 M 106 M

4.2

4.1

4.0

3.9

3.8 0.0

0.2

0.4

0.6

0.8

1.0


Sgr A : M = (4.2 ± 0.2) 106M,
1 -error region
2000 Th

a = 0.65 ± 0.05 !

1500 T QPO2


Ts

1000

QPO1

500

Sgr A a 0.65 0.05 M 4.2 0.2 106 M

0 0.0 0.2 0.4 a 0.6 0.8 1.0


Results and Conclusions
Mass of the nearest supermassive black in the Galactic center M = (4.2 ± 0.2) 106 M
Previously known value: M = (4.1 ± 0.4) 106 M


The nearest supermassive black hole rotates not very fast a = 0.65 ± 0.05

Identification of quasi-perio dic oscillations from Sgr A* Rotation perio d of black hole horizon Th = 11.5 in Latitude oscillation perio d of hot spots in the accretion flow T = 19 min Mo derately fast rotation is in agreement with the black hole evolution due to accretion of stars from the central cluster Black hole Sgr A* in the Galactic center is a mo derately effective cosmic rays generator