4. Bondi accretion rate (spherical inflow).
Ý  4 
M
acc
(GM )2
 c3

• The accretion rate when the object is moving through
the ISM with speed V is:

Ý  4 
M
acc
• X-ray transients:

(GM )2
 (V 2 c 2 )3/ 2

Rocket Propulsion
Equations governing rocket engine Thrust
are same as the accretion problem!
Thrust:
ÝV
FM
exh
Ý  AV,
Equations: M
dV
2 d ln 
V
 cs
0
dx
dx

Adding a nozzle (the hour glass shape) increases the
thrust
by about 60% (for adiabatic index 5/3 gas)

and by a much larger factor as the index gets
closer to 1.
The gas speed at the Throat must be Mach 1
otherwise the Thrust will not increase!
Low-mass x-ray binary (LMXB)
Corona
Low mass star filling
Roche lobe
Accretion disk
Binary classification
Roche Potential
Potential in a rotating frame:
R  
GM 1
r r1
Roche Lobe


GM 1
r r1
  r sin 
1
2
2 2
2
Accretion disk
(order of magnitude estimate)
Theory for geometrically thin and optically thick disk
1. Disk thickness:
H  c s1
2. Temperature as a function of r
 spectrum
3. Emergent
Keith Horne, 1993, in “Accretion disks in
Compact stellar systems”, Ed. J.C.Wheeler;
World Scientific.
T r
3 / 4
During outburst

Quiescence data
Low-mass x-ray binary (LMXB)
Corona
Low mass star filling
Roche lobe
Accretion disk
Chandrashekhar-Balbus-Hawley mechanism
Balbus & Hawley, 1998, Rev. Mod. Phys., Vol. 70, p. 1-53
(sections III A, B, & IV B in particular)
Analysis of a mechanical model consisting of a
spring and two point mass on an orbit.
Schwarzschild Black-hole
(Shapiro & Teukolsky: BH, WD & NS; chapter 12)
Metric:
Effective potential:


2
dr
2
2
d 2  dt 2 (1 2Mr ) 

r
d

(1 2Mr )
Veff (r)  (1 )(1 2M
2 )
r
l2
r2
Effective potential for several different L
The binding energy of the
last stable circular orbit
is: 5.72%.
Kerr Black-hole
Metric in Boyer-Lindquist coordinate:
d  (1
2
2Mr

)dt 
2
4 aMr sin 2 

dtd   dr 2  d 2
 [r  a 
2
a  MJ ,
2
  r 2  2Mr  a2,
2Mra 2 sin 2 

]sin 2 d 2
  r 2  a2 cos2 
|a|<M
Horizon occurs where:
  0 or rH  M  M 2  a 2
Ergosphere
Angular speed allowed for a particle at a fixed r and theta:

d
dt

Ý
tÝ
2
2
Ý
1  t (gtt  2gt   g  )
The quantity in the bracket must be negative. This implies
That the allowed range for rotation speed is:

 
gt
gt2  gtt g
g
  0 when gtt  0, or rs  M  M 2  a 2 cos 2 
For r<rs the particle cannot remain stationary and is
forced to rotate with the black-hole. The region between
rH and rs is called the Ergosphere.
Ergosphere of a Kerr black hole
Semenov et al., 2004, Science 305, 978
QuickTime™ and a
Microsoft Video 1 decompressor
are needed to see this picture.
Semenov et al., 2004, Science 305, 978
QuickTime™ and a
Microsoft Video 1 decompressor
are needed to see this picture.
Semenov et al., 2004, Science 305, 978
QuickTime™ and a
Microsoft Video 1 decompressor
are needed to see this picture.
It can be shown that the energy (per unit mass) for
the last stable circular orbit is 1 - 42.3%. Therefore,
we one can extract about 42.3% energy from the
accretion around a maximally rotating BH.
X-ray Bursts (type I bursts)
Review: Strohmayer & Bildstein, 2003, astro-ph/0301544
• These busts are seen once every few hours to ~ 1 day in LMXBs
70 of the ~160 LMXBs show bursting behavior).
• They last for 10--100 s; the spectrum is thermal & the area of
the emission region is consistent with n-star radius of ~10km.
• The energy observed is produced by thermonuclear
conflagration -- nuclear burning of a thin H-He layer
of accreted material on the surface of n-star.
• Thermally unstable H-burning occurs in-10a H-He layer
when the accretion rate is less than 2x10
Mo/year.
Strohmayer & Bildstein, 2003, astro-ph/0301544
Strohmayer & Bildstein, 2003, astro-ph/0301544
Strohmayer & Bildstein, 2003, astro-ph/0301544
Superbursts: once in a while (perhaps one in a few years)
a burst is seen with total energy release of ~1042 erg; these
last for ~ an hour and are likely produced by unstable c-burning
in deeper layers.
Advection Dominated Accretion Flow (ADAF)
(Narayan et al, 1998, astro-ph/9803141; Narayan astro-ph/0201260)
• The mass of BH at the center of our galaxy is 2.6x106 Mo
(The Eddington luminosity is 3x1044 erg s-1)
•
The observed luminosity:
at the peak (sub-mm) is 3x10-9 LEdd
in the infrared band is < 3x10-10 LEdd
and in quiescent x-ray < 10-11 LEdd
•
Chandra finds gas at 1kev with density ~ 30 cc in an area of
size 10 arcsec surrounding the BH (Baganoff et al. 2001);
the capture radius for this gas is ~ 1 arcsec.
•
The Bondi accretion rate is ~ 3x1020 g s-1. If the Bondi flow
terminates in a thin accretion disk, the luminosity expected
is ~ 3x1040 erg s-1 (must of it in infrared and optical); this is
far larger than the observed luminosity.
• Many of the nearby (low z) SMBHs have very low luminosity.
• This cannot be explained as a result of low gas supply alone.
What might be going on in these systems is that at low
accretion rate the gas does not radiate efficiently and
much of the energy produced by viscous dissipation is
advected into the BH (hence the name ADAF).
Structure of ADAFs
• Two temperature plasma – p+s are much hotter than e-s.
• Thick disk – because cs ~ vk (vk is the Keprelian speed).
• vr ~ vk (vr ~  csH/r) and   r-3/2
• Positive Bernoulli parameter; so gas can flow outward
•
Possible detection of polarized radio emission
from the Galactic center (if confirmed) suggests
that the density near the center is much less than
in the ADAF model (Faraday rotation will depolarize
the radiation).
•
Convection dominated accretion flow (CDAF)
has a lower density (  r-1/2) and solve the
polarization problem.