Stars & Singularities: Notes
001. Stars and Singularities
BHs in themselves are simple objects, characterized by three number
only: mass, angular momentum and charge. It is their interaction with
their surroundings that results in a wealth of physical phenomena.
These talks will focus on the interaction of the central BH in the
Galactic Center with the stars very close to it. We will discuss
processes for which there is already some observational evidence, as
well as processes that are suggested by theory and may yet be
discovered by future observations.
002. Singularities
In addition to the BH singularity, there are 3 other "effective
singularities" associated with the BH.
A stellar density singularity: This is predicted to occur in most
scenarios for the evolution of a stellar system around a BH. A
density distribution that (formally) diverges at the origin is called
a cusp. In practice, infinite density is not reached. Stars cannot
exist closer than the event horizon, and in fact are destroyed well
before this point either by collisions or by the BH tidal field.
A velocity singularity: Close to the BH the velocity field is Keplerian
and so formally diverges as r^-1/2, The velocity cusp is also limited
in practice - by the speed of light.
An optical singularity: Any mass bends light and amplifies the flux of
background sources. Behind the BH (or any other sufficiently
concentrated mass) there is a small region (a caustic) where the
amplification formally diverges to infinity. This divergence is
truncated by the finite size of the source.
003. 1: Stars near a BH
This is an 8"x8" field around the BH, taken in the K-band with a
resolution of 1" (Krabbe et al. 1995). The disks are the resolution
limit and not the actual size of the giant stars, whose angular
diameter is only ~10 micro arcseconds. None of the IR sources is
associated with the radio source SgrA*. For comparison, a region of a
similar size around the Sun contains no other stars but the Sun.
004. Why stars?
Unlike gas, whose dynamics are affected by non-gravitational forces
such as thermal pressure, radiation pressure and magnetic fields,
stars are clean gravity probes. We are specifically interested in
stars near the BH, where near means in the region where stars can
exist (beyond the tidal radius) but where the potential is completely
dominated by the hole. For the GC, the event horizon is much smaller
than the tidal radius (for a solar type star) and so General
Relativistic effects can neglected to 1st approximation.
The properties of stars are well known from other (normal)
environments, and their observed luminosity and spectrum can be
translated into mass and maximal age, both very important quantities
for understanding the dynamics of the system. In particular, processes
that operate on time-scales much longer than the maximal age are not
relevant for the star.
Stars very near the BH are also connected to BH growth through tidal
disruption, mass loss from stellar winds and from stellar collisions.
Finally, the region near the BH may provide a laboratory for studying
stellar phenomena under extreme conditions: high density, velocity and
strong tidal fields.
005. Young stars - old stars
IR spectroscopy is possible for the brighter, well separated stars in
the field. Spectroscopy indicates that the stellar population is a mix
of old (red) stars and young (blue) stars.
The old stars (red giants) seen near the BH in the Galactic Center are
in the mass range ~1-8 M0 and are older than 1 Gyr. The faintest
observable young stars (blue giants) may be main sequence O stars with
masses of ~40 M0 and lifetimes < 5 Myr. The brightest young stars
(the He stars) are Wolf Rayet-like stars with masses of >20 M0 and
lifetimes of < 10 Myr. The blue stars are too young to have relaxed,
and their orbits (position, velocity) still reflect the initial
conditions of their formation (e.g. the emission line stars are
observed to counter rotate relative to the galactic rotation).
The faint stars in the cluster around SgrA* have blue featureless
spectra, which are typical of young stars.
006. Nature vs Nurture
The observation of seemingly young stars very close to the BH and a
mixture of young and old stars farther out raises a key question. Are
we seeing essentially random variations in the stellar population,
whch are explainable in terms of normal star formation processes
("Nature"), or is this a result some systematic effects of the unique
extreme environment very near the BH ("Nurture")?
If these are the effects of the environment, there are two options to
consider. First, this could the result of an unusual from of star
formation, in which case the stars are indeed young and dynamically
unrelaxed, and so do not convey direct information on the dynamical
processes in near the BH. Second, this could be a results of unusual
evolution, so that the stars only appear young, but are in fact old
and dynamically relaxed.
In these talks I focus on the second possibility.
007. Collisions: dynamics & evolution
One way of systematically surveying the dynamical processes in a
gravitating stellar system is by considering collisional processes as
function of scale.
008. Physical scales: time
N/A
009. Physical scales: length
N/A
010. Relaxed stellar system with MBH
N/A
011. The stellar distribution function
The Jeans equation is essentially a re-statement of the continuity
equation of the stellar orbits in phase space in terms of averaged
quantities: the mean stellar density and velocity dispersion. Here it
is given for the simplest case: A steady state, isotropic,
non-rotating system. The steady-state assumption is justified because
the dynamical timescale is much shorter than the relaxation
timescale. The assumptions of istropy and non-rotation are
observationally justified.
THe Jeans equation very near the BH ties the density and the velocity
dispersion together. The steeper the cusp (large alpha), the larger
the ratio between the circular velocity and the velocity dispersion,
and so the fraction of loosely bound stars or unbound stars is
smaller. This means that stars in steep cusps have are more local
than those in flat cusps, where a larger fraction of the stars that
are observed near the BH in any instant are merely "passing
by". Therefore, it is less likly to have a long-lived locally distinct
population of stars without a density cusp.
012. Summary: Stars near a BH
N/A
013. 2: The stellar collider
N/A
014. Cusp or core?
The distribution of the stellar mass around the BH at distances not
much smaller than the radius of influence can be derived from the
change in the total enclosed mass (measured from the stellar
velocities) with radius. This becomes progressively more difficult
nearer to the BH, since the enclosed stellar mass there is just a
small fraction of the BH mass. Close to the BH one has to count stars
directly in order to probe the density distribution. This is
problematic, because it not clear whether the observed giant stars
faithfully trace the unobserved, underlying population of old, relaxed
stars.
Over the years, many density models were suggested, differing in core
size and core density, depending on the data set and analysis method
(usually a problem of small number statistics). One of the more
recent density models with a relatively small core (r_c = 0.4 pc,
central density = 4e6 Mo/pc^-3) is compared to a relaxed cusp
normalized to have the same total mass within 0.4 pc). The cusp
reaches densities ~50 times higher than the core model).
015. The collision rate
N/A
016. Star counts
N/A
017. The stellar distribution
N/A
018. Modes of collisional destruction
The K luminosity of a star of radius R increases with T_eff roughly as
T_eff, so that the K luminosity scales as R^2*T_eff, while the
bolometric luminosity scales as R^2*T_eff^4. A giant with R=100 R0 and
T_eff=3x10^3 K that is reduced by a collision to 1 R0 will maintain
its bolometric luminosity by raising its effective temperature to
3x10^4 K, but its K luminosity will fall by a factor of 10^3 (7.5
mag).
It is not easy to destroy a giant's enevelope - a star passing through
it will simply "punch a hole" that will be quickly filled. Th enevelope
can be destroyed by a 3-body collision that leaves a tight binary in
the envelope, which, churns up the enevelope until it evaporates, or by
a collision that knocks the core out off the enevelope.
019. Collisional destruction of giants
N/A
020. Smoothed Particle Hydrodynamics
Smoothed Particle Hydrodynamics (SPH): A numeric method to calculate
gravity and hydrodynamics. The star is represented by discrete point
masses, whose mass is spread within a sphere (smoothing length). The
mass is centrally concentrated, and falls smoothly towards the
edge. The size of the sphere reflects the local density of mass
points: high point density -> small sphere -> high mass density. The
superposition of the spheres defines a continuous density field, that
together with an assumed equation of state (here, that of an ideal
gas), allows to calculate the thermodynamic quantities(pressure,
internal energy, entropy) and model the hydrodynamics of the system.
(R) A collision between two n=3/2 equal mass polytropes with
vp/vesc=1.7. The mass loss is 2% of the total, and the spin up 15% of
break-up.
(L) A collision between a MS star and a giant of equal mass, with
vp/vesc=2.5. The giant mass loss is 1% and the spin-up 10% of
break-up.
021. Exotic object formation
Thorne-Zytkow objects (TZO) are NS that have sunk to the core of a
massive star (O-star, 10-15 Mo), thus forming a hybrid, extremely red
super-giant with strong winds and unusual surface abundances.
022. Tidal spin-up by collisions
Collisional destruction requires an almost head-on collision, and is
therefore rare. Near-misses are much more frequent, but their effects
are more subtle. When two stars pass very close to each other, they
undergo a tidal interaction, where energy and angular momentum are
transferred from the orbit to the stars. Tidal encounters near the BH
are strongly hyperbolic (E>>0), because the Keplerian velocity is much
higher than the escape velocity from the stellar surface. The orbital
energy lost in such encounters is rarely enough to lead to tidal
capture (E-dE>0). Most of the effects of the encounter are
transient. The energy and the tidal oscillations are dissipated on the
dynamical and thermal relaxation timescales, which are much shorter
than the stellar lifetime or the mean time between collisions. It is
however harder for the star to lose the excess angular momentum, since
magnetic breaking typically operates on timescales comparable to the
stellar lifetime. Over its lifetime, the star will undergo many
collisions, whose effects will increase the angular momentum in a
random walk fashion.
High rotation can affect the appearance and subsequent evolution of the
star by processes such as meridional circulation, which replenishes
the Hydrogen in the core and changes the surface abundances, and by
adding rotational support to the core, which extends the main sequence
lifetime of the star and leads to a more luminous giant phase. Unusual
surface abundances indicating very high rotation have been observed in
spectra of giants in the GC.
023. BH-MS collision
N/A
024. MS-MS collision
N/A
025. Stellar spin near the BH
N/A
026. Tidal scattering by the BH
N/A
027. MBH-MS tide
N/A
028. MBH-MS tide (zoom)
N/A
029. Summary: Stellar collisions
N/A
030. 3: The Gravitational Telescope
N/A
031. Gravitational lensing in the GC
N/A
032. Lensing by a point mass
Difference between gravitational lens and a glass lens:
In glass lens bending angle DECREASES with smaller impact parameter.
In grav. lens bending angle INCREASES with smaller impact parameter.
033. 2 lensed images
N/A
034. Pinpointing the BH
Constraints are in terms of observables only! No need to assume M_bh
or R0. Area covered by error circle + error polygon is ~0.05 of
search area. Probability of ML in random anywhere in area is ~0.01.
Combined probabilityi of result in random: 1/2000. ML at best fit
position dominated by the 2 best image pairs: of a blue, variable
supergiant (M_K=-8.7) ~3 kpc behind the GC, and of a red supergiant
(M_K=-8.4) ~16 kpc behind the GC.
035. Modes of detection
N/A
036. Microlensing lightcurve
Lensed images will appear as variable sources with anomalous accelerations
and high velocities.
037. X-section, duration, amplification
N/A
038. Optical depth - Lensing probability
Here n_\star is the density of SOURCES, not of LENSES as is the case
in estimates of the optical depth in Galactic Microlensing searches.
039. Lensing event rate
N/A
040. Were lensing flares observed?
N/A
041. Magnification bias
N/A
042. Beyond the point lens assumption
N/A
043. Enhanced BH-star lensing
N/A
044. Summary: Gravitational lensing
N/A
045. Summary: Stars very near the GC BH
N/A