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Homework #12
11/30/15
Due 12/7/15
Chapter 14
Review questions 4, 6, 12
Problem 8
PHYS 3380 - Astronomy
Black Holes
Just like white dwarfs (Chandrasekhar limit: 1.4 M), there is a
mass limit for neutron stars (neutron degeneracy):
Neutron stars cannot exist with masses
> 3 M
We know of no mechanism to halt the collapse of a
compact object with > 3 M.
It will collapse into a single point – a singularity:
=> A Black Hole!
PHYS 3380 - Astronomy
Black Holes
Black holes are completely
collapsed objects - radius of the
“star” becomes so small that the
escape velocity approaches the
speed of light
Escape velocity for particle from
an object of mass M and radius R
2GM
v esc 
R
If photons cannot escape, then
vesc>c. Schwarzschild radius is

2GM
M
R  RS  2
 3 km
c
MSol
Nothing (not even light) can escape from inside the Schwarzschild radius
- we have no way of finding out what’s happening inside the
Schwarzschild radius - the “event horizon”
PHYS 3380 - Astronomy
Size of black holes determined by mass. Example Schwarzschild
radius for various masses given by:
Object
M (M)
Rs
Star
10
30 km
Star
3
9 km
Sun
1
3 km
Earth
3x10-6
9 mm
The event horizon is located at
Rs - everything within the event
horizon is lost. The event
horizon hides the singularity
from the outside Universe.
If the entire mass of the Earth was confined to 9mm, it would be a
black hole - can’t collapse spontaneously into black hole because
mass < 3 M
PHYS 3380 - Astronomy
Black Holes in Supernova Remnants
Some supernova remnants with no pulsar / neutron star in the center may
contain black holes.
Composite X-ray/optical/radio image of the supernova remnant W49B. The
structure and composition of this remnant hints that it was a gamma ray burst
and likely harbors a black hole at its center.
PHYS 3380 - Astronomy
“Black Holes Have No Hair”
Matter forming a black hole is losing almost all
of its properties.
Black Holes are completely determined
by 3 quantities:
Mass
Angular Momentum
Electric Charge
PHYS 3380 - Astronomy
Types of Black Holes
Schwarzschild - Non-rotating black hole
-simplest black hole, in which the core does not rotate
- only has a singularity and an event horizon
Kerr - Rotating black hole
-probably the most common form in
nature,
- rotates because the star from which
it was formed was rotating. When the
rotating star collapses, the core
continues to rotate, and this carried
over to the black hole (conservation of
angular momentum).
-Has an Ergosphere - egg-shaped
region of distorted space around the
event horizon
-caused by the spinning of the
black hole, which "drags" the
space around it.)
-Static limit - The boundary between
the ergosphere and normal space
PHYS 3380 - Astronomy
Black Hole Gravity Well
At a distance, the
gravitational fields of a black
hole and a star of the same
mass are virtually identical.
At small distances, the
much deeper gravitational
potential will become
noticeable.
PHYS 3380 - Astronomy
General Relativity Effects Near Black Holes
An astronaut descending down
towards the event horizon of the
BH will be stretched vertically
(tidal effects) and squeezed
laterally - friction would heat the
astronaut to millions of degrees
emitting x-rays and gamma rays.
“Spaghettification”
PHYS 3380 - Astronomy
General Relativity Effects Near Black Holes
Time dilation
Clocks starting at 12:00
at each point.
After 3 hours (for an
observer far away from
the BH):
Clocks closer to the BH
run more slowly.
Time dilation becomes
infinite at the event horizon.
Event Horizon
PHYS 3380 - Astronomy
General Relativity Effects Near Black Holes
Gravitational Red Shift
All wavelengths of emissions from
near the event horizon are stretched
(red shifted).
 Frequencies are lowered.
Event Horizon
PHYS 3380 - Astronomy
Remember: General Theory of Relativity predicted gravity could bend space confirmed during a solar eclipse when a star's position was measured before,
during and after the eclipse.
An object with immense gravity (like a galaxy or black hole) between the Earth and
a distant object could bend the light from the distant object into a focus, much like a
lens can.
Einstein ring -the deformation of the light from a source into a ring through gravitational
lensing of the source's light by an object with an extremely large mass (such as
another galaxy, or a black hole).
PHYS 3380 - Astronomy
Lensing by a Black Hole
Animated simulation of gravitational lensing caused by a going past a
background galaxy.
- secondary image of the galaxy can be seen within the black hole
Einstein ring on the opposite direction of that of the galaxy.
- secondary image grows (remaining within the Einstein ring) as the
primary image approaches the black hole.
- surface brightness of the two images remain constant, but their
angular size varies
- produces an amplification of the galaxy luminosity as seen from a
distant observer. The maximum amplification occurs when the
background galaxy (or in the present case a bright part of it) is
exactly behind the black hole.
PHYS 3380 - Astronomy
Brightening of MACHO-96-BL5 happened when a gravitational lens passed between
it and the Earth. When Hubble looked at it, it saw two images of the object close
together - indicated a gravitational lens effect - intervening object was unseen.
- conclusion that a black hole had passed between Earth and the object.
PHYS 3380 - Astronomy
Stephen Hawking showed that black holes are not entirely black but emit small
amounts of thermal radiation.
- applied quantum field theory in a static black hole background.
- result - a black hole should emit particles in a perfect black body spectrum.
- Hawking radiation.
Virtual particle pairs constantly created near the horizon of the black hole, as they
are everywhere - quantum fluctuations. Normally, they are created as a particleantiparticle pair and they quickly annihilate each other. But near the horizon of a
black hole, it's possible for one to fall in before the annihilation can happen, in which
case the other one escapes as Hawking radiation.
- removes energy from black hole - evaporation
Temperature of the emitted black body spectrum is proportional to the surface
gravity of the black hole.
- large black holes are very cold and emit very little radiation
- black hole of 10 solar masses would have a Hawking temperature of
several nanokelvin, much less than the 2.7K produced by the Cosmic
Microwave Background.
- micro black holes on the other hand could be quite bright producing high
energy gamma rays.
PHYS 3380 - Astronomy
Compact Objects with Disks and Jets
Black holes and neutron stars can be part of a binary system.
Matter gets pulled
off from the
companion star,
forming an
accretion disk.
=> Strong X-ray source!
Heats up to a few million K.
PHYS 3380 - Astronomy
Cygnus X-1
Wide-field radio image of the environment of
the black hole system Cygnus X-1. The cross
marks the location of the black hole. The bright
region to the left (East) of the black hole is
a dense cloud of gas existing in the space
between the stars, the interstellar medium. The
action of the jet from Cygnus X-1 has 'blown a
bubble' in this gas cloud, extending to the north
and west of the black hole.
PHYS 3380 - Astronomy
Observing Black Holes
No light can escape a black hole
=> Black holes can not be observed directly.
If an invisible compact
object is part of a
binary, we can estimate
its mass from the orbital
period and radial
velocity - the same way
we have discovered
extra-solar planets.
Mass > 3 M
=> Black hole!
PHYS 3380 - Astronomy
How to Determine Compact Object Masses
P = orbital period
Kc = semiamplitude of
companion star
i = inclination of the
orbit to the line of
sight (90o for orbit
seen edge on)
MBH and Mc = masses
of invisible object and
companion star
Keplers Laws give:
3
PKc3
M BH
sin 3 i

2G M BH  M c 2
This gives us a firm lower limit on BH mass from relatively simple measurements

PHYS 3380 - Astronomy
Candidates for Black Hole
These candidates are all members of X-ray binary systems in which the
compact object draws matter from its partner via an accretion disk.
Name
BHC Mass (solar
masses)
Companion Mass
(solar masses)
Orbital period (days)
Distance from Earth
(light years)
A0620-00/V616 Mon
11 ± 2
2.6–2.8
0.33
about 3500
GRO J1655-40/V1033 Sco
6.3 ± 0.3
2.6–2.8
2.8
5000−11000
XTE J1118+480/KV UMa
6.8 ± 0.4
6−6.5
0.17
6200
Cyg X-1
11 ± 2
≥18
5.6
6000–8000
GRO J0422+32/V518 Per
4±1
1.1
0.21
about 8500
GRO J1719-24
≥4.9
~1.6
possibly 0.6
about 8500
GS 2000+25/QZ Vul
7.5 ± 0.3
4.9–5.1
0.35
about 8800
V404 Cyg
12 ± 2
6
6.5
about 10000
5–6
1.75
about 15000
0.43
about 17000
GX 339-4/V821 Ara
GRS 1124-683/GU Mus
7.0 ± 0.6
XTE J1550-564/V381 Nor
9.6 ± 1.2
6.0–7.5
1.5
about 17000
4U 1543-475/IL Lupi
9.4 ± 1.0
0.25
1.1
about 24000
XTE J1819-254/V4641 Sgr
7.1 ± 0.3
5–8
2.82
24000 – 40000
GRS 1915+105/V1487 Aql
14 ± 4.0
~1
33.5
about 40000
XTE J1650-500
9.7 ± 1.6 [17]
.
0.32[18]
Compact object with > 3 M
must be a black hole!
PHYS 3380 - Astronomy
Black Hole and Neutron Star Masses from Binary Systems
From J. Caseres, 2005, astro-ph/0503071
PHYS 3380 - Astronomy
Black Holes at the Center of Galaxies
A black-hole-powered jet of
electrons and other subatomic particles streaming out
from the center of M87 at
nearly the speed of light
-the blue jet contrasts with the
yellow glow from the
combined light of billions of
unseen stars and the yellow,
point-like clusters of stars that
make up this galaxy.
- the monstrous black hole at
center of M87 has swallowed
up matter equal to 2 billion
times our Sun's mass. M87 is
50 million light-years from
Earth.
PHYS 3380 - Astronomy
Relativistic Jets
Jets streaming out from the center of
M87 observed at various wavelengths
-
Active Galactic Nuclei (AGN) often
have black-hole-powered jets - highly
collimated and fast outflows that
emerge from close to the disc - of
electrons and other sub-atomic
particles
• production mechanism and jet
composition are not known at
present
observations cannot
distinguish between the
various theoretical models
that exist.
• have the most obvious
observational effects in the radio
waveband but radiate in all
wavebands from radio to gammaray
synchrotron radiation
PHYS 3380 - Astronomy
Superluminal speed illusion created by the finite speed of light and rapid motion
• clouds move towards Earth at speeds very close to that of light, in this case,
more than 98 percent of the speed of light - nearly keep pace with the light they
emit as they move towards Earth. When the light finally reaches us, the motion
appears much more rapid than the speed of light.
PHYS 3380 - Astronomy
Apparent superluminal motion when the radiating source
is moving so fast that it nearly “catch up” with its own
radiation.
A source component is moving at velocity v and at
an angle θ(  relative to the line-of-sight. Consider
the emission of photons at two different times t =0
and t = tc. Photons emitted at t= tc will reach the
observer at Δt = tc(1 − βcos ) later than those
emitted at t=0.
The apparent separation of
the two source components
then is Δr = vtcsin
Yielding an apparent velocity on the sky of: vapp = Δr/Δt = vsin /(1 − βcos 
PHYS 3380 - Astronomy
PHYS 3380 - Astronomy
Death Star Galaxy
Jet from a black hole at the center of a galaxy striking the edge of another galaxy
• composite image - X-rays (purple), optical and ultraviolet (red and orange), and
radio (blue)
• jet from the main galaxy on the lower left is striking its companion galaxy to the
upper right.
• jet impacts the companion galaxy at its edge and is then disrupted and deflected
PHYS 3380 - Astronomy
M84 - a massive elliptical galaxy in the Virgo Cluster, about 55 million light years
from Earth.
• composite X-ray (blue), radio (red), and visible (yellow and white)
• number of bubbles visible in hot gas, outlined with blue X-ray emission.
 blown by relativistic particles generated by the central supermassive
black hole in M84 - travel outwards in the form of a two-sided jet.
• smaller bubbles are found inside large bubbles
 provide clear evidence for repeated outbursts from the central black hole.
PHYS 3380 - Astronomy
Spectroscopic Observations of Black Hole at the Center of M84
Indicate a rapidly swirling disk of trapped material encircling the black hole. Rapid
dramatic swing to the left (blueshifted or approaching gas) and then an equivalent
swing from the right (redshifed)
Measured velocity 400 km/s within 26 LY of the galaxy's center, where the black
hole dwells - gives a black hole mass of at least 300 million solar masses.
PHYS 3380 - Astronomy
M31 Andromeda
PHYS 3380 - Astronomy
40 Light - Years
Two nested disks in
Keplarian rotation
around a
supermassive black
hole.
Triple nucleus of M31
PHYS 3380 - Astronomy
Zoom into the nucleus of the M31 then dissolve into animation of a
concentration of red stars and a disk of young blue stars swirling around a
black hole.
-revealed by Hubble's Space Telescope Imaging Spectrograph
(STIS)
-Astronomers not sure how the pancake-shaped disk of stars
could form so close to a giant black hole
- tidal forces should tear matter apart, making it difficult for gas and
dust to collapse and form stars.
PHYS 3380 - Astronomy
PHYS 3380 - Astronomy
high-resolution spectrum obtained with the Faint-Object Camera on board HST,
• change in Doppler shift of the [O II] emission line at 372.7 nm clear from both the
spectral images and intensity crosscuts.
• plot of the measured velocities compared to a model with a massive central object
shows very good fit with a central mass of 3 billion solar masses confined within a
radius of 3.5 parsecs or less - almost completely requires that this mass be a black
hole.
PHYS 3380 - Astronomy
3,700 light-year-diameter dust disk encircles a 300 million solar-mass black hole in the
center of the elliptical galaxy NGC 7052 - possibly a remnant of an ancient galaxy
collision. Disk rotates 155 kilometers per second at 186 light-years from the center.
strong source of radio emission and has two oppositely directed `jets' emanating from
the nucleus.
PHYS 3380 - Astronomy
The Hubble Expansion Law
In 1929, Edwin Hubble announced that almost
all galaxies appeared to be moving away from
us.
• observed as a redshift of a galaxy's
spectrum.
• appeared to have a larger displacement
for faint, presumably further, galaxies.
 the farther a galaxy, the faster it is
receding from Earth.
Hubble constant H - one of the most important numbers in cosmology
• may be used to estimate the size and age of the Universe
• indicates the rate at which the universe is expanding
• changes with time
H0 = v/d
where v is the galaxy's radial outward velocity, d is the galaxy's distance from
earth, and H0 is the current value of the Hubble constant.
PHYS 3380 - Astronomy
The Hubble Constant
Most recent calculation H0 used 2003 data from the Wilkinson Microwave
Anisotropy Probe (WMAP) satellite
• combined with other astronomical data yielded a value of:
H0 = 70.1 ± 1.3 km/s/Mpc.
• agrees well with that of obtained in 2001 by using NASA's Hubble
Space Telescope:
H0 = 72 ± 8 km/s/Mpc
• less precise figure obtained independently in August, 2006 using data
from Chandra X-ray Observatory:
H0 = 77 km/s/Mpc with an uncertainty of ± 15%.
NASA summarizes existing data to indicate a constant of
H0 =70.8 ± 1.6 km/s/Mpc
if spacetime is assumed to be flat, or
H0 =70.8 ± 4.0 km/s/Mpc
otherwise
In 2012, Freedman et al. found H0 = 74.3 ± 2.1 (km/s)/Mpc after
a re-calibration of the Cepheid distance relation based on Spitzer
infrared data, combined with WMAP7 and Hubble cosmological
data. Because Cepheid variables are used to determine galactic
distances, this recalibration had a significant effect on the galactic
distance scale and hence H0.
PHYS 3380 - Astronomy
The Age of the Universe
The age of the universe has been estimated by a number of methods:
Lower limit provided by observations - must be at least as old as the oldest
thing in it.
• temperature of the coolest white dwarfs
• turnoff point of the red dwarfs in globular clusters
 typically yield ages in the range 14-18 billion years
Inverse of Hubble constant
1 d
 T
H v
1
3.09  1019 km
1yr
T


 13.8 1010 yr
7
70.8km /s / Mpc
Mpc
3.15 10 s

PHYS 3380 - Astronomy
Quasars
Certain objects emitting radio waves observed in 1960s had very unusual optical
spectra.
• named Quasistellar Radio Sources (meaning "star-like radio sources") contracted to quasars.
• finally realized that the reason optical spectra were so unusual is that the lines
were Doppler shifted by a very large amount, corresponding to velocities away
from us that were significant fractions of the speed of light.
• objects were thought to be relatively nearby stars, no one had any reason to
believe they should be receding from us at such velocities.
Quasar 3C273
First and brightest Quasar discovered
• left image shows radio and optical composite of quasar and its jet.
• right image superposes contours of radio frequency intensity.
• sharp radial lines are optical spike artifacts because of its brightness
PHYS 3380 - Astronomy
Quasars have very large redshifts, indicating by the Hubble law that they are at great
distances.
• the fact that they are visible at such distances implies they emit enormous
amounts of energy and are certainly not stars.
Now thought to be AGNs powered by supermassive rotating black holes at their
centers.
• most luminous objects known in the universe - objects that have been observed
at the greatest distances from us - most distant are so far away that the light we
see coming from them was produced when the Universe was only one tenth of its
present age.
Images show the three most distant quasars known.
• redshift parameters are 4.75, 4.90, and 5.00 respectively, which places them at
distances of about 15 billion light years