Black Holes and Close Binary Star Systems
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Black Holes (continued…)
Gravity in a close binary system
Accretion Disks
White Dwarfs in semi-detached
binaries
Type IA Supernovae
Final Exam
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Take-home will be handed out in class on Tuesday April 23.
To be submitted on Wednesday May 1 at 10:30AM in 210 JFB (regularly
schedule final exam meeting time).
Comprehensive , Slight emphasis on material in chapters 13-18.
A study “guide/review” is on the website as Lecture 27.
The Schwarzschild metric
•Consider a sphere of radius R and
mass M placed at the origin of the
coordinate system. The coordinate r
does NOT represent distance from the
origin. A concentric sphere whose
surface is at r would have a surface
area 4r2.
•A Flamm paraboloid helps “visualize”
this curvature. Remember that “you”
would be contained in the curved
spacetime and you cannot directly
“view” the curvature into the 4th
dimension….
•Proper distance along a radial line
•Proper Time at radial coordinate r
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http://en.wikipedia.org/wiki/Schwarzschild_
metric
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http://casa.colorado.edu/~ajsh/schwp.html
http://people.hofstra.edu/Stefan_Waner/diff_geo
m/Sec15.html
http://channel.nationalgeographic.com/episode/journey-to-theedge-of-the-universe-3023/Overview#tab-interactive
Black Holes
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In 1783 John Mitchell pondered that
the escape velocity from the surface
of a star 500 times larger than the
sun with the same average density
would equal the speed of light.
v esc = 2GM /r = 2G(500MÄ ) /7.93RÄ = c
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Light would not be able to escape
from such a star!!!!
Naïve solution of Newtonian escape
velocity equation for c gives a radius
of R=2GM/c2 for a star whose
escape velocity equals the speed of
light.
R=2.95(M/M) km….kinda small!!!
In 1939 J. Robert Oppenheimer and
Hartland Snyder described the ultimate
gravitational collapse of a massive star
that has exhausted its sources of nuclear
fusion. They pondered what happened to
the cores of stars whose mass exceeded
the limit of neutron stars..
In 1967 the term “black hole” was coined
By John Archibald Wheeler
The Schwarzschild Radius
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Consider the Schwarzschild metric
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When the radial coordinate of the star’s
surface has collapsed to RS=2GM/c2 the
square roots in the metric go to zero. RS is
known as the Schwarzschild Radius.
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At r=RS the behavior of space and time is
“remarkable”….
The proper time measured by a clock here is
d=0.
Time has slowed to a complete stop! As
measured from a vantage point far away.
From this viewpoint nothing ever happens at
the Scwarzschild radius
Does this mean that even light is frozen in
time???
The speed of light by an observer
suspended above the star must always
be c. But from far away we can determine
that light is delayed as it moves through
curved spacetime…
•The apparent speed of light, the rate at
which the spatial coordinates of a photon
change, is called the coordinate speed of
light. For light ds=0.
• For dd=0, we have
•In flat spacetime dr/dt~c, however at
r=RS dr/dt=0
Light does appear frozen in time at the
Schwarzschild radius!!!
The Schwarzschild Radius
and Event Horizon
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A star that has collapsed within the
Schwarzschild radius is called a
A non-rotating black hole would have a
particularly simple structure!!!
Black Hole.
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It is enclosed by the Event
Horizon located at r=RS.
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The region inside of the event
horizon is forever hidden from view
of the rest of the universe!!!
However the properties of the Black
Hole may be calculated.
Its gravitational effects are still felt…
Note that the event horizon is a
“mathematical surface” and does
not necessarily coincide with any
physical surface
•At the center is a Singularity. A
point of zero volume and infinite
density where all of the black hole’s
mass is located.
•Spacetime is infinitely curved at the
singularity.
•Cloaking the central singularity is the
event horizon, so the singularity can
never be observed
•“Law of Cosmic Censorship” that
forbids a naked singularity from
appearing unclothed…
A Trip into a Black Hole
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Imagine an attempt to investigate a
black hole from a safe distance and
reflecting a radio photon (or any
photon) from an object at the event
horizon.
Travel time between radial
coordinates r1<r2.
If r1=RS…t=infinity
Any object falling to the horizon will
appear frozen there (in the infinite
future). Even the collapsing star.
What happens to an object
(indestructible person) that free falls into
a black hole?
•She shines a monochratic flashlight
back at us once per second.
•These light signals arrive farther and
farther apart for several reasons…
•Subsequent signals must travel
farther as she accelerates
•Her proper time is running slower
at ther location due to gravitational
time dilation and her motion special
relativity time dilation
•The coordinate speed of light
become slower as she approaches
the black hole
•The frequency of light becomes
increasingly redshifted
•Due to her acceleration away from
us
•And gravitational redshift
•The light is redshifted and dimmed to
invisibility
A Trip into a Black Hole
A Trip into a Black Hole
The Discomforts Associated with a Trip into a
Black Hole
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Differential Tidal Forces…
– Rip and Squeeze…
– Ripped apart several hundred
kilometers from the Black Hole
In just 2 milliseconds she would fall
the final few hundred kilometers to
the event horizon …and crosses
it!!!
Her proper time continues normally
and she encounters no frozen stellar
surface since it has fallen through
long ago.
However once inside the event
horizon…..
– (ds)2=(c dt)2(1-Rs/r)<0
– Spacelike interval
– Drawn to the singularity within
6.6 x 10-5 s
Mass Range of Black Holes
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Stellar-Mass Black holes: 3-15 M
may form directly or indirectly as a
consequence of the core collapse of a
sufficiently massive supergiant star.
Could also form when a neutron star
stips away enough mass from a
companion.
Intermediate Mass Black Holes: may
exist in a range of 100 to in excess of
1000 M . Ultraluminous X-ray
Sources.
•Supermassive Black Holes: are known
to exist at the center of many (probably
most) galaxies. Range in mass from
105-109M. The milky way has one that is
about 3.7 x 106 M.
•Primordial Black Holes: from 10-8kg to
105M. The only criterion for a black hole
is that its entire mass must lie within the
Schwarzschild radius.
•Schwarzschild radius of proton:
RS=2Gmp/c2=2.47x10-54m
•Schwarzschild radius of electron:
RS=2Gmp/c2=1.34x10-57m
Black Holes Have No Hair!
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Completely described by
– Mass
– Angular Momentum
– Electric Charge
Upper limit to black hole’s angular
momentum
– Lmax=GM2/c
Kerr Metric
Kerr-Newman Metric
Spacetime Frame Dragging
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Rotation distorts the central singularity
from a point to a flat ring.
The event horizon assumes the shape of
an ellipsoid.
As a massive object spins, it induces a
rotation in the surrounding spacetime
Spacetime Frame Dragging
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Ergosphere: a nonspherical region
outside the event horizon where any
particle must move in the same direction
that the black hole rotates. Spacetime
within the ergosphere is so rapidly rotating
that a particle would have to move more
rapidly than c to remain at the same
angular coordinate.
Even earth generates frame dragging..!!!
Tunnels in Spacetime: “Wormholes”
• There are some vacuum
solutions to Einstein’s
equations that admit
tunnels in space-time.
• Also the realm of
science-fiction….
Stellar-Mass Black Hole Candidates
• Evidence for Stellar
Mass Black holes
– X-ray binaries with
massive compact
objects of M>3 M
• Cygnus X-1
• LMC X-3
• V404 Cygni
Hawking Radiation
Hawking Radiation
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Hawking predicted that black holes
can radiate away energy - evaporate
by a process where particle antiparticle pairs appear as a fluctuation
out of the vacuum near an event
horizon. Normally the pair would
annihilate and disappear. However
near the horizon one of the pair can
disappear within the event horizon
alllowing the other to escape…
Slow Evaporation!!!!
The End….
Super Massive Black Holes
http://en.wikipedia.org/wiki/Supermassive_black_hole
http://en.wikipedia.org/wiki/File:A_Black_Hole%E2%80%99s_Dinner_is_Fast_Approaching_-_Part_2.ogv
Super Massive Black Holes
Super Massive Black Holes
Gravity in a close binary system
Lagrangian Points and Equipotential Surfaces
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Lagrange points: no net force on a
test mass
Classes of Binary Systems
Roche Lobes
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Tear drop shaped region of space of
the equipotential surface
Mass Transfer Rates
r vA
M =
x » Rd
v RMS =
3kT
m
M » r vRMS p x 2
M » p Rd r
3kT
mH
Mass Transfer Rates
Energy Release through “falling”…
Energy Release through “falling”…
Accretion Disks
• What is the temperature of Disk?
• What is its Luminosity?
Accretion Disks
Accretion Disks
Accretion Disks
Accretion Disks
Evolution of a
Binary System
Types of Interacting Binaries
Types of Interacting Binaries
Types of Interacting Binaries
White Dwarfs in Semidetached Binaries
Cataslysmic Variables
Dwarf or Classical Nova: Survives its
Energy Release
Supernova Type IA
White Dwarf is destroyed…
Dwarf Novae
Dwarf Novae
Dwarf Novae
Type IA Supernovae
Observations
“Remarkably” consistent in their light output
Standard Candle…
Type IA Supernovae
Models
• Double degenerate: two white
dwarfs
• Single Degenerate: One
white dwarf
– Complete destruction of
white dwarf
– Accumulation of helium
deposit on surface
triggers a helium flash.
Shock wave triggering
ignition of Carbon and
Oxygen ignition….
Type IA Supernovae
Models
Type IA Supernovae
Applications
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Map the Universe!!!
Determine that expansion of
Universe is accelerating
Neutron Stars and Black Holes in Binaries
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X-ray systems !!!
Binary X-ray Pulsars
Alfven Radius and infall channeling towards
poles
Binary X-ray Pulsars
Neutron Star Binaries
Lab for General Relativity
Short Hard Gamma Ray Bursts
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GRBs of < 2 second duration
believed to be caused by the merger
of 2 neutron stars
“Nearby” GRB as deadly as an
asteroid impact…
The End