Last Section of AY4

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Last Section of AY4

• Last quiz, Thursday, March 13
• Optional Final on March 17, 12-3pm
• Neutron stars, pulsars, x-ray binaries
• Relativity
• Black Holes
• The Big Bang and cosmology

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Neutron Stars

• There is a last test of SNII theory
• If the scenario is correct, there should be a
  VERY dense, VERY hot ball of neutrons left behind
  and the explosion.
• This is called neutron star

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Neutron Star

• Neutron star mass gt 1.4Mo
• Neutron star radius 10 - 80 km
• Neutron star density 1014 grams/cm3
  100 million
  tons/thimble
• Initial Temperature gt2,000,000k
• Neutron star remnant will be spinning rapidly and
  have a huge magnetic field

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Neutron Star Spins

• The reason n-stars are predicted to be rapidly
  spinning is another Law of Physics called
  Conservation of Angular Momentum.
• Linear momentum is a property of a moving object
  and is a vector quantity of a moving object to
  remain in motion.
• To change linear momentum
• you need to exert a force on an object.

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Conservation of Angular Momentum

• Any spinning object has angular momentum which
  depends on how fast it is spinning and how the
  objects mass is distributed.
• how fast -gt w (greek letter omega)
• mass distribution -gt Moment of inertia (I)

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Conservation of Angular Momentum

• Conservation of angular momentum means
• Moment of
  Angular
• Inertia
  velocity

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Conservation of Angular Momentum

• Think about those ice skaters. With arms out, a
  skater has a large moment of inertia. Pulling
  his/her arms in reduces the moment of inertia.
• Arms out large I, low spin rate
• Arms in small I, high spin rate

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Conservation of Angular Momentum

• The moment of inertia for a solid sphere is
• If a sphere collapses from a radius of 7x105km to
  a radius of 10km, by what factor does its spin
  rate increase?

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• Conservation of angular momentum means
• Sun rotates at 1 rev/month. Compress it to 10km
  and conserve L, it will spin up to 1890
  revolutions/second (and fly apart)

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Magnetic Fields

• Magnetic field lines are also conserved. When the
  core collapses, the field lines are
• conserved, and the
  density
• of the field lines
  goes way
• up . This is the
  strength
• of the magnetic
  field.

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Neutron Stars

• The possibility of n-stars was discussed way back
  in the 1930s but for many decades it was assumed
  they would be impossible to detect.
• But, in 1967, Jocelyn Bell and Tony Hewish set up
  a rickety barbed-wire fence in the farmland near
  Cambridge England to do some routine radio
  observations.

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LGMs

• Bell and Hewish discovered a source in Vela that
  let out a pulse every 1.3 seconds. Then they
  realized is was accurate to 1.337 seconds, then
  1.3372866576 seconds. They soon realized that the
  best clocks of the time were not accurate enough
  to time the object. They called it LGM.

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First Pulsar

• Bell was a graduate student at the time. The
  source was assumed to be man made, but when no
  terrestrial source could be identified, they
  briefly considered an artificial
  extra-terrestrial source.
• When a second source was discovered (Cass A) they
  announced the discovery as a new phenomenon.

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• The discovery led to a year of wild speculation,
  but explanations involving neutron stars quickly
  rose to the top.
• A pulsing source with period of 0.033 seconds was
  discovered in the Crab nebula.
• Big clue! Spin the Sun or Earth or a WD 30 times
  per second and they will be torn apart.
• Need a small object with very large material
  strength.

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Pulsars

• The new objects were named pulsars and is was
  soon discovered that they were slowly slowing
  down -- this provided the answer to the mystery
  of why the Crab Nebula was still glowing.
• There are now more than 1000 known pulsars in the
  Galaxy.

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Pulsars The Lighthouse Model

• So, what is the pulsing all about?
• The key is to have a misalignment of the nstar
  magnetic and spin axes?
• What do you call a rotating powerful magnetic
  field?

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Lighthouse model

• A rotating magnetic field is called a generator.
  The pulsar is a dynamo which is typically about
  1029 times more powerful than all the powerplants
  on Earth.
• The misalignment of the magnetic and spin axes
  results in a lighthouse-like effect as the beam
  sweeps past the Earth once per rotation period.

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Pulsars

• The period of the Crab pulsar is decreasing by 3
  x 10-8 seconds each day. The rotational energy is
  therefore decreasing and the amount of the
• 
  decrease in rotation
• 
  energy is equal to
• the
  luminosity of
• 
  nebula. Old pulsars
• 
  spin more slowly.

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• There is a mysterious cutoff in pulsar periods at
  4 seconds. The Crab will slow to this in about 10
  million years. The pulsar will turn off. Although
  the n-star will still be there, it will be
  essentially invisible.
• Most pulsars have large space velocities. This is
  thought to be due to asymetric SNII explosions.

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Pulsars

• Do all SN remnants have pulsars?
• No - some SN remnants are from SNI
• No - some rotating neutrons stars will have beams
  that dont intersect the Earth

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Milli-sec Pulsars and X-ray Binaries

• Since the first x-ray telescopes went into space
  on rockets it has been known that there are
  Luminous X-ray stars.
• In 1982, the first of many milli-second pulsars
  was discovered

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• The two phenomenon are connected.
• When a neutron stars has a close companion, it
  pulls material through the L1 point. This
  material flies down to the surface of the n-star
  and crashes onto the surface, releasing LOTS of
  gravitational potential energy. This energy comes
  out mostly as x-rays and is modulated with the
  n-stars spin.

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Mass-transfer and N-stars

• Some of the x-ray binaries have allowed a
  measurement of the neutron star mass
• In 10 of 11 cases, M1.44Mo
• This is good! Neutron stars are all supposed to
  be more massive than the Chandrasekar limit and
  there is even reason to expect them to be close
  to this limit as that is what initiated the core
  collapse in a SNII

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Millisecond Pulsars

• The discovery of pulsars that were spinning more
  than 100 times per second (the first was spinning
  640 times per second) threw the field for a loop.
  When some millisecond pulsars were discovered in
  old star clusters it was even more confusing.
• Eventually it was determined that all millisecond
  pulsars were in close binary systems and were
  spun up by accreting material.

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Detecting Neutron Stars

• Detecting n-stars via their photospheric emission
  is difficult.
• N-stars are VERY hot, but have a tiny surface
  area so have low luminosity.
• Initial temperature may be greater than
  3,000,000k so a very young n-star will emit most
  of its Planck radiation in X-rays.

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• First isolated n-star observed in photospheric
  light was discovered in 1997.
• Tsurface700,000
• Estimated age is 106 years.
• This is combined x-ray through visible light image

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• In 2002 there are about 6 isolated n-stars known
  that are seen in the light of their Plank
  radiation.
• Most are very nearby (lt300 pc) and traveling VERY
  fast.

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Puppis A remnant with 2 millionK n-star racing
away at 600 km.sec. Estimated age is 6000 years.
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• Sun R105km
• density6 gram/cm3
• 
  Neutron star R20km
• 
  density1014
• 
  Mass gt 1.4Mo
• White Dwarf R6000km
• 
  density106
• 
  Mass lt 1.4Mo

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Is there a limit to neutron degeneracy?

• Yes! Gravity wins the final battle. The current
  best estimate for the maximum mass of a
  neutron-degenerate star is 3Mo.
• If a neutron star exceeds this mass it will
  collapse into an infinitely small volume called a
  black hole.
• But, this story starts with Einsteins theories
  of special and general relativity.

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Special Relativity

• Various experiments starting in the late 1800s
  suggested that the speed of light was constant,
  independent of the motion of the observer.
• This is very counter-intuitive.

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• The spaceship traveling in the same direction of
  a photon measures the photon zooming away at the
  speed of light NO MATTER how fast the spaceship
  is traveling!

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Special Relativity

• Einstein (and others before him) decided to take
  the speed of light as an invariant and not make
  any assumptions about the two properties that go
  into determining speed
• Space and Time

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Time Dilation and Length Contraction

• The invariance of the measured speed of light
  independent of the motion of the observer can be
  understood if
• (1) Clocks run more slowly as speed
  increases.
• (2) Metersticks shrink as speed increases.
• Say what?

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Time Dilation

• As your speed with respect to another observer
  increases, your watch runs more slowly than the
  observers. This is called time dilation

Note, when vltltc, TT0
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Time Dilation

• As v approaches c, v/c -gt 1 and the denominator
  goes to zero. Dividing by zero gives infinity so
  as v-gtc, time grinds to a halt

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• Q. Suppose you measure an event that lasts for 1
  second by your watch. What will your friend in a
  spaceship moving at 0.98c measure as the duration
  of the event?
• Time has been stretched by a factor of 5 for your
  friend.

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Length Contraction

• In the same way, metersticks (space) contracts in
  the direction of motion.
• But wait, theres more!

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Mass

• Mass grows with speed.

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Constant Speed of Light

• The shrinking rulers and slowing clocks conspire
  to let observers in any moving frame measure the
  same speed of light.

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The Reason Travel to other Galaxies will be
Difficult

• The slowing clocks and increasing mass conspire
  to make it impossible for objects with mass to
  ever reach the speed of light.
• The increasing mass requires an ever-larger force
  to accelerate to larger speed and the force need
  would become infinite.
• Even if you could find the force, your clock
  would slow and slow and the last step would take
  and infinitely long time

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Is this right?

• Yes! There are many tests of Special Relativity.
• In particle accelerators mass increase and time
  dilation effects are routinely measured
• There have been tests flying very accurate clocks
  in high-speed jets that show time dilation
  directly.
• We might not be here if not for time dilation in
  the frame of cosmic rays called muons.

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General Relativity

• Einsteins theory of General Relativity is a
  theory of gravity
• The basic idea is to drop Newtons idea of a
  mysterious force between masses and replace it
  with the 4-dimensional
• SpaceTime Continuum

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General Relativity

• In GR, mass (or energy) warps the spacetime
  fabric of space.
• Orbits of planets around stars are not due to a
  central force, but rather the planets are
  traveling in straight lines through curved space

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Imagine tossing a shotput onto your bed and
rolling marbles at different speeds and distances
from the shotput. (also imagine that you have a
frictionless blanket on the bed).
The marbles that are moving slowly or close will
fall down toward the shotput. If you look from
above, it will appear as if the marbles were
attracted to the shotput.
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Fabric of Space

• This is a RADICALLY different view of the
  Universe and gravity
• In regions where space is not strongly curved, GR
  reduces to Newtons law of gravity
• Einstein pointed out his new theory would explain
  the Precession of the Perihelion of Mercury

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The Deflection of Starlight

• There were several other predictions of GR, one
  important one was that light rays would also
  follow straight lines through curved space.

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Tests of GR

• In 1919, during a total eclipse of the Sun, the
  predicted deflection of starlight for stars near
  to the limb of the Sun was measured and Einstein
  became a household name.
• GR also predicted that time would slow in
  strongly curved space. This was verified
  experimentally in 1958.

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Tests of GR

• There is another long list of predictions made by
  GR -- in every case to date, they have verified
  the theory perfectly.
• One of the more useful predictions was for
  gravitational lenses.

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On to Black Holes

• One important difference between Newtonian
  gravity and General Relativity is that photons
  are affect by gravity in GR.
• This is what leads to the idea of Black Holes. It
  starts with the concept of escape velocity.

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Escape Velocity

• Imagine feebly tossing a rocketship up in the
  air. It falls back to Earth because its kinetic
  energy was less than its gravitational potential
  energy.
• However, toss it with a larger and larger
  velocity and it will go higher and higher before
  falling back to Earth.
• There is a velocity above which it will not
  return to Earth -- this is the escape velocity.

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Escape Velocity

• To determine the escape velocity from Earth you
  set the gravitational potential energy equal to
  kinetic energy and solve for velocity

Mass of the object from which you want to escape
Radius from which you want to escape
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Escape Velocity

• Note that the escape velocity doesnt depend on
  the mass of the escaping body.
• For the Earth, put in the mass and radius of the
  Earth (for escape from the surface of the Earth)
  and you get
• Vesc 11 km/sec 25,000 miles/hr

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Escape Velocity

• Now suppose you shrink the Earth to 1/100 of its
  current radius (at constant mass). What happens
  to Vesc?
• As R goes up, Vesc
  goes down
• As R goes
  down, Vesc goes up
• Dont forget
  the square root
• For this
  case, Vesc increases by 10x

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Escape Velocity

• Reduce the radius of the Earth to 1cm and
  
• Vescc (speed of light)
• In this new theory of Gravity, where photons are
  affected by gravity, if the escape velocity
  equals or exceeds the speed of light, that object
  can no longer be observed. This is a Black Hole

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Black Holes

• The critical radius for which an object of a
  particular mass has an escape velocity of c is
  called the Schwarzschild Radius.
• This is also called
• the Event Horizon.

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Schwarzschild Radius

• You can easily calculate the Schwarzschild radius
  for any mass by setting Vescc
• Every object has a radius at which it becomes a
  Black Hole

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Black Holes

• But, it is VERY, VERY difficult to compress an
  object to its Schwarzschild radius.
• For the Sun, you would have to somehow overcome
  thermal pressure, then e- degeneracy, then
  neutron degeneracy. We know of no cosmic vice
  that can do that.

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Black Holes

• But, go back to a neutron star and we are
  building a pretty big vice. Thermal pressure has
  already been overcome as has e- degeneracy
  pressure.
• There is a limit to the pressure that can be
  generated by neutron degeneracy. Its hard to
  calculate, but is probably between 2Mo and 3Mo

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Black Holes

• Think about the n-star core of a SNII explosion.
  If say 1.6Mo of material falls back, the core
  will exceed the neutron degeneracy limit and
  undergo collapse to zero volume (what?) zero
  volume.

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Black Holes

• What is left behind?
• The gravitationally force (i.e. a warp in
  spacetime) including a singularity at the
  center of the warp
• An Event Horizon with radius given by
• RSch8.9km

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Hawking radiation
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Black Hole FAQs

• What would happen if the Sun collapsed into a
  Black Hole, would the Earth be dragged in?
• No, the gravitational force at the distance of
  the Earth would not change.

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• Is the Event Horizon a physical boundary?
• No, it is simply the distance from the center
  where the escape velocity of c.

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• What happens if a Black Hole absorbs some mass?
• As M increases, the Schwarzschild radius also
  increases.

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• Is there any reason to believe that Black Holes
  exist?
• You Bet!

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This would be great. But not too likely
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Black Hole Evidence

• The best stellar-mass cases are binary x-ray
  sources.
• Cygnus X-1 is a
  good
• 
  example.

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Black Hole Evidence

• Cyg X-1 is a bright x-ray source. Look there in
  the visual part of the spectrum, we see a 30Mo
  blue main-sequence star which is a spectroscopic
  binary with a period of 5.6 days.
• The companion has a mass of between 5 and 10Mo.
  What is it?

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Cygnus X-1

• There is no sign of the companion at any
  wavelength (but, remember the x-rays) so what is
  it?
• 1) A red giant would be easily seen
• 2) A main-sequence star would be seen with
  a little effort
• 3) Cant be a WD because Mgt1.4Mo
• 4) Cant be a n-star because Mgt3Mo

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Cygnus X-1

• By elimination, we are left with a black hole
• The x-rays back this up. In an accreting WD we
  see UV radiation, in an n-star we see soft
  x-rays, in Cyg X-1 we see hard x-rays because
  the accreting material falls into a deeper
  potential well.

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Stellar-mass Black Holes

• We now have a few dozen excellent stellar-mass
  black hole candidates and few people doubt that
  such objects exist.
• There was a microlensing event in 1996 that
  was ascribed to a blackhole gravitationally
  lensing a background star.
• There are various claims that x-ray transients
  are black holes accreting little bits of stuff.

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Supermassive Black Holes

• Since the early 1960s extraordinarily energetic
  objects called qsos or quasars have been
  identified a large distances and lookback times.
• The only explanation astronomers could come up
  with for their energy source was accreting mass
  onto a large (gt105Mo) black hole.

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Supermassive Black Holes

• QSOs had large radio jets emitted at enormous
  velocities.
• Eventually it becamse clear that QSOs were all
  located in the cores of galaxies and nearby
  counterparts were identified.

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• Cen A radio jets
• The nearby systems allowed observations much
  closer to the central engine and over time the
  evidence for the black holes has become more
  direct

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The Galactic Center

• After years of speculation about a possible
  supermassive black hole in the center of the
  Milky Way, work at Keck by Andrea Ghez at UCLA
  demonstrated convincingly in 1999 that we have a
  2 million solar mass black hole at the center of
  the Galaxy.

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Supermassive Black Hole in the Galaxy

• 2002 observations pretty much cinch the case for
  a 2.6 million solar mass black hole in the center
  of the galaxy.
• See the movie!