Download Stellar mass Black Holes

ASTR 200 : Lecture 20
Stellar mass Black Holes
1
High-mass core collapse
• Just as there is an upper limit to the mass of a white
dwarf (the Chandrasekhar limit), there is an upper limit
to the mass of a neutron star
• The maximum mass that it is felt neutron degeneracy
pressure could hold up the star is called the
Oppenheimer-Volkov limit, estimated to be ~3 M solar
2
• This mass is uncertain because the equation of state of
extremely dense matter is modified by nuclear forces,
and there is several equations of state for degenerate
neutron matter exists (uncertainties ~factor 2).
• BUT, if the mass exceeds this, the relativistic neutron
core will collapse, and no known force can stop it.
A Black hole
• To the current knowledge of physics, such an object
will collapse to a 'point' of zero radius, called a
`singularity', because the finite mass occupies zero
volume and the density becomes infinite.
• In astronomy and physics, the fundamental property
of a `black hole' is that there is a place where the
escape speed exceeds the speed of light.
• Although doing all of this precisely requires general
relativity, there are aspects that can be understood
with just Newtonian gravity.
3
Escape speed
• Return to our friend, the vis-viva equation, relating
orbital speed v to the mass of the central object M, the
current distance r and the orbit's semimajor axis a
2 1
v =GM ( − )
r a
2
• If we are at some distance r=R where all the mass is
contained inside that distance and is spherically
symmetric, then M includes all the mass.
– What speed does one need to travel to get arbitrarily far
away? That would correspond to a → infinity, and thus
√
2GM
v esc =
R
4
which we derived before
The Schwarzschild radius
• So given the mass M, there is some radius rs that if you
pack all the mass interior to that radius, the escape
speed reaches the speed of light. Its expression:
2 GM
M
r s=
≃ 3
km
2
M solar
c
• If a star of mass M falls below this radius, nothing from
inside rs could be moving fast enough to escape to great
5
distance, not even a light beam
• This is called the Schwarzschild radius, after Karl
Schwarzschild, who first solved Einstein's field
equations for a spherically symmetric mass. Note that
this depends only on M...
Note that incredible density NOT required
• Even in the 18th century, Laplace and Michell realized, after
Newton's corpuscular (photons, in modern term) that since light
traveled at a finite speed, it might be possible to have vesc > c .
• Michell (english scientist) suggested there could be 'dark stars'
whose gravitational pull would be strong enough that light could
not escape
• Laplace suggested however a simpler idea. Imagine a larger and
larger uniform density object; then
v esc =
√
2GM
=
R
√
3
8πG ρR
≃√ 8 ρ G R
3
R
so even for `normal' densities, a large-enough object has vesc --> c
– Eg., for water's density, R>3000 AU would have vesc> c
6
The event horizon
• The surface of radius rs is called the event horizon of the black
hole. It is called such because an outside observer cannot `see'
(receive photons) from inside this surface; they cannot escape.
• Even if some unknown quantum mechanical effect were to
prevent the central singularity, there is STILL an event horizon,
and light and any material object cannot escape.
• Any matter which enters the event
horizon cannot escape; it is cut off
forever from the rest of the
universe.
• In principle, any object can form a
black hole if you force it into a
small enough volume, but stellar
evolution provides a natural path
7
V404 Cygni
• A binary system of period 6.5 days, with X-ray emission.
• The visible star is roughly Sun-like (0.6 times the mass), but its periodic
Doppler shift (of amplitude ~210 km/sec) shows it orbits an invisible
object
• Minimum mass (because of sin i effect) of invisible companion is 6 solar
masses; best estimate is 10-12 solar masses
– Larger than a neutron star can be
• Clearest case in our galaxy of a
stellar-scale black hole
– Distance of ~2400 pc
• X-rays are coming from an `accretion
disk' surrounding the black hole. Gas
drawn off the companion spirals
down and heats up in a disk around
black hole, emitting X-rays before it
8 spirals inside the event horizon
Black hole accretion disks
• The black hole can be
surrounded by a disk that feeds
matter into it, just like a
protostar
• Local energy dissipation at
each radius results in material
losing orbital energy and
spiraling slowly towards the
black hole, getting very hot
• Because ionized, make a
magnetic field which some
charged particles escape along
• `Seeing' this tricky, because
black hole bends light and you
9 can see the far side of the disk!
A brief intro to General Relativity
• When gravitational fields become very strong,
newtonian gravity is only an approximation to how
objects accelerate
• General relativity (GR) is a geometrical description of
gravity; it and quantum mechanics are the two pillars of
modern physics. There are no known phenomena that
are not explained by them.
• In GR, space and time are considered simultaneously
and on an equal footing. The 3 spatial dimensions and
one time dimension are combined to form 4d spacetime
• As time progresses, an object traces out a path through
spacetime. If there are no forces, the path is a straight
10 line – the shortest distance between two points.
Motion in curved spacetime
11
• Objects always move 'locally straight' in spacetime
• But the presence of a massive
objects warps space itself,
causing an object's path to
appear to `bend'
• Newton says this bending
is because there is a `force'
pulling the object
• In GR, the object is going along a path that is the
shortest distance between the two points, along a
curve called a geodesic, and a freely moving body is
simply following the most logical path
Geodesics on a sphere
• We can think of motion in curved space
• Example: motion on a sphere is
along geodesics which are
great circles
-Shortest path between 2 points
(why airliners to Europe cross N pole)
•
Here imagine two paths leaving A,
on paths b and c.
• At first they separate, slowing and reaching a maximum
distance (marked a here) at B and C; as they continue to the other side
of the sphere, they re-converge.
• If you think of motion 'along the geodesic' as the time dimension, this
separation and then re-convergence would be you'd see if two
12 gravitationally attracting objects where initially thrown apart. No forces!
• Note that away from the event horizon, motions
behave just like before in newtonian physics
- attraction to center, orbits, and deflection
13
Supplemental slides
14
Einstein's field equations
• Not presented
15
Motion in spacetime and constant speed
• Can be drawn on paper if one imagines one spatial
dimension (motion left/right, for eg) and time going up.
• Traditionally, motion at the speed of light is drawn at a
45-deg angle on these spacetime diagrams
16
Binary Pulsars
• Two neutron stars in
binary orbits
• Only the beam of
one of the pulsars
will intercept the
line of sight to Earth
• The binary orbit involves so much gravitational
energy that general relativity predicts the mutual
semimajor axis will decay as the system emits
gravitational waves
17
Binary Pulsar
• The orbital decay
(and gravity waves)
can be detected
indirectly by
monitoring the
arrival times of the
pulses, over decades
• Precise agreement with the prediction of general relativity.
Earned Hulse and Taylor the Nobel prize and very strong
confirmation of Einstein's theory of general relativity
• DIRECT detection of gravitational waves has come recently
(see next lecture)
18