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Transcript
```General Relativity
Physics Honours 2008
A/Prof. Geraint F. Lewis
Rm 560, A29
[email protected]
Lecture Notes 6
Gravitational Lensing
There are many astrophysical applications using what we have
already learnt. Here we will examine two; gravitational lensing
and relativistic accretion disks.
Given that massive
objects can deflect the
path of a beam of light,
this means that objects
can behave as a
gravitational lens. Such
lenses resemble glass
lenses in many ways,
giving multiple images
which can be magnified.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Chapter 11
Gravitational Lensing
How “strong” is a gravitational lens? Remember that the
deflection angle is given by
It is straightforward to define a lens
equation of the form;
Given astronomical distance scales, and
the deflection angle being small, then
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Gravitational Lensing
The gravitational lensing equation for a point mass can then
be written as;
where
This is known as Einstein radius (or angle). A source at =0
will result in a circular image on the sky, and radius of =E.
But how big is E?
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Gravitational Lensing
Let’s consider a star in our Galaxy, with the mass of the Sun.
All of the distances are ~10kpc = 1017m. For this
But for galaxies (M~1041 kg and D~1025 m) then
And for clusters of galaxies, then
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Gravitational Lensing
Given the lens equation, we can
find the image positions to be
For the mathematically minded, the lensing equation is one-tomany (one source -> several images); this makes
gravitational lensing tricky.
If the source is not a point,
then it is magnified
(stretched).
It should be clear that the
angular (polar) extent is
the same for the images
as the source.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Gravational Lensing
We can differentiate  and so
So it is straightforward to calculate the area of an image. How
bright an image? We can use the fact that surface brightness
is conserved through lensing (be it a glass lens or gravitational
lens). Therefore the brightness of the image compared to the
source is simply the ratio of the areas;
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Gravitational Lensing
Hence we can write the magnification of an image as
Hence, we have two images, the outer one brighter than the
inner one. The total magnification is simply
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Microlensing
We know that our
galaxy is surrounded
by a halo of dark
matter, but we don’t
know what it is. If it is
stars, then every-sooften, one will pass in
front of a distant star,
microlensing it.
The resultant images are 10-3-10-6 arcsecs apart, and are
unresolvable. We should just expect to see the distant star
brighten and fade with a characteristic shape. Also, the
variations should be achromatic.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Microlensing
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Multiple Lenses
What if we have more than one microlens? We can write the
lens equation as
Where ,  &  are all 2-dimensional vectors
For a particular source position, solving for  becomes very
tricky as there will be several images (and the number
changes as you change ). When the number of lenses
exceeds 2, then numerical methods are used.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Microlensing
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Smooth Matter Lenses
It is straightforward to go from many individual lenses to
smooth matter distributions. The deflection angle becomes
Where  is the surface mass density.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Multiply Imaged Quasars
Quasars can be multiply
imaged and exhibit the
Shapiro time delay.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
Accretion disks are important
in astrophysics as they
efficiently transform
gravitational potential energy
Accretion disks are seen
around stars, but the most
extreme disks are seen at the
centre of quasars.
These orbit black holes with masses of ~106-9 M, and radiate
up to 1014 L, outshining all of the stars in the host galaxy.
If we assume the black hole is not rotating, we can describe its
spacetime with the Schwarzschild metric and make predictions
for what we expect to see.
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
Suppose we have radiating material orbiting a black hole. The
photons leave the disk and are received by a distant observer.
How much redshifting of the photons does the observer see?
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
How fast is the material orbiting? Assuming circular orbits
So we write the 4-velocity of the emitting matter to be
And using the normalization of the 4-velocty
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
We can now us the Killing vector conserved quantities;
Remember, the conserved quantities and the impact
parameter are related via
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
Remember we are in a plane where =/2. What redshift do
we see? Let’s consider the matter moving transversely to the
line-of-sight. For these, b=0 and so
So, there is no Doppler component. What about the matter
moving directly towards or away from the observer? Now we
explicitly need to consider the value of b
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
Remembering that photons are null
And so
The resultant frequency ratio is
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
For small values of M/r then
where
Lecture Notes 6
http://www.physics.usyd.edu.au/~gfl/Lecture
Accretion Disks
Accretion disks are hot an
emit in highly ionized species.