Today’s class
• The optical Doppler effect
• Excursion to the universe!
The Doppler Effect
We are all familiar with the acoustic Doppler effect. It
relies on the speed of sound in air. For light: No ether! We
expect some differences. (Also must include time dilation!)
How about the optical Doppler effect?
Dt: proper time between
crests @ source
l = c · Dt
Time between crests:
Dt’ = γ·Dt (time dilation)
v
l’ = ?
2
v·Dt’
1 was emitted from here
2 was emitted from here
1
How about the optical Doppler effect?
Time between crests:
Dt’ = γ·Dt (time dilation)
v
l’ = ?
2
v·Dt’
1 was emitted from here
2 was emitted from here
1
How about the optical Doppler effect?
Time between crests:
Dt’ = γ·Dt (time dilation)
v
l’ = ?
2
v·Dt’
1 was emitted from here
2 was emitted from here
1
How about the optical Doppler effect?
Time between crests:
Dt’ = γ·Dt (time dilation)
v
l’ = ?
1
2
c·τ
v·Dt’
c·(τ - Dt’)
l’ = c·τ - c·(τ - Dt’)-v·Dt’
= c·Dt’ - v·Dt’
= Dt’ · (c - v)
= γ ·Dt · (c - v)
1 was emitted from here
2 was emitted from here
How about the optical Doppler effect?
l’ = c·τ - c·(τ - Dt’)-v·Dt’
= c·Dt’ - v·Dt’
= Dt’ · (c - v)
= γ ·Dt · (c - v)
How about the optical Doppler effect?
l’ = c·τ - c·(τ - Dt’)-v·Dt’
= c·Dt’ - v·Dt’
= Dt’ · (c - v)
= γ ·Dt · (c - v)
The observed frequency (from the track) is:
c
c
1
v
f obs = =
=
with b =
l ' g (c - v )Dt g (1 - b )Dt
c
Note that the frequency of the source in its rest-frame is:
c
c
1
f source = =
=
l cDt Dt
Therefore:
f source
f obs =
g (1 - b )
The optical Doppler effect
1+ b
f obs =
f source
1- b
v
with b =
c
If source is approaching
≥ 1 à fobs ≥ fsource, if source and observer
are approaching each other
1- b
f obs =
f source
1+ b
If source is receding
≤ 1 à fobs ≤ fsource, if source and observer
are receding from each other
Questions?
Applications in astronomy
• How far are galaxies away?
• Is the universe inflating or
collapsing?
• Are there planets outside our solar
system? (aka. “exoplanets”)
• Is there dark energy & dark matter?
How many stars in this picture?
This spiral galaxy probably contains about
200 billion stars (= 200,000,000,000 stars!)
There are galaxies that contain ~1000 times
more stars than this spiral galaxy!
There are several hundred galaxies in this picture!
So there are at least 100 trillion stars in this picture!!
~100,000,000,000,000 stars!
(The visible universe contains approximately 1024
stars: 1,000,000,000,000,000,000,000,000)
• How far away are these galaxies?
• How fast are they moving?
• Where are they going?
• What holds them together?
Edwin Hubble: Use the Doppler effect!
If a star is receding, it’s light should
look reddish (longer wavelength)
If a star is approaching, it’s light should
look bluish (shorter wavelength)
v
v
1889 - 1953
But how can we measure this?
Quantum mechanics comes to the rescue!!
Spectra of elements are discrete
Hydrogen has very well
known lines.
Seen through diffraction
grating we can distinguish
individual, narrow lines. (More
about this in the second half
of this course!)
Now we just have to measure
the position of these lines.
Each of these hydrogen atoms is like having an
accurate clock on a distant star!
Spectra of elements are discrete
Atoms on earth
Atoms on star
Measure by how much the lines
have shifted due to Doppler effect.
Q34
Velocity of stars in a galaxy
L
Let’s investigate the internal
working of a galaxy
L
R
R
longer wavelength
lower frequency
Which side is moving towards us?
a) Left
b) Right
How fast are the stars at the edges?
Dl = 0.5 nm
lsource = 664.5 nm
f source
f source
f obs =
»
g (1 - b ) 1 - b
for v << c
How fast are the stars at the edges?
Dl = 0.5 nm
lsource = 664.5 nm
f source
f obs »
1- b
f =
c
l
Now we need to convert this
to an expression in l! Which
of the following is the correct
relation?
Q35
lsource
A) lobs »
1- b
B) lobs » lsource (1 - b )
C) lobs » c × lsource (1 - b )
D) lobs »
E) None of the above
lsource
c
(1 - b )
How fast are the stars at the edges?
Dl = 0.5 nm
lsource = 664.5 nm
Þv»
Dl
lsource
for v << c
lobs » lsource (1 - b )
» lsource - Dl
with Dl » lsource b
v
» lsource
c
c
Numeric value:
0.5nm
v»
c = 226 km / s
664.5nm
Dark matter required for this much
!"
&$
centripetal acceleration! $ =
#
#
Hubble and the big bang
Edwin Hubble, PNAS March 15, 1929 vol. 15 no. 3 168-173
Red-shift
The big bang: Far-away objects are
receding faster: All space is expanding!
Far-away stars are
receding faster!
à All of space is
expanding!
Distance from earth
Next step: Is there dark energy, i.e. is the expansion accelerating?
Exoplanets (extra-solar planets)
Are there any solar systems (i.e. stars with planets
around them) besides our own? Of course there are!
(It would be a bit foolish not to assume this, considering that
there are ~1024 suns out there!)
But how to prove it?!
We typically cannot see the planet directly. (Imagine you try to
see something really tiny and dark next to a huge and bright
object.)
à Most methods focus on the star, instead of the planet.
Brightness
Shadowing due to the planet
Time
Brightness
The planet shadows off
part of the star light as it
passes in front of it.
Time
Doppler shift
Doppler shift due to re-coil
1
2
Time
[planet years]
That’s the end of our little
excursion to the vast universe.
(And all I have on relativistic spacetime)
Questions?
Chapter 2
Relativistic mechanics
Previously:
Ideas of space and time
Simple algebra
Next:
Ideas of momentum and energy
Slightly more involved algebra
Momentum
The classical definition of the momentum p of a
particle with mass m is: p=mu.
In absence of external forces the total momentum is
conserved (Law of conservation of momentum):
n
å p = const .
i
i =1
Due to the velocity addition formula, the definition p=mu is not
suitable to obtain conservation of momentum in special
relativity!!
à Need new definition for relativistic momentum!
y
Conservation of Momentum
y'
Frame S’ moves along x with v = u1x
m
m
u2
u1
u'2
u1y
u'1
m u1x
m
S
S'
x
If u1 = -u2 we find:
ptot,before = 0
ptot,after = 0
x'
System S' is moving to the
right with the velocity v = u1x.
We will use relativistic velocity
transformations here.
Classical momentum
y
m
u2 = (-ux,-uy)
p1, before = m(ux , uy)
p2, before = m(-ux , -uy)
ptot , before = m(0 , 0)
p1, after = m(ux , -uy)
p2, after = m(-ux , uy)
m
u1=(ux,uy)
ptot , after = m(0 , 0)
S
à ptot , before = ptot, after
Galileo (classical):
y'
p1, before = m ( 0 , uy)
p2, before = m (-2ux , -uy)
u'2
m
p1, after = m ( 0 , -uy)
p2, after = m (-2ux , uy)
u'1
S'
ptot , before = m (-2ux , 0)
ptot , after = m (-2ux , 0)
m
x'
à ptot , before = ptot, after
Velocity transformation (3D)
Classical:
u'x = ux – v
u'y = uy
u'z = uz
Relativistic:
ux - v
u' x =
2
1 - ux v / c
u' y =
uy
g (1 - ux v / c )
2
uz
u' z =
2
g (1 - ux v / c )
Lorentz transformation
y'
Use:
u'2
m
u'1
S'
ux - v
u' x =
1 - ux v / c 2
uy
u' y =
2
g 1 - ux v / c
(
Algebra
Part of HW#4
m
x'
à ptot , before ≠ ptot, after
)