Post on 19-Dec-2015
transcript
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cosθ =cos ′ θ + β
1+ β cos ′ θ
c
v
)(
) v(
2v xtt
zz
yy
txx
c
21
1
Lorentz Transformation
Transformation of angles, From formulae for transformOf velocities:
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tanθ =sin ′ θ
γ cos ′ θ + β( )
Stellar Aberration
Discovered by James Bradley in 1728
Bradley was trying to confirm a claimof the detection of stellar parallax,by Hooke, about 50 years earlier
Parallax was reliably measuredfor the first time by Friedrich Wilhelm Bessel in 1838
Refn:A. Stewart: The Discovery of Stellar Aberration, Scientific American,March 1964Term paper by Vernon Dunlap, 2005
Because of the Earth’s motion in its orbit around the Sun, the angle atwhich you must point a telescope at a star changes
A stationary telescope
Telescope moving at velocity v
As the Earth moves around the Sun, it carries us through a succession of reference frames, each of which is an inertial referenceframe for a short period of time.
Bradley’s Telescope
With Samuel Molyneux, Bradley had master clockmaker George Graham (1675 – 1751) build a transit telescope with a micrometer which allowed Bradley to line up a star with cross-hairs and measure its position WRT zenith to an accuracy of 0.25 arcsec.
Note parallax for the nearest stars is ~ 1 arcsec or less, so he would not have been able to measure parallax.
Bradley chose a star near the zenith to minimize the effects of atmospheric refraction.
.
The first telescope was over 2 stories high,attached to his chimney, for stability. He later made a more accurate telescope at hisAunt’s house. This telescope is now in theGreenwich Observatory museum.
Bradley reported his results by writing a letter to the Astronomer Royal, Edmund Halley.Later, Brandley became the 3rd Astronomer Royal.
Is ~40 arcsec reasonable?
The orbital velocity of the Earth is about v = 30 km/s
410v c
Aberration formula:
coscoscos
)cos1)((cos
cos1
cos'cos
22
2sincoscos (small β) (1)
Let aberration of angle
Then
sinsincoscos
)cos(cos
α is very small, so cosα~1, sinα~α, so
sincoscos (2)
Compare to (1): 2sincoscos
we get sin Since β~10^-4 radians 40 arcsec at most
BEAMING
Another very important implication of the aberration formula isrelativistic beaming
cos1
cos'cos
cos
sintan
Suppose 2 That is, consider a photon emitted at
right angles to v in the K’ frame.
Then
1tan
1sin
small is sin ,1 For 1
So if you have photons being emitted isotropically in the source frame, they appear concentrated in the forward direction.
The Doppler Effect
When considering the arrival times of pulses (e.g. light waves)we must consider - time dilation - geometrical effect from light travel time
K: rest frame observerMoving source: moves from point 1to point 2 with velocity vEmits a pulse at (1) and at (2)
The difference in arrival times between emission at pt (1) and pt (2) is
where
cos1 tc
dttA
2
t
’w
ω` is the frequency in the source frame.ω is the observed frequency
cos1
2
At
Relativistic Doppler Effect
1
term: relativistic dilation
cos1
1
classicalgeometric term
Proper Time
Lorentz Invariant = quantity which is the same inertial frames
One such quantity is the proper time
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c 2dτ 2 = c 2dt 2 − dx 2 + dy 2 + dz2( )
It is easily shown that under the Lorentz transform
dd
is sometimes called the space-time interval between two eventscd
• dimension : distance
• For events connected by a light signal:
0cd
Space-Time Intervals and Causality
Space-time diagrams can be useful for visualizing the relationshipsbetween events.
ct
x
World line for light
future
past
The lines x=+/ ct representworld lines of light signals passingthrough the origin.
Events in the past are in the regionindicated.Events in the future are in the regionon the top.
Generally, a particle will have some world line in the shaded area
x
ct
The shaded regions here cannotbe reached by an observer whose worldline passes through the origin since toget to them requires velocities > c
Proper time between two events:
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( )2= Δct( )
2− Δx( )
2
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( )2= Δct( )
2− Δx( )
2> 0 “time-like” interval
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ct( )2
= Δx( )2
“light-like” interval
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( )2= Δct( )
2− Δx( )
2< 0 “space-like” interval
Superluminal Expansion Rybicki & Lightman Problem 4.8
- One of the niftiest examples of Special Relativity in astronomy is the observation that in some radio galaxies and quasars, and Galactic black holes, in the very core, blobs of radio emission appear to move superluminally, i.e. at v>>c.
- When you look in cm-wave radio emission, e.g. with the VLA, they appear to have radio jets emanating from a central core and ending in large lobes.
DRAGN = double-lobed radio-loud active galactic nucleus
Superluminal expansion
Proper motion
μ=1.20 ± 0.03 marcsec/yr
v(apparent)=8.0 ± 0.2 c
μ=0.76 ± 0.05 marcsec/yr
v(apparent)=5.1 ± 0.3 c
VLBI (Very Long BaselineInterferometry) or VLBA
HST WFPC2 Observations of optical emission from jet, over course of 5 years:
v(apparent) = 6c
Birreta et al
Recently, superluminal motions have been seen in Galactic jets,associated with stellar-mass black holes in the Milky Way – “micro-quasars”.
+ indicates position of X-ray binary source,which is a 14 solar massblack hole. The “blobs”are moving with v = 1.25 c.
GRS 1915+105 Radio Emission
Mirabel & Rodriguez
Most likely explanation of Superluminal Expansion:
vΔtθ
v cosθ Δt
(1)
(2)
v sinθ Δt
Observer
Blob moves from point (1) to point (2)in time Δt, at velocity v
The distance between (1) and (2) is v Δt
However, since the blob is closer to the observer at (2), the apparent time difference is
cos
c
v1tt app
The apparent velocity on the plane of the sky is then
coscv
1
sin v
sin v v
app
app t
t