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BeamingBeaming
BeamingBeaming
LHC LHC ~7 TeV protons ~7 TeV protons = 7000= 7000
1 TeV1 TeVTeV blazarsTeV blazars
10102020 eV = 10 eV = 1088 TeV = 10 TeV = 101111 m mppcc2 2 = =
tennis ball at 100 km/stennis ball at 100 km/s
Cosmic raysCosmic rays
A few milligrams per decade?A few milligrams per decade?
Radio-loud Radio-loud AGNsAGNs
Gamma Ray Gamma Ray BurstsBursts
~ 0.1 M~ 0.1 Moo yr yr-1-1
~20~20
~ 10~ 10-5-5 M Moo
in a few in a few sec sec ~300~300
Lorentz transformations: v Lorentz transformations: v along x along x
x’ = x’ = (x – vt) (x – vt)
y’ = yy’ = y
z’ = zz’ = z
t’ = t’ = (t – v (t – v x/cx/c22))
for for t = 0t = 0 x = x = x’/x’/ContractionContraction
for for x’ = 0x’ = 0 t = t = t’t’ time time dilationdilation
Text book special Text book special relativity relativity
x = x = (x’ + vt’) (x’ + vt’)
y = y’y = y’
z = z’z = z’
t = t = (t’ + v (t’ + v x’/cx’/c22))
To remember: mesons created at a height of ~15 km can reach the earth, even if their lifetime is a few microsec ct’life=hundreds of meters.
v=0 v=0 =1=1
v=0.866c v=0.866c =2=2
vv
Can we see contracted Can we see contracted spheres?spheres?
Einstein: Einstein: Yes!Yes!
James Terrel 1959James Terrel 1959
Roger Penrose Roger Penrose 19591959
v=0 v=0 =1=1
vv
NO!NO!
v=0.866c v=0.866c =2=2
Rotatio
n, not
Rotatio
n, not
contra
ctio
n!
contra
ctio
n!
Relativity with Relativity with photonsphotons
From rulers and clocksFrom rulers and clocks
to photographs and to photographs and frequenciesfrequencies
Or:Or:
from elementary particles to extended from elementary particles to extended objectsobjects
The moving squareThe moving square
==00=0.5=0.5
Your Your camera, very camera, very far awayfar away
The moving squareThe moving square
t=l’/ct=l’/c
vt=vt=l’l’
l’/l’/
lltottot = l’ ( = l’ (+1/+1/))
max:2max:21/21/2l’ (diag)l’ (diag)
min: l’ (for min: l’ (for =0)=0)
l’l’
l’cosl’cos = = l’ l’ coscos==
coscossinsin
) )
TimeTime
CD = cCD = cttee – c – ctteecoscos
ttAA= = tte e (1-(1-coscos) )
ttAA= = ttee’ ’ (1-(1-
coscos) )
tte e = emission time in lab = emission time in lab
frame frame ttee’ = emission time in ’ = emission time in
comov. frame comov. frame tte e = = ttee’ ’
Relativistic Doppler Relativistic Doppler
factor factor ttAA= = ttee’ ’ (1-(1-coscos) ) = = ’ / ’ / (1-(1-coscos) )
==
11
(1-(1-coscos))
StandarStandard d relativitrelativityy
Doppler Doppler effecteffect
You change frame
You remain in lab frame
Relativistic Doppler Relativistic Doppler
factor factor
==
11
(1-(1-coscos))
22 for for =0=0oo for for =1/=1/foforr
==At small angles, Doppler wins over Spec. At small angles, Doppler wins over Spec. Relat.Relat.
25 light y
ears in
3 years… th
e
25 light y
ears in
3 years… th
e
velocity is
8.3 c
velocity is
8.3 c
Nucleo
v=0.99c
Core
appapp = = sinsin
1-1-coscos
==vvappapp
= =
vvtteesinsinttee (1- (1-coscos))
ssappapp
ttAA
=0=0oo appapp=0=0
coscos==; ; sinsin=1/=1/
appapp==
=90=90oo appapp==
There is no There is no Correct?Correct?
app ~ 30
Aberration of lightAberration of light
Gravity bends Gravity bends spacespace
Aberration of lightAberration of light
sinsin = sin = sin’/’/
dd==dd’/’/22
sinsin = = sinsin’/’/
Aberration of lightAberration of light
KK’’
dd= = dd’/’/22
KK
vv
Observed vs intrinsic Observed vs intrinsic IntensityIntensity
33I’(I’(’’))
I(I())
I’(I’(’)’)
’’== ==invariainvariantnt
I(I())==
I(I())cmcm2 2 s Hz s Hz steradsterad
==ergerg
==dAdA dt ddt d d d
EE
Observed vs intrinsic Observed vs intrinsic IntensityIntensity
33I’(I’(’’))
I(I())
I’(I’(’)’)
’’== ==invariainvariantnt
I(I())==
I(I())cmcm2 2 s Hz s Hz steradsterad
==ergerg
==dAdA dt ddt d d d
EE
Observed vs intrinsic Observed vs intrinsic IntensityIntensity
33I’(I’(’’))
I(I())
I’(I’(’)’)
’’== ==invariainvariantnt
I(I())==
I(I())cmcm2 2 s Hz s Hz steradsterad
==ergerg
==dA’dA’ dd’’//22
E’E’33I’(I’(’’))
==
II 44I’I’==
FF 44FF’’
==
v=0
L=100 W
v=0.995c =10
L=16MW
L=10mW
L=0.6mW
v=0.995c =10
blazars
radiogalaxies
…….?
v=0.995c =10
blazars
radiogalaxies
blazars!
jet
counterjet (invisible)
v
v
A questionA question
• Some blob is moving at >>1, above a black hole of mass M. It is optically thin.
• It moves within a region full of radiation produced by the accretion disk.
• What is the Eddington luminosity?
UUradrad
U’U’radrad ~ ~ 22
UUradrad
Little help….Little help….
Radiation Radiation processesprocesses
Radiation processesRadiation processes
Line emission and radiative transitions in atoms Line emission and radiative transitions in atoms and moleculesand molecules
Breemstrahlung/BlackbodyBreemstrahlung/Blackbody
Curvature radiationCurvature radiation
CherenkovCherenkov
AnnihilationAnnihilation
Unruh radiationUnruh radiation
Hawking radiationHawking radiation
SynchrotronSynchrotron
Inverse ComptonInverse Compton
V=0V=0EE
V
((=2))
Charge at Charge at time 9.00time 9.00
Contracted sphere…Contracted sphere… E-field lines at time E-field lines at time 9.00 point to… where 9.00 point to… where the charge is at 9.00the charge is at 9.00
EE
Breaking news: what happens with the gravitational field?Breaking news: what happens with the gravitational field?
dP = edP = e22aa22 sinsin22
dd44cc33
VV
http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html
Stop at Stop at 8:008:00
SynchrotronSynchrotron
Synchrotron Synchrotron Ingredients: Magnetic Ingredients: Magnetic field and relativistic field and relativistic chargescharges
Responsible: Lorentz Responsible: Lorentz forceforce
Curiously, the Lorentz Curiously, the Lorentz force doesn’t work.force doesn’t work.FFLL = = dd
dtdt ((mmv)v)
== eecc
v x Bv x B
Total losses Total losses
PPee = P’ = P’ee
Please, PPlease, Pee is not is not
PPreceivedreceived!!!!
P=E/t and E and t Lorentz P=E/t and E and t Lorentz transform in the same waytransform in the same way
Total losses Total losses
a’ = a’ = 33a a
a’a’ = = 22a a
PPee = P’ = P’ee
==
2e2e22
3c3c33((22aa22 + a + a22 ) )44
2e2e22==
3c3c33a’a’22
==
2e2e22
3c3c33(a’(a’22 + a’ + a’22 ) )PPee = =
P’P’ee
BigBig??
NO! a is smallNO! a is small
Why?Why?
FFLL = = dd
dtdt ((mmv)v)
== eecc
v x Bv x B
aa|| || = = 00
a = a =
e v B e v B sinsin
mcmc
PPSS(() )
==
2e2e44
3m3m22cc33
BB222 2 2 2
sinsin22
PPSS(() )
==22TTcUcUBB
2 2 2 2
sinsin22
rr00=e=e22/m/meecc22
TT = 8 = 8rr00/3/322
<P<PSS> >
==4 4 TTcUcUBB
2 2 22
33If pitch angles If pitch angles are isotropicare isotropic
=pitch =pitch angleangle
~constant, at ~constant, at least for one least for one gyroradius gyroradius
FFLL = = dd
dtdt ((mmv)v)
== eecc
v x Bv x B
aa|| || = = 00
a = a =
e v B e v B sinsin
mcmc
PPSS(() )
==
2e2e44
3m3m22cc33
BB222 2 22
sinsin22
PPSS(() )
==22TTcUcUBB
2 2 22
sinsin22
rr00=e=e22/m/meecc22
TT = 8 = 8rr00/3/322
<P<PSS> >
==4 4 TTcUcUBB
2 2 22
33If pitch angles If pitch angles are isotropicare isotropic
Log ELog E
Log
Log
PP
SS
vv2 2 ~ E~ E
~ ~
EE22
Why Why 22????
PPSS(() )
==22TTUUBB
2 2 2 2
sinsin22What happens when What happens when 0 ?0 ?Sure, but what happens Sure, but what happens to the to the receivedreceived power if power if you are in the beam of you are in the beam of the particles?the particles?
mcmc22 sin sineBeB
rrLL ==vv22
aa==
e e BB mmcc
==BB = 1/T T = 1/T T = 2= 2 r rLL/v/v
Synchrotron Synchrotron SpectrumSpectrumCharacteristic Characteristic
frequencyfrequency
This This is notis not the characteristic the characteristic frequencyfrequency
e v B sine v B sin
mcmca =a =
v<<cv<<c
v ~ cv ~ c
ttAA = ? = ?
SS = =
1
ttAA
== 22
eeBB22mcmc
Compare with Compare with B. B. S S = = BB 33
The real stuffThe real stuff
x=
x1/3
The real stuffThe real stuff
x=
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) dd= =
11
44N(N() P) PSS
dd
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44KK-p-p B B2222 d d
dd
Emission is Emission is peaked! peaked!
SS ==22eeBB22mm
cc
dd dd
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44K BK B2 2 (2-p)/2(2-p)/2 -1/2-1/2
BB1/2 1/2 BB(2-p)/2 (2-p)/2
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44K BK B(1+p)/2 (1+p)/2 (1-p)/2(1-p)/2
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44 K BK B+1 +1 --
==p-1p-1
22
power
power
lawlaw
power power lawlaw
(() ) ~ ~ 11
44 K BK B+1 +1 --
So, what?So, what?
44Vol Vol (() ) ~~ss
2 2 R R K BK B+1+1 --
F(F() ) ~ ~ 44dd22
Log
Log
F
)
K K
BB+1+1 If you know s and R
Two unknowns, one equation… we need another one
Synchrotron self-Synchrotron self-absorptionabsorption
• If you can emit you can also absorbIf you can emit you can also absorb• Synchrotron is no exceptionSynchrotron is no exception• With Maxwellians it would be easy With Maxwellians it would be easy
(Kirchhoff law) to get the absorption (Kirchhoff law) to get the absorption coefficientcoefficient
• But with power laws?But with power laws?• Help: electrons able to emit Help: electrons able to emit are also are also
the ones that can absorb the ones that can absorb
A useful trickA useful trick
-p-p
Many Many Maxwellians Maxwellians with with kT=kT=mcmc22
I(I() = 2 ) = 2 kTkT 22/c/c22
= 2 = 2 mcmc2222/c/c22
Log Log
Log
Log
N
(N
(
==22eeBB22mm
cc
~ (B)B)1/21/2 5/25/2
BB1/21/2~~ There is no K There is no K
!!
From data to physical From data to physical parametersparameters
get Bget B insert B insert B and get Kand get K
t belongs to thick
and thin part. Then in principle one observation is enough
Inverse Inverse ComptonCompton
Inverse ComptonInverse Compton• Scattering is one the basic interactions Scattering is one the basic interactions
between matter and radiation.between matter and radiation.• At low photon frequencies it is a classical At low photon frequencies it is a classical
process (i.e. process (i.e. e.m. wavese.m. waves))• At low frequencies the cross section is At low frequencies the cross section is
called the Thomson cross section, and it called the Thomson cross section, and it is a peanut.is a peanut.
• At high energies the electron recoils, and At high energies the electron recoils, and the cross section is the Klein-Nishina the cross section is the Klein-Nishina one.one.
= scattering = scattering angleangle
00
11
Thomson scatteringThomson scattering
• hvhv00 << m << meecc22
• tennis ball against a wall tennis ball against a wall
• The wall doesn’t moveThe wall doesn’t move
• The ball bounces back with the same The ball bounces back with the same speed (if it is elastic)speed (if it is elastic)
11= = 00
Thomson cross sectionThomson cross section
ddTT
dd==
rr0022
22(1+cos(1+cos22))
TT == rr0022
3388
==rr00mmeecc22
ee22
a a peanutpeanut
Electromagnetic mass of the electron: Electromagnetic mass of the electron: See Vol. 2, chapter 28.3 of See Vol. 2, chapter 28.3 of The Feynman Lectures on The Feynman Lectures on PhysicsPhysics
Why a peanut?Why a peanut?
Why a peanut?Why a peanut?
Why a peanut?Why a peanut?
E
B
Why a peanut?Why a peanut?
dd
dP dP ee22aa22
44cc33
sinsin22==RemembRemember:er:
EE
BB
ddTT
dd==
rr0022
22(1+cos(1+cos22))
1122
100%
100%
P
ol
Pol
no no PolPol
Direct ComptonDirect Compton
xx11 = = xx00
1+x1+x00(1-(1-
coscos))
x = x = hh
mmeecc22
xx00
xx11
Klein-Nishina cross sectionKlein-Nishina cross section
Klein-Nishina cross Klein-Nishina cross sectionsection
~ E~ E-1-1
Klein-Nishina cross Klein-Nishina cross sectionsection
Inverse Compton: typical Inverse Compton: typical frequencies frequencies Thomson regimeThomson regime
Rest frame K’
x’x’11=x’=x’
xxx’x’
xx11
Lab frame K
Min and max frequenciesMin and max frequencies
==180180oo 11=0=0o o
xx11=4=422
xx
==00oo 11=180=180o o
xx11=x/4=x/422
Total loss rateTotal loss rate
vtvtTT
Everything in the lab frameEverything in the lab frame
n(n() = density of seed photons of energy ) = density of seed photons of energy =h=h
vvrelrel = “relative velocity” between photon and = “relative velocity” between photon and
electron velectron vrelrel = c-vcos = c-vcosc(1-c(1-coscos))
Total loss rateTotal loss rate
vtvtTT
There are many There are many 11, because there are many , because there are many
11.. We must average the term 1-.. We must average the term 1-coscos11, ,
getting getting
Total loss rateTotal loss rate
There are many There are many 11, because there are many , because there are many
11.. We must average the term 1-.. We must average the term 1-coscos11, ,
getting getting TheThenn
UUradrad
{{
Total loss rateTotal loss rate
If seed are isotropic, average over If seed are isotropic, average over and and take out the power of the incoming take out the power of the incoming radiation, to get the net electron losses:radiation, to get the net electron losses:
UUradrad
{{
<P<Pcc> => = 4 4 TTccUUradrad2 2 22
33
<P<PSS> >
==4 4 TTccUUBB
2 2 2 2
33
Compare with Compare with synchrotron synchrotron losses:losses:
If the seeds are not If the seeds are not isotropic….isotropic….
Inverse Compton Inverse Compton spectrum spectrum
The typical frequency The typical frequency is: is:
Going to the rest frame of the e- we see Going to the rest frame of the e- we see 00
There the scattered radiation is There the scattered radiation is isotropizedisotropized
Going back to lab we add another Going back to lab we add another --factor.factor.
The real stuffThe real stuffdowdownn
upscatteriupscatteringng
The real stuffThe real stuffdowdownn
upscatteriupscatteringng
75%75%
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) dd= =
11
44N(N() P) PCC
dd
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44KK-p-p U Uradrad22 d d
dd
Emission is Emission is peaked! peaked!
dd dd
44== 2 2 00
33
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44KUKUradrad (2-p)/2(2-p)/2 -1/2-1/2
Emission from many Emission from many particlesparticles
N(N() = K) = K-p-p The queen of The queen of relativistic relativistic distributionsdistributions
Log
N
()
Log Log
Log
)
(() ) ~ ~ 11
44 KUKUradrad --
==p-1p-1
22
power
power
lawlaw
power power lawlaw
Synchrotron Self Compton: Synchrotron Self Compton: SSCSSC
Due to synchro, Due to synchro, then proportional then proportional to: to:
cc B B+1 +1 --
cc(() )
~ ~ 22
cc B B+1 +1 cc
-- Electrons work Electrons work twice twice
EndEnd??
World recordsWorld records
Frequency??Frequency??1/t1/tPlanck Planck ~ 10~ 104343 Hz, but… Hz, but…
Power??Power??MMPlanckPlanckcc
22/t/tPlanck Planck ~~ 3.6x103.6x105959 erg/s erg/s
EndEnd
The moving barThe moving bar
==00