Beaming. Beaming LHC ~7 TeV protons = 7000.

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

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


sinsin = sin = sin’/’/

dd==dd’/’/22

sinsin = = sinsin’/’/


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

==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



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



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



Log

N

()

Log Log

Log

)

(() ) ~ ~ 11

44K BK B(1+p)/2 (1+p)/2 (1-p)/2(1-p)/2



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?


E

B


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))


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


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

{{


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%



Log

N

()

Log Log

Log

)

(() ) dd= =

11

44N(N() P) PCC

dd



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



Log

N

()

Log Log

Log

)

(() ) ~ ~ 11

44KUKUradrad (2-p)/2(2-p)/2 -1/2-1/2



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

Beaming. Beaming LHC ~7 TeV protons = 7000.

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