Mitch Begelman

JILA, University of Colorado

SPECIAL RELATIVITY FOR JET MODELERS

2 DISTINCT EFFECTS:• Lorentz transformation

– Connects observers in different frames • “Rest” (jet) frame → ”Lab” (observer) frame

– Depends on relative speed but not location of sources

– For radiation use Doppler factor

• Light travel-time effects– Connects different observers in same frame

– Depends on location of sources (nearer, further)

2/1222

)/1(factor Lorentz whereetc.,,'cos

'

),(),(

cvxc

vtt

txtx

frequency ,/ wherecos1 11 cv

emittedreceived tc

vt

cos1

VARIABILITY TIMESCALE COMPRESSION: A LIGHT TRAVEL-TIME (LTT) EFFECT

• Suppose source emits flashes 1 day apart, while moving toward you at c8.0

0.8c

Flash 1

1 lt-day

0.8 lt-day

Flash 2

Flash 1

Flash 2

Flash 10.2 lt-day

Day 0 Day 1

Flashes emitted 1 day apart, received 0.2 days apart.

SUPERLUMINAL MOTION: THE MOST FAMOUS LIGHT TRAVEL-TIME EFFECT

• Consider continuously glowing blob, moving almost directly toward you at c8.0

0.8c

0.8 lt-day

Day 0

Actual dist. traveled: 0.8 lt-day Apparent travel time: (1- 0.8 cos) daySideways dist: 0.8 sin Apparent sideways speed:

0.8 sin / (1- 0.8 cos ) c

Day 1

0.8 sin

0.8 cos

ANOTHER LTT EFFECT: ASYMMETRIC EXPANSION OF A SYMMETRIC SOURCE

Receding Approaching

cos1

sin

vvapp

cos1

sin

vvrec

... as seen in Sco X-1! (Fomalont et al. 2001)

ULTRARELATIVISTIC LIMIT γ>>1•Doppler factor

•Light travel-time factor

LTT effect is more intense – why?

Because Doppler factor has extra γ-1 factor, due to time dilation: “transverse Doppler shift” (not present in Newtonian Doppler shift)

for 2 ,2

11 1

2

emittedreceived tt 22

1

SS 433: Mixture of LTT + Doppler

precessing jets – 0.26 c

“skywriting” with LTT asymmetry (VLA: Blundell & Bowler 2004)

jets in plane of sky – offset from rest wavelength due to transverse Doppler effect

ABERRATION OF LIGHT

Aberration of rain (Galilean effect)

Aberration of light (Newtonian idea, corrected by Einstein)

direction. forwardin beaming i.e., ,frameobserver in 1 gives framejet in any almost 1,For

cos1

coscos

as transformangles Therefore

)cos1( and )cos1( where,1/ so , then , If

special. is reference of frame No1111

2 as transformangles Solid

Synchrotron emission from a single electron: another combination of Doppler+LTT

frequency criticaln synchrotro 2

~

:frequencyFourier dominant

n)compressio (LTT beaming)(Doppler ~t

:for time beamin Observer 2

frequency orbit Electron

23

2-11

1

mc

eB

mc

eB

g

g

g

DOPPLER BEAMING

0.5 c 7x brighter

0.75 c 30x brighter

0.94 c 440x brighter

0.98 c 3100x brighter

Amazing fact: power radiated (over all ν and all directions) is Lorentz-invariant!

Doppler boost of each photon’s energy exactly compensates decrease in rate of photon emission due to time-dilation.

Doppler Beaming effect primarily due to transformation of solid angles!!

Ptd

Ed

td

Ed

dt

dEP

... but it’s more complicated if one looks at spectral flux or surface brightness

dddd

tddtEddE

IdddAtd

Ed

dddAdt

dEIv

2

1

321

effect) LTT dilation (time

))(()(

:Intensity

RADIATIVE TRANSFER

invariant is that so :tcoefficien Extinction

:Emissivity

)frequency! shiftedDoppler at measure(must :Intensity

:ray alongLength

:equation transfer Radiative

1

2

3

dsd

jj

III

sdds

Ijds

dI

Observer’s view Same ray as viewed by jet

Complication: conditions in jet frame can change in time it takes ray to cross emitting region (simultaneity different in different frames).

JET-COUNTERJET RATIOS

Steady jets: path length through jet and counterjet the same. Surface brightness and flux ratios both proportional to

Expanding hotspots: add LTT effect – emission from near-side is received sooner, faster (and decay is sooner)

Ij

j

cj

j for cos1

cos12

,

,

3

,

,

,

,

cos1

cos1

)cos1(

)cos1(

cj

j

cj

j

j

j

S

S

obs.

BRIGHTNESS TEMPERATURES•Direct observation of surface brightness (resolved source):

•Deducing brightness temp. from variability (unresolved source):

•Applications: synchrotron self-absorption, induced Compton scattering, intraday variability (scintillation vs. intrinsic) (radio, sub-mm); “compactness” to pair production (X-ray, gamma, TeV); synchrotron “efficiency” (cooling times, energy requirements) (X-ray, gamma)

preserved is spectrumbody black of shape,1bb TT

tbb

Ltb

TT

SStc

tc

Sd

k

czT

tcS

,3

13

2

2

2

2

,

, , size max. :effects icRelativist

temp.brightnessapparent , )(2

)1(

size source max.infer t, in time changeflux Measure

COMPTON SCATTERING•Compton power radiation energy density in jet frame. 2 generic sources of seed photons:

– Synchrotron self-Compton:

– External radiation Compton:

~isotropic ambient radiation density boosted by factor γ2 (γ for photon density γ for photon energy)

•Applications: gamma-ray blazars, large-scale X-ray jets, Compton emission from compact radio lobes

boosting-de strong

lity)by variabi estimated size source (if )(

flux n synchrotro Measure

2

26

4

tc

SdU

SSS

Lsynch