mitch begelman jila, university of colorado special relativity for jet modelers
Post on 15-Jan-2016
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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