mitch begelman jila, university of colorado special relativity for jet modelers
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![Page 1: Mitch Begelman JILA, University of Colorado SPECIAL RELATIVITY FOR JET MODELERS](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649d425503460f94a1cc33/html5/thumbnails/1.jpg)
Mitch Begelman
JILA, University of Colorado
SPECIAL RELATIVITY FOR JET MODELERS
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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
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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.
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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
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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)
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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
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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
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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
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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
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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
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... 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
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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).
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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.
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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
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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