Mitch Begelman

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SPECIAL RELATIVITY FOR JET MODELERS

• Mitch Begelman
• JILA, University of Colorado

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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)

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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

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SUPERLUMINAL MOTION THE MOST FAMOUS LIGHT
TRAVEL-TIME EFFECT

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

Actual dist. traveled 0.8 lt-day Apparent
travel time (1- 0.8 cos?) day Sideways dist 0.8
sin? Apparent sideways speed 0.8 sin ? / (1-
0.8 cos ?) c
0.8c
0.8 lt-day
Day 0
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ANOTHER LTT EFFECT ASYMMETRIC EXPANSION OF A
SYMMETRIC SOURCE
Approaching
Receding
... as seen in Sco X-1! (Fomalont et al. 2001)
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ULTRARELATIVISTIC LIMIT ?gtgt1

• Doppler factor
• Light travel-time factor
• LTT effect is more intense why?
• Because Doppler factor has extra ?-1 factor, due
  to time dilation transverse Doppler shift (not
  present in Newtonian Doppler shift)

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SS 433 Mixture of LTT Doppler
precessing jets 0.26 c
jets in plane of sky offset from rest
wavelength due to transverse Doppler effect

skywriting with LTT asymmetry (VLA Blundell
Bowler 2004)
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ABERRATION OF LIGHT
Aberration of rain (Galilean effect)
Aberration of light (Newtonian idea, corrected by
Einstein)




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Synchrotron emission from a single electron
another combination of DopplerLTT
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DOPPLER BEAMING
Amazing fact power radiated (over all ? and all
directions) is Lorentz-invariant! Doppler boost
of each photons energy exactly compensates
decrease in rate of photon emission due to
time-dilation. Doppler Beaming effect primarily
due to transformation of solid angles!!
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• ... but its more complicated if one looks at
  spectral flux or surface brightness

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RADIATIVE TRANSFER
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)
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)

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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 ?2 (? for photon density ? ? for photon
  energy)
• Applications gamma-ray blazars, large-scale
  X-ray jets, Compton emission from compact radio
  lobes