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Synchrotron radiation - some observational aspects

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Theory of synchrotron radiation (summary)

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Synchrotron radiation

B

21

ea

2

1

e

a

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Synchrotron emission

1) Opening angle of emission cone ~ 2/gamma

2) Contraction of sweep time of the cone by 1-v~1/(2gamma^2)

3) Net result ~ (1-v)Delta t~1/gamma^3

t

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Synchrotron radiation power

is the pitch angle

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Emission spectrum

B

21

ea

Total power emitted:

Derives from EM radiation integral

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Typical photon energies

Area=

Shape function F

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Spectrum from radiation and self-absorption

Transition frequency condition:

Above, medium is optically thin:

Below, medium is self-absorbed:

(Rayleigh-Jeans!)

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Brightness temperature

Black body temperature: temperature of a grey body emitting the observed intensity (at a given frequency)

One brightness temperature at all frequencies only for a genuine black body source

Emission spectrum of grey bodies is different: brightness temperature depends on choice of frequency bin

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At low frequencies, source is optically thick (to own synchrotron photons)

The spectrum satisfies Rayleigh-Jeans law – just like a black body spectrum

Assign effective temperature of e’s (typically with a power law distribution):

Rayleigh-Jeans regime

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Electron temperature

Dominant frequency of emission

Electron energy

EOS (relativistic fluid)

Electron temperature

(defines effective temperature)

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Electron temperature

From the observed dominant frequency:

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Brightness temperature Rayleigh-Jeans

Large optical depth

At thermal equilibrium

Full thermodynamic equilibrium

Brightness temperature

Rayleigh-Jeans at low frequencies

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~

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At low frequencies

Rayleigh-Jeans relativistic e’sLarge optical depth, self-absorbed (sum of BB-spectra,

one for each energy of the electron population)

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Typical synchrotron spectrum

p=2.4

Independent of p

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Observed radio spectra

Optically thin

Steepening (e.g. by cooling)

self-absorbed

Opacity increasing?

self-absorbed

opaque

self-absorbed?

Kellerman & Owen, 1998, in Galactic and Extragalactic radius astronomy, eds. Verschuur & Kellermann (Springer Verlag)

Optically thin

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How much energy is in e’s or B?

e-energy density to synchrotron power

e-energy density associated with a given radiation frequency range

B-energy density

(keeping L fixed)

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Total energy in particles and fields http://www.cv.nrao.edu/course/astr534/SynchrotronSrcs.html

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Minimum total pressure in particle and fields

Total pressure

Total pressure minimum

Equilibrium p

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Minimum pressure in particles and fields Total pressure

The minimum pressure is attained at approximate equipartition

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Minimum pressure (2nd derivation)

Consider p=2 as before

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Minimum pressure (2nd derivation)

Pressure in relativistic electron population

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Total pressure

Total pressure minimum

“Minimum near equipartition”

Minimum pressure (2nd derivation)

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Equipartition

Equipartition is natural if relaxation time is short

Here, equipartition argument leads to minimum energy

Useful to eliminate one parameter

e.g.http://www.cv.nrao.edu/course/astr534/SynchrotronSrcs.html

Thus,So, the “

So, the “4/3” relation between pressure of e’s and B holds at the minimum of the total pressure at given L or Tb.

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Equilibrium magnetic field strength

Total pressure

Total pressure minimum

Total pressure minimumEquilibrium value depends on R

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Results for moving blobs Transform by Doppler factor

Baryon number density, is zero in a leptonic jet

If electron-proton plasma, then

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Results for moving blobs

Energy densities

(cold)(relativistic)

Lorentz factor of blob

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Cyg A

(p=2.6)

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Cyg A

Equivalent to the luminosity of a galaxy, at the size R of a small galaxy!

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Cyg A

Indeed

Can show for each of the two lobes based on L:

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Questions

1. In true thermal equilibrium with a thermal source, the e’s assume a Maxwell energy distribution. Can their temperature be larger than the brightness temperature? [Hint: What is the role of synchrotron self-absorption?]

2. Based on the cooling time of a few million years, the lobes in Cyg A are “light bulbs” powered instantaneously by energy transport from jets emanating from the nucleus. The jets are hardly visible. What does this say about the efficiency of the jets as power carriers?

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