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20. The Compton catastrophy
20.1 The brightness temperature limit It is generally assumed that he radio emission from radio-loud AGN is suynchrotron emission of isotropic electrons in the jet, that is optically thick during the early phases out outbursts. If that is correct, then the electrons must also undergo inverse Compton scattering of their own synchrotron photons. We have seen that the energy loss rates – and hence the luminosities – scale as the energy densities of the magnetic field and the soft radiation field. Let us now calcu- late the emissivities using the synchrotron photons as input to the inverse Compton scattering, because we know them to be produced co-spatially and hence know their density. We assume the synchrotron emission is optically thin above a critical frequency νm and has the spectral index α. In a spherical emission region of radius R the differential energy density in the center of the emission zone is then (see notes to Astro 505 ”radiation transport”)
uν = (^) dV dνdE =^4 c πIν = (^4) π R^3 Lν (^2) c (20. 1 .1)
If we for a moment assume isotropic emission, i.e. no relativistic jets, then the differential luminosity relates to the brightness temperature via the radiation flux Sν for a source distance D.
⇒ uν =
3 D^2 Sν R^2 c =
8 πν m^2 kTm c^3
( (^) ν νm
)−α (20. 1 .2)
where the brightness temperature Tm must be measured at the transition frequency νm. We are dealing with very low-energy photons, so the Thomson regime should apply. Then the energy loss rate for inverse Compton scattering scales with the total energy density of soft photons, which we calculate by integrating (20.1.2) up to a maximum frequency νt.
uph =
∫ (^) νt dν uν =^8 πν
(^3) m kTm (1 − α)c^3
( (^) νt νm
) 1 −α α 6 = 1 (20. 1 .3)
Now we only need to know the magnetic energy density, which we will express as function of the brightness temperature and the transition frequency νm. The electron spectrum should be N(γ) = N 0 γ−s^ with s = 2α + 1. We know the spectral power per electron can be written as P (ν) = p 0 x^1 /^3 exp(−x) with x = ν/(ν 0 γ^2 ). The synchrotron emission coefficient therefore is in approximation
jν = (^41) π
∫ dγ N(γ)P (ν) ' N 40 πp^0
( (^) ν ν 0
) 1 / 3 ∫ √ (^) ν ν 0
dγ γ−s−^2 /^3
' (^4) π(sN −^0 p 01 /3)
( (^) ν ν 0
) 1 − 2 s (20. 1 .4)
Likewise we find for the absorption coefficient
αν ' N^08 pπ m ν^0 (s^ + 2) 2
( (^) ν ν 0
) 1 / 3 ∫ √ (^) ν ν 0
dγ γ−s−^5 /^3 = (^8) π m νN^0 p^02 ((ss^ + 2)+ 2/3)
( (^) ν ν 0
)− s 2
' ( 2 sν^ + 2)( (^2) m (ss + 2−^1 //3)3)
( (^) ν ν 0
)− (^12) jν (20. 1 .5)
We are still assuming a spherical emission zone with radius R. Furthermore we recall the relations used in (20.1.1) and (20.1.2)
8 π kT ν
2 c^3
= 3 Lν 4 π c R^2
=^4 πR jν c
⇒ R jν =^2 ν
(^2) kT c^2
As criterion for the transition frequency we can use αν (νm) R = 1, for which the average optical depth is about unity. Inserting yields
(s + 2) (s − 1 /3) (s + 2/3)
k Tm m c^2
( (^) νm ν 0
)− (^12) = 1 (20. 1 .7)
We know from (6.2.1)
ν 0 = (4 · 1010 Hz)
( B
T
)
so we finally obtain for the magnetic field strength
B = 10^9 νm(s^ + 2/3)
2 (s + 2)^2 (s − 1 /3)^2 T (^) m^2 T
⇒ umf = B
2 2 μ 0 '^4 ·^10
23
( νm(s + 2/3)^2 (s + 2)^2 (s − 1 /3)^2 T (^) m^2
) 2 J m−^3 (20. 1 .8)
The ratio of luminosities for inverse Compton scattering and synchrotron radiation is equal to that of the respective energy loss rates, for which we find (for s = 2 or α = 0.5)
uph umf^ '^2.^5
( (^) νm GHz
) ( T
1012 K
) 5 √^ νt νm^ (20.^1 .9)
To be noted from this equation is that for isotropic emission the power in inverse Compton scattering will drastically exceed that in synchrotron emission, as soon as the brightness tem- perature exceeds Tm = 10^12 K. Eq.(20.1.8) indicates that for that brightness temperature and νm = 10^9 Hz the magnetic field strength would be B ≈ 0. 2 μT, so the radiating electrons would have a Lorentz factor around 400 and the upscattered inverse Compton emission could be upscattered a second time in the Thomson limit, so the right-hand side of (20.1.9) effectively squares. This is the famous Compton catastrophy. It has two aspects:
A phase shift of a plane wave is equivalent to scattering by an angle
δΨ = δ 2 Φπ Lλ ' (0.1 milli − arcsec)
( (^) ne cm−^3
) (^ λ m
) 2 (20. 2 .5)
The compact regions of AGN have an angular extent much smaller than a milli-arcsec, so the effect even of moderately dense plasma structure can be immense. Generally these density fluctuations are not static, but move around, which leads to scintillation, i.e. the rapid flickering that can falsely suggest a short variability timescale.
