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by: Solon Leuschke


Solon Leuschke
GPA 3.81


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Class Notes
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This 4 page Class Notes was uploaded by Solon Leuschke on Sunday September 27, 2015. The Class Notes belongs to ASTRO 505 at Iowa State University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/214519/astro-505-iowa-state-university in Astronomy and Astrophysics at Iowa State University.

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Date Created: 09/27/15
20 The Compton catastrophy 201 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 7 and hence the luminosities 7 scale as the energy densities of the magnetic eld and the soft radiation eld 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 um and has the spectral index 04 In a spherical emission region of radius R the di erential energy density in the center of the emission zone is then see notes to Astro 505 77radiation transport dE i 47r 3L dVdV T c V 47139RQC uy 2011 If we for a moment assume isotropic emission ie no relativistic jets then the di erential luminosity relates to the brightness temperature via the radiation ux S for a source distance D 3D2 SV 87w2 kTm V quotX 7 m 2012 i UV 7 R20 03 um where the brightness temperature Tm must be measured at the transition frequency um 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 2012 up to a maximum frequency Vt Vt 87w3 kT Vt 1 u du uym m i or 1 2013 ph 1io c3 um i Now we only need to know the magnetic energy density which we will express as function of the brightness temperature and the transition frequency um The electron spectrum should be Ny N07 5 with 3 204 1 We know the spectral power per electron can be written as PV pal3 exp7 with VVO 72 The synchrotron emission coef cient therefore is in approximation 1 Nope V 13 7 7 V77 d N P g 7 d s 23 J 47r 7 7 V 4 V0 70 M 2 2014 1 Likewise we nd for the absorption coef cient aygNopos2ltVgt13 mfgs Nopos2 87rmu2 VT 787Tmu2323 V O 1 2 71 3 2 w 1 2 2015 2V2m 3 23 V0 We are still assuming a spherical emission zone with radius R Furthermore we recall the relations used in 2011 and 2012 3L 7 MR 2 2CT 7 gt RjVV7 C CZ 2 87rkTV77 7 2016 03 47139CR2 As criterion for the transition frequency we can use any um R 17 for which the average optical depth is about unity lnserting yields 323713kTm um 7 323 mo2 lt70 1 20 We know from 621 B V0 4101 HZ so we nally obtain for the magnetic eld strength Vm8 232 8 2W8 132T3 32 u 3 23 2 7241023 quotquot7 J 3 2018 2 M 2M0 lt8 2gt2lts 71mm m l l The ratio of luminosities for inverse Compton scattering and synchrotron radiation is equal to B109 that of the respective energy loss rates7 for which we nd for 3 2 or or 05 5 uph um T gt Vt 7 2 25 7 7 7 2019 umf ltGHZgt 1012 K um To be noted from this equation is that for isotropic emission the power in inverse Compton scattering will drastically exceed that in synchrotron emission7 as soon as the brightness tem perature exceeds Tm 1012 K Eq2018 indicates that for that brightness temperature and um 109 HZ the magnetic eld strength would be B m 02 MT7 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 limit7 so the right hand side of 2019 effectively squares This is the famous Compton catastrophy It has two aspects 2 o The energy loss time of electrons radiating at a GHz with T 1012 K is about 10000 years At a brightness temperature of 1013 K it would be considering secondary scattering only three days At higher energies frequencies it would be even shorter o The observed luminosity of radio loud AGN in the optical and X ray band is only about 2 orders of magnitude higher than that in the radio Both arguments imply that the brightness temperature cannot be signi cantly higher than 1012 K When in the Sixties the then new VLBl technique made measurement of the brightness temperature of sources they were found not to exceed that limit This is understandable because the average source ux per resolution element gave a brightness temperature just below the limit However if we know the distance to the source D and use the variability timescale 739 as indicator for the size then the angular extent of the source would be 9 2 713962 7392 D so 3 D2 s D 2 u 2 T 2 T 2 7 1019 K 7 7 7 20110 27139 VQkTQ ltJygt ltGpcgt GHz day The brightness temperature thus determined can exceed 1020 K 202 Solutions to the Compton catastrophy We have argued before that the emission regions are probably located in the relativistic jets so the emission properties need to be corrected via the Doppler factor D In Eq20110 the transformations of time and frequency compensate each other and only the ux needs to be considered S7 D3103 with y 10 2021 Even jets with Lorentz factors around 100 would can not explain brightness temperatures around 1020 K Measurements indicate that much of the low frequency radio variability of AGN may actually be caused by interstellar scattering which introduces a rapid ickering of sources and thus leads to wrong estimates of the variability timescale The dispersion relation of electromagnetic waves in an ionized medium see notes to Astro 505 77waves reads 0 1 W2k202w2 i 717 7 2022 7 vph 1jL Vii where the plasma frequency depends on the density of thermal free electrons up 2 104 Hzxne 2023 Any density uctuation in the interstellar medium will then act like a lens Suppose a gas structure of size L The phase shift of a wave upon passage through that gas structure is 70L 0 64gt 27139vph77 neLreA 2024 3 A phase shift of a plane wave is equivalent to scattering by an angle 64gt A me A 2 6 1 7 7 01 milli 7 arcsec arts lt gt 2025 H1 The compact regions of AGN have an angular extent much smaller than a milli arcsec7 so the e ect even of moderately dense plasma structure can be immense Generally these density uctuations are not static7 but move around7 Which leads to scintillation7 ie the rapid ickering that can falsely suggest a short variability timescale


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