Physics 777 Plasma Physics and Magnetohydrodynamics (MHD)

1. Physics 777Plasma Physics and Magnetohydrodynamics (MHD) Instructor: Gregory Fleishman Lecture 6. Transport of Radiation 14 October 2008

2. Plan of the Lecture • General Definitions; Equation of Radiation Transfer • Optically Thick and Thin Emission • Negative Absorption • Einstein Coefficients • Gyrosynchrotron Radiation

3. Section 1. General Definitions; Equation of Radiation Transfer (credit: Dulk, 1985)

8. Section 2. Optically Thick and Thin Emission Radiation transfer equation for the specific intensity For surface: - blackbody radiation Solution for uniform source: Optically thick emision: Optically thin emision:

9. Section 3. Negative Absorption & are always positive, however, can be negative If we obtain: AND which describes exponential growth of the radiation intensity. How long this growth can happen? Could it happen infinitely long? Could you suggest any factor limiting this growth?

10. Section 4. Einstein coefficients Probability of spontaneous radiation Probability of stimulated radiation Probability of absorption Assume thermodynamic equilibrium - balance of emission and absorption Considering we find and using obtain: The ratio Can easily be found from the Rayleigh-Jeans law

11. Calculation of the absorption coefficient with the Einstein coefficient method Dividing by yields:

12. Section 5. Bremsstrahlung

15. Section 6. Gyrosynchrotron radiation

16. Gyrosynchrotron radiation from power-law distributions

17. Dulk and Marsh approximation These expressions are very simple, however, they are not accurate enough: the associated error can be as large as 30% or even larger. Also, they do not work for dense plasma when the Razin-effect is important. However, for tenuous plasma they give appropriate parametric dependences. Much more exact results are provided by Klein’ numeric code, which is very fast (callable DLL is available).

18. Synchrotron (relativistic) approximation

20. Effect of the pitch-angle anisotropy

21. Synchrotron radiation from a supernova remnant X-rays (0.5-6.6 keV) Radio (84 GHz)

22. Section 7. Homework • Written report topic (WRT) # 1. Assume stochastic acceleration at the loop top with a power-law spectrum of magnetic turbulence. Calculate the DSR spectrum from power-law electron distributions with different power-law indices. Express the result in Solar Flux Units (sfu). Normalize the source volume by 1027cm3 and the turbulent magnetic energy by 103 erg cm-3. Review the papers on the betatron and Fermi acceleration in the collapsing traps. • WRT # 2. Express the DRL spectrum, calculated at the first step, in Solar Flux Units (sfu). Normalize the Langmuir energy density by 103 erg cm-3. Consider cases of large-scale and small-scale Langmuir turbulence. • WRT # 3. Using Step 1 results, calculate intensity of the synchrotron radiation taking into account the free-free absorption in the warm interstellar medium and in the local cloud. Assume that the local cloud is isotropic spherical layer with some thickness (a few pc). • WRT # 4. Assume global sausage mode oscillations of the magnetic loop. Calculate modulation amplitude (or power), phase, and degree of polarization as a function of frequency. • WRT # 5. Using Step 1 results present the radio emission flux in Solar Flux Units (sfu). Review and describe different models suggested for the Masuda source in the literature. • WRT # 6. Using Step 1 results calculate synchrotron spectra from a few locations including the location of the spatial peak of the emission. • For all WRT: expand the introduction for the written report. Use Latex, e.g., ApJ style (preferable).