Transition radiation in turbulent astrophysical medium. Application to solar radio bursts
aa r X i v : . [ a s t r o - ph ] O c t Transition radiation in turbulent astrophysical medium.Application to solar radio bursts.
Gregory D. Fleishman , , Dale E. Gary , Gelu M. Nita New Jersey Institute of Technology, Newark, NJ 09102 Ioffe Physico-Technical Institute, St. Petersburg, 194021 Russia
Summary.
Modern observations and models of various astrophysical objects suggest that many oftheir physical parameters fluctuate substantially at different spatial scales. The rich variety of theemission processes, including Transition Radiation but not limited to it, arising in such turbulentmedia constitutes the scope of Stochastic Theory of Radiation. We review general approaches appliedin the stochastic theory of radiation and specific methods used to calculate the transition radiationproduced by fast particles in the magnetized randomly inhomogeneous plasma. The importanceof the theory of transition radiation for astrophysics is illustrated by one example of its detailedapplication to a solar radio burst, including specially designed algorithms of the spectral forwardfitting.
The phenomenon of transition radiation was discovered theoretically by two Nobel Prize winning(2003 and 1958 respectively) physicists [1]. Ginzburg and Frank (1946) considered a simplest casewhen a charged particle passed through a boundary between two dielectrically different media andso generated waves due to a variation of the dielectric constant at the boundary. Remarkably, noacceleration of the particle is necessary to produce the emission due to transition through theboundary.It is easy to understand that a similar effect of electromagnetic emission will take place ifa medium is uniformly filled by turbulence that produces fluctuations of the dielectric constantthroughout the whole volume rather than at an isolated boundary. Many astrophysical sources,especially those under strong energy release, are believed to be filled by turbulent, randomly in-homogeneous plasma and fast, nonthermal particles. In this situation, an efficient contribution ofthe transition radiation to the overall electromagnetic emission should be produced. Therefore,distinguishing this contribution from competing mechanisms is important. Below we describe thefundamentals of the transition radiation produced in a magnetized turbulent plasma, and demon-strate its high potential for astrophysical applications.
Stochastic theory of radiation now represents a broad field of physics with many applications.In astrophysical sources the nonthermal radiation arises as charged fast particles move througha turbulent plasma with random fluctuations of plasma density, electric, and magnetic fields. Animmediate consequence of the random magnetic and electric fields at the source is that the trajectoryof a charged particle is a random function of time. This means that calculating the emission requiressome appropriate averaging of the relevant equations and parameters over the possible particlepaths.The presence of the density inhomogeneities acts differently. Indeed, these inhomogeneities havelittle effect on the fast particle trajectories; rather they give rise to fluctuations of the plasma dielec-tric tensor. These fluctuations allows for the plasma current stimulated by the fast particle field toemit powerful transition radiation. Since the plasma has a resonance around the plasma frequency,the intensity of the transition radiation is extremely large around this plasma frequency. The cor-responding peak in the transition radiation spectrum is referred to as resonant transition radiation(RTR) and is of exceptional importance for astrophysical applications. Indeed, the microturbulenceaccompanied by the density fluctuations is likely to exist in various cosmic objects from geospace todistant cosmological sources of gamma ray bursts. In many instances, however, the number densityof the plasma in the astrophysical object and the corresponding plasma frequency is so low thatthe radiation cannot be observed at the Earth because of ionosphere opacity and absorption of thisradiation in the interstellar medium. Gregory D. Fleishman, Dale E. Gary, Gelu M. NitaIn some cases, nevertheless, the plasma frequency of a source is large enough for the corre-sponding radiation to be observable. For example, this is true for radio bursts produced in the solarcorona. In our previous publications [2, 3] we demonstrated that about 10% of all microwave solarcontinuum bursts are accompanied by decimetric resonance transition radiation (RTR) and pre-sented ample evidence in favor of transition radiation for one of the events, 06 April 2001, includingdetailed study of spatially resolved observations.Specifically, Nita et al. (2005) summarized and checked against observations the following mainproperties of RTR, expected in the case of solar bursts. The emission (1) originates in a denseplasma, f pe ≫ f Be , where f pe and f Be are the electron plasma- and gyro-frequencies; (2) has arelatively low peak frequency in the decimetric range, and so appears as a low-frequency compo-nent relative to the associated gyrosynchrotron spectrum; (3) is co-spatial with or adjacent to theassociated gyrosynchrotron source; (4) varies with a time scale comparable to the accompanyinggyrosynchrotron emission (assuming a constant or slowly varying level of the necessary microtur-bulence); (5) is typically strongly polarized in the ordinary mode (o-mode), since the extraordinarymode (x-mode) is evanescent, as for any radiation produced at the plasma frequency in a magne-tized plasma; (6) is produced by the lower-energy end of the same nonthermal electron distributionthat produces the gyrosynchrotron emission, with intensity proportional to the instantaneous totalnumber of the low-energy electrons in the source; (7) has a high-frequency spectral slope that doesnot correlate with the spectral index of fast electrons (in contrast to gyrosynchrotron radiation,which does). Here we analyze this event in even more detail and apply the observations to the flareplasma diagnostics. Fig. 1 gives an overview of the event under study. It displays the dynamic spectrum of the radioburst 06 April 2001 recorded by Owens Valley Solar Array (OVSA) in the frequency range 1-18GHz (Nita et al. 2005) in total intensity and polarization, as well as images of the dm and cmsources superimposed on a Transition Region and Coronal Explorer (TRACE) 171 ˚A image. Thisfigure shows that the event indeed consists of two distinct spectral components, whose sourcescoincide spatially with each other and with a dense loop visible in the TRACE UV image. The dmcomponent is highly right-hand circularly (RCP) polarized.
Fig. 1.
Overview of the event. Left: dynamic spectra of the total power (top) and circular polariza-tion (bottom). The period of RTR is the highly polarized (red) emission in the lower panel. Right:OVSA maps of radio emission at 2 GHz (red) and 7.4 GHz (blue) superimposed on the TRACE171 ˚A image.Fig. 2 displays contour levels of the dm emission at 95, 97 and 99% of the peak intensity at eightdm OVSA frequencies in RCP (thick lines) and LCP (thin lines) superimposed on the photosphericmagnetogram (left) and on the SXR SOHO image of a dense hot loop (right). A number of thingscan be noted in this figure. The stronger, RCP, component of the dm emission originates from theregion of the negative magnetic polarity throughout all the dm frequency channels; therefore, itis O-mode radiation. All the RCP contours coincide spatially with the brightest part of the hotdense loop visible in SXR, i.e., this emission goes from a region with relatively large plasma density,although exact position of the peak brightness changes with frequency. The trend of this change issuch that the lower frequency radiation tends to originate at the loop-top, while higher frequencyransition radiation in turbulent astrophysical medium. Application to solar radio bursts. 3radiation tends to originate from the loop legs. The behavior of the weaker LCP component bearsboth similarities and differences with the RCP component. First, the LCP emission at 1.2-2 GHzcomes from the region of positive magnetic polarity, so it is also O-polarized radiation. Second,the lower frequency LCP sources are displaced relative to the brightest part of the SXR loop. Andfinally, the higher frequency LCP sources (at 2.4-3.2 GHz) are located in a region with negativemagnetic polarity; therefore, it is X-mode polarized radiation, unlike other dm radiation. Thesespatial relationships are in excellent agreement with expectations derived from the spectral behav-ior of this radio burst: indeed, since the LCP contribution of the RTR component is very weak,the LCP emission at 2.4-3.2 GHz is interpreted as a continuation of the X-polarized microwavegyrosynchrotron radiation, rather than RTR contribution.
Fig. 2.
OVSA maps of the RCP (thick lines) and the LCP (thin lines) dm emission at 8 distinctfrequencies superimposed on the magnetic field distribution (left) and SXR SOHO map (right).In addition to previously established properties of the dm continuum component (Nita et al.2005) and spatial relationships discussed above, we consider here the characteristic decay timeconstants at both dm and cm spectral components. Specifically, we looked into the decay phasesof the light curves at all frequencies and determined the range of time when the decay profile canbe approximated by an exponential function. Then, the characteristic decay constant at each lightcurve was considered as a characteristic decay time at this frequency. Fig. 3 displays these time decayconstants vs frequency for the dm and cm components separately. The dependences are remarkablydifferent for these two emission components, although both of them can be easily understood if thedecay constants are specified by the fast electron life times against the Coulomb collisions at thesource. Indeed, for the gyrosynchrotron emission, higher emission frequency means greater meanenergy of the emitting electrons. These higher energy electrons have a longer life time against theCoulomb collisions, which explains the observed increase of the decay constant with frequency. Forthe RTR contribution, however, higher emission frequency corresponds to a higher plasma frequencyin the corresponding source level, therefore, denser plasma and, consequently, shorter life time ofthe fast electrons, which is really observed. Thus, the more detailed study performed here confirmsfurther the interpretation of the dm continuum component as an RTR contribution.
Now, as we have a solid interpretation of the dm continuum component as produced by RTR byfast electrons moving in a dense plasma, we can make a next step and apply the observationsfor the flaring plasma diagnostics. We assume that the inhomogeneity of the flare volume can bedescribed by a Gaussian distribution of the source volume over the plasma frequency with somemean plasma frequency (mean plasma density) and dispersion (provided by scatter of the plasmadensity through the source), F ( f pe ) = A exp( − ( f pe − f ) /∆f ). Depending on how other relevantparameters change with the plasma density at the source, the RTR intensity can be parameterized Gregory D. Fleishman, Dale E. Gary, Gelu M. Nita τ , s e c f, GHz τ =27 f τ =100/ f τ , s e c f, GHz Fig. 3.
The decay constant of the radio emission vs frequency for the dm component (left) and cmcomponent (right).as proportional to f a exp( − ( f − f ) /∆f ) with different values of the parameter a . For clarity, weuse two distinct values a =0 (model 1 – no dependence on frequency besides the Gaussian factor)and a =2 (model 2 – rather strong frequency dependence).Then, we fit the sequence of the observed dm spectra with these two model functions to obtaina sequence of the fitting parameters, which are the peak flux, the mean plasma density f , and thedispersion ∆f . Fig. 4 displays the sequence of the recorded spectra and corresponding fits. It isclear that fits with both values of a are comparably good and even indistinguishable for most ofthe instances.Fig. 5 gives the sequence of the fitting parameters together with the chi-square plot. Again, wecannot select between the two models based on the chi-square criterion, since they are very similarto both considered models. Both models give the same peak flux values and similar bandwidthof the distributions. However, the central frequency of the distribution behaves differently in thetwo models: it is almost constant with 10% variations in model 1, while it varies substantially inmodel 2. Thus, using the requirement of reasonable smoothness of the derived physical parameterswe conclude the model 1 (which assumes that all other parameters besides the plasma density areconstant through the source) is preferable. Nita et al. (2005) proved that the dm continuum component of this solar radio burst is producedby RTR and derived the level of the microturbulence in the plasma to be (cid:10) ∆n (cid:11) /n = 10 − . Thisfinding is potentially very important for other cosmic objects. Indeed, the obtained microturbulencelevel is not particularly strong and much stronger turbulence is expected in many cases, especially,when there is a strong release of the energy at the source. Sometimes, such energy release givesrise to a relativistic expansion of the source, so the emission spectrum is Doppler-boosted and RTRproduced at the local plasma frequency can be observed at the Earth even from relatively tenuoussources with low plasma frequency.In this study we present more evidence in favor of RTR generation at the dm continuum solarbursts and use this emission component to derive additional plasma parameters. In particular,we determine the mean plasma frequency and its dispersion at the source in the course of time.Interestingly, these two parameters do not change much during the time of the dm burst. We notethat these parameters are obtained from the total power spectra recorded without spatial resolution.Fig. 2 demonstrates that the radio sources at various frequencies do not coincide exactly. Therefore,in cases where a sequence of spatially resolved spectra are available we would be able to studythe structure of the flaring plasma density in much greater detail as well as the distribution of themicroturbulence over the source.Generally speaking, the RTR contribution is also informative about the fast electrons producingit. In the example presented in Fig. 3 we show the emission decay constants, which can be associatedransition radiation in turbulent astrophysical medium. Application to solar radio bursts. 5 Fig. 4.
Sequence of the dm spectra with two fits: S(f) fa exp(-(f - f0)2/?f2). Model 1 (a=0, red)and model 2 (a=2, blue).with the fast electron life times. In our case we used exponential fragments of the light curves atthe late decay phase of the emission, since no exponential phase was found in the early decay phase.We found the life time to be within 10-40 sec, which corresponds to the electrons of 300 keV orlarger in the case of dense flare plasma available in this event. On the other hand, we can expectthat most of the RTR emission (around the peak of the burst) is produced by the electrons withE=100-200 keV (Nita et al. 2005). This apparent contradiction can be easily resolved if we recallthat the lower energy electrons have the life time of only a few seconds in the given dense plasma,so they die even before the light curves reach the exponential decay stage, and we observe the RTRcontribution from preferentially higher energy electrons late in the event.
Conclusions.
RTR represents a distinct emission mechanism, which is observable in many solar burst and probablyrelevant to many other cosmic sources. Application of the available theory to observations allows foradvanced plasma diagnostics, including study of the plasma density and the turbulence distributions,and fast particle kinetics. The theory of RTR is currently developed for usual plasmas, but is notavailable for the case of relativistic plasmas. Given that the turbulence level in the relativistic Gregory D. Fleishman, Dale E. Gary, Gelu M. Nita P ea k F l u x [ s f u ] C en t r a l F r equen cy [ G H z ] B and w i d t h [ G H z ] CH I S Q Fig. 5.
Evolution of the fitting parameters and chi-square measure for model 1 (red) and model 2(blue).plasmas is expected to be very high, the extension of the RTR theory to the relativistic case ishighly desirable.
Acknowledgments.
GDF is grateful to organizers of the Baikal School for FundamentalPhysics for invitation to deliver a lecture on transition radiation. This work was supported in partby NSF grant ATM-0707319 to New Jersey Institute of Technology, and by the Russian Foundationfor Basic Research, grants No. 06-02-16295, 06-02-16859, 06-02-39029.
References
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