Detecting fast radio bursts at decametric wavelengths
aa r X i v : . [ a s t r o - ph . H E ] N ov Mon. Not. R. Astron. Soc. , (2016) Printed 8 November 2016 (MN LaTEX style file v2.2)
Detecting fast radio bursts at decametric wavelengths
K. M. Rajwade , ⋆ and D. R. Lorimer , , † Department of Physics and Astronomy, West Virginia University, Morgantown,WV 26506, USA Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA Green Bank Observatory, Green Bank, WV 24944, USA
ABSTRACT
Fast radio bursts (FRBs) are highly dispersed, sporadic radio pulses that are likelyextragalactic in nature. Here we investigate the constraints on the source populationfrom surveys carried out at frequencies < ∼
30 or more FRBs per day, even in the most extreme scenarios where free-free absorp-tion and scattering can significantly impact the fluxes below 1 GHz. We also show thatUTMOST will detect 1–2 FRBs per month of observations. For CHIME and HIRAX,the detection rates also depend greatly on the assumed FRB distance scale. Some ofthe models we investigated predict an increase in the FRB flux as a function of red-shift at low frequencies. If FRBs are truly cosmological sources, this effect may impactfuture surveys in this band, particularly if the FRB population traces the cosmic starformation rate.
Key words: radiation mechanisms: general — radiative transfer — scattering —cosmology: theory
The origin of fast radio bursts (FRBs) remains anunanswered question since their discovery a decade ago(Lorimer et al. 2007). FRBs are millisecond duration, highlysporadic and dispersed radio pulses which follow thesame dispersion relation seen in radio pulses from neu-tron stars. Of the 20 FRBs known so far, 18 havebeen found at Parkes (Thornton et al. 2013; Lorimer et al.2007; Petroff et al. 2015; Keane et al. 2016; Champion et al.2016), one at Arecibo (Spitler et al. 2014, 2016) and one atGreen Bank (Masui et al. 2015). With the exception of thelatter, FRB 110523, which was detected at 800 MHz, all theother FRBs have so far been seen in the 1–2 GHz band.FRB dispersion measures (DMs) are substantially greaterthan that expected from free electrons in our Galaxy, sug- ⋆ E-mail: [email protected] † E-mail: [email protected] gesting that FRBs are extragalactic in origin. There havebeen arguments about local origin of FRBs but the modelscannot explain all the observed characteristics (for a review,see Katz 2016).Broadly speaking, the FRB source models fall intotwo categories: those of a catastrophic nature which wouldonly be seen once (e.g., prompt emission from a gamma-ray burst; Yamasaki et al. 2016) or those with the possi-bility to repeat (e.g., giant pulses from Crab-like pulsars;Cordes & Wasserman 2016; Cordes et al. 2004). So far, theonly source known to repeat is FRB 121102 (Spitler et al.2016). In the light of these recent discoveries, and totry to shed light on the origins of FRBs a number ofgroups are carrying out extensive radio surveys at sub-GHzfrequencies (Karastergiou et al. 2015; Caleb et al. 2016;Deneva et al. 2016). To date, however, the 0.7–0.9 GHz de-tection of FRB 110523 remains the only source found below1 GHz (Masui et al. 2015).Lyutikov et al. (2016) argues that a lack of detections
Rajwade & Lorimer could be due to absorption in an ionized medium along theline of sight. Recent discoveries suggest low scattering inall FRBs which precludes a local plasma in the vicinityof the progenitor to explain their high DMs (Masui et al.2015; Macquart & Koay 2013). Kulkarni et al. (2015) arguefor a young magnetar model with circum-dense mediumaround the star which can explain the high DM and thenon-detections at lower frequency due to free-free absorp-tion. The non-detections can also be explained by youngneutron star progenitor within an expanding supernova rem-nant shell with hot ionized filaments (Piro 2016).In this paper, we present a detailed analysis of the afore-mentioned absorption and scattering models. We use theapproach to investigate the significance of non-detections inthree recently completed surveys to constrain the spectralindex of the burst for each model. Based on these constraintswe make predictions for FRB detections from CHIME, UT-MOST and HIRAX. Connor et al. (2016) make optimisticpredictions for these upcoming low frequency surveys basedon single FRB detection in the 0.7–0.9 GHz band. Here, wepresent predictions on the FRB detection rates based ondifferent models of flux mitigation in the ISM. The plan forthis rest of this paper is as follows. We describe our analysismethods in §
2. In §
3, we describe our results and discusstheir implications in § This section describes the methodology used for obtainingupper limits on FRB predictions with CHIME under differ-ent astrophysical scenarios. Our study is motivated by ourrecent work on modeling gigahertz peaked spectrum pul-sars via free-free absorption (Rajwade et al. 2016). Here,we investigate what could happen to an FRB that is ab-sorbed or scattered and how that affects detectability withCHIME and UTMOST. We will begin by making use of therecent null results of FRB detections in the ongoing UT-MOST survey (Caleb et al. 2016), the Arecibo drift scansurvey (AO327; Deneva et al. 2016) and the 145 MHz LO-FAR survey (Karastergiou et al. 2015). We also consideredthe 155 and 182 MHz surveys with the Murchison WidefieldArray (MWA) (Tingay et al. 2015; Rowlinson et al. 2016)in our analysis. However, since the flux limits for those sur-veys are higher than the LOFAR survey, the spectral indexconstraints are less stringent than the LOFAR survey. Wedo not include results from single-pulse searches in the on-going Green Bank North Celestial Cap (GBNCC) survey(Stovall et al. 2014) in this analysis. A paper describing theconstraints from these results will be presented elsewhere(Chawla et al., in prep).
Our methodology builds upon that usedby Karastergiou et al. (2015) in their LOFAR survey,to include the effects of free-free absorption and scattering.Following these authors, we assume that FRBs are standardcandles and the energy output from the source follows apower law with respect to frequency (see, e.g., Lorimer et al. 2013). Then, the peak flux density S peak = L R ν (1+ z ) ν (1+ z ) E ν ′ dν ′ πD ( z ) ( ν − ν ) R ν ′ high ν ′ low E ν ′ dν ′ , (1)where L is the bolometric luminosity, the pulse energy E ν ′ ∝ ν ′ α for some spectral index α and source framefrequency ν ′ = (1 + z ) ν at redshift z and luminosity dis-tance D ( z ). Also in the source frame, ν ′ low and ν ′ high arethe frequency bounds in which the source emits. Follow-ing Lorimer et al. (2013), we assumed ν ′ high =10 GHz and ν ′ low =10 MHz. The observed frequency band is defined by ν and ν and is different for each survey under considera-tion. We will discuss the implications of this standard-candleassumption in § z max = 0 .
75 (see, e.g.,Lorimer et al. 2013); (ii) an “extragalactic model” for whichthe characteristic distance is 100 Mpc (i.e. z max = 0 . z max . At the cho-sen value of z max this rate matches, by definition, the ratespublished by Crawford et al. (2016) based on FRB surveysat Parkes. Using this curve, for each of the other surveysunder consideration (i.e. LOFAR, AO327 and UTMOST),we can compute the number of FRBs expected as a functionof redshift by multiplying the rate–redshift relationshipby the appropriate survey sky and time coverage. Theresulting number versus redshift curves then lead to alimiting redshift z lim for each survey. This limiting redshiftis defined to be that at which < α and, using Eq. 1, findthe corresponding value of L such that S peak = 1 Jy at z max .The 1 Jy reference flux is approximate, and motivated by theresults of Thornton et al. (2013). Our results turn out to beinsensitive to the exact value adopted here. For each of thesurveys under consideration, we calculate the correspondingflux at the survey’s redshift limit, i.e. S peak ( z lim ) and iterateuntil the spectral index is found where S peak ( z lim ) equals thesurvey flux limit. This spectral index is, by definition, thelimiting value appropriate to the assumptions of that par-ticular model and distance scale, and we refer to this lowerlimit as α lim .Our baseline model, which follows this process using asimple power-law spectral behaviour amounts to a repeat ofthe analysis of Karastergiou et al. (2015). We refer to thiscase as model “A” henceforth and, as necessary, distinguishbetween the cosmological and extragalactic cases in the text.The relevant parameters used for each of these models andconstraints obtained from them are given in Table 2 anddiscussed further in the sections below. RBs at decametric wavelengths Figure 1.
Flow diagram showing the logical flow of our analy-sis procedure for placing constraints on the spectral index. Forfurther details, see § From radiometer noise considerations, if W is the width ofthe FRB then, for a search in which the pulse is optimallymatch filtered by a top-hat pulse of peak flux density S , thesignal-to-noise ratioS / N =
S G p W n p ∆ νT sys , (2)where T sys is the system temperature, ∆ ν is the bandwidth, n p is the number of polarizations summed and G is the gain.In all current FRB surveys, where incoherent dedispersiontechniques are used to process the data, and in the con-text of our models DM depends on redshift, then there isa dispersive broadening effect that results in a dependencebetween survey sensitivity and redshift. To model this effect,we compute the effective width of the pulse W eff = q W + W + W τ , (3)where W int is the intrinsic pulse width of the FRB, W DM is the intra-channel dispersion smear and W τ is the addi-tional broadening due to the finite sampling interval of thesurvey. To calculate W DM , we adopted a DM-redshift scal-ing from (Inoue 2004) where DM = 1200 z cm − pc. Usingthe standard expression for dispersion broadening (see, e.g.,Lorimer & Kramer 2004), we have W DM = 99 . (cid:18) zn chan (cid:19) (cid:18) ∆ ν MHz (cid:19) (cid:16) ν GHz (cid:17) − , (4)where n chan is the number of frequency channels used fordedispersion. Future FRB surveys may well introduce high-speed algorithms to implement coherent dedispersion (see,e.g. Zackay & Ofek 2014), in which case W DM will not benecessary. To model the degradation due to incoherent dedis-persion of current and near-future surveys, consider an “op-timal survey” signal-to-noise ratio, S/N which is obtainedfrom Eq. 2 for the case for a top-hat pulse with height S andwidth W int . For a broadened pulse of width W eff , energy con-servation means that its peak flux density is S W int /W eff .It is straightforward to show that the S/N of the broadenedpulse is lower than S/N by a factor of p W int /W eff . For anactual survey with a constant S/N threshold, this amountsto an increase in the limiting peak flux density for detectionby the reciprocal of this factor, so that the resulting limitingsensitivity S lim = S r W eff W int = S / N lim T sys G W int s W eff n p ∆ ν . (5)This expression is used when calculating the sensitivitycurves throughout this paper (see, e.g., the right panel ofFig. 2). Here S/N lim is the limiting signal-to-noise ratio re-quired for a detection in a given survey. Table 1 summarizesthe essential observing parameters for each of the surveysconsidered in this paper. Radio signals propagating through the ISM are modulatedby free electrons in the intervening medium. These interac-tions leave observational signatures in the received radiation
Rajwade & Lorimer
Survey Centre frequency Bandwidth Flux limit Reference(MHz) (MHz) mJyUTMOST 843 31.5 11000 Caleb et al. (2016)AO327 327 57 83 Deneva et al. (2016)LOFAR 145 6 62000 Karastergiou et al. (2015)CHIME 600 400 125 Newburgh et al. (2014)HIRAX 600 400 24 Newburgh et al. (2016)
Table 1.
Table showing various parameters of different surveys. The system parameters of CHIME and HIRAX are estimated values(see text for details). at the earth. Some of these signatures (e.g. scattering, free-free absorption and scintillation) have been observed in var-ious radio sources. FRBs, being astrophysical in nature, aresubject to the same phenomena. It is therefore importantto model these effects in detail before we draw any infer-ences about their intrinsic spectral indices and make predic-tions for future surveys. Below, we describe our mathemati-cal models to characterize effects of scattering and free-freeabsorption.
As discussed by other authors (Kulkarni et al. 2015;Lyutikov et al. 2016), but not taken into account byKarastergiou et al. (2015), thermal absorption can signifi-cantly reduce FRB fluxes at lower frequencies. For this anal-ysis, following our earlier work (Rajwade et al. 2016), weassume E ν ′ ∝ ν ′ α exp (cid:0) − τ ν ′− . (cid:1) , (6)where, as described further by Rajwade et al. (2016), theoptical depth of the absorber τ = 0 . T − . e EM . (7)Here T e is the electron temperature and EM is the emis-sion measure of the absorber. Then the peak flux iscomputed by combining Eq. 1 and Eq. 6. We considertwo cases for absorption: (i) cold, molecular clouds withionization fronts for which T e = 200 K and EM =1000 cm − (Lewandowski et al. 2015) (hereafter, model B);(ii) hot, ionized magnetar ejecta/circum-burst medium forwhich T e = 8000 K and EM = 1 . × cm − (hereafter,model C). The value of EM for model C has been chosenfrom a range of values reported in Rajwade et al. (2016),Kulkarni et al. (2015) and Lewandowski et al. (2015). Multi-path scattering due to free electrons in the ionizedmedium along the line of sight to the observer can causea reduction in the measured flux at the telescope. Scatter-ing manifests itself as an exponential tail in the radio pulseof the FRB. FRBs that have been discovered so far, showonly a modest amount of scattering: for the 17 FRBs, 10 ofthem have scattering measurements and 7 have them haveupper limits (Cordes et al. 2016). Hence, we computed thescattering timescale by taking the average of the publishedvalues (estimates and upper limits) of these 17 sources. Forsources with upper limits, conservatively, we assumed those values as measured values when taking the average. We ob-tained a mean scattering timescale of ∼ ∼ τ s can be computed for any frequency ν viathe ν − scaling law (Bhat et al. 2004) as opposed to ν − . .The non-Kolmogorov scaling exponent is due to fact thatthe diffraction length scale is smaller than the inner scale ofthe wavenumber spectrum (see Bhat et al. 2004, , and ref-erences therein). Assuming that energy of the burst is con-served, if the pulse scatters with a timescale of τ s , the widthincreases and hence, the measured flux reduces by a factorof p τ s /W eff ) where W eff is the effective pulse widthdefined in the preceding section. Including this effect intoour analysis, we introduce three final models. Model D hasscattering with no free-free absorption, while models E andF have scattering in addition to the respective absorptionparameters adopted for models B and C. Taking into account all the factors discussed in the previoussection, the results of our analysis are collected for modelsA–F in in Table 2. For each of these models, we constrainedthe spectral indices assuming each of the two distance con-straints in turn. A graphical illustration of this process isshown for model E as an example in Fig. 2 where we showthe constrained spectral index for one of the models for theUTMOST survey (Caleb et al. 2016). As mentioned previ-ously, our baseline model (A) is an update on the results ofKarastergiou et al. (2015) using the more recent rate FRBrate estimates from Crawford et al. (2016). In our analy-sis, which also includes the non-detections in UTMOST andAO327, the most constraining power-law spectral index forthis model is α > . α > .
25) isobtained from the AO327 survey if the extragalactic distancescale is applied to this survey.In model B, where we go beyond the simple power-lawspectral dependence and include free-free absorption withcold molecular clouds, we find only a modest change in theresults for model A for AO327 and UTMOST but as ex-pected a greater deviation at the LOFAR frequency bandwhere spectral turnover effects are more severe. The LOFARsurvey does not in fact provide any constraints on the spec-
RBs at decametric wavelengths Model T e EM α lim z lim (K) cm − pc UTMOST LOFAR AO327 CHIMEcosmo exgal cosmo exgal cosmo exgal cosmo exgalA — — –0.70 –1.30 0.0 –0.50 0.70 1.25 1.54 0.10B 200 1000 –0.80 –1.30 –1.0 –2.10 0.50 1.10 1.56 0.10C 8000 1 . × –1.50 –2.50 — — –0.30 –2.85 1.64 0.09D — — –2.10 –3.30 –3.0 –4.0 –3.30 –2.20 0.84 0.06E 200 1000 –2.20 –3.30 –4.10 –5.70 –3.50 –2.50 0.85 0.06F 8000 1.5 × –2.70 –4.50 — — –4.50 –6.45 0.82 0.05 Table 2.
Model parameters and resulting spectral constraints from the various surveys considered. From left to right, we list the model,assumed electron temperature ( T e ) and emission measure (EM), limiting spectral index ( α lim ) for the three published surveys (LOFAR,AO327 and UTMOST). For the future CHIME survey, we list the limiting redshift ( z lim ) predicted by our models. The “cosmo” and“exgal” columns give results from the two different distance scales assumed: “cosmological” ( z = 0 .
75) and “extragalactic” ( z = 0 . Figure 2.
Left: Number of FRBs versus redshift for the UTMOST survey parameters Caleb et al. (2016). The curves indicatethe Crawford et al. (2016) rates with 99% bounds. The horizontal line corresponds to one FRB. The intersection of the horizontalline and the upper bound of the curve is shown by the red cross at z = 0 .
28. Right: peak flux versus redshift for UTMOST surveyfor model E (dashed curve). The solid line shows the flux limit of the UTMOST survey. The intersection of the two curves is denotedby z lim . Note the non-linear dependence of flux limit with redshift for both surveys shown here is due to the impact on intra-channeldispersion broadening upon sensitivity (see § tral index for models C and F, where a hot ionized mediumis assumed. These models predict flux densities below thesurvey threshold for essentially all values of α > −
10. Thecorresponding α lim values are therefore not listed in Table2. The spectral index constraints become much weakerwhen the effects of interstellar scattering are incorporatedin models D, E and F. For model D, with scattering but nofree-free absorption is assumed, the UTMOST null resultsonly bound α > − . α > − . Fig 3 shows the predicted detection rates for UTMOST,CHIME and HIRAX for the two distance scales considered.The vertical line corresponds to the redshift limit of thesurvey for all models A–F. These predictions were obtainedfrom the spectral constraints on each model obtained in the previous section, and computing the sensitivity of each sur-vey as described below.In modeling the sensitivity of CHIME, we assume thatthe gain G = 2 K Jy − and system temperature T sys = 50 Kremain constant over the band. We also assumed a singleCHIME beam of width 1.5 by 90 degrees (Bandura, privatecommunication). Using Eq 2, we obtained the optimum fluxlimit of 0.125 Jy for a 5 ms duration burst. For the scatteringscenario, we used the frequency weighted average value of τ s over the whole CHIME band. We obtained τ s = 92.2 ms. Foreach of the models described in Table. 2, and the using theconstraint on the spectral index from the UTMOST sur-vey, we plotted the peak flux versus redshift using Eq. 2.For each model at the constrained spectral index, we ob-tained the z lim which is the redshift where the peak fluxof the FRB is equal to the flux sensitivity limit of CHIMEas shown in Fig. 4. Then, using the expected sky coverageof CHIME and scaling the Crawford et al. (2016) rate withthe comoving volume, we obtained the predicted numberof FRB detections per day versus redshift as shown in leftpanel of Fig. 3. The ordinate of the point at which the z lim for each model intersects the curve and the bounds gives the Rajwade & Lorimer
Figure 3.
The number of FRBs predicted per day/month as a function of redshift for various surveys. The black dashed curve is thenumber of FRBs per day based on the (Crawford et al. 2016) rates. The blue curves are the 99% upper and lower bounds on the blackdashed curve. Left panels show predictions for the cosmological case while the right panels show predictions for extragalactic case. Inpredictions for CHIME (cosmological case), from left to right, the vertical lines correspond to models F,D,E,A,B and C respectivelywhile they correspond to models F,E,D,C,B and A respectively for the extragalactic case. Similarly for HIRAX, the vertical lines fromleft to right correspond to models D,E,F,A,B and C respectively for the cosmological case and F,E,D,C,B and A respectively for theextragalactic case. In case of UTMOST, the single vertical line corresponds to all models for their respective constrained spectral indexat the limiting redshift of the survey. The ordinate of intersection of the vertical line and the curves gives the predicted number for eachmodel.
RBs at decametric wavelengths Figure 4.
Peak flux versus redshift predicted for model E assum-ing the nominal parameters of CHIME. The intersection of thesurvey limit and the curve gives the limiting redshift probed byCHIME for this model. predicted number of FRB detections per day for that givenmodel. We investigated the yield for HIRAX surveys withidentical parameters as the ones used for CHIME except for G = 10.5 K Jy − . The analysis suggests that CHIME willbe able to detect from 30–100 FRBs per day depending onthe model for the cosmological case while the yield increasesby an order of magnitude ( ∼ Our analysis has a number of simplifying assumptions aboutthe nature of FRBs. In this section, we investigate the sensi-tivity of our results to these assumptions. A key simplifica-tion we have made is to assume that FRBs are standard can-dles. Recent models and surveys for FRBs suggest that theremight be distribution of luminosities for these bursts (see,e.g., Caleb et al. 2016; Vedantham et al. 2016). Hence, weinvestigated the effect of FRBs having a range of luminosi-ties. By definition, for a population of standard candles, allsources are detected out to a survey’s redshift limit. Thismeans that, for a distribution of luminosities, only thoseFRBs that are fainter than the currently assumed value willhave any impact on the results. To investigate this, we re-peated our analysis by reducing the luminosities by a factorof 10 from the value assumed above. This factor is motivatedby the approximate distribution of energies in the study ofCaleb et al. (2016). This exercise resulted in weaker con-straints on the spectral index values for each model suchthat the α lim values reported in Table 2 are reduced by fac-tor of anywhere between 1.5 and 2 . Therefore, for a popula-tion with a range of luminosities in general, we would expectthe constraints given in Table 2 to be reduced slightly. Wealso note that lowering the luminosities assumed necessar-ily results in lower predicted yields for future FRB surveys.For example, we found that our predictions for CHIME werereduced by up to a factor of 2. In summary, a range of lu-minosities for the FRB population will tend to reduce the constraints on spectral index and lead to different surveyyields. This complication only further highlights the valuethat future surveys will have in probing the FRB population.The recent discovery of a repeating FRB (Spitler et al.2016) provides some evidence that a neutron star scenariois the most plausible model for these bursts. If FRBs dooriginate from neutron stars, we detect the brightest pulsesfrom them in the local universe. This constrains the dis-tance to these sources to z = 0 .
025 (i.e. 100 Mpc). We alsoinvestigated the effect of such an assumption and results areshown in Table 2 and Fig. 3. One would assume that givena smaller distance to the sources, CHIME would see more ofthem. The results agree with this conjecture. Fig. 3 suggeststhat even with models including scattering and free-free ab-sorption, CHIME would see ∼
100 FRBs per day if they werein the local universe.In all of our calculations, we have implicitly assumedthat the FRB rate is constant per unit comoving movingprobed by the surveys. If the FRB rate traces the cosmolog-ical star formation rate (SFR), then we would expect themaximum number of sources to be found at z = 2 (5.3Gpc) (Madau & Dickinson 2014). Caleb et al. (2016) com-pared a sample population of FRBs based on the SFR tothe observed sample and found a good match with differ-ent parameters of the observed sample although the pulsewidths could not be accounted for. Given the current size ofthe FRB population, and difficulties in ascribing a distancescale, we regard this as a subtlety that is currently not wellprobed by the observations. We do, however, comment ona related factor that may impact future observations in thediscussion below. Our results suggest that telescopes in the 0.4–1.0 GHz bandwill make vital contributions to our understanding of FRBs.Even with free-free absorption and scattering playing a vi-tal role in flux mitigation of FRBs, CHIME will be able todetect these bursts on a daily basis by the virtue of its ex-tensive bandwidth and vast instantaneous sky coverage. Wealso looked into the possible caveats in the analysis and theeffects those would have on the predictions for CHIME. Ourinvestigation suggests that with all the caveats considered,the lowest yield for a future CHIME survey is ∼
30 FRBsper day which is very optimistic compared to expected yieldfrom other surveys. For example, the corresponding yield forfuture UTMOST observations is about 1–2 FRBs per monthfor future observations which makes it difficult to differenti-ate between the two models at the moment.We also discussed certain caveats in our analysis ( § Rajwade & Lorimer
Figure 5.
Left: peak flux versus redshift for the AO327 survey for model F illustrating the effect of absorption and Doppler shift ofthe observed frequency as described in the text. The black dashed line is the flux of the FRB. The different vertical lines correspondto different redshifts. In this case, we assumed α = −
2, EM = 3 × cm − pc and T e = 8000 K. Right: the different regions of theabsorption spectrum probed by the survey at different redshifts. The different shaded regions correspond to the rest frame frequencyprobed by the survey at different redshifts. gests that a large contribution to the DM comes from thelocal plasma around the source which favours models C andF as the most plausible scenarios describing these events.Assuming the parameters in model C, we can estimate thelinear size of the absorber around the source in order toproduce the high DMs observed for FRBs. If we take theFRB with the highest known DM (FRB 121002) and placeit at z = 0 .
025 then, assuming model C, we obtain a lin-ear size of ∼ < § ACKNOWLEDGMENTS
We thank Jayanth Chennamangalam for providing code tomake plots for Fig. 3, and Joeri van Leeuwen for pointing outthe flux degradation effect shown in Figs. 2, 3, 4 and 5. Weacknowledge the assistance of Kevin Bandura who provided