------------------------------------------------------------------------ LaRosa_preprint.tex ApJ, 2001 December 10, in press X-Mailer: VM 6.43 under 20.4 "Emerald" XEmacs Lucid Message-ID: <15237.50628.9101.852067@exeter.nrl.navy.mil> Reply-To: lazio@rsd.nrl.navy.mil X-URL: http://rsd-www.nrl.navy.mil/7213/lazio/ X-Attribution: TJWL Content-Length: 36995 %astro-ph/0108360 \documentclass[10pt,preprint2,longnamesfirst]{aastex} % % \received{2001 June~1} \revised{2001 August~6} \accepted{2001 August~13} \journalid{563}{2001 December~10} %\articleid{START PAGE}{END PAGE} \paperid{54166} %\cpright{PD}{2000} %\ccc{} \newcommand{\mjybm}{\mbox{mJy~beam${}^{-1}$}} \shortauthors{LaRosa et al.} \shorttitle{Parallel Nonthermal Filaments Near the Galactic Center} \begin{document} \title{A New System of Parallel Isolated Nonthermal Filaments Near the Galactic Center: Evidence for a Local Magnetic Field Gradient} \author{T.~N.~LaRosa\altaffilmark{1}} \affil{Department of Biological and Physical Sciences, Kennesaw State University, 1000 Chastain Rd., Kennesaw, GA 30144} \email{ted@avatar.kennesaw.edu} \altaffiltext{1}{Navy-ASEE Summer Faculty Fellow, Naval Research Laboratory} \author{T.~Joseph~W.~Lazio, Namir E.~Kassim} \affil{Code~7213, Remote Sensing Division, Naval Research Laboratory, Washington, DC 20375-5351} \email{lazio@rsd.nrl.navy.mil} \email{kassim@rsd.nrl.navy.mil} \begin{abstract} We report the discovery of a system of isolated nonthermal filaments approximately 0.5\arcdeg\ northwest (75~pc in projection) of \hbox{Sgr~A}. Unlike other isolated nonthermal filaments which show subfilamentation, braiding of subfilaments, and flaring at their ends, these filaments are simple linear structures and more closely resemble the parallel bundled filaments in the Galactic center radio arc. However, the most unusual feature of these filaments is that the 20/90~cm spectral index uniformly decreases as a function of length, in contrast to all other nonthermal filaments in the Galactic center. This spectral gradient may not be due to simple particle aging but could be explained by a curved electron energy spectrum embedded in a diverging magnetic field. If so, the scale of the magnetic gradient is not consistent with a large scale magnetic field centered on Sgr~A$^{\star}$ suggesting that this filament system is tracing a local magnetic field. \end{abstract} \keywords {Galaxy: center --- radio continuum} \section{Introduction}\label{sec:intro} The isolated nonthermal filaments (hereafter NTFs) observed in the Galactic center (hereafter GC) are unique to that region. They are characterized by extreme length to width ratios (from~10 to $>$100), highly polarized emission (30--70\%), strong magnetic fields ($\sim 1$~mG) aligned along their long axis, and nonthermal spectra (e.g., the 20/90 cm spectral indices, $S \propto \nu^\alpha$, range from $\alpha= -0.4$ to~$-0.6$, typically steepening above~5~GHz to $\alpha\sim-1.5$), for a review see \cite{ms96}. Subsequent work has revealed that many isolated NTFs consist of subfilaments braided around each other \citep[e.g.,][]{y-zwp97, laklg99}, and that the surface brightness appears to be a maximum at the intersection of the subfilaments \citep{lklh00}. Lastly, \cite{lme99} and \cite{lklh00} found that the spectral indices in several well studied filaments (the southern and northern threads and the Sgr~C filament) are constant with length. The exception is the Snake filament which exhibits a spectral gradient in the region surrounding its major \lq\lq kink\rq\rq \citep{gnec95}. The relationship between the NTF phenomenon and the parallel bundled filaments in the GC Radio Arc is not clear. The filaments in the Radio Arc show little substructure, have nearly flat spectra and can be observed at frequencies as high as 150~GHz \citep{rsm00}. Several theoretical models suggest that the Arc filaments and the NTFs are tracing a large-scale magnetic field that pervades the GC region \citep[e.g,][]{sm94}. At present, there is no consensus interpretation of these structures and new models are under development \citep[e.g.,][]{sl99,bl01,desl01}. A recent wide-field image of the GC region at~90~cm \citep{lklh00} revealed several extended sources well away from the GC itself. One of these, \objectname[]{G358.85$+$0.47}, which is 225~pc in projection from \objectname[]{Sgr~A$^{\star}$}, was found to be the first NTF that is parallel to the Galactic plane \citep{laklg99}. This paper reports a detailed study of \objectname[]{G359.85$+$0.39}, another source discovered on the wide-field image. It is a system of three parallel NTFs located approximately 0.5\arcdeg\ ($\approx 75$~pc in projection) northwest of \objectname[]{Sgr~A}. In \S\ref{sec:observe} we present 90, 20, and 6~cm observations of this source, made with the Very Large Array (VLA) of the National Radio Astronomy Observatory.\footnote{% The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated under a cooperative agreement with the Associated Universities, Inc.% } In \S\ref{sec:theory} we describe alternate scenarios to explain the morphology and spectral index gradient of the source; we present our conclusions in \S\ref{sec:conclude}. \section{Observations and Analysis}\label{sec:observe} \subsection{Observations}\label{sec:observations} The extended source \objectname[]{G359.85$+$0.39} was discovered originally on a wide-field, 90~cm image of the GC region \citep{lklh00}. Figure~\ref{fig:90} is a subimage of the original wide-field image and shows this source to consist of a linear structure approximately 8\arcmin\ long that curves into a semi-circular shape with a radius of~2\arcmin. The surface brightness over the linear region rises uniformly from about~20~\mjybm\ at the southeastern end to a peak of~49~\mjybm\ near the center before merging into the circular part in the northwest. The surface brightness in the circular region is somewhat patchy and ranges from~15 to~25~\mjybm. The rms noise on this image is about 5~\mjybm\ so we regard both the linear and semi-circular structures to be significant. At the distance of the GC (8.5~kpc) the 90~cm length of~8\arcmin\ corresponds to a physical length of about 20 pc. In order to determine the spectrum and search for substructure in the source, we obtained additional, higher-frequency observations. \begin{figure} \vspace*{-1cm} \epsscale{0.9} \rotatebox{-90}{\plotone{LaRosa_f1.eps}} \caption[]{Grey scale representation of \protect\objectname[]{G359.85$+$0.39} at~90~cm \citep{lklh00}. The resolution is $43\arcsec \times 23\arcsec$, the rms noise level is 5~\mjybm, and the gray scale is linear.} \label{fig:90} \end{figure} We observed the source at~6 and~20~cm with the VLA; Table~\ref{tab:log} summarizes the observations. The 20~cm observations were made in 1999 February with the VLA in the DnC configuration. This configuration provides a resolution comparable to that of the 90~cm observations. Standard processing techniques were used to calibrate and image the visibility data. Figure~\ref{fig:20} shows the 20~cm DnC array image of the region centered on \objectname[]{G359.85$+$0.39}. This image shows both the northern and southern threads, as well as the NTFs \objectname[]{G359.79$+$0.17} and \objectname[]{G359.54$+$0.18}. At this resolution \objectname[]{G359.85$+$0.39} appears to be another isolated NTF with a similar orientation as the other NTFs. Figure~\ref{fig:20sub} shows a subimage centered on \objectname[]{G359.85$+$0.39}. The 20~cm surface brightness is similar to the 90~cm brightness but peaks at a different location. The source is also approximately 50\arcsec\ shorter than at~90~cm, and the semi-circular emission is not detected. \begin{deluxetable}{cccccc} \tablewidth{0pc} \tabletypesize{\small} \tablecaption{VLA Observing Log\label{tab:log}} \tablehead{ & & \colhead{VLA} & & \colhead{Integration} & \colhead{Synthesized} \\ \colhead{Wavelength} & \colhead{Epoch} & \colhead{Configuration} & \colhead{Bandwidth} & \colhead{Time} & \colhead{Beam} \\ \colhead{(cm)} & & & \colhead{(MHz)} & \colhead{(hr)} & \colhead{(arcsecond)}} \startdata 6 & 2000 July~6 & DnC & 50 & 1.5 & $13 \times 9.5$ \\ 20 & 1999 February~27 & DnC & 50 & 2.9 & $43 \times 38$ \\ \enddata \tablecomments{We summarize only the new observations. The 90~cm observations are described in \cite{lklh00}.} \end{deluxetable} \begin{figure} \epsscale{1} \plotone{LaRosa_f2.eps} \caption[]{Grey scale representation of the region surrounding \protect\objectname[]{G359.85$+$0.39} at~20~cm. The resolution is $42.3\arcsec \times 38.5\arcsec$, the rms noise level is 2~\mjybm, and the gray scale is linear.} \label{fig:20} \end{figure} \begin{figure} \begin{center} \epsscale{0.6} \rotatebox{-90}{\plotone{LaRosa_f3a.eps}} \rotatebox{-90}{\plotone{LaRosa_f3b.eps}} \end{center} \vspace{-0.9cm} \caption[]{\protect\objectname[]{G359.85$+$0.39} at~20~cm. The resolution is $42\farcs3 \times 38\farcs5$, and the rms noise level is 2~\mjybm. \textit{Top}: The gray scale is linear. \textit{Bottom}: The contour levels are 1~\mjybm\ $\times$ $-3$, 3, 6, 9, \ldots.} \label{fig:20sub} \end{figure} The semi-circular structure visible at~90~cm (Figure~\ref{fig:90}) is notably absent at~20~cm (Figure~\ref{fig:20}). Given the confused nature of the GC, we cannot rule out conclusively the possibility that this semi-circular emission is the superposition of another source, but we regard it as unlikely. First, its morphology is unlike that of most extragalactic sources. While its morphology is similar to that of a (portion of a) supernova remnant, its spectrum is too steep. Our 20~cm noise level is approximately 1~\mjybm, so a 3$\sigma$ upper limit on its flux density implies a 20/90~cm spectral index steeper than $-1.1$. Second, the surface brightness from the linear part merges smoothly with that of the semi-circular part at~90~cm. Finally, the fact that the source is longer at 90~cm and that the emission peaks at different locations at~20 and 90~cm suggests a spectral index gradient. Below we show that this source does indeed have a spectral index gradient (from north to south) and the spectral index at the north end of the the object is $-1.1$, consistent with the spectral index of the semi-circular part being steeper than $-1.1$. In~2000 July, 6~cm continuum observations were made in the VLA DnC configuration. These observations were made in dual polarization mode at~4515 and~4765 MHz. Standard calibration and imaging techniques were used to process the visibility data. Because both the 6 and~20~cm observations were obtained in the DnC configuration, the 6~cm image has higher angular resolution. Figure~\ref{fig:6} shows the resulting image. The increased resolution reveals that the linear structure consists of three \emph{parallel} filaments. As with the 20 and~90~cm emission the surface brightness in all three filaments rises uniformly from the southeastern end, peaks near the mid point, and declines toward the northwestern end. However, the ultra thin, top filament, although detectable to the eye, is at best only a $3\sigma$ detection at some points along its length. \begin{figure} \begin{center} \epsscale{0.6} \rotatebox{-90}{\plotone{LaRosa_f4a.eps}} \vspace{-0.25cm} \rotatebox{-90}{\plotone{LaRosa_f4b.eps}} \end{center} \vspace{-0.75cm} \caption[]{\protect\objectname[]{G359.85$+$0.39} at~6~cm. The resolution is $12\farcs5 \times 9\farcs3$, and the rms noise level is 0.08~\mjybm. \textit{Top}: The gray scale is linear between~$-0.31$ and~0.73~\mjybm. \textit{Bottom}: The contour levels are 0.1~\mjybm\ $\times$ $-0.225$, 0.225, 0.325, 0.425, \ldots.} \label{fig:6} \end{figure} Figure~\ref{fig:6pol} shows the linear polarization image. This image indicates that filaments are strongly polarized but also shows considerable polarized emission that does not correspond to any features in total intensity. A similar phenomonen has been found at other wavelengths and in other directions \citep{gldt98,hkd00,gdmgwh00} and can be explained by a foreground Faraday screen inducing polarization in a diffuse background emission. Analysis of the polarization is beyond the scope of this work and will be discussed elsewhere. \begin{figure} \epsscale{0.8} \rotatebox{-90}{\plotone{LaRosa_f5.eps}} \caption[]{Total linear polarization of \protect\objectname[]{G359.85$+$0.39} at~6~cm. The resolution and noise level are similar to that of Figure~\ref{fig:6}. The polarized emission from \protect\objectname[]{G359.85$+$0.39} is evident at the center of the image, but there is considerable polarized flux that has no correspondence in total intensity (see \S\ref{sec:observations})} \label{fig:6pol} \end{figure} We stress that the sub-filaments are parallel since that term has be used to describe many isolated NTFs. However, in all other isolated NTF systems the filaments appear to overlap and cross \citep{y-zwp97,lme99}. Several also show flaring at their ends. Even at this fairly high resolution no substructure is detectable. Thus, from a morphological perspective these filaments more closely resemble the bundled NTFs in the Galactic center Radio Arc \citep{y-zmc84} than the do the isolated NTFs. To summarize we have observed polarized nonthermal emission from several linear structures with aspect ratios exceeding 10. These characteristics indicate that these filaments should be classified as GC NTFs. \subsection{The Spectral Index and Its Gradient}\label{sec:index} The spectral index of \objectname[]{G359.85$+$0.39} must change along its length because the peak in emission occurs at different locations at the different wavelengths and because it is longer at 90~cm than at~6 or~20~cm. Figure~\ref{fig:6vs90} compares the 6 and~90~cm emission. \begin{figure} \epsscale{0.65} \rotatebox{-90}{\plotone{LaRosa_f6.eps}} \caption[]{Grey scale of 6~cm emission (Figure~\ref{fig:6}) overlayed on contours of the 90~cm emission (Figure~\ref{fig:90}).} \label{fig:6vs90} \end{figure} The similar image resolutions at 90 and 20~cm make it is possible to estimate the 20/90~cm spectral index with only a minimal convolution. After convolving the 90~cm image to the 20~cm resolution, we made cross-cuts perpendicular to the linear part and determined the peak fluxes by fitting a two-point linear baseline at each cross-cut. The length of the cross-cuts was over~300\arcsec\ on each side of the filament. In almost all cross-cuts the endpoints were used to establish the baseline. However, it was also possible to choose reasonable baselines using other points. This leads to a range of possible values for the peak flux at each cross-cut and hence a range in spectral index at each position. Figure~\ref{fig:si} shows the 20/90~cm spectral index plotted as a function of length along \objectname[]{G359.85$+$0.39} with the uncertainties on the points reflecting the uncertainties in the baselines. The spectral index varies smoothly from $\alpha = -0.15$ to~$-1.1$ from south to north, i.e., away from the \hbox{GC}. \begin{figure} \epsscale{1} \plotone{LaRosa_f7.eps} \caption[]{The 20/90~cm spectral index as a function of length along \protect\objectname[]{G359.85$+$0.39}. The origin is at the southern end of the filament. Error bars were determine from basline uncertainties in the individual cross-cuts.} \label{fig:si} \end{figure} There are two possible sources of systematic error in our determination of the spectral index. However, one source of systematic error serves to \emph{flatten} the derived spectral indices, i.e., the actual spectral index gradient may well be larger than Figure~\ref{fig:si} shows. First, the images from which the spectral indices were determined do not have matching spatial frequency coverage. Even though the 20 and~90~cm images have similar resolutions (see Figures~\ref{fig:90} and~\ref{fig:20sub}), the 90~cm visibility data cover more of the inner $u$-$v$ plane than do the 20~cm visibility data. We obtain a rough estimate of the effect of the mismatch of spatial frequency coverage in the following manner: The shortest antenna spacings in the VLA's DnC configuration are approximately 35~m. At~20~cm, these baseline lengths correspond to angular scales of approximately 20\arcmin, meaning that structures on scales of~10\arcmin--20\arcmin\ are poorly sampled or missing entirely from Figure~\ref{fig:20} (and~\ref{fig:20sub}). These angular scales are comparable to or larger than the length of \objectname[]{G359.85$+$0.39} at~20~cm, so we can approximate the effect of the missing short baselines as a constant offset to the brightness of \objectname[]{G359.85$+$0.39}. Thus, the spectral indices shown in Figure~\ref{fig:si} may be too steep, but, if so, the systematic bias should apply approximately equally to all. The second possible systematic effect also occurs because of missing short spacings, in particular the ``zero spacing,'' and is evident as the large negative regions surrounding \objectname[]{Sgr~A} in Figure~\ref{fig:20}. Because \objectname[]{Sgr~A} is so bright, it produces a large ``negative bowl'' surrounding it \citep[for an additional illustration see][]{bv79}. This negative bowl may extend well past \objectname[]{G359.85$+$0.39}, but its gradient is directed toward \objectname[]{Sgr~A}. That is, the largest negative regions occur near \objectname[]{Sgr~A}, and their magnitudes become progressively smaller farther away from \objectname[]{Sgr~A}. Thus, this negative bowl would tend to depress the brightness at the south end of \objectname[]{G359.85$+$0.39}, where the spectral index is flattest, more so than at the north end. We conclude that the 20/90~cm spectral index gradient shown in Figure~\ref{fig:si} is robust and may actually be \emph{steeper} than shown. Furthermore there is no evidence of a negative bowl in the region of the semi-circular emission. Recall that the semi-circular emission would not be detectable if the spectral index was steeper than -1.1. Given the spectral index is -1.1 in the linear structure and assuming the gradient continues we would not expect to detect the semi-circular emission. We also attempted to use the same method to determine the 6/20~cm spectral index along the length of \objectname[]{G359.85$+$0.39}. The differing spatial frequency coverage of the 6 and~20~cm observations is far more problematic in this case. The missing short spacings mean that angular scales larger than about 25--35\% of the filament's length are poorly sampled or missing. Furthermore the wide difference in resolution requires a large convolution. We find that the 6/20~cm spectral index is steeper than the 20/90~cm spectral index, with $\alpha$ ranging from~$-0.9$ to~$-1.3$. This result must be regarded as an upper limit to the spectral curvature. We also found no systematic variation with position. However, given the caveats mentioned above we can not attach much significance to this result. Additional observations are required to make definitive statements about the 6/20~cm spectral index. \section{A Spectral Index Gradient: Discussion}\label{sec:theory} Previous studies of several NTFs \citep{lme99,lklh00} have found that their spectral indices are constant with length. For example, the \objectname[]{Sgr~C} filament exhibits a constant 20/90~cm spectral index $\alpha \approx -0.5$ over its entire length, about~27~pc. One prominent exception is the \objectname[]{Snake} filament, which shows a spectral index gradient at the location of the major kink \citep{gnec95}. The index is steep at the kink with $\alpha \approx -0.5$ and flattens to~0 moving away from the kink. By contrast, \objectname[]{G359.85$+$0.39} exhibits a 20/90~cm spectral index gradient, with the flattest spectral index ocurring at the southeastern end of the filament, \emph{not} at the midpoint of the filament where the flux peaks. One might expect the location of the peak flux to coincide with the acceleration site and the flattest spectral index. Clearly the electron acceleration in this object is more complicated than this simple scenario. In general, the coherence of the NTFs over many parsecs in a region with a strong ram pressure from the surrounding molecular gas suggests that the magnetic field in these structures is quite strong \citep{y-zm87a,y-zm87b}. Equating ram pressure with magnetic pressure these authors estimate that the magnetic field of an NTF is of order \hbox{1~mG}. We adopt this value for \objectname[]{G359.85$+$0.39}. The synchrotron emissivity corresponds to a number density of relativistic particles of order $10^{-5}$~cm${}^{-3}$. The background particle density from observations of X-ray emitting gas in the GC region is of order 1~cm${}^{-3}$ \citep{kmstty96}. The electrons responsible for the 90~cm emission have an energy of~0.14~GeV and a synchrotron lifetime of $6 \times 10^4$~yr while the 20~cm emission is from~0.29~GeV electrons with a lifetime of $2.9 \times 10^4$~yr. As discussed above, although our estimate of the 6/20~cm spectral index suggests spectral curvature between~6 and~20~cm, it is not conclusive since we are missing 6~cm flux. However, curved spectra have been found by \cite{lme99} for both the Southern and Northern NTFs and for the \objectname[]{Snake} filament by \cite{gnec95}. \emph{We therefore argue that spectral curvature is a general property of NTFs.} Curvature is most commonly interpreted in terms of particle aging. Our spectral index measurements indicate a break in the frequency spectrum between~6 and~20~cm. Assuming an initial power-law spectrum, the time~$t$ for synchrotron losses to produce the observed curvature is given by $t = 3.3 \times 10^4\,\mathrm{yr}\,B^{-3/2}\nu_b^{-1/2}$, where the magnetic field~$B$ is in mG and the break frequency~$\nu_b$ is in GHz. Using a magnetic field of~1~mG, a break frequency between~6 and 20~cm corresponds to a source age of order $2 \times 10^4$~yr. For comparison in a 1~mG field the synchrotron lifetime of a 0.1~GeV electron emitting at a wavelength of~90~cm is of order $6 \times 10^4$~yr. Although we cannot discount the possibility the acceleration mechanism itself generated a curved spectrum and that this is a young source, we will, for purposes of illustration, assume an age of a few tens of thousands of years. The origin and structure of the NTF magnetic fields is still a matter of debate \citep[e.g.,][]{sl99}. The central question is whether the filaments are tracing a large-scale, globally-ordered magnetic field or are independent local magnetic structures. If part of an ordered large-scale field, we would expect the NTFs are static equilibrium magnetic structures. Alternatively, if they are local, they are more likely evolving dynamic structures. In addition to the magnetic field, the key questions surrounding the NTF phenomenon are the location of the acceleration region and the subsequent transport electrons. There are no detailed models for the acceleration of electrons in the NTFs, and it is often assumed that electron acceleration occurs in a region considerably smaller than the size of the filament \citep[e.g.,][]{sm94,rb96}. The accelerated electrons subsequently expand along the magnetic field and illuminate the rest of the structure. Particle transport is likely to be very different in static structures than it is in dynamic ones. We begin by showing that this filament cannot be considered to be a static equilibrium structure. We then discuss the scenario we favor in which the filament represents a localized enhancement in the GC magnetic field. \subsection{Magnetic Field Structure and Particle Transport}\label{sec:transport} We assume acceleration occurs locally and particles stream along the (static) magnetic field. Such anistropic velocity distributions are unstable and generate MHD waves which in turn scatter the particles. Such resonant scattering \citep[e.g.,][]{w74,m82,dbm94} impedes electron streaming and results in diffusive motion. Resonant scattering by low-frequency plasma waves produces electron pitch-angle scattering without changing the energy of the electrons. Pitch-angle scattering randomly changes the direction of the electrons resulting in diffusion parallel to the large-scale magnetic field. For strong pitch-angle scattering the electrons change direction rapidly and can be confined to a small region of space \citep{dbm94}. The magnitude of the spatial diffusion coefficicient, $D_{xx}$ is inversely proportional to the pitch angle diffusion coefficient $D_{\alpha \alpha}$ as \begin{equation} D_{xx} = \frac{1}{6}\frac{V^2}{D_{\alpha\alpha}}, \label{eqn:diffuse} \end{equation} where $V$ is the particle velocity. The pitch angle diffusion coefficient for electron scattering by whistler or Alfv{\'e}n waves (ignoring angular factors) is \citep{m82} \begin{equation} D_{\alpha\alpha}=\pi^2 e^2\frac{W(k_r)}{p\epsilon}, \label{eqn:adiffuse} \end{equation} where $k_r = eB/m_ec$ is the resonant wave number, $W(k_r)$ is the wave energy density per unit wave number, $p$ is the particle momentum, and $\epsilon$ the particle energy. Expressing this coefficient in terms of the background magnetic energy density $B^2/8\pi$ gives \begin{equation} D_{\alpha\alpha} \approx 4 \left(\frac{B}{1\,\mathrm{mG}}\right)\left(\frac{\epsilon}{1\,\mathrm{GeV}}\right)^{-1} \frac{k_rW(k_r)}{B^2/8\pi}. \label{eqn:adiffuse2} \end{equation} For diffusive motion the root mean square displacement of a particle is $\langle\Delta X^2\rangle = D_{xx}t$. The parameter that determines the transport is the ratio of the wave energy density to the background magnetic energy density. According to \cite{m82} the maximum in the wave energy level occurs when the scattering reduces the velocity anisotropy to the threshold anisotropy in one scattering time. For $(V_A/c) \sim 10^{-2}$, $k_r W(k_r)/(B^2/8\pi)$ is of order $10^{-8}$ \citep [equation~7.76]{m82}. We conclude that electrons with an energy of 0.14~GeV diffuse about 13~pc in a magnetic field of~1~mG in $4 \times 10^4$~yr. Given that the lengths of \objectname[]{G359.85$+$0.39} and other NTFs are of the same order of magnitude, the age indicated by the spectral curvature is consistent with this length if the acceleration occurs in a region whose length is small compared to the length of the filament. However, such diffusion acting alone does not produce a spectral gradient. The parallel diffusion coefficient due to scattering by self-generated waves depends on the first power of the energy. Since the synchrotron lifetime of an electron depends inversely on the first power of the energy, the diffusive length associated with a given timescale is independent of energy. Spectral steepening could be produced by diffusion that depends on a power of the energy that is less than 1. For $D_{xx} \propto E^{\beta}$ with $\beta < 1$, then $\Delta X \propto E^{-(1-\beta)/2}$. For example, if $\beta =1/2$, then $\Delta X\propto E^{-1/4}$ resulting in fewer higher energy particles at large distances from the acceleration site. A full mathematical treatment, which is beyond the scope of this paper, is required to establish if diffusion by pre-existing plasma turbulence could generate the observed spectral steepening. However, in a static magnetic field configuration there is no compelling physical basis to introduce a spectrum of turbulence. We therefore explore transport in a dynamic configuration. In a dynamic, evolving magnetic field configuration both plasma and MHD turbulence are expected. Furthermore the magnetic field in a turbulent system could be tangled with a significant random component. Polarimetric studies of other NTFs \citep[e.g.,][]{y-zwp97,lme99} show that on the large-scales the magnetic field is aligned along the long axis of the NTF suggesting an ordered structure. However, on scales much smaller than a beam (i.e., subparsec), the field could have a significant random component. Cross-field transport would dominate in this situation. Although classical theory predicts that the ratio of parallel to cross field diffusion is of order $10^{-6}$, observations of cosmic rays indicate a ratio of order $10^{-2}$ \citep{j99}. A reduction in our previous diffusion coefficient by this factor reduces the diffusion length to~1~pc, which would certainly require acceleration to occur along the entire length of the filament. Electrons would clearly age before they could fill the filament, but it is not at all clear the transport and distributed acceleration will conspire to produce a uniform spectral gradient. It seems in either the static or dynamic configuration something more is required to explain a uniform gradient. \subsection{Magnetic Field Variation}\label{sec:bfield} As an alternate possibility to explain spectral index variation, we consider a combination of a curved particle spectrum and a spatially varying magnetic field \citep{rk-sa92}. In general particles of energy~$E$ will emit synchrotron radiation most intensely at a frequency $\nu_{\mathrm{ob}}=cB^2E$. In a region of lower magnetic field, higher energy particles will emit at a given $\nu_{ob}$. If the particle energy spectrum is curved, the result is a gradient in the spectral index. Assuming a smoothly declining energy spectrum between two locations with magnetic field strengths of $B_{1,2}$, the change in spectral index is $\delta\alpha =a_c\log(B_1/B_2)$, where $a_c$ is the curvature of the frequency spectrum \citep{br00}. Unfortunately, in our case we have only three measured frequencies, and it is not meaningful to fit a second order polynomial to the spectrum and quantitately determine a curvature. If a magnetic field gradient is responsible for the spectral gradient in this source, the much larger northwestern extant of the source at 90~cm compared to~6 and~20~cm suggests a fairly rapid decrease in the magnetic field at the this end of the filaments. Such a rapid divergence is \emph{not} consistent with a globally ordered magnetic field. \cite{mm94,m96} suggests that the magnetic field in the GC region is, to first order, a dipole centered on Sgr~A$^{\star}$ with a radius of curvature comparable to the size of the central molecular zone, 100 to~200~pc. The ratio $B/\nabla B$ is an estimate for the scale of variation. At a projected distance of~75~pc, this implies variation on a scale of~25~pc. However, the spectral index changes can be measured on a scale of a few parsecs. The region over which the spectral index was measured is 13~pc. Over this region 15 cross-cuts were made, and changes in $\alpha$ occur over a few cross-cuts. If a magnetic field gradient is the cause for the spectral gradient, it must be a local field. We conclude that \objectname[]{G359.85$+$0.39} is not tracing a large-scale magnetic field. \section{Summary}\label{sec:conclude} We have discovered a system of three parallel, nonthermal filaments about~75~pc in projection from \hbox{Sgr~A}. The key observational feature of this system is a uniformly decreasing 20/90~cm spectral index as a function of length. It is also noteworthy that the peak flux occurs about midway along the length of the filament, but the flattest spectral index occurs at the end closest to the Galactic center. There are several explanations for the observed spectral gradient. These include energy dependent diffusion in a turbulent plasma magnetic field configuration and electron acceleration distributed along the length of the filament. However, independent of the electron acceleration and transport, a simpler and more natural scenario is that the magnetic field is varying along the length of the filament. A spectral gradient is a natural consequence of a curved electron energy spectrum radiating in a decreasing magnetic field. If so, the gradient scale length of a few parsecs is not consistent with a large-scale field centered on \hbox{Sgr~A}. This result suggests that this nonthermal filament system is not necessarily tracing a large-scale magnetic field. Observations at additional frequencies could be used to determine the spectral curvature and provide specific constraints on the magnetic field structure. \acknowledgements We thank S.~Shore for several stimulating discussions, C.~Lang for assistance with the observations and discussions in the early stages of this work, and the referee for several suggestions that improved the presentation of this work. Basic research in radio astronomy at the NRL is supported by the Office of Naval Research. 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Joseph W. Lazio, Ph.D. voice: +1 202 404 6329 Remote Sensing Division fax: +1 202 404 8894 Naval Research Lab, Code 7213 lazio@rsd.nrl.navy.mil Washington, DC 20375-5351 USA http://rsd-www.nrl.navy.mil/7213/lazio/