\documentclass[journal]{rmaa}

\usepackage{paralist}
\usepackage{psfrag,color}
\usepackage[latin1]{inputenc}


\title{Temperature, brightness and spectral index\\ of the Cygnus radio loop}

\author{
V. Borka Jovanovi\'{c}\altaffilmark{1} 
and D. Uro\v{s}evi\'{c}\altaffilmark{2}}

\altaffiltext{1}{Vin\v{c}a Institute of Nuclear Sciences, University of Belgrade, Serbia.}
\altaffiltext{2}{Faculty of Mathematics, University of Belgrade, Serbia.}


\shortauthor{Borka Jovanovi\'{c} \& Uro\v{s}evi\'{c}}

\shorttitle{The Cygnus radio loop}


\fulladdresses{
\item V. Borka Jovanovi\'{c}: Atomic Physics Laboratory (040), Vin\v{c}a Institute of Nuclear Sciences, University of Belgrade, 
P.O. Box 522, 11001 Belgrade, Serbia (vborka@vinca.rs)
\item D. Uro\v{s}evi\'{c}: Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 
Belgrade, Serbia (dejanu@matf.bg.ac.rs).}


\listofauthors{V.~Borka~Jovanovi\'{c} \& D.~Uro\v{s}evi\'{c}}

\indexauthor{Borka Jovanovi\'{c}, V.}
\indexauthor{Uro\v{s}evi\'{c}, D.}

\SetVolume{47}
\SetFirstPage{159}
\SetYear{2011}
\ReceivedDate{2010 October 18} 
\AcceptedDate{2011 February 10}


\abstract{The estimated brightness of the Cygnus loop supernova remnant (SNR) at 2720, 1420, 820, 408 and 34.5~MHz is presented. The 
observations of the continuum radio emission are used to calculate the mean brightness temperatures and surface brightnesses of this 
loop at the five frequencies in a wide spectral range, using the method we have previously developed for large radio loops. The spectrum 
for mean temperatures versus frequency between the five frequencies is estimated and the spectral index of the Cygnus loop is obtained. 
Also, from our results it can be concluded that the Cygnus loop evolves in a low density environment and that the initial energy of the 
supernova explosion was relatively low. The results obtained confirm the non-thermal origin of the Cygnus radio loop and show that our 
method is applicable to almost all remnants.\smallskip}


\resumen{Se presenta el brillo estimado del remanente de supernova del Velo del Cisne en 2720, 1420, 820, 408 y 34.5~MHz. Mediante el 
m\'etodo que hemos desarrollado anteriormente para anillos grandes, utilizamos las observaciones de radioemisi\'on en el continuo para 
calcular las temperaturas de brillo medias y los brillos superficiales del Velo del Cisne en las frecuencias mencionadas. Estimamos el 
espectro de las temperaturas medias versus frecuencia entre las cinco frecuencias y obtenemos el \'indice espectral del Velo del Cisne. 
Nuestros resultados muestran que el Velo evoluciona en un medio de baja densidad, y que la energ\'ia inicial de la explosi\'on de supernova 
fue relativamente baja. Confirmamos el origen no-t\'ermico de la radiaci\'on del anillo del Cisne y mostramos que nuestro m\'etodo es 
aplicable a la mayor\'ia de los remanentes de supernova.}


\addkeyword{ISM: supernova remnants}
\addkeyword{radiation mechanisms: non-thermal}
\addkeyword{radio continuum: general}
\addkeyword{surveys}


\RescaleTitleLengths{0.94}

\begin{document}


\maketitle


\section{Introduction}
\label{sec01}

In our investigation, we are supposing the radio loops to be evolved supernova remnants (SNRs). Supernova remnants represent shells of 
approximately spherical shape which are spreading, and depending of the interstellar matter density, they change their shape. We can see 
only parts of these spherical contours which represent radio spurs, i.e., areas of more intense emission in the sky, spur-like and of huge 
dimensions. Radio loops consist of several spurs lying approximately on the same small circle of the celestial sphere. Their material probably 
expands inside of bubbles of low density. Bubbles are made by former SNR explosions or by strong stellar winds (Salter 1983; McKee \& 
Ostriker 1977, and references therein). The radio emission from SNRs is generally understood to be synchrotron emission from relativistic 
electrons moving in the magnetic field.

A star in the constellation of Cygnus exploded and its remnant is the Cygnus loop. It is classified as a mid-aged SNR of type S, is located 
below (but near the plane of) the Galactic equator, and is less than 1~kpc away from us. It is listed in Green's catalogue of SNRs as G74.0-8.5 
\citep{gree04,gree06,gree09}. As shown in \citet{gree84}, this remnant has decelerated considerably by its interaction with the 
surrounding interstellar medium.

Leahy, Roger, \& Ballantyne (1997) presented high resolution 1420~MHz total intensity and polarization maps of the Cygnus loop and derived 
rotation measures. \citet{asch99} made a comparison between the radio and X-ray emission from supernova remnants. The X-ray emission traces 
the hot gas behind the shock front, while the radio emission comes from the relativistic electron population emitting synchrotron radiation 
in the ambient magnetic field. They found that there are significant differences in the distribution of X-ray and radio brightness in the 
Cygnus loop. Also, significant temperature variations, seen along the rim and in the bright filaments interior to the rim, indicated that 
the Cygnus loop was caused by a supernova exploding in a cavity.

\citet{uyan02} suggested that the Cygnus loop may consist of two overlapping remnants. Substantial differences between the northern and southern 
part as well as in their emission characteristics have been observed \citep{uyan04,patn02}. Some authors proposed that the Cygnus loop consists 
of two likely interacting SNRs: G74.3-8.4 and G72.9-9.0 \citep{uyan02,leah02}. In \citet{sun06}, from 4800~MHz observations, it is explained 
that the polarization maps (the difference in the polarization characteristic between the northern and southern part) support previous ideas 
that the Cygnus loop may consist of two SNRs. Besides, several compact radio sources are located within the boundary of the remnant. The main 
characteristics of the Cygnus loop in different spectral bands are: (a) in the radio band it is shell, brightest to the north-east, with fainter 
breakout regions to the south, with spectral variations; (b) in the optical it is a large filamentary loop, brightest to the north-east, not well 
defined to the south and west; (c) it is a shell in soft X-rays (see e.g., Green 2006).

Some radio maps which include this remnant are the following: the map by \citet{kund72} at 4940~MHz, \citet{keen73} at 2695~MHz, \citet{uyan04} 
at 2675~MHz, \citet{leah97} at 1420~MHz, \citet{dick80} at 610~MHz, \citet{gree84} at 408~MHz. We found observations of the continuum radio emission 
at 2720~MHz \citep{reif87}, 1420~MHz \citep{reic86}, 820~MHz \citep{berk72}, 408~MHz \citep{hasl82} and 34.5~MHz \citep{dwar90}, in electronic form 
and used them in this paper.

In our previous paper \citep{bork09a} we only calculated the temperatures and brightnesses at three frequencies. In this paper 
we expand the scope of our investigation to: brightness temperatures and surface brightnesses at five frequencies, as well as 
spectrum, $T-T$ graphs, radio spectral indices, an estimation of environment density and the initial energy of supernova explosion, 
and the flux density spectrum.

In our calculations, we used brightness temperatures over the whole area of the loop, so the mean temperature we estimated refers to the 
northern and southern parts together. In this research the average brightness temperatures and surface brightnesses of the Cygnus radio 
loop are calculated at the five frequencies: 2720, 1420, 820, 408 and 34.5~MHz. Then we study how these results compare with previous 
results (Roger et al. 1999; Reich, Zhang, \& F\"{u}rst 2003; Uyaniker et al. 2004) and with current theories of SNR evolution. These theories 
predict that loops are non-thermal sources which are spreading inside the hot, low density bubbles made by former supernova explosions or 
by strong stellar winds (see Salter 1983; McKee \& Ostriker 1977, and references therein).

Our aim is also to apply the method for the determination of the brightness temperature given in \citet{bork07} which was developed for 
large radio loops, and to show that it is rather efficient in the case of much smaller radio loops, e.g., the Cygnus loop. We also check 
whether the results obtained using this method agree with the method of $T-T$ graphs and with results obtained with other methods. Our 
method is quite simple because we are using brightness temperature isolines to define borders of the Cygnus loop at the wide range of 
frequencies ($\nu_\mathrm{max}/\nu_\mathrm{min}=2720~\mathrm{MHz}/34.5~\mathrm{MHz} \approx 79$). Other authors use different square or 
rectangular areas to determine the area of the loop and to calculate spectral indices, the brightness temperature and the flux of the loop 
(Uyaniker et al. 2004, page 917, and Leahy \& Roger 1998, page 786). Also, we calculate the flux from the Cygnus loop and compare our results 
with results of other authors in different frequency ranges.


\section{Analysis}
\label{sec02}

\subsection{Data}
\label{sec02a}

We used the following radio-continuum surveys as the basic source of data in this paper: at 2720~MHz \citep{reif87}, 1420~MHz \citep{reic86}, 
820~MHz \citep{berk72}, 408~MHz \citep{hasl82} and 34.5~MHz \citep{dwar90}. These surveys are available in electronic form, in Flexible Image 
Transport System (FITS) data format\footnote{Available at the site \url{http://www.mpifr-bonn.mpg.de/survey.html}.}. This online survey sampler 
of the {\it Max-Planck-Institut f\"{u}r Radioastronomie} (MPIfR) near Bonn, Germany, allows users to pick a region of the sky and obtain images 
and data at different frequencies. The 2720-MHz Stockert survey has an angular resolution of $0^{\circ}.35$, the 1420-MHz Stockert survey 
\citep{reic86} $0^{\circ}.59$, the 820-MHz Dwingeloo survey \citep{berk72} $1^{\circ}.2$, the 408-MHz all-sky survey \citep{hasl82} $0^{\circ}.85$, 
and the 34.5-MHz Gauribidanur survey \citep{dwar90} $0^{\circ}.7$. The corresponding observations are given at the following rates (measured data) 
for both $l$ and $b$: $1^{\circ}/8$ at 2720~MHz, $1^{\circ}/4$ at 1420~MHz, $1^{\circ}/2$ at 820~MHz, $1^{\circ}/3$ at 408~MHz and $1^{\circ}/5$ at 
34.5~MHz. The effective sensitivities are about 5~mK~$T_b$ ($T_b$ is an average brightness temperature), 50~mK, 0.20~K, 1.0~K and about 700~K, 
respectively.


%%%%% FIGURA 1

\begin{figure}[!t]
\centerline{\includegraphics[width=0.98\columnwidth]{vborkafig1a.eps}}
\vspace{0.35cm}
\centerline{\hspace{0.65cm}
\includegraphics[width=0.8\columnwidth]{vborkafig1b.eps}}
\vspace{0.35cm}
\centerline{\includegraphics[width=0.98\columnwidth]{vborkafig1c.eps}}
\caption{Top: the 2720~MHz map of a region in Cygnus, in new Galactic coordinates ($l$, $b$), showing 
contours of brightness temperature. This radio loop has position: $l=[76.5^{\circ},~71.5^{\circ}]$; 
$b=[-10.5^{\circ},~-7^{\circ}]$. The HPBW (Half Power Beam Width) for this frequency is $0^{\circ}.35$. The 
contours plotted represent the temperatures $T_\mathrm{min}$ and $T_\mathrm{max}$ from Table~\ref{tab01} 
and nine contours in between. The contours are plotted every 0.265~K, starting from the lowest temperature 
of 0.395~K and up to 0.66~K. The corresponding temperature scale is given (in K). Bottom: the 2720~MHz area
map of Cygnus. The color figure can be viewed online.}
\label{fig01}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 2

\begin{figure}[!t]
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig2a.eps}}
\vspace{0.3cm}
\centerline{\hspace{0.65cm}
\includegraphics[width=0.8\columnwidth]{vborkafig2b.eps}}
\vspace{0.3cm}
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig2c.eps}}
\caption{The same as Figure~1, but for 1420~MHz. The HPBW for this frequency is $0^{\circ}.59$. The contours are plotted 
every 0.07~K, starting from the lowest temperature of 4.2~K and up to 4.9~K. The color figure can be viewed online.}
\label{fig02}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 3

\begin{figure}[!t]
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig3a.eps}}
\vspace{0.4cm}
\centerline{\hspace{0.65cm}
\includegraphics[width=0.8\columnwidth]{vborkafig3b.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig3c.eps}}
\caption{The same as Figure~1, but for 820~MHz. The HPBW is $1^{\circ}.2$. The contours are plotted every 0.39~K, 
starting from the lowest temperature of 10.1~K and up to 14~K. The color figure can be viewed online.}
\label{fig03}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 4

\begin{figure}[!t]
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig4a.eps}}
\vspace{0.4cm}
\centerline{\hspace{0.65cm}
\includegraphics[width=0.8\columnwidth]{vborkafig4b.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig4c.eps}}
\caption{The same as Figure~1, but for 408~MHz. The HPBW is $0^{\circ}.85$. The contours are plotted every 
3.1~K, starting from the lowest temperature of 41~K and up to 72~K. The color figure can be viewed online.}
\label{fig04}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 5

\begin{figure}[!t]
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig5a.eps}}
\vspace{0.4cm}
\centerline{\hspace{0.65cm}
\includegraphics[width=0.8\columnwidth]{vborkafig5b.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[width=0.99\columnwidth]{vborkafig5c.eps}}
\caption{The same as Figure~1, but for 34.5~MHz. The HPBW is $0^{\circ}.7$. The contours are 
plotted every 3550~K, starting from the lowest temperature of 34500~K and up to 70000~K. The 
color figure can be viewed online.}
\label{fig05}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


FITS data format stores the multidimensional arrays and 2-dimensional tables containing rows and columns of scientific data sets. We 
extracted observed brightness temperatures from this data format into ASCII data files, and afterwards, using our programs in C and 
FORTRAN, we obtained the results presented in this paper.


\subsection{Method}
\label{sec02b}

As there is a great influence of background radiation over the Cygnus loop area, it is very difficult to determine its area precisely. The 
area of this loop is enclosed with brightness temperature contours. The maps of a region in Cygnus, in new Galactic coordinates ($l$, $b$), 
with contours of the brightness temperatures $T_b$ are plotted in Figures~\ref{fig01}--\ref{fig05}. In these figures, among all the brightness 
temperature contours, the most important are the outer contour and inner one, which represent the loop borders. The outer one (which corresponds 
to the minimum temperature of the loop) separates loop and background, while the inner one (corresponding to the maximum temperature) 
separates the loop and some superposed source. The interval of Galactic longitude of this loop is $l$=[76.5$^{\circ}$, 71.5$^{\circ}$], and of 
latitude it is $b$=[$-10.5^{\circ}$, $-7^{\circ}$]. We used the same method of calculation as given in \citet{bork07} for Galactic radio loops 
I--VI, Borka, Milogradov-Turin, \& Uro\v{s}evi\'{c} (2008) for Loops V and VI and in  Borka Jovanovi\'{c} \& Uro\v{s}evi\'{c} 2009b for the 
Monoceros loop. Our aim is to apply our method, which we developed for main Galactic loops I-VI and described in \citet{bork07} and \citet{bork08}, 
to smaller remnants and to show that it is applicable to almost all remnants. To confirm that our method works well, we also provide the area 
maps of Cygnus at the five frequencies (Figures~\ref{fig01}--\ref{fig05}) and some examples of the temperature profiles at 2720~MHz (Figure~\ref{fig06}). 
In Figure~\ref{fig06} it should be noticed that the bottom panel shows only background radiation (not including the loop), where for $b=-10^{\circ}$ 
the brightness temperature is not higher than 0.38 K. It is in agreement with the temperature intervals given in the first row of Table~\ref{tab01}, 
where we posit 0.395~K for the minimum temperature of the loop with background, at 2720~MHz.

The mean temperatures and surface brightnesses of this radio loop are computed using data taken from radio-continuum surveys at 2720, 
1420, 820, 408 and 34.5~MHz. We have subtracted the background radiation in order to derive the mean brightness temperature of the 
SNR alone. First, the temperature of the loop plus background was determined. For every bin in the survey we have some value for temperature 
$T_\mathrm{sum}$. We take all bins within the loop border and obtain the average temperature $<$$T_\mathrm{sum}$$>$ of the loop plus background. 
Then, the background alone near the loop was estimated. We determine the temperature of the background near the outer border of the loop 
(average value from all data near the outer border of the loop) and that is taken as the background temperature $T_\mathrm{back}$. And finally, 
the difference of these values is calculated to obtain the average temperature of the loop at each frequency: average temperature of 
the loop is $<$$T_\mathrm{sum}$$>-~T_\mathrm{back}$. The areas over which an average brightness temperature is determined at each of the five 
frequencies are taken to be as similar as possible within the limits of the measurement accuracy. $T_\mathrm{min}$ in Table~\ref{tab01} 
means the lower temperature limit between the background and the loop, and $T_\mathrm{max}$ means the upper temperature limit of the 
loop. So we used all measured values between $T_\mathrm{min}$ and $T_\mathrm{max}$, inside the corresponding regions of $l$ and $b$, to 
calculate the brightness temperature of the loop including the background. The mean brightness temperature for the loop is found by 
subtracting the mean value of the background brightness temperature from the mean value of the brightness temperature over the area of the loop.

After deriving the mean brightness temperatures $T_{\mathrm{b},\nu}$, we have converted these values into surface brightness $\Sigma_{\nu}$ by:
\begin{equation}
\Sigma_\nu = \left( {2k\nu ^2 /c^2 } \right) T_{\mathrm{b},\nu}\,,
\label{equ01}
\end{equation}
\noindent where $k$ is the Boltzmann constant and $c$ the speed of light. Results are given in Table~\ref{tab01}.


%%%%% FIGURA 6

\begin{figure*}[!t]
\centering
\includegraphics[width=0.62\textwidth]{vborkafig6.eps}
\caption{Temperature profiles at 2720~MHz for Galactic longitude from $76^{\circ}.5$ to $71^{\circ}.5$ 
and for the following values of Galactic latitude: $b=-7^{\circ}.5$ (top left), $b=-8^{\circ}$ (top right), 
$b=-8^{\circ}.5$ (middle left), $b=-9^{\circ}$ (middle right) and $b=-10^{\circ}$ (bottom). Temperatures 
are given in K, and Galactic longitudes in degrees. Notice that the bottom figure shows the temperature 
profile at a latitude which lies outside (but close enough to) the loop, and from that profile we can see 
what is the highest temperature of the background at $b=-10^{\circ}$, at 2720~MHz (not of the loop with 
background).}
\label{fig06}
\vspace{0.535cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 1

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Temperatures and brightnesses of the Cygnus radio loop\\ at 2720, 1420, 820, 408 and 34.5 MH\lowercase{z}}
\begin{tabular}{cccc}
\toprule
Frequency & Temperature limits                  & Temperature & Brightness \\
(MHz)     & $T_\mathrm{min}$, $T_\mathrm{max}$ (K) & (K)         & [$10^{-22}$~W/(m$^2$~Hz~Sr)] \\
\midrule
2720  & 0.395, 0.66  & 0.160 $\pm$ 0.005 & ~3.64 $\pm$ 0.12 \\
1420  & 4.2, 4.9     & 0.49~ $\pm$ 0.05~ & ~3.04 $\pm$ 0.30 \\
~820  & 10.1, 14.0   & 2.30~ $\pm$ 0.20~ & ~4.75 $\pm$ 0.40 \\
~408  & 41, 72       & 15.3~ $\pm$ 1.0~~ & ~7.83 $\pm$ 0.50 \\
~34.5 & 34500, 70000 & 13960 $\pm$ 700~~ & 50.97 $\pm$ 2.56 \\
\bottomrule
\end{tabular}
\label{tab01}
\vspace{0.5cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 2

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Spectral indices for Cygnus loop from $T-T$ plots,\\ between 2720, 1420, 820, 408 and 34.5~MH\lowercase{z}}
\begin{tabular}{cccccc}
\toprule
Frequency (MHz) & 2720 & 1420 & 820 & 408 & 34.5 \\
\midrule
2720  & $\cdots$        & 1.52 $\pm$ 0.35 & 1.96 $\pm$ 0.53 & 2.35 $\pm$ 0.26 & 2.57 $\pm$ 0.23 \\
1420  & 2.22 $\pm$ 0.76 & $\cdots$        & 2.27 $\pm$ 0.64 & 2.60 $\pm$ 0.28 & 2.67 $\pm$ 0.25 \\
~820  & 2.64 $\pm$ 0.61 & 3.10 $\pm$ 0.64 & $\cdots$        & 2.82 $\pm$ 0.31 & 2.82 $\pm$ 0.10 \\
~408  & 2.67 $\pm$ 0.26 & 2.79 $\pm$ 0.28 & 3.10 $\pm$ 0.31 & $\cdots$        & 2.78 $\pm$ 0.15 \\
~34.5 & 2.86 $\pm$ 0.23 & 2.95 $\pm$ 0.25 & 2.92 $\pm$ 0.10 & 2.94 $\pm$ 0.15 & $\cdots$ \\
\bottomrule
\end{tabular}
\label{tab02}
\vspace{0.5cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Results}
\label{sec03}

The radio continuum maps are used for determining the area of the Cygnus loop and for deriving brightness temperatures over it. At 
each of the five frequencies, the areas are determined to be as similar as possible within the limits of measurement accuracy. There 
are still some differences between these areas and we think that the major causes of differing borders between the five frequencies 
are small random and systematic errors in the data. The surface brightnesses of SNRs must be above the sensitivity limit of the 
observations, and must be clearly distinguishable from the Galactic background emission \citep{gree91}. As it is very difficult to 
resolve the fainter parts of the loop from the background, they are not taken into account. For evaluation of the brightness temperatures 
over the area of the loop we had to take into account background radiation (see Webster 1974). Borders enclosing the spurs are defined 
to separate the spur and its background. For the method of calculation see also \citet{bork07}, \citet{bork08} and \citet{bork09b}. 
As mentioned in these papers, if the value of $T_\mathrm{min}$ is changed by a small amount, the brightness contours become significantly 
different. If $T_\mathrm{min}$ is too small, the area of the spur becomes confused with the background and it becomes obvious that the 
border has been incorrectly chosen.

The results are given in Tables~\ref{tab01}, \ref{tab02} and \ref{tab03}. $T_\mathrm{min}$, given in the second column of Table~\ref{tab01}, 
is the lower temperature limit, while $T_\mathrm{max}$ is the upper temperature of the loop and it is also the upper temperature limit (because 
there are no other superposed sources). These temperature limits enable us to distinguish the loop from the background. Then we derived the 
surface brightnesses using equation~(\ref{equ01}) for each frequency.

The values for the brightnesses in units of $\mathrm{10^{-22}~W/}$ $\mathrm{(m^2~Hz~Sr)}$ can be compared with results for the flux densities 
in Jy. The flux densities can be transformed into brightnesses or vice versa, taking into account that the frequencies have to be the same. 
Knowing the loop size $\Omega$, the flux densities $S_\nu$ given in Jy can be transformed to brightnesses given in $\mathrm{10^{-22}~W/(m^2~Hz~Sr)}$ 
by:
\begin{equation}
\Sigma_\nu = S_\nu \times 10^{-26} / \Omega\,.
\label{equ02}
\end{equation}

By use of the spectral indices, the brightnesses can be reduced to 1000~MHz according to the relation:
\begin{equation}
\Sigma_{1000} = \Sigma_\nu (1000 / \nu)^{(2 - \beta)}\,,
\label{equ03}
\end{equation}
\noindent where the temperature spectral index $\beta=\alpha + 2$, and $\alpha$ is the spectral index defined by $S_\nu\propto \nu^{-\alpha}$.


%%%%% FIGURA 7

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth,height=6.5cm]{vborkafig7.eps}
\caption{Cygnus loop spectrum: temperature versus frequency, for three measurements (at 408, 
820 and 1420~MHz).}
\label{fig07}
\vspace{0.4cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 3

\begin{table}[!t]
\centering
\caption{Brightness of Cygnus radio\\ loop reduced to 1000 MH\lowercase{z}\tabnotemark{a}}
\tablecols{2}
\setlength{\tabnotewidth}{0.85\columnwidth}
\setlength{\tabcolsep}{1.2\tabcolsep}
\begin{tabular}{cc}
\toprule
Frequency & Brightness at 1000~MHz \\
(MHz) & (10$^{-22}$ W/(m$^2$~Hz~Sr)) \\
\midrule
2720  & 7.05 $\pm$ 0.86 \\
1420  & 3.84 $\pm$ 0.52 \\
~820  & 4.17 $\pm$ 0.29 \\
~408  & 4.33 $\pm$ 0.07 \\
~34.5 & 5.52 $\pm$ 1.40 \\
\bottomrule
\tabnotetext{a}{Using spectral index derived in this paper: $\beta_5=2.66\pm 0.09$ 
(from the spectrum of all five frequencies 2720, 1420, 820, 408 and 34.5~MHz).}
\end{tabular}
\label{tab03}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{Spectrum}
\label{sec03a}

The spectrum was generated using mean temperatures at five different frequencies. The best-fit straight line spectrum enables the calculation 
of the spectral index as the negative value of the line's direction coefficient. In Figure~\ref{fig07} we show the spectrum for the three middle 
frequencies: 1420, 820 and 408~MHz because they are very well fitted with a straight line (see Figure~\ref{fig07}). Frequencies 2720 and 34.5~MHz, 
lie on the very high and very low ends of the spectrum, as presented in Figure~\ref{fig08}. If for the calculation of the spectral index we take 
only three frequencies 1420, 820 and 408~MHz, the result would be $\beta_3=2.76\pm 0.03$. From a linear fit to all five frequencies 2720, 1420, 820, 
408 and 34.5~MHz, we obtain $\beta_5=2.66\pm 0.09$. It is a steeper spectral index in comparison to the average value for Galactic SNRs $\beta=2.5$ 
(in Green's catalogue (Green 2009) the adopted value for the spectral index is not given because of its variability). The Cygnus loop is relatively 
old SNR. The shock wave has to be weak for evolved SNRs. The steeper spectral indices of SNRs should be expected for the smaller shock wave velocities. 
This is a result obtained from the diffuse particle acceleration theory \citep{bell78a,bell78b}. Additionally, the Cygnus loop probably expands in 
a low density environment. It looks like the large Galactic radio loops -- evolved SNRs with steep spectral indices, immersed in a low density environment 
(see Borka 2007; Borka et al. 2008, and references therein).

The values we obtain, $\beta_3=2.76 \pm 0.03$ (from three frequencies), $\beta_5=2.66 \pm 0.09$ (from all five frequencies), are greater 
than 2.2 and confirm the non-thermal origin of the Cygnus loop emission. From Figure~\ref{fig08} it can be noticed that a linear fit is 
quite satisfactory. The value for the brightness temperature spectral index of the Cygnus loop is rather steep. This is at the high end 
of the spectral index distribution for SNRs as suggested in \citet{clar76}.


%%%%% FIGURA 8

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth,height=6.5cm]{vborkafig8.eps}
\caption{Cygnus loop spectrum: temperature versus frequency, for five measurements (at 34.5, 
408, 820, 1420 and 2720~MHz).}
\label{fig08}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{$T-T$ plot}
\label{sec03b}

The measured data have different resolutions for different frequencies (see \S~2.1), and therefore in order to obtain $T-T$ plots 
the data were retabulated: the higher resolution maps are convolved to the resolution of the lowest resolution map. In this way, 
we convolved data at 2720, 1420, 408 and 34.5~MHz to $0^{\circ}.5 \times 0^{\circ}.5$ resolution, which is the sampling rate of the 
820~MHz survey. These retabulated data are presented in Figure~\ref{fig09} for the following frequencies: 2720~MHz (top left), 
1420~MHz (top right), 820~MHz (middle left), 408~MHz (middle right) and 34.5~MHz (bottom). Then, for each frequency pair we used 
only the common points [with the same $(l,b)$] which belong to the loop area at both frequencies. In this way, we reduced the loop area 
to the same area for different frequencies. The obtained $T-T$ plots for five pairs of frequencies enabled the calculation of the spectral 
indices. We calculated two $\beta$ values for each of these pairs: between 2720--1420, 2720--820, 2720--408, 2720--34.5, 1420--820, 
1420--408, 1420--34.5, 820--408, 820--34.5 and 408--34.5~MHz. They are presented in Table~\ref{tab02}. For each of the ten frequency 
pairs, by interchanging the dependent and independent variables we obtained two $\beta$ values for each pair and the mean value of 
these results is adopted as the radio spectral index, as suggested in \citet{uyan04}. Regarding only three frequencies (because 
their spectrum lies on straight line, see Figure~\ref{fig07}) 1420, 820 and 408~MHz, the average value of spectral index from $T-T$ 
is $<$$\beta_{TT}$$>_3~=2.78 \pm 0.41$. Taking into account all five frequencies, we get $<$$\beta_{TT}$$>_5~=2.63\pm 0.30$. It can be noticed 
that, as expected, this value agrees well with the corresponding value obtained from the spectrum (see Uyaniker et al. 2004).


%%%%% FIGURA 9

\begin{figure*}[!t]
\centering
\includegraphics[width=0.8\textwidth]{vborkafig9.eps}
\caption{The data retabulated to $0^{\circ}.5 \times 0^{\circ}.5$ resolution, for the 
following frequencies: 2720~MHz (top left), 1420~MHz (top right), 820~MHz (middle left), 
408~MHz (middle right) and 34.5~MHz (bottom). The HPBWs for these frequencies are 
$0^{\circ}.35$, $0^{\circ}.59$, $1^{\circ}.2$, $0^{\circ}.85$ and $0^{\circ}.7$, respectively. 
The gray scales of temperatures are also given.} 
\label{fig09}
\vspace{0.3cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Then we calculated the mean value of the spectral index: regarding spectrum and $T-T$ graphs. Between 1420, 820 and 408~MHz we obtained 
$<$$\beta$$>_3~=2.77\pm 0.22$, and between 2720, 1420, 820, 408 and 34.5~MHz we obtained $<$$\beta$$>_5~=2.64 \pm 0.20$.


\section{Discussion}
\label{sec04}

The spectral index variations with position of more spatial features within the Cygnus loop can be found in the paper by \citet{leah99}. 
He studied radio spectral indices by the $T-T$ plot method between 2695, 1420 and 408~MHz and found that the bright radio filaments all 
show negative curvature (steeper at higher frequency), and that regions dominated by diffuse emission show positive curvature (flatter 
at higher frequency).

It can be noticed that the mean value of spectral index for the Cygnus loop is a little higher then the corresponding value obtained from 
the literature (see Uyaniker et al. 2004 or Leahy \& Rogers 1998). The reason for this is the different areas used for data of the Cygnus 
loop in these papers. In both papers they used square areas (see page 917 from Uyaniker et al. 2004, and page 786 from Leahy \& Roger 1998). 
In these square areas they took into account parts of the Cygnus loop and parts of the background radiation near the loop (these parts we 
do not take into account). We take in account only higher intensity regions from these areas (see Figures~\ref{fig01}--\ref{fig05}).

In order to compare our values for the brightnesses with results for the fluxes given in other papers, we transform their calculated fluxes 
into brightnesses at 1000~MHz. Knowing the loop size $\Omega$ we reduce the flux densities given in Jy to brightnesses given in 
$\mathrm{10^{-22}\, W/(m^2~Hz~Sr)}$ by means of equation~(\ref{equ02}) and then, using the spectral indices, the brightnesses are extrapolated 
to 1000~MHz according to relation~(\ref{equ03}).

In our previous paper \citep{bork09a} we transformed the flux densities for the Cygnus Loop given in \citet{roge99} at 22~MHz ($S_\nu=1378$~Jy) 
and \citet{reic03} at 863~MHz ($S_\nu=184$~Jy) into $\Sigma_{1000}$ using the loop size $\Omega=240' \times 170'$ from Reich et al. (2003) 
and the spectral index $\beta=2.49$ from \citet{trus02}. We obtained that the values from \citet{bork09a} agree with previous data. In this paper, 
we use our derived value for the spectral index $\beta_5=2.66 \pm 0.09$, and obtain even better agreement. From the flux densities given in 
the paper mentioned, we calculated the following values for radiation intensities: $\Sigma_{1000}=3.22 \times \mathrm{10^{-22}\, W/(m^2~Hz~Sr)}$ 
from the flux given in \citet{roge99} and $\Sigma_{1000}=4.84 \times \mathrm{10^{-22}\, W/(m^2~Hz~Sr)}$ from the flux given in \citet{reic03}.

Using our calculated $\beta_5$ and applying relation~(\ref{equ03}) to our calculated values of $\Sigma_\nu$ at 2720, 1420, 820, 408 and 
34.5~MHz, for $\Sigma_{1000}$ we obtain the values given in Table~\ref{tab03}. Absolute errors for brightnesses at 1000~MHz are calculated 
in this way: $\Delta \Sigma_{1000}=\left({\Sigma_{1000} / \Sigma_{\nu 1}}\right) \left({\Delta \Sigma_{\nu 1} + \Sigma_{\nu 1} \cdot \ln \left( {\nu_1 / 1000} \right) \cdot \Delta \beta} \right)$,~where $\nu_1$ takes values 2720, 1420, 820, 408 and 34.5~MHz.

Information about variations of the brightness and the spectrum over the radio image is presented by \citet{leah98} and by \citet{uyan02}. 
The average values reduce the information imprinted in variations of brightness and spectrum over the radio image but give us other 
interesting information, like the explosion energy, the surface brightness and the distance to the Cygnus radio loop.

With regard to the loop borders in Figures~\ref{fig01}--\ref{fig05}, we derived the loop area at each frequency, as well as the corresponding 
angular radii (see Table~\ref{tab04}). An angular radius is obtained in this way: our derived loop area, when approximated with a circle of 
the same area, gives the possibility of determining the angular radius as $\theta=\sqrt{\Omega / \pi}$. These areas can be compared to the 
areas calculated by other authors, e.g., $9(^{\circ})^{2}.80$ at 1420~MHz \citep{leah02} and $11(^{\circ})^{2}.33$ at 863~MHz \citep{reic03}. When 
their areas are recalculated into angular radii in the way we described, the result are: $1^{\circ}.77$ at 1420~MHz \citep{leah02} and $1^{\circ}.90$ 
at 863~MHz \citep{reic03}. It can be noticed that we obtained somewhat smaller values, but it has to be taken into account that other authors 
estimated only the rectangular map size while we estimated the loop size (inside its contour borders) exactly.


%%%%% TABLA 4

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{The Cygnus loop areas $\Omega$\\ and angular radii $\theta$\\ at the five frequencies}
\begin{tabular}{ccc}
\toprule
Frequency (MHz) & $\Omega$ [$(^{\circ})^2$] & $\theta$ ($^{\circ}$) \\
\midrule
2720  & 7.52 & 1.547 \\
1420  & 8.64 & 1.658 \\
~820  & 9.46 & 1.735 \\
~408  & 9.88 & 1.773 \\
~34.5 & 8.44 & 1.639 \\
\bottomrule
\end{tabular}
\label{tab04}
\vspace{0.5cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The relation between surface brightness ($\Sigma$) and diameter ($D$) for supernova remnants (SNRs) --so-called the $\Sigma-D$ relation-- 
is appropriate for the description of the radio brightness evolution of these sources. The empirical relations should be used with caution 
because of the limited usefulness due to selection effects (see Green 2009; Uro\v{s}evi\'{c} 2010, and references therein). The updated 
theoretical relations were derived by \citet{duri86}, based on the \citet{bell78a,bell78b} diffuse shock particle acceleration theory, and by Berezhko \& V\"{o}lk (2004) based on the non-linear kinetic theory of the diffuse shock acceleration mechanism. The $\Sigma-D$ diagram at 1~GHz with the theoretically derived evolutionary tracks taken from \citet{bere04} and with our value for the Cygnus loop superposed, is shown in Figure~\ref{fig10}. To superpose the Cygnus position, we used the following results of our calculations: the mean value of the brightnesses at 1~GHz and the diameter, calculated as $D=2 r \sin \bar \theta$, where $\bar \theta=1^{\circ}.67$ is the mean angular radius for all five frequencies. The distance $r=0.44$~kpc is taken from Green's (2009) catalogue of SNRs. So we added this point to the diagram: ($D$, $\Sigma$) = (25.7~pc, $4.98 \times 10^{-22}$~W/(m$^2$~Hz~Sr)). From its position in this diagram it can be concluded that the Cygnus loop evolves in the low density environment and the initial energy of the supernova (SN) explosion was relatively low (see Figure~\ref{fig10}).


%%%%% FIGURA 10

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{vborkafig10.eps}
\caption{The surface brightness versus diameter diagram from Berezhko \& V\"{o}lk 
(2004), with the value for Cygnus loop added. Three different densities for the 
interstellar matter (ISM) ($N_\mathrm{H}=3$, 0.3 and 0.003~cm$^{-3}$) are presented, 
plus three values for the (SN) explosion energy ($E_\mathrm{SN}=0.25$, 1 and 
$3 \times 10^{51}$~erg).}
\label{fig10}
\vspace{0.4cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Flux density spectrum}
\label{sec04a}

On the basis of the values for brightnesses given in the fourth column of Table \ref{tab01} and our calculated values for the loop size 
$\Omega$ (Table~\ref{tab04}), we derived flux values in Jy. The calculated flux densities are given in Table~\ref{tab05}. As the Cygnus 
loop is well studied SNR, it is possible to perform a multi-frequency spectral study. In Figure~\ref{fig11} we present a summary of some 
results obtained from other papers and from our study. Our results are labeled with asterisks. In this figure, we have seventeen results 
for the flux density values $S_{\nu}$ of Cygnus loop derived in several papers (values are given in Table~3 of \citet{uyan04} and five 
results from our paper.

It can be seen from Figure~\ref{fig11} that our added flux values fit very well among other fluxes, which shows the correctness of our method, 
and also that we determined the loop area and its brightness well.



%%%%% TABLA 5

\begin{table}[!b]
\vspace{0.3cm}
\centering
\caption{Flux densities (J\lowercase{y}) that we calculated at the five frequencies}
\setlength{\tabcolsep}{2.8\tabcolsep}
\begin{tabular}{cr@{$\pm$}l}
\toprule
Frequency (MHz) & \multicolumn{2}{c}{Flux density (Jy)} \\
\midrule
2720  & 83.37   &  2.60 \\
1420  & 80.05   &  8.13 \\
~820  & 139.35  & 12.10 \\
~408  & 235.44  & 15.36 \\
~34.5 & 1309.04 & 65.65 \\
\bottomrule
\end{tabular}
\label{tab05}
%\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conclusions}
\label{sec06}

The main result is to use a method for determining the brightness temperature given in \citet{bork07} which is developed for large radio 
loops, and to show here that this method is applicable for much smaller loops, e.g., the Cygnus loop. We check the method by applying it 
to the Cygnus radio loop over a wide range of frequencies. It is in good agreement with the method of $T-T$ graphs and with results obtained 
with other methods. This method is quite simple because we use a brightness temperature isoline to define the border of the Cygnus loop. Other 
authors are using different square or rectangular areas to determine the area of the loop and to calculate spectral indices, brightness 
temperatures and the flux density of the loop (Uyaniker et al. 2004, page 917, and Leahy \& Roger 1998, page 786). Also, we calculate 
the flux for the Cygnus loop and compare our results with those of other authors in different ranges of frequencies. We show that our 
results are in good agrement with other work.


%%%%% FIGURA 11

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{vborkafig11.eps}
\caption{Spectrum of Cygnus Loop: flux versus frequency for additional different frequencies, obtained 
from all available flux density values (values are given in Table~3 in Uyaniker et al. 2004) and from 
values calculated in this paper. We identify our values by asterisks.}
\label{fig11}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We estimate the temperatures and brightness of the Cygnus loop SNR on the basis of observations of the continuum radio emission at the 
frequencies: 2720, 1420, 820, 408 and 34.5~MHz (this paper and Borka Jovanovi\'{c} \&  Uro\v{s}evi\'{c} (2009a). The sensitivity of the 
brightness temperatures are: 5~mK for 2720~MHz, 50~mK for 1420~MHz, 0.2~K for 820~MHz, 1.0~K for 408~MHz and about 700~K~$T_\mathrm{b}$ 
for 34.5~MHz. At a frequency of 2720~MHz the measurements are the most precise (they have the smallest relative errors) so positions 
of the brightness temperature contours of the loop are the most realistic for this frequency.

The borders between the five frequencies are somewhat different for this SNR, probably due to small, random and systematic errors in the 
calibrated data. Also, we suppose that there are uncertainties of about ($2 \times \Delta T$) 10~mK for 2720~MHz, 100~mK for 1420~MHz, 0.4~K 
for 820~MHz, 2.0~K for 408~MHz and about 1400~K $T_\mathrm{b}$ for 34.5~MHz, in the border due to measurement errors, and there is a tiny 
difference in the absorption of radio emission in the interstellar medium at different wavelengths \citep{pach70}.

We determine the average brightness temperature for the Cygnus radio loop region, after subtraction of a background level. Our values 
(when all reduced to 1000~MHz for comparison) are in good agreement with the earlier results. We present the radio continuum spectrum of 
the Cygnus loop using average brightness temperatures at five different frequencies. As can be seen from Figures~\ref{fig07} and \ref{fig08}, 
the linear fit provides a reliable spectral index. We present the $T-T$ plots which also enable the calculation of spectral index.

Also, from our results it can be concluded that the Cygnus loop evolves in a low density environment and that the initial energy of SN explosion 
was relatively low. This can be read off after superposing the position of this loop onto a theoretical $\Sigma-D$ diagram from \citet{bere04}, 
which was derived from non-linear kinetic theory of diffuse shock acceleration mechanism.

We show that our method for defining a loop border and for determining the values of temperature and brightness, which was developeded for 
main Galactic loops I-VI, could be applicable to all SNRs.


\acknowledgments

The authors are grateful to the referee whose suggestions substantially improved the paper. This research is supported by the Ministry of 
Science of the Republic of Serbia through project No. 176005.

\vspace{0.2cm}

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\end{document}
