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$\textit{Herschel}$ discovery of far-infrared emission from the hot spot D in the radio galaxy Cygnus A

The far infrared counterpart of hot spot D, the terminal hot spot of the eastern jet hosted by the radio galaxy Cygnus A, is detected with \textit{Herschel} Aperture photometery of the source performed in 5 photometric bands covering the wavelength range of $70\unicode{x2013}350$ $\mathrm{μm}$. After removing the contamination from another nearby hot spot, E, the far-infrared intensity of hot spot D is derived as $83\pm13$ and $269\pm66$ mJy at $160$ and $350$ $\mathrm{μm}$, respectively. Since the far-infrared spectrum of the object smoothly connects to the radio one, the far-infrared emission is attributed to the synchrotron radiation from the radio-emitting electron population. The radio-to-near-infrared spectrum is confirmed to exhibit a far-infrared break feature at the frequency of $ν_\mathrm{br}=2.0^{+1.2}_{-0.8} \times10^{12}$ Hz. The change in energy index at the break ($Δα=0.5$) is interpreted as the impact of radiative cooling on an electron distribution sustained by continuous injection from diffusive shock acceleration. By ascribing the derived break to this cooling break, the magnetic field, $B$, in the hot spot is determined as a function of its radius, $R$ within a uniform one-zone model combined with the strong relativistic shock condition. An independent $B$-$R$ constraint is obtained by assuming the X-ray spectrum is wholly due to synchrotron-self-Compton emission. By combining these conditions, the two parameters are tightly determined as $B=120\unicode{x2013}150$ $\mathrm{μG}$ and $R=1.3\unicode{x2013}1.6$ kpc. A further investigation into the two conditions indicates the observed X-ray flux is highly dominated by the synchrotron-self-Compton emission.