# Solar Radio Emissions and Ultralight Dark Matter
## 5 Detection
We average the signal over the bandwidth and obtain the signal per unit of frequency as shown in Equation (34). \(\mathcal{B}\) should be chosen as the larger one between the telescope resolution \(\mathcal{B}_{\rm res}\) and the natural bandwidth \(\mathcal{B}_{\rm DM}\). Due to the velocity dispersion of dark matter, the converted photons have a natural bandwidth of frequency [67; 82]:  
\[\mathcal{B}_{\rm DM}\simeq\frac{1}{2\pi}m_{X}v_{p}^{2}\simeq 0.15\;{\rm kHz} \times\Big{(}\frac{m_{X}}{10^{-6}\;{\rm eV}}\Big{)}. \tag{35}\]  
On the other hand, we need to know the telescope's ability to detect weak signals. The minimal detectable spectral flux density of a telescope is [67; 83]  
\[S_{\rm min}=\frac{\rm SEFD}{\eta_{s}\sqrt{n_{\rm pos}\mathcal{B}t_{\rm obs}}}. \tag{36}\]  
where \(\eta_{s}\) is the system efficiency of a telescope, which is 0.9 and 1 for SKA and LOFAR, respectively, [83; 84]; \(n_{\rm pol}=2\) is the number of polarizations of photons; and \(t_{\rm obs}\) is the observation time. SEFD is the system equivalent (spectral) flux density:  
\[v_{\theta}=v_{0}\frac{b}{r},\ \ v_{r}=\sqrt{v_{0}^{2}-v_{\theta}^{2}+\frac{2 GM_{ \odot}}{r}},\ \ \ v^{2}=v_{r}^{2}+v_{\theta}^{2}. \tag{28}\]0  
where \(T^{\rm sys}\) is the system temperature of the radio telescope antenna; \(T^{\rm nos}_{\odot}\) is the noise temperature induced on the antenna when pointing towards the Sun, which can be estimated as the black-body temperature of the quiet Sun [67]; and \(A_{\rm eff}\) is the effective area of the radio telescope array such as SKA and LOFAR. Then, taking the signal (Equation (34)) equal to the telescope detection ability (Equation (36)),  
\[v_{\theta}=v_{0}\frac{b}{r},\ \ v_{r}=\sqrt{v_{0}^{2}-v_{\theta}^{2}+\frac{2 GM_{ \odot}}{r}},\ \ \ v^{2}=v_{r}^{2}+v_{\theta}^{2}. \tag{28}\]1  
one can find the upper limit of the coupling \(g_{\sigma\gamma\gamma}\) for the axion case or \(\epsilon\) for the dark photon case that can be detected by the telescope. Equation (38) reflects how large the parameter space of dark photon/axion dark matter can be explored via observing solar radio emissions. Ref. [67] used the SKA phase 1 (SKA1) and LOFAR as benchmark telescopes. To be specific, the following frequency ranges have been considered: the low-frequency band, SKA1-Low (50-350 MHz); two middle-frequency bands, SKA1-Mid B1 (350-1050 MHz) and B2 (950-1760 MHz); and two LOFAR frequency bands, (10-80 MHz) (low-band antenna) and (120-240 MHz) (high-band antenna). Note that LOFAR does not operate in the gap frequency range (80-120 MHz) due to the contamination from the human-made frequency-modulation (FM) broadcast band that spans over \(\sim\)(87-108 MHz) [47; 85]. The effective area \(A_{\rm eff}\) for SKA1 is about \(10^{4}\)-\(10^{5}\) m\({}^{2}\), while it is about \(10^{3}\) m\({}^{2}\) for LOFAR. The antenna system temperatures \(T^{579}\) are also given in [67] based on the data of [47; 83], ranging from \(\sim\)10 to \(10^{4}\) K for the five frequency bands. Lastly, the telescope resolution \(\mathcal{B}_{\rm res}\) for SKA1-Low, SKA1-Mid and LOFAR are, 1, 3.5 and 197 kHz, respectively. All of them are larger than the natural bandwidth of dark matter (Equation (35)) for the radio frequency bands we are interested in. Therefore, one should choose \(\mathcal{B}=\mathcal{B}_{\rm res}\) when computing the received signal (Equation (34)). With the above properties of the radio telescopes known, the sensitivity for the dark photon case based on Equation (38) is plotted in \[v_{\theta}=v_{0}\frac{b}{r},\ \ v_{r}=\sqrt{v_{0}^{2}-v_{\theta}^{2}+\frac{2 GM_{ \odot}}{r}},\ \ \ v^{2}=v_{r}^{2}+v_{\theta}^{2}. \tag{28}\]2 The observation time is assumed to be 1 and 100 h, respectively. As shown in the plot, the sensitivity of the kinetic mixing parameter \(\epsilon\) based on the SKA and LOFAR radio telescopes could be several orders of magnitude better than the existing constraints from the cosmic microwave background (CMB) [8; 86] and the cavity experiment WISPDMX [87].