# Solar Radio Emissions and Ultralight Dark Matter
## 5 Detection  
For dark matter particles with typical initial velocity \(v_{0}\simeq\) 220 km/s, the solar gravitational effect cannot be neglected when they travel close to the Sun since \(GM_{\odot}/(R_{\odot}v_{0}^{2})\simeq\) 3.45 is an \(\mathcal{O}(1)\) number. The trajectory of a dark matter particle from far away is hyperbolic in the solar gravitational potential. In the polar coordinates \((r,\theta)\) centred at the Sun, the dark matter trajectory can be expressed as  
\[r=\frac{v_{0}^{2}b^{2}}{GM_{\odot}}\frac{1}{1+e\cos\theta},\ \ \ e=\sqrt{1+b^{2} \Bigg{(}\frac{v_{0}^{2}}{GM_{\odot}}\Bigg{)}^{2}} \tag{27}\]  
where \(b\) is the impact factor and \(e\) is the eccentricity of the trajectory. The tangential velocity \(v_{\theta}\) and radial velocity \(v_{r}\) of dark matter are, respectively  
\[v_{\theta}=v_{0}\frac{b}{r},\ \ v_{r}=\sqrt{v_{0}^{2}-v_{\theta}^{2}+\frac{2 GM_{ \odot}}{r}},\ \ \ v^{2}=v_{r}^{2}+v_{\theta}^{2}. \tag{28}\]  
The dark matter velocity \(v_{0}\) follows the Maxwellian distribution in the Galactic frame [79; 80; 81]:  
\[f_{G}(v_{0})=\frac{4}{\sqrt{\pi}}\frac{v_{0}^{2}}{v_{p}^{3}}\exp\Bigg{(}- \frac{v_{0}^{2}}{v_{p}^{2}}\Bigg{)}, \tag{29}\]  
and we have \(\int_{0}^{\infty}dv_{0}f_{G}(v_{0})=1\). \(v_{p}\) is the most probable speed, which is taken as the speed of local standard of rest, i.e., the circular speed about the Galactic centre at the solar position, \(v_{p}\simeq v_{\odot}\simeq\) 220 km/s. The distribution in Equation (29) should be truncated at \(v_{\rm esp}\) which is the Galactic escape speed at the solar position, \(v_{\rm esp}\simeq\) 544 km/s. However, this modification is exponentially tiny as \(\int_{v_{\rm esp}}^{\infty}dv_{0}f_{G}(v_{0})\approx\) 0.66%, which can be safely ignored. Next, we make a Galilean boost to obtain the velocity distribution in the rest frame of the Sun (e.g., [80]):  
\[f(v_{0})=\frac{1}{\sqrt{\pi}}\frac{v_{0}}{v_{p}v_{\odot}}\Bigg{\{}\exp\Bigg{[} -\frac{(v_{0}-v_{\odot})^{2}}{v_{p}^{2}}\Bigg{]}-\exp\Bigg{[}-\frac{(v_{0}+v_{ \odot})^{2}}{v_{p}^{2}}\Bigg{]}\Bigg{\}}. \tag{30}\]  
Dark photons or axions in the dark matter wind can reach the resonant-conversion layer \(r_{c}\) if their impact parameters \(b\) are within  
\[b_{\rm max}=r_{c}\frac{v(r_{c})}{v_{0}}=r_{c}\sqrt{1+\frac{2 GM_{\odot}}{v_{0}^ {2}r_{c}}}. \tag{31}\]Then, the total power of the converted photons per unit of solid angle flowing out of the sphere \(r_{c}\) is [67]  
\[\frac{d\mathcal{P}_{0}}{d\Omega}=2\times\frac{1}{4\pi}\rho_{\rm DM}v_{0}\cdot \int_{0}^{b_{\rm max}}db2\pi b\cdot P_{X\rightarrow\gamma}=r_{c}^{2}P_{X \rightarrow\gamma}(v_{0})\rho_{\rm DM}v(r_{c}). \tag{32}\]  
The prefactor 2 accounts for the fact that the dark matter particles entering and leaving the resonant sphere can both resonantly convert into photons with equal probability. The converted photons propagating inward will be totally reflected as the plasma frequency is larger in deeper regions. Then, with the Maxwellian velocity distribution (Equation (30)) taken into consideration, the emitted power (Equation (32)) is corrected as [78]  
\[\frac{d\mathcal{P}}{d\Omega}=\int_{0}^{\infty}dv_{0}f(v_{0})\frac{d\mathcal{P }_{0}}{d\Omega}. \tag{33}\]  
The converted photons will generate a spike-like feature superimposed on the normal solar radio spectrum. Such signals can be detected by radio telescopes, such as SKA and LOFAR, when they reach Earth. For Earth-based radio telescopes, the spectral flux density (in \(\rm W/m^{2}/Hz\)) received is thus [67; 78]  
\[S_{\rm sig}=\frac{1}{d^{2}}\frac{1}{\mathcal{B}}\frac{d\mathcal{P}}{d\Omega} \cdot P_{\rm sur}\cdot\beta[g(\theta)]. \tag{34}\]  
The factor \(\beta\) represents the suppression on the final detected signal due to the scattering effect. It depends on the angular distribution \(g(\theta)\) that requires a detailed ray-tracing numerical simulation [78]. If there is no scattering effect, then \(\beta=1\). \(\mathcal{B}\) is the frequency bandwidth over which the signal distributes.