# Solar Radio Emissions and Ultralight Dark Matter
## 3 Conversion in Solar Plasma  
The equations of motions corresponding to the Lagrangians (1) and (4) for the axion and dark photon cases can both be written in the form [71]  
\[\left(\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial r^{2}} +\tilde{M}\right) \begin{pmatrix}A\\ X\end{pmatrix} =0,\ \ \ \tilde{M}= \begin{pmatrix}\tilde{\Delta}_{AA}&\tilde{ \Delta}_{AX}\\ \tilde{\Delta}_{AX}&\tilde{\Delta}_{XX}\end{pmatrix} . \tag{6}\]  
where \(X\) denotes either axion \(a\) or dark photon \(A^{\prime}\). For the axion case, we have  
\[\tilde{\Delta}_{AA}=m^{2}_{A}=\omega_{p}^{2},\ \ \ \tilde{\Delta}_{XX}=m^{2}_{a},\ \ \ \tilde{\Delta}_{AX}=-g_{\alpha\gamma\gamma}|B_{\gamma}|\cdot\omega, \tag{7}\]  
and for the dark photon case, we have  
\[\tilde{\Delta}_{AA}=m^{2}_{A}=\omega_{p}^{2},\ \ \ \tilde{\Delta}_{XX}=m^{2}_{A^{ \prime}},\ \ \ \tilde{\Delta}_{AX}=-\epsilon m^{2}_{A^{\prime}}. \tag{8}\]  
Since the external environment in the solar plasma mostly depends on the radial direction \(r\) (this is true for the dominant term \(\Delta_{AA}\) which is determined by electron density according to Equation (5)), we only keep the spatial derivative in the radial direction in the equation of motion (Equation (6)) while ignoring trivial dynamics in the other two spatial dimensions. Some differences between the axion case and the dark photon case are as follows. In comparison with the dark photon case, the conversion between axions and photons requiresthe presence of a magnetic field \(B_{T}\) transverse to the axion trajectory. The solar magnetic field provides the source of \(B_{T}\). Due to the pseudoscalar feature of axions, the converted photons \(A\) in Equation (6) is polarized in the direction of \(B_{T}\)[56; 71]. For the dark photon case, Equation (6) describes converted photons in two transverse modes perpendicular to the propagation trajectory. We assume the solutions to the equation of motion (Equation (6)) are in the wave form:  
\[A(t,r)=\tilde{A}(r)\exp(-i\omega t+ik_{r}r),\ \ \ X(t,r)=\tilde{X}(r)\exp(-i \omega t+ik_{r}r) \tag{9}\]  
where \(\omega\) is the energy of the wave, \(k=\sqrt{\omega^{2}-m_{X}^{2}}\) is the momentum, and \(k_{r}\) is the momentum component in the radial direction. For resonant conversion to happen, we should have \(m_{A}=m_{X}\) such that both \(\omega\) and \(k\) of \(A\) and \(X\) match before and after the conversion. In the case we consider here where the dark matter velocity is non-relativistic \(v\sim\)\(10^{-3}c\), we have \(\omega\simeq m_{X}\). Then, near the resonant layer in solar plasma, we have the following relation [71]:  
\[ \begin{split}\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{ 2}}{\partial r^{2}}&=-\omega^{2}-\frac{\partial^{2}}{\partial r ^{2}}=-\left(k_{r}+i\frac{\partial}{\partial r}\right)\left(k_{r}-i\frac{ \partial}{\partial r}\right)-m_{X}^{2}-k_{T}^{2}\\ &\simeq-2k_{r}\left(k_{r}+i\frac{\partial}{\partial r}\right)-m_ {X}^{2}-k_{T}^{2}.\end{split}  \tag{10}\]  
The last approximated relation holds because the parameters of solar plasma, e.g., plasma frequency, vary slowly in a length scale much larger than the wavelength \(1/k_{r}\) such that  
\[\left|\frac{\partial}{\partial r}\tilde{A}(r)\right|\ll k_{r}|\tilde{A}(r)|, \ \ \ \left|\frac{\partial^{2}}{\partial r^{2}}\tilde{A}(r)\right|\ll k_{r}\left| \frac{\partial}{\partial r}\tilde{A}(r)\right|. \tag{11}\]  
Therefore, using the relation in Equation (10), the equation of motion (Equation (6)) can be simplified to  
\[\left(i\frac{\partial}{\partial r}-M\right) \begin{pmatrix}\tilde{A}(r)\\ \tilde{X}(r)\end{pmatrix} =0 \tag{12}\]  
where  
\[ \begin{split}& M=\frac{1}{2k_{r}}(\tilde{M}-m_{X}^{2})\equiv M_ {0}+M_{1},\\ & M_{0}=\begin{pmatrix}\frac{1}{2k_{r}}(\omega_{p}^{2}-m_{X}^{2}- k_{T}^{2})&0\\ 0&-\frac{1}{2k_{r}}k_{T}^{2}\end{pmatrix},\ \ \ M_{1}=\begin{pmatrix}0&\Delta_{AX}\\ \Delta_{AX}&0\end{pmatrix},\ \ \ \Delta_{AX}\equiv\frac{\tilde{\Delta}_{AX}}{2k_{r}}. \end{split}