# Solar Radio Emissions and Ultralight Dark Matter
## 4 Propagation of the Converted Photons  
After the radio photons are converted from axions or dark photons, they will be refracted, absorbed, and scattered by thermal electrons in the plasma. We discuss these effects during the photon propagation in the following. **Refraction effect**--The behaviour of converted photons propagating in the solar plasma obeys the rule of refraction:  
\[\frac{n(r_{c})}{n(r)}=\frac{\sin\theta(r)}{\sin\theta(r_{c})}. \tag{22}\]  
where \(\theta(r)\) is the angle of incidence or refraction with respect to the radial direction. \(n(r)\) is the refraction index as a function of distance, which is equal to the inverse of phase velocity:  
\[n(r)=\frac{k}{\omega}=\sqrt{1-\frac{\omega_{p}^{2}(r)}{\omega^{2}}}. \tag{23}\]At the position of resonant conversion, \(n(r_{c})\simeq vs.\sim 10^{-3}\). Once the converted photons leave the resonant layer, the refraction index \(n(r)\) quickly increases to values close to 1 as the number density of electrons \(n_{e}\) (and thus the plasma frequency, according to Equation (5)) abruptly decreases with the distance. For example, as we can see from Figure 1a, \(n_{e}\) drops by about two orders of magnitude from the height \(10^{4}\) km to the height \(10^{6}\) km. Therefore, the converted photons will quickly propagate nearly radially \(\theta\simeq\pi/2\) after travelling a distance based on the refraction law (Equation (22)). **Absorption effect**--In addition to the refraction effect discussed above, the converted photons could be absorbed by interacting with the plasma along the propagation. The absorption effect is mainly due to the inverse bremsstrahlung process, the rate of which is estimated as [67; 72]  
\[\Gamma_{\rm invB}\simeq\frac{8\pi\alpha^{3}n_{e}n_{I}}{3m_{e}^{2}\omega^{3}} \sqrt{\frac{2\pi m_{e}}{T}}\ln\left(\frac{2T^{2}}{\omega_{p}^{2}}\right)\left( 1-\mathrm{e}^{-\omega/T}\right). \tag{24}\]  
where \(n_{I}\) is the number density of ions in the plasma. This equation is derived under the condition \(\omega\ll T\ll m_{e}\), which is satisfied in both the solar corona and solar wind plasma, as can be seen in Figure 1. Taking into consideration the long-range Coulomb force cut off by the plasma screening effect ([PERSON] screening length \(\lambda_{d}\sim\sqrt{T/(m_{e}\omega_{p}^{2})}\)), we have the ln factor in Equation (24). At first glance, it seems that the inverse bremsstrahlung rate \(\sim\omega^{-3}\) will diverge for low-energy photons, but the numerator factor \(n_{e}n_{I}\simeq n_{e}^{2}\propto\omega_{p}^{4}\propto\omega^{4}\) at the resonant layer actually makes \(\Gamma_{\rm invB}\) decease quickly as \(\omega\to 0\). **Scattering effect**--The converted photons also interact with plasma via Compton scattering, the rate of which is  
\[\Gamma_{\rm Com}=\frac{8\pi\alpha^{2}}{3m_{e}^{2}}n_{e}. \tag{25}\]  
Such interactions can shift the photon energy and propagation direction. We will elaborate more on this effect in the following. To estimate how the converted photons are affected by the effects of absorption and scattering, one can define an attenuation factor by simply adding the two effects together, \(\Gamma_{\rm att}=\Gamma_{\rm invB}+\Gamma_{\rm Com}\)[67]. Then, the survival probability of photons after conversion can be estimated as [67]  
\[P_{\rm sur}\simeq\exp\left(-\int_{r_{c}}^{\infty}[\Gamma_{\rm att}(r)]/v_{r} dr\right). \tag{26}\]  
It denotes the probability that a converted photon is neither absorbed nor scattered during propagation. Compared with the inverse bremsstrahlung process (Equation (24)), the Compton scattering plays a minor role. This can be seen by taking the ratio of Equation (25) to Equation (24) which gives \(\sim\)\((m_{e}/T)^{-3/2}\ll 1\). In addition, the propagating photons can scatter with random electron density fluctuations (e.g., [73]), which is labelled as 'irregular refractions' in [74]. Ref. [73] treated the density fluctuations as effectively static, and therefore the scatterings do not change the photon energy but only the photon propagating direction.