Parameter Effects on the Total Intensity of H i Ly $\alpha $ Line for a Modeled Coronal Mass Ejection and Its Driven Shock

The combination of the H i Ly\(\alpha \) (121.6 nm) line formation mechanism with ultraviolet (UV) Ly\(\alpha \) and white-light (WL) observations provides an effective method for determining the electron temperature of coronal mass ejections (CMEs). A key to ensuring the accuracy of this diagnostic technique is the precise calculation of theoretical Ly\(\alpha \) intensities. This study performs a modeled CME and its driven shock via the three-dimensional numerical magneto-hydrodynamic simulation. Then, we generate synthetic UV and WL images of the CME and shock within a few solar radii to quantify the impact of different assumptions on the theoretical Ly\(\alpha \) intensities, such as the incident intensity of the solar chromospheric Ly\(\alpha \) line (\(I_{disk}\)), the geometric scattering function (\(p(\theta )\)), and the kinetic temperature (\(T_{ \boldsymbol{n}}\)) assumed to be equal to either the proton (\(T_{p}\)) or electron (\(T_{e}\)) temperature. By comparing differences of the Ly\(\alpha \) intensities of the CME and shock under these assumptions, we find that: (1) Using the uniform or Carrington maps of the disk Ly\(\alpha \) emission underestimates the corona Ly\(\alpha \) intensity (with relative uncertainties below 10%) compared to the synchronic map, except for a slight overestimate (<4%) observed in the partial CME core. The Carrington map yields lower uncertainties than the uniform disk. (2) Neglecting the geometric scattering process has a relatively minor impact on the Ly\(\alpha \) intensity, with a maximum relative uncertainty of no more than 5%. The Ly\(\alpha \) intensity is underestimated for the most part but overestimated in the CME core. (3) Compared to the assumption \(T_{\boldsymbol{n}}=T_{p}\), using \(T_{\boldsymbol{n}}=T_{e}\) leads to more complex relative uncertainties in CME Ly\(\alpha \) intensity. The CME core and void are both overestimated, with the maximum relative uncertainty in the core exceeding 50% and in the void remaining below 35%. An appropriate increasing proton-to-electron temperature ratio can reduce the uncertainty in the CME core and void. In the CME front, both overestimates and underestimates exist with relative uncertainties of less than 35%. The electron temperature assumption has a smaller impact on the shock, with an underestimated relative uncertainty of less than 20%.