Surface-integrated quantities S_r and $S\cdot {\hat{c}}_A$ shown as functions of radius for the system with an orbital radius of 5 R_⋆. The surface integration is performed over spherical shells of varying radii centered on the star, following the approach illustrated in Figure 1 and using the formulations in Equations (13) and (14). The translucent portion of the solid red $(S\cdot {\hat{c}}_A)$ curve corresponds to integration radii where the chosen surfaces are suboptimal for evaluating the integral (see explanation in Appendix A). Consequently, the solid portion of the red curve highlights the range over which the integrated quantity is considered most reliable. The dotted and dashed black horizontal lines represent the corresponding analytical predictions from the Saur and Lanza models, as given by Equations (9) and (10), respectively.