“…There are several widely used, analytical distribution functions of this kind (e.g., Chen et al, 2018; Horne et al, 2000; Liu et al, 2011). Here, consistent with our previous studies (Min, Boardsen, et al, 2018; Min, Liu, Denton, & Boardsen, 2018; Min et al, 2019; Min, Liu, Wang, et al, 2018), we use the partial shell velocity distribution given by where is the velocity modulus; α is the pitch angle; A is the effective temperature anisotropy, ; v s , and θ s are the ring (or shell) speed and the thermal spread of the shell, respectively; n s , eq is the number density, and C ( x ) is the normalization constant given by …”