This paper investigates the number of scatterings a photon undergoes in random walks before escaping from a medium. The number of scatterings in random walk processes is commonly approximated as τ + τ 2 in the literature, where τ is the optical thickness measured from the center of the medium. However, it is found that this formula is not accurate. In this study, analytical solutions in sphere and slab geometries are derived for both optically thin and optically thick limits, assuming isotropic scattering. These solutions are verified using Monte Carlo simulations. In the optically thick limit, the number of scatterings is found to be 0.5 τ 2 and 1.5 τ 2 in a sphere and slab, respectively. In the optically thin limit, the number of scatterings is ≈ τ in a sphere and ≈ τ (1 − γ − ln τ + τ ) in a slab, where γ ≃ 0.57722 is the Euler-Mascheroni constant. Additionally, we present approximate formulas that reasonably reproduce the simulation results well in intermediate optical depths. These results are applicable to scattering processes that exhibit forward and backward symmetry, including both isotropic and Thomson scattering.

Introduction
Analytical Approximations
    Sphere
    Slab
Monte Carlo Simulation
    Simulation Methods
    Comparison with the Analytic Approximations
Summary and Discussion
Acknowledgments
References

• Kwang-Il Seon(Korea Astronomy & Space Science Institute, Daejeon 34055, Republic of Korea, Astronomy and Space Science Major, University of Science and Technology, Daejeon 34113, Republic of Korea) Corresponding author
• Hyung-Joe Kwon(Korea Astronomy & Space Science Institute, Daejeon 34055, Republic of Korea, Astronomy and Space Science Major, University of Science and Technology, Daejeon 34113, Republic of Korea)
• Hee-Gyeong Kim(Korea Astronomy & Space Science Institute, Daejeon 34055, Republic of Korea, Astronomy and Space Science Major, University of Science and Technology, Daejeon 34113, Republic of Korea)
• Hyeon Jeong Youn(Korea Astronomy & Space Science Institute, Daejeon 34055, Republic of Korea, Astronomy and Space Science Major, University of Science and Technology, Daejeon 34113, Republic of Korea)