import numpy as np


def model_density_distribution(r_bins: np.ndarray, M: float = 5, a: float = 5) -> np.ndarray:
    """
    Generate a density distribution for a spherical galaxy model, as per the Hernquist model.
    Parameters:
    - r_bins: The radial bins to calculate the density distribution for.
    - M: The total mass of the galaxy.
    - a: The scale radius of the galaxy.
    See https://doi.org/10.1086%2F168845 for more information.
    """
    rho = M / (2 * np.pi) * a / (r_bins * (r_bins + a)**3)
    return rho


def model_particle_count(r_bins: np.ndarray, M: float = 5, a: float = 5, mi: float = 1) -> np.ndarray:
    rho = model_density_distribution(r_bins, M, a)
    v_shells = 4/3 * np.pi * (r_bins[1:]**3 - r_bins[:-1]**3)
    v_shells = np.insert(v_shells, 0, 4/3 * np.pi * r_bins[0]**3)
    n_shells = rho * v_shells / mi
    return n_shells