Resonant angle of the libration centers (σ_k) for L₄ (top) and L₅ (bottom) of Venus from different solutions. The positions of the libration centers for L₃ and L₁ obtained from the numerical solutions to the equations of motion are also presented in the upper and lower panels, respectively. “SIMP” represents the results calculated from the simplified approximation formulae Eqs. (20) and (21). The results from approximation formulae for L₅ are only shown for 0 < β < 0.01 (we insert a window to zoom in) because their deviations from the numerical solutions are too large for β > 0.01.