For bound orbits, the radial turning points of the orbit is characterized by specific energy E = 1 2 v 2 + Φ Φ ( r ) and specific angular momentum L = | r → → × × v → → | are given by the positive roots of the cubic equation R 3 + G M 0 E R 2 − − ( L 2 2 E + a 2 ) R − − G M 0 a 2 E = 0 , where R = r 2 + a 2 , so that r = R 2 − − a 2 . This equation has three real roots for R : two positive and one negative, given that L < L c ( E ) , where L c ( E ) is the specific angular momentum for a circular orbit for the same energy. Here L c can be calculated from single real root of the discriminant of the cubic equation, which is itself another cubic equation E _ _ L _ _ c 3 + ( 6 E _ _ 2 a _ _ 2 + 1 2 ) L _ _ c 2 + ( 12 E _ _ 3 a _ _ 4 + 20 E _ _ a _ _ 2 ) L _ _ c + ( 8 E _ _ 4 a _ _ 6 − − 16 E _ _ 2 a _ _ 4 + 8 a _ _ 2 ) = 0 , where underlined parameters are dimensionless in Henon units defined as E _ _ = E r V / ( G M 0 ) , L _ _ c = L c / G M r V , and a _ _ = a / r V = 3 π π / 16 .