Due to gravitational attraction, the earth revolves round the sun. Assume that the sun and the earth are point particles with their existing masses, and that Bohr's quantization rule for angular momentum holds true for gravitation. Determine the minimum radius the earth can have for its orbit. (Given: mass of the earth = 6.0 10²⁴ kg , mass of the sun = 2.0 10³⁰ kg , earth–sun distance = 1.5 10¹¹ m ).

Step-by-step solution

According to Bohr's quantization rule, the angular momentum of the earth is quantized as: M_e v r = nh 2 The gravitational force provides the necessary centripetal force for the earth's orbit: G M_s M_e r^2 = M_e v^2 r v = G M_s r Substituting the expression for v into the quantization rule: M_e G M_s r r = nh 2 M_e G M_s r = nh 2 Squaring both sides, we get the radius of the n -th orbit: r = n^2 h^2 4 ^2 G M_s M_e^2 For the minimum radius, n = 1 . Substituting the standard values ( h = 6.63 10⁻³⁴ J s , G = 6.67 10