2. The earth-sun system can be considered as a gravitational analog of the hydrogen atom. The potential energy function (let m be the mass of the earth and M the mass of the sun) replacing: V(r) = e² 1 Απεργ GmM V(r) = r (a) What is the "Bohr radius", ag, for the system? Work out the actual number. (b) Write down the gravitational “Bohr formula” and, by equating En to the classical energy of a planet in a circular orbit of radius ro. show that: n = √ro/ag (c) Hence estimate the quantum number, n, of the earth. The earth/sun distance is Recall: a= ħ² (4π€0) me² = 0.529 × 10-10m 11 To=149 x 106km G = 6.67 × 10 ¹¹m³kg¯¹s¯ -2 M = 1.99 × 1030 kg m = 5.97 × 1024 kg

2. The earth-sun system can be considered as a gravitational analog of the hydrogen atom. The potential energy function (let m be the mass of the earth and M the mass of the sun) replacing: V(r) = e² 1 Απεργ GmM V(r) = r (a) What is the "Bohr radius", ag, for the system? Work out the actual number. (b) Write down the gravitational “Bohr formula” and, by equating En to the classical energy of a planet in a circular orbit of radius ro. show that: n = √ro/ag (c) Hence estimate the quantum number, n, of the earth. The earth/sun distance is Recall: a= ħ² (4π€0) me² = 0.529 × 10-10m 11 To=149 x 106km G = 6.67 × 10 ¹¹m³kg¯¹s¯ -2 M = 1.99 × 1030 kg m = 5.97 × 1024 kg