Consider the earth-sun system as a gravitational analog to the hydrogen atom. You may need to know that the mass of the earth is m = 5.97219 × 1024 kg, the mass of the sun is the distance between the sun and the earth is ework 08 M = 1.98847 × 1030 kg, R = 1.49598 × 1011 m, 1 2 and the force and potential energy distribution that the earth experiences due to the gravitational field of the sun are GmM F 7.2 GmM U(r) = - == r where G is the gravitation constant G = 6.67428 × 10-11 N m²/kg². [Some standard calculators have issues with the numbers in this problem. If you get overflow or underflow errors, try using MATLAB] (a) Using the fact that the centripetal force must have magnitude mv2 F = r what is the centripetal velocity of the earth at an arbitrary radius r? (b) Using the centripetal velocity, find the total energy of this system as E=T+U 1 = √mv² + U(r). (c) Use Bohr's assertion that mvr = nh, to find the radii rn as a function of the principle quantum number n for the earth-sun system. Compute the numerical value for the Bohr radius ag = 11. (d) From the results to part (c), show that T n = ag and estimate the n for the earth, which has radius R given above.

Consider the earth-sun system as a gravitational analog to the hydrogen atom. You may need to know that the mass of the earth is m = 5.97219 × 1024 kg, the mass of the sun is the distance between the sun and the earth is ework 08 M = 1.98847 × 1030 kg, R = 1.49598 × 1011 m, 1 2 and the force and potential energy distribution that the earth experiences due to the gravitational field of the sun are GmM F 7.2 GmM U(r) = - == r where G is the gravitation constant G = 6.67428 × 10-11 N m²/kg². [Some standard calculators have issues with the numbers in this problem. If you get overflow or underflow errors, try using MATLAB] (a) Using the fact that the centripetal force must have magnitude mv2 F = r what is the centripetal velocity of the earth at an arbitrary radius r? (b) Using the centripetal velocity, find the total energy of this system as E=T+U 1 = √mv² + U(r). (c) Use Bohr's assertion that mvr = nh, to find the radii rn as a function of the principle quantum number n for the earth-sun system. Compute the numerical value for the Bohr radius ag = 11. (d) From the results to part (c), show that T n = ag and estimate the n for the earth, which has radius R given above.