In this problem you will determine whether the magnetic field in the Sun might affect the loss of angular momentum from the Sun. For this question you may assume that the solar wind is measured to have a density of np~10cm-3 and a velocity of v = 4001cm/sec at the Earth.
(i) Calculate the ratio of angular momentum of the Sun to Mat of Jupiter. Consider both sources of angular momentum: orbital and rotational, for each body.
(ii) Draw up a table with the orbital and rotational angular momentum for each of the other planets. Do any of the other planets have comparable angular momentum to Jupiter and the Sun?
(iii) Re-calculate the total angular momentum loss from the Sun, assuming the Sun loses all its angular momentum through the solar wind. How long would it take for the Sun to lose all its angular momentum? What assumptions have you made? Comment on your answer. You can assume the age of the Sun is 5x109 ears.
(iv) The magnetic field of the Sun is rooted in the dense interior of the Sun, and co-rotates with it. Since the particles in the solar wind are charged, they are compelled to move along the magnetic field lines and will (approximately) co-rotate with the Sun. Flow will angular momentum of these particles change as they move away the Sun? Qualitatively, how will this affect the angular momentum calculated in (iii)?
(v) The kinetic energy density of the solar wind is given by ½ρv2. How will this quantity change as the wind travels away from the Sun? You can assume that the change in the gravitational field has a negligible effect on the velocity of the wind.
(vi) At some point, the solar wind must de-couple from the Sun's magnetic field, and continue flowing along radial lines from the Sun. This occurs at the Alfven point or radius, where the magnetic energy density is equal to the kinetic energy density of the solar wind. Outside this radius, the angular momentum of each particle remains constant Write down the equation for magnetic energy density. How does this quantity change with radius, assuming that the magnetic B field is purely radial? - be careful to get the right geometry for the magnetic field.
(vii) Write down the equation for the total kinetic energy density in the solar wind in terms of the mass loss from the Sun Me, and the velocity v of the wind.
(viii) Equate the terms in (v) and (vi) to determine an expression for the Alfven radius. Assuming a magnetic field strength of ~10-3 T for the Sun, calculate the Alfven radius.