I have trouble with an exercise that I need help with. Here it is:
The magnetic field lines for the average interplanetary magnetic field (IMF) follow Archimedean spirals.
i) Find the heliocentric distance r in Astronomical Units (AU), where a field line has wrapped itself around the Sun once. Assume that the solar wind speed is Usw = 400 km/s.
ii) Also determine the number of times the magnetic field has wound around the Sun by a heliocentric distance of 100 AU.
The Attempt at a Solution
i) The plasma flows radially from the Sun as I've understood it. Therefore, the flow of the plasma shall have a constant radial velocity Usw with respect to the heliocentric distance.
We have to look at the velocities in polar coordinates I believe, so vφ = −ΩR in the φ-direction and Vr=Usw. Here, Ω is the angular rotation rate of the Sun at its equator and R is the radial distance from the Sun.
Further, an equation for Archimedian sprials is: R − R⊙ = − VSW Ω (φ − φ0) , where R⊙ is the solar radius and φ0 is the angular position of a particular magnetic field line at the surface of the Sun.
Now, how should I apply this to the problem? It is said that the field line has wrapped itself around the Sun once, is that related to the angular rate (Ω maybe)?
ii) I actually have no idea here. I think that we need to consider that it has a circular move, so something with the angular velocity I guess, where the radius is involved maybe.
Hope that you can help me