Parker Spiral

What is modelled

The magnetic field is frozen into the solar wind, and the wind leaves the Sun nearly straight out at a roughly constant speed. The foot of each field line is fixed on the turning Sun, so while a bit of wind travels outwards its source rotates ahead. The field line through all the wind that left the same spot is therefore an Archimedean spiral, wound more tightly the farther out and the slower the wind. Because the radial flux is conserved the radial field falls as the inverse square of distance, while the sideways part falls only as the inverse of distance.

Equations

• Winding angle tan ψ = Ω r / v, about 45 degrees for 400 km/s at 1 AU.
• Fields Br = B1 (1 AU / r)2, Bφ = − Br Ω r / v, |B| = Br √(1 + tan2ψ).
• Footpoint longitude offset Δφ = Ω r / v = rotation rate times travel time.
• Field line length (v / 2Ω) [s √(1 + s2) + asinh s], s = Ω r / v.
• A fast stream (speed v₂) leaving the same longitude one rotation P after a slow one (v₁) catches it at r = v1 v2 P / (v2 − v1).

Checks run in the page

• Winding angle measured on a curve built from released plasma against the Parker formula.
• Field line length by adding short steps against the closed form.
• Field-line integration inwards against the rotation rate times travel time.
• Step-by-step overtaking of a slow parcel by a fast one against the collision formula.

Honest simplifications

The wind speed is constant and the same everywhere, the Sun turns as a solid body, the picture is the solar equatorial plane only, and turbulence, the finite size of the source and the pile-up of fast and slow streams are left out. The measured field at Earth varies a lot around the value given here.

• Press Slow wind and watch the spiral tighten.
• Press Fast wind to open the spiral up.
• Press Out at Jupiter and read the winding angle far from the Sun.