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# Solar Wind: Parker Spiral Model

This module allows you to examine the dependence of the Parker spiral model on parameters such as velocity.

## Solar Wind

The solar wind is the expanding outer atmosphere of the Sun that fills interplanetary space. It is a primarily hydrogen plasma that flows approximately radially from the region in the chromosphere or corona where it is accelerated. Typical solar wind speeds are around 400 km/s, although speeds as low as 250 km/s and over 1000 km/s have been observed. The speed does not vary with heliocentric distance r but the density of particles falls off as 1/r2 due to the radial expansion. The solar wind at 1 AU has a density of about 10 particles per cm3.

The existence of the solar wind was predicted by E.N. Parker in the late 1950s. Parker also predicted that the highly-conducting solar-wind plasma would carry the Sun’s magnetic field with it into interplanetary space. Parker showed that this interplanetary magnetic field (IMF) should have a spiral geometry due to the combination of radial outflow and solar rotation. His model for the IMF has since been called the Parker spiral model.

## Parker Spiral Model

The Parker spiral concept is based on the idea that the solar wind stretches the solar magnetic field out into space. It would be essentially radial in the absence of solar rotation. However, because the Sun is rotating (at a speed of about once a month as seen from Earth) the field winds up into the shape of an Archimedean spiral. In the solar equatorial plane, the angle a that the IMF makes with respect to the radial is given by

a = arctan(rω / V)

where ω is the angular rotation rate of the Sun and V is the solar wind speed. This angle is known as the “garden hose” angle because of the similarity in appearance of the IMF lines and the pattern of water jets produced by a rotating garden sprinkler. At 1 AU the average garden hose angle is about 45°.

## Graph Options

The graph in the upper-right corner as well as the two lower graphs are fixed, but you can choose which of two graphs to see displayed in the upper-left corner:

• Plasma Motion Animation - xy-Plane

When you chose this option, the upper-left graph shows elements of plasma emitted radially from the Sun from two regions on opposite sides of the Sun. As the Sun rotates (about once every 27 days) the radially moving plasma elements line up in a spiral pattern. If these radially moving plasma elements carried the magnetic field of the “active” regions with them, the magnetic field lines would form spirals as on the upper-right graph.

• Magnetic Field Lines - xz-Plane

When you chose this option, you can examine how the model field lines behave out of the equatorial plane. In three dimensions, the Parker spiral field is described by the equations

Br = B0 (r0 / r)2
Bθ = 0.0
Bazi = B0 r02 ω sin(latitude) / rV

Here r0 is the radius of some “source surface” at which the field is radial and has magnitude B0. ω is the rotation rate of the Sun, r is the radial distance of the observer, and V is the solar wind velocity. The spherical coordinate system is referenced to the solar rotation axis. The three orthogonal directions of the field are: radial, Br; colatitude, Bθ; and the (clockwise) azimuthal, Bazi.

The top two graphs show the projections of the spiraling magnetic field lines along cones of constant latitude, projected into two orthogonal planes: that containing the solar rotational axis (left) and that into the rotational equatorial plane (right). The spiral angle of the magnetic field can be adjusted by changing the solar wind velocity and the rotation rate of the Sun.

Note that in reality the rotation rate features on the solar surface do vary with latitude. A value of unity here is equivalent to the normal equatorial rotation rate of 27 days as seen from the Earth.

The bottom two graphs show the radial variation of the angle between the magnetic field and the radial direction (garden hose or spiral angle) and the radial variation of the magnetic field strength (Btot), the azimuthal field (Baz), and the radial field (Br) in nanoteslas.

You can show or hide the positions of the planets on the bottom graphs by setting or clearing the Show planet positions check box. The positions of the planets are indicated on the graphs by dashed vertical lines labeled Mercury, Venus, Mars, Earth, Jupiter, Saturn, Uranus, Neptune, and Pluto.