The Law of Field Line Preservation

When a spacecraft breaks away from the influence of the Earth's magnetic field into interplanetary space, it finds there a weak magnetic field. The field may be weak, but it extends over huge distances, and can have important effects. From the observed direction of interplanetary magnetic field lines, we believe this field comes from the Sun, carried by magnetic field lines dragged out by the solar wind.

But if the field is weak, then the plasma rules and pushes the field lines around. A rule which is fairly well obeyed states then that:

    (For any space physicist reading this: the deformation process also involves electric fields.)

Drawing Interplanetary Field lines

Using this "law of field line preservation", we will now derive the shape of interplanetary magnetic field lines.

1. In the middle of the bottom (short side) of a sheet of paper, draw a small circle, about one inch across: that will represent the Sun, viewed from far above its north pole. If the Sun rotates once in 27 days, then each day it rotates by
   360°/27 = 13.3°

   Draw from the center of the Sun a line perpendicular to the bottom of the page, extending most of the way to the top. Using a protractor and a ruler, draw on each side of that line 3 additional radial "spokes" from the center of the Sun, each making angles of 13.3° with its neighbors.

(As a shortcut, download the drawing here, or with higher resolution here.)

(Alternative method: Let center of the Sun be the origin of a system of cartesian coordinates, with the x-axis parallel to the bottom of the page. Draw the two axes, with the y-axis extending to near the top of the page. With a pencil, draw faintly the line y=4, parallel to the x axis but 4" (4 inches) above it.
    On that line mark points at distances 15/16", 2" and 3 3/8" on both sides of the y-axis, then draw radial lines from the center of the Sun through those points, extending them until they are 1/2" from the sides of the sheet or 1" from the top.
    For those used to metric units, let the radius of the Sun be 1 cm (diameter=2cm), the pencil line follows y=10 cm and the marks on it are at distances of approximately 23.7, 50.2 and 83.9 millimeters from the y-axis. Extend the lines until they reach within 1 cm of the sides or 3 cm of the top.)

2. On each of the spokes, mark the point where it emerges from the Sun, and mark from there, along each spoke, additional points at intervals of 1.5 inches. Each interval marks the distance the solar wind covers in one day.

   (Yes, the Sun is drawn much too big on this scale, but we will ignore the difference this makes. Besides, the solar wind does not start moving from the Sun's surface, but from some greater distance.)

   The magnetic field at all these points is already so weak that the solar wind overpowers it and shifts its field lines, while its own motion--radially outward--remains unchanged. We will now derive the shape of those lines.

3. Since the Sun is viewed here from north, and it rotates in the same sense as the Earth, on this scale drawing, does it rotate clockwise... or counterclockwise?
       Decide for yourself and write it down--no peeking! (The sense of rotation depends on the point of view: if the POV is far to the north it rotates one way, if far to the south it is in the opposite direction.)

Marking the Spokes

4. Mark with the number 1 the point where the line furthest to your right emerges from the Sun. You are told that 7 ions are located at that point, close to each other and on the same magnetic field line. You are also told that in the coming week, all 7 are destined to join the solar wind, one day apart. On day one, however, they are all still at the starting point, although one ion has just started moving outwards

5. Go to the next "spoke" on the left of the first one. The Sun is drawn here as rotating counterclockwise (yes!), so on day 2 all of them are at the base of that line--except for the one which started the previous day, which has moved radially and is now at "first base," the next point on the first line. Also, still another ion has just started moving outwards from the starting point. Mark both points with the number 2.

   On day 3, the ion which started out first is at "second base," and the one which started on day 2 is at the first point out on its spoke. All others are at the base of the 3rd spoke, to which the Sun has now rotated, and one more ion has just begun to move. Mark all three point with the number 3.

   On day 4, the Sun has rotated to the 4th spoke and 4 ions remain at the base point of that spoke, including one which is just starting to move. The other three, in the order they were released, are at 3rd, 2nd and 1st "base." Mark all four points with the number 4.

   And so on, day after day. The points marked 5, for instance, are where the particles are on the 5th day. Obviously, you must give up on marking any ions which have gone past the limits of the paper.

   Each of the radial lines is now marked with the day on which "its" particle reaches each marked point. If any unmarked points are left, you may, if you with, extend the marking further (8, 9, ...), to days for which we have no starting particle.

Spiral field lines

6. Now connect--preferably with a red pen, or in some color different from that of the rest of the drawing--all points with the number 2, also the ones with 3, 4, 5, 6 or 7, and perhaps also those with 8, 9 and 10. Take those marked 6: they show where the ions are after 5 days have passed--about the time the first of them reaches Earth's orbit. Since at the beginning they were all on the same magnetic field line, after 5 days they still are. The line you have drawn therefore gives the expected shape of an interplanetary magnetic field line.

   You may use a straight ruler for the connection: the actual lines curve smoothly, but even with lines composed of straight sections it becomes clear that the shape is a spiral. This agrees with observations at the Earth's orbit, where the average interplanetary magnetic field is found to make an an angle of 45° with the flow of the solar wind, similar to what the drawing shows. In other words--after being 5 days on their way, and reaching 150,000,000 km from the Sun, the magnetic field lines still "remember" the Sun's rotation.

7. They "remember" it for months afterwards, even as the solar wind speeds past the orbits of Saturn and Uranus. If you continue this graphic excercise to such large distances, you will find that the spiral gets wound tighter and tighter, until the field line direction is very close to that of circles around the Sun. The space probe Voyager 2 has shown that this indeed does happen.

It was also noted that with increasing distance from the Sun, the spiral shape of interplanetary magnetic field lines becomes more and more tightly wound, until their shape differs little from circles.

Both points were well illustrated by the phenomena that followed intense solar activity in April-May 1998, reported by Robert Decker of the Applied Physics Lab of the Johns Hopkins University in Maryland. That activity created a disturbance in the solar wind, as well as a rarefied cloud of protons with energies about 1000 times that of the solar wind. Both were observed by a number of spacecraft--ACE at the L1 Lagrangian point (near Earth, distance from the Sun about 1 AU), by Ulysses (5 AU), by Voyager 2 at 56 AU and by Voyager 1 at 72 AU.

The disturbance in the solar wind arrived at Voyager 1 about 7.5 months later, propagating radially at the velocity of the solar wind in which it was embedded. The protons, on the other hand, although moving much faster, were relatively few in number, which forced them to spiral along field lines. They were observed by Voyager 1 after 6 months--1.5 months before the disturbance in the solar wind reached that distance--and Dr. Decker calculated that their spiral path took them 10 times around the Sun, a total distance of about 2000 AU.