(Temporary relocation)

Plumes of Smoke

A steamship moves across the ocean with velocity u, in a wind with velocity v perpendicular to u. If u defines the x-axis and v the y-axis--in which direction does the plume of smoke trailing behind the ship move?

This problem becomes much easier if we work in the ship's frame of reference. In that frame and in the absence of wind, the smoke's velocity is -u. Adding the wind to the motion, the smoke's velocity becomes v-u (=v+(-u)), and that vector defines the direction of the plume of smoke trailing behind the ship. It will be in the (-x,y) quarter of the coordinate plane, and the angle a between it and the angle a between it and the (-x) axis satisfies

tan a = v/u         or else     sina = v/w, w = u + v

Trying to solve this problem in the frame of the ocean on which the ship is moving can get confusing, because we are not adding velocities but displacements. After each puff of smoke is released, it no longer shares the ship's velocity. Only in the ship's frame do motions become simple.

Comet Tails and the Solar Probe

At locations closer to the Sun, bigger aberrations of the solar wind can occur, because objects speed up as they approach the Sun (see Kepler's 2nd law!). This has an interesting effect on comet tails, which act very much like the plume of smoke in the preceding example.

Typically 1-20 km across, a comet is a collection of frozen gases and dust (a "dirty snowball") which had accumulated in the outer reaches of the solar system and was nudged into an sunward orbit. As the comet approaches the Sun, its frozen gases evaporate in the heat and a long tail is formed, pushed away from the Sun by sunlight and by the solar wind. The tail thus points behind the comet as it approaches the Sun but ahead of it when it recedes again.

Actually, two distinct tails are often observed--a dust tail pushed by the pressure of sunlight, and a plasma tail pushed by the magnetic field embedded in the solar wind (direct collisions between particles of the comet and solar wind are rare). The colors of the two tails differ: on comet Hale-Bopp in 1997 (see picture) the plasma tail was blue, a color produced by the ions which formed it, while the dust tail was white, the color of scattered sunlight.

The dust tail pointed in the direction of sunlight, as observed from the comet: it was probably shifted like the starlight observed by Bradley, but at an angle too small to be noted by eye. The plasma tail, on the other hand, pointed in the direction of the solar wind--again, as sensed by the comet. As seen above, Earth sees the solar wind shifted by about 4°, but for the comet the shift was appreciably larger, because it moved faster than the Earth, especially when it got closer to the Sun. The two tails therefore formed a distinct angle, as shown in the picture.