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Note: In this section, vector quantities are given in bold face letters.
Imagine yourself sitting in a boat, traveling over water with velocity u on a windless day. A small flag is attached to the boat's mast: in what direction will it point?
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Seen from the boat, the flag always points to the rear, because in the boat's frame of reference, a wind with velocity -u seems to be blowing. It always points in the same direction. What if the day is not windless? If a wind is blowing with velocity v? In the frame of the boat, the air now moves with velocity v - u , a vector sum, with the direction of v specified relative to the boat ("in the boat's frame"). The flag now points neither downwind, nor to the rear, but somewhere in between. Suppose the boat now changes direction. In the frame of the boat u is still directed to the rear, but the wind now seems to come from a different direction, relative to the boat. Consequently, the direction of v changes, and the flag, pointing alongthe new v - u, has a different direction as well.
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The Apparent Displacement of StarsThis section is about starlight, not about boats and flags. From Newton's days, astronomers have tried to find how far the stars were by the parallax method, using the diameter of the Earth's orbit as a baseline. They carefully measured the positions of stars at times half a year apart--representing two positions of the Earth separated by 300,000,000 km--and then checked whether the positions of stars in the sky changed. They soon found that, indeed, the positions did change. The trouble was that the observations did not make much sense.
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Jean Picard, one of the early French astronomers, made possible precise observations by introducing crosshairs in the telescope eyepiece. With this instrument he noted around 1680 that the observed positions of stars were not always the same. John Flamsteed, the astronomer royal of Britain--head of the Royal Observatory in Greenwich--confirmed those shifts. For instance Polaris, the pole star, seemed to travel annually around an ellipse whose width was 40", 40 seconds of arc. As discussed in the section on parallax, that might suggest that the distance to Polaris was 1/40 of a parsec or less than 0.1 light year. However, the shifts in position did not occur at the times they were expected . The greatest shift of Polaris in any given direction occured not when the Earth's was at the opposite end of its orbit, as it should have been, but 3 months later.
For instance, in the drawing above, the apparent position of Polaris should have been shifted the furthest in the direction of "December" when Earth was in its "June" position, which is as far as it can go in the opposite direction. Instead, it happened in September, when the Earth had moved 90° from its position in June. In hindsight, the important quantity was not the displacement of Earth, but its velocity, which in September pointed towards the direction towards which Polaris was displaced.
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Bradley's explanationAstronomers were greatly puzzled, the more so when it turned out that all other stars near Polaris were shifted the same way. Then in 1729 the British astronomer royal, James Bradley, took a boat trip on the river Thames near London and noted the strange behavior of the flag on top of the boat's mast: it pointed neither downwind nor to the back of the boat, but in some direction in between, and when the boat changed course, that direction changed, too.
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A sudden inspiration came to Bradley. The flag sensed a combination of two air flows, as seen in the frame of the boat: one due to the wind, the other due to the boat's motion. In a similar way, he reasoned, the velocity of the light coming from Polaris was modified in our own frame, by the added velocity of the Earth! The velocity of light--today universally denoted by the letter c, as in "E=mc2"-- had been estimated in 1675 by Ole Romer, a Dane working at the Paris observatory, from a study of the eclipses of a moon of Jupiter. The velocity u of Earth in its orbit was also approximately known. Viewed from the frame of the moving Earth, the rest of the universe had a velocity -u, perpendicular to the velocity c of light coming from Polaris. Add these two up, as vectors, and you get the observed displacement.
When we are on the outside and observe an object moving with velocity u--e.g., an airplane in a cross-wind--we add u to its other motions. But when u is the velocity with which we, the observers, move, the outside world is moving relative to us with velocity (-u). Then (-u) must then be added to any other motion observed in the outside world.] In the drawing shown here, let AP be the direction towards Polaris--the star itself is much more distant--and AB is along -u, opposed to the motion of the Earth. To simplify the calculation we will assume AP is perpendicular to AB, in which case the apparent motion is a circle; actually the angle is usually less than 90° making the apparent motion not a circle but an ellipse. . The vector PA will represent c, the velocity of light coming from Polaris at c=300,000 km/s, while AB is the velocity (-u) to be added to c, of magnitude 30 km/s. The length of each side in the triangle ABP is proportional to the velocity it represents, so that if that triangle is viewed as a pie-slice from a circle and the angle at P is denoted by a, we get
a= 10800/60000p = 5.7296/1000 degrees. Each degree contains 60 minutes of arc (60') and each minute has 60 seconds of arc (60"), units unrelated to the minutes and seconds of time. Each degree thus equals 3600", giving
Half a year later the direction of u is reversed and the displacement is in the opposite direction, giving an annual range of about 40", as observed.
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Aberration of the Solar WindA somewhat similar process is evident in the solar wind, a fast outflow (~400 km/s) of hot gas from the Sun. It originates in the Sun's corona, the highest and most rarefied layer of the solar atmosphere. That layer is so hot (about 1,000,000° C) that it does not achieve a stable static equilibrium, but boils off into a constant flow of rarefied, hot gas.
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Strictly speaking, the solar wind is a plasma, a mixture of free electrons and of positive ions, atoms which have lost electrons in the violent collisions experienced in a 1,000,000 degree gas. Being a plasma, it can conduct electric currents and its particles can be steered by magnetic fields. The Earth's magnetism, in particular, deflects the solar wind flow, creating an elongated cavity known as the magnetosphere, from which the solar wind is excluded (see picture). On the side facing the Sun, the solar wind only reaches within 10-11 Earth radii of the Earth's center (65-70,000 km) before it is deflected sideways. On the night side, facing away from the Sun, a long "magnetic tail" extends to great distances, along the flow direction of the solar wind. But what is that direction? In the reference frame of the Sun, the solar wind on the average streams radially outwards, with a velocity v of about 400 km/s (it does not change with distance). The Earth, however, orbits the Sun with a velocity u perpendicular to v, of about 30 km/s. Viewed from the frame of the Earth, a velocity -u is then added to v, so to us the solar wind appears to move with v' = v-u, as the drawing shows. The magnitude v' of the new velocity is found by applying the theorem of Pythagoras to the triangle ABC, which gives
If b is the angle by which the solar wind is shifted in the Earth's frame, then
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b = 4.289° With a spacecraft exploring the distant nightside tail, as the Japanese "Geotail" did (at distances around 200 Earth radii), this effect must be taken into account if we wish to place the spacecraft in the tail and not next to it. Unfortunately, the direction of v also varies randomly by a few degrees, so that, while taking b into account helps, sometimes a point calculated to be inside the tail still misses it.
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Plumes of SmokeA 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?
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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
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.
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Comet Tails and the Solar ProbeAt 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.
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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. An even greater greatest aberration effect is expected aboard the solar probe, planned by NASA for observing the solar wind near the Sun. That spacecraft is expected to approach the Sun within 4 solar radii, avoiding melt-down in the intense heat by hiding behind a specially designed heat shield. How then, one may well ask, can one shield out the sunlight and yet observe the solar wind, which like sunlight flows radially outwards? That is where aberration helps. At closes approach (perihelion) the solar probe would move at about 300 km/s, so that the solar wind moving at 400 km/s (but not sunlight) would be aberrated by about 37°, allowing it to reach detectors protected behind the heat shield.
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Optional: #22b The Theory of Relativity
Next stop: #22c Airplane Flight
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