Adapted from Jim Kaler's S P E C T R A

Planetary Discovery

Exoplanets are those found in orbit around other stars. The large majority of exoplanets are found through the back-and-forth movement they produce on their parent stars, while a small few are located via stellar side-to-side motions. Some others are detected when the planets cross in front of their parent stars, slightly dimming their light, yet others through gravitational lensing, and in the ultimate form of discovery, by direct observation. The following explains the principal means of location, that through back-and-forth movement.

The Doppler (Radial Velocity) Technique

Stars move along the line of sight, some going away from us, some coming toward us. If a star moves toward us, its light waves seem to come more frequently and the wavelengths seem shortened; if the star is moving away from us, we see the reverse, the wavelengths seeming to be longer. This "Doppler effect" is easily seen in water waves and heard in sound waves, the latter affecting the pitch of a moving car or airplane. If the speed were high enough, a good fraction the speed of light, a star would actually change color, seeming too blue if coming at us, too red if going away.

Stellar speeds along the line of sight (the radial velocities), however, are usually measured only in tens of kilometers per second or less, so the changes are not at all visible directly to the eye. However, the Doppler effect also causes changes in the wavelengths of spectrum lines that ARE readily detectable. At the modern limit, astronomers can measure shifts produced by line-of-sight motions that are only a few meters per second, less than the speed of good runner.

Double Stars and the Doppler Effect

A great many stars are readily seen through the telescope to be double. However, if the stars are too close together, the observer will see them as one, the two images forever blurred together. We can still separate them by means of the spectrum. If the stars of a double are of comparable brightness, the spectrum will be the composite of the two. As the pair orbit each other, the two alternately move toward and then away from the observer (unless we are looking right down the orbital axis). As a result, the spectrum of each star is Doppler shifted first one way and then the other. As one spectrum is shifted to the blue (to shorter wavelengths), the other is shifted to the red (to longer), and vice versa.

From the doubled lines that shift back and forth, we know that there are two stars in the system. From the degree of shift, we can derive the back-and-forth speeds of the stars. Since the orbit is probably tilted to the line of sight, these observed speeds are lower limits to the orbital velocities, from which we can find lower limits to the stars' masses through gravitational theory. If the star is an eclipsing double, then we know the orbit's tilt and can derive actual masses.

If the components of a very close double star system are very different in brightness, then only one set of absorption lines will be seen. We will still see the one set shift back and forth as the stars orbit, however, and can still tell that our star is double. Such "single-line" stars (so-called because there is only one SET of lines, that is, the observed lines are not doubled) provide limited information on masses, but if we can estimate the mass of the star we see from its nature and brightness, then we can derive a lower limit to the mass of the invisible companion.

On to the Planets

Discovery of an extrasolar planet, one orbiting a star other than the Sun, differs from that of a faint companion in a single- line double star only in that planets are not very massive and do not move the star very much. Though Jupiter does not have enough mass to be a star (missing by a large factor), it and the Sun rather behave like a double: as Jupiter orbits the Sun, the Sun must also make a small orbit about Jupiter. The two actually go around a common center of mass. (Because Jupiter is so massive compared with the other planets, except for Saturn, they make little difference.) Since The Sun is 1000 times more massive than Jupiter, the Sun's "orbit" is only 1/1000 the size of Jupiter's. Given the size of the orbit and the twelve year period, the Sun moves at a speed of about 3 meters/second, similar to modern detection ability. We therefore have the basis for discovering small, low mass bodies -- planets -- orbiting other stars.

The orbital giveaway is purely periodic motion, which other sources of Doppler shift (motions in a star's atmosphere, for example) cannot produce. Adopt the simplest kind of orbit, a perfect circle. Kepler's third law (as generalized by Newton) says that the orbital period P squared equals (multiplied by appropriate constants) the orbital radius of the planet around the star (not the center of mass) cubed, divided by the sum of the masses (planet plus star). However, the huge mass difference between the star and the planet means that the orbit of the planet around the star is about the same as the orbit of the planet around the center of mass, and that the sum of the masses of the pair is about equal to that of the star alone. Therefore, period squared equals the orbital radius of the planet cubed divided by the stellar mass, which is known from the kind of star. The measured period and the stellar mass thus allows calculation of the planet's orbital radius.

The ratio of masses (planet-to-star) equals the inverse of the ratio of the orbital radii (star-to-planet). The radius of the stellar orbit is known from the star's measured velocity and the orbital period. With the planetary radius also known, we can calculate the ratio of masses, and therefore (since the star's mass is known), the planet's mass.


There are complications. The orbit may be elliptical, which means that the "radius" becomes the ellipse's semimajor axis, and that the velocity varies over the orbital period (Kepler's second law). This matter is easily handled, and along with the planetary mass, we also find the orbital eccentricity, or the degree of orbital flattening.

The second complication is difficult to treat. We observe the true velocity of the star only if the plane of the orbit is in the line of sight. Except under unusual circumstances (the planet eclipsing the star, for example), the degree of tilt cannot easily be observed. If the orbit is tilted, we observe only a lower limit to the velocity, and thus find only a lower limit to the planet's mass. (If the orbit is tilted perpendicular to the line of sight, we see no motion at all!) Statistics can come to the rescue, as we can calculate the average expected tilt, and given the number of planets there are, there is no question as to their low average masses.

However, the problem of orbital tilt means that we cannot be sure that any given planet is not really much more massive than it appears, and that it might not be a planet at all, but a "brown dwarf," a "substar" with a mass less than that required to fuse hydrogen to helium (8 percent the mass of the Sun). The nominal limit between planets and brown dwarfs is 13 times the mass of Jupiter, as which point the bodies are hot enough inside to fuse their deuterium (heavy hydrogen) into helium. Many real brown dwarfs are being found by the same Doppler technique, but no one knows if there is any overlap, whether real brown dwarfs (made whole from the interstellar gases) can be smaller than planets and whether planets (made from accretion of dust and gas within disks that circulate around new stars as part of the mechanism of star birth) can be bigger than brown dwarfs. At the end, the surprise is the observation of big Jupiter-like planets tucked very close to their stars, some with periods of only a few days. Since giant planets are thought to form far from the stars, where it is cold and the planets can accrete a lot of hydrogen, this positioning presents something of a mystery. Quite likely such a planet migrated inward after it was formed, the result of turbulent "friction" within a thick disk of debris left over from the star's birth.

Discovery favors big planets close to their stars, as these give the stars the biggest velocities. Astronomers are far from working their way down to systems like ours, so we do not yet how common our kind of planetary system might be, or how many "earth's" there are. There is no doubt, however, that the dedicated work of research astronomers will someday find out. And no doubt that someday we will get a look at the planets themselves.
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