Astrometry refers to the method of detecting planets around other stars discussed in the previous notes. One looks for a star (these would be only near stars) that has, not just straight line motion, but a wobbly path. This indicates that an invisible companion is in orbit around it. By observing the wobble we can deduce the mass of the companion. Astronomers tried this for many decades, but were hampered by turbulence in the earth's atmosphere. Because of the refraction of light as it goes through air, the image of a star on a photograph is subject to random variation that can easily cover up the small motions of the wobbly path.

This problem was finally solved with the launch of the Hubble Space Telescope. Photographing the sky from above the atmosphere, Hubble began in 1995 providing clear evidence of planets around near stars. The first objects seen were (and are) ambiguous and mysterious. They lie in the region around one-tenth of the mass of the least massive star; thus they are much more massive than Jupiter.

Hubble also makes possible a second method of detecting exoplanets: direct observation. A planet's light output might be one-billionth (10-9) of that of the star that it orbits. Therefore the planet cannot be directly seen, unless one has a way of blocking out the light of the star itself. This can be done by placing a circular mask in front of the starlight entering the telescope. But if the light has passed through the atmosphere, random variations in the path of the light rays make it impossible to block them with a fixed mask. With the telescope mounted above the atmosphere this problem is eliminated, and it becomes possible to "see" the planet as a bright point of light reflected from the masked star.

The method of direct observation doesn't give us a measurement of the mass of the planet. We can get an estimate of the mass though, because the brightness of the planet is a measure of it size (actually its surface area), since it's reflecting starlight. Since size usually correlates with mass, we have an approximate determination of the mass.


A third method of detecting exoplanets became productive also toward the end of the twentieth century. This method uses the Doppler effect to detect, in a different way, the wobbly path of the star around which the planet moves.

The Doppler effect is a general and very important property of any kind of a wave (sound waves, light waves, etc.): If the source of the waves is moving toward or away from the observer, the wavelength gets shifted. If the source moves toward the observer the wavelengths are shorter than they would be if the source were stationary. If the source moves away the wavelengths are longer. In the case of the visible spectrum, the shorter wavelengths are at the blue-violet end, the longer wavelengths at the red end. So for a source moving toward you there is a "blue shift;" for source moving away there is a "red shift."

In light from a star (and the sun), if you look carefully at the spectrum -- particularly in a photograph -- you see many dark lines where the light has been blocked.

This link shows the sun's spectrum with many dark lines. The spectrum is produced by a diffraction grating (a plate that acts like a prism to spread out the wavelengths of the incoming white light). The wavelengths, in nanometers (1 nm = 10-9 meters), range from about 400 nm in the violet to about 700 nm in the red. The linked site also lists some of the important lines with their wavelengths and the elements which produce them. Click the small rectangles below the right end of the spectrum to enlarge the photograph.

It is understood that this absorption of light occurs because of gases in the outer atmosphere of the star. Each element in the atmosphere absorbs a particular set of wavelengths, and the exact value of these wavelengths (for hydrogen, helium, nitrogen, even for elements like iron and sodium that are vaporized at star temperatures) is known. Now if the star is moving toward the earth these lines are blue shifted; if the star is moving away the lines are red shifted. From the amount of the shift we can determine the speed of that motion.

Figure 1 shows the geometry of this method. The star's straight line path is along the y-axis. The line from the earth to the star is along the x-axis. And the path wobbles left and right, toward and away from the earth. From the variation in the wavelength, changing periodically from the blue shift to the red shift and back, we deduce the orbit of the star, and hence the mass of the invisible companion. Note that this method does not depend on light from the star going in an absolutely straight line as it comes to the observer, and so it works for an observer based on the earth's surface.

Note that the Doppler method and the astrometric method allow you to look at two different kinds of orbital motion. Suppose in the same figure the straight path of the star is still in the +y direction, but the wobble is not from side to side, but up and down. (Let there be a z-axis coming up out of the paper, and suppose the wobbly motion is alternating plus z to minus z to plus z, etc. In this kind of motion the star would not be moving closer or further from the observer, and so there would be no Doppler shift. But the observer (at least above the atmosphere) would "see" a wobbly path. Figure 2 shows the same thing from another perspective:


In the figure above the straight line path of the star (the y-axis) is perpendicular to the paper. The star is going "into" the paper. It's wobbly motion might be left and right, toward and away from the observer (along the x-axis), and so it would be detected via the Doppler effect. Or, its motion might be up and down (the z-axis), and this up-down motion would be seen directly on the telescope's photo.

This figure also shows that the wobbly motion doesn't have to be along the x-axis or the z-axis, but could be on an angle as shown by the doubled line with arrows. In fact, most of the orbital motion we detect is probably like this. In this case we would detect wobbly motion with either the astrometric method or the Doppler method. It also shows that if we use one of these methods, we are probably detecting only part of the actual motion. If I observe the wobble with the astrometric method I am probably finding just a lower bound on the size of the orbit; the orbit could be bigger than what I detect. Similarly, if I observe with just the Doppler method I am also finding just a lower bound on the size of the orbit. This means when I use these methods to find the mass of the invisible companion, I am really finding a lower bound. The mass could be what I calculate, or it could be larger.

Both the wobble methods (astrometry and Doppler) are easier to observe if the mass of the invisible companion is larger. Hence they are, you might say, prejudiced against smaller masses. If we don't see many smaller masses it's not so much that they are not there, but that our system of detection doesn't find them as easily.

The wobble methods are also prejudiced in favor of planets that are close to their stars, since the gravitational force, which produces the wobble, is greater if the planet is closer. The gravitational lensing method (below) however, works the other way.


A new and very exciting method for detecting exoplanets began to be used in 2006. It's based on the idea that the path of a light ray is bent when it passes near a massive object, in the same way that the path of a physical object, a space probe for example, is bent when it goes near a massive object. (See Figure 3.)

The space probe's path is bent by the force of gravity, according to Newton's theory of universal gravitation. The bending of the light ray is not in Newton's theory, but it is part of a more sophisticated theory of gravity developed by Einstein around 1916 (and well-corroborated by observations since).

In Figure 4 there is a distant and stationary star S2, and a ray of light is shown coming to the observer on earth. S1 is a near star with a planet; the distance from S1 to the planet is fairly large (exaggerated in the diagram).

We're watching the earth move in a direction perpendicular to the line from S2. Over the course of a few hours it moves from point (a) to point (g). At (a) the light ray from S2 to earth does not go very near S1; the ray is straight and detected in a telescope on earth. When the earth gets to (b) the straight line from S2 goes very near S1 and so the light ray is bent downwards. (The dotted line shows where the ray would have gone if there were no mass present.) The observer on earth at (b) doesn't see S2 where it is expected. A little later the earth gets to (c), and there the ray that passes near S1 comes into the telescope. At position (c) the observer wouldn't see S2 at all (because it would be blocked by S1), if it weren't for the gravitational bending. Now the earth moves to (d), and at that point you also would not expect to see S2 because it is blocked; but you do see it because there is a light ray that goes on the other side of S1 and is bent up into the telescope.

Up to this point, you've seen what gravitational lensing would look like for a single star. The ray is bent first in one direction and then the other. Now consider the presence of planet P, a distance from S1. At point (e) in the earth's path the light ray from S2 may be not too close to S1 and not too close to P, and so it is pretty straight. When the observer gets to point (f) he expects to see the ray that comes in a straight line from S2. But he doesn't because the ray is bent downward by the planet. He does see the ray when he gets down to point (g). This secondary downward bending indicates that there are two objects out there. The amount of the bends depends on the masses of the two objects. Thus the mass of the planet is determined.

Again, this method works best when the separation between the planet and the star is larger, in contrast with the wobble methods. This is important because many of the objects found in the early years, with the wobble method, were much closer to the star than is the case for the planets in our solar system. These would be planetary systems not very much like the solar system. The planets would be hotter, and too hot to allow the existence of life forms such as there are on earth.


Another method that has come unto use recently is based on the time variation of the total light from a star-planet system. This works when the plane of the planet's orbit is along the line from the star to the viewer. (In the figure below the orbit is in a plane perpendicular to the paper.)

At the time of (a) the planet is to the right of the star and the light we collect is the total of star + planet. (The assumption is that we cannot see the two separate objects, but only collect a certain total quantity of light from that direction in space.) The planet's light is really reflected starlight. Only the half of the planet facing toward the star is lit (just as when we see a half moon). At (b) the planet has gone behind the star and we collect only the light from the star. The graph shows that the quantity of light is reduced. At (c) the planet comes out at the left, and we collect the total star + planet light again. At (d) the planet is in front of the star. The planet is dark, and a part of the starlight is blocked (eclipsed). So the light in (d) is less than the light in (a) and (c), as shown in the graph.

FIG. 5
This kind of graph is called a "light curve." From it astronomers can deduce various properties of the planet, including its diameter, but not its mass. The eclipsing technique is now being employed by a dedicated satellite-based telescope, launched in 2010, to study a large sample of 156,000 stars.