The eclipse or transit method has been enormously successful in boosting the numbers of exoplanets, in particular in the last few years using the Kepler satellite. It's also an indirect method because it looks for the slight shadowing of a star caused by a planet passing in front of it in its orbit. It requires the orientation of the orbit to be almost perfectly in the line of sight, such that the planet will cross the face of the star. Calculating the depth of the eclipse is quite simple. It's basically just the area of the exoplanet compared to the area of the star. For Jupiter compared to the sun, this is one percent. That effect is easily detectable from the ground and from space with the stability photometry possible above the Earth's atmosphere, substantially smaller transits or eclipses can be detected down to Earth size planets. If we see an animation of a transiting exoplanet, we can also see that the effect is transient. It only occurs for a short amount of time as the planet moving in its orbit passes in front of the star. It is of course a repeating signal that will repeat with the period of the planet's orbit. In fact, confirming a transiting exoplanet requires multiple period observation. Typically, three consecutive transits are required to detect an exoplanet and declare confidently a detection. A single transient event is not sufficient. We can use Jupiter and the Sun as an example of the difficulty of detecting a transit. Jupiter moves in its orbit of the Sun at 13 kilometers per second. If you work that out as seen from afar, it will take less than a day for it to transit the sun in a 12-year orbit. In other words, blink and you miss it. So any given planet will only transit for a tiny fraction of the time. The solution is that thousands or tens or even hundreds of thousands of stars have to be observed either sequentially or better in parallel so that the rare transits can be detected. Exoplanets are transiting at random orientations in space. The probability that we will see one is quite small for any given system. In fact, the probability of an exoplanet transit increases with the size of the star, with the size of the planet, and with the proximity of the planet. The probability of detecting a Jupiter around a sun-like star is actually about a tenth of a percent, but because many of the early exoplanets were hot Jupiters on close orbits, the probability of them showing transits is much higher, several percent. In fact, transits have been observed around many of the first exoplanets discovered. More than 150 of the planets that were first detected by the Doppler method have now been followed up with transits, with that detection giving you the size of the planet to go with a mass from the Doppler method. In principle, if the transit can be followed carefully enough, we could observe the shadowing of the exoplanet as it crosses the limb of the star, and as the light passes through the atmosphere of the exoplanet, we can get a sense of what kind of atmosphere it has. So the depth and shape of the transiting light curve can give information on the atmosphere of the exoplanet. In fact, there are two kinds of eclipses. There are primary eclipses when the exoplanet passes in front of its star, that gives us the size of the exoplanet, and if light filters through the atmosphere of the exoplanet, we can in principle learn what the atmosphere is made of, but when the exoplanet passes behind a star, if we look in the infrared, we can get a measure of its temperature because the thermal emission from the exoplanet will disappear momentarily as it passes behind the star. Both types of information are useful in characterizing exoplanets. A final niche way of detecting exoplanets is microlensing. It's an elegant technique but has only been used to detect a border a dozen exoplanets. Its importance comes from the fact that in principle it can detect extremely low mass objects, perhaps even down to the moon's mass. In microlensing, a nearby star passes almost directly in front of a more distant star. General relativity says that the light will be bent around the nearby star and cause lensing. This is called microlensing because the lensing effect of a star and another star induces an angular deflection of only 1000 or millionth of an arc second, not detectable from the ground. The image splitting and distortion is not detectable, but the momentary magnification caused by the lensing event is. Microlensing has been observed dozens of times with one star passing in front of another star. The use of this method for detecting exoplanets depends on tracking the brightening caused by the lensing of the foreground star from the background star, and then looking for a secondary spike on the light curve caused by the fact that the foreground star has an exoplanet which causes a little extra bit of brightening. The limitation of microlensing as a method is that it's statistically rare, and a worst limitation is the fact that the foreground star continues its passage across the sky and so the event is not repeatable. Detailed gravitation and radiation physics gives us the detection sensitivity of the various techniques, microlensing, direct imaging, the Doppler method, and transits for any particular exoplanet of a given size and mass. Each of the techniques has its own merits and deficits. There are trade-offs and selection effects for each one, and all have been useful in characterizing the exoplanet population. Transits or eclipses are a very effective way of detecting exoplanets and determining their size. Mass does not come from this measurement that requires the Doppler method. Exoplanet transits are also only possible when the orbit of the star and the planet are such that the planet passes directly in front of the star as seen from our perspective, and that only occurs for a small fraction of the situations. So transit surveys have to observe thousands or tens or hundreds of thousands of stars to detect the rare transits. Nonetheless, this method is very sensitive, has been used to detect Earth-sized planets or even smaller.
