The first planet found around a star like the sun, 51 Pegasi b, was announced in 1995. More than 5{,}000 planets outside the solar system are known today, yet fewer than a hundred of them have been seen directly in an image. The difficulty is not that planets are small, but that each one is next to a star.
A planet gives off almost no visible light of its own; it only reflects the light of its star. Seen from far away, Jupiter sends us about 10^{-9} of the light of the sun, and the earth about 10^{-10}. When a telescope points at such a star, the light of the star spreads across the part of the image where the planet would appear.
The planet is also very close to its star in the sky. From 10 parsecs, a short distance for stars, Jupiter would be 0.5'' from the sun and the earth only 0.1''. Separating a point that faint from a point that bright at such a small angle takes the largest telescopes and special optics that block the starlight.
For this reason most planets have been found without seeing them at all. A planet changes two things that can be measured in the light of the star itself: the star moves a little, and the star becomes slightly fainter whenever the planet passes in front of it.
A planet does not go around the center of its star. The star and the planet both go around their common center of mass, as the two stars of a binary do, and they take the same time to do it. Because the star is much heavier, the center of mass lies close to the star, and the star moves around it in a small circle.
The speeds are in inverse ratio to the masses, v_\ast = v_p \dfrac{m_p}{M_\ast}. Jupiter goes around the sun at 13 km/s and has about a thousandth of the sun's mass, so the sun goes around the center of mass of the pair at about 12.5 m/s. The earth moves the sun at only about 9 cm/s.
The circle itself is far too small to see, but the part of the motion along our line of sight can be measured in the spectrum. While the star comes toward us its spectral lines move to shorter wavelengths, and while it goes away they move to longer ones, by \Delta\lambda / \lambda = v / c. For 12.5 m/s this is about 4 \times 10^{-8} of the wavelength, and the best spectrographs now measure speeds below 1 m/s.
This is how 51 Pegasi b was found. Its star moves back and forth at about 56 m/s every 4.2 days, which means a planet with about half the mass of Jupiter, much closer to its star than Mercury is to the sun. Only the motion along the line of sight is measured, so the method gives m \sin i, where i is the tilt of the orbit, and the true mass can only be larger.
If we see the orbit almost edge on, the planet passes in front of the star once every orbit. During this transit it covers a small part of the star's disk, and the light we receive drops for a few hours and then returns to its former level.
The fraction of light lost is the area of the planet's disk divided by the area of the star's disk, \delta = \left(\dfrac{R_p}{R_\ast}\right)^2. Jupiter's radius is about a tenth of the sun's, so a Jupiter in front of the sun would lower its light by 1 percent. The earth's radius is about a hundredth of the sun's, and its transit lowers the light by only 84 parts per million.
A transit can be seen only when the orbit is lined up with our line of sight. The chance of that is roughly the radius of the star divided by the radius of the orbit, about 0.5 percent for an earth at 1 AU from a sun. To find the few that line up, the Kepler space telescope measured the brightness of about 150{,}000 stars every half hour for four years and found more than 2{,}600 planets.
The time between transits is the period of the orbit. With the mass of the star, Kepler's third law then gives the size of the orbit, and from that the amount of starlight the planet receives.
The two methods measure different things. Radial velocity gives the mass, and the transit gives the radius. For a planet that shows both, the transit also shows that the orbit is close to edge on, so \sin i is close to 1 and m \sin i is the mass itself.
From the mass and the radius we get the mean density, \rho = \dfrac{m}{\frac{4}{3}\pi R^3}. The earth's is 5.5 g/cm³, the value of rock around an iron core. Jupiter's is 1.3 g/cm³ and Saturn's only 0.7 g/cm³, since they are made mostly of hydrogen and helium.
On a graph of radius against mass, planets of one density lie on a straight line, because the radius is proportional to the cube root of the mass. Rocky planets lie near the earth's line. A planet that lies well above it, larger than a rocky planet of its mass would be, has a thick envelope of light gas.
Many of the planets found so far are between the earth and Neptune in size, a kind of planet the solar system does not have. Their densities show two groups: planets up to about 1.5 times the earth's radius are mostly rocky, and those above about 2 times have thick envelopes of gas. Few planets have radii between the two.
Whether water on a planet freezes, stays liquid or boils depends mostly on how much starlight each square meter receives. The light of a star spreads over a sphere, so the flux at a distance d is L / 4\pi d^2, which falls as the square of the distance.
The habitable zone is the range of distances at which a planet like the earth could keep liquid water on its surface. For the sun it runs from about 0.95 to 1.7 AU. A planet around a star of luminosity L receives the same flux as the earth at d = \sqrt{L / L_\odot} AU, so the zone moves out with the square root of the luminosity.
A star 100 times as luminous as the sun has its habitable zone about 10 times farther out. A red dwarf with a hundredth of the sun's luminosity has it at about 0.1 AU, much closer than Mercury is to the sun. The red dwarf TRAPPIST-1, with about 0.0005 of the sun's luminosity, has seven planets the size of the earth within 0.07 AU, several of them in or near its habitable zone.
Planets this close to a small star usually rotate once per orbit, so the same side always faces the star. Being in the habitable zone also does not mean that a planet has water or life. It means only that the starlight it receives would let water stay liquid, if the planet has water and a suitable atmosphere.