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A planetary nebula is an astronomical object consisting of a roughly spherical glowing shell of gas formed by certain types of stars at the end of their lives.
The first planetary nebula discovered was the Dumbbell Nebula in the constellation of Vulpecula, observed by Charles Messier in 1764 and listed as M27 in his Messier Catalogue. To early observers with low-resolution telescopes, M27 and subsqequently discovered planetary nebulae somewhat resembled the gas giants, and William Herschel, discoverer of Uranus, eventually coined the term 'planetary nebula' for them.
In fact, planetary nebulae are very different objects to planets. A typical small star (less than a few times the mass of the Sun) spends most of its lifetime shining as a result of nuclear fusion reactions converting hydrogen to helium in its core. After several billion years, the star runs out of hydrogen, and the core contracts and heats up, eventually reaching temperatures at which helium fusion begins. Helium fusion is extremely temperature sensitive (reactions rates vary as T40!), which makes the star unstable. Huge pulsations eventually kick off the outer layers of the star.
While the central progenitor star dwindles to a white dwarf, the thrown-off gases form a cloud of material around it. It is this cloud which is designated as a planetary nebula. The remains of the star which produced the nebula is also responsible for the energy which causes it to glow. The Sun is predicted to become a planetary nebula at least 5 billion years in the future.
Interestingly, there may after all be a link between planets and planetary nebulae -- it has been suggested that the intricate shape of the Cat's Eye nebula may have been caused by a planetary system being engulfed by the nebula when it was born.
Some of the more famous examples of this type of object are the Ring Nebula, the Helix nebula, the Eskimo Nebula and the Cat's Eye nebula.