Are planetary nebulae the result of supernovae?
No, they are two entirely different things. A planetary nebula is born when a low mass star dies (low mass means less than about 8 times the mass of the Sun), while the supernova is the death of a massive star.
In low mass stars, the outer envelope of the star is ejected out while the core of the star becomes a "white dwarf". The ejected envelope expands away from the central star and creates the nebula that we see. So, if you look at the pictures of classic planetary nebulae like the Ring nebula, you will be a ring of material with a star in the center. The ring of material was once part of the star in the center, but now has been ejected. The central star is now a white dwarf.
In high mass stars, the story is entirely different. Nuclear fusion in the core produces elements all the way to iron. Once the mass of the iron core exceeds a certain threshold, it collapses causing shock waves to propagate outwards. The end result is the titanic explosion called the supernova, leaving behind a neutron star or a black hole. The exploded remains of the star form a "supernova remnant". Examples are the Crab nebula (M1 in Taurus) and the Veil nebula in Cygnus. Hence, a supernova produces a supernova remnant and not a planetary nebula.
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