Are white dwarfs classified as stars without nuclear fusion?

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Puzzling, isn't it? IAU stands for "international astronomical union", which translates literally to "group involving many countries that is concerned with the laws pertaining to stars." You'd think a group like that would have a clear definition of what a star is, wouldn't you? But this is the whole problem-- cut-and-dried definitions are not actually very conducive to research efforts. Instead, what is effective is having a somewhat flexible working meaning of these terms. That's what flies everywhere except in the tests they give in elementary school classes, so beware of "authoritative definitions."
 
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Ken G said:
Puzzling, isn't it?
Yes, that is why I asked.
 
http://iopscience.iop.org/article/10.1086/307414/pdf

"For objects this cool, the presence of lithium proves that they are substellar."

"Some of these dwarfs certainly are substellar but such a designation cannot be tied uniquely to any particular spectroscopic trait."

"Because some of these objects are substellar and thus not truly stars, the entire collection should be referred to as 'L dwarfs,' not 'L stars.'"

https://arxiv.org/pdf/astro-ph/0608359.pdf

"3. Definitions:
(1) A “primary” body is a star or substar formed by core accretion from an
interstellar cloud, not by secondary accretion from a disk.
(2). A “substar” is a body with less than 80 Jupiter masses, the lower limit for
stellar hydrogen fusion."
 
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Yes, so none of that tells you if a white dwarf is a star. But as said above, the answer to that is, no if you look in a dictionary or encyclopedia, yes if you go to an astronomy meeting.