QUESTION: Although we've long established that the planets travel around our sun, is it physically possible for a small star to travel around a large planet?
ANSWER: No, it is not. The mass required for an object to become even a very small star is much larger than the mass of any planet. If a planet had more mass than such a star, it would become a star itself, so instead of a star going around a planet you'd end up with a binary star system.
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QUESTION: I assumed a star is a star due to burning of gases and its general composition? Not it's size and density?
A star is distinguished from a planet by being hot enough to radiate a substantial amount of heat and light (planets can radiate substantial amounts of heat but are not hot enough to radiate substantial amounts of light). However, the heat and light generated by a forming object depends on its mass. If it has less than about 10 to 20% the mass of the Sun, it will not become hot enough to be classified as a star (the exact boundary between small stars and large planets is unknown, so that is just an estimate, as discussed below). A different definition involves whether the star becomes hot enough to ignite nuclear reactions in its core, allowing it to shine (very faintly for low mass stars) for a very long time; if not hot enough, the star cannot replace the heat and light it is radiating and eventually becomes too faint to give off a significant amount of radiation. This generally leads to several classifications:
Normal planets, which no matter how large, have too little mass to give off significant amounts of heat or radiation, and are definitely planets.
Planets which are substantially heavier, and give off significant amounts of heat and radiation while forming, but soon cool off and become just as "faint" as normal planets.
Sub-dwarf stars, which can ignite some types of nuclear reactions (particularly involving isotopes of elements other than hydrogen and helium), but quickly run out of the fuels involved and also cool off relatively soon.
Red dwarfs, which can ignite hydrogen nuclear reactions, and shine (very faintly) for trillions of years.
Normal planets are thought to extend up to 5 to 10% the mass of the Sun (Jupiter, by comparison, is only 1/1000th the mass of the Sun; so "normal" planets can be much heavier than Jupiter).
Super-planets are thought to extend up to 8 to 15% the mass of the Sun.
Sub-dwarf stars are thought to have masses from about 10 to 20% the mass of the Sun.
Red dwarfs are thought to have masses from about 15 to 25% the mass of the Sun at the low end, to about half the mass of the Sun at the high end.
It has been quite a few years since I last went over this in detail, so the range of masses may have been narrowed down a bit in various theoretical calculations, but different calculations probably still give different results, so the range of results may still be just as uncertain as shown above.