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I don't know much about planets outside our solar system. Do we even know the limit of the size of how large a planet or satellite can be? The largest planet I know of is Jupiter, but could there be planets as large as our Sun or even larger? Is there a theoretical limit to the size of a planet?

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Yes, there is a limit. Anything with a mass larger than about 13 times that of Jupiter would be called a brown dwarf (a failed star), though whether such an object would consist entirely of gas, or had a rocky/icy core as is probable for most giant planets, is not presently observable. Any larger than about 75 Jupiter masses and we would just call it a star. The exact definition of what a planet is (especially, the 13 Jupiter mass boundary) is still disputed.

Of the bonafide planets that have been detected and confirmed, the catalogue at exoplanets.org lists Kepler-435b as the one with the largest measured radius (although its radius error bar overlaps with that of other planets). The quoted radius is $1.99 \pm 0.18$ times that of Jupiter.

Most giant planets have very similar radii for masses between about 0.5 and 10 times that of Jupiter. The reason for this is that they are largely supported by electron degeneracy pressure. The diversity in the radii (between about 0.7 and 2 times that of Jupiter) of such planets is not yet fully understood.

The plot below shows mass vs radius for "planets". The smaller (probably rocky/icy) planets do show a trend of increasing radius with mass (the solid line is where a theoretical relationship for rocky/icy planets has been used to estimate the mass from the radius). The gas giants above about 0.5 Jupiter masses show no trend and a small scatter.

Mass versus radius

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    $\begingroup$ Good answer. Are the two sections ($<0.01M_J<M$ and $0.01M_J<M<0.1M_J$) because of data extrapolated from mass-radius relationships, or are they actual observed data points? $\endgroup$ – HDE 226868 May 21 '16 at 15:52
  • $\begingroup$ @HDE226868 You know, I cannot locate any information about what mass-radius relation has been assumed for those stars without directly measured mass estimates. I will make further enquiries. $\endgroup$ – Rob Jeffries May 22 '16 at 9:12

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