In May, I did my own amateur analysis of Kepler data that was supplied here. This was a table of all confirmed "Kepler planets" to date (April 1, 2015). This table had some unbelievably high masses/densities listed in the first column after the planet designation (ie: planets of Kepler- 23, -24, -25, -27, -28, -32, -39, -48, -52, -54, -57, -58, -59, -60 and others). I didn't take these seriously, but would like to know if there was a "units" error or something I am missing. To give you an idea, Kepler-23b is given a radius of $1.9 R_{\text{Earth}}$ and a mass of $254.3 M_{\text{Earth}}$ - and that's nothing compared to some others. I checked just now and the data is the same.

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    $\begingroup$ I'll leave this as a comment rather than an answer because it isn't exhaustive, but it looks to me like the mass column is potentially an upper limit. Though the table shows upper limits explicitly elsewhere, I had a look through the paper and the planet masses in Kepler-23 seem to be principally derived from stability arguments (see the end of Section 6.1). $\endgroup$ – Warrick Sep 14 '15 at 6:20
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    $\begingroup$ I think you're right, those numbers are impossible and have to be errors for those lines. If you click on the planets (I only clicked on a few), the errors don't remain in the details, just in the chart. It looks like somebody multiplied by an extra Jupiter mass to earth mass for a those entries. All of the "way off" numbers appear to be off by about a factor of about 300. $\endgroup$ – userLTK Sep 14 '15 at 6:34
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    $\begingroup$ Earth has a density of 5.5 g/cm^3. Quadruple the volume and that suggests a density for these planets of 63.5 g/cm^3: no known element. Osmium only runs 22.6 g/cm^3. $\endgroup$ – Wayfaring Stranger Sep 14 '15 at 12:39
  • $\begingroup$ kepler.nasa.gov/Mission/discoveries/kepler68c has a density of 28, even in the chart, not just the table. Neutron stars are much denser, so I think the upper limit for osmium only applies at standard temperature/pressure. $\endgroup$ – user21 Sep 14 '15 at 16:35
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    $\begingroup$ The density at the centre of the Sun is about $150\,\mathrm{g}\cdot\mathrm{cm}^{-3}$, and some super-Jupiters probably have partially degenrate cores, which could be very dense. Even so, stars and massive planets have puffy envelopes that bring the average down. The Sun's mean density is only about $1.4\,\mathrm{g}\cdot\mathrm{cm}^{-3}$. $\endgroup$ – Warrick Sep 14 '15 at 17:01

Starting from the index you mentioned, I clicked through the links for some individual planets, which in turn link to discovery papers or other relevant observations. For planets around Kepler-23, -24, -25, -26, -27, and -28, the relevant papers are Ford et al. (2012) and Steffen et al. (2012), two out of a series of papers. Both papers used transit timing variations (TTV) and something called dynamical stability, which gives an upper limit to the mass of the planets such that the relevant n-body system is stable over long timescales. This leads to plots like the following (Fig. 3, Ford et al.):

enter image description here

The arrows indicate simulations where the system was stable over timescales greater than $\sim10^7$ years.

Additionally, as Ford et al. note,

We caution that the uncertainty in the masses and sizes of the host stars directly translates into uncertainties in the planet masses and sizes.

Furthermore, shorter observations by Kepler also lead to larger uncertainties. Thus, the maximum masses of the planets as given in these simulations may very well be much greater than their actual masses. Steffen et al. have similar results, and they use a very similar method.

After more reading, it's clear that the same is true for planets around Kepler-32, Kepler-48, Kepler-52, Kepler-54, and Kepler -57 to -60, inclusive.

Kepler-39b is interesting. The paper NASA cites, Bouchy et al. (2011), states that the object is larger than one would expect, even taking into account that there is a large uncertainty in radius. It may be a fast rotator (see Zhu et al. (2014)), which would explain its oblateness, but that still isn't enough to explain the discrepancy.

Now, I referred to Kepler-39b as an "object", not a planet. NASA lists its mass as over 20 Jupiter masses; Bouchy et al. placed it at about 18 Jupiter masses, with a very low uncertainty. This could mean that it's more like a brown dwarf than a massive gas giant, and remember, the distinction is fuzzy when it comes to sub-brown dwarfs. This, then, may be the most likely explanation.

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