Kipping and Teachey discovered it among 300 exoplanets in Kepler's catalogue, all of which produce predictable dips in starlight that occur as an orbiting body passes in front of its sun - a phenomenon called a "transit".
The researchers found one instance, in Kepler 1625b, that had intriguing anomalies.
That's where Hubble was able to help with an extra 40 hours' worth of observations.
In 2017 NASA's Kepler Space Telescope detected hints of an exomoon orbiting the planet Kepler-1625b.
There are two ways to potentially identify an exomoon. At more than three times the radii of the Earth, these gaseous planets are less common than terrestrial planets, but each is expected to host several large moons.
Kipping said that's how the Earth and moon would appear from far away. "It was definitely a shocking moment to see that light curve - my heart started beating a little faster and I just kept looking at that signature", David Kipping described his feelings. "But we knew our job was to keep a level head testing every conceivable way in which the data could be tricking us until we were left with no other explanation".
One of the researchers behind the work, Columbia's Alex Teachey, was involved in a previous study that looked for exomoons despite the long odds (Teachey is joined by David Kipping for this new paper).
However, before getting too excited about the prospect of finding Ewoks, it should be noted that the exomoon (formally named as Kepler-1625b-i) has a radius of around four times that of the Earth and a mass of around 16 times that of our planet, so is in fact similar in size and mass to the planet Neptune. This catalog included 284 planets found by Kepler with wide orbits around their host stars. "But we did think it was interesting enough to try to get more data on".
"It's the unknown unknowns which are ultimately uncharacterizable", Kipping said. Their results were published Wednesday in the journal Science Advances.
In principle this anomaly could also be caused by the gravitational pull of a hypothetical second planet in the system, but the Kepler Space Telescope found no evidence for additional planets around the star during its four year mission.
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The moon is estimated to be only 1.5% the mass of its companion planet, which is close to the mass-ratio between the Earth and its moon. The star was probably cooler in the past, so this heat could be a reason for the size of the moon, inflating the gas giant as the temperature rises.
Now the researchers say that what Hubble saw seems to confirm the idea that this planet has a moon.
How did this moon form in the first place?
The candidate moon, with the designation Kepler-1625b-i, is unusual because of its large size - comparable to the huge, icy planet Neptune in our solar system. Kepler-1625b stood out. There are no indications of tidal capture among our Solar System's moons.
Given that both the planet and its potential moon are gas giants, no one is suggesting conditions that might support life.
The exomoon is exponentially larger than our solar system's biggest moon.
Based on the observations of the orbital timing of the planet and the depth of the dip in the light curve, Kipping and Teachey estimate the mass of this exomoon at around 1.5 per cent of the mass of the planet. "We can expect to see really tiny moons", Kipping said.
So, in theory, detecting an exomoon is hard but not impossible.
The gas giants in our own solar system have rocky and icy moons, but no examples of having gas-giant moons.
Those observations could come from another go-round with Hubble, or they could come from NASA's James Webb Space Telescope, which is due for launch in 2021.