With the planet m measured in Earth masses, M is measured in solar masses, and a in au.. And k for those units is 833.
The planet "clears its orbit" if this number is > 1
This is an update to my article: How A "Dwarf Planet" Gas Giant Could Challenge IAU Definition - Pluto, Ceres, Haumea Etc Can All Be Planets. That article pointed out that we could discover a gas giant in our own solar system that satisfies the IAU definition of a "dwarf planet" as it wouldn't clear its orbit if it was far enough away.
First, I should have pointed out there that the WISE search has not ruled out gas giants in the remote parts of our solar system. It ruled out a Saturn sized gas giant out to 10,000 au and a Jupiter sized gas giant out to 26,000 au, but the hypothetical Nemesis would have orbited 1.5 light years away or 100,000 au. We know now that we don't even have a red dwarf at that distance, and even a brown dwarf is unlikely as WISE found them ten light years away. But we could have a very cold brown dwarf and we certainly could have a large Saturn sized gas giant over 0.16 light years away orbiting our Sun or a Jupiter over 0.41 light years away, and we'd never spot it. Even a Jupiter at that distance would easily count as a dwarf planet according to the IAU definition.
So anyway in that article I asked if any of the exoplanets might be dwarf gas giants if we use the same definition. Anyway I had a go at calculating Margot's Π for some exoplanets since then. The various measures are all are in pretty good agreement in the discrimination between planets and non planets - and that's the easiest to calculate.
Here is the formula:
With the planet m measured in Earth masses, M is measured in solar masses, and a in au.. And k for those units is 833.
The planet "clears its orbit" if this number is > 1
The M of the parent star dominates the equation, so we want a heavy star with the planet far from it, so to find it I sorted the Wikipedia list of exoplanets with the most massive star first.
The wikipedia table is in Jupiter masses for the planet. Jupiter is 317.83 Earth masses, and 833/317.83 = 2.62. So then the formula is
2.62*m / ((M(5/2)*( a (9/8))
m= planet mass in Jupiters, M= star mass in sun masses, a= distance in AU.
Sort with the most massive host star first,
The very first star in that list is promising already, HD 13189 b is a planet or a brown dwarf of 14 Jupiter masses planet orbiting at a distance of 1.85 AU. Minimum mass calculated as between 8 to 20 so using the highest mass in that range. 2.62*20 / ((4.55/2*( 1.85 9/8) gives it a Margot PI of 0.61, so it would count as a dwarf planet according to the IAU definition. Yet it orbits its star at a distance similar to that of Mars and its mass is probably at least 8 times the mass of Jupiter. Details of its parameters here:
But those are minimum masses. It could be more massive and clear its orbit (and be a brown dwarf too). Also the host star's mass is 4.5 +- 2.5. At the lower end of that range its Margot's PI is 4.6 so it could just about squeeze in as an IAU definition exoplanet.
So it may well be a dwarf exoplanet, but we need to look a bit further.
We want to find a planet that
For another example:
M =2.49
m = 3
a = 2.1
2.62*3 / ((2.225/2*( 2.1 9/8) = 0.46
However as often the case with planets found by the doppler shift method, they only have a minimum mass. They just say the mass is at least 3 Jupiters - the mass multiplied by the sin i is 3. So it depends on the angle of view. Even at 6 Jupiters it would not count as an exoplanet. But it could potentially be much more massive.
Let's try another one Fomalhaut b. This one has actually been imaged :). Or at least a cloud or dust around it.
Fomahaut b is in a 2,000 year highly elliptical orbit from these observations by Hubble.
It orbits at a distance of 115 au around a star that has mass at least 1.9 times the mass of the sun. It is probably a planet of mass at most twice that of Jupiter though probably much less. So putting this into the equation:
M =1.9
m = 2
a = 177 ± 68
gives a Margot PI of 0.0054. Even if its mass was 20 times that of Jupiter it would have a value of 0.054. So this surely is a dwarf planet. It is "shrouded in dust but very plausibly a planet identified from direct imaging"
Artist's impression of Fomalhaut b (courtesy NASA). If it is indeed a gas giant, then it is definitely a Gas giant dwarf according to the IAU definition. And it can't possibly be a planet if their definition is applied to exoplanets.
There are lots of other candidates there, if you use the minimum mass or even double the minimum mass, they are "dwarf exoplanets" according to the IAU definition. The problem is finding one with a well determined maximum mass. Ones found by the transit method would be a good bet there.
Does anyone reading this know of any other good exoplanet candidate with a maximum mass, which orbits at a fair distance from a massive star, say twice the mass of our Sun or more, and which is too small to be a brown dwarf?
If we can find such a star, it has a decent chance of being a dwarf gas giant exoplanet according to the IAU definition if applied to exoplanets.
Anyway Fomalhaut b probably is one. And if not, surely we will find such an exoplanet eventually. As I said in the other article, I think this shows that the terminology of the IAU can't work with "dwarf planet" replaced by "dwarf exoplanet". Of course it makes no difference to HD 13189b if humans choose to call it a gas giant "dwarf exoplanet" non exoplanet but surely it's distorting our language so much as to be uncomfortable and confusing?
As for whether we find a gas giant dwarf planet non planet in our solar system, well that could happen at any time. It could be just on the edge of detection. For instance by chance the search for a new "Planet X" could easily accidentally turn up a more distant and larger object than expected.
I think this means that the IAU definition is both
My earlier article is here: How A "Dwarf Planet" Gas Giant Could Challenge IAU Definition - Pluto, Ceres, Haumea Etc Can All Be Planets