How long time does it take before three planets achieve the same relative position?
The answer is never, except for the case when their orbital periods can be expressed with low integers, like the 4:2:1 resonance of Io, Europa and Ganymede
However, what you are asking about is when they are going to be in almost the same position again, a quazi-period.
To find those periods, we are pretty much only left by brute forcing as our method. A nice little detail about the case with three planets is that the inner planet is always aligned with one of the other ones at the closest three-planet alignments. That allows us to calculate accurate solutions. In the cases of four or even five planets I simply give up.
To check all the possibilities, we can use a program. Here is an example of a function in JavaScript returning a list of quazi-periods and alignment error:
sameLine = function (period1,period2,period3,limit){
results = [];
newMargin = 1;
synodic1 = 1/(1/period1 - 1/period2);
anomaly1 = (synodic1/period1) % 1;
synodic2 = 1/(1/period1 - 1/period3);
anomaly2 = (synodic2/period1) % 1;
alert(synodic1+","+synodic2);
for (i = synodic2; i < limit; i+=synodic2){
numb1 = i/synodic1 - (i/synodic1) % 1;
numb2 = i/synodic2 - (i/synodic2) % 1;
err1 = Math.abs((numb1 * anomaly1 - numb1 * synodic1/period3) % 1);
err2 = Math.abs((numb2 * anomaly2 - numb2 * synodic2/period2) % 1);
if (err1 > 1 - err1){
err1 = 1 - err1;
};
if (err2 > 1 - err2){
err2 = 1 - err2;
};
if ((err1 < newMargin) && (numb1 > 0)){
results.push([numb1 * synodic1,err1]);
newMargin = err1;
};
if ((err2 < newMargin) && (numb2 > 0)){
results.push([numb2 * synodic2,err2]);
newMargin = err2;
};
};
return results;
};
For Jupiter, Saturn and Uranus, I get the following output:
Time,,,error
13.81170069444156,,,0.30449020900657225
39.71676854387252,,,0.12441143762575813
41.43510208332468,,,0.08652937298028318
139.00868990355383,,,0.06455996830984656
138.1170069444156,,,0.04490209006572288
179.55210902774027,,,0.041627282914560304
317.6691159721559,,,0.0032748071511630172
3991.581500693611,,,0.002329597100612091
4309.250616665767,,,0.0009452100505313865
The first of this periods is of no use, as the error in alignment is almost a third of an orbit. Note that the one you found (that is really impressive you did,actually) gives an error in the alignment of less than a percent. We have to look at periods more than a thousand years long to find any better alignment.
Be sure to feed this function with accurate orbital periods.
Best Answer
Short answer: It might make statistical sense, but statistics don't mean it applies every time. The 4 inner planets in our solar system very likely did NOT begin as mini Neptunes. There's a number of theories of the origins of planets in our solar-system that suggest they didn't begin that way.
Long answer:
Breaking down the article into parts, and I'm going to add some terms just for clarity.
Gas giant refers to Jupiter or Saturn types of planet, those aren't discussed in the article.
Smaller gas planet or "gas planet" or Neptune like planet, refers to Uranus or Neptune types. "Smaller" is in relation to the gas giants. Uranus and Neptune are sometimes called ice giants, but ice implies cold and this article specifically is taking about hot planets, so I don't like the term ice giant in this instance.
Super Earth, like it sounds, refers to a rocky world larger than Earth but without a thick gaseous outer layer like Uranus and Neptune, presumably rocky, but many "super-earths" lack the expected density of a rocky planet, suggesting either much smaller iron cores or water worlds. We don't have enough information to know much about super-earths.
The problem with exoplanet study is that we can see very little and the border between mini-neptune planets and super-earth planets is a pretty fuzzy border. Estimating the density of those planets helps, but it's hard to get an accurate read on density of exoplanets because they're so hard to see at all.
Also, I want to discuss diameter, because the article addresses diameter specifically (in the 1-4 Earth range). Jupiter and Saturn are about 10 times the diameter of Earth. Gas giants are very large and a size about 10 Earth diameters is pretty standard.
Uranus and Neptune are each about 4 times the diameter of Earth. The article mentions planets 4 times the size of Earth specifically, so that ratio is important. Likewise, mini-neptunes would be smaller, perhaps 3 Earth diameters. The article mentions that size as well.
"Super Earths", might or might not be Earth-like. (is Venus Earth like? is Mars? From many light years away with our current telescopes, Mars, Venus and Earth all qualify as Earthlike, though Venus might be considered a little too close to it's star to be in the habitation zone)
Finally, using our solar-system as a model, we don't know how much material would be left from the 4 outer planets if they lost their gaseous outer layers. It's very hard to see inside gas planets, even in our solar-system, so not much is known about their interiors. Estimates can be made based on their density, but it's difficult to know precisely what would be left if they drifted close to our sun and managed to lose their gas. There might be a rocky body with size and mass similar to Earth inside Uranus or Neptune, but it's hard to say with any accuracy. We know that they are primarily made up of lighter elements like Water, CO2, Ammonia and Methane.
Moving onto the article:
This is entirely possible. Planets, especially hot planets, can and often do lose gas. Gas giant planets could, theoretically, lose their entire gaseous outer layers, though it would probably take some time. Planets lose gas more easily if they are close to their star, if they are hot, if they lack a magnetic field and if their star is very active in ejecting solar material. Smaller stars often have more active coronal mass ejections. (I've read that, I'll try to post a source).
So hot planets around smaller stars (the article mentions orbits varying from 2 to 100 days implying smaller stars), would suggest planets that could lose their gas fairly rapidly.
The more massive the planet, the better it is at retaining it's atmosphere. It's not surface gravity but escape velocity that actually matters. Saturn, Uranus and Neptune have similar gravity to Earth but greater escape velocity due to greater mass. More massive planets are better at retaining their gas and the mass difference between 2 and 3 or 3 and 4 Earth diameters can be fairly significant.
The article has a paragraph that I don't like.
This ignores the frost-line theory, which is largely accepted. Young stars can emit a lot of heat shortly after formation. This heat and an active young star solar wind is thought to blow much of the lighter material further out in the forming solar system so planets that form close in are rocky, planets that form further out have more ices/gases.
This paragraph also doesn't address that planets can migrate and that's very important. The discovery of hot jupiters suggests that it's not at all uncommon for planets to migrate. We also suspect that Jupiter migrated in our solar-system, it moved in towards the sun, then moved back out again.
Our telescopes can tell us about where the planets we see planets are, but we can't see where when they formed.
What the study comes down to is that there is a gap in planet size. Now, planet size is kind of funny. If we assume rocky planets like Earth with similar composition, 1 Earth mass = 1 Earth Diameter. You'd need about 9 Earth masses (with gravitational contraction) to have a super earth with 2 Earth diameters. The 4 inner planets combined have only about 2 Earth mass and their combined diameter (if we ignore density variations and contracting), would be about 1.25 Earths. The point is, you need a lot of material to build Earth-like planets approaching 2 or more Earth diameters and the majority of these planets in this study are orbiting smaller stars.
We don't know how much silicate and Metallic-oxide material smaller solar systems are likely to have during planet formation, but rocky super-earths with diameters approaching 2 Earths might be somewhat rare, especially around smaller stars due to a lack of material - now, we don't know that for certain, because we don't know what a standard amount of planet forming material is but it's possible that very large rocky worlds are somewhat rare around smaller stars.
The other problem is, when you have 8 or 9 Earth's worth of mass in one planet, is that such an object should be much better at accumulating and holding onto an atmosphere. Mass accrues more mass in planet formation. Bigger planets grow faster. This puts a soft limit on how large a rocky planet is likely to grow, unless it grows in a region that is nearly devoid of ices and gases, which might occur around very large-hot stars.
Looking at hot planets close to their star, in the 1-4 Earth diameter range, there's a valley where certain sizes haven't been found. The study suggests that many of those 1-4 Earth diameter planets are cores of former gas planets, both Neptunes and mini-neptunes where the atmosphere has been stripped. That's an idea I'd not heard before, but it's entirely reasonable.
A gap isn't surprising because of how much more material it takes to have a rocky planet 2 earth diameters compared to one, and there's the accumualte gases and grow even larger problem. A gap between rocky worlds and smaller gas planets shouldn't be too surprising,
Similarly a gap between gas planets that retain their gas and gas planets that lose their gas shouldn't be surprising either. That's the conclusion that the people who made this study seem to have reached.
As a sidebar, the big difference between Neptune and Jupiter/Saturn is that Jupiter/Saturn have lots of hydrogen and helium. They're large enough that they've accumulated lots of the most abundant gases in the galaxy, while Neptune and Uranus have only small percentages of hydrogen and helium. It's thought that when planets reach a certain mass, they begin to accrue hydrogen and helium and grow quite a bit larger, so like the rocky to gas planet gap, there's a range between gas planet and gas giant where planet sizes are more rare.
We also know that planetary migration happens fairly often and we know that planets can be stripped of their atmospheres/outer gaseous layers, so there's nothing that surprising in this study.
What the article says which I haven't heard before, is they suggest that many "super-earths" are de-gassed Neptunes, which is a new idea to me, but entirely reasonable. They reach that conclusion based on the gap in planet size. I'm not smart enough to critique their conclusions, but it sounds entirely possible. It fits with planetary migration and atmospheric escape.
It's worth noting that those planets, while devoid of atmospheres, and with hot surface temperatures, could still have internal oceans like Ceres, which is dry on it's surface but light enough that an internal ocean is possible.
It's been suggested before that "earth-like" planets around red-dwarf stars may well be barren of atmospheres because the red-dwarf's are more active in generating solar wind and that in combination with closer orbits might make those planets unlikely to retain atmospheres, so losing their atmosphere is nothing new. The only thing that's new to me in this study is the proposal that many super-Earth-like planets are de-gassed Neptunes.
This article, however, doesn't apply to our solar-system. While it's unknown how thick the early atmospheres of the inner planets were and they might have been a lot thicker than they currently are (Certainly Mars' was thicker in the past). The frost-line theory makes it unlikely that the 4 inner planets ever resembled anything close to Neptune. It's more likely that the the exoplanets in the study were migrating planets.
A Neptune like planet is not likely to form close to it's star in a hot environment. Our solar-system, based on what we've observed of other solar-systems so far, is a bit of an odd-ball. Just because many "Earth-like" planets may be degassed neptunes doesn't mean that the planets in our solar-system formed that way.
All that said, there's still a lot we don't know about planet formation and what types of planets are most common. We've only gotten a preliminary glimpse of other solar-systems.
(too long?). I felt it was worth it to go into atmospheric escape and frost line and planetary migration and planet size to give a more complete picture, made the answer kind of long.