Most of the known gas planets (Jupiter, Saturn, etc.) are huge. They are even called "gas giants".
is it possible to exist a gas planet around the size of Earth? If yes, why; if no, why?
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According to the article "Minimum planetary size for forming outer Jovian-type planets - Stability of an isothermal atmosphere surrounding a protoplanet" (Sasaki, 1989), not only is this possible, but is suggested to have been an evolutionary stage for the development of Uranus and Neptune. But there are a few conditions to allow this to happen, primarily due to distance from the sun, hence temperature.
The article's findings suggest that the critical mass for 'atmospheric collapse' is 0.2 of the current mass of the Earth for distances of 5 AU or greater for our solar system (this value of course would vary, depending on the parent star). This critical mass increases the closer you are to the parent star - the article explicitly states that Mercury and Mars are too small for this to occur.
There are a few major problems with calling something a small gas planet and of course depends on your definition of a planet. By currently accepted classification (that excludes Pluto), a planet has to be:
All of these points are largely argumentative, but two major problems that a small gas planet would have to sustain itself for longer time during and after the formation within the protoplanetary disk are:
Another problem that we didn't yet discuss is that we have no observational proof that such small gas planets actually exist, whatever you might end up calling them. Problem of course lies in the fact there simply aren't any smaller gas planets than Uranus or Neptune (you pick your candidate for the smallest one; former is least massive, and latter is least voluminous of the four gas giants) in our own Solar system, while extrasolar small gas planets would likely be too small to be detectable over enormous distances to even our nearest stars by any methods of detecting such extrasolar planets, or we wouldn't be observing enough into the system's past (i.e. distance) to see it still forming and it would have already lost grip to its former planets now exoplanets, or erode the smaller protoplanets.
So we have this conundrum between whether they exist (likely they do), are in theory possible (I have no doubt), and how we'd end up calling them, because neither of the two possibilities I can think of for their existence would be considered planets by mentioned current convention of what actually constitutes a planet. Protoplanet wouldn't have cleared its neighborhood yet, and exoplanet wouldn't orbit a star.
KOI-314c is small even when compared to Uranus or Neptune.
Also, look for "Mini-Neptune" on Wikipedia.
So, it seems those "dwarf-giant" planets are possible.
Although I do not have the required knowledge, I wonder if it is possible to exist a planet smaller than the earth that still could be considered a gas planet.
For example, just think of something mars-sized or between mars and the earth in size and mass (by size I mean not only the "solid" part, but including the atmosphere outer layers, just as one would see at the eyepiece - Venus, Titan). Now, think of it as a heftier version of Titan with a much extended and denser atmosphere... could that be called a Gas planet? Besides, nature don't have to fit exactly to our classifications/expectations.
I know Hydrogen and Helium are very volatile and have a small atomic mass, but I guess in a very cold environment, for example few AUs around a red/brown dwarf, a small planet with a small mass (that means low escape velocity) could retain all of its original atmosphere for aeons.
I dare say that even the most creative sci-fi authors coudn't predict the variety of planets that are being discovered nowadays!
It appears that some small gas planets do exist, there is even a hypothesis (to explain its high iron comp) that Mercury was once one such small gas planet "a hot neptune" that 'fell' to a closer orbit and over the course if the last billion years or so had its atmosphere blasted away. In recent years, two exo planets were discovered that can be classified as small gas planets, however they are still larger than the earth (as this is what our present tech allows us so far) these planets are called 'Gas Dwarfs' or Mini-Neptunes with Kepler 138 being the smallest known so far being estimated at about 1.7-3 earth radii or something like that. It wouldnt surprise me if transitional planets could be as small as Venus, or at the very least for there to be Venus-sized planets that superficially appear to be gas planets but are technically mostly rocky planets with an liquid/gas atmosphere composed of hydrogen, ammonia, helium(the usual stuff) as thick percentile-wise as Europa's undersurface ocean.
As of what's mathematically possible, that should vary from au, temperature, mass, and composition. The further away from a star and thus lower the temperate, the smaller it is possible for a planetary body to be and remain stable, and the more likely it is to be smaller given available material at formation.