Why doesn't the ammonia and methane in Jupiter's atmosphere sink into the core?
All planets show a diferentiation in their structure: Denser elements sink to the center of the planet and lighter elements "float" to the outer layers of a planet. Models of Jupiter show that it has a rocky core surrounded by an icy mixture of water and amonia that is surrounded by hydrogen and helium.
Now comes my question: If gases like amonia, methane, etc. are denser than hydrogen, why are they present in Jupiter's atmosphere? Shouldn't they have "sunk" a long time ago?
Your question is a good one. The basic reason that there are gasses heavier than hydrogen in the upper part of Jupiter's atmosphere is that Jupiter is hot enough that the methane, ammonia, etc. in the atmopshere are gaseous (they don't condense to a liquid), and the atmosphere is well mixed by convection.
As Jupiter was accreting material during its formation, it grew hot enough that water and other molecules became gaseous and formed an envelope around the growing core. As time went on and more and more material fell onto the planet, the gaseous envelope became larger. Planetessimals that impacted the planet late in its formation had trouble penetrating through the surrounding envelope which was quite thick. The impactors sublimated in the atmopshere and added their heavy elements to the upper part of the planet. The heat caused convection which distributed the heavy elements uniformly through the envelope. They weren't able to sink into the core.
Another way to picture this is to think of Earth. We have N2 and O2 in our atmosphere, and yet we don't have a layer of N2 on the bottom and then a layer of O2. Atmospheric gasses don't seperate out like this because they are constantly mixed by convection caused by thermal gradients.
Also, Saturn and Jupiter will still differentiate more over time. We think this is happening in Saturn right now. Saturn has cooled to a temperature where the helium becomes immiscible (unable to stay dissolved) in the liquid metallic hydrogen layer. The helium "rains" out onto the core, which releases energy. This process can explain the extra energy we see Saturn giving off, as well as the depletion of helium in the atmosphere that we observe. This same process should happen in Jupiter as it cools, but we think Jupiter is still too hot. (Although some astronomers believe that it has actually started recently, and this explains the slightly lower than expected helium abundance in Jupiter's atmosphere.)
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