How is a day measured on a gas giant?

Question

After watching this JPL video of a Cassini time lapse of Saturn, I wondered since being mostly gases, how do we know how long a day—one rotation period—is?

Is there a usable relation between how fast the upper most layer of gases is and the planet's solid core?

Answer 1

That the rotation period of the bulk mass of a planet is estimated through something with the magnetic field is true.

But let me maybe elaborate a bit in-depth on that.

No planetary magnetic field is a dipole. Earth has strong magnetic anomalies over the Atlantic, there may be ionic disturbances by moons like Io, or your magnetic field is just really crazy, like Uranus and Neptunes.

(A simple, low-order model of Earth's magnetic field, from GFZ Potsdam)

But no matter how it looks, it will usually be constant on lines of latitude and longitude. This in particular signifies that the field and its distortions are created somehow by the bulk mass of the planet, else the field would move relative to a surface map, in the case of Earth. So all this talk just to establish the following: We can take the shapes of the magnetic field as if they were surface shapes, and by following them determine a rate of rotation.

Now the story doesn't end there. While this sounds nice in principle, to do this in practice, we would need to send a probe and do in-situ measurments of this changing magnetic landscape.

But what can be done from afar, and from Earth is observe ionospheric patterns. Because the ionosphere of a planet is well, ionized gas, at super-low pressures (roughly on the order of microbar to nanobar) it is frozen-in with the magnetic field that makes the particles of the plasma gyrate. This means, the ionospheric plasma follows the magnetic landscape, now translating into a plasma landscape. This plasma landscape is then usually observable from Earth by the radio signals they emit. It is also possible to see this in the ultraviolet, if there is strong plasma activity through auroral bombardment of a planet's upper atmosphere.

For your second question, that's a simple no. First of all there is no 'uppermost' layer of atmosphere, as the atmosphere is continously decreasing in density with increasing distance from the planetary body in question.
What you're possibly thinking about are the upper cloud layers, as they visibly end at some height.
But wind speeds are much more determined by pressure gradients created by the central star heating the planet, and the atmospheres ability to cool.
There is one class of planet, however where a relation between the wind speeds at roughly 0.1 bar pressure and magnetic dynamo might exist. Those are Hot Jupiters which host atmospheres that might be sufficiently ionized for its winds to be braked down by the planetary magnetic field.
Then there would exist a limiting speed, given a certain planetary field. However this is still a field of active research, but it would give you 'a' relation between the two.

Answer 2

I thought for a long time that day's on gas giants were measured by waiting for a certain feature come around, like Jupiter's great red spot. However, looking it up just now I learned that that it is done by measuring the magnetic field. This article is specifically about Jupiter, but I imagine measuring Saturn is much the same,

Scientist were finally able to use radio emissions from Jupiter’s magnetic field to calculate the planet’s rotational period and speed. While other parts of the planet rotate at different speeds, the speed as measured by the magnetosphere is used as the official rotational speed and period.

I'm afraid I don't have an answer to your second question.

Answer 3

There's already good answer provided, but it is worth noting that the rotation rate of a planet is ill defined, as many planets are unlikely to be uniformly rotating. For example, Hot Jupiters present zonal flows, so should the rotation rate of the planet be measured taking into account the flows or not?

We don't even know if there's a solid core in gas giants, so the question of the rotation rate should be 'is there a way to measure the rotation rate profile of a planet' and the answer is yes, but it's really hard. This is for example one of the long-term objectives of the Juno Mission. By measuring the pressure, density and composition of the atmosphere at different radius, you can try to stick a general circulation model that would give you the local speed hence the rotation rate. Of course, the picture is made harder when you consider magnetic fields.