Turns out the way they're going to do that is quite a bit different than my initial thoughts, all neatly explained by Bill Nye in one of his Why With Nye videos he did for NASA's Juno mission:
Gravity perturbed orbit of the Juno spacecraft as it orbits planet Jupiter (video: Bill Nye Asks Does Jupiter Have A Core?)
The video is quite fun, so click on the image above and all will be revealed. For the textophiles among you though, the explanation goes like this:
Core or not
We don't know what's inside Jupiter. Jupiter could have a core made of
heavy metal, that's sunk to the center. Or it might be, that Jupiter
has this exotic material that's hydrogen, squeezed under such a
pressure, that the hydrogen acts like a metal. Metallic hydrogen. Now,
we can't cut Jupiter in half. We don't have that capability. Instead,
we have a happy little spacecraft in orbit around Jupiter.
[I'm a little spacecraft ...]
Now, as it orbits, we strongly believe there will be slight variations
in its path due to tiny variations in the gravity of Jupiter. And that
will help us figure out what's inside. Now, how would we measure the
tiny variations of the orbit of a spacecraft around a great big
planet, millions of kilometers from here? We would do it by measuring
the frequency of radio waves. Radio waves coming from the spacecraft,
and radio waves coming from Earth. We can measure those variations
using the Doppler effect.
[Sets alarm to go off on an old alarm clock and spins it around on a string]
Listen to the sound, as the alarm clock moves towards you. Now listen
to the sound as the alarm clock moves away. As it's moving towards
you, the sound gets higher-pitched. As it's moving away, the sound
gets lower-pitched. Now this effect, or this phenomenon was described
at first by a man named Christian Doppler. And to this day, my
brother would like to have a rock'n'roll group called Christian
Doppler and The Effects. But my brother's first name isn't Christian.
So, we believe that precisely measuring the path of Juno as it orbits
Jupiter, using the Doppler effect, we'll be able to determine from
Jupiter's gravity the nature of what's inside. Is it heavy metal? Is
it metallic hydrogen? Or is it ...
Yes, I just transcribed the video!! Raaaaaaaaaaaaa!!!!!!
(A bit more conventional explanation is also available here)