<p>I certainly don't know the details of these kinds of calculations, but as my thought is a bit too long for a comment I'll write it up as an answer.</p>
<p>If you measure the flattening of a planet due to rotation (e.g. by measuring its rotation period) and the gravity (measured while keeping the orbiter at constant height above the planet), then you have everything you need to calculate the moment of inertia I/(MR<sup>2</sup>) using <a href="https://en.wikipedia.org/wiki/Darwin%E2%80%93Radau_equation" rel="noreferrer">Radau's approximation</a> (not the most informative link, but I couldn't find a better one). After that, it's a matter of taking a model for a planet, e.g. a two-layer model with a core and a mantle, which has free parameters x, the fractional core radius, and f the difference in density between both layers of the planet. Compare your measured moment of inertia with the model and you've got x and f for that planet, which allows you to calculate the mass of the core (if you also have the total mass of the planet from gravity measurements). The figure below shows the two-layer model and the measurements for some solar system bodies. The value 0.4 on the y-axis is the value for a homogeneous planet. From that you can deduce that Ganymede is a very differentiated planet. Hope that helps a little bit.</p>
<p><a href="https://i.sstatic.net/jG58vm.png" rel="noreferrer"><img src="https://i.sstatic.net/jG58vm.png" alt="enter image description here" /></a></p>
<p>A note on how this graph was created: first of all, this isn't my graph, I took it from my lecture notes. The moment of inertia factor for a homogeneous body is calculated as</p>
<p><span class="math-container">$$I = \frac{8\pi}{3}\int_0^R \rho r_0^4 dr_0 = \frac{8\pi}{15}R^5 $$</span></p>
<p><span class="math-container">$r_0$</span> is the mean radius of a shell within the planet (because planets generally aren't round).
Then you take a simple two-layer model for a planet where <span class="math-container">$\rho(r) = f\rho_0$</span> for <span class="math-container">$0\leq r \leq xR$</span> and <span class="math-container">$\rho(r) = \rho_0$</span> for <span class="math-container">$xR &lt; r \leq R$</span>,  where <span class="math-container">$R$</span> is the radius of the planet.
The moment of inertia factor is then
<span class="math-container">$$I/MR^2 = 2[1 + (f - 1)x^5]/5[1 + (f - 1)x^3] $$</span>
which is the curve that you see in the figure.</p>

CC BY-SA 4.0
2021-03-15T09:00:43.763Z
Can you tell us how you created that graph?
PM 2Ring
CC BY-SA 4.0
2021-03-15T10:07:03.417Z
@PM2Ring, I added a note to my post. I didn't want to go too much into detail, partly for fear of making mistakes, because as I said, not an expert.
theWrongAlice
CC BY-SA 4.0
2021-03-15T10:46:29.037Z
Thanks, that's a great addition to your answer.
PM 2Ring
CC BY-SA 4.0
2021-03-15T10:52:21.637Z
Wonderful answer! There are bits of discussions on Jupiter's gravity in Space SE in answers to [How, and How Well will Juno measure the effects of frame dragging? [GR effect due to Jupiter's mass & rotation)](https://space.stackexchange.com/q/30556/12102) and also [Better understanding of the “Rivers of metallic hydrogen” in the atmosphere of Jupiter?](https://space.stackexchange.com/q/21649/12102)
uhoh
CC BY-SA 4.0
2021-03-16T13:29:32.620Z
Thank you. Could this method be used to account for the fuzzyness of Jupiter's core, and the sharpness of Saturn's?
zabop
CC BY-SA 4.0
2021-03-29T15:12:19.213Z
@zabop, I just wanted to add that the sharpness of a core or other layer can also be deduced from seismology. If seismic waves show sudden changes in speed, then you know the boundary is rather sharp. Moreover, sharp boundaries are usually expected, but in the case of Jupiter, calculations are able to predict a certain "fuzzy-ness" (the core is diffusing into the mantle).
theWrongAlice
CC BY-SA 4.0
2021-03-29T18:54:49.490Z
Thank you. How do you measure seismic waves from orbit then?
zabop
CC BY-SA 4.0
2021-03-13T19:35:16.760Z
2021-03-15T10:00:05.017Z
2021-03-15T10:00:05.017Z
theWrongAlice
<p>I am told in an astrophysics lecture the following.</p>
<p>The mass of Saturn's core was measured by Cassini when it completed its final flyby between the rings and the planet itself. It was also found that its core has a sharp boundary.</p>
<p>Juno measured the mass of Jupiter's core &amp; found that its boundary is a bit fuzzy.</p>
<p>How were these measurements done in principle?</p>
<p>(Ie what was measured from which we can deduce core mass?)</p>

CC BY-SA 4.0
2021-03-13T19:11:16.560Z
Related: https://en.wikipedia.org/wiki/Gravity_science_(Juno)
PM 2Ring
<planet><solar-system><saturn><cassini><outer-solar-system>
How to measure mass of planets' core from orbit
zabop
# How to measure mass of planets' core from orbit

(Asked by: username_0 on 2021-03-13)

I am told in an astrophysics lecture the following.

The mass of Saturn's core was measured by Cassini when it completed its final flyby between the rings and the planet itself. It was also found that its core has a sharp boundary.

Juno measured the mass of Jupiter's core & found that its boundary is a bit fuzzy.

How were these measurements done in principle?

(Ie what was measured from which we can deduce core mass?)

**Tags:** planet, solar-system, saturn, cassini, outer-solar-system
> **Comments:**
> **username_1**: Related: https://en.wikipedia.org/wiki/Gravity_science_(Juno)  

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## 1 answers

### Answer by username_2 on 2021-03-13. Score: 10

I certainly don't know the details of these kinds of calculations, but as my thought is a bit too long for a comment I'll write it up as an answer.

If you measure the flattening of a planet due to rotation (e.g. by measuring its rotation period) and the gravity (measured while keeping the orbiter at constant height above the planet), then you have everything you need to calculate the moment of inertia I/(MR2) using [Radau's approximation](https://en.wikipedia.org/wiki/Darwin%E2%80%93Radau_equation) (not the most informative link, but I couldn't find a better one). After that, it's a matter of taking a model for a planet, e.g. a two-layer model with a core and a mantle, which has free parameters x, the fractional core radius, and f the difference in density between both layers of the planet. Compare your measured moment of inertia with the model and you've got x and f for that planet, which allows you to calculate the mass of the core (if you also have the total mass of the planet from gravity measurements). The figure below shows the two-layer model and the measurements for some solar system bodies. The value 0.4 on the y-axis is the value for a homogeneous planet. From that you can deduce that Ganymede is a very differentiated planet. Hope that helps a little bit.

[![enter image description here](https://i.sstatic.net/jG58vm.png)](https://i.sstatic.net/jG58vm.png)

A note on how this graph was created: first of all, this isn't my graph, I took it from my lecture notes. The moment of inertia factor for a homogeneous body is calculated as

$$I = \frac{8\pi}{3}\int\_0^R \rho r\_0^4 dr\_0 = \frac{8\pi}{15}R^5 $$

$r\_0$ is the mean radius of a shell within the planet (because planets generally aren't round).
Then you take a simple two-layer model for a planet where $\rho(r) = f\rho\_0$ for $0\leq r \leq xR$ and $\rho(r) = \rho\_0$ for $xR < r \leq R$, where $R$ is the radius of the planet.
The moment of inertia factor is then
$$I/MR^2 = 2[1 + (f - 1)x^5]/5[1 + (f - 1)x^3] $$
which is the curve that you see in the figure.

> **Comments:**
> **username_1**: Can you tell us how you created that graph?  
> **username_2**: @PM2Ring, I added a note to my post. I didn't want to go too much into detail, partly for fear of making mistakes, because as I said, not an expert.  
> **username_1**: Thanks, that's a great addition to your answer.  
> **username_3**: Wonderful answer! There are bits of discussions on Jupiter's gravity in Space SE in answers to [How, and How Well will Juno measure the effects of frame dragging? [GR effect due to Jupiter's mass & rotation)](https://space.stackexchange.com/q/30556/12102) and also [Better understanding of the “Rivers of metallic hydrogen” in the atmosphere of Jupiter?](https://space.stackexchange.com/q/21649/12102)  
> **username_0**: Thank you. Could this method be used to account for the fuzzyness of Jupiter's core, and the sharpness of Saturn's?  
> **username_2**: @zabop, I just wanted to add that the sharpness of a core or other layer can also be deduced from seismology. If seismic waves show sudden changes in speed, then you know the boundary is rather sharp. Moreover, sharp boundaries are usually expected, but in the case of Jupiter, calculations are able to predict a certain "fuzzy-ness" (the core is diffusing into the mantle).  
> **username_0**: Thank you. How do you measure seismic waves from orbit then?  

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