Determination of mass density distribution of an object This is a follow-up to a previous question How can you weigh your own head in an accurate way?.
My purpose is not to restart the flurry of more or less humorous
jokes (which are not such a bad thing when infrequent) but to try to draw
some conclusions, as I found the event instructive, and hopefully to
get a real answer.
Beyond the jokes, I think the question was also interesting from a
more social point of view: how fallacies operate, how people vote,
what is the effect how the fast answer inducement by site rules.
It is clear now, at least to me, that the real physical problem was to
determine the mass density distribution of a (partly ?) solid object
without using destructive means. Of course the concept of
destructiveness may depend on the concerned body, and that was one
source of jokes.
Still, it is a relevant topic to review the means, if any, for
achieving such a purpose, as it seems that there are not that many.
This is actually the question I am asking.
But I would like also, at the same time, to review some of the answers
given to the previous qiestion.
One that attracted my attention was the micro-satellite solution.
Does it really work ? So we could precisely ask: given a precise
knowledge of the gravity field created by a solid object, can we deduce
from it the mass density distribution in the solid.  The answer seems
to be no. Take the trivial case of a sphere with a uniform radial
density distribution. The gravity outside the sphere depends only on
the total mass, so that it tells us nothing about the internal density
distribution. Is that an exceptional degenerate case, or is it a
strong hint that the analysis of the gravity field is not enough ?
The slicing solution I proposed was an interesting mistake, certainly
a silly one. But it is ever so tempting to believe the solution is
near when you have a set of equations, and easy ones to boot. A cute
trap. It becomes more obvious when you try to do it in continuous
rather than discrete form.
It should have been obvious, as we know that the torque produced by
weighing a massive object depend only on total mass and distance of
the center of mass to the axis.
What was less obvious, at least to me (but I have not done much
physics since college), was the use of the moment of inertia.  One
could also think of measuring moments of inertia after virtual slicing
of the object as previously with torque.  Unfortunately the moment of
inertia depends only on 3 values, the previous two and a reference
moment of inertia (for a given direction of the rotation axis). It is
a useful remark for computing moments of inertia, but shows that there
is no hope there for solving our problem.
This was remarked by @Ben Crowell in another question (see below).
More generally, it seems there is no hope from any discretized
measurement of a quantity that depends polynomially on the distance. One
cannot get more unknowns than the degree of the polynomial.
I will not comment on Compton scattering, and other uses of indirect
physical phenomena, if only for lack of competence. There is also the
fact that I would like to know of a solution involving only mechanics.
One technique I heard of is measuring wave propagation across the
object.  I hinted to that in a comment on the Compton scattering
solution. I have no expertise on that, but I think that is how the
structure of the planet is analyzed by geophysicists (earthquake
waves).  I also heard of underground analyses by similar means, using
explosives.  But I am not sure about the type of information that can
be obtained in this way. Does one get density distribution. Does it
always work ?
The other technique I thought of is ultrasound based medical imagery.
Could it measure densities.
Now since we are considering waves, could we get something out of the
mesurement of gravity wave propagation, assuming it is possible to do
the necessary measurements.  But I have no idea what this actually
entails or means.
Interestingly, a simpler version of the problem has already been
discussed to some extent two months ago in
How can I determine whether the mass of an object is evenly distributed?
Some ideas were repeated for the recent question, but new ideas
emerged too (gravity field analysis) even if they have weaknesses, and are somewhat hard to use in most situations.
Then, what techniques have been used to know the density distribution
of Earth, the Sun or possibly other bodies. Is there anything
systematic that we overlooked.
So, are there means of solving the problem purely with mechanics and gravity?
A related question is whether it is easier, or as hard, to determine
whether the mass density distribution is uniform.
More generally, I am wondering whether there is a way of
characterizing the properties of phenomena that can help determine the
density distribution inside a solid object.
Alternatively, could one prove it is not achievable by purely mechanical means.
 A: You are interested in the density field of an object which is a scalar valued (just a number) function over 3 dimensional region of space (shape of the object under study). 
Your wish is to study this function using just 2 forms of interactions - mechanics and gravity in most simple form without destroying object. 
I agree that we need to try to understand this question from very general point of view first. 
Mechanics and gravity exhibit properties of superposition - which means that properties of objects spread over space are substituted with point-like properties (one example you already gave - gravitational field of a sphere does not depend on its radial density distribution - so function of density distribution is substituted with just one number of total mass).
Hence using just gravity (without scanning inside with a waves - which for gravity waves is impossible at the moment) gives a negative answer - not possible.
Same is true for mechanics because mechanics also reduces 3 dimensional distributions to point-like ones.
But! The funny part is that THAT is why mechanics and gravity are so successful theories because they allow us to abstract from inner properties of objects and reduce dynamics to only global properties. 
Logically it is also clear that to know 3 dimensional distribution you need to offer algorithm which will give you that amount of information, and this information can only be obtained by means of physical interaction with each point inside of object or observation of waves emitted from each point.
Finally you already gave counterexample to your goal - the radially not uniform sphere. There is no way to find out this distribution using just mechanics and gravity without destroying it or using waves.
A: If I take a heavy lead object of arbitrary shape and mount it in a cardboard sphere such that the center of mass of the object coincides with the center of the sphere, you would never determine the lead object's shape nor mass density to be anything other than that of a point source using Newtonian mechanics or external field measurements.
Referencing your comment about the gravitational field of a cube being different to that of a sphere, this may be true when comparing measurements of g field strength at points on the surface of these objects, where distances to center of mass are arbitrarily different depending on the shape or location measured. But if we compare values at points of equal distance to their respective center of mass, then a cube is indistinguishable from a sphere (center of mass may or may not be in the geometric center depending on uniformity). 
Due to the above examples allowed by the superposition principle, by mapping the g field strength at arbitrarily many points around an object in 3D space we can find a unique point in an unmapped area around which in all directions, equal distances yield equal field strength values--essentially the point denoting the location of the object's COM. This is the only point in the unmapped area where we know what the value is (it is zero). Meanwhile, every other point in the unmapped area is unknown, but only because we are simply forbidden by your question to use invasive procedures. 
The boundary of mappable space is simply the surface of the object. If simple inspection shows that the shape is a regular polygon, we can make comparisons between values on this boundary and values we would expect on the surface of the same type of polygon of equal average density as the object. Any deviations would represent a deviation in mass distribution inside the object under the surface. This phenomenon is called a gravity anomaly and has been used to find interesting stuff undergound here on earth. Note that higher than expected gravity on one point on the surface of a spherical object doesn't imply denser material just under that surface--it could also be due to a void on the opposite side of the sphere. 
This is why when analyzing anomalies, it is important to map a large area of the surface, which after accounting for the center of mass of an ideal uniform object, can reveal the center of mass of an object which would solely account for the anomaly. Again, using external measurements only reveal COM's which can be very misleading since any number of objects can have COM's nowhere near their actual volume (horseshoes, shells, etc)
