# 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.

• I think the body of this question is too long. Jul 15, 2013 at 21:40
• @metacompactness I know. I was concerned about that. But I did not see where and how else summarize what had been done, as it was not all in jest. I thought a summary was needed if we did not want to waste the information. And I am really curious about this problem, even if I am not the one who will solve it. Also I do not think the connection had been made with the previous question on this topic. Jul 15, 2013 at 23:04
• @babou Did you never heard of GRAIL? nasa.gov/mission_pages/grail/overview/index.html#.UeSH4Y0zP_k nasa.gov/mission_pages/grail/main/index.html There does exist a way to determine density distribution, or at least most of it, using measurements of external gravity field. The micro-satellite thing could work, in short! Jul 15, 2013 at 23:40
• I was aware of the project, though not of the name or details. Thanks. I wrote a question on the role of assumptions, with zero success. I am under the impression that unwarranted assumptions are made for this problem too. Gravity field has a lot of information, but not all of it. Analysis of the moment of inertia gives some information missing in the gravity field, for example that the density of a sphere is not radially homogenous. Can object motion in non-uniform gravity give information not to be found in its gravity field ? Jul 16, 2013 at 12:42
• It is a really good question. I was searching for it as well but in a different field. The topic I am working on now is Limited resource allocation and in the algorithm, I am developing I need to know the mass density distribution of Electrical vehicles so then I will be able to allocate limited resources available to them. Would you mind please if you got the answer for it, share your way? Many thanks I was thinking of calculating the integrated range. Oct 15, 2021 at 18:30

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.

• I do not master enough these issues to dispute any opinion. However, I am under the impression that knowledge of the gravity field brings more constraints than classical mechanics. The gravity field of a cube is not like the field of a sphere. The question is then how much is missing. Can the field of a sphere be produced by an object without spherical symetries? Also, could it be possible to get more information by analyzing motion in a known non uniform gravity fields. Can you then still reduce the mechanics of an object to a few parameters ? Jul 15, 2013 at 23:28
• The field of any object always produces a field of a sphere (more accurately a point source, but sphere, shell, etc are applicable ) and that sphere is concentric to the object's center of mass. Nov 13, 2013 at 3:32

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)

• "But if we compare values at points of equal distance to their respective center of mass, then a cube is indistinguishable from a sphere" This is not strictly true. It is strictly true of objects that have spherical symmetry, but the non-spherical shape of the cube introduces small variations in both strength and direction of the field at distances not a lot larger than the size of the object. Nov 13, 2013 at 6:02
• To choose a limiting case hammer the sphere into a thin, flat disk many times the radius of the sphere and then find the field above the center at the radius of the sphere; it is both lower than the gravity of the sphere at the same distance and nearly uniform. The difference due to a cube is much smaller, but still nominally present. Nov 13, 2013 at 6:06
• what you did is cause one dimension of the object to exceed the dimension of the original sphere of measurement, so this represents a violation of "external measurements only rule" (some of the values you measure are actually IN the disc and are forbidden from measurement). the limit of measurement becomes the circumderence of the hammered object (now a disc). g field strength at all points from the disc's COM with distance equal to the disc's radius are equal.. Nov 13, 2013 at 11:56
• I chose a limiting case to make the difference striking rather than subtle, but there is still a difference. You can only neglect the particulars of the distribution in two cases (a) the distribution is spherically symmetric of (b) you are in the far field regimes (not just "outside" the object but many time as far away as the largest dimension). Nov 13, 2013 at 15:06
• The higger multipoles of a not-spherically-symmetric mass distribution are not zero. You can neglect them in the far-field regime because their effects fall by higher powers of the distance, but if you are only a <size of body> or two away then you don't get much help there. Jackson has an extensive discussion in the context of electrostatics but it applies just as well to classical gravity. Nov 13, 2013 at 16:05