# How can an orbiting spacecraft's motions yield the orbitee's deep structure?

I've always been a fan of satellites like GRACE and GRAIL, where two satellites orbit a body, precisely measuring the changing distance between them in order to sense gravitational anomalies caused by near-surface density changes. I also understand how you can do the same with a single satellite such as Dawn, by doing sensitive Doppler measurements on the radio telemetry link to precisely measure its movements.

However, Dawn is reported to have used this technique to determine that Vesta has a dense iron core. This surprises me, because by the shell theorem, a spherically symmetric body's gravity should be identical to that of a different body whose mass is concentrated at its center. As long as the mass is spherically symmetric, you shouldn't be able to determine its distribution, because spherically symmetric changes in the mass distribution shouldn't change the resulting gravitational field.

Somehow, my understanding is wrong. How can the motions of an orbiting spacecraft be used to measure the deep structure of mass in the orbited body?

Gravimetric Doppler measurements can actually confirm that Vesta has an iron core, as it eliminates the alternatives. This works in the following way:

1. The mass of Vesta is first measured accurately, for example using the orbital period of a satellite, for instance the Dawn spacecraft.

2. Then, based on observed geological features, its average density, the oblateness of Vesta compared to its rotation rate, the location of the poles, and various assumptions about its internal structure, based on other objects in hydrostatic equilibrium, you can give a good estimate on its moment of inertia.

3. From that, you can conclude that the inner structure of Vesta must contain one or more regions with higher density, and get a relatively good impression on how they are located relative to each other.

4. If this more dense matter is arranged in some unsymmetrical way (e.g. not spherical), Doppler observations will then detect the non-uniformity of Vesta’s gravitational field, based on small changes in Dawn's orbit, as the orbit then does not follow that of a spherical body. The observations by Dawn indicated that the dense matter must be assembled in a spherical core in the middle.

That confirms the existence of a core. But as you state, Newtons shell theorem makes it difficult to measure accurately the size of the core, or if it is layered in regions with different densities. To achieve more data on Vesta's internal structure, you have to land probes on the surface and measure how the shock waves from an earthquake propagate through the planetoid. Exact gravimetry has its limitations, and good seismic data is definitely going to improve the geological assumptions.

Here are some useful references:

• Ah, so there are assumptions made about the different materials that make up Vesta, and how the density can vary (e.g. there's ice, there's rock, there's iron, but there isn't much that's in-between). – Daniel Griscom Jan 1 '16 at 17:12
• @DanielGriscom Both the volume and mass of Vesta is now well know, therefore, we know the average density. The materials on the surface has a lower density than this. The separation of light and heavy materials are what characterize an evolved object. Hence the importance of a core. – Hohmannfan Jan 1 '16 at 17:17
• See The Dawn Gravity Investigation at Vesta and Ceres (PDF) or for a general description of principles involved Astronomy Notes - Planet Interiors and here for some example figures. – TildalWave Jan 1 '16 at 19:35
• Thanks for both of your work on this. An ideal answer would include a list of additional information needed beyond gravimetry to determine a body's deep structure. – Daniel Griscom Jan 1 '16 at 20:24
• Well, there's the assumption that it's iron, based on our general knowledge of composition of celestial bodies. This could be a different metal or alloy, adding up to similar mass and size. – SF. Jan 1 '16 at 21:32

I'm going to give an opposing answer.

As long as the mass is spherically symmetric, you shouldn't be able to determine its distribution, because spherically symmetric changes in the mass distribution shouldn't change the resulting gravitational field.

If the Sun, the planets, and the other bodies in the solar system truly were spherically symmetric, and that could be determined about those objects by observing how objects orbit another would be their masses. (More precisely, what would be determined are the standard gravitational parameters $\mu=GM$, the product of the universal gravitational constant and mass. The large uncertainty in $G$ means that mass can only be determined to four decimal places of accuracy.)

Masses in the solar system aren't spherically symmetric. The Sun is the closest thanks to its large size and its slow rotation rate. The giant planets have a marked spherical bulge, but also apparently have some gravity anomalies due to uneven mass distribution. The terrestrial planets are even less spherical than are the giant planets, the asteroids, even less so. It's best to describe asteroids and moons with a radius smaller than 200 to 300 kilometers as lumpy potatoes rather than spherical cows. Vesta is right in the middle of that range. While not near as lumpy as 25143 Itokawa (visited by JAXA's Hayabusa a decade ago), Vesta is still far from spherical. This gives a handle for measuring characteristics within Vesta, just by observing how Dawn orbits Vesta.

Numerical orbit determination and propagation techniques account for the non-spherical nature of solar system bodies and for perturbing effects from other bodies. The Earth-based tracking stations used to communicate with Dawn provided measurements that give clues regarding how Dawn was orbiting Vesta, which in turn gives clues to Vesta's gravitational field. The most precise measurement is the Doppler shift in signals sent from Earth to Dawn which Dawn simply relays back to Earth. (Every deep space probe launched by the US and Europe is outfitted with a device for just that purpose.) Enough of these observations, spread out over time, coupled with models of the solar system, models of how gravity works, plus lots of computing horsepower, gives a picture of the interior of a solar system body.

One way to model the gravitational field of not-quite spherical objects is with spherical harmonics (or sometimes elliptical harmonics). The mathematics is very well established. This approach works nicely for planets, somewhat nicely for Moon-sized object (but the Moon has five large mascons (mass concentrations) on the near side and a two kilometer offset between its center of mass and geometrical center). They might work even for a Vesta-sized object.

Another approach is to use the body's geometrical shape to model the gravitational field. Dawn took lots of pictures of Vesta. These pictures collectively yield a 3D shape model of Vesta. A commonly used approach in computer graphics is to represent the surface of some object via polygons (typically triangles). These shape models can be used as a gravitational model by projected those polygons to some well-chosen center to create polyhedra. The gravitational field for a constant density polyhedron is easy to calculate (with a computer, that is), and from that, you have a polyhedral gravity model of the object in question.

The orbits from the harmonics-based and polyhedra-based gravity models inevitably won't agree with one another, or with the observations. The biggest problem is almost certain to be the assumption of a constant density in the polyhedral gravity model. A Vesta-sized body should have some amount of differentiation. Polyhedral gravity models can be augmented with a density profile. Modifying that model so it is more consistent results with the observations gives a clue as to what's inside Vesta.

## References:

Robert A. Werner and Daniel J. Scheeres, "Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia," Celestial Mechanics and Dynamical Astronomy 65.3 (1996): 313-344.