Hi there! Thanks for coming on the Planetary Mechanics Blog.

Today I will tell you about new results on the interior of the planet Mercury, by Ashok Kumar Verma and Jean-Luc Margot.

Mercury has been orbited during 4 years by the spacecraft MESSENGER, and gravity data have been derived from the deviations of the spacecraft. These data tell us how the mass is distributed in the planet.

**Planet Mercury facts**

Mercury is the innermost planet of the Solar System. Its radius is about one third of the one of the Earth, and its closeness to the Sun associated with the absence of an atmosphere induces large temperature variations between the day and the night. Another consequence is its very slow rotation, i.e. a Hermean (Mercurian) day lasts 58 terrestrial days, while its revolution around the Sun lasts 88 days, which is exactly 50% longer! This phenomenon is called a 3:2 spin-orbit resonance state, it is a unique case in the Solar System but is somehow analogous to the spin-orbit synchronization of our Moon. It is a consequence of the Solar tides, which despin the planet.

A last interesting fact I would like to mention is that Mercury is too dense for a such a small planet. This suggests that in the early ages of the Solar System, the proto-Mercury was much bigger, and differentiated between a core of pretty heavy elements and a less dense mantle. And then, Mercury has been stripped from this mantle, either slowly, or because of a catastrophic event, i.e. an impact.

**The missions to Mercury**

Sending a spacecraft to Mercury is a challenge, once more because of the proximity of the Sun. Not only the spacecraft should be protected from the Solar radiations, heat,… but it also tends to fall on the Sun instead of visiting the planet. For these reasons, only two spacecrafts have visited the Mercury up to now:

- the US spacecraft Mariner 10 made 3 flybys of Mercury in 1974-1975. It mapped 45% of the surface and measured a magnetic field,
- the US spacecraft MESSENGER orbited Mercury during 4 years between March 2011 and April 2015. It gave us invaluable information on the planet, including the ones presented here,
- and let me mention the European-Japanese mission Bepi-Colombo, which should be launched to Mercury in April 2018.

**The rotation of Mercury**

The rotation of Mercury is in a resonant state, known as 3:2 spin-orbit resonance. This is a dynamical equilibrium reached after dissipation of its rotational energy, in which

- Mercury rotates about one axis,
- this axis is nearly perpendicular to its orbit, the deviation, named obliquity, being a signature of the interior,
- the rotation and orbital periods are commensurate, here with a ratio 3/2. Around this exact commensurability are small librations, due to the periodic variations of the Solar gravitational torque acting on Mercury. The main period of these librations is the orbital one, i.e. 88 days, which is a direct consequence of Mercury’s eccentric orbit. They are supplemented by smaller oscillations, at harmonics of the orbital period (44 d, 29 d, 22 d, etc…), and at the periods of the other planets, meaning that they result from the planetary perturbations on the orbit of Mercury. The largest of these perturbations is expected to be due to Jupiter, but it has not been measured yet.

**What the rotation can tell us**

An issue in the pre-MESSENGER era was: does Mercury have an at least partially molten (outer) core? We now know that it has, thanks to Peale’s experiment, due to the late Stan Peale. The idea was this: the viscous core responds like a fluid to short-period excitations, and like a rigid body for long-period (secular) excitations. And the good thing is that the librations (called longitudinal physical librations) are due to a 88 d-oscillations, while the obliquity is due to a secular one (actually an oscillation which is some 200 kyr periodic, i.e. the rotation of the orbital plane of Mercury). So, in measuring these 2 quantities, one should be able to invert for the size of the core. This was achieved in 2007 thanks to radar measurements of the rotation of Mercury, and confirmed from additional Earth-based measurements, and MESSENGER data, since.

We now know that Mercury has a large molten core, which does not rule out the presence of a solid inner core. For that, additional investigations should be conducted.

## The gravity field

The most basic model of gravity is the point-mass, which just gives us a mean density of the planet. This can be obtained from planetary ephemerides, i.e. in studying how Mercury affects the motion of the other planets, and with more accuracy from the deviations of the spacecraft. We know since Mariner 10 that Mercury has a density of 5.43 g/cm^{3}, while 1g/cm^{3} is expected for ice, 3.3 g/cm^{3} for silicates, and 8 g/cm^{3} for iron.

A more accurate model is to see Mercury has a triaxial ellipsoid. This requires to add two parameters in the gravity field: J_{2} and C_{22}, also know as Stokes coefficients. A positive J_{2} means that the body is flattened at its poles, while C_{22} represents the equatorial ellipticity of the planet. A positive polar flattening is expected as a consequence of the rotation of the planet, while the equatorial ellipticity can result from differential gravitational action of the Sun, i.e. the tides.

Knowing these two Stokes coefficients is possible from gravity data, and this would give us the triaxility of the mass distribution in Mercury. But something is missing: we do not know its radial distribution, i.e. heavier elements are expected to be in the core. For that, we need the polar momentum C, which could be derived from the obliquity, knowing the Stokes coefficient.

For a spherical homogeneous body, C=2/5 MR^{2}, M being the mass and R the radius, and is smaller when heavier elements are concentrated in the core.

## The tidal Love coefficient k_{2}

The tides tend to alter the shape of the planet. In addition to a mean shape, there are periodic variations, which are due to the variations of the distance between Mercury and the Sun.

The amplitude of these variations depend on the Love parameter k_{2}, which characterizes the response of the material to the periodic excitations. Actually, k_{2} depends on the frequency of excitation, in the specific case of Mercury k_{2}@88d and k_{2}@44d affect the gravity field. But distinguishing these two quantities requires a too high accuracy in the data, this is why k_{2} is often mentioned without precising the frequency involved.

If Mercury were spherical and fluid, k_{2} would be 1.5, while it would be null if Mercury were fully rigid. Actually, all the natural bodies are somewhere between these two end-members.

The frequency-dependence of the tides is based on the assumption that if you impose a slow deformation of a viscous body, it will not loose any internal energy and slowly recover its shape after (elastic deformation). However, rapid solicitations induce permanent deformations. The numbers associated with these two different regimes depend on the interior of the planet.

## In this paper

This study, *Mercury’s gravity tides, and spin from MESSENGER radio data*, by A.K. Verma and J.-L. Margot, has been accepted for publication in *Journal of Geophysical Research – Planets*. It presents

- an updated gravity field for Mercury,
- an updated Love number,
- an updated spin orientation.

These results are based on measurements of the instantaneous gravity field of Mercury. This is particularly interesting for the determination of the spin, since classical methods are based on the observation of the surface, while the gravity field is ruled by the whole planet. This means that here, the rotation of the whole planet is observed, not just its surface. This allows to constrain the possible differential rotation between the surface and the core.

For the gravity field of Mercury, a 40^{th} order solution is considered, because Mercury is something more complicated than a triaxial ellipsoid. The second order Stokes coefficients are consistent with previous studies, which were also based on MESSENGER data. Some higher-order coefficients are identified as well.

This is the second determination of the Love number k_{2} = 0.464, which implies than the mantle of Mercury is pretty hot.

## Some perspectives

We are some years away from the orbital insertion of the European / Japanese mission Bepi-Colombo, which is expected to be ten times more accurate than MESSENGER. So, results like the ones presented here are in some sense preparing the Bepi-Colombo’s measurements. This mission will also secure the results, and providing independent determinations.

Knowing Mercury is also a way to understand planetary formation. There are many discoveries of exoplanets, which orbit close to their parent star, but are so far from us that we cannot hope to send spacecrafts orbiting them. So, understand the way Mercury has been formed helps understanding the other planetary systems.

I hope that one day we will be able to measure the frequency-dependence of the Love numbers, this would be very helpful to constrain the tidal models.

## To know more

- The study, freely available on arXiV (thanks to the authors for sharing this)
- The web site of Jean-Luc Margot , see in particular the page on the rotation of Mercury
- The page of Ashok Kumar Verma on ResearchGate
- The mission MESSENGER
- The mission Bepi-Colombo
- Another resource on the rotation of Mercury

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