Category Archives: Planets

Hinting for Planet Nine in the orbits of Trans-Neptunian Objects

Hi There! Today I will present you a paper by Matthew Holman and Matthew Payne, entitled Observational constraints on Planet Nine: Astrometry of Pluto and other Trans-Neptunian Objects, which aims to derive constraints on the hypothetical Planet Nine from the orbits of small bodies, which orbit beyond the orbit of Neptune. For that, the authors investigate how an unknown, distant and massive planet, could improve the ephemerides of the known Trans-Neptunian Objects (TNOs). This study has recently been accepted for publication in The Astronomical Journal.

The quest for Planet Nine

Here is a longstanding pending question: is there a ninth planet on the Solar System? Some will answer: Yes, and its name is Pluto. But as you may know, Pluto has been reclassified in 2006 as a dwarf planet by the International Astronomical Union. So, is there another ninth planet, still to be discovered? In January 2016, Konstantin Batygin and Michael Brown, answered “probably yes” to this question, from the orbits of TNOs. They discovered that the clustering of their orbits could hardly be due to chance, and so there should be a cause, which has a gravity action. Since this study, several groups try to constrain its orbit and mass, while observers try to detect it.

The purpose of this post is to discuss the study of one of these groups. Let me briefly cite other ones (sorry for oblivion):

  • In 2014, Chad Trujillo and Scott Sheppard discovered a TNO, 2012VP113, whose apparent orbit seemed to be too difficult to explain with the known planets only. This made a case for the existence of the Planet Nine.
  • In 2015, a team led by the Brazilian astronomer Rodney Gomes, showed that a Planet Nine could explain an excess of bright object in the population of the most distant TNOs.
  • In January 2016, Batygin and Brown published their result, which triggered a bunch of other studies.
  • Hervé Beust, from Grenoble (France), showed from a statistical analysis that resonant effects with Neptune could explain the observed clustering,
  • Renu Malhotra, Kat Volk and Xianyu Wang, from the University of Arizona, considered that the largest TNOs could be in mean-motion resonance with the Planet Nine, i.e. that their orbital periods could be commensurate with the one of the Planet Nine. Such a configuration has a dynamical implication on the stability of these bodies. In such a case, the TNO Sedna would be in a 3:2 resonance with the Planet Nine.
  • A team led by Agnès Fienga, from the Observatoire de la Côte d’Azur (France), has suggested that a signature of the Planet Nine could be found in the deviation of the Cassini spacecraft, which currently orbits Saturn. The JPL (Jet Propulsion Laboratory, NASA) does not seem to believe in this option, and indicates that the spacecraft does not present any anomaly in its motion.
  • Gongjie Li and Fred Adams, based respectively at the Harvard-Smithsonian Center for Astrophysics, and at the University of Michigan, show that the orbit of the Planet Nine is pretty unlikely to be stable, because of passing stars close to the outer Solar System, which should have ejected it.
  • de la Fuente Marcos and de la Fuente Marcos, from Spain, reexamined the statistics, and concluded that there should be at least two massive perturbers beyond the orbit of Pluto
  • Matthew Holman and Matthew Payne, from the Harvard-Smithsonian Center for Astrophysics, tried to constrain the orbit of the Planet Nine from the orbits of the TNOs.

All this should result in the present architecture for the Solar System (AU stand for Astronomical Unit, i.e. ≈150 million km:

  • 1 AU: the Earth,
  • 5.2 AU: Jupiter,
  • 9.55 AU: Saturn,
  • 19.2 AU: Uranus,
  • 30.1 AU: Neptune,
  • 39.5 – 48 AU: the Kuiper Belt,
  • 39.5 AU: Pluto,
  • >50 AU: the scattered disk,
  • 67.8 AU: Eris
  • 259.3 AU: 2012VP113
  • 526.2 AU: Sedna,
  • 300 – 1500 AU: the Planet Nine,
  • 50,000 AU: the Oort Cloud,
  • 268,000 AU: Proxima Centauri, which is the closest known star beside the Sun.


The astrometry consists to measure the position of an object in the sky. Seen by a terrestrial observer, the sky is a spherical surface. You can determine two angles which will give the direction of the object, but no distance. These two angles are the right ascension and the declination.

Determining the right ascension and the declination of an object you observe is not that easy. It involves for example to have good reference points on the sky, whose positions are accurately known, with respect to which you will position your object. These reference points are usually stars, and their positions are gathered in catalogs. You should also consider the fact that an object is more than a dot, it appears on your image as a kind of a circle. To be accurate, you should determine the location of the center of the object from its light circle, due to light diffraction. You should in particular consider the fact that the center of the light is not necessarily the center of this object.

When all this is done, you have a right ascension and a declination with uncertainties, at a given date. This date is corrected from the light travel time, i.e. the position of an object we observe was the position of the object when the Solar light was refracted on its surface, not when we observe it. Gathering several observations permits to fit ephemerides of the considered body, i.e. a theory which gives its orbit at any time. These ephemerides are very convenient to re-observe this object, and to send a spacecraft to it…

Fitting an orbit

Ephemerides give you the orbit of a given body. Basically, the orbit of a Solar System body is an ellipse, on which the body is moving. For that, a set of 6 independent orbital elements shall be defined. The following set is an example:

  1. the semimajor axis,
  2. the eccentricity (a null eccentricity means that the orbit is circular; an elliptical orbit means that the eccentricity is smaller than 1),
  3. the inclination, usually with respect to the ecliptic, i.e. the orbital plane of the Earth,
  4. the pericentre, at which the distance Sun-body is the smallest,
  5. the ascending node, locating the intersection between the orbital plane and the ecliptic,
  6. the longitude, which locates the body on its orbit.

The first 5 of these elements are constant if you have only the Sun and an asteroid; in practice they have a time dependence due to the gravitational perturbations of the other bodies, in particular the giant planets, i.e. Jupiter, Saturn, Uranus and Neptune. This study aims at identifying the gravitational influence of the Planet Nine.

A numerical simulation gives you the orbit of an asteroid perturbed by the Sun and the giant planets. But for that, you need to know initial conditions, i.e. the location of the body at a given date. The initial conditions are derived from astrometric positions. Since the astrometry does not give exact positions but positions with some uncertainty, you may have many solutions to the problem. The best fit is the solution which minimizes what we call the residuals, or the O-C, for Observed Minus Calculated. All the O-C are gathered under a statistical quantity known as χ2. The best fit minimizes the χ2.

This study

The purpose of this study is to use 42,323 astrometric positions of TNOs with a semimajor axis larger than 30 AU, 6,677 of them involving Pluto. For that, the fitting algorithm not only includes the gravitational influence of the giant planets, but also of 10 large TNOs, and of the hypothetical Planet Nine, in considering two models: either the Planet Nine is moving on a circular orbit, or it is a fixed point-mass. Its expected orbital period, i.e. several thousands of years, is so large that no significant difference between the two models is expected, given the time span covered by the observations.

Indeed, the two models give pretty the same result. The authors split the sky into several tiles, to check the preferred locations for the Planet Nine, and it appears that for some locations the fit is better, while it is worse for some others.

They also find that if the Planet Nine has a mass of 10 Earth masses, then the distance of the Planet Nine to the Sun should be between 300 and 1,000 AU, while Batygin and Brown found it to be between 400 and 1,500 AU. This discrepancy could be explained by the presence of an another planet at a distance of 60 to 100 AU. In addition to that, the node of the Planet Nine seems to be aligned with the one of Pluto, which had already been noticed by other authors. This could reveal an enhanced dynamical interaction between them.

Finally the authors acknowledge that the astrometric positions have some inaccuracy, and that further observations could affect the results.

The quest for Planet Nine is very exciting, and I am pretty sure that new results will come in a next future!

To know more…

  • The study, made freely available by the authors here, thanks to them for sharing!
  • The webpage of Matthew Holman
  • The profile of Matthew Payne on ResearchGate
  • The press release relating the likely existence of the Planet Nine
  • The study by Trujillo and Sheppard
  • The study by Gomes et al.
  • The study by Batygin and Brown, freely available here
  • The study by Beust, also freely available here
  • The study by Malhotra et al., also freely available here
  • The study by Fienga et al., also freely available here
  • The study by de la Fuente Marcos and de la Fuente Marcos, also freely available here


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New clues on the interior of Mercury

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/cm3, while 1g/cm3 is expected for ice, 3.3 g/cm3 for silicates, and 8 g/cm3 for iron.

A more accurate model is to see Mercury has a triaxial ellipsoid. This requires to add two parameters in the gravity field: J2 and C22, also know as Stokes coefficients. A positive J2 means that the body is flattened at its poles, while C22 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 MR2, M being the mass and R the radius, and is smaller when heavier elements are concentrated in the core.


The tidal Love coefficient k2

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 k2, which characterizes the response of the material to the periodic excitations. Actually, k2 depends on the frequency of excitation, in the specific case of Mercury k2@88d and k2@44d affect the gravity field. But distinguishing these two quantities requires a too high accuracy in the data, this is why k2 is often mentioned without precising the frequency involved.

If Mercury were spherical and fluid, k2 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 40th 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 k2 = 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.


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