Modeling the shape of a planetary body

Hi there! Do you know the shape of the Moon? You say yes of course! But up to which accuracy? The surface of the Moon has many irregularities, which prompted Christian Hirt and Michael Kuhn to study the limits of the mathematics, in modeling the shape of the Moon. Their study, entitled Convergence and divergence in spherical harmonic series of the gravitational field generated by high-resolution planetary topography — A case study for the Moon, has recently been accepted for publication in Journal of Geophysical Research: Planets.

The shape of planetary bodies

If you look at a planetary body from far away (look at a star, look at Jupiter,…), you just see a point mass. If you get closer, you would see a sphere, if the body is not too small. Small bodies, let us say smaller than 100 km, can have any shape (may I call them potatoids?) If they are larger, the material almost arranges as a sphere, which gives the same gravity field as the point mass, provided you are out of the body. But if you look closer, you would see some polar flattening, due to the rotation of the body. And for planetary satellites, you also have an equatorial ellipticity, the longest axis pointing to the parent planet. Well, in that case, you have a triaxial ellipsoid. You can say that the sphere is a degree 0 approximation of the shape, and that the triaxial ellipsoid is a degree 2 approximation… but still an approximation.

A planetary body has some relief, mountains, basins… there are explanations for that, you can have, or have had, tectonic activity, basins may have been created by impacts, you can have mass anomalies in the interior, etc. This means that the planetary body you consider (in our example, the Moon), is not exactly a triaxial ellipsoid. Being more accurate than that becomes complicated. A way to do it is with successive approximations, in the same way I presented you: first a sphere, then a triaxial ellipsoid, then something else… but when do you stop? And can you stop, i.e. does your approximation converge? This study addresses this problem.

The Brillouin sphere

This problem is pretty easy when you are far enough from the body. You just see it as a sphere, or an ellipsoid, since you do not have enough resolution to consider the irregularities in the topography… by the way, I am tempted to make a confusion between topography and gravity. The gravity field is the way the mass of your body will affect the trajectory of the body with which it interacts, i.e. the Earth, Lunar spacecrafts… If you are close enough, you will be sensitive to the mass distribution in the body, which is of course linked to the topography. So, the two notions are correlated, but not fully, since the gravity is more sensitive to the interior.

But let us go back to this problem of distance. If you are far enough, no problem. The Moon is either a sphere, or a triaxial ellipsoid. If you get closer, you should be more accurate. And if you are too close, then you cannot be accurate enough.

This limit is given by the radius of the Brillouin sphere. Named after the French-born American physicist Léon Brillouin, this is the circumscribing sphere of the body. If your planetary body is spherical, then it exactly fills its Brillouin sphere, and this problem is trivial… If you are a potatoidal asteroid, then your volume will be only a fraction of this sphere, and you can imagine having a spacecraft inside this sphere.

The asteroid Itokawa in its Brillouin-sphere. Credit: JAXA.
The asteroid Itokawa in its Brillouin-sphere. Credit: JAXA.

The Moon is actually pretty close to a sphere, of radius 1737.4±1 km. But many mass anomalies have been detected, which makes its gravity field not that close to the one of the sphere, and you can be inside the equivalent Brillouin sphere (if we translate gravity into topography), in flying over the surface at low altitude.

Why modeling it?

Why trying to be that accurate on the gravity field / topography of a planetary object? I see at least two good reasons, please pick the ones you prefer:

  • to be able to detect the time variations of the topography and / or the gravity field. This would give you the tidal response (see here) of the body, or the evolution of its polar caps,
  • because it’s fun,
  • to be able to control the motion of low-altitude spacecrafts. This is particularly relevant for asteroids, which are somehow potatoidal (am I coining this word?)

You can object that the Moon may be not the best body to test the gravity inside the Brillouin sphere. Actually we have an invaluable amount of data on the Moon, thanks to the various missions, the Lunar Laser Ranging, which accurately measures the Earth-Moon distance… Difficult to be more accurate than on the Moon…

The goal of the paper is actually not to find something new on the Moon, but to test different models of topography and gravity fields, before using them on other bodies.

Spherical harmonics expansion

Usually the gravity field (and the topography) is described as a spherical harmonics expansion, i.e. you model your body as a sum of waves with increasing frequencies, over two angles, which are the latitude and the longitude. This is why the order 0 is the exact sphere, the order 2 is the triaxial ellipsoid… and in raising the order, you introduce more and more peaks and depressions in your shape… In summing them, you should have the gravity field of your body… if your series converge. It is usually assume that you converge outside the Brillouin sphere… It is not that clear inside.

To test the convergence, you need to measure a distance between your series and something else, that you judge relevant. It could be an alternative gravitational model, or just the next approximation of the series. And to measure the distance, a common unit is the gal, which is an acceleration of 1 cm/s2 (you agree that gravity gives acceleration?). In this paper, the authors checked differences at the level of the μgal, i.e. 1 gal divided by 1 million.

Methodology

In this study, the authors used data from two sources:

  • high-resolution shape maps from the Lunar Orbiter Laser Altimeter (LOLA),
  • gravity data from the mission GRAIL (Gravity Recovery And Interior Laboratory),

and they modeled 4 gravity fields:

  1. Topography of the surface,
  2. Positive topographic heights, i.e. for basins the mean radius was considered, while the exact topography was considered for mountains,
  3. “Brillouin-sphere”, at a mean altitude of 11 km,
  4. “GRAIL-sphere”, at a mean altitude of 23 km.

In each of these cases, the authors used series of spherical harmonics of orders between 90 (required spatial resolution: 60.6 km) and 2,160 (resolution: 2.5 km). In each case, the solution with spherical harmonics was compared with a direct integration of the potential of the body, for which the topography is discretized through an ensemble of regularly-shaped elements.

Results

And here are the results:

Not surprisingly, everything converges in the last two cases, i.e. altitudes of 11 and 23 km. However, closer to the surface the expansion in spherical harmonics fails from orders 720 (case 1) and 1,080 (case 2), respectively. This means that adding higher-order harmonics does not stabilize the global solution, which can be called divergence. The authors see from their calculations that this can be predicted from the evolution of the amplitude of the terms of the expansion, with respect to their orders. To be specific, their conclusion is summarized as follows:

A minimum in the degree variances of an external potential model foreshadows divergence of the spherical harmonic series expansions at points inside the Brillouin-sphere.

 

My feeling is that this study should be seen as a laboratory test of a mathematical method, i.e. testing the convergence of the spherical harmonics expansion, not on a piece of paper, but in modeling a real body, with real data. I wonder how the consideration of time variations of the potential would affect these calculations.

To know more…

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

A polar resonant asteroid

Hi there! Did you know that an asteroid could be resonant and in polar orbit? Yes? No? Anyway, one of them has been confirmed as such, i.e. this body was already discovered, known to be on a polar orbit, but it was not known to be in mean-motion resonance with Neptune until now. This is the opportunity for me to present you First transneptunian object in polar resonance with Neptune, by M.H.M. Morais and F. Namouni. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

Polar asteroids

The planets of the Solar System orbit roughly in the same plane. In other words, they have small mutual inclinations. However, asteroids are much more scattered, and can have any inclination with respect to the ecliptic, i.e. the orbital plane of the Earth, even if low inclinations are favored.

Two angles are needed to orientate an orbit:

  • the ascending node, which varies between 0 and 360°, and which is the angle between a reference and the intersection between the ecliptic and the orbital plane,
  • the inclination, which is the angle between the ecliptic and the orbital plane. It varies between 0° and 180°.

So, an almost planar orbit means an inclination close to 0° or close to 180°. Orbits close to 0° are prograde, while orbits close to 180° are retrograde. However, when your inclination is close to 90°, then you have a polar orbit. There are prograde and retrograde polar orbits, whether the inclination is smaller (prograde) or larger (retrograde) than 90°.

There are 7 known Trans-Neptunian Objects with an eccentricity smaller than 0.86 and inclination between 65 and 115°, hence 7 known polar TNOs. You can find them below:

Semimajor axis Eccentricity Inclination Ascending node Period
(471325) 2011 KT19 (Niku) 35.58 AU 0.33 110.12° 243.76° 212.25 y
2008 KV42 (Drac) 41.44 AU 0.49 103.41° 260.89° 266.75 y
2014 TZ33 38.32 AU 0.75 86.00° 171.79° 237.20 y
2015 KZ120 46.07 AU 0.82 85.55° 249.98° 312.70 y
(127546)2002 XU93 67.47 AU 0.69 77.95° 90.39° 554.18 y
2010 WG9 52.90 AU 0.65 70.33° 92.07° 384.77 y
2017 CX33 73.97 AU 0.86 72.01° 315.88° 636.21 y

These bodies carry in their names their year of discovery. As you can see, the first of them has been discovered only 15 years ago. We should keep in mind that TNOs orbit very far from the Earth, this is why they are so difficult to discover, polar or not.

The last of them, 2017 CX33, is so recent that the authors did not study it. A recent discovery induces a pretty large uncertainty on the orbital elements, so waiting permits to stay on the safe side. Among the 6 remaining, 4 (Niku, Drac, 2002 XU93 and 2010 WG9) share (very) roughly the same orbit, 2 of them being prograde, while the others two are retrograde. This happened very unlikely by chance, but the reason for this rough alignment is still a mystery.

Orbits of the polar TNOs, in the x-y plane.
Orbits of the polar TNOs, in the x-y plane.
Orbits of the polar TNOs, in the y-z plane.
Orbits of the polar TNOs, in the y-z plane.

The study I present you today investigated the current dynamics of these bodies, and found a resonant behavior for one of them (Niku).

Behavior of the resonant asteroids

By resonant behavior, I mean that an asteroid is affected by a mean-motion resonance with a planet. This means that it makes a given (integer) number of revolutions around the Sun, while the planet makes another number of revolutions. Many outcomes are possible. It can slowly enhance the eccentricity and / or the inclination, which could eventually lead to a chaotic behavior, instability, collision… it could also protect the body from close encounters…

It usually translates into an integer combination of the fundamental frequencies of the system (orbital frequencies, frequencies of precession of the nodes and pericentres), which is null, and this results in an integer combination of angles positioning the asteroid of the planet, which oscillates around a given number instead of circulating. In other words, this angle is bounded.

Another point of interest is how the asteroid has been trapped into the resonance. A resonance is between two interacting bodies, but the mass ratio between an asteroid and a planet implies that the planet is insensitive to the gravitational action of the asteroid, and so the asteroid is trapped by the planet. The fundamental frequencies of the orbital motion are controlled by the semimajor axes of the two bodies, so a trapping into a resonance results from a variation of the semimajor axes. Models of formation of the Solar System suggest that the planets have migrated, this could be a cause. Another cause is close encounters between planets and asteroids, which result in abrupt changes in the trajectory of the asteroid. And this is probably the case here: Niku got trapped after a close encounter.

Numerical and analytical study

The authors used both numerical and analytical methods to get, understand, and secure their results.

Numerical study

The authors ran long-term numerical simulations of the orbital motion of the 6 relevant asteroids, perturbed by the planets. They ran 3 kinds of simulations: 2 with different integrators (algorithms) over 400 kyr and 100 Myr and 8 planets, and one over 400 Myr and the four giant planets. With less planets, you go faster. Moreover, since the inner planets have shorter orbital periods, removing them allows you to increase the time-step, and thus go further in time, inward and backward. In each of these simulations, the authors cloned the asteroids to take into consideration the uncertainty on the orbital elements. They used for that a well-known devoted code, MERCURY.

Analytical study

Numerical studies give you an idea of the possible dynamical states of a system, but you need to write down equations to fully understand it. Beside these numerical simulations, the authors wrote a dynamical theory of resonant polar orbits, in another paper (or here).

This consists in reducing the equations to the only terms, which are useful to reproduce the resonant dynamics. For that, you keep the secular variations, i.e. precessions of the nodes and pericentres, and the term involving the resonant argument. This is a kind of averaged dynamics, in which all of the small oscillations of the orbital elements have been dropped. To improve the relevance of the model, the authors used orbital elements which are based on the barycenter (center of mass) of the whole Solar System instead on the Sun only. This is a small correction, since the barycenter is at the edge of the Sun, but the authors mention that it improves their results.

Results

Niku, i.e. (471325) 2011 KT19, is trapped into a 7:9 mean-motion resonance with Neptune. In other words, it makes 7 revolutions around the Sun (sorry: the barycenter of the Solar System) while Neptune makes 9. More precisely, its resonant argument is φ=9λ-7λN-4ϖ+2Ω, where λ and λN are the mean longitudes of the asteroid and of Neptune, respectively, ϖ is the longitude of its pericenter, and Ω is the one of its ascending node. Plotting this argument shows a libration around 180°. Niku has been trapped in this resonance after a close encounter with Neptune, and should leave this resonance in 16±11 Myr. This means that all of the numerical simulations involving Niku show a resonant object, however they disagree on the duration of the resonance.
Their might be another resonant object: a few simulations suggest that Drac, i.e. 2008 KV42 is in a 8:13 mean-motion resonance with Neptune.

To know more

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

A new contact binary

Hi there! Today I will tell you on the discovery that an already known Trans-Neptunian Object is in fact probably a contact binary. This is the opportunity for me to present you 2004 TT357: A potential contact binary in the Trans-Neptunian Belt by Audrey Thirouin, Scott S. Sheppard, and Keith S. Noll. This study has recently been published in The Astrophysical Journal.

2004 TT357‘s facts

As suggested by its name, 2004 TT357 was discovered in 2004. More precisely in August by a team led by Marc W. Buie, at Kitt Peak Observatory, Arizona, USA, on the 4-m Mayall telescope. From its magnitude, its radius is estimated to be between 87 and 218 km, depending on the albedo of the asteroid, i.e. the fraction of Solar light which is reflected by its surface. This albedo is unknown. You can find below its orbital elements.

Orbital elements of 2004 TT357
Semimajor axis 54.97 AU
Eccentricity 0.43
Inclination
Orbital period 408 y

These elements show that 2004 TT357 is in a 5:2 mean-motion resonance in Neptune, i.e. it performs 2 revolutions around the Sun while Neptune makes 5. This makes 2004 TT357 a Scaterred Disc Resonant Object. Its high eccentricity is probably at least partly due to this resonance.

Contact binaries

In astronomy, a binary object is a group of two objects, which are so linked together that they orbit around a common barycenter. Of course, their separation is pretty small. There are binary stars, here we speak about binary asteroids.
A contact binary is a kind of extreme case, in which the two components touch each other. In some sense, this is a single object, but with two different lobes. This was probably a former classical binary, which lost enough angular momentum so that the two objects eventually collided, but slowly enough to avoid any catastrophic outcome. It is thought that there is a significant fraction of contact binaries in the Solar System, i.e. between 5% and 50%, depending on the group you are considering.

Characterizing a known object as a contact binary is not an easy task, particularly for the Trans-Neptunian Objects, because of their distance to us. Among them, only (139775) 2001 QG298 is a confirmed contact binary, while 2003 SQ317 and (486958) 2014 MU69 are probable ones. This study concludes that 2004 TT357 is a probable one as well.

Observations at Lowell Observatory

Lowell Observatory is located in Flagstaff, Arizona, USA. It has been founded by Percival Lowell in 1894, and among its achievements is the discovery of the former planet Pluto in 1930, by Clyde Tombaugh. Currently, the largest of its instruments is the 4.3-m Discovery Channel Telescope (DCT), which has been partly funded by Discovery Communications. This telescope has its first light in April 2012, it is located in the Coconino National Forest near Happy Jack, Arizona, at an altitude of 2,360 meters.

The Discovery Channel Telescope. © Lowell Observatory
The Discovery Channel Telescope. © Lowell Observatory

The authors used this telescope, equipped with the Large Monolithic Imager (LMI). They acquired two sets of observation, in December 2015 and February 2017, during which they posed during 600 and 700 seconds, respectively. 2004 TT357 had then a mean visual magnitude of 22.6 and 23, respectively.

The Large Monolithic Imager. © Lowell Observatory
The Large Monolithic Imager. © Lowell Observatory

Analyzing the data

You can find below the photometric measurements of 2004 TT357.

The first set of observations. The measurements are represented with the uncertainties.
The first set of observations. The measurements are represented with the uncertainties.
The second set of observations. The measurements are represented with the uncertainties.
The second set of observations. The measurements are represented with the uncertainties.

We can see pretty significant variations of the incoming light flux, these variations being pretty periodic. This periodicity is the signature of the rotation of the asteroid, which does not always present the same face to the terrestrial observer. From these lightcurves, the authors measure a rotation period of 7.79±0.01 h. From the curves, the period seems twice smaller, but if we consider that the asteroid should be an ellipsoid, then its geometrical symmetries tell us that our line of sight should be aligned twice with the long axis and twice with the short axis during a single period. So, during a rotation period, we should see two minimums and two maximums. This assumes that we are close to the equatorial plane.

Another interesting fact is the pretty high amplitude of variation of the incident light flux. If you are interested in it, go directly to the next section. Before that, I would like to tell you how this period of 7.79±0.01 h has been determined.

The authors used 2 different algorithms:

  • the Lomb periodogram technique,
  • the phase dispersion minimization (PDM).

Usually periodic signals are described as sums of sinusoids, thanks to Fourier transforms. Unfortunately, Fourier is not suitable for unevenly-spaced data. The Lomb (or Lomb-Scargle) periodogram technique consists to fit a sinusoid to the data, thanks to the least-squares method, i.e. you minimize the squares of the departure of your signal from a sinusoid, in adjusting its amplitude, phase, and frequency. PDM is an astronomical adaptation of data folding. You guess a period, and you split your full time interval into sub-intervals, which duration is the period you have guessed. Then you superimpose them. If this the period you have guessed is truly a period of the signal, then all of your time intervals should give you pretty the same signal. If not, then the period you have guessed is not a period of the signal.

Let us go back now to the variations in the amplitude.

Physical interpretation

The authors assume that periodic magnitude variations could have 3 causes:

  • Albedo variations
  • Elongation of the asteroid
  • Two bodies, i.e. a binary.

The albedo quantify the portion of Solar flux, which is reflected by the surface. Here, the variations are too large to be due to the variations of the albedo.

The authors estimate that, if 2004 TT357 were a single, ellipsoidal body, then a/b = 2.01 and c/a = 0.38, a,b, and c being the 3 axis of the ellipsoid. This is hardly possible if the shape corresponds to an equilibrium figure (hydrostatic equilibrium, giving a Jacobi ellipsoid). Moreover, this would mean that 2004 TT357 would have been ideally oriented… very unlikely

As a consequence, 2004 TT357 is probably a binary, with a mass ratio between 0.4 and 0.8. Hubble Space Telescope observed 2004 TT357 in 2012, and detected no companion, which means it is probably a contact binary. Another way to detect a companion is the analysis of a stellar occultation (see here). Fortunately for us, one will occur in February 2018.

A star occultation in February 2018

On 5 February 2018, 2004 TT357 shall occult the 12.8-magnitude star 2UCAC 38383610, in the constellation Taurus, see here. This occultation should be visible from Brazil, and provide us new data which would help to determine the nature of 2004 TT357. Are you interested to observe?

To know more

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

How rough is Mercury?

Hi there! Today I will tell you on the smoothness of the surface of Mercury. This is the opportunity for me to present The surface roughness of Mercury from the Mercury Laser Altimeter: Investigating the effects of volcanism, tectonism, and impact cratering by H.C.M. Susorney, O.S. Barnouin, C.M. Ernst and P.K. Byrne, which has recently been published in Journal of Geophysical Research: Planets. This paper uses laser altimeter data provided by the MESSENGER spacecraft, to measure the regularity of the surface in the northern hemisphere.

The surface of Mercury

I already had the opportunity to present Mercury on this blog. This is the innermost planet of the Solar System, about 3 times closer to the Sun than our Earth. This proximity makes space missions difficult, since they have to comply with the gravitational action of the Sun and with the heat of the environment. This is why Mercury has been visited only by 2 space missions: Mariner 10, which made 3 fly-bys in 1974-1975, and MESSENGER, which orbited Mercury during 4 years, between 2011 and 2015. The study of MESSENGER data is still on-going, the paper I present you today is part of this process.

Very few was known from Mercury before Mariner 10, in particular we just had no image of its surface. The 3 fly-bys of Mariner 10 gave us almost a full hemisphere, as you can see below. Only a small stripe was unknown.

Mercury seen by Mariner 10. © NASA.
Mercury seen by Mariner 10. © NASA.

And we see on this image many craters! The details have different resolutions, since this depends on the distance between Mercury and the spacecraft when a given image was taken. This map is actually a mosaic.
MESSENGER gave us full maps of Mercury (see below).

Mercury seen by MESSENGER. © USGS
Mercury seen by MESSENGER. © USGS

Something that may be not obvious on the image is a non-uniform distribution of the craters. So, Mercury is composed of cratered terrains and smooth plains, which have different roughnesses (you will understand before the end of this article).
Craters permit to date a terrain (see here), i.e. when you see an impact basin, this means that the surface has not been renewed since the impact. You can even be more accurate in dating the impact from the relaxation of the crater. However, volcanism brings new material at the surface, which covers and hides the craters.

This study focuses on the North Pole, i.e. latitudes between 45 and 90°N. This is enough to have the two kinds of terrains.

Three major geological processes

Three processes affect the surface of Mercury:

  1. Impact cratering: The early Solar System was very dangerous from this point of view, having several episodes of intense bombardments in its history. Mercury was particularly impacted because the Sun, as a big mass, tends to focus the impactors in its vicinity. It tends to rough the surface.
  2. Volcanism: In bringing new and hot material, it smoothes the surface,
  3. Tectonism: Deformation of the crust.

If Mercury had an atmosphere, then erosion would have tended to smooth the surface, as on Earth. Irrelevant here.

To measure the roughness, the authors used data from the Mercury Laser Altimeter (MLA), one of the instruments of MESSENGER.

The Mercury Laser Altimeter (MLA) instrument

This instrument measured the distance between the spacecraft and the surface of Mercury from the travel time of light emitted by MLA and reflected by the surface. Data acquired on the whole surface permitted to provide a complete topographic map of Mercury, i.e. to know the variations of its radius, detect basins and mountains,… The accuracy and the resolution of the measurements depend on the distance between the spacecraft and the surface, which had large variations, i.e. between 200 and 10,300 km. The most accurate altimeter data were for the North Pole, this is why the authors focused on it.

Roughness indicators

You need at least an indicator to quantify the roughness, i.e. a number. For that, the authors work on a given baseline on which they had data, removed a slope, and calculated the RMS (root mean square) deviation, i.e. the average squared deviation to a constant altitude, after removal of a slope. When you are on an inclined plane, then your altitude is not constant, but the plane is smooth anyway. This is why you remove the slope.

But wait a minute: if you are climbing a hill, and you calculate the slope over 10 meters, you have the slope you are climbing… But if you calculate it over 10 km, then you will go past the summit, and the slope will not be the same, while the summit will affect the RMS deviation, i.e. the roughness. This means that the roughness depends on the length of your baseline.

This is something interesting, which should be quantified as well. For this, the authors used the Hurst exponent H, such that ν(L) = ν0LH, where L is the length of the baseline, and ν the standard deviation. Of course, the data show that this relation is not exact, but we can say it works pretty well. H is determined in fitting the relation to the data.

Results

To summarize the results:

  • Smooth plains: H = 0.88±0.01,
  • Cratered terrains: H = 0.95±0.01.

The authors allowed the baseline to vary between 500 m and 250 km. The definition of the Hurst exponent works well for baselines up to 1.5 km. But for any baseline, the results show a bimodal distribution, i.e. two kinds of terrains, which are smooth plains and cratered terrains.

It is tempting to compare Mercury to the Moon, and actually the results are consistent for cratered terrains. However, the lunar Maria seem to have a slightly smaller Hurst exponent.

To know more

That’s it for today! The next mission to Mercury will be Bepi-Colombo, scheduled for launch in 2018 and for orbital insertion in 2025. Meanwhile, please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

The dynamics of the Quasi-Satellites

Hi there! After reading this post, you will know all you need to know on the dynamics of quasi-satellites. This is the opportunity to present you On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited, by Alexandre Pousse, Philippe Robutel and Alain Vienne. This study has recently been published in Celestial Mechanics and Dynamical Astronomy.

The 1:1 mean-motion resonance at small eccentricity

(see also here)

Imagine a pretty simple case: the Sun, a planet with a keplerian motion around (remember: its orbit is a static ellipse), and a very small third body. So small that you can neglect its mass, i.e. it does not affect the motion of the Sun and the planet. You know that the planet has no orbital eccentricity, i.e. the static ellipse serving as an orbit is actually a circle, and that the third body (let us call it the particle) has none either. Moreover, we want the particle to orbit in the same plane than the planet, and to have the same revolution period around the Sun. These are many conditions.
Under these circumstances, mathematics (you can call that celestial mechanics) show us that, in the reference frame which is rotating with the planet, there are two stable equilibriums 60° ahead and astern the planet. These two points are called L4 and L5 respectively. But that does not mean that the particle is necessary there. It can have small oscillations, called librations around these points, the resulting orbits being called tadpole orbits. It is even possible to have orbits enshrouding L4 and L5, this results in large librations orbits, called horseshoe orbits.

All of these configurations are stable. But remember: the planet is much less massive than the Sun, the particle is massless, the orbits are planar and circular… Things become tougher when we relax one of these assumptions. And the authors assumed that the particle had a significant eccentricity.

At high eccentricities: Quasi-satellites

Usually, increasing the eccentricity destabilizes you. This is still true here, i.e. co-orbital orbits are less stable when eccentric. But increasing the eccentricity also affects the dynamical structure of your problem in such a way that other dynamical configurations may appear. And this is the case here: you have an equilibrium where your planet lies.

Ugh, what does that mean? If you are circular, then your particle is at the center of your planet… Nope, impossible. But wait a minute: if you oscillate around this position without being there… yes, that looks like a satellite of the planet. But a satellite is under the influence of the planet, not of the star… To be dominated by the star, you should be far enough from the planet.

I feel the picture is coming… yes, you have a particle on an eccentric orbit around the star, the planet being in the orbit. And from the star, this looks like a satellite. Funny, isn’t it? And such bodies exist in the Solar System.

Orbit of a quasi-satellite. It follows the planet, but orbits the star.
Orbit of a quasi-satellite. It follows the planet, but orbits the star.

Known quasi-satellites

Venus has one known quasi-satellite, 2002 VE68. This is a 0.4-km body, which has been discovered in 2002. Like Venus, it orbits the Sun in 225 days, but has an orbital eccentricity of 0.41, while the one of Venus is 0.007. It is thought to be a quasi-satellite of Venus since 7,000 years, and should leave this configuration in some 500 years.

The Earth currently has several known quasi-satellites, see the following table:

Known quasi-satellites of the Earth
Name Eccentricity Inclination Stability
(164207) 2004 GU9 0.14 13.6° 1,000 y
(277810) 2006 FV35 0.38 7.1° 10,000 y
2013 LX28 0.45 50° 40,000 y
2014 OL339 0.46 10.2° 1,000 y
(469219) 2016 HO3 0.10 7.8° 400 y

These bodies are all smaller than 500 meters. Because of their significant eccentricities, they might encounter a planet, which would then affect their orbits in such a way that the co-orbital resonance would be destabilized. However, significant inclinations limit the risk of encounters. Some bodies switch between quasi-satellite and horseshoe configurations.

Here are the known quasi-satellites of Jupiter:

Known quasi-satellites of Jupiter
Name Eccentricity Inclination Stability
2001 QQ199 0.43 42.5° > 12,000 y
2004 AE9 0.65 1.6° > 12,000 y
329P/LINEAR-Catalina 0.68 21.5° > 500 y
295P/LINEAR 0.61 21.1° > 2,000 y

329P/LINEAR-Catalina and 295P/LINEAR being comets.

Moreover, Saturn and Neptune both have a confirmed quasi-satellite. For Saturn, 2001 BL41 should leave this orbit in about 130 years. It has an eccentricity of 0.29 and an inclination of 12.5°. For Neptune, (309239) 2007 RW10 is in this state since about 12,500 years, and should stay in it for the same duration. It has an orbital eccentricity of 0.3, an inclination of 36°, and a diameter of 250 km.

Understanding the dynamics

Unveiling the dynamical/mathematical structure which makes the presence of quasi-satellites possible is the challenge accepted by the authors. And they succeeded. This is based on mathematical calculation, in which you write down the equations of the problem, you expand them to retain only what is relevant, in making sure that you do not skip something significant, and you manipulate what you have kept…

The averaging process

The first step is to write the Hamiltonian of the restricted planar 3-body problem, i.e. the total energy of a system constituted by the Sun, the planet, and the massless particle. The dynamics is described by so-called Hamiltonian variables, which allow interesting mathematical properties…
Then you expand and keep what you need. One of the pillars of this process is the averaging process. When things go easy, i.e. when your system is not chaotic, you can describe the dynamics of the system as a sum of sinusoidal contributions. This is straightforward to figure out if you remember that the motions of the planets are somehow periodic. Somehow means that these motions are not exactly sinusoidal, but close to it. So, you expand it in series, in which other sinusoids (harmonics) appear. And you are particularly interested in the one involving λ-λ’, i.e. the difference between the mean longitude of the planet and the particle. This makes sense since they are in the co-orbital configuration, that particular angle should librate with pretty small oscillations around a given value, which is 60° for tadpole orbits, 180° for horseshoes, and 0° for quasi-satellites. Beside this, you have many small oscillations, in which you are not interested. Usually you can drop them in truncating your series, but actually you just average them, since they average to 0. This is why you can drop them.
To expand in series, you should do it among a small parameter, which is usually the eccentricity. This means that your orbit looks pretty like a circle, and the other terms of the series represent the difference with the circle. But here there is a problem: to get quasi-satellite orbits, your eccentricity should be large enough, which makes the analytical calculation tougher. In particular, it is difficult to guarantee their convergence. The authors by-passed this problem in making numerical averaging, i.e. they computed numerically the integrals of the variables of the motion over an orbital period.

Once they have done this, they get a simplified system, based on one degree-of-freedom only. This is a pair of action-angle variables, which will characterize your quasi-satellite orbit. This study also requires to identify the equilibriums of the system, i.e. to identify the existing stable orbits.

Perspectives

So, this study is full of mathematical calculations, aiming at revisiting this problem. The authors mention as possible perspective the study of resonances between the planets, which disturb the system, and the proper frequency of the quasi-satellite orbit. This is the oscillating frequency of the angle characterizing the orbit, and if it is equal to a frequency already present in the system, it could have an even more interesting dynamics, e.g. transit between different states (quasi-satellite / horsehoe,…).

To know more…

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