Tag Archives: numerical methods

Triton: a cuckoo around Neptune

Hi there! Did you know that Neptune had a prominent satellite, i.e. Triton, on a retrograde orbit? This is so unusual that it is thought to have been trapped, i.e. was originally an asteroid, and has not been formed in the protoneptunian nebula. The study I present you today, Triton’s evolution with a primordial Neptunian satellite system, by Raluca Rufu and Robin M. Canup, explains how Triton ejected the primordial satellites of Neptune. This study has recently been published in The Astronomical Journal.

The satellites of Uranus and Neptune

We are tempted to see the two planets Uranus and Neptune as kinds of twins. They are pretty similar in size, are the two outermost known planets in the Solar System, and are gaseous. A favorable orbital configuration made their visitation possible by the spacecraft Voyager 2 in 1986 and 1989, respectively.

Among their differences are the high obliquity of Uranus, the presence of rings around Uranus while Neptune displays arcs, and different configurations in their system of satellites. See for Uranus:

Semimajor axis Eccentricity Inclination Radius
Miranda 5.12 Ru 0.001 4.338° 235.8 km
Ariel 7.53 Ru 0.001 0.041° 578.9 km
Umbriel 10.49 Ru 0.004 0.128° 584.7 km
Titania 17.20 Ru 0.001 0.079° 788.9 km
Oberon 23.01 Ru 0.001 0.068° 761.4 km
Puck 3.39 Ru 0 0.319° 81 km
Sycorax 480.22 Ru 0.522 159.420° 75 km

I show on this table the main satellites of Uranus, and we can see that the major ones are at a reasonable distance (in Uranian radius Ru) of the planet, and orbit almost in the same plane. The orbit of Miranda is tilted thanks to a past mean-motion resonance with Umbriel, which means that it was originally in the same plane. So, we can infer that these satellites were formed classically, i.e. the same way as the satellites of Jupiter, from a protoplanetary nebula, in which the planet and the satellites accreted. An exception is Sycorax, which is very far, highly inclined, and highly eccentric. As an irregular satellite, it has probably been formed somewhere else, as an asteroid, and been trapped by the gravitational attraction of Uranus.

Now let us have a look at the system of Neptune:

Semimajor axis Eccentricity Inclination Radius
Triton 14.41 Rn 0 156.865° 1353.4 km
Nereid 223.94 Rn 0.751 7.090° 170 km
Proteus 4.78 Rn 0 0.075° 210 km
Larissa 2.99 Rn 0.001 0.205° 97 km

Yes, the main satellite seems to be an irregular one! It does not orbit that far, its orbit is (almost) circular, but its inclination is definitely inconsistent with an in situ formation, i.e. it has been trapped, which has been confirmed by several studies. Nereid is much further, but with a so eccentric orbit that it regularly enters the zone, which is dynamically perturbed by Triton. You can also notice the absence of known satellites between 4.78 and 14.41 Neptunian radii. This suggests that this zone may have been cleared by the gravitational perturbation of a massive body… which is Triton. The study I present you simulates what could have happened.

A focus on Triton

Before that, let us look at Triton. The system of Neptune has been visited by the spacecraft Voyager 2 in August 1989, which mapped 40% of the surface of Triton. Surprisingly, it showed a limited number of impact craters, which means that the surface has been renewed, maybe some 1 hundred of millions of years ago. Renewing the surface requires an activity, probably cryovolcanism as on the satellite of Saturn Enceladus, which should has been sustained by heating. Triton was on the action of the tides raised by Neptune, but probably not only, since tides are not considered as efficient enough to have circularized the orbit. The tides have probably been supplemented by gravitational interactions with the primordial system of Neptune, i.e. satellites and / or disk debris. If there had been collisions, then they would have themselves triggered heating. As a consequence of this heating, we can expect a differentiated structure.

Moreover, Triton orbits around Neptune in 5.877 days, on a retrograde orbit, while the rotation of Neptune is prograde. This configuration, associated with the tidal interaction between Triton and Neptune, makes Triton spiral very slowly inward. In other words, it will one day be so close to Neptune that it will be destroyed, and probably create a ring. But we will not witness it.

A numerical study with SyMBA

This study is essentially numerical. It aimed at modeling the orbital evolution of Triton, in the presence of Nereid and the putative primordial satellites of Neptune. The authors assumed that there were 4 primordial satellites, with different initial conditions, and considered 3 total masses for them: 0.3, 1, and 3 total masses of the satellites of Uranus. For each of these 3 masses, they ran 200 numerical simulations.

The simulations were conducted with the integrator (numerical code) SyMBA, i.e. Symplectic Massive Body Algorithm. The word symplectic refers to a mathematical property of the equations as they are written, which guarantee a robustness of the results over very long timescales, i.e. there may be an error, but which does not diverge. It may be not convenient if you make short-term accurate simulations, for instance if you want to design the trajectory of a spacecraft, but it is the right tool for simulating a system over hundreds of Myrs (millions of years). This code also handles close encounters, but not the consequences of impacts. The authors bypassed this problem in treating the impacts separately, determining if there were disrupting, and in that case estimated the timescales of reaccretion.

Results

The authors found, consistently with previous studies, that the interaction between Triton and the primordial system could explain its presently circular orbit, i.e. it damped the eccentricity more efficiently than the tides. Moreover, the interaction with Triton caused collisions between the primordial moons, but usually without disruption (hit-and-run impacts). In case of disruption, the authors argue that the reaccretion would be fast with respect of the time evolution of the orbit of Triton, which means that we could lay aside the existence of a debris disk.

Moreover, they found that the total mass of the primordial system had a critical role: for the heaviest one, i.e. 3 masses of the Uranian system, Triton had only small chances to survive, while it had reasonable chances in the other two cases.

Something frustrating when you try to simulate something that happened a few hundreds of Myrs ago is that you can at the best be probabilistic. The study shows that a light primordial system is likelier to have existed than a heavy one, but there are simulations with a heavy system, in which Triton survives. So, a heavy system is not prohibited.

The study and its authors

  • The study, which is available as free article. The authors probably paid extra fees for that, many thanks to them! You can also look at it on arXiv.
  • A conference paper on the same study,
  • The ResearchGate profile of Raluca Rufu,
  • The Homepage of Robin M. Canup.

Before closing this post, I need to mention that the title has been borrowed from Matija Ćuk (SETI, Mountain View, CA), who works on this problem as well (see these two conference abstracts here and here).

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.

Rotation and activity of a comet

Hi there! We, Earthians, are regularly visited by periodic comets, the most famous one being probably 1P/Halley, which will visit us in 2061. Since we cannot wait, we study others of that kind. Today I tell you about 49P / Arend-Rigaux. This is the opportunity for me to present you The rotation and other properties of Comet 49P/Arend-Rigaux, 1984 – 2012, by Nora Eisner, Matthew M. Knight and David G. Schleicher. This study has recently been published in The Astronomical Journal.

The comet 49P / Arend-Rigaux

The comet 49P / Arend-Rigaux has been discovered in February 1951 at the Royal Observatory of Belgium, by Sylvain Arend and Fernand Rigaux. It is a periodic comet of the Jupiter family, i.e. with a period smaller than 20 years. Its period is actually 6.71 years, its semimajor axis 3.55 AU (astronomical units, 1 AU being 150 millions km, i.e. the Sun-Earth distance), its eccentricity 0.6, and its orbital inclination 19°, with respect to the ecliptic. These numbers are extracted from the JPL Small-Body Database Browser, and are calculated at the date Apr 6, 2010. I have plotted below the distances Sun-comet and Earth-comet.

Distance to the Sun.
Distance to the Sun.
Distance to the Earth.
Distance to the Earth.

The distance to the Sun clearly shows the periodic variations. The orbit of the Earth is at 1 AU, the one of Mars at 1.5 AU, and the one of Jupiter at 5.2 AU. Every 6.71 years, the comet reaches its perihelion, i.e. minimizes its distance to the Sun. This proximity warms the comet and provokes an excess of cometary activity, i.e. sublimation of dirty ice. At these occasions, the distance with the Earth is minimized, but with variations due to the orbital motion of the Earth. We can see for instance that the comet gets pretty close to the Earth in 1951 (when it was discovered), in 1984, and in early 2032. These are favorable moments to observe it. The paper I present you today is mainly (but not only) based on photometric observations made between January and May 2012, at Lowell Observatory.

Observations at Lowell Observatory

Lowell Observatory is located close to Flagstaff, AZ (USA). It was founded by the famous Percival Lowell in 1894, and is the place where Clyde Tombaugh discovered Pluto, in 1930. Among its facilities is the 4.28 m Discovery Channel Telescope, but most of the data used in this study were acquired with the 1.1 m Hall telescope, which is devoted to the study of comets, asteroids, and Sun-like stars. The authors also used a 79 cm telescope. The observations were made in the R(ed) band.

The data

Besides these 33 observation nights during the first half of 2012, the authors used data acquired close to the 1984 and 2005 perihelion passages, even if the 2005 ones revealed unusable. The observations consists to measure the magnitude (somehow, the luminosity) of the comet, in correcting for atmospheric problems, so as to be able to detect the variations of this magnitude. You can find below an example of data:

Magnitude of 49P / Arend-Rigaux measured in April 2012.
Magnitude of 49P / Arend-Rigaux measured in April 2012.

Of course, the data have holes, since you cannot observe during the day. Moreover, the comet needs to be visible from Arizona, otherwise it was just impossible to observe it and make any measurements.

We can see a kind of periodicity in the magnitude, this is a signature of the rotation of the comet.

Measuring the rotation

Most of the planetary bodies are kinds of triaxial ellipsoids. Imagine we are in the equatorial plane of one of them. We see an alternation of the long and short axes of its equatorial section. If the albedo of the surface element we face depends mainly on its curvature (it depends on it, but mainly may be an overstatement), then we should see two peaks during a period. As a consequence, the period of the lightcurve we observe should be half the rotation period of the comet.

In combining all the measurements, the authors managed to derive a rotation period of 13.45 ± 0.01 hour. For that, they used two different algorithms, which gave very close results, giving the authors confidence in their conclusions. The first one, Phase Dispersion Minimization (PDM), consists to assume a given period, split the measurements into time intervals of this period, and overlap them. The resulting period gives to the best overlap. The other algorithm is named Lomb-Scargle, following its authors. It is a kind of Discrete Fourier Transform, but with the advantage of not requiring uniformly sampled data.

In addition to this rotation period, the authors detected an increasing trend in the 2012 data, as if the spin of the comet accelerated. This is in agreement with an alteration of the measured rotation from the Earth, which moves, and reveals a retrograde rotation, i.e. an obliquity close to 180°. In other words, this is an illusion due to the motion of the observer, but this illusion reveals the obliquity.

Moreover, in comparing the 2012 data with the ones of 1984, the authors managed to detect a variation in the rotation period, not larger than 54 seconds. This is possible regarding the fact that the comet is altered by each perihelion passage, since it outgasses. In this case, that would imply a change of at the most 14 seconds of the rotation period between two passages. Such variations have also been detected for at least 4 other comets (2P/Encke, 9P/Tempel 1, 10P/Tempel 2, and 103P/Hartley 2, see Samarinha and Mueller (2013)).

Comet Period (h) Variation (s)
2P/Encke 11 240
9P/Tempel 1 41 -840
10P/Tempel 2 9 16.2
103P/Hartley 2 18 7200
49P/Arend-Rigaux 13.45 -(>14)

Finally, since the lightcurve is a signature of the shape as well, the authors deduced from the amplitude of variation that the axial ratio of the nucleus, i.e. long axis / short axis, should be between 1.38 and 1.63, while an independent, previous study found 1.6.

Cometary activity

49P / Arend-Rigaux has a low activity. Anyway, the authors detected an event of impulse-type outburst, which lasted less than 2 hours. The analysis of the coma revealed an excess of cyanides with respect to the 1984 passage. Moreover, 49P / Arend-Rigaux is the first comet to show hydroxyde.

The study and its authors

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.

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.

Our water comes from far away

Hi there! Can you imagine that our water does not originally come from the Earth, but from the outer Solar System? The study I present you today explains us how it came to us. This is Origin of water in the inner Solar System: Planetesimals scattered inward during Jupiter and Saturn’s rapid gas accretion by Sean Raymond and Andre Izidoro, which has recently been published in Icarus.

From the planetary nebula to the Solar System

There are several competing scenarios, which describe a possible path followed by the Solar System from its early state to its current one. But all agree that there was originally a protoplanetary disk, orbiting our Sun. It was constituted of small particles and gas. Some of the small particles accreted to form the giant planets, first as a massive core, then in accreting some gas around. The proto-Jupiter cleared a ring-shaped gap around its orbit in the disk, Saturn formed as well, the planets migrated, in interacting with the gas. How fast did they migrate? Inward? Outward? Both? Scenarios diverge. Anyway, the gas was eventually ejected, and the protoplanetary disk was essentially cleared, except when it is not. There remains the telluric planets, the giant planets, and the asteroids, many of them constituting the Main Belt, which lies between the orbits of Mars and Jupiter.
If you want to elaborate a fully consistent scenario of formation / evolution of the Solar System, you should match the observations as much as possible. This means matching the orbits of the existing objects, but not only. If you can match their chemistry as well, that is better.

No water below this line!

The origin of water is a mystery. You know that we have water on Earth. It seems that this water comes from the so-called C-type asteroids. These are carbonaceous asteroids, which contain a significant proportion of water, usually between 5 and 20%. This is somehow the same water as on Earth. In particular, it is consistent with the ratios D/H and 15N/14N present in our water. D is the deuterium, it is an isotope of hydrogen (H), while 15N and 14N are two isotopes of nitrogen (N).

These asteroids are mostly present close to the outer boundary of the Main Belt, i.e. around 3.5 AU. An important parameter of a planetary system is the snow line: below a given radius, the water cannot condensate into ice. That makes sense: the central star (in our case, the Sun) is pretty hot (usually more than pretty, actually…), and ice cannot survive in a hot environment. So, you have to take some distance. And the snow line of the Solar System is currently lose to 3.5 AU, where we can find these C-type asteroids. Very well, there is no problem…

But there is one: the location of the snow line changes during the formation of the Solar System, since it depends on the dynamical structure of the disk, i.e. eccentricity of the particles constituting it, turbulence in the gas, etc. in addition to the evolution of the central star, of course. To be honest with you, I have gone through some literature and I cannot tell you where the snow line was at a given date, it seems to me that this is still an open question. But the authors of this study, who are world experts of the question, say that the snow line was further than that when these C-types asteroids formed. I trust them.

And this raises an issue: the C-types asteroids, composed of at least 5% of water, have formed further than they are. This study explains us how they migrated inward, from their original location to their present one.

Planet encounter and gas drag populate the Asteroid Belt

The authors ran intensive numerical simulations, in which the asteroids are massless particles, but with a given radius. This seems weird, but this just means that the authors neglected the gravitational action of the asteroids on the giant planets. The reason why they gave them a size in that it influences the way the gas drag (remember: the early Solar System was full of gas) affects their orbits. This size actually proved to be a key parameter. So, these asteroids were affected by the gas and the giant planets, but in the state they were at that time, i.e. initially Jupiter and Saturn were just slowly accreting cores, and when these cores of solid material reached a critical size, then they were coated by a pretty rapid (over a few hundreds of kyr) accretion of gas. The authors considered only Jupiter in their first simulations, then Jupiter and Saturn, and finally the four giant planets. Their different parameters were:

  • the size of the asteroids (planetesimals),
  • the accretion velocity of the gas around Jupiter and Saturn,
  • the evolution scenario of the early Solar System. In particular, the way the giant planets migrated.

Simulating the formation of the planet actually affects the orbital evolution of the planetesimals, since the mass of the planets is increasing. The more massive the planet, the most deviated the asteroid.

And the authors succeed in putting C-type asteroids with this mechanism: when a planetesimal encounters a proto-planet (usually the proto-Jupiter), its eccentricity reaches high numbers, which threatens its orbital stability around the Sun. But the gas drag damps this eccentricity. So, these two effects compete, and when ideally balanced this results in asteroids in the Main-Belt, on low eccentric orbits. And the authors show that this works best for mid-sized asteroids, i.e. of the order of a few hundreds of km. Below, Jupiter ejects them very fast. Beyond, the gas drag is not efficient enough to damp the eccentricity. And this is consistent with the current observations, i.e. there is only one C-type asteroid larger than 1,000 km, this is the well-known Ceres.

However, the scenarios of evolution of the Solar System do not significantly affect this mechanism. So, it does not tell us how the giant planets migrated.

Once the water ice has reached the main asteroid belt, other mechanism (meteorites) carry it to the Earth, where it can survive thanks to our atmosphere.

Making the exoplanets habitable

This study proposes a mechanism of water delivery, which could be adapted to any planetary system. In particular, it tells us a way to make exoplanetary planets habitable. Probably more to come in the future.

To know more…

  • The study, presented by the first author (Sean N. Raymond) on his own blog,
  • The website of Sean N. Raymond,
  • The IAU page of Andre Izidoro.
  • And I would like to mention Pixabay, which provides free images, in particular the one of Cape Canaveral you see today. Is this shuttle going to fetch some water somewhere?

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.