Switching Orbits - Greg Laughlin, 2005-2007, Systemic: Characterizing Extrasolar Planetary Systems


  • Types of Coorbits
  • Lagrange Orbits
  • Formation of Lagrange Orbits
  • Horseshoe Orbits
  • Formation of Horseshoe Orbits
  • Eccentricity Switching Orbits
  • Formation of Eccentricity Switching Orbits
  • 1:1 Eccentric Resonance
  • Sidebar: Listening to Coorbital Interaction
  • Sidebar: Negative Heat Capacity

    Types of Coorbitals
    greg posted in systemic faq on July 25th, 2006

    The strongest 2-planet mean-motion resonances occur when the pair of planets share a common period and engage in a one-to-one resonant motion.

    There are a variety of different one-to-one resonances, including binary planet orbits (e.g. Earth and Moon; Pluto and Charon), trojan configurations (e.g. Jupiter's trojans), and generalizations of retrograde satellite orbits. In this last category, one can have two planets with the same semi-major axis. The horseshoe orbit (e.g. Janus and Epimetheus) results when the orbits of these two planets have the same eccentricity, typically near-concentric. The eccentricity switching orbit results when the orbits have different eccentricities.

    The following sections describe these one-to-one resonances.

    I Wish I had an Evil Twin
    greg posted in worlds on December 21st, 2005

    A hundred, or even fifty-five years ago, it was thought that Mars and Venus might both harbor complex life, and the aspirations of science fiction writers and adventurers were pinned squarely on those two worlds. With the advent of space probes, however, we visited these planets, and the dream of lush sister worlds orbiting our own Sun was shattered. Mariner 2 reported the hot sulfurous truth about Venus; the crushingly poisonous atmosphere has no water and is hot enough to melt lead. Mars, when brought into focus by the Mariner and Viking probes, was only somewhat less disappointing. Aside from flood channels that have been bone-dry for billions of years, and the faint possibility that microbial life clings to the fringes of hypothesized hot springs, Mars has little to offer in the way of luxurious alien romance. For this, we must turn to other planets around other stars.

    But we can still speculate. What would have happened if our solar system harbored a second, truly Earthlike, truly habitable world? What if there had been a genuine marquee destination for the cold war rockets?

    Is such a planetary configuration dynamically feasible? We know that the continuously habitable zone around a star like the Sun may be relatively narrow. Is it possible to fit two Earth-mass planets within? More specifically, what would happen if we placed an exact copy of the Earth in the Earthís orbit, with Earthís orbital elements, and with the only difference being a 180 degree advance in the mean anomaly. In other words, what would the dynamical consequences be if Earth had a twin on the other side of the Sun?

    In 1906, the German astronomer Maximillian Franz Joseph Cornelius Wolf discovered an asteroid at roughly Jupiterís distance from the Sun which was orbiting roughly 60 degrees ahead of Jupiter, and thus forming a point of an Equilateral triangle with Jupiter and the Sun. It was soon realized that the orbit of this asteroid was very stable, since it is positioned at the so-called Jovian L4 point, one of the five stable Lagrangian points associated with the Sun and Jupiter. These points represent special solutions to the notorious three-body problem, and were discovered in 1772 by the Italian-French mathematician Joseph Louis Lagrange. The following diagram was lifted from the wikipedia:

    Wolf named his Jupiter-L4 asteroid 588 Achilles, after the sulky Greek hero of Homerís Illiad. A year later, August Kopff discovered a similar asteroid, this one orbiting at the so-called L5 point, 60 degrees behind Jupiter. In keeping with the Homeric tradition launched by Wolf, Kopff named his asteroid 617 Patroclus, after Achillesí gentleman friend and fellow greek warrior. Thus, with fitting cosmic symmetry, the two heroes were immortalized in the heavens to either side of mighty Jupiter.

    Later in 1907, Kopff discovered a third co-orbital asteroid of Jupiter, this one near L4, which he named 624 Hektor, in honor of Achillesí Trojan nemesis. Hubble Space Telescope observations indicate that Hektor is actually a contact binary, in which two asteroids are effectively glued together by their weak gravity. In 1908, Wolf discovered yet another object (659 Nestor) near Jupiterís L4 point. It was clear that a whole class of Trojan asteroids existed, and in order to keep things straight, it was decided that asteroids found near L4 would be named after Greeks (the Greek camp), whereas asteroids near L5 would be named after Trojans (the Trojan camp). Hektor and Patrocles, who were thus orbiting in the camps of their respective enemies, were given the unique status of spies.

    Nearly two thousand trojan asteroids are now known. Even minor figures such as Hektorís infant son (1871 Astyanax) are now attached to asteroids, and the Illiadís roster is nearly completely exhausted. Trojan asteroids of recent province, such as 84709 2002 VW120 are relegated to the status of neutral observers.

    The US Navy maintains a website that charts the orbital motion of a number of trojan asteroids. (The Navyís involvement seems rather appropriate, as it was Helen, whose beauty was sufficient to launch a thousand ships, who touched off the Trojan War.) As a Trojan asteroid orbits the Sun, it also orbits about its Lagrange point by executing two essentially independent librations. The combination of the two librational motions leads to an intricate motion when viewed in a frame that rotates along with Jupiter.

    Back to our hypothetical Doppelganger of the Earth.

    In the language of Lagrange, when we place a new world on the opposite side of the Sun from the Earth, we have populated the L3 point. A linear perturbation analysis shows that if an object at L3 is perturbed, then the orbit will drift steadily away from the initial L3 location. That is, the orbit is linearly unstable, in contrast to the the orbits at L4 and L5, which are linearly stable, and hence stick around in the vicinity of trojan points, even when they are subjected to orbital perturbations.

    A computer is required to find out what would happen to the orbits of the Earth and our hypothetical twin planet. It turns out that the motion is nonlinearly stable. The Earth and its twin would be perfectly content, and, in a frame rotating with a 365 day period, the motion of the two planets over a period of years would look like this:

    As one planet tries to pass the other one up, it receives a forward gravitational pull. This forward pull gives the planet energy, which causes it to move to a larger-radius orbit, which causes its orbital period to increase, which causes it to begin to lag behind. Likewise, the planet which is about to be passed up receives a backward gravitational pull. This backward pull drains energy from the orbit, causes the semi-major axis to decrease, and causes the period to get shorter. The two planets are thus able to toss a bit of their joint orbital energy back and forth like a hot potato, and orbit in a perfectly stable variety of a 1:1 orbital resonance, known as a horseshoe configuration. The horseshoe orbit is an example of the negative heat capacity of self-gravitating systems, which is one of the most important concepts in astrophysics: If you try to drain heat away from a self gravitating object, it gets hotter.

    Hereís a thought. It is dynamically possible that 51 Peg b (or any of the other extrasolar planets that do not transit within the predicted window) is actually two planets participating in a stable 1:1 orbital resonance Ö

    While weíre on the topic of far-out planetary configurations, another type of allowed 1:1 configuration is the 1:1 eccentric resonance, an example of which is shown below. In this situation, two Jupiter-mass planets share the same period, but have very different eccentricities.

    Over time, the planets pass their eccentricity back and forth in an endless resonant cycle. If one of these configurations is found orbiting a sun-like star, it will induce a very distinctive radial velocity curve which will allow an unambiguous determination of the planetary masses and inclinations. And you can rest assured that the code that generates the systemic database is fully aware of the different flavors of one-to-one resonance.

    Good Librations
    greg posted in systemic faq, worlds on May 29th, 2006

    Last week, I wrote a post about the negative heat capacity of self-gravitating systems. I never cease to find it remarkable that if you drain energy out of a system that is held together by its own gravity (such as a giant planet, or a cluster of stars), then that system gets hotter. There really is such a thing as a free lunch, brought to you courtesy of the attractive gravitational force.

    A collection of bodies orbiting a larger body is a self-gravitating system, and therefore will also display a negative heat capacity. We illustrated this with the idea of a satellite running through a cloud of dust. Friction between the satellite and the dust heats both bodies up, and they radiate energy away to space. The satellite simultaneously spirals into an orbit with higher velocity, and hence a higher kinetic energy, or temperature.

    A family of orbital trajectories known as horseshoe orbits present a riff on this basic principle. A horseshoe orbit occurs when two bodies, with slightly different orbital periods, start off in near-circular orbits on opposite sides of a large central mass. The body with the shorter orbital period eventually attempts to overtake the body with the longer orbital period.

    As the short-period body catches up with the long-period body, an attractive gravitational force is exerted between the pair. This force pulls the short-period body forward, and pulls the long-period body back. That is, the gravitational interaction leads to an exchange which drains orbital energy from the long-period (leading) body, and gives energy to the short-period (trailing) body. This exchange causes the bodies to swap orbital periods. The long-period body gets a shorter period, and the short-period body gets a longer period. In a frame that rotates with the average orbital velocity of the pair, the two bodies eventually come in to contact again on the opposite side of the star, and the process is repeated. Again and again in an mindlessly delicate cycle.

    The orbital trajectory in the above figure is lifted and adapted from a paper in the Astronomical Journal that I wrote with John Chambers. In that paper, we studied a number of weird co-orbital planetary configurations, and speculated that they might eventually be observed using the radial velocity method. If you canít fit a particular data set with the console, the horseshoe configuration is always a good thing to check.

    In our own solar system, there are two small Saturnian moons, Janus and Epimetheus, which are caught in a horseshoe-like orbit. The following picture shows a Cassini photograph of these moons taken near the time during which they exchange periods.

    Janus and Epimetheus Source: JPL

    One of the most useful features of the Systemic console is its ability to sonify radial velocity waveforms. The soundfiles are produced by making a full integration of the equations of motion, hence all of the nonlinear gravitational interactions between the bodies are incorporated into the sound. When the console is used as a nonlinear digital synthesizer, the horseshoe orbits provide a method for producing amplitude modulation of a tone. To see how this works, launch the downloadable console, and set up the following system (just ignore the radial velocity data, since weíre not interested in fitting, but rather just in waveform generation):

    That is, set up two 0.2 Jupiter mass planets with mean anomalies of 0 and 180 degrees. Make the period of one planet 10.1 days, and the other 10.0 days. For simplicity, keep the eccentricities at zero. Clicking the integration box shows the resulting radial velocity waveform. When the planets are on opposite sides of the star, their radial velocity influences on the star cancel. When they are on the same side of the star, their radial velocity influences are additive. This gives an overall modulation envelope on top of the fundamental ~10.05 day period. Use the sonify button to create a 220 hz tone out of this system:

    Hereís a link to the resulting .wav file. The amplitude modulation (or tremolo) can clearly be heard.

    Try building some more complex sounds by nesting horseshoe orbits, and using unequal masses. If you get something cool, e-mail me at laughlin ucolick edu.

    Downloadable console

    greg posted in systemic faq on May 11th, 2006

    Sonification takes the N-body initial condition corresponding to the current positions of the console sliders and performs an integration of the equations of motion to produce a self-consistent radial velocity curve for the star. The radial velocity curve is then interpreted as an audio waveform and the resulting audio signal is written to the .wav format. You, the user, choose the duration of the integration and the audio frequency to which the innermost planetís orbital frequency is mapped (440 Hertz, for example, corresponds to the A below middle C). A simple envelope function is also provided in order to avoid strange-sounding glitches associated with sharp turn-on and turn-off transients.

    A single planet in a circular orbit produces a pure sine-wave tone. Very boring. The introduction of orbital eccentricity adds additional frequency content to the single-planet signal, and produces a variety of buzzing hornlike timbres, depending on the chosen values for the eccentricity and longitude of periastron.

    Systems in 2:1 mean-motion resonances can generate some very weird audio waveforms. Oklo favorite GJ 876 was the first (and is still by far the best) example of a 2:1 resonant configuration. GJ 876ís audio signal, however, is pretty lackluster (the .wav file is here). This is because the system is so deeply in the resonance that the waveform has a nearly invariant long time-baseline structure. Much more interesting from an audio standpoint, are the newly discovered 2:1 resonant systems HD 128311 and HD 73526.

    With the console, one can work up a quick fit to the HD 128311 data set which has one 2:1 resonant argument in circulation and the other in libration. The long-term orbital motion is completely bizarre (as shown by this .mpeg animation) and the corresponding audio file has a certain demented quality. The signal definitely evolves on longer timescales than shown in this snapshot of the fit:

    Sounding Interaction Resonance
    greg posted in systemic faq on July 25th, 2006

    The strongest 2-planet mean-motion resonances occur when the pair of planets share a common period and engage in a one-to-one resonant motion.

    In the case of eccentricity switching orbits, if one starts the planets in the following configuration, then the motion is dynamically stable, and evolves in a complicated way over time. The motion leads to an interesting audio wave-form, in which you can hear the system cycling between configurations in which both planets are modestly eccentric and configurations in which one orbit is nearly circular while the other one is highly eccentric. As a specific example, set the console to the following configuration: P1=P2=10 days, M1=M2=0.3 Mjup, MA1=180., MA2=190., e1=0.9, e2=0.1, long1=0.0, long2=0.0. If you increase MA2 to about 225 degrees while keeping the other parameters fixed, youíll hear the system go unstable.

    Evolving, high-eccentricity orbits tend to have an insect-like quality, which brings to mind the 1986 album, The Insect Musicians, by Greame Revell (formerly of SPK). From the album jacket:

    For the two years 1984-85, Graeme Revell travelled from Australia to Europe, to Africa, Indonesia and North America recording and negotiating copyrights of insect sound recordings. It took another full year sampling and metamorphosing some fourty sounds thus gathered using the Fairlight Computer Musical Instrument, to produce this record. The only sounds used are those of insects, altered digitally and combined into a unique orchestra of instruments, an orchestra of strange and delicate timbres, music of natural rhythm and texture.

    Switching Eccentricities
    greg posted in systemic faq, exoplanet detection on October 9th, 2006

    Take a system of two equal-mass planets with masses of 1.04 times that of Jupiter, a common period of 365 days, and eccentricities of 0.7 and 0.2.

    This system is an example of a one-to-one eccentric resonance. It is based on a system that was discovered by UCSC physics student Albert Briseno in one of the simulations that he ran for his undergraduate thesis, and it was formed as the result of an instability in a system that originally contained more planets. The system experienced a severe dynamical interaction, which led to a series of ejections. After the last ejection, two planets remained. They share a common orbital period, and gradually trade their eccentricity back and forth. Their interaction gives a strong non-Keplerian component to the resulting radial velocity curve for the star, which makes this a tricky system to fit. While the system might seem absurdly exotic, itís recently been suggested by Gozdziewski and Konacki that HD 82943 and HD 128311 might have their planets in this configuration (you can of course try investigating this hypothesis for yourself with the console). Their paper is here.

    The system above is an example of a general class of co-orbital configurations in which the two bodies constitute a retrograde double planet. If you stand on the surface of either world, the other planet appears to be making a slow retrograde orbit around your moving vantage as the libration cycle unfolds over several hundred orbits.

    In tomorrowís post, weíll stay on the topic of co-orbital planets, and look at some interesting new work by Eric Ford on the possibility that we might soon be able to observe planets in Trojan configurations. Two planets in a Trojan orbit librate around the points of an equilateral triangle in the rotating frame. Indeed, when such an arrangement occurs, itís possible that a particularly interesting dataset might have the capacity to launch a thousand fits.

    How the Synthetic Eccentric System Data Set was Produced

    There appear to be two sets of possible Jacobi elements. My explanation is below.

    Here are the Jacobi elements (i=90 deg):

    Parameter Planet 1 Planet 2
    Period (days) 352.499079 364.885297
    Mass (M_Jup) 1.04735500 1.04735500
    Mean Anomaly (deg) 0.00000000 0.00000068
    Eccentricity 0.69372876 0.20000000
    Omega (deg) 135.000000 134.999999

    Thereís something odd about the Jacobi elements, because I listed the e=0.2 planet first using the following astrocentric elements:

    Parameter Planet 1 Planet 2
    Period (days) 365.067512 365.067512
    Mass (M_Jup) 1.04735500 1.04735500
    Mean Anomaly (deg) 0.00000000 0.00000000
    Eccentricity 0.20000000 0.70000000
    Omega (deg) 135.000000 135.000000

    I reran the simulation just now, swapping the planet ordering. Here are the resulting Jacobi elements:

    Parameter Planet 1 Planet 2
    Period (days) 361.398709 364.885297
    Mass (M_Jup) 1.04735500 1.04735500
    Mean Anomaly (deg) 0.00000069 0.00000000
    Eccentricity 0.19489662 0.70000000
    Omega (deg) 134.999999 135.000000

    (I think it might be best to go with the second set of Jacobi parameters, if you only have to use one set.)

    Ultimately, I had to generate the system, but the e=0.2 and 0.7 and the planetary alignment were based on Albertís system.

    Extrasolar Trojans
    greg posted in exoplanet detection on October 11th, 2006

    Last week, during my visit to Harvard CfA, I talked to Eric Ford, who has been exploring the idea of searching for trojan companions to extrasolar planets. He pointed out that the discovery of a body in a trojan configuration with a known extrasolar planet would provide an important test of theories of hot Jupiter formation. Hereís a link to his paper.

    One way to make a hot Jupiter is to form the planet through the standard core-accretion method at a large radius in the protostellar disk. In this scenario, a newborn gas giant planet starts as a core of rock and ice, which grows to a size of 5-10 Earth masses and begins to rapidly accrete gas from the surrounding nebula. As the planet increases in size, it begins to clear a annular gap in the parent disk. Hydrodynamical simulations (such as the one reproduced in the illustration below) show that L4 and L5, the so-called trojan points located 60 degrees ahead and 60 degrees behind the forming planet, are the last regions of the gap to be cleared out.

    Itís possible that co-orbital planets can form from the slow-to-clear material at L4 and L5. When the gas is gone, these objects will remain in stable trojan orbits.

    If a pair of planets is caught in a trojan configuration, then they will migrate inward together through the disk, and the migration process will not cause them to become dynamically unstable. Eric points out that the observed presence of a trojan companion to a hot Jupiter would thus be evidence that the hot Jupiter arrived at its short-period orbit via migration. Other possibilities for forming hot Jupiters, such as dynamical instability followed by orbital circularization, do not allow for trojan companions.

    A trojan pair of planets presents an interesting conundrum for planet hunters. Normally, a single planet on a circular orbit goes through its radial velocity zero point at the moment when the planet lies on the plane containing the line of sight from the Earth to the parent star. If we have a trojan, however, a planetary transit will be offset from the radial velocity zero point, which is associated with the orbit of a ďghost bodyĒ that combines the gravitational effect of the primary planet and its trojan companion. Using the console, try obtaining a two-planet perfect trojan (60 degree separation) fit to a well-known hot Jupiter data set such as that for HD 187123. Youíll find that itís perfectly possible. The resulting configurations have utterly indistinguishable radial velocity signatures.

    Trojans can be detected, however, if the primary planet happens to transit. The presence of the trojan companion can be inferred by measuring the lag between the center of the transit and the zero crossing of the radial velocity curve. For planets with an equal mass ratio, this would amount to a full 1/12 of an orbital period (6 hours for a 3-day orbit).

    Most of the known hot Jupiters have been checked photometrically for transits. These transit searches, however, are performed in the time window surrounding the radial velocity zero point. In the (admittedly unlikely) case that some of these objects are trojan pairs with near-equal mass ratios, the transits would have been missed using this approach. To fully rule out transits, one should cover the full 1/6th of an orbital period surrounding the nominal predicted transit time Ö

    Because of the essentially threshold nature of Jovian planet formation, youíre much more likely to have L4 or L5 companions winding up much smaller than the primary.

    Among the fourteen known transiting planets, there are clearly no Jovian-mass trojan companions, so itís at least clear that near-equal mass trojan ratios are not the usual mode of formation for hot Jupiters.

    Using Fordís timing approach, we will soon have good limits on Earth-mass trojan companions to transiting giant planets.

    Negative Heat Capacity
    greg posted in worlds on May 16th, 2006

    Imagine leaving the front door open on a cold day, and having the inside of your house grow warmer as a result. Curiously, thatís exactly how self-gravitating systems such as stars, nascent giant planets, accretion disks, and globular clusters behave. Drawing energy from any of these systems causes them to heat up. The negative heat capacity of self-gravitating systems is one of the most central concepts in astrophysics.

    The dynamics of the Keplerian orbit can be used to understand how this works. Imagine a particle initially on a circular orbit around a central star. The particle slams into a cloud of dust. As a result, the dust and the particle both heat up and radiate energy. The particle decreases its velocity and drops into an eccentric orbit with a smaller semi-major axis.

    Hereís the key point: the smaller semi-major axis means that the average squared speed of the particle (averaged over an orbit) has increased. The fact that the particle is slow near apastron is more than compensated by the high speed near periastron. Since the particleís kinetic temperature is proportional to its speed squared, the temperature of the system goes up. In effect, the reserve of gravitational potential energy gets double billed: once to provide the radiated energy, and a second time to increase the kinetic energy of the particle.

    Itís a lot like taking a cash advance on your credit card and using half to pay late bills and the other half to buy a set of 22 inch rims for your Escalade. Itís a little sad to observe Nature operating on such a dissolute and spendthrift principle.

    Beyond Coorbital Basics - Greg Laughlin, 2008 +, Systemic: Characterizing Extrasolar Planetary Systems

    More on 1:1 Eccentricity
    greg posted in exoplanet detection on March 30th, 2008

    The range of planetary orbits that are observed in the wild is quite a bit more varied than the staid e < 0.20 near-ellipses in our own solar system. For regular oklo readers, the mere mention of Gl 876, 55 Cancri, or HD 80606, is enough to bring to mind exotic worlds on exotic orbits.

    Non-conventional configurations involving trojan planets have been getting some attention recently from the cognescenti. Even hipper, however, is a configuration that Iíll call the 1:1 eccentric resonance. Two planets initially have orbits with the same semi-major axis, but with very different eccentricities. Conjunctions initially occur close to the moment of apoastron and periastron for the eccentric member of the pair.

    Hereís a movie of two Jupiter-mass planets participating in this dynamical configuration.

    At first glance, the system doesnít look like itíll last very long. Remarkably, however, itís completely stable. Over the course of a 400-year cycle, the two planets trade their angular momentum deficit back and forth like a hot potato, and manage to orbit endlessly without anyone getting hurt.

    Hereís an animation which shows a full secular cycle. The red and the blue dots show the planet positions during the two orbit crossings per orbit made by one of the planets. Itís utterly bizarre.

    These animations were made several years ago by UCSC grad student Greg Novak (whoíll be getting his PhD this coming summer with a thesis on numerical simulations of galaxy formation and evolution). As soon as we can get the time, Greg and I are planning to finish up a long-dormant paper that explores the 1:1 eccentric resonance in detail. In short, these configurations might be more than just a curiosity. When planetary systems having three or more planets go unstable, two of the survivors can sometimes find themselves caught in the 1:1 eccentric resonance. The radial velocity signature of the resulting configuration is eminently detectable if the planets can be observed over a significant number of orbital periods.

    The behaviour of the 1:1 solutions for the HD 128311 and HD 82943 systems proposed by Gozdziewski and Konacki seem to be doing a similar kind of thing (but with mutual inclination being involved in all the fun as well). They are a specific example of the 1:1 eccentric resonance. Thereís a huge variety of available configurations once mutual inclination is allowed. It would be interesting to watch them vibrate in 3D over secular time scales. In the rotating reference frame, one sees that the 1:1 resonance is indeed a retrograde quasi-satellite.

    In the HD 128311 system, one of the arguments is librating, while the other is circulating. In this state of affairs, the apsidal line acts like a pendulum that is swinging over the top. In addition, the orbital eccentricities are higher, and the sum of planet-planet activity is strikingly greater. (See this animation of the evolution of the HD 128311 system.)