GJ 876 — cracked with the console!

Users familiar with console tutorial #3 will have noticed that the self-consistent 2-planet fit to the remarkable multi-planet system orbiting GJ 876 is presented as a fait accompli. We are currently implementing an “epoch” slider for the console which will greatly smooth the transition from Keplerian to Newtonian fits for interacting systems, but amazingly, it turns out to be possible to obtain a competitive 3-planet fit to the Rivera et al (2005) GJ 876 data set using only the current version of the systemic console. This post gives the details, and gets a bit technical, so if you are interested in following it closely, we suggest that you first work through tutorials 1, 2, and 3.

Also, a cautionary remark. The 3-planet integrated fit requires patience. I was able to get the fit described below in about 2 hours on a machine with two 3.4 GHz Intel Xeon CPU’s (with hyperthreading turned on). Thus, I was able to use the other CPU’s to do other work. On single-core, single-processor machines, the systemic console will hog the CPU (unless it’s niced and put in the background).

In any case, here’s the 411:

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If the suit fits…

Five radial velocity datasets (published last year by Marcy et al. 2005) have just been added to the systemic console: HD 183263, HD 117207, HD 188015, HD 45350, and HD 99492. Each of these more-or-less sunlike stars is too faint to be seen with the naked eye, and each is accompanied by (at least) one detectable planet. The periods range from 17 days to several years. None of these planets were extraordinary enough to warrant much fanfare in the popular press. (Ten years ago, however, the announcement of 5 planets would have been front page news. Ahh, those were the days!)

When you use the console to obtain orbital fits to these systems, you’ll notice that several of the stars have a long-term radial velocity trend superimposed on the variations that arise from the much more readily detectable shorter-period planet. These velocity trends are likely caused by as-yet undetected massive planets lying further out in the systems, and as these stars are monitored over the long term, the orbits of these distant, frigid giants will gradually reveal themselves.

In the meantime, the residual velocity trends underscore an interesting general property of extrasolar planets. The presence of a known planet is the best indicator that a given star harbors detectable (but as-yet undetected) planetary companions. That is, if you want to find new planets, then look at stars that already have known planets. Indeed, six of the first twelve planet-bearing stars that were monitored for more than two years at Lick Observatory were subsequently been found to harbor additional bodies. This impressive planetary six-pack includes luminaries such as Upsilon Andromedae, 55 Cancri, and 47 UMa, in addition to the more pedestrian Tau Boo, HD 217107, and HD 38529. (See Fischer et al. 2001).

Planets at the AAS Meeting

Frequent visitors to oklo.org will have noticed a definite fall-off in the number of recent posts. This was a direct result of the start of the winter quarter here at UCSC, but now things are rolling, and the systemic team is working hard to prepare the next phase of the collaboration.

Last week was also the 207th meeting of the American Astronomical Association. I took a one-day trip to Washington in order to give a talk at Tuesday’s extrasolar planets session entitled, “From Hot Jupiters to Hot Earths“. I teach class on both Monday and Wednesday mornings, so the trip was more of a lightning raid.

a departure lounge at dulles international

I arrived at Dulles Airport at 6 am, after an overnight flight. My talk wasn’t finished, so I sat in an empty departure lounge for several hours and worked on the slides. By mid-morning, I realized that I had better head to the venue. I took a cab to the conference hotel, tapping on the laptop for most of the way.

Hundreds of astronomers were thronging in the hallways. I studied the posters that had been set up in a large exhibition hall, and then went to hear NASA Administrator Griffin give a keynote address, the gist of which was clearer than this snapshot (taken under low-light conditions).

NASA Administrator Griffin

Two of the things he said stuck in my mind.

Like it or not, NASA has been charged to fly to the moon for reasons that are completely divorced from astronomy, and this means that there will be opportunities to use a lunar platform for observations of extrasolar planets. Transit photometry of nearby stars, especially M-type stars, jumped to mind, but clearly, there is a serious opportunity right now to start thinking outside the box.

He also said that the primary education and outreach mission of NASA should be to inspire by doing “cool things”. I do remember watching the last Saturn V’s blast off for a lunar destination, and I remember, a few years later, learning in grade school science class, about the Space Shuttle, that “pickup truck into orbit”, and feeling distinctly less inspired. In the intervening years, my list of the coolest NASA things runs along the lines of, Voyager, HST, WMAP, Cassini-Huygens, and Spitzer. And there’s also the BPP project. (For more detail on interstellar missions, Paul Gilster’s Centauri Dreams is always the place to go).

Uh, my talk wasn’t all it could have been. In order to facilitate rapid transitions between the session speakers, everyone’s slides were uploaded to a central server. The server was running Windows, and all the Powerpoint presentations looked exactly like they were supposed to. Full screen ahead. As a Keynote user, however, my slides were in the form of .pdfs. They looked just fine in the speaker ready room, but then, when I stepped up to the podium, I was aghast to see that my .pdfs were displaying on only on a small portion of a screen containing an acrobat viewer, complete with a sneak “preview” and a sneak “review” of the next and previous slides. The resolution was too low to see any detail. Score one for Mr. Bill Gates.

For the record, though, here are the slides (full resolution .pdfs).

orbital

Let a pebble slip from your hand and it falls straight to the ground. Toss the pebble sideways, and it traces a parabolic arc through the air. Imagine throwing the pebble sideways with even more speed. It lands further away. Imagine throwing the pebble with such great velocity that the surface of the Earth begins to curve away beneath it as it falls. In the absence of air friction, a pebble thrown sideways with sufficient velocity will fall in such a way that the Earth curves continuously out from underneath. The pebble falls endlessly without ever touching the ground. It is in orbit.

Cassini view of Mimas, Dione and Rhea near the ring plane (credit: NASA/JPL)

The idea that an orbit is the state of a body in continual free-fall can be traced to the 1600s, and was first stated in print by Robert Hooke, whose paper entitled, “The Inflection of a Direct Motion into a Curve by a Supervening Attractive Principle” was read to the Royal Society on May 23rd 1666. Robert Hooke’s fame and reputation have spent the last three hundred and twenty years in Newton’s shadow, but he was a tremendously inventive scientist, and indeed, was one of the founders of what we now consider the scientific method. (See, for example, the recent Hooke biography, “The Forgotten Genius” by Stephen Inwood). Hooke, drawing on the earlier ideas of William Gilbert and Jeremiah Horrocks, and profiting from conversations with fellow Royal Society member Christopher Wren, realized that if the Sun exerts an attractive force on bodies in space, then “all the phenomena of the planets seem possible to be explained by the common principle of mechanic motions.” Hooke had an intuitive (but non-mathematical) understanding of the the orbit in the sense described in the paragraph that opens this post.

Robert Hooke was shouldered with a bewildering variety of interests and responsibilities. One of his many jobs was to produce weekly demonstrations for the entertainment and edification of the Royal Society. In order to illustrate his concept of the planetary orbit, he devised a demonstration that provided a suggestive analogy. A bob was placed on a long string pendulum. Tension from the string provided a central attractive force, and a sideways push provided the requisite tangential motion. When given a sideways push of exactly the correct speed, the bob would swing in a circle. When started at other speeds, it traced an elliptical path. With this simple device, Hooke was able to illustrate how an orbit is a compound of tangential motion and an attractive radial force.

Hooke then made the analogy more elaborate by attaching two bobs to the end of the string. Once set in motion, the two bobs would orbit each other, while their center of mass orbited the center of attraction:

compound pendulum

In 1670 , Hooke delivered a Cutler lecture at Gresham College, entitled, “An Attempt to Prove the Motion of the Earth by Observations”. The written version of this lecture contains three remarkable postulates, including, (1) a specification of the concept of universal gravitational attraction, that is, that mass attracts mass, (2) the assertion that all bodies “that are put into a direct and simple motion would continue to move in a straight line unless deflected”, and (3) the hypothesis that the attractive gravitational force falls off with distance. Taken together, these ideas are a remarkably correct qualitative formulation of the foundations of gravitational dynamics. Had Hooke been equipped with the mathematical skill to express his three ideas quantitatively, he would have gone very far indeed.

At the same time that Hooke was demonstrating his pendulum analogy to the Royal Society, Isaac Newton was nearing the close of his Anni Mirabiles. By 1666, Newton, who was working in total isolation, had found a quantitative model that explained the circular orbit, and also showed that gravity is manifested by an inverse square law of attraction.

I began to think of gravity extending to the orb of the Moon, & (having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere) from Kepler’s rule of the periodical times of the Planets being in sequialterate proportion of their distances from the center of their Orbs, I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squarres of their distances from the centers about which they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them to answer pretty nearly. (All of my Newton quotes are drawn from Richard Westfall’s “Never at Rest — A Biography of Isaac Newton“)

Here’s what Newton is saying. Kepler’s Third Law holds that the orbital period of a planet is proportional to the semi-major axis of its orbit to the 3/2 power, that is,

For the simplified case of a satellite in a circular orbit, the semi-major axis, a, is just the orbital radius, i.e. a=r. In Newton’s state of understanding in 1666, the “centrifugal” outward force an orbiting satellite must cancel the inward force exerted by gravitational attraction from the central body. The gravitational attraction is assumed to be spherically symmetric and to fall of with some power of the distance. That is,

where x needs to be determined. The fact that distance is rate multiplied by time implies that

and therefore

This means that

and if Kepler’s third law is to be satisfied, then x=2. Newton had realized that Kepler’s third law implies that gravity is an inverse-square force.

Newton had thus found a workable mathematical model for the circular orbit in 1666, but at that time, he was behind Hooke in terms of his intuitive understanding of the actual physical situation. Newton’s initial conception of the orbit was one of a mechanical equilibrium, in which an innate tendency to recede during circular motion is balanced by a gravitational attraction. In reality, Hooke’s concept of the orbit as the state of continual free-fall, a state of disequilibrium, is the correct notion.

On Nov. 24, 1679, Hooke, in his capacity as the secretary of the Royal Society, wrote a letter to Newton in order to solicit a discussion of orbital dynamics. Hooke was likely quite proud of his theories concerning orbital motion, and he may well have been eager to bring his ideas to Newton’s attention.

Let me know your thoughts of that of compounding the celestaill motions of the planets of a direct motion by the tangent and an attractive motion towards the centrall body.

Hooke had no way of knowing that Newton had already thought carefully about orbits. It is likely that as soon as Newton saw Hooke’s phrase, he immediately saw that it represented an improved qualitative conception of orbital motion. He quickly wrote back to Hooke, and politely declined the offer of an extended dialog. He, was, he said, too busy with other studies.

And having thus shook hands with Philosophy, & being also at present taken of with other business, I hope it will not be interpreted out of any unkindness to you or the R. Society that I am backward in engaging my self in these matters…

Nevertheless, the correspondence between the two continued through several more letters. By January 1680, Hooke had managed to guess (on the basis of two incorrect arguments that combine by chance to give a correct formula) that the gravitational force obeys an inverse square law. What he could not prove, however, was what the general path of an orbiting planet is an ellipse (that is, he could not go beyond the special case of the circular orbit). The fact that the planetary orbits have elliptical figures was then known empirically from Kepler’s First Law, which states that planetary orbits are ellipses with the Sun at one focus.

On January 6th, 1680, Hooke took the liberty of informing Newton of the inverse square law, “My supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall…”, and in a letter on the 17th of January, he further urged Newton to find the general mathematical description of a planetary orbit,

I doubt not but that by your excellent method you will easily find out what that Curve must be, and its proprietys, and suggest a physicall Reason of this proportion.

Hooke’s letters of November through January of 1679-80 seem to have greatly annoyed Newton. With his formidable intuition and dismal mathematical skills, Hooke was steadily blundering toward the pedestal where he could claim the renown of explaining the “System of the World”. Without informing Hooke, Newton carried out (in early 1680) a marvelous derivation that showed that the path of a planet is an ellipse, and simultaneously proved Kepler’s Second Law, which states that the radius vector of a planet sweeps out equal areas in equal times, and which is a statement of the principle of the conservation of angular momentum.

Years later, soon after the appearance of the Principia, when Hooke claimed that Newton had plagiarized his ideas, Newton lashed out at Hooke in a letter to Edmund Halley:

Should a man who thinks himself knowing, & loves to shew it in correcting & instructing others, come to you when you are busy, & notwithstanding your excuse, press discourses upon you & through his own mistakes correct you & multiply discourses & then make use of it, to boast that he taught you all he spake and oblige you to acknoledge it & cry out injury and injustice if you do not, I beleive you would think him a man of a strange unsociable temper.

Robert Hooke, that man of strange unsociable temper, is nevertheless a man after my own heart. Newton is so far out on the curve that I can’t relate to him at all. I have no concept of how his mind operated, but Hooke, Hooke would be a fantastic person to have a beer with. I can appreciate the way he thought by analogy. I greatly admire his demonstrations, even if they aren’t fully rigorously correct.

With Hooke’s approach in mind, let’s look at some elliptical orbits. When an ellipse has zero eccentricity, the two foci come together, and the ellipse is a circle. A planet on a circular orbit travels at constant speed, and its positions at a hundred equally spaced time intervals are equally spaced:

e=0 ellipse

When the eccentricity reaches 0.1, the orbit looks very much like a shifted circle. When the planet is closest to the Sun (the point known as perihelion), it is moving faster. If you look carefully, you can see that the dots are spaced just a bit more sparsely on the right side than on the left. The e=0.1 orbit just below is very similar to the orbit of Mars, which has an eccentricity e=0.0935. In all of the following figures, the e=0 circular orbit is also shown for comparison:

e=0 ellipse

Among the eight major planets in our Solar System, Mercury, with e=0.205, has the most eccentric figure. Mercury’s orbit is almost identical to the orbit in this plot, which has e=0.20:

e=0 ellipse

Planets “c” and “d” of the Upsilon Andromedae system have eccentricities of e=0.27 and e=0.28 respectively. Their orbital figures are quite close to this orbit, which has e=0.3:

e=0 ellipse

70 Vir b was one of the first extrasolar planets to be discovered. It has a very well defined orbit with e=0.4:

e=0 ellipse

When e=0.5, it’s impossible to mistake the ellipse for an off-center circle. The extrasolar planet GJ 3021 b has an eccentricity e=0.511, which is close to e=0.5:

e=0 ellipse

At an eccentricity e=0.6, the brevity of the periastron swing is highly pronounced:

e=0 ellipse

When e=0.7, the planet is 5.667 times closer to the star at periastron than at apastron.

e=0 ellipse

e=0 ellipse

e=0 ellipse

The extrasolar planet with the highest known eccentricity, HD 80606 b, has an 111 day orbital period and e=.938. See my article on this strange world posted last month. Because it dives in so close to its parent star, HD 80606 b is an interesting transit candidate, but a transitsearch.org campaign carried out last year did not detect transits. We’ll try again this year during the upcoming transit opportunities.

Do you give talks on extrasolar planets? High-resolution .pdf files of the above orbital plots are free here for the taking: e=0.0, e=0.10, e=0.20, e=0.30, e=0.40, e=0.50, e=0.60, e=0.70, e=0.80, e=0.90.

2006

On Dec. 12, 2005, we arrived at Kansai with the Sun low on the horizon, casting orange shafts through the plane. Whitecaps were frothing on the Inland Sea. The airport is built on two 4 kilometer by 1 kilometer artificial islands, and is connected to Honshu by a 3 kilometer bridge that cost 100 billion yen. Beneath the vast new terminal, an attendant with a pressed shirt and tie helped us navigate the ticket machines to buy two Haruka Ltd. express tickets to Kyoto. The bullet train pulled away as soon as we stepped on, gliding glass-smooth through the night blur of an endless city.

That night, in a room in the hyper-modern Hotel Granvia, I lay awake, jet lagged and alert, listening to the faint rush of warm air flowing from a network of unseen ducts. Outside, the lights of the city were a panoply of mysterious characters and sparkling complexity, illuminating blocks of buildings that stretched away in all directions to the dark mountainous horizon. I was suddenly brought around to a simple fact that I always find startling:

No one arrived from outer space to build all this. In a very real sense, the planet Earth has done has done all this itself.

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systemic 001

saturn as seen by the approaching cassini probe (nasa/jpl)

The goal of the systemic research collaboration is to improve our statistical understanding of the galactic planetary census. This will be accomplished through a large-scale simulation in which the public is invited to participate.

At the core of the systemic simulation, we have generated a realistic catalog that contains 100,000 stars, and we have created planetary systems in orbit around some of these stars. As the collaboration unfolds, the systemic catalog of stars will be “observed” using a realistic model of the radial velocity technique, and a radial velocity data set for each star will be made available. Participants will use the systemic console (or their own software if they choose) to discover and characterize planets within the data sets.

The measured orbital properties and distributions of the planets that are uncovered in the systemic data sets will eventually be compared with the known properties of the planets that were placed into orbit around the systemic catalog stars.

Why the name systemic?

We have four answers: (1) The collaboration utilizes a planetary system integrator console. (2) We are seeking to better understand the statistical distribution of planetary system initial conditions in the galaxy. (3) We hope that the collaboration will make the analysis of extrasolar planetary systems more evident, “Ahh, now I see!” (4) Finally, and most importantly, the planetary systems that we have designed are fully internally consistent. (More on this later.)

The project will officially start in early 2006. In the meantime, we have released a beta version of the systemic console, along with three tutorials (1, 2, and 3). The www.oklo.org site is also a weblog where we’ve been posting a variety of articles on the topic of extrasolar planets and their detection and characterization.

Currently, the systemic console has access to a number of published radial velocity data sets for real stars containing known planetary systems. We have also added the first star of the systemic catalog (which coincidently shows definite indications of harboring a planetary system). Launch the console, choose systemic001 from the system menu, and use the comment space for this post to let us know what you find!

— The Systemic Team,

Greg Laughlin — UC Santa Cruz

Stefano Meschiari — University of Bologna

Eugenio Rivera — UC Santa Cruz

Paul Shankland — US Naval Observatory

Aaron Wolf — UC Santa Cruz

i wish i had an evil twin

mars

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 Earth’s orbit, with Earth’s 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 Jupiter’s 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:

Lagrange points (from Wikipedia)

Wolf named his Jupiter-L4 asteroid 588 Achilles, after the sulky Greek hero of Homer’s 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 Achilles’ 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 Achilles’ Trojan nemesis. Hubble Space Telescope observations indicate that Hektor is actually a contact binary, HST image of asteroid 624 Hektor
in which two asteroids are effectively glued together by their weak gravity. In 1908, Wolf discovered yet another object (659 Nestor) near Jupiter’s 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 Hektor’s infant son (1871 Astyanax) are now attached to asteroids, and the Illiad’s 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 Navy’s 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.

orbit of 1437 Diomedes in a frame that rotates 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:

horseshoe orbit for equal mass planets

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.

Here’s 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…

While we’re 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.

1:1 eccentric resonance

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.

This post isn’t yet complete, so check back later if it has caught your interest…

The black cloud

We came from a black cloud.

barnard 68

Stated with such conviction and simplicity, the theory of planet formation is as remote and dogmatic as, say, the creation myth of the ancient Greeks: “In the beginning, there was chaos”

Going about everyday life, with the flip-top cell phone, the busy schedule, the cars with soft leather seats, traffic reports on the radio, blogs dissecting politics, the idea of planet formation, the fact that the Earth hasn’t always existed has no chance for a foothold. You must shut everything off and stare at a dark night sky.

Easier said than done. Almost certainly, your night sky does not induce much wonder. Part of my own view, for example, is blocked by the neighbor’s garage. The nearer streetlights are brighter than the full moon, and they suffuse the air in prismatic halos of light. Some stars are visible, even the best-known constellations. Orion in the winter. The Big Dipper. But on the whole, the stars hardly concern us because we can hardly see them.

To see the stars as they are really meant to be seen, you probably need to plan a trip. Look at a satellite image of the country taken at night, or better yet, use a dark sky finder java applet, and drive to the spot on a clear, warm, windless and moonless night. It is a strange condition of our state of affairs that an applet can help us obtain what was once obvious to anyone who simply looked up. A price paid for a modern world of ease and comfort. A perfectly dark clear night is now, quite literally, a commodity.

earth at night

Let your eyes dark-adapt, and then look up. The effect is overwhelming. Stars. So many of them that the bright ones hardly matter. On a truly dark night, the Milky Way is unmistakable. It spills a swath of patchy luminosity across the dome of the sky. A barred spiral galaxy, seen edge-on, and from within. One hundred billion intensely glowing stars, like sand grain jewels, each separated by miles. Faced with this firmament, the idea that we came from a black cloud seems somehow more within the realm of the possible.

Black clouds, giant billowing masses of molecular hydrogen and helium laced with dust of the consistency of cigarette smoke congregate in the spiral arms of the Milky Way. Their centers are frigid, ten degrees Kelvin, and if you could watch a time-lapse movie of a million years compressed into a minute, you would see them billow and boil.

Within a cloud, the cold dense gas is always poised to collapse in on itself under its own weight. Disaster is staved off by the roiling currents in the gas, and the magnetic field lines that thread the cloud. The cloud contains a tiny fraction of charged particles, ions and free electrons that are outnumbered by neutral atoms and molecules by a factor of a billion or more.

Charged particles are tied to magnetic field lines. Motion of charges drags the magnetic field lines along, and vice versa. Magnetic field lines, however, don’t like being compressed or twisted. They have a tendency — verging on insistence — to spring back into shape. This prevents the ions and electrons from joining the gravitational collapse of the cloud. The ions and the electrons, in turn, bounce continuously against the neutral particles in the cloud, and in so doing, delay the great inward crush.

Most of the time, the frantic collisions of the ions are decisive. The great unwieldy black cloud is torn apart by the tidal forces of the galaxy before it can collapse under its own weight. The cloud dissipates like Arizona thunderheads in the face of approaching night. Occasionally, however, the ions and electrons are overwhelmed. Neutral gas slips past their efforts and pools in the centers of the clouds. This process gains momentum, the ions lose their effectiveness, and vast gulfs of the cloud begin to collapse.

HST image of IC 2944

Picture the scene 4.56 billion years ago when the solar system began to form. The atoms that now constitute the Earth have already been forged. Every atom of hydrogen in the molecules of the pre-solar cloud has already seen 9 billion years of history.

For some of that hydrogen, the past was uneventful. Atoms born in regions of the big bang that, by dint of the role of quantum dice, were a few parts in a million less dense than their surroundings were left cool and marooned in the stretches of space between the Milky Way and nearby galaxies. For billions of years, they fell through intergalactic space, to land, by chance, on the disk of the Milky Way just prior to the assembly of the giant molecular pre-solar cloud.

Other hydrogen atoms, some of them now vibrating in molecules massed in aqueous solution in your blood, or indentured to the long-chain monomers that form the polycarbonate shells of laptop computers, have experienced more colorful histories, having flowed, in some cases, in the oceans of now-dead terrestrial planets that orbited ancient generations of stars. Heavy atoms have less ancient pedigrees. The Earth’s carbon comes mostly from soot blown off of red giant stars. The gold was created in supernovae.

As the scene of the formation of the solar system unfolds, a gigantic volume of gas settles gradually into the core of the giant molecular cloud. where increasingly, the magnetic fields are losing their grip. At the center of the cloud, the view of the stars has long since been blotted out. It is utterly black and frigidly cold, but for ears pitched 24 octaves below middle C, it is not silent. The cloud rumbles and groans. The sound is like the ocean, like an earthquake, but it is also beyond simple description, and it permeates the vast stygian gulf.

Eventually, the cloud begins to collapse in earnest. Not all together, but from the center. As the gas in the center begins to form a protostar, the layer just above the center loses its support against gravity and begins to career inward as well. A wave of rarefaction radiates upward, triggering a downward avalanche of gas.

As the collapse picks up speed, a new effect becomes apparent. Gas that has fallen from large distances does not land on the central protostar. Rather, it falls onto a differentially spinning disk, a platter of gas and dust that orbits the actual center. The original protosolar molecular cloud harbored an ever-so-slight random component of rotation, and this rotation is eventually expressed in the form of the spinning protostellar disk. The basic principle — conservation of angular momentum — is what causes the skater to spin so fast when the arms are pulled in. For the protostellar disk, the originally outstretched arms of the cloud extended for a fraction of a light year from from the core.

The idea that the Sun and planets arose from a spinning disk of gas and dust dates to the eighteenth century. Isaac Newton, whose theory of universal gravitation explains the motion of the planets (and is thus the basis of the systemic console), was dismissive of a natural origin for the Sun and Planets. He issued a firm rejoinder against the whole idea of a natural cosmogony.

Where natural causes are at hand, God uses them as instruments in his works, but I do not think them alone sufficient for the creation.

The idea of creation as the product of natural law stems from the free-thinking spirit of the enlightenment. Georges Louis Leclerc, Comte de Buffon, formulated one of the first natural cosmogonies. His idea is that a comet struck the sun, throwing out the material that later condensed into the planets. This theory accounted for the fact that the planets all orbit the sun in the same direction. Immanuel Kant postulated that the planets arose via condensation from a spinning cloud of gas. Kant’s idea was developed later, independently, by Laplace, who imagined that the disk contracted as it cooled, leaving behind a succession of rings that fragmented to form the planets.

The idea that the Earth originally arose from a disk of gas and dust is tough to accept now (other than simply believing it because one has been told) and it was even harder to accept in the 1700s, when evidence was scarce. Laplace, in 1802, explained his theory to Napoleon, who didn’t like it. Napolean angrily exclaimed,

And who is the author of all this?

Thomas Jefferson, furthermore, celebrated as one of our more erudite presidents, had this to say:

Dreams about the modes of creation, inquiries whether our globe has been formed by the agency of fire or water, how many millions of years it has cost Vulcan or Neptune to produce what the fiat of the creator could affect by a single act of will are too idle to be worth even a single hour of any man’s time.

The eighteenth century cosmogonies are couched in quaint language and are not fully correct, but they are nevertheless surprisingly close to the mark. They stand up particularly well when compared to other theories of genesis that were motivated by new technologies of observation. (see for example Regnier de Graaf’s observations and theory of the homunculus). William Herschel and others mistakenly believed that the galaxies and nebulae that they saw through their telescopes were actually solar systems in the process of formation. Saturn’s rings provided another observable manifestation of a disk orbiting a central object. And although the scales are vastly different (a galactic disk is 100 million times larger than a protostellar disk, which is in turn 30,000 times larger than Saturn’s rings) the disks themselves are a ubiquitous phenomenon. They arise whenever material (gas, rocks, dust) crowds into orbit around a central object. By observing how galaxies and planetary rings behaved, but without an understanding of the length scales that were being observed, it was still possible to make reasonable inferences about the behavior of protosteller disks.

Astronomy is inherently simpler than biology.

55’s the limit

55 Cancri is an ordinary nearby star, barely visible to the naked eye. Through a modest telescope (or, more practically, with the use of the Goddard Skyview) one sees that it is actually a binary pair.

Goddard Skyview Image of 55 Cancri

55 Cancri “A” (the bright star in the middle of the above photo) harbors an extraordinary planetary system. Indeed, it was the subtlety and the depth of the 55 Cancri radial velocity data set that motivated us to develop the systemic console. The fact that the 55 Cancri system continues to defy easy categorization gives us confidence that the systemic collaboration will be a worthwhile project.

Where to begin?

Click on the system menu on the console, scroll down, and select 55 Cancri. (If you’re unfamiliar with the console, and if you’re the methodical type, there are three tutorials available on the menu bar to the right. Otherwise, just follow along!) The published radial data for 55 Cancri now appears in the main console window. The sweeping spray of points, with its curiously non-uniform distribution, contains a fascinating narrative in its own right.

The very first point in the data set has a timestamp of JD 2447578.73 A Julian Date Converter tells us that this was 9:31 PM on Monday Feb. 20, 1989 (Pacific Standard Time). The observation was obtained by Geoff Marcy at the Shane 3-meter telescope at Lick Observatory on Mt. Hamilton, and the velocity error is 9.7 m/s. Back in 1989, Geoff and his colleague Paul Butler were laboring to improve their iodine cell technique, and were struggling to get enough telescope time to adequately track the motion of about 70 nearby solar type stars with the eventual hope of detecting giant planets.

The first 10 radial velocity points were obtained at a rate of 1 to 3 per year. With hindsight, it is easy to see that these 10 points are ample cause for a planet-hunter to be optimistic. The radial velocity variation in the first 10 points spans more than 100 meters per second, suggesting a signal with a signal-to-noise of at least five. The periodogram of these ten points shows a strong peak at 14.65 days, indicating that the data could be explained by a planet with 80% of Jupiter’s mass, circling on an orbit lasting just over two weeks.

Today, if such a planet were discovered, the announcement would not make the news, and the major excitement would be among amateur transit hunters, who would likely have a new high-priority follow-up candidate with a ~5% transit probability. (A two-week period is right at the borderline where transits can be reliably confirmed or ruled out by the photometric collaborators working with the RV-discovery teams prior to announcement of the planet).

In 1993, however, nobody was expecting to find Jovian planets in 14-day orbits. Conventional wisdom at the time was informed by the architecture of our own solar system, and held that gas giant planets should be found beyond the so-called snowline (located at r=4-5 AU) of the protostellar disk. Although the theory of orbital migration had been studied in considerable detail, nobody had proposed that giant planets might regularly spiral in and then be marooned on very short-period orbits. I don’t know whether Geoff and Paul even considered the possibility that the 14.65 day peak in their data was real. If they saw the peak, it is more likely that they would have ascribed it to an alias, an artifact of their uneven hard-won sampling.

During 1994, the velocities suddenly started to trend upward. This would have seemed rather disconcerting, and may even have raised alarm. Was some unaccounted-for instrumental or astrophysical process affecting the newer radial velocity data? Certainly, at the end of 1994, the case for a planet orbiting 55 Cancri would have been weaker than it had been a year earlier.

Nevertheless, the 55 Cancri campaign was at an important turning point. The last measurement of 1994 (JD 2449793.80) has a remarkably lower error (3.3 m/s) than any of the earlier radial velocities. In November of 1994, the Schmidt camera optics on the “Hamilton” spectograph at Lick Observatory had been upgraded, and the resulting improvement effectively tripled the intrinsic resolution to which the spectral lines could be discerned. With the ability to measure radial velocities to a precision of 3 m/s, the planet search had suddenly entered an entirely new realm. When one is in the business of detecting Jupiters, a velocity measurement with 3 m/s precision is literally 10 times as valuable as a velocity with 10 m/s precision.

In October 1995, Mayor and Queloz announced their discovery of a Jupiter-like planet in a 4.5 day orbit around the nearby star 51 Peg. Due to a catalog error that misclassified 51 Peg as a subgiant, it had not been included in Geoff and Paul’s survey, but they were able to rapidly confirm the Swiss discovery.

All at once, the idea of a gas giant with a 2-week orbit was no longer outlandish at all. The telescopes on Mt. Hamilton, which had been slipping inexorably in worldwide prestige as larger telescopes were built on higher mountains, were suddenly at the forefront of relevance. The Lick 3-meter telescope-iodine-cell-spectrograph combination was the best instrument in the world for obtaining precision doppler velocities of bright stars such as 55 Cancri. Extrasolar planets were front page news. Alotments of telescope time increased dramatically. In the six months running from December 1995 through May 1996, 55 Cancri was observed 41 times at Lick. This drastic increase in the cadence of observations is easily visible in the radial data:

1996 RVs

With the 41 high-quality observations, the presence of the 14.65 day planet was obvious in the power spectrum.

RV powerspectrum

In October 1996, Paul, Geoff, and several other collaborators announced the discovery of the 14.65 day planet, and in January 1997, they published the discovery in a now classic paper that also introduced the world to the inner planetary companions of Tau Bootes and Upsilon Andromedae.

With eight years of data, it was clear that other bodies were present in the system. In the discovery paper, Butler et al. wrote:

The residuals exhibit a long-term trend, starting at -80 m/s in 1989 and climbing to +10 m s-1 by 1994 (the velocity zero point is arbitrary). The velocities appeared to decline toward 0 m s/1 during the past year, although at least another year of data will be required for confirmation. This trend and the possible curvature in the velocity residuals are consistent with a second companion orbiting HR 3522 [aka 55 Cancri] with a period P > 8 yr and M sin i > 5MJUP.

This speculation proved to be correct. Use the console to get a best-fit for the 14.65 day planet, and compute the periodogram of the residuals to the fit:

RV residuals powerspectrum

The strongest remaining peak is at 4260 days, corresponding to an 11.7 year orbit (very similar to Jupiter’s 11.8 year orbital period). Keeping the orbits circular, use the “polish” button to produce a Levenberg-Marquardt optimized fit. Zoom in and scroll to show the time interval between 1996 and 2002. The gaps each year when the star is behind the Sun as seen from Earth are easily visible:

Lick RVs 1996-2002

The two planet system does quite a reasonable (but by no means perfect) job of reproducing the observed radial velocities. After the announcement of the first planet at the end of 1996, interest in the star died down to some degree. The number of target stars being observed at Lick was being increased as Debra Fischer stepped in to manage the Lick Survey, and other systems, especially Upsilon Andromedae, were clamoring for telescope time. During the 1998 season, 55 Cancri was observed only twice. By 1999, however, the Upsilon Andromedae system had been sorted out, and renewed attention was focused on 55 Cancri. During 2000 and 2001, it became clear that the system likely contained at least three planets. With the 14.65 and (in my fit) 5812 day planets removed from the radial velocity curve, the residuals periodogram shows a peak at 44.3 days:

residuals of the residuals

The signal from the 44.3 day planet is not as strong as for the other two planets, but a large number of velocities from 2002 seemed to clinch the case for this third planet:

residuals of the residuals

Use the console to optimize the three planet fit using circular Keplerian orbits. When I do this, the chi-square statistic is reduced to 6.4, and the rms scatter is 12.5 m/s. The fit is still not perfect. Either the planets are eccentric, or there are additional planets in the system.

Why does HD 209458 b wear an XXL?

extrasolar planetary transit

In 1999, the sun-like star HD 209458 was discovered to harbor a transiting planet on a 3.52 day orbit. This was a big deal. The recurring occultations permitted, for the first time, an accurate measurement of both the radius and the mass of an extrasolar planet, and there have been a huge number of follow-up observations of the transits using a variety of telescopes and techniques. The most impressive result came from Brown et al. (2001), who used the (now defunct) STIS instrument on the Hubble Space Telescope to obtain a photometric light curve that has precision of about one part in ten thousand per 80-second sample:

extrasolar planetary transit

The plot above can be found in the Astrophysical Journal , or, alternately, the paper containing the plot is available for free at the arXiv preprint server. A careful analysis of the photometric curve and the radial velocity data (which can be explored using the systemic console), combined with estimates for the size, mass and other properties of the parent star, indicates that the planet, HD 209458 “b”, has a radius about 1.35 times larger than the radius of Jupiter, and a mass of 0.69 times Jupiter’s mass. The temperature on the surface of the planet should be a toasty ~1200 K.

Various teams of scientists, including a group led by Peter Bodenheimer here at Santa Cruz, and independent groups led by Tristan Guillot and Gilles Chabrier in France, and Adam Burrows’ group in Arizona have all developed detailed computer programs that can predict how planets respond when placed in different physical environments. Everybody agrees that a gas giant planet with a standard hydrogen-helium composition and the mass and surface temperature of HD 209458 b should have a radius (corresponding to the 1-Atm pressure level) that is about 5-10% larger than Jupiter. The observed size of the planet is thus far out of agreement with the theoretical models. The planet is too large!

As soon the size problem became clear, a number of explanations for HD 209458 b’s large radius were put forward. The Burrows group (2003) pointed out that the planet may appear large during the transit because we are looking obliquely through long path lengths in the planetary atmosphere. Tristan Guillot and Adam Showman (2002) suggested that the ferocious winds on the planetary surface are transferring energy into the deeper layers of the planet, and that this extra source of energy is enough to bloat the planet to its observed size. These two phenomena don’t require anything special about HD 209458 b, and so both hypotheses predict that other planets with similar masses and temperatures should have similarly inflated radii. This doesn’t seem to be the case, however. In August 2004, a transiting planet of very similar mass and temperature was found in transit around an 11.8 magnitude star known as Tres-1 (Alonso et al. 2004). This planet has exactly the size (~1.05 Jupiter radii) predicted by the baseline theories. It thus appears that there is something unusual about HD 209458 b.

One intriguing possibility, suggested by Peter Bodenheimer, Doug Lin and Rosemary Mardling in 2001, is that another planet exists further out in the HD 209458 system. This planet would be exerting gravitational perturbations on HD 209458 b, which would cause its orbit to maintain a small eccentricity. If a planet like HD 209458 is in an eccentric (non-circular) orbit, then it experiences significant tidal stretching and squeezing which generate heat in the planetary interior. In a follow-up paper published in 2003, Bodenheimer et al. calculated that an orbital eccentricity, e=0.03 would likely be sufficient to generate enough tidal heating to inflate HD 209458 b to the observed size.

At that time, there were only 30 high-precision radial velocity measurements of HD 209458, and it was easily possible to find 2-planet fits to the radial velocity data which had (1) a small non-zero eccentricity for HD 209458 b, as well as (2) a second planet with a period of order 80 days, and a mass of ~0.12 Jupiter Masses. In the following diagram, the orbits are to scale, but the star and especially the planets are grossly too big.

a perturbing body

Over the past two years, the California-Carnegie Planet Search Team used the Keck telescope to obtain a number of additional radial velocity measurements of HD 209458, and these have been published in a new paper. The full set of (out-of-transit) measurements have been loaded into the system menu of the systemic console. In our paper, our conclusion was that HD 209458 b is likely the only RV-detectable planet in the system, and that its orbit is most likely circular (more on this in a future post). See, however, if you can use the console to find viable 2-planet fits that have the correct period for the inner planet P=3.52474541 d, and which have a required RMS jitter for the star of less than 5 m/s. (Technically, you should also apply a simultaneous constraint on Mean Anomaly and eccentricity that arises because the time of central transit is known very accurately, but the console doesn’t yet have this capability. If you find a good fit, and post it here, we can likely fold in the additional timing constraint without greatly changing the basic orbital parameters).

The number of known transiting planets has been increasing steadily, and the total now stands at nine. Using the results of Peter Bodenheimer’s planetary structure code, we can compare the planets predicted sizes with their observed sizes:

the properties of the known transiting planets

(Here’s a larger-size .pdf of the above table, which will appear in an upcoming PPV review article). Three of the planets in the table, HD 209458b, HD 149026b, and HD 189733b, have radii that do not agree at all with the predictions. HD 209458b (and to a slightly lesser extent) HD 189733b are both larger than predicted, whereas HD 149026b is too small, likely because it has a huge rocky core:

a size comparison

These discrepancies indicate that the bulk properties of the transiting planets must depend significantly on factors other than their mass and estimated effective temperatures. Like the planets of our solar system, the extrasolar planets are imbued with interesting individual personalities.