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We’ve noticed that fresh content encourages regular return visits to oklo.org.

With that sentiment in mind, here’s a stop-action .mpeg4 animation of the newly discovered 2:1 resonant planetary system orbiting HD73526. The planets are represented by red and green peppercorns, and a kumquat stands in for the central star:

hd73526.mov

If the version above won’t load in your browser, try this one. Rest assured that the systemic team is hard at work on more substantive posts (including some very interesting new exoplanet-related results), so check back frequently!

It’s been unseasonably cold in Santa Cruz. Last night, a freak hailstorm left drifts of icy planetesimals lodged between the leaves of the banana tree outside the bedroom window.

My office, however, is nice and warm. This is because two 2.5 Ghz G5 processors are running mercury.f at full tilt to simulate the formation of habitable planets orbiting low-mass red dwarf stars. The calculations are being done in preparation for Ryan Montgomery’s presentation at AbSciCon in Washington D.C. Two weeks to go.

Two Spheres Within A Sphere Alexander Calder, 1931

Most of the runs have already been carried out using a linux-based beowulf cluster, which is able to run more than 100 individual simulations at once. Each of these simulations starts with 1000 low-mass planetesimals, and calculates the final stages of terrestrial planet formation by allowing the orbiting planetesimals to interact under their mutual gravitational influence. Collisions and ejections gradually winnow the initial swarm down to a few surviving terrestrial mass planets. By doing many simulations, we build up a statistical picture of what the distribution of red dwarf planetary systems should look like.

The desktop computer contains the fastest individual processors to which we have full access. We’ve therefore harnessed it for a single test-case run to investigate the overall sensitivity of our results to the number of initial particles. The processors have spent the last six weeks evolving a system that had an initial distribution of 10,000 small planetesimals. At the projected rate of evolution, it should just manage to finish up just in time. Indeed, if you see Ryan hunched over his laptop at the conference, you’ll know what he’s up to.

John Chambers’ Mercury code (like the Systemic Console in integrator mode) is based on the method of direct summation. At each timestep, each particle in the simulation experiences a gravitational attraction of the form GM/r^2 from every other body in the system. For a 10,000 particle system, that means of order ((10,000)x(9,999))/2=49,995,000 square roots must be computed every timestep (and the actual number is higher, because each timestep consists of a considerable number of substeps). As the number of particles increases, the cost of the calculation increases as the number of particles squared. Our 10,000 particle simulation is 100 times more expensive than the 1,000 particle production runs, and thus pushes the limits of what we can currently readily do.

Many problems in gravitational N-body dynamics can be solved without resorting to direct summation. In essence, this is because to a high degree of approximation, the gravitational attraction from distant particles depends only on the the rough location and total mass of the distant particles. One gets nearly the same result by lumping distant particles into a single, equivalent, large-mass particle:

Using clever variations of this basic idea, one can speed up an N-body (or equivalently, an SPH) calculation enormously. Competitive N-body codes for large-N problems, such as the collisions of galaxies or the formation of structure in the early universe, generally scale as N log(N). For large N, the difference between N^2 and N log(N) is profound. With a million particles, for example, an N log(N) calculation is a cool 72,382 times faster than the brute-force N^2 approach.

One might ask, is there a way to further speed up the computation of the gravitational forces so that finding the accelerations becomes an order N process?

Remarkably, an order-N computational N-body method was employed by Erik Holmberg of the Lund Observatory in Sweden in 1941. Instead of integrating the equations of motion with a computer, Holmberg modeled a two-dimensional system of gravitating particles as an actual physical distribution of movable light bulbs laid out on a gridded sheet of dark paper! Because the intensity of light from a point source diminishes as 1/r^2, one can directly relate the intensity of the light at a particular spot to the gravitational acceleration. The order-N^2 process of computing the gravitational force on a given particle from all of the other particles reduces to a measurement of the total intensity of light in two perpendicular directions using a photocell and a galvanometer. Since one set of measurements is required for the location of each light bulb, the method scales as N. Here is a link to Holmberg’s paper. It’s one of my all-time favorites.

With his analog method for computing the net gravitational acceleration on each of his light bulb “point masses”, Holmberg could compute the change in trajectories which would occur over a time interval using a simple integration scheme such as Euler’s method. A timestep would then be completed by moving all of the light bulbs to their updated positions, at which point a new estimate of the gravitational acceleration could be made. Holmberg’s scheme allowed him to gain a better understanding of important aspects of the dynamics of close encounters between disk galaxies, including the phenomena of orbital decay and the formation of tidal tails:

There is an interesting lesson to be drawn. Use of an analog method reduces an N^2 direct summation computation to order N, foreshadowing a time when quantum computation will similarly reduce the computational time for direct summation from N^2 to N. Until that time, however, the light-bulb method beats all others as the number of particles approaches an arbitrarily large value.

In honor of analog methods, here’s a home-made mpeg-4 stop-action animation of a peppercorn orbiting a kumquat in an ellipse with e=0.90. (Try this version if the other one loads a screen of gibberish).

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The black cloud post describes how the formation of a star and a planetary system can be traced back to the moment when a dense core within a giant molecular cloud begins to suffer an inside-out collapse. The gas at the center of the cloud collapses first, and congregates into the beginnings of a hydrostatically supported protostar. The overlying regions thus lose their support and begin to career inward as well. A wave of collapse radiates outward from the center of the cloud, triggering a downward avalanche of gas and dust. Computer simulations show how gas that has fallen from large distances comes together to form a protostellar disk in orbit around the nascent central protostar. In the image shown below, a simulation of the earliest phases of our own solar system show a region (viewed edge-on) that is several hundred astronomical units across, and plotted 40,000 years after the collapse has started. At this stage of the simulation, roughly half of a solar mass of material has collected in the central protostar, and another half a solar mass or so is orbiting in a very massive protostellar disk.

As the disk grows in mass, it begins to feel its own self-gravity, and some regions begin to collapse under their own weight. At the same time, the pressure of the gas in the disk resists the tendancy to collapse, and the differential rotation of the disk acts to sheer out fragments as they grow. This process can also be simulated, and the result is spiral waves (viewed here from above):

The presence of the spiral waves causes angular momentum to be transferred outward through the disk, while allowing the majority of the mass to flow inward to eventually join with the central protostar. Even after the spiral waves have dissipated, there must exist continuing source(s) of angular momentum transfer through the disk. The identification of these mechanisms is still an active area of research. Possible mechanisms that might operate after the disk is no longer massive enough to support self-gravitating spiral waves include the magneto-rotational instability, as well as convection-driven turbulence in the disk. One way or another, angular momentum transport was extremely effective. The initial cloud that formed our solar system was rotating more or less uniformly, wheras at the present day, there is nearly a complete separation between mass and angular momentum in the solar system. The Sun contains more than 99.8% of the mass, and the planets carry more than 98% of the system angular momentum.

When I give public talks on planet formation, I like to show the above image (taken by HST, and released in 1995) of a protostellar disk, or proplyd, in the Orion star-forming region. We see the cold proplyd from an edge-on vantage, against a diffuse background of hot glowing gas. This disk is at a somewhat later phase of evolution than the ones pictured in the above simulations. It’s roughly 1400 AU across, which is more than 15 times the diameter of Neptune’s orbit, and considerably larger even than the orbits of the newly discovered Kuiper belt objects 2003 UB313, and Sedna. To give an idea of scale, I’ve integrated both Sedna and 2003 UB313 for one Sedna orbit (12,050 years) and plotted their positions relative to the plane of our solar system and superimposed (to scale) on the proplyd. Sedna, is currently in the portion of its orbit where it is speeding (in its rather lazy, loosely bound fashion) through perihelion, and hence the dots plotted at 120.5 year intervals in that region are spaced widely apart. Seen from above, Sedna’s eccentric (e=0.855) orbit would have a aphelion point considerably beyond the radial edge of the proplyd.

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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.

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).

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).

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.

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:

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:

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:

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:

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:

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

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:

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

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

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.