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:

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,
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.

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.

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.

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…

We came from a black cloud.

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.

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.

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.

Of all the photographs that our robot emissaries have radioed back to Earth, my vote for the most stunning is the Hubble ACS image of the “Sombrero Galaxy”, M104. The glow of its halo makes the the idea of 100 billion stars seem comprehensible.

It’s important to remember, however, that the Hubble image is actually a long CCD time-exposure to light gathered by a 240 cm mirror. If you could be somehow transported to a location in space where M104 looms large in the sky, you would see that HST imparts a severely inflated expectation. From a distance, say, of 300,000 light years, M104 would be so dim that you would see only a faintly ominous, faintly glowing flying saucer.

Indeed, the great Andromeda Galaxy, M31, subtends an angle larger than the full Moon in the sky, and it is literally almost directly overhead right now (9:36 PM, Dec 3, latitude 36.97 deg N). The storms from earlier this week have blown through. The sky sparkles with brilliant clarity. Yet when I step outside and look up, I can’t see the Andromeda Galaxy at all. It’s too faint. In a 1:10,000,000,000,000 scale model of M31, the stars are like fine grains of sand separated by miles. Our Galaxy, the Andromeda Galaxy, and the Sombrero Galaxy are all essentially just empty space. To zeroth, to first, to second approximation, a galaxy is nothing at all.

A Hot Jupiter, on the other hand, seen at similar angular size, is undeniably impressive.

The dayside, blindingly illuminated by the scorching proximity of the star, is roughly 500 times brighter than desert sand dunes on a midsummer day. In order to look at the illuminated side of the planet at all, you need extremely dark wraparound sunglasses, or better yet, an eyeshield made from #10 welders glass (where #14 welder’s glass is recommended for those who stare at the sun).

With the brilliance of the dayside cut to a manageable level, what would you see? The majority of the light coming from the planet is simply reflected starlight. If the planet uniformly reflects the light that strikes it, then you simply see a blank white surface if the parent star is similar to the Sun, and a yellow-orange to orange-red expanse if the parent star is a cooler K-type or M-type dwarf star.

The gases that make up the outer layers of the planet do not reflect all frequencies of light equally, however. The air of the outer layers of a hot Jupiter is a scaldingly toxic witches brew of hydrogen, helium, steam, methane, ammonia, cyanide, acetylene, hydrogen sulfide, soot, and a whole host of other hardy, reactive, and generally unpleasant compounds.

In our solar system, for example, Uranus and Neptune have distinctive blue-green casts because at the level in their atmospheres where light is primarily reflected, the ambient methane gas is highly effective at absorbing red frequencies. The originally white sunlight is reflected with a blue-green hue by the selective removal of red.

The photo (mosaic) below was obtained by the Cassini spacecraft as it was flung past Jupiter on its way to Saturn. The images were processed to give the same view that the naked eye would see. Jupiter reflects an enormous amount of detail from its cloudy face.

Across the swathes of Jupiter where the visible clouds tower to great heights, the eye sees regions that are frigid, eighty degrees colder than the depths of an Antarctic winter (-200 F). In such a cold environment, icy compounds of Ammonia are stable, and their presence lends the clouds a reddish hue. Jupiter’s Great Red Spot is an example of just such a topographic high.

On other regions of Jupiter’s visible surface, the atmosphere is transparent to greater depths. As on Earth, where clear skies are associated with dry air, so too on Jupiter. When we look down into the drier Jovian regions, we see to lower lying decks of cloud where the temperature is about the same as a chilly Arctic night. Here, the chemistry in the clouds causes their color to tend toward lighter shades, whites, beiges, ochers.

Like any non-transparent object, Jupiter glows with its own radiation. Because the outer layers of Jupiter are so cold, this intrinsic light lies in the infrared. Seen with an infrared detector (such as this view made at 5 microns with the NASA IRTF) Jupiter is a dramatic sight.

In the rattlesnakes-eye view, the Red Spot forms an oval of relative darkness. The high clouds act like a blanket that blocks the warmer underlying layers from view. In the infrared, the dry areas, where we see the deepest, glow the brightest. In an ironic twist of fate, the Galileo atmospheric probe parachuted into one of the driest regions of the Jovian atmosphere, a so-called 5 micron hot spot (circled in the image above).

On a hot Jupiter, the surface gas is heated to temperatures in the 1000-1500 K range on the dayside. Computer simulations show that winds of hellacious strength tear continually around the planet, carrying heat from the dayside and disgorging it into the night. The atmosphere on nightside glows brilliantly. Turbulent brick-red whorls merge into fiery tendrils of orange braided with dazzling white.