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Planetary System Formation

March 21, 2012

Scientists continually try to make sense of their experimental and observational data and to generalize what they see to a broader scope of nature. They've found that mathematics is a wonderful tool to do this. It works so well that Eugene Wigner was inspired to write a paper entitled, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."[1]

Galileo, who is known as the father of modern physics, was among the first to apply mathematics to experimental data, and one of the first to believe that nature obeys mathematical laws. He expressed this belief in The Assayer:
"Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth."[2]

Johannes Kepler is known as much for the Kepler conjecture, that the face-centered cubic packing of spheres is the most efficient at filling space, as his astronomy. Having such geometry on his mind would explain his model of the spacing of the planets in the solar system shown in the figure.

Figure captionAn illustration of Kepler's planetary model, as it appears in the 1910 book by Francis Rolt-Wheeler, "The Science-History of the Universe," as figure 37 on page 112.

(Scan by Sue Clark, via Wikimedia Commons).

This figure, which appears in his 1596 book, Mysterium Cosmographicum, has a nesting of regular polyhedra with Saturn, the farthest known planet at his time, exists at a distance set by a sphere circumscribed around an outer cube. The orbits of the other planets are represented by inner circumscribed and inscribed spheres. I haven't found a reference to how well these distances scale with their actual values, but someone with enough interest can easily do the calculations themselves.

A more mathematical model of planetary distance from the Sun is the Titius-Bode Law that dates to the eighteenth century. This law, which expresses the distance of the planets from the Sun a in astronomical units, is given as,
a = 0.4 + 0.3(2n), n = -∞, 0, 1, 2, 3...

As the graph shows, the equation quite well matched the distances to the planets known at the time. Ceres was included in the law, since it seemed to fit, and it stood as a placeholder for the entire asteroid belt. The asteroids would have formed a planet were it not for the gravitational effects of Jupiter.
Figure caption
Actual distances (orange), and distances predicted by the Titius-Bode Law (green), for the planets. In this case, Ceres and Pluto are considered planets. (Rendered by author using Gnumeric).

The discovery of Neptune in 1846 ended Titius-Bode Law's reign of accuracy. When Pluto was discovered in 1930, the deviation from the formula was nearly 100%. Of course, there was no physical explanation as to why the law should be true, so it's just an example of scientific numerology.

Deviation from Bode's LawDeviation from Bode's Law.

(Rendered by author using Gnumeric).

Nowadays, with continued discovery of exoplanets, we have more than our solitary solar system to study. Observations of planets in orbit around young stars have shown "planet pileups," in which many planets orbit very near to each other; and "planet deserts" that contain no planets in a wide swath of a stellar system. An explanation for this finding is contained in a paper scheduled for the Monthly Notices of the Royal Astronomical Society and appearing as an arXiv preprint.[3-4]

This paper, and a companion presentation at the 43rd Lunar and Planetary Science Conference by Ilaria Pascucci of the University of Arizona Lunar and Planetary Laboratory and Richard Alexander of the Department of Physics & Astronomy, University of Leicester, describes computer simulations of planetary formation and evolution in such systems.[3-4]

What the simulations show is that there is a competition in these systems between gravity and a process, called photo-evaporation, by which protoplanetary material is evaporated by the star's intense radiation and is lost into space. Says coauthor, Alexander, "The disk material that is very close to the star is very hot, but it is held in place by the star's strong gravity... Further out in the disk where gravity is much weaker, the heated gas evaporates into space."[3] Somewhat further out, there isn't enough radiation for evaporation.

For such stars a gap results from these competing processes at a distance of about 1-2 astronomical units. As explained by coauthor, Pascucci, "The planets either stop right before or behind the gap, creating a pile-up... The local concentration of planets leaves behind regions elsewhere in the disk that are devoid of any planets. This uneven distribution is exactly what we see in many newly discovered solar systems."[3]

Their research was funded by the US National Science Foundation and the UK Science and Technology Facilities Council.[3]

Artist's conception of a protoplanetary disk.Artist's conception of a protoplanetary disk around the brown dwarf, OTS 44.

(Image: NASA Spitzer Space Telescope Collection, NASA/JPL-Caltech/T. Pyle).

References:

  1. E.P. Wigner, "The unreasonable effectiveness of mathematics in the natural sciences," Communications on Pure and Applied Mathematics, vol. 13, no. 1 (February, 1960). pp. 1-14. A PDF file is available, here.
  2. Galileo quotations on Wikiquotes; Selections of a translation of The Assayer; Stillman Drake, "Discoveries and Opinions of Galileo," Doubleday & Co.(New York, 1957), pp. 231-280.
  3. Daniel Stolte, "Popular Solar System Orbits Result in 'Planet Pileups'," University of Arizona Press Release, March 19, 2012
  4. R.D. Alexander and I. Pascucci, "Deserts and pile-ups in the distribution of exoplanets due to photoevaporative disc clearing," arXiv Preprint Server, February 24, 2012.