Explaining the Mass-Period Relations of Exoplanets

Computer modeling of the formation and subsequent history of planetary system formation—which now takes into account changes in protoplanetary disk viscosity and tracks the movement of disk inhomogeneities (irregularities)—reveals several additional factors that make our solar system’s array of planets so special. It adds substantially to the accumulating evidence explaining why Earth may be alone in manifesting all the attributes necessary for sustaining advanced life.


The first extrasolar planet discovered was found in 1995 orbiting the star 51 Pegasi.1 At that time, most astronomers presumed we would eventually discover many planets manifesting characteristics virtually identical those in our solar system.

To date, astronomers have discovered 854 planets outside the solar system for which they have also measured orbital and physical parameters. Not even one of these planets matches the characteristics of any of the solar system’s planets. The only one that comes close is Upsilon Andromedae e. Its mass compared to Jupiter’s is 1.06+ (because the inclination of Andromedae e’s orbit is not known, only a minimum mass has been determined). Its distance from its host star compared to Jupiter’s distance from the Sun is 1.01.

Planets in outer space.However, Andromedae e is accompanied by two much larger planets with closer, highly eccentric orbits. Upsilon Andromedae c has a mass 14.57+ times greater than Jupiter’s and orbits its host star 6.1 times closer that Andromedae e. Meanwhile, Upsilon Andromedae d’s mass is 10.19+ times greater than Jupiter’s and the planet orbits 2.1 times closer than Andromedae e. Moreover, their host star is 1.28 times more massive and 3.4 times more luminous than the Sun. This system also includes a dim star orbiting the main star 750 times farther away than Earth orbits the Sun. The characteristics of both the main star and the two heavy planets accompanying Upsilon Andromedae e effectively rule out the possibility that another planet in the same system could possibly support advanced life.

The availability of such a large number of well-measured exoplanets indicates to astronomers that the solar system’s planets very likely experienced a unique formation history. Researchers are learning that the diversity of the observed exoplanets’ properties reflects the range of protoplanetary disks from which planets arise. However, observed planet pileups (congregations into closely packed convoys at certain distances from the host stars) and observed mass-period relation (mass of the planet increases in proportion to the duration of its orbital period) present a challenge.

In the second of two papers, astronomers Yasuhiro Hasegawa and Ralph Pudritz show how inhomogeneities (irregularities) in the protoplanetary disks can account for the observed trends seen in the statistics of the 854 detected and measured exoplanets.2 They developed computer models that enabled them to compute how planets grow and radially migrate, either inward or outward relative to the host star, in response to certain protoplanetary disk inhomogeneities.

Hasegawa and Pudritz found that the most significant factors determining the growth and migration rates of protoplanets were the disk viscosity and the rate of disk-gas photoevaporation by the host stars’ light and heat. The inhomogeneities that most affected disk viscosity and disk-gas photoevaporation were (1) dead zones, (2) ice lines, and (3) heat transitions. Dead zones are portions of the protoplanetary disk’s inner regions where poor coupling between the magnetic field and the weakly ionized disk suppresses magnetorotational instabilities. The ice line refers to the distance from the host star where the disk’s temperature drops enough for gases (e.g., water vapor) to freeze into ices. Heat transitions refer to those distances from the host stars at which the main heat source changes from viscous heat to stellar irradiation. By tracking the movement of such disk inhomogeneities Hasegawa and Pudritz were able to produce computer simulations that explain, at least qualitatively, the observed planet pileups and mass-period relation.

Hasegawa and Pudritz’ model also made a number of important predictions that they believe accumulating extrasolar planet discoveries and measurements will help confirm. First, they predict that exoplanet subpopulations will reflect the properties of planet formation traps—regions in the protoplanetary disk where the direction of migration can switch from inward to outward in response to inhomogeneities.5 Second, Hasegawa and Pudritz expect that the biggest planet pileups will occur among gas giants with orbital periods approximating 500 days in duration and super-Earths and Neptune-sized planets orbiting their host stars much closer than Earth orbits the Sun. Third, they anticipate that inward migrations from the ice lines will fill in the “planet desert.” The planet desert refers to the scarcity—predicted by other formation models—of planets between 5–50 times the mass of Earth that orbit their host stars between 4–50 percent of the distance that Earth orbits the Sun.

Exoplanet surveys, and now Hasegawa and Pudritz’s model, reveal that extrasolar planetary systems are characterized by super-Earths and Neptune-sized planets inhabiting the inner planetary system regions and Saturn-, Jupiter-, and super-Jupiter-sized planets inhabiting orbital distances from their host stars akin to those manifested by Venus, Earth, and Mars. By comparison, the solar system contains no planets larger than Earth residing interior to the orbit of Mars. Furthermore, whereas the known extrasolar systems are filled with many high-eccentricity planets, our solar system contains only one—diminutive Mercury (Jupiter’s moon Ganymede is 5 percent larger in diameter and half as massive), which resides so close to the Sun as to pose no risk of disturbing any of the other seven solar system planets.

Time will tell whether Hasegawa’s and Pudritz’s predictions continue to be borne out. What is evident is that observations and measurements on the 854 confirmed exoplanets and the theoretical work of Hasegawa and Pudritz add to the weight of evidence for the rare-Earth and rare-solar system doctrines. These doctrines state that while planets the size and mass of Earth may prove to be abundant, planets with the just-right host stars, just-right planetary partners, and just-right characteristics to support advanced life will prove either rare or nonexistent. Such doctrines are consistent with the Bible’s message that God supernaturally designed Earth, its life, and its cosmic habitat for the specific benefit of human beings.


1. Michel Mayor and Didier Queloz, “A Jupiter-Mass Companion to a Solar-Type Star,” Nature 378 (November 23, 1995): 355–59.

2. Yasuhiro Hasegawa and Ralph E. Pudritz, “Evolutionary Tracks of Trapped, Accreting Protoplanets: The Origin of the Observed Mass-Period Relation,” Astrophysical Journal 760 (December 1, 2012): 117.

3. F. S. Masset et al., “Disk Surface Density Transitions as Protoplanet Traps,” Astrophysical Journal 642 (May 1, 2006): 478–87.

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