Wednesday, May 31, 2017

The Small Mars Problem

Figure 1. All planets and dwarf planets orbiting within 6 astronomical units (AU) of our Sun, shown at their relative diameters.

From afar, our Solar System looks regular and well-organized. At its center is a large, massive sphere of incandescent gases (the Sun) surrounded by eight smaller and less massive spheres of heavier elements (the planets) distributed in concentric orbits out to a distance of about 4.5 billion km/2.8 billion miles.

The orbital distribution of the eight planets also seems regular, at least at first glance. Smaller, rocky worlds are confined to the inner system, while larger, gaseous worlds dominate the outer system. Planet sizes follow a curve, rising from the inner to the middle planets and then declining again from the middle to the outer planets.

In more specific terms, mass and radius increase along with distance among the three planets closest to the Sun (Mercury through Earth). Both parameters peak at the orbit of the fifth planet, Jupiter, which is almost a dozen times the radius and more than 300 times the mass of Earth. Then, from Jupiter through Uranus, the seventh planet, both mass and radius decline substantially along with distance from the Sun.

But this orderly progression of planet sizes has two notable interruptions: Mars and Neptune. If the distribution of planets were truly regular, Mars would be larger and more massive than Earth, and Neptune would be smaller and less massive than Uranus. Instead, the Red Planet has only 53% of Earth’s radius (0.53 Rea) and 11% of its mass (0.11 Mea), while the Azure Planet, at 17.2 Mea and 3.9 Rea, has about 98% of the radius of Uranus but 119% of its mass.

How did that happen?

In the present post I’m going to ignore the oddity of Neptune and concentrate on the Martian half of the question. My rationale is that Mars occupies our system’s classical habitable zone, and therefore – along with Earth and Venus – plays a critical role in theories of the habitability of extrasolar planets. If mass had been more uniformly distributed in the inner Solar System, Mars would be more massive than it is. If its mass were in the range of 1 to 2 Mea, Mars would likely be able to sustain a magnetic field, plate tectonics, surface water, and long-term habitability. Therefore, if we want to understand the potential system architectures that might support life-bearing planets, we need to understand why Mars is so small.

Figures 1 and 2 highlight the Small Mars Problem and the Great Martian Gap, which is the name I just invented for the general depletion of mass between Earth and Jupiter. The planet Mars and the dwarf planet Ceres orbit within this gap at 1.52 AU and 2.77 AU, respectively. With a little more than 1% of the mass of our Moon, Ceres accounts for fully one-third of all mass in the Asteroid Belt, which is concentrated between 2.2 and 3.3 AU (the latter boundary provided by the 2:1 resonance with Jupiter's orbit; Jewitt et al. 2009). The entire region between the orbits of Earth and Jupiter contains less than 0.12 Mea, with little Mars accounting for 99% of the total. By contrast, the region extending inward from Earth’s orbit to the Sun contains 1.87 Mea, yet Earth, the most massive object, accounts for only 53% of the total.

Figure 2. The Great Martian Gap

Blue numbers along the bottom refer to astronomical units (AU), where the Earth/Sun separation = 1. Planets are shown at their relative sizes and relative distances from the Sun, with separate scales for radius and distance. As astronomers have long noted, mass is severely depleted between the orbit of Jupiter at 5.2 AU and the orbit of Earth at 1 AU (see Weidenschilling 1977).

zigzag migration

Recent studies by Konstantin Batygin & Greg Laughlin (2015) and by Sean Raymond & colleagues (2016) have presented conflicting scenarios to explain the Small Mars Problem and the Great Martian Gap. Both involve zigzag migratory paths for Jupiter during the primordial phase of system evolution.

Batygin & Laughlin based their approach on earlier models by Kevin Walsh & colleagues (2011) and Pierens & Raymond (2011), in which Jupiter formed in the outer Solar System (somewhere beyond 3 AU) and then migrated first inward and then outward again. The popular name for this scenario (blogged here and here) is the Grand Tack. According to Batygin & Laughlin, these maneuvers not only swept most solid mass out of the region exterior to Earth’s present orbit, but also created an instability that emptied the region interior to 0.7 AU.

Raymond & colleagues took a very different approach based on the “inside out” model of planet formation presented by Chatterjee & Tan (2014). Contrary to his own earlier work, Raymond’s group proposed that Jupiter formed in the inner Solar System near the Sun, and then migrated first outward, then inward, and finally outward again, depleting the region inward of Venus and wreaking havoc beyond Earth.

sweeping secular resonances

While the Grand Tack has been more widely discussed and endorsed than the inside-out scenario, both explanations have been faulted. Now Benjamin Bromley & Scott Kenyon (2017) present an alternative approach in which “sweeping secular resonances” with Jupiter’s orbital motion, rather than any migratory scenario, become the mechanism for clearing the Great Martian Gap. Their model implies a less dramatic but equally consequential role for Jupiter, and I suspect that it can be extended to explain similar gaps observed in the architecture of multiplanet systems around other stars.

To develop their model, Bromley & Kenyon (hereafter BK17) conducted extensive numerical simulations based on earlier work by and with their collaborators Makiko Nagasawa and Edward Thommes (Nagasawa et al. 2007, Thommes et al. 2008). They also note recent work on the same problem by Xiaochen Zheng & colleagues (2017).

BK17 begin with the familiar theoretical construct of the Minimum Mass Solar Nebula (blogged here). They assume that a dusty gas nebula (generally known as a protoplanetary disk) is present at the outset of their simulations. Jupiter is fully formed at its current semimajor axis of 5.2 AU (Figure 2), having cleared a gap in the disk for 1 AU on either side of its orbital path. A swarm of planetesimals orbits inward of this gap, while Saturn orbits well beyond it. Both gas giants exert gravitational effects on their surroundings, and the disk itself has gravity. In addition, the orbit of Jupiter is slightly eccentric, but probably less so than its present value of 0.05. BK17 assume an eccentricity of 0.03 in their simulations.

The key factor in their approach is the n5 resonance (“nu-5,” Greek letter nu with superscript 5), a “secular” or “very long-term” resonance between the motion of the protoplanetary disk and Jupiter’s orbital period. BK17 define the nu-5 resonance as the location “where the local apsidal precession rate matches Jupiter’s rate of precession,” and note that a planetesimal or protoplanet at this location will be perturbed by Jupiter’s gravitational influence onto a highly eccentric orbit. The likely result will then be either collision with another planet or protoplanet, engulfment by the Sun, or ejection from the Solar System.

In the early Solar System, when the gas disk was still present, the nu-5 resonance was located in the vicinity of the present Asteroid Belt (Zheng et al. 2012). As the gas dissipated, the resonance moved inward, destabilizing (“shaking up”) the orbits of protoplanets and planetesimals and effectively clearing out a substantial mass in solids. After the gas was completely depleted, the nu-5 resonance reached its present position inside the orbit of Venus. This sweeping shake-up created the Great Martian Gap while leaving behind enough mass to build Earth and Venus, as well as their two by-blows, Mercury and Mars.

BK17 discovered that several different factors were critical to reproducing the mass of Mars and the present-day Asteroid Belt within the time constraints provided by the known age of Mars. These include the mass of the perturbing planet, its distance from the system habitable zone, and the timing and speed of the sweeping secular resonance generated by its orbital motion.

Regarding mass, BK17 find that only a “Jupiter-mass planet” can produce the magnitude of perturbation required to induce a shake-up in the protoplanetary disk of a Sun-like star. Unfortunately, they don’t provide a precise value for the necessary mass – for example, would an object of Saturn’s mass (95 Mea) be sufficient? They also note that a “super-Earth” would be massive enough to produce sweeping secular resonances in an M dwarf system, likely referring to an object in the range of 1-10 Mea (see, e.g., Kenyon & Bromley 2009).

Regarding orbital location, they find that the masses of Earth and Mars depend sensitively on the semimajor axis of Jupiter at the time of the sweeping resonance. If Jupiter had been substantially farther from the Sun, the resonance would never have reached the orbit of Mars, and Mars would have grown much bigger than it actually did – presumably massive enough to support a habitable environment. But if Jupiter had been substantially closer to the Sun, and thus closer to the system habitable zone, the resonance would have inhibited the formation of Earth in the same way that it stunted the growth of Mars when Jupiter was at 5.2 AU. Instead of one living planet, our system would have none at all.

Regarding the timing of the sweeping secular resonance, BK17 note that its schedule is determined by the lifetime of the protoplanetary disk. As we saw in an earlier post, the system age when gas dissipation commences can fall anywhere between 1 and 10 million years. At the early end of that range, according to BK17, dissipation accompanied by shake-up would have extremely negative consequences for rocky planet formation, as it would destroy planetesimals before they had time to accrete into protoplanets. At the latter end, however, the effects would be modest, since accretion would already be well advanced, potentially permitting the growth of Earth-size planets out to a distance of 3 AU. In the case of our Solar System, we can assume that the shake-up happened before a system age of about 4 million years, given radiometric evidence that Mars was fully formed by then.

The rate of disk dispersal also matters. Although many studies have found that gas dissipation happens rapidly, requiring less than half a million years from start to finish (Williams & Cieza 2011), variation is inevitable: some disks take longer than others to disperse. BK17 find that the relative speed of dissipation strongly affects the outcomes of secular resonance sweeping. If the gas dissipates quickly, the resonance sweeps inward at the same rate, resulting in minimal disruption of the planetesimal population. If the gas dissipates more slowly, the resonance becomes increasingly more destructive, clearing larger and larger quantities of solid mass from the system.

Figure 3. Calaveras street sweepers on the Day of the Dead
From a print by José Guadalupe Posada (1852-1913)

extrasolar asteroids and orbital gaps

Although BK17 are interested primarily in the evolution of our Solar System, they attempt to generalize some of their results to extrasolar locales. Their chief concern is the occurrence of extrasolar analogs of the Asteroid Belt. They argue that most systems with a gas giant in Jupiter’s approximate location (i.e., just outside the system ice line, where accretion is maximized) will experience a sweeping secular resonance whose outcome will be a ring of rocky debris in the inner system. While they concede that few such structures have been discovered to date (HD 69830 is most familiar), they attribute these limited findings to the difficulty of discerning modest aggregations of warm debris even around nearby stars. In the future, they predict, more sensitive searches will be more successful.

I suggest that the implications of their model are much broader than their relevance to extrasolar asteroid belts, and far more dispiriting. If sweeping secular resonances are common in systems with cool gas giants, then the outlook for habitable planets is even less promising than I thought. Here’s why.

An important focus of this blog is the possibility of Solar System analogs – that is, exoplanetary systems containing cool giants whose orbital parameters would permit the survival of Earth-mass planets (0.5-2 Mea) in the local habitable zone. (For recent posts on this topic, see here and here.) My January search of the Extrasolar Planets Encyclopaedia identified 17 such systems located within 60 parsecs/196 light years. All center on Sun-like stars in the range of 0.85-1.15 Solar masses, so their habitable zones have boundaries similar to those proposed for our own system (0.99-1.70 AU; Kopparapu et al. 2013).

Among the Jupiter analogs in these systems, semimajor axes range from 3 AU to 5.2 AU, and more than half orbit inside 4 AU. According to the findings of BK17, virtually all these systems will have experienced a sweeping secular resonance very similar to the one they propose for the Solar System. Because all but one of the 17 confirmed Jupiter analogs orbits closer to the local habitable zone than does our own Jupiter, the depletion of mass in this favored region is likely to be even more extreme than it was at home. Therefore, habitable planets appear to be less likely in the existing sample of Solar System analogs than they are in the Solar System.

To put it another way: BK17 have just shown that gas giant planets are even more unfriendly to the formation and survival of habitable planets than we already suspected. It’s not enough for the giant to reside outside the system ice line in a configuration that permits an Earth-like planet to maintain an Earth-like orbit. The giant must also be distant enough from the central star that the sweeping secular resonance generated by its orbit was insufficient to clear solid mass from the local habitable zone. Even Jupiter managed to evacuate mass from more than half of the radial extent of our own habitable zone, drastically reducing our system’s potential to produce life-bearing planets. Now it looks like extrasolar Jupiters might be still more likely to foreclose the possibility of life around other stars.

Batygin K, Laughlin G. (2015) Jupiter’s decisive role in the inner Solar System’s early evolution. Proceedings of the National Academy of Sciences 112, 4214-4217. Abstract: 2015PNAS..112.4214B
Bromley BC, Kenyon SJ. (2017) Terrestrial planet formation: Dynamical shake-up and the low mass of Mars. Astronomical Journal 153, 216. Abstract: 2017AJ....153..216B
Chatterjee S, Tan JC. (2014) Inside-out planet formation. Astrophysical Journal 780, 53.
Jewitt D, Moro-Martín A, Lacerda P. (2009) The Kuiper Belt and Other Debris Disks. In Astrophysics in the Next Decade, edited by Harley A. Thronson, Massimo Stiavelli, Alexander Tielens. Springer. Abstract: 2009ASSP...10...53J
Kopparapu R, Ramirez RM, Kasting JF, Eymet V, Robinson TD, Mahadevan S, Terrien RC, Domagal-Goldman S, Meadows V, Deshpande R. (2013) Habitable zones around main-sequence stars: New estimates. Astrophysical Journal 65, 131.
Kenyon SJ, Bromley BC. (2009) Rapid formation of icy super-Earths and the cores of gas giant planets. Astrophysical Journal 690, L140-L143.
Nagasawa M, Thommes EW, Kenyon SJ, Bromley BC, Lin DNC. (2007) The diverse origins of terrestrial-planet systems. In Protostars and Planets V, edited by B. Reipurth, D. Jewitt, K. Keil. University of Arizona Press, pages 639-654. Abstract: 2007prpl.conf..639N
Pierens A, Raymond SN. (2011) Two phase, inward-then-outward migration of Jupiter and Saturn in the gaseous solar nebula. Astronomy & Astrophysics 533, A131. Abstract: 2011A&A...533A.131P
Raymond SN, Izidoro A, Bitsch B, Jacobson SA. (2016) Did Jupiter’s core form in the innermost parts of the Sun’s protoplanetary disk? Monthly Notices of the Royal Astronomical Society 458, 2962-2972. Abstract: 2016MNRAS.458.2962R
Thommes E, Nagasawa M, Lin DNC. (2008) Dynamical shake-up of planetary systems. II. N-body simulations of Solar System terrestrial planet formation induced by secular resonance sweeping. Astrophysical Journal 676, 728-739. Abstract: 2008ApJ...676..728T
Weidenschilling JS. (1977) The distribution of mass in the planetary system and solar nebula. Astrophysics and Space Science 51, 153-158.
Williams JP, Cieza LC. (2011) Protoplanetary disks and their evolution. Annual Review of Astronomy and Astrophysics 49, 67-117. Abstract: 2011ARA&A..49...67W
Zheng X, Lin DNC, Kouwenhoven MBN. (2017) Planetesimal clearing and size-dependent asteroid retention by secular resonance sweeping during the depletion of the Solar Nebula. Astrophysical Journal 836, 207.


  1. The conclusion doesn't seem optimistic. But the HIP 11915 system might still have a chance. Its "Jupiter analog" (with around the same mass) is only slightly closer to the star. I guess if it was more massive (like the other planets from your list), it'd have more influence.

  2. Have you seen the new models to explain the small Mars problem?