# Jupiter: Friend or Foe?

Διὸς δ᾽ ἐτελείετο βουλή — Homer, Iliad, I 5

Jupiter was once thought to play a protective role in our Solar System, preventing comets from the outer Solar System from reaching the region of the terrestrial planets, and Earth in particular, where they pose a threat to the survival of complex life by colliding with the planet. The destructive power of such collisions is evident from the Chixulub crater in Yucatán, created by the impactor that probably caused the extinction of the dinosaurs. A large enough impactor would be sufficient to destroy all the biosphere save perhaps a few hardy micro-organisms, and a planetary system where such impacts were frequent would be inimical to the survival or even emergence of complex life.

The existence of Jupiter as a “shield”, protecting our planet from dangerous bodies, is one of the pillars of the “Rare Earth Hypothesis” promulgated by Ward & Brownlee. This states that planets with complex multi-cellular life are extremely rare in the cosmos, because the conditions that allow such life to develop and survive are hard to fulfil. For example, a planet must support liquid water on its surface, and hence cannot be too close to nor too far from its star (This region is known as the Habitable Zone). Its spin axis must be fairly well-aligned with its orbital axis, so as to avoid excessive seasonal temperature variations; our own Moon stabilises the Earth’s rotation axis, keeping it only moderately misaligned with the orbital axis. And the planet must not be subjected to too large a flux of dangerous asteroids and comets; it was thought that Jupiter plays a protective role in our Solar System by clearing out dangerous comets. Hence, Ward & Brownlee argued that the suitability of a planet for hosting complex life is exquisitely sensitive to the properties of other planets in its solar system. In particular, they argued that a Jupiter-like gas giant must exist for a terrestrial planet to support complex life, and that such giant planets are not common, so nor will complex life-bearing planets be.

In a series of papers, Horner & Jones have set out to test this latter argument: the hypothesis that Jupiter does actually protect our planet from impactors. They is reason to be skeptical of this claim, since while Jupiter may throw some bodies out of the Solar System or remove them by colliding with them, it may equally well destabilise others and send them onto Earth-crossing orbits. “What Jupiter gives with one hand, it may take away with the other.” The authors test the hypothesis by numerically integrating the orbits of hypothetical comets and asteroids under the gravitational influence of the planets, and counting how many hit the Earth (The size of Earth is artificially `inflated’ to ensure good statistics, since the real Earth is a very small target). There are two related questions that need answering: first, does the very existence of Jupiter enhance or reduce impact flux, and second, does changing the mass and orbit of Jupiter change the impact flux. Horner & Jones’ first three papers examined the role of changing Jupiter’s mass on three populations of impactors: asteroids from the Main Belt between Mars and Jupiter; Centaurs, which have unstable orbits crossing the giant planets’; and long-period comets from the Oort cloud. Their newest paper looks at the role of changing Jupiter’s orbital eccentricity and inclinations on the Asteroid Belt and Centaurs. Let us, like the authors, take each of these in turn.

## I: The Asteroids

The effect of Jupiter on a hypothetical Asteroid Belt. The plots show histograms of the number of asteroids per semi-major axis bin. The initial population is shown in the bottom panel. The population remaining after 10 million years is shown in the upper two panels. The top panel shows the effects of the real Jupiter, while the bottom shows the effects of a "Jupiter" whose mass is only a quarter that of the real Jupiter. Notice the depletion of bodies at resonant locations in the Belt.

The first paper looks at the efficiency of Jupiter-type planets at destabilising bodies in the Asteroid Belt. A hypothetical primordial Asteroid Belt was placed between the orbits of Mars and Jupiter (shown in the lower panel of the above plot), and the evolution of the asteroids’ orbits followed for 10 million years. The shape of the belts that remained at the end of the integration, for the real Jupiter and a “Jupiter” reduced to a quarter of its real mass, are shown in the upper two panels. Both belts show cleared regions associated with mean motion resonances with “Jupiter”, where the asteroids’ orbital periods are close to an integer ratio with “Jupiter’s” and the asteroids experience strong perturbations and are destabilised. These are known as Kirkwood Gaps after their discoverer. There is also a broader cleared region at the inner edge of the belt, at 2 AU for the Jupiter and 2.5 AU for the quarter-Jupiter. This is due to another type of resonance called a secular resonance, which again destabilises the asteroids.

The location of the secular resonance moves closer to “Jupiter’s” location, where there are more asteroids, at lower masses of the “Jupiter”. This means that, somewhat counterintuitively, the lower-mass “Jupiters” may destabilise more asteroids. The numbers of bodies hitting Earth for a whole range of “Jupiter” masses are shown below:

The total number of asteroids hitting the (inflated) Earth, as a function of "Jupiter's" mass. The lines show the cumulative number of impactors at 1, 2, 5 and 10 million years. The real Jupiter is more dangerous than very large or very small "Jupiters", but less dangerous than an intermediate-sized "Jupiter".

It is clear that very small “Jupiters” do not result in many impactors since they do not perturb the Asteroid Belt significantly. Larger “Jupiters”, up to around 0.3 Jupiter masses, result in significantly more disruption to the Belt, while as “Jupiter’s” mass is increased beyond this the number of Earth impactors falls again. Hence, the hypothesis the Jupiter acts as a shield is indeed only partly true: while the real Jupiter provides more protection than one only half or a third of the size, more protection would be afforded by one either more massive or significantly less massive.

## II: The Centaurs

The Centaurs are a population of bodies whose orbits in the outer Solar System intersect the giant planets’. As such they are highly unstable, and many are sent into the inner Solar System to become short-period comets. The population is thought to be ultimately replenished by objects from the Kuiper Belt beyond Neptune.

In their second paper the authors looked at the number of Earth impactors coming from the Centaur population as a function of Jupiter’s mass. The same pattern is seen as for the Main Belt Asteroids: the impact risk is small for small masses of “Jupiter”, rises to a maximum at around 0.2 Jupiter masses, and then falls as “Jupiter’s” mass is increased further:

The number of Short-Period Comets hitting Earth as a function of "Jupiter's" mass. Lines show the cumulative number after 2, 4, 6, 8 and 10 million years.

In this case, the danger posed by “Jupiter” is due to a balance between its ability to destabilise the Centaur bodies and its ability to remove them from the Solar System. Planets around a quarter of Jupiter’s mass are good at the former but bad at the latter, explaining why they are most dangerous. The fact that the impact flux peaks at about the same mass for both Asteroids and Centaur populations appears to be a coincidence.

## III: The Oort Cloud

Long-Period Comets hail from the Oort Cloud, the swarm of bodies on very wide (many thousands or tens of thousands of AU) orbits which surrounds the planetary regions of the Solar System. Bodies in the Oort Cloud suffer perturbations from extra-Solar sources such as nearby stars, and the changes to their orbits can bring their pericentres to within a few AU of the Sun where they can interact with the planets.

The these objects, the cause of injection onto Earth-crossing orbits is effects from outside the Solar System, while the role of Jupiter and the other giant planets is simply to eject ones that encounter them, an outcome which is more likely for higher planetary masses. Hence, this population is the only one from which Jupiter acts unambiguously as a shield, since there is a decreasing number of Earth-crossing comets as “Jupiter’s” mass is increased. Indeed, the efficiency of Jupiter removing such comets was the origin of the idea that Jupiter acts as a shield in the first place.

Number of Long-Period Comets from the Oort Cloud that cross Earth's orbit, as a function of time. The different lines show different values of "Jupiter's" mass: from top to bottom, the masses are 0, 0.25, 0.5 1 and 2 times Jupiter's mass. Here Jupiter is unambiguously a shield: the impact flux would be much greater if it were absent or smaller.

So far we have seen that Jupiter definitely acts as a shield from Long-Period Comets, but for both Main Belt Asteroids and Centaurs its role is more ambiguous: while a slightly decreased Jovian mass would result in a significantly higher impact flux, either a larger or a very small Jovian mass, or no Jupiter at all, would result in fewer impactors. In the past it was thought that Long-Period Comets posed the greatest impact risk to Earth. If true, this would mean that Jupiter on the whole acts as a shield. However, the greatest impact threat is now thought to come from the Asteroids, a threat which would be much lower if Jupiter were much smaller.

## IV: The Jovian Eccentricity and Inclination

As well as varying Jupiter’s mass, one should ask what are the effects of varying its orbital eccentricity and inclination, to see whether our own Jupiter has a particularly fortuitous combination of these elements or not. This the authors did in their latest paper. They tested the effects of varying these parameters on the impact flux from the Asteroid Belt and Centaurs. Increasing Jupiter’s eccentricity and inclination has a strong destabilising effect on the Asteroid Belt, resulting in noticeably more impacts:

The effects of varying "Jupiter's" mass and eccentricity on Earth impactors from the Asteroid Belt. The upper line shows a high-eccentricity "Jupiter" with e=0.1. The middle line shows the real Jupiter with e=0.049. The lower line shows a low-eccentricity "Jupiter" with e=0.01.

This is largely through the destabilising effects of the stronger mean motion and secular resonances at higher eccentricity. However, the effect on the Centaur population is rather weak. Similarly increasing “Jupiter’s” inclination also increases the number of impactors.

The conclusion of this study then is that Jupiter’s current eccentricity and inclination are not optimal for protecting the Earth from impactors, but the situation could be a lot worse.

## Conclusions

Taken together, these papers show that the old idea of Jupiter being a protector of the Earth is somewhat naïve. Jupiter only plays an unambiguous protective role in the case of Oort Cloud comets, which are not now thought to constitute the major impact hazard.

The implications of this for the Rare Earth hypothesis are not entirely clear. While it is the case that, if Jupiter were to not exist, the impact flux suffered by Earth would be much less, it is also the case that Jupiter could be much more hostile to life on Earth, if its mass were a little lower or eccentricity a little higher. Knowledge of the proportion of Earth-like planets with impact regimes suitable for sustaining complex life will doubtless have to await a thorough census of the numbers and orbital properties of both Earth-like planets and their giant planet companions.

An additional complication is that, if “Jupiter” were much different from the real Jupiter, the populations of small bodies in the Solar System may be very different, since their present locations are determined by the formation and evolution of the Solar System as a whole. Properly the vulnerability of a planet to impactors should be determined within the context of a full model of Solar System evolution, but as Horner & Jones say, we are a long way from the conceptual knowledge and computational power required to simulate this…

# Kozai and the Crab

The star 55 Cancri hosts a system of five known planets of diverse nature, ranging from the small 55 Cancri e close to the star (3% of Jupiter’s mass, around 10 times Earth’s, at 0.016 AU) to the massive and distant 55 Cancri d (4 times Jupiter’s mass at 5.7 AU). When many planets are detected in a system, it is important to verify that the planets are in a stable configuration that will not be disrupted on relatively short time-scales. While this is true for 55 Cancri there is a hitherto overlooked complication in this system: the existence of a binary companion to the primary star. While distant, the companion could still potentially disrupt the planetary system. A recent paper by Kaib, Raymond & Duncan attempts to determine whether the planetary system is stable to perturbations from this companion.

The particular dynamical effect that could destabilise the planets is known as the Lidov-Kozai effect, after its discoverers. In the simplest case, this occurs in systems comprising a star, a companion such as a planet or binary star, and a massless test particle such as a comet or asteroid, when the ratio of semi-major axes is very large, and the mutual inclination of their orbits exceeds a certain critical value. If the conditions are met, the test particle experiences very large oscillations in eccentricity and inclination. There is a constant of motion, $\cos I \sqrt{1-e^2}$, where $I$ is the inclination and $e$ is the eccentricity, so the eccentricity and inclination oscillations are in phase. The time-scale for them to occur is approximately $P_\mathrm{Kozai}=P_\mathrm{Kep}\frac{m_\star}{m_\mathrm{b}}\frac{a_\mathrm{b}^3}{a_\mathrm{pl}^3}$, where $P_\mathrm{Kep}$ is the orbital period of the planet, $m_\star$ and $m_\mathrm{b}$ are the masses of the primary star and binary companion, and $a_\mathrm{pl}$ and $a_\mathrm{b}$ are the semi-major axes of the planet and binary companion.

Eccentricity evolution of a planet experiencing Lidov-Kozai cycles. Note how the eccentricity is driven to very high values (0.8, in this case). From Malmberg, Davies & Chambers (2007).

In the 55 Cancri system, there is a massive binary companion star, and the planets, being much less massive, would effectively behave as test particles. Since there is no reason to assume that the companion’s orbit is in the same plane as the planets’, their mutual inclination could be large, and the planets would then undergo Lidov-Kozai cycles. Such cycles would be devastating to the 55 Cancri planetary system, since the planetary eccentricities would increase to very large values, their orbits would intersect, and the planets could collide or scatter each other onto very different orbits. In contrast, in the system as currently observed, the planets’ eccentricities are all fairly low, less than 0.1 , and the system has remained stable for around 10 billion years. This suggests that the Lidov-Kozai effect is not at work in this system.

The authors investigate why this is so, integrating the equations of motion for the planets and the companion star interacting gravitationally. Since the companion star’s orbit is unknown (only the distance from the star projected onto the plane of the sky can be measured), they integrate many different systems with different orbits of the binary companion. They also reduce the five planet system to a four or two planet system in order to follow the evolution for longer, since the time that the equations can be integrated for is limited by finite computing power. The qualitative behaviour is, however, the same in all cases.

They find that the system is stable in the majority of runs (84%), and the planets do not undergo the large eccentricity oscillations associated with the Lidov-Kozai effect. However, there are still large inclination oscillations, as shown below:

Inclination evolution of planets in the 55 Cancri system. the points show the inclinations of four planets in the 4-planet integrations. The line shows the inclination of the fifth planet in the 5-planet integrations. All planets evolve in phase, and the addition of the fifth planet has no effect on the nature of the solution. From Kaib, Raymond & Duncan (2011).

The planets’ inclinations vary enormously, but all change in phase: effectively, the system behaves as a rigid body, locked together. The inclined binary companion excites inclination oscillations in the planets. If each planet were experiencing Lidov-Kozai cycles alone, the periods of the oscillations would all be different, since the period depends on the planet’s semi-major axis (see the equation above). Here, however, the planets are sufficiently close that they transfer their change in inclination to each other on much shorter time-scales, and thus all behave the same way under the companion’s perturbations. The eccentricity oscillations are suppressed because the Lidov-Kozai effect depends on a sensitive resonance between the precession rates of the orbital plane and the pericentre, which is broken by the additional perturbations from the other planets. For a detailed analysis of this stabilisation mechanism, see this paper.

While the inclination shown in the above figure reaches a maximum of around 120 degrees, this is dependent on the unknown orbital parameters of the companion star. The authors therefore integrate 500 systems with different binary orbits, to determine statistically the distribution of planetary inclinations that would result. The results are shown below:

Distribution of planetary inclinations, summed over all possible perturber orbits. The triangles show the true inclinations, while the circles show the observable values, projected onto the plane of the sky. Most configurations result in very large inclinations for the planetary system, some being retrograde (greater than 90 degrees). From Kaib, Raymond & Duncan (2011).

This paper therefore makes a startling prediction: the planetary system is overwhelmingly likely to be misaligned with its host star. This is in contrast to our own Solar System, where all the planets’ orbits are within a few degrees of the Sun’s equator. For 55 Cancri, the median value is around 60 degrees, and the planets can often be driven onto retrograde orbits, going backwards relative to their star, as shown above.

Since planet e transits the star, it is possible to determine the inclination observationally, by means of the Rossiter-Mclaughlin effect. Thus, the authors’ prediction that the system is misaligned with the host star will be able to be tested in the near future.