Floating free

Recent results from planet-hunting surveys suggest that there is a sizeable population of planets which are not bound to any star. The origin of these free-floating planets is not yet understood. Did they form in protoplanetary discs like other planets, later to be removed by some dynamical process? Or were they created directly from the collapse of gas clouds, in the same way as stars?

A leading hypothesis to explain their origin postulates that they were formed in systems of several giant planets. After the damping effect of the protoplanetary disc ended when the disc disappeared, the planets interacted with each other strongly, throwing some out of the system and into interstellar space. This scattering process explains the distribution of exoplanet eccentricities, as well as explaining the existence of some planets on highly inclined orbits.

A recent paper by Veras & Raymond asks how effective such scattering can be at creating free-floating planets. The central problem is that there are, on average, two free-floating giant planets for every star in the Galaxy, while less than half of stars appear to have even one gas giant planet orbiting them. This suggests that each scattering event must throw out several planets to account for this discrepency in numbers.

Veras & Raymond quantify this more thoroughly as follows. The ratio of free-floating planets to stars must be equal to the fraction of stars that host giant planets, multiplied by the fraction of those that undergo scattering events, multiplied by the average number of planets ejected in each scattering event. All of these numbers can be quantified.

According to Sumi et al’s microlensing results, there are 1.8^{+1.7}_{-0.8} free-floating giant planets for every star. While the errors on this estimate are large, it is clear that the number of such planets is at least comparable to and probably greater than the number of stars.

The fraction of stars hosting giant planets is not known for certain, but radial velocity surveys (sensitive to planets close to their stars) suggest at least 15% of Sun-like stars host giant planets, while microlensing results suggest the fraction may be higher. Note however that not all of these stars will either currently or in the past have hosted more than one planet, which would be required for scattering to take place.

The fraction of planetary systems undergoing scattering events has been estimated at around 75%, in order to reproduce the large numbers of eccentric giant planets in the radial velocity surveys.

Putting these together allows the average number of planets ejected from each system to be estimated. Taking mid-range values from the above estimates, Veras & Raymond find that around twelve Jupiter-mass planets must have been ejected from each planetary system, an astonishingly high number. Assuming extreme values gives a range of between 2 and 50 ejections per system. Even the lower bound may be too large to credit: our own Solar System currently contains only two gas giants.

How many planets can be ejected from a system? There must to at least as many planets initially as are later ejected, so the authors answer this by performing numerical simulations of systems containing up to 50 gas giant planets. The results are shown below. Typically, between 20% and 70% of the total number of planets are ejected. Hence, to explain the figure of 12 ejections per system, systems of giant planets must form with on average several dozen planets.

Fraction of planets ejected from multiple planet systems, assuming all the planets have the same mass (left) or different masses (right). From Veras & Raymond (2012).

Forming this many planets seems implausible, because there is simply not enough space in a typical planet-forming disc to form them all. The maximum number of planets that might be formed is estimated by the authors to be around 8–13, far lower than the number needed for 12 ejections.

Hence, it is likely that some other means of creating free-floating planets must be at work, as well as endogenous scattering. The authors postulate disruption by other stars coming close to planetary systems, effects on planets’ orbits when stars reach the ends of their lives, rare collisions between protoplanetary discs, and the effects of the Galaxy on planets’ orbits. However, all of these may meet with the same objection: that since there are more free-floating planets than stars, and not all stars form giant planets, each star must supply a large number of planets to the free floating population, by whatsoever means. Perhaps the most likely explanation is the authors’ final suggestion: that the free-floating planets form directly from interstellar gas clouds, in the same way as small stars. In which case, free-floating planets would be born free, rather than liberating themselves in violent instabilities.

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 under the Lidov-Kozai mechanism

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

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

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.

Anatomy of an orbit

Since I’m going to be discussing orbital dynamics fairly extensively on this ‘blog, I thought I’d put together a post describing how orbits are described, since the terminology can be fairly confusing, even to professional astronomers.

First, recall that particles orbiting under an inverse-square force such as gravity follow orbits which are conic sections — circles, ellipses, parabolae or hyperbolae. Particles which are bound to the body they are orbiting follow circular or elliptical orbits, while unbound particles follow parabolic or hyperbolic orbits. I’ll confine my attention to bound orbits here, since most objects of interest (planets, moons, asteroids…) are on bound orbits.

Such orbits are confined to a single plane. Restricting attention to this plane, an elliptical orbit is described algebraically by r=\frac{a(1-e^2)}{1+e\cos f}, while geometrically it looks like this:

Eccentric orbit

An orbit with an eccentricity of 0.3. The orbiting particle is located at the filled symbol, having moved through an angle f.

The orbit has one parameter describing its size, the semi-major axis a, and one describing its shape, the eccentricity e. The higher the eccentricity, the less circular the orbit is. The particle’s location on the orbit, measured from the point of closest approach to the central body, is measured by the angle f, the true anomaly. This point of closest approach is located at a distance  q=a(1-e) from the central body. It is known variously as the pericentre, periapse (pl. periapses) or periapsis (pl. periapsides). For orbits around specific bodies, it may be called the perigee (Earth), perihelion (Sun), etc.

Note that the symbols marking the orbit in the above Figure are not evenly spaced. They in fact represent the particle’s position at equal intervals in time. The particle moves fastest at periapsis; this is a simple consequence of angular momentum conservation.

While this description of an orbit is quite simple, it is often necessary to describe an orbit with respect to a different reference frame. For example, when observing an extra-solar planetary system it is convenient to align the reference frame with the line of sight and the plane of the sky. The orientation of the orbital plane with respect to the reference plane is then given by two angles, the inclination I and the longitude of ascending node \Omega. The former describes the angle between the x-y plane of the reference frame and the orbital plane, while the latter describes the line of intersection of the two planes (this is known as the line of nodes). These angles are shown below:

Orbital angles

We also see the familiar angle f, the true anomaly, making its appearance. One final angle is needed to completely specify the orientation of the orbit: this is the argument of periapsis \omega, which describes which way the periapsis points within the orbital plane.

To summarize, there are now six parameters, called orbital elements, describing the particle’s position in space:

  • I, inclination, describing the angle between a reference plane and the orbital plane
  • \Omega, longitude of ascending node, describing the orientation of the intersection of these two planes
  • \omega, argument of periapsis, describing the direction the periapsis points relative to the line of nodes.
  • f, true anomaly, describing the particle’s position on its orbit relative to the periapsis.
  • a, semi-major axis, the average distance of the particle from the central body
  • e, eccentricity, a measure of how non-circular the orbit is

If the particle only experiences an inverse square force from the central body, then all these apart from the true anomaly are constant, and the orbit remains unchanged in time. However, when more bodies are introduced, they can cause all the orbital elements to change. The goal of celestial mechanics is to determine what these changes are, which is a very difficult problem for which there is no general solution.

Finally, there are alternative orbital elements. It is common to use the longitude of periapsis \varpi=\omega+\Omega to express the direction of periapsis relative to the reference direction, and the true longitude \theta=f+\omega+\Omega to express the particle’s position on its orbit relative to this direction. Remember:

  • Longitudes are measured relative to a fixed reference direction
  • Arguments are measured relative to the line of nodes
  • Anomalies are measured relative to the periapsis


The space surrounding Earth’s orbit is far from empty. Small meteoroids can be seen as they enter Earth’s atmosphere and disintegrate, leaving wakes of incandescent gas. Comets leave extensive debris trails as they approach the Sun. There is also, unseen to the naked eye, a large population of asteroids whose orbits are in the vicinity of Earth’s.

Last week, one such asteroid, 2010 TK7was announced to be the first known member of a special class of asteroid known as Trojans. While Trojans relating to other planets such as Jupiter were known previously, this is the first example of one associated with Earth. The distinguishing feature of Trojan asteroids is that they are located close to one of the Lagrange points of the planet-Sun system, shown below:

Lagrange points

Image credit: Wikipedia

The contours show the gravitational potential due to the Earth and Sun. This is in a rotating reference frame, so the Earth remains fixed at the right as it orbits the Sun. Since it is a rotating reference frame, particles will feel fictitious forces, since they want to remain on a straight trajectory in the inertial frame. One of these forces is the centrifugal force, which is included in the contours. The behaviour of a particle due to the gravitational and centrifugal forces can be visualised by imagining that the contours represent a system of hills and valleys, with particles rolling “downhill” under the action of gravity. There are 5 points where the gravitational and centrifugal forces exactly balance, labelled L1 to L5. If a particle were located exactly at one of these points, there would be no net force, just as if it were located exactly at the top of a hill, and it would remain there.

However, the existence of such equilibrium points does not tell the whole story. Equilibria may be stable or unstable, depending on whether a particle placed close to one will move towards or away from it. Thinking again in terms of a landscape, the top of a hill and the bottom of a valley are both equilibria, but only the second is stable.

2010 TK7 is located close to but not at the L4 Lagrange point. Despite being a “hill” in the contour plot, suggesting instability, the L4 point is actually stable. This is due to the effects of another fictitious force, the Coriolis force. This is the force that governs the rotation of weather systems on Earth. The Coriolis force acts only on moving bodies, and so cannot be captured in a static description of forces as shown in the above figure. The effects are best shown in an animation:

Coriolis effect

Animation credit: Wikipedia

On the left the trajectory of a particle moving from one side of the circle to the other in an inertial frame is shown, and on the right the trajectory in the rotating frame (The details don’t correspond exactly to the motion about the Lagrange point, but it suffices to demonstrate qualitatively what’s going on). Note that the particle follows a small curved path in the rotating frame, with the particle returning to its starting point. This effect only occurs for particles moving in the rotating frame: if a particle were fixed (say, at a boundary between light and dark on the edge of the circle) it would not experience such a force.

Now we can piece together the motion of a particle close to L4. If initially at rest, it rolls “downhill” from L4 under combined gravitational and centrifugal forces. However, before it gets too far, the Coriolis force forces it to curve round, back towards L4. As it approaches, it slows, the Coriolis force weakens, and gravitational and centrifugal forces force the particle away from L4 again. The process repeats, with the particle moving towards and away from the Lagrange point. Qualitatively similar behaviour can be seen in the following animation from the discoverers of 2010 TK7, showing the asteroid’s orbit (although note that the large size of the oscillations makes the actual motion rather more complicated):

You have probably noticed that there are actually two components to the motion: an annual epicyclic oscillation, and a long-term libration which slowly varies the centre of the epicyclic oscillation. The large amplitudes of both components mean that 2010 TK7 is only weakly attached to the L4 point. Indeed, the discoverers integrated the asteroid’s orbit backwards in time and found that before AD 500 it was actually librating about the L5 point on the opposite side of Earth, but the particle’s high libration amplitude enabled it to cross the L3 point to enter the L4 point’s sphere of influence.

What of future evolution? Due to the effects of chaos–small errors in the asteroid’s exact position blowing up exponentially–it’s impossible to exactly predict the orbit more than a few thousand years in advance. Nevertheless, it appears that the object will continue transitioning between the Lagrange points, although precisely what will happen is unknown.

Could there be more such asteroids? Since the L4 and L5 points are located only 60 degrees from the Sun as seen from Earth, detecting faint objects there is challenging. 2010 TK7 was detected by the satellite WISE, and then followed up from the ground. The large libration amplitude and epicyclic oscillations, and the fact that the asteroid was discovered at the near-Earth end of its libration, carry it further from the Sun, and hence make it easier to detect. Trojans bound closely to the Lagrange points may well have escaped detection, and it’s intriuguing to think that more may be waiting to be discovered in this region of space so close to us.

Further reading:
  • The paper published in Nature
  • The discoverers’ website, with more visualisations.
  • Murray & Dermott, Solar System Mechanics, CUP, 1999. Not for the mathematophobic!