Solar System Formation & Dynamics

In Newtonian gravity, the two body problem is the hardest problem that can be solved exactly. Two Newtonian point masses in a bound orbit describe closed Keplerian ellipses, and are stable for all time.

For more complex N-body systems we would like to be able to assess their stability. There are various definitions of stability applicable to planetary systems, but a simple definition is that a planetary system is stable if there are never any close approaches between the planets (if there are close approaches, then usually collisions, ejections, or similarly violent rearrangements of the orbits follow soon after). As many famous mathematicians over the ages have found, rigorously proving that a complex planetary system like the Solar System is stable is extremely difficult, and it is also hard to even identify the precise dynamical mechanisms that lead to instability. Nonetheless, a few exact results are known for simple systems, and these allow us to develop heuristic understanding (backed up by numerical calculations) of the behavior of more complex systems.

The circular restricted three body problem is the simplest case of three body dynamics. We consider a star and a planet (of arbitrary mass) on circular obits, together with a “test particle” whose mass is so low as to be negligible for the dynamics (physically, something like an asteroid would qualify). In this setup, it is quite easy to derive the condition for the system to be Hill stable, which means that the test particle is guaranteed never to have a close approach to the planet for all time. If we consider, for example, test particles on initially circular exterior orbits, with,

,

then Hill stability is assured provided that the dimensionless measure of the separation,

.

A similar result holds for two planets, whose masses can be comparable to one another but which are both much smaller than that of the star. Stability requires,

.

Notice the dependence of the stability boundary on the planet masses to the one third power. This is the same dependence as the Hill radius, and it follows from essentially the same physics – gravitational perturbations between the planets are destabilizing and the strength of those perturbations is measured by the separation of the planets in units of their Hill radii.

These exact mathematical results apply only to systems of two planets. More complex planetary systems are invariably chaotic, and their stability or instability can only be assessed by means of numerical calculations. An example of the results of such calculations is show below. To make this plot, we placed three planets (of equal masses, but increasing from left to right through the panels) on initially circular obits, separated by a fixed number of Hill radii, and integrated their orbits numerically until the first close approaches occurred. We then plotted the median time scale to the first close approach as a function of the initial separation.

The details of the plot are fairly complex, but (especially for the low mass planets plotted in the left-most panel) the trends are simple. The time scale for instability to develop is a steep function of the initial separation of the planets, measured in Hill radii units. Roughly, the dependence is exponential in the initial separations. This has important implications. Even if we cannot prove that a multiple planet system is permanently stable, it may be very securely stable for practical purposes if the planets are well enough separated. This is the case for the Solar System, whose long term stability has been studied in great detail. Although the orbits of the planets in the inner Solar System are chaotic, and neighboring trajectories diverge on a surprisingly short time scale (just millions of years!), the probability of close approaches between planets is extremely small. Dynamically it appears possible that Mercury could be perturbed into a Venus-crossing orbit, but the chances of this happening during the remaining five billion years of the Sun’s lifetime is only around a percent.

Keplerian orbits are solutions to the two body problem in Newtonian gravitational dynamics. Under Newtonian gravity, two point masses that are gravitationally bound to each other describe closed elliptical orbits in space. Their mutual separation vector also describes an ellipse. The Keplerian orbital elements that describe the shape of the orbit are,

• The semi-major axis a of the ellipse. Geometrically, this is half of the longest distance across the ellipse.
• The eccentricity e of the ellipse.
• The inclination i of the orbit relative to some plane.

(There are also angles which describe the orientation of the orbit in space, but these are less important for our purposes.) For our purposes, a number of basic properties of Keplerian orbits are important.

The period of the orbit is solely a function of the masses and semi-major axis. For a Solar mass star, this is given by Kepler’s third law in the simple form,

,

where the period is measured in years and the semi-major axis in AU. More generally, the mean angular velocity is given by

and the period is .

The pericenter distance (the distance of closest approach between the body and the star) is given by . The apocenter distance (the greatest separation) is .

The total energy of the orbit per unit mass is,

.

Energy is conserved, so at any point around the orbit the sum of the kinetic and potential energies must add up to the total energy,

.

This equation allows us to calculate the speed of a body on a Keplerian orbit at any distance from the star.

In the standard model of terrestrial planet formation there are three stages to growth:

• Runaway growth
• Oligarchic growth
• Giant impact stage

The first two stages are distinguished by the importance of dynamical feedback from the growing protoplanets on the orbits of nearby planetesimals. In the runaway growth phase, the protoplanet grows from planetesimals whose random velocities are set by some other process (normally, by a balance between excitation due to planetesimal-planetesimal encounters, and residual aerodynamic damping against the gas disk). If is fixed, the gravitational focusing term increases as the protoplanet grows, making it easier for the protoplanet to grow still faster. Hence, “runaway” growth.

Runaway growth ends when the assumptions that underlie it break down, specifically when the assumption that the planetesimal swarm maintains a fixed velocity dispersion fails. Once a protoplanet gets massive enough, its own gravitational perturbations are enough to excite the velocity dispersion of nearby planetesimals. This leads to a negative feedback effect, as the protoplanet gets more massive it reduces the amount of gravitational focusing of collisions, leading to slower growth. The result is that, in neighboring regions of the disk, we expect oligarchic growth to set in. A series of oligarchs form, all much more massive than the surviving planetesimals, but none able to grow much faster than the others. These oligarchs might eventually attain masses comparable to the Moon or Mars, so they are well on the way to being terrestrial planets.

The final stage of terrestrial planet formation involves giant impacts among the oligarchs, which develop crossing orbits due to longer term dynamical instabilities. This phase is best studied with numerical simulations. Shown here is a movie of a “standard” simulation done by my collaborator Sean Raymond, in which the Earth assembles within about 100 Myr from a smoothly distributed population of smaller bodies. (The colors here represent the assumed water fraction of the bodies.)

Simulations of this kind do a generally reasonable job at explaining the properties of the Solar System’s terrestrial planets. The most discussed problem is a possible discrepancy in the mass of the real Mars compared to the mass of planets that form at the orbital distance of Mars in simulations. Almost always, the real Mars is lower in mass than one would expect based on theory. This problem remains open. One possible explanation is that the Mars region of the inner Solar System was dynamically depleted of planetesimals during an epoch when Jupiter moved temporarily closer to the Sun. This idea, known as the “Grand Tack”, is discussed in more detail by Sean Raymond on his web pages if you’re interested.

The rate at which planets grow can be estimated from a statistical argument. Let us consider a protoplanet of mass , radius and surface escape speed , orbiting within a very large number of smaller planetesimals. The planetesimals have a surface density and a velocity dispersion .

The idea for calculating the rate of growth is to assume that the planetesimals, rather than being treated individually, can be considered as a “gas” of very many “molecules” that is accreted by the planet. The first step is thus to calculate the mid plane density of the planetesimal swarm. If we consider a single planetesimal, most of its velocity is circular motion about the star. However, it also has a small out-of-plane velocity, of the order of , which results in an inclination,

.

The vertical excursions of the planetesimal above and below the mid plane are thus,

,

with the Keplerian angular velocity. We conclude that the disk of planetesimals should have a vertical thickness , and a volume density,

.

The statistical estimate of the growth rate of the planet is then just the density of the planetesimal disk, multiplied by the velocity at infinity and by the collision cross-section. Accounting for gravitational focusing, we have,

.

Substituting for the density, we find,

.

This calculation already reveals a number of important points, some obvious and some not so obvious:

• The growth rate depends linearly on the surface density of the planetesimal population
• Growth is faster where the angular velocity is higher, i.e. in the inner disk
• The velocity dispersion enters only via the gravitational focusing term

We can get an idea for how growth proceeds by considering the limit where the velocity dispersion of the planetesimals (recall that this is the random component of their motion) is small compared to the escape speed of a growing protoplanet. In this limit the gravitational focusing enhancement to the cross section is large, and the term in parenthesis simplifies to,

.

(Here we have substituted for the escape velocity in terms of the mass and radius.) We can simplify this by noting that , so that . The formula for the rate of planetary growth then takes the form (considering only the mass and radius dependent terms),

.

The predicted rate of growth increases more rapidly than linearly with the mass! This result, which follows directly from the role that gravitational focusing plays in increasing the collision cross section, is important. It implies that if we start from a uniform population of small bodies, one object that happens to get “lucky” and grow faster initially is thereafter able to consistently accrete more rapidly than the average (smaller) object. This “rich get richer” mode of growth is described as runaway growth.

As we discussed previously, how planetesimals form remains a substantial mystery of planet formation. If we allow ourselves to stipulate that they do form, however, we can develop a theory for how planetesimals collide and grow into larger objects. In the simplest case, we make three assumptions,

• Planetesimals form with a smooth radial distribution across some relatively wide range of disk radii. We denote the surface density of the planetesimals (smoothed out over the area of the disk) as . As with other surface densities, it has units of .
• Planetesimals form with sizes that are large enough that gravity, rather than aerodynamic forces, dominates their subsequent evolution. This is often taken as a definition of “planetesimal”, though one should note that the dominance of gravity does not mean that aerodynamic forces suddenly become zero – to the contrary aerodynamics continues to play an important role in damping the random motions of planetesimals which are continually being excited by gravitational perturbations from other planetesimals and from growing planets.
• Collisions between planetesimals, and between planetesimals and protoplanets, lead to accretion. As we showed in class, this is generally a good assumption as long as the random velocities of the bodies are small compared to the escape speed from their surfaces. Provided this is the case, the energy of collisions is controlled primarily by the acceleration the two bodies experience as they fall into their gravitational potential wells. In this limit, even a modest amount of dissipation during the impact renders the two bodies gravitationally bound, and even if pieces fly off during the collision they will tend to reimpact later on (forming a rubble pile).

We can develop a theory for the early phases of planet formation that is quite analogous to the statistical theory of a gas. If we consider a gas, the molecules are moving around “randomly”, in the sense that the motions of neighboring molecules are un-correlated. For a gas with number density , collision cross-section , and random velocities , the collision rate of one molecule with others is,

.

Here , the collision cross-section, can be thought of as a measure of the physical size of the colliding molecules.

To apply the same ideas to planet formation there is one crucial difference. Planets and planetesimals are gravitating objects, which attract each other when close approaches happen. If we sketch the geometry of a close encounter, it is clear that the effect of gravity is to bring the bodies closer together at closest approach than they would have been if gravity had been negligible. This effect, called gravitational focusing, increases the collision cross-section to a value that is larger than the physical area of the colliding bodies.

To compute the enhancement to the collision rate caused by gravitational focusing, we set up the problem as shown above. We consider two equal mass bodies , moving with relative velocity at infinity on trajectories that would result in an impact parameter (this is the distance of closest approach ignoring gravity). Suppose that the distance of closest approach (with gravity) is , and the velocity of each body at closest approach is . We can solve for these two unknowns using conservation of energy and angular momentum. Equating the total energy in the initial state (just kinetic) to that when the bodies are at the moment of closest approach (kinetic plus potential) we have,

.

Noting that there is no radial velocity component to the relative motion at closest approach, angular momentum conservation gives,

.

With two equations in two unknowns, we can solve for the distance and velocity at closest approach. If we then note that a physical collision happens if , where is the sum of the physical radii of the bodies, we obtain that physical collisions occur provided that,

.

In terms of the escape velocity from the point of contact, , we have,

.

Equivalently, the cross-section for collisions is,

.

The first term is here the usual physical cross-section, the second term is the boost to the collision cross-section due to gravitational focusing. Gravitational focusing dominates collisions if the random velocity of the colliding bodies is small compared to the escape velocities from their surfaces. if this condition is met, it can result in a much higher rate of collisions than would be the case for non-self-gravitating bodies.

For this analysis, we assumed that the two colliding bodies were the only objects that matter. It is generally safe to ignore the serendipitous presence of a third planetesimal (three-body “collisions” are rare events), but the effects of the Sun need more careful attention. If the two planetesimals are at different orbital radii, there will be a differential (“tidal”) gravitational force from the Sun acting on each of them. The Sun will only be negligible for the dynamics of collisions if the tidal forces are small compared to the pairwise gravitational forces between the bodies.

We can estimate the role of tidal forces with a simple time scale argument. Consider a planet, mass , orbiting the star at distance $a$. A nearby planetesimal has a radial separation from the planet of distance . If the planetesimal orbits the star, its angular velocity is,

,

whereas if it orbits the planet the angular velocity is,

.

Setting these frequencies equal defines the distance from the planet within which the planet’s gravity dominates over the Solar tidal forces. This is known as the Hill radius (after George Hill). A more precise version of the argument given above gives a formula for the Hill radius,

,

that depends only on the one third power of the planet mass. This means that the gravitational “reach” of planets – the region of the disk within which the dynamics is controlled by the planet – increases more slowly with mass than one might expect. If we compare Jupiter to the Earth, for example, the former is 300 times more massive. The size of the Hill sphere as a fraction of the orbit, , on the other hand, is only larger by a factor of 6.7.

The size of the Hill sphere defines the region of space within which a planet can hold on to permanently bound satellites. However, the concept has much more general importance. The stability of a planetary system, for example, can be roughly measured in terms of the planetary separation measured in units of the Hill radii of the planets involved.

Although there is vastly more gas than solid material in protoplanetary disks (typically by a factor of the order of 100), it is the evolution of the solid component that is critical to the formation of both terrestrial planets and the cores of gas giants. As long as the solid particles are relatively small (anywhere between microns in size up to tens or hundreds of meters) their dynamics is closely coupled to that of the gas via aerodynamic forces. These forces comes in two flavors, depending upon the particle size,

Protoplanetary disks are extremely tenuous gases. The mean free path at the mid plane of the disk may be of the order of tens of cm or meters in size, depending upon the distance from the star and the surface density of the disk. As a result, the earliest phases of solid growth are always securely in the Epstein regime of drag. Consider a solid particle modeled (probably not very well) as a sphere of radius . The particle has a velocity relative to the local gas of (note that this is not the orbital velocity of either the gas or the particle, but generally a much smaller number). The gas has density , and a mean thermal speed . Under these conditions, the Epstein drag is given by,

.

As you might guess, the drag depends upon the frontal area of the particle (i.e. it scales with the square of the particle radius), but as you might not guess the force is only linear in the relative velocity. It is this feature of Epstein drag which is distinct from the Stokes drag that we’re more familiar with in terrestrial situations.

The aerodynamic interactions between solid particles and the gas are particularly important for the vertical and radial mobility of solids with the disk. Let us first consider a dust particle suspended in the disk above (or equivalently below) the mid plane. The gas within the disk is maintained in hydrostatic equilibrium by a balance of forces: the vertical component of stellar gravity, acting downward, is balanced by an upward-acting vertical pressure gradient. A particle, however, does not feel the pressure force within the gas. With the only force acting on the particle being gravity, there is a tendency for dust to settle toward the mid plane, potentially forming a dense particle layer.

We can calculate how quickly dust settling occurs in a laminar (non-turbulent) disk quite easily. We begin by defining a quantity known as the friction time (or stopping time),

.

Since is the linear momentum associated with the relative motion of the particle against the gas, this time scale is the time on which aerodynamic drag tries to slow down the particle to match the local gas speed. Often, it is useful to use a dimensionless version of the same quantity, which we accomplish by multiplying by the local Keplerian orbital frequency,

.

Substituting for the drag in the Epstein regime (and remembering that , with the material density of the particle) we find,

.

For small particles this time scale is very small indeed. For micron sized particles at 1 AU, for example, a few seconds is a representative estimate. As a consequence, it is generally reasonable to assume that small dust particles attain terminal velocity as they settle toward the mid plane, in much the same way as a skydiver jumping from a plane accelerates downward only up to the point where air resistance matches gravity. In our situation, we have the same two forces acting, gravity, acting vertically downward toward the mid plane,

,

and aerodynamic drag, given by the Epstein formula as before,

.

Equating these forces defines the settling speed,

,

from which we could also compute a rough estimate of the settling time scale,

.

Even for quite small particles (micron sized dust grains, which are coupled the gas very strongly) this time scale is generally shorter than the lifetime of the disk. At 1 AU, the settling time scale, ignoring both collisions with other particles (which may lead to coagulation and faster settling) and turbulence (which by stirring up particles opposes settling), typically may be of the order of 100,000 yr.

What about the effects of aerodynamic drag on the radial motion of solid particles? It is not immediately obvious that there are any effects, as a solid body such as a planet orbits a star in a balance between a radial gravitational force and an outward centrifugal force. There are thus no “unbalanced” forces of the kind that we appealed to when discussing settling. This is too simple, however, The gas in a protoplanetary disk is subject to additional non-gravitational forces, which cause it to orbit the star at a very slightly different speed from the Keplerian value, . This difference means that solid bodies do have a velocity differential with respect to the gas, and suffer aerodynamic effects as a consequence.

Quantitatively, we consider a gas disk that is circular and static (for the curious, the terms we are about to neglect that arise from radial gas flows are, indeed, generally negligible). Working in the disk mid plane, the orbital velocity is determined by the sum of the radial gravitational force and the pressure gradient,

.

If we neglect pressure, then this simplifies to the usual formula for the Keplerian orbital velocity. The pressure term, however, contributes a small but important correction. If – as is normally the case – the disk mid plane pressure is high at small radius and lower further out, then the pressure gradient term is negative. The pressure gradient provides an outward “push” that partially opposes gravity, and as a result the gas does not have to rotate as fast to maintain radial force balance.

The key result is thus: gas in protoplanetary disks typically rotates at less than the local Keplerian velocity.

For many purposes, this effect is small and can be safely ignored. If we write the pressure as a power law in radius,

,

then substituting we find,

.

Since , the gas velocity differs from the Keplerian value by a small amount, normally less than 1%. This is still, however, quite a large absolute velocity – at 1 AU of the order of 100 meters per second – and it is has a profound effect on the dynamics of solid bodies. Since the gas rotates slower than Keplerian velocity, while the solid particles want to orbit at the Keplerian velocity, the solids feel a headwind as they orbit faster than the local gas. The aerodynamic effect of this headwind causes the solids to lose angular momentum, and spiral in toward the star.

The striking fact is how fast this radial drift is. Weidenschilling (1977) showed that the rate of aerodynamic radial drift (not to be confused with the gravitational migration we will discuss later) is a function of the particle size. The effects of drift slow down for very massive bodies (obviously, as those have a lot of inertia relative to their surface area), but they are also small for extremely small particles that are effectively frozen into the gas by tight aerodynamic coupling. The strongest effects occur for a dimensionless friction time . Plotted against the friction time, the radial velocity has the following form,

The values on the y-axis depend upon the disk properties (especially on , which sets the velocity differential that drives radial drift) – here I have assumed . The peak drift speeds in this case exceed 0.1% of the Keplerian orbital velocity. This implies very short decay times,

For the same disk model (the middle line in the plot), particles with are predicted to be lost into the star on a time scale that is only of the order of 100 or 1000 years in the planet forming region! What size these most vulnerable particles are again depends upon the disk model, but in the terrestrial planet forming region values of tens of centimeters to around a meter are typical.

Radial drift, due to aerodynamic forces against a sub-Keplerian gas disk, is the cause of the radial drift problem or meter-sized barrier in planet formation. Clearly, if meter-sized bodies only survive for a brief period within the disk, either growth through the adjacent size scales must proceed very quickly, or the disk must possess “traps” within which radial drift is halted or slowed. In class we discussed a number of possible ways to get around the radial drift problem, but this is still an open research problem. Indeed, how to form planetesimals (the km-sized bodies that are the smallest objects immune to radial drift) out of smaller solid particles is often considered to be the most serious unsolved problem of planet formation.

Protoplanetary disks – like those seen above in HST images of the Orion Nebula – are a natural consequence of the broader star formation process. Stars within the Galaxy form within molecular cloud cores, relatively dense regions of gas that are embedded within much larger structures known as giant molecular clouds. The Orion Nebula, which can readily be seen through a small telescope, is part of one of the nearest GMCs.

Observationally, molecular cloud cores have scales of around 0.1 pc, and masses of perhaps 1 to a few Solar masses (these are obviously very rough numbers). It is thus plausible that one such core forms one star, or perhaps a binary or small multiple system. We can estimate how long collapse of a core to much higher densities ought to take, under the assumption that there is no persistent source of support against gravity (e.g. by magnetic fields). By equating kinetic energy to gravitational potential energy,

,

we obtain the escape velocity,

.

The collapse time scale can then be estimated as,

.

By noting that the mass of the cloud, , with the mean density, this can be rewritten in the more transparent form,

,

i.e. the time to collapse under gravity depends solely on the density of the collapsing cloud. Numerically, for a Solar mass cloud that is 0.1 pc in radius, we estimate that the collapse time scale is about 300,000 yr. This is the characteristic time scale on which the formation of low mass stars like the Sun proceeds.

One tenth of a parsec is about 20,000 AU, or 4 million times the current radius of the Sun. Since the specific angular momentum,

,

the very large disparity between molecular cloud core scales and those of stars means that even vanishingly small amounts of core rotation translate into too much angular momentum to form a star directly. Instead, we expect a disk to form. To be a bit more quantitative, we can equate the specific angular momentum of a core, with scale and rotation speed ,

,

to the specific angular momentum at distance from a protostar of mass ,

.

Doing so allows us to estimate a characteristic disk size, which works out to be,

.

If we take for a core of radius 0.1 pc and mass one Solar mass, we get,

.

In other words, even a very small rotational component on cloud core scales implies that infalling gas will form a disk much larger than stellar scales. There are a couple of caveats to this conclusion. If a binary forms, then much of the angular momentum of the initial core could end up in the orbital angular momentum of the binary, with correspondingly less ending up in disks. If angular momentum is lost on the collapse time scale, for example because magnetic fields act to brake the rotation as the cloud collapses, then again disks may be smaller or in principle even non-existent.

Once a star and a disk form, the specific angular momentum of gas orbiting within the disk,

,

is an increasing function of orbital radius. This has important consequences. It implies that absent either,

• Angular momentum transport, i.e. redistribution of angular momentum within the disk, or
• Angular momentum loss from the disk as a whole

the disk ought to be stable. Since both angular momentum transport and loss processes are thought to be rather slow, protoplanetary disks are able to persist for a very large number of orbital times, and in particular to outlast the dynamical collapse phase of the cores that form them.

These theoretical considerations are closely related to the scheme used to observationally classify Young Stellar Objects. Four basic stages are envisaged:

• Class 0 – this is the dynamical collapse phase, characterized by a deeply embedded protostar that is so shrouded by surrounding dust that it is only visible in the far-IR and sub-mm wavebands.
• Class I – at this stage a star and disk (perhaps accompanied by an outflow or jet) have formed, but the remnants of the cloud core are still being accreted. The system becomes visible in the near-IR.
• Class II – a star and disk only… by now the envelope has all been accreted. The stellar photosphere is now visible even in the optical.
• Class III – the disk is dispersed, leaving a pre-main-sequence star only

A cartoon of these stages looks like this,

How long the disk-bearing phases of YSO evolution last is an observational question that is not easy to answer. Young stars do not display clocks, and estimating their ages is notoriously tricky. The standard approach is to measure the disk fraction in clusters, by looking for the excess near-IR emission indicating warm dust in a surrounding disk, and then estimate the age of the cluster by fitting the position of the stars in a Hertzprung-Russell diagram to evolutionary tracks for young stars. Following this procedure, one obtains a plot of disk fraction vs age,

…from which one can read off the median disk lifetime. The standard estimate is about 3 Myr, though very recent work suggests that this may be a modest underestimate… 5 Myr may be a better number. In any event, the estimated lifetimes of protoplanetary gas disks have two important implications for planet formation,

• We have less than 10 Myr to form gas giants, before the gas is gone.
• If the Moon formation is correctly dated at 50-100 Myr after the formation of the earliest solids in the Solar Nebula, the final assembly of the terrestrial planets in the Solar System must have occurred in a gas-free environment.

The transit method is conceptually the simplest way to detect extrasolar planets. The “night side” of a planet emits essentially no optical radiation, so if the planet passes directly in front of the disk of the star as seen from Earth it acts as a black disk that temporarily blocks some fraction of the star light. Transits can be seen in the Solar System. In June 2012 the transit of Venus was visible from Earth:

For other stars, of course, we are generally unable to resolve the stellar disk. The signature of an extrasolar planetary transit is thus not an image of a small black circle crossing the face of the star, but rather a temporary drop in the brightness of the star as some of its light is blocked by the planet. (A more subtle difference is that the Earth is quite close to Venus, so the disk of Venus transiting the Sun is larger than it would appear for a distant observer seeing the same event from interstellar distances.)

The fractional drop in stellar brightness during a planetary transit can be calculated from simple geometry. The stellar disk has an area , while the planetary disk has area . The fraction of star light that is blocked is thus,

.

Jupiter has a radius that is about 10% that of the Sun (i.e. ). The same is true, roughly speaking, for all gas giant planets with masses between about a Saturn mass and up to 10 Jupiter masses. For Jupiter and other gas giant, then, we expect,

.

For the Earth (radius 6600km), the depth of a transit as it crosses the Sun (radius 696,000km) is,

.

The cartoon below shows what we expect the light curve to look like in the specific case of a gas giant that produces a roughly 1% dip in the stellar flux:

The most basic observables of a transit survey are:

• The depth of the transit – this is a direct measure of the relative size of the planet as compared to its host star.
• The period of the orbit , which can be measured directly if a series of successive transits are observed. For a planet orbiting a star of mass at semi-major axis , .

Inspection of the above formulae shows that measuring the period and transit depth are not sufficient to tell us all that much about the planet and its orbit. If, however, we have a good estimate of the stellar properties – its mass and radius – then the transit measurements give us the physical size of the planet and the semi-major axis of its orbit. The importance of good knowledge of the host star is relevant for the Kepler mission, whose ultra-precise photometry means that uncertainty in stellar parameters is a significant source of overall uncertainty in measured planetary properties.

For ground-based observatories, the primary limitation on the planets that can be detected via the transit method comes from atmospheric fluctuations, which limit the precision to which the stellar flux can be measured. The attainable precision is not a strong function of the size of the telescope, and the most successful surveys to date have used dedicated arrays of small telescopes to scan wide areas of the sky in search of planets orbiting relatively bright stars. (These offer the best targets for follow-up once planets have been found.) The SuperWASP search, for example, uses an array of 8 telephoto camera lenses, which have a field of view of 8 degrees on a side:

The atmospheric limitations mean that ground-based transit surveys are able to detect the 1% dips caused by gas giants, but are unable to detect the two orders of magnitude smaller signals produced by transiting Earth-like planets. (At least in the visible for Solar-type stars – the MEarth project targets smaller stars, around which even terrestrial planets can produce transit depths accessible from the ground.)

From space there are no atmospheric limitations, and the main barrier to detecting transits of small planets is instead the intrinsic variability of the stars themselves. Even stars that we don’t think of as being “variable” – like the Sun – vary in brightness by small amounts as star spots and other surface features form and rotate through the visible disk. NASA’s Kepler mission was designed to be able to detect transits of Earth-like planets at around 1 AU (i.e. in the habitable zone) around stars whose intrinsic noise properties were similar to those of the Sun. In practice, the realized transit detection efficiency for such planets around the actual sample of Kepler stars was not quite as good as predicted, and with the failure of the Kepler spacecraft’s pointing it is not clear whether any true extrasolar Earth analogs will be recovered from the data. Nonetheless, Kepler discovered a large number of small – probably terrestrial – planets, including Kepler-37b that has a size comparable to the Moon.

The radial velocity (or Doppler) method relies on the fact the orbital motion of a planet induces a counter-balancing (or “reflex”) motion in the star. In general, for a binary system the two orbiting bodies describe elliptical orbits about the center of mass of the binary. If the binary is made up of two equal mass stars, the center of mass will lie midway between them. In the case of a star-planet system, the center of mass will typically be inside the physical radius of the star, but not at the center. As the planet orbits, the star will then execute a much smaller orbit around the center of mass,

The idea of the radial velocity method is to indirectly detect the planet by observing the reflex motion of the star. This can be done by measuring the line of sight (“radial”) component of the stellar motion by observing the Doppler shift of spectral lines produced in the atmosphere of the star. We can estimate the size of the effect by calculating the orbital velocity of the star as it executes its small orbit about the center of mass. We first note that the orbital velocity of a planet on a circular orbit at distance from a star of mass is,

.

Conservation of linear momentum in the binary implies that when the planet is moving toward us with this velocity, the star must be moving away at a velocity such that the star and the planet have equal and opposite momenta, i.e.

.

Substituting, we get the stellar velocity as,

.

This would be the maximum radial velocity we would see if the orbit was exactly edge-on to us, at the moment when the star was coming directly toward us. If instead, the orbit is inclined at an angle (as shown in the figure above) we see only the component of the stellar orbital velocity that is projected along our line of sight. The semi-amplitude of the radial velocity curve is then,

.

Substituting numbers, we find that the radial velocity signature that the planets in the Solar System induce on the Sun is,

(Jupiter)

(Earth).

Currently, the most precise measurements of stellar radial velocities attain roughly 1 m/s precision. A wide range of planets can be detected via this technique, though it is not yet possible to find Earths within the habitable zone of Solar-type stars.

The basic observables of a radial velocity survey are,

• The period of the orbit. As with a transit survey, this gives us the semi-major axis of the planetary orbit provided we have good knowledge of the stellar mass.
• The semi-amplitude of the radial velocity oscillation . If we know the orbital inclination , then together with the stellar mass this gives us the planetary mass. In the more normal case where we don’t know the inclination, we can only deduce a minimum mass for the planet.
• The time-dependence of the radial velocity oscillation. If the orbit of the planet is circular this will be a simple sinusoid. For larger eccentricities, the curve will look more like a sawtooth, reflecting the fact that both the planet and the star are moving fastest during the brief period of closest approach. Some examples of curves for eccentric orbits are shown below,

  

  By fitting the shape of such curves, the eccentricity can be measured.

Transits and radial velocity measurements furnish independent information about extrasolar planetary systems. The transit method gives the most direct information on planetary sizes, while the radial velocity method gives information about planetary masses. If, fortuitously, we can observe transits and measure the radial velocity signal for the same planet, then combining the information gives the mass, the radius, and hence the density.

The identification of extrasolar planetary systems is a relatively recent development. The first broadly accepted detection of a planetary system outside of our own was made by Wolszczan and Frail in 1992 around a millisecond pulsar (a rapidly rotating neutron star). Shortly afterwards, in 1995, 51 Peg b, the first confirmed planet around a main-sequence star, was announced by Michel Mayor and Didier Queloz.

A variety of observational techniques are now used to either find or characterize extrasolar planets. We will discuss three of the most important: direct imaging, transits and radial velocity surveys. Gravitational lensing and astrometry are other methods that we will not consider here.

Direct imaging

Direct imaging is the simplest planet-search technique to describe, but the most difficult to successfully execute. To see why, suppose we have a planet of radius that orbits its host star at a distance . The planet has an albedo (the albedo is the fraction of incoming star light that the planet reflects back into space – for the Earth this number is something like 0.3 to 0.35). How much fainter is the planet than the star? This is purely a geometric problem. As seen from the star, the planet presents a disk on the sky with an area,

.

The fraction of star light that hits this disk is just the area of the disk divided by the area of the whole sphere that has a radius equal to that of the planet’s distance from the star,

.

Taking that ratio, and accounting for the fact that only a fraction of the star light hitting the planet gets reflected back out to space, we estimate the contrast ratio between star and planet to be,

.

Substituting some numbers gives an idea of the difficulty of the task. For the Earth, , while . We find,

,

for an albedo . A similar calculation for Jupiter, at orbital radius of 5.2 AU, and assuming an albedo of 0.5, gives,

.

We conclude that no matter whether we’re hoping to image extrasolar Earths or extrasolar Jupiters, planets are extremely faint sources. They’re about one billionth or one ten billionth as bright as their host stars in reflected star light!

What about the fraction of the incident star light that is not reflected? This energy is absorbed by the planet, and – since the planet cannot simply keep gaining energy and getting hotter – it must end up in balance with energy lost from the planet in thermal radiation. In general, for thermal or black body radiation, the peak of the emitted radiation is emitted at wavelengths that scale inversely with the temperature of the body, . For a star like the Sun, with a temperature of about 6000 K, the bulk of the energy is radiated in the optical and near-infrared bands, say at around 1 micron. For the Earth, with a surface temperature of about 300 K, the above relation implies that the wavelength of peak emission is a factor 6000 / 300 = 20 times longer. So we expect the thermal emission of planets to peak at 20 microns, in the mid-infrared (or even longer wavelengths for cooler planets further from their stars). This has both pluses and minuses if we want to directly image planets. On the plus side, if we look at a star + planet system in the mid-infrared, we’re focusing on a wavelength where relatively the planet is brighter as compared to its star. It’s still not absolutely bright – going from the optical (reflected star light) the the infrared (thermal emission) reduces the contrast ratio from to perhaps – but the difference is still substantial. On the minus side, telescopes and instrumentation become more challenging to design in the mid-infrared. Our Earthly surroundings are, not coincidentally, at the same temperature as the planets we’re trying to detect, and they emit at the same wavelengths as the faint extrasolar planets we’re seeking. Going into space, and cooling the telescope and instruments to low temperatures, is one solution, but an expensive one.

A final consideration for direct imaging searches for extrasolar planets comes from the fact that a faint source is much harder to detect when it’s close to a very bright one that when it’s well away from any other sources. At the simplest level, this problem can be quantified by considering the theoretical resolution limit of a perfect telescope, which is set by the “smearing” of the image due to diffraction. Two equal point sources can be distinguished (“resolved”) by a telescope of diameter , working at wavelength , if their angular separation on the sky (measured in radians),

.

In practice, a search for planets will do nowhere near as well as naive application of this formula would suggest. A star-planet system is very very far from being two equal brightness sources, and we will need to go to several times before we have any hope at spotting the planet against the overwhelming glare of the host star. Direct imaging surveys typically quote an “inner working angle”, which is the smallest angular separation from the star where there is sensitivity to planets of some specified brightness, and a great deal of technical ingenuity goes into designing instruments and observational techniques to reduce the inner working angle. If you’re interested in the gory details of how this is done, it’s worth reading the description of the SPHERE instrument installed on the European Southern Observatory’s 8m diameter Very Large Telescope.

Despite these difficulties a number of planets have now been securely identified via direct imaging. The most interesting system is HR 8799, discovered by Christian Marois and collaborators in 2008. A more recent image (showing an additional planet in the system, discovered later) looks like:

(Note that light from the star, which in this case is younger and somewhat more massive than the Sun, has been suppressed in the image.) Four planets are detected, all with masses that are probably around 5-10 times that of Jupiter, at orbital radii that extend out to 70 AU in projection. As we will discuss later, how these planets formed is a substantial mystery. We expect on quite general grounds that planet formation should become harder beyond 10-20 AU. That far from the star, the orbital velocity is low, so gravitational interactions between growing bodies have an increasing tendency to eject bodies from the system before they have a chance to collide. This is one of the reasons why we think that Neptune probably formed closer to the Sun than its current orbital separation. In HR 8799, though, we have four very massive planets orbiting out to radii where, in the Solar System, there are only the puny bodies of the Kuiper Belt. Did the planets (or perhaps just their solid cores) migrate there from smaller radii, or did they form in situ via a different mechanism? The answer is not known, though probably most theorists incline toward the first possibility.

The goal of this class is to provide an introduction to the formation and dynamical evolution of planetary systems (“dynamical evolution” here basically means “evolution due to the action of gravity”). For the most part, we don’t directly observe planets forming, and the dynamics we’re most interested in plays out over time scales far longer than any human observer (on observable time scales, we generally see a very good approximation to simple Keplerian motion). Our indirect knowledge derives from three main sources:

• Observations of protoplanetary disks around young stars – the initial conditions or raw material for planet formation.
• Observations of the Solar System.
• Observations of extrasolar planetary systems.

We will be discussing each of these during the course of the semester. Together with theory (and, to a limited extent, lab experiments) we will try to outline what we know about planet formation and what are the open questions that remain to be answered. We will start close to home.

A great deal is known about the Solar System. Spacecraft have visited all of the planets, and we have samples that originate from the Moon, perhaps 10-100 different asteroids, and Mars (the last two from meteorites). I hardly ever attend a research seminar on planetary science without learning new facts about the Solar System. For example, surprisingly recently (about a decade ago) it was discovered that the distribution of rayed craters on the surface of the Moon is notably asymmetric, with more craters on the “leading” side of the Moon along its orbit than on the “trailing” side. The reasons are not fully understood. For planet formation, the key is to identify what are the key facts that should inform our theories, and which are merely incidental.

An incomplete list of “interesting” properties of the Solar System might include:

• The planets orbit in approximately the same plane. Historically, this observation motivated the nebular hypothesis that the planets formed from a flattened disk – the Solar Nebula. That the planets formed from a disk is no longer in doubt, as we will see such disks are observed around the majority of sufficiently young stars. A curiosity, perhaps an important one, is that the orbital plane of the planets is not exactly the same as the equatorial plane of the Sun as defined by its rotation. The misalignment between these planes is about 7 degrees.
• There are two broad classes of planets, the giants and the terrestrial planets. The giant category includes the true gas giants (Jupiter and Saturn) which are primarily composed of light elements (hydrogen and helium), and the ice giants (Uranus and Saturn) which have cores made up of a mixture of water, ammonia, methane and rocks, atop which sit substantial envelopes of H and He. The terrestrial planets likewise split into two large terrestrial planets (the Earth and Venus) and two much smaller bodies in Mercury and Mars.
• None of the planets have the same composition as the Sun. This is obviously true of the terrestrial planets, but even Jupiter is enormously enriched in heavy elements (i.e. not H and He) as compared to the Sun.
• The terrestrial planets all lie interior to the giant planets.
• The “major” planets (let’s exclude Mercury and Mars for the time being) all have very nearly circular orbits.
• There are two main reservoirs of smaller bodies, the main asteroid belt, between Mars and Jupiter, and the Kuiper belt, beyond Neptune. The Kuiper belt includes many objects with interesting orbital properties, including Pluto (and many other objects) that occupy a 3:2 resonance with Neptune (i.e. Neptune orbits the Sun three times while Pluto orbits twice). In the case of Pluto, the orbit in fact crosses that of Neptune.
• The long period comets arrive in the inner Solar System on trajectories that suggest they originated from a reservoir at very large distances, known as the Oort cloud.
• Most of the planets have satellites, and some of the satellite systems are very extensive. Around the giant planets there are both regular satellites, which orbit in their planets’ equatorial plane (e.g. the Galilean satellites of Jupiter: Io, Europa, Ganymede and Callisto), and irregular satellites whose orbital planes are randomly distributed. There is also the Earth’s Moon, which is only a few times less massive than Mercury.
• The Sun is not part of a binary system.

A couple of other properties of the Solar System, which are less obviously relevant, are also worth mentioning:

• None of the planets are in mean-motion resonances with any of the others. A mean-motion resonance occurs when the orbital periods of two planets are close to an integer ratio, i.e. for two planets with orbital periods and , , with and integers. There are, on the other hand, numerous examples of such resonances among satellite orbits.
• The Solar System is “packed”, a loose term that in this context means (a) that almost all locations where bodies could in principle orbit stably for billions of years are, in fact, occupied, and (b) that we could not, in most cases, add another planet without destroying the long term stability of the system. One exception: we probably could add another terrestrial planet as long as its orbit was well inside that of Mercury.

With the benefit of the hindsight afforded us by the discovery of extrasolar planets, some of these properties now seem more surprising and interesting than they once did. The formerly unremarkable circularity of the gas giant planets’ orbits, for example, is an uncommon feature of known extrasolar planetary systems. Likewise, I don’t recall anyone ever finding it noteworthy that there are no Solar System planets with orbital periods measured in mere days, but we now know (after NASA’s Kepler mission) that many stars that seem similar to the Sun have such short-period systems of super-Earths or mini-Neptunes.

The mass budget of the Solar System

The mass of the Sun is . The mass of Jupiter, much the most massive planet, is . Obviously, most of the mass is in the Sun. A little less trivially, most of the mass of heavy elements (i.e. all those that are not H or He) is also to be found in the Sun. One interpretation of this fact is that, if we assume that most of the current mass of the Sun was once in a disk around a smaller protostar, the planet formation process need not be terribly efficient at converting the heavy elements present within the disk into planets.

The angular momentum budget of the Solar System

Although most of the Solar System’s mass is in the Sun, most of the angular momentum is in the orbital motion of the planets. Recall that the angular momentum of a particle with mass , moving with tangential velocity at distance , is,

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The velocity of a circular Keplerian orbit around a star of mass is,

,

where is Newton’s gravitational constant. Combining these formulae, we can calculate the angular momentum associated with Jupiter’s orbital motion about the Sun,

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This is a large but meaningless number, until we put it into context by comparing it to the angular momentum associated with the rotation of the Sun. That’s about , i.e. a hundred times smaller. The large “lever arm” of the planets’ orbits, together with the rather slow rotation rate of the Sun, mean that despite their low masses the planets have the lion’s share of the Solar System’s angular momentum.

When we discuss protoplanetary disks, we’ll discuss how it can be that the process of star and planet formation results in most of the mass going to the Sun, while the angular momentum ends up in the planetary orbits.

The Minimum Mass Solar Nebula (MMSN)

Knowing the masses, orbital radii and compositions of the planets (which we do, at least roughly), it’s possible to take a stab at estimating the surface density distribution of the gas in the Solar Nebula that would have just sufficed to form the planets in their current orbital configuration. This is called the Minimum Mass Solar Nebula. The basic procedure is:

• For each planet, we estimate the mass of some heavy element (e.g. iron) within that body. We then multiply that mass by the ratio of the mass of light elements to iron in the Sun. This gives us, for each planet, the mass the planet would have if it had its current mass of iron but the Solar composition.
• We then imagine spreading this augmented mass across an annulus that extends inward halfway to the orbit of the next planet in, and outward halfway to the next planet out. We divide the mass by the area of this annulus to get a surface density (units g per square cm) at the location of each planet.
• Finally we plot as a function of orbital radius.

Following this recipe, we get a version of the famous plot from Weidenschilling’s paper The distribution of mass in the planetary system and solar nebula:

This may not look like much of a power-law to you, but if we ignore Mars and the asteroid belt the result is that between Venus and Neptune,

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This radial distribution of gas (with a specified normalization, often taken to be at 1 AU) is called the Minimum Mass Solar Nebula. If we integrate to find out how much mass is enclosed within radius we have,

which means that most of the mass in the MMSN is in the outer part of the disk.

As we will discuss in class, what the MMSN really means is a bit unclear. It’s probably not the real mass distribution of gas in the Solar Nebula. It’s important, however, both for historical reasons and as a benchmark or fiducial mass profile that’s often used in discussion of planet formation. Even if it’s not the “right” profile, it sometimes helps for different people thinking about planet formation to compare their models using the same assumptions for how the gas is distributed with distance from the star.