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,

$a_{\rm test} = a_p ( 1 + \Delta)$ ,

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

$\Delta > 2.4 \left( \frac{M_p}{M_*} \right)^{1/3}$ .

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,

$\Delta > 2 \times 3^{1/6} \left( \frac{M_1}{M_*} + \frac{M_2}{M_*} \right)^{1/3}$ .

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.

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Lecture #13: Keplerian orbits

Leave a comment Posted by philarmitage on November 10, 2013

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,

$P_{\rm yr}^2 = a_{\rm AU}^3$ ,

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

$\Omega = \sqrt{G (M_1 + M_2) / a^3}$

and the period is $P = 2 \pi / \Omega$ .

The pericenter distance (the distance of closest approach between the body and the star) is given by $r_p = a ( 1 - e )$ . The apocenter distance (the greatest separation) is $r_a = a (1 + e)$ .

The total energy of the orbit per unit mass is,

$E = - \frac{G M}{2 a}$ .

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,

$\frac{1}{2} v^2 - \frac{GM}{r} = -\frac{GM}{2a}$ .

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

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Lecture #12: Giant planet formation

Leave a comment Posted by philarmitage on November 10, 2013

The accepted theory for the formation of most giant planets, both in the Solar System and in extrasolar planetary systems, is core accretion. In one sentence, the basic idea of core accretion is that a massive solid core (probably with a mass of several to several tens of Earth masses) forms in the same way as do terrestrial planets, and only subsequently accretes a massive gaseous envelope from the surrounding disk.

The key to understanding core accretion lies in the behavior of the envelope as the core mass grows. A core of several Earth masses embedded with the gas of the protoplanetary disk has strong enough gravity that it is able to retain a dense atmosphere. This atmosphere or envelope is initially in hydrostatic equilibrium, meaning that the inward force of gravity on the envelope (coming primarily from the gravity of the core) is balanced by an outward pressure gradient. Somewhat surprisingly, however, there is a maximum core mass above which it is not possible to have an envelope in hydrostatic equilibrium. We can give a hand waving justification for the existence of such a critical core mass. As the core grows more massive, the mass of the envelope itself also increases. If the envelope is lower in mass than the core, this poses no problem: the extra gas is denser and can sustain the larger pressure gradient needed to support the larger envelope. Once the mass of the envelope approaches that of the core, however, a crisis develops. Adding extra gas still increases the pressure, but it also significantly increases the total gravity of the planet and makes the envelope harder to support. Above some limit hydrostatic equilibrium cannot be maintained, and the envelope contracts rapidly allowing more gas from the disk to be captured by the planet. Schematically,

A slightly more detailed calculation shows that hydrostatic equilibrium solutions for planets composed of a core plus a surrounding envelopes fall on to tracks in a plot of total (core plus envelope) planet mass versus core mass. For typical disk parameters, the maximum mass of a core that can maintain a hydrostatic envelope might be 5-10 Earth masses (though this depends on where in the disk the planet is growing, how much dust there is in the atmosphere, and various other hard-to-quantify effects). A representative plot looks like,

Based on these ideas, the general scenario for forming giant planets has four phases,

Core formation – the solid core is formed by accretion of smaller bodies (in traditional models these are planetesimals, though recent work has also considered the possibility that substantial mass is gained by accretion of much smaller pebbles instead). Because of the boost in surface density outside the snow line, and because the Hill radius of a planet of fixed mass is larger at larger distance from the star, it is thought to be easiest to grow a massive core at radii between a few AU out to maybe 10 AU.
Hydrostatic phase – once the core becomes massive enough it accretes an envelope, which is initially maintained in hydrostatic balance. Energy (provided either by bombardment of the planet by planetesimals, or by slow gravitational contraction of the envelope) steadily leaks out of the envelope, which contracts allowing more gas to be accreted. If the gas disk was dissipated during this phase, the resulting planet might resemble a Solar System ice giant.
Rapid envelope growth – at some point the critical core mass is exceeded. Hydrostatic equilibrium is lost, and a much more rapid “runaway” phase of gas accretion ensues. This phase can see several hundred Earth masses of gas being captured, so that in the end the mass of the planet is totally dominated by gas rather than by the mass of the solid core that catalyzed the whole process.
Gas starvation – accretion ceases either when the gas disk around the star is dissipated, or when the planet creates a local gap around its location that starves it of further gas supply.

Pollack et al. (1996) presented the first detailed calculations of the formation of the gas giants via core accretion. The plot below, based closely on their work, shows how the core, envelope and total mass of the planet increase over time.

The exact time scales and prerequisites for core accretion to work are debated. Current thinking is that it is possible to form a giant planet via core accretion on scales of a few AU within 1-3 Myr, consistent with the estimated life times of protoplanetary gas disks. It becomes harder to form a massive core at large orbital radii (because gravitational encounters between solid bodies lead more often to ejection from the shallower potential well of the star rather than accretion), and hence the formation channel of massive planets beyond a few tens of AU is especially contentious. It is not thought possible to form giant planets on very small scales, and hence standard explanations for extrasolar hot Jupiters appeal to orbital migration of these planets from formation sites further out in the disk.

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Lecture #11: Terrestrial planet formation

Leave a comment Posted by philarmitage on November 10, 2013

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 $\sigma$ 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 $\sigma$ is fixed, the gravitational focusing term $v_{\rm esc}^2 / \sigma^2$ 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.

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Lecture #10: Rates of planet growth

Leave a comment Posted by philarmitage on November 10, 2013

The rate at which planets grow can be estimated from a statistical argument. Let us consider a protoplanet of mass $M$ , radius $R_s$ and surface escape speed $v_{\rm esc}$ , orbiting within a very large number of smaller planetesimals. The planetesimals have a surface density $\Sigma_p$ and a velocity dispersion $\sigma$ .

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 $\sigma$ , which results in an inclination,

$i \sim \sigma / v_K$ .

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

$h \sim i a \sim \sigma / \Omega$ ,

with $\Omega$ the Keplerian angular velocity. We conclude that the disk of planetesimals should have a vertical thickness $\sim 2 h$ , and a volume density,

$\rho_{sw} \simeq \frac{\Sigma_p}{2 h}$ .

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,

$\frac{{\rm d}M}{{\rm d}t} = \rho_{sw} \sigma \pi R_s^2 \left( 1 + \frac{v_{esc}^2}{\sigma^2} \right)$ .

Substituting for the density, we find,

$\frac{{\rm d}M}{{\rm d}t} = \frac{1}{2} \Sigma_p \Omega \pi R_s^2 \left( 1 + \frac{v_{esc}^2}{\sigma^2} \right)$ .

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,

$\left( 1 + \frac{v_{esc}^2}{\sigma^2} \right) \approx \frac{v_{esc}^2}{\sigma^2} = \frac{2GM}{\sigma^2 R_s}$ .

(Here we have substituted for the escape velocity in terms of the mass and radius.) We can simplify this by noting that $M \propto R_s^3$ , so that $R_s \propto M^{1/3}$ . The formula for the rate of planetary growth then takes the form (considering only the mass and radius dependent terms),

$\frac{{\rm d}M}{{\rm d}t} \propto M R_s \propto M^{4/3}$ .

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.

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Lecture #9: Gravitational dynamics

Leave a comment Posted by philarmitage on November 10, 2013

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 $\Sigma_p$ . As with other surface densities, it has units of $g \ cm^{-2}$ .
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 $\sigma$ 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 $n$ , collision cross-section $A$ , and random velocities $v$ , the collision rate of one molecule with others is,

$\Gamma = n A v$ .

Here $A$ , 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 $m$ , moving with relative velocity at infinity $\sigma$ on trajectories that would result in an impact parameter $b$ (this is the distance of closest approach ignoring gravity). Suppose that the distance of closest approach (with gravity) is $R_c$ , and the velocity of each body at closest approach is $v_{\rm max}$ . 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,

$\frac{1}{4} m \sigma^2 = m v_{\rm max}^2 - \frac{Gm^2}{R_c}$ .

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

$v_{\rm max} = \frac{1}{2} \frac{b}{R_c} \sigma$ .

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 $R_c < R_s$ , where $R_s$ is the sum of the physical radii of the bodies, we obtain that physical collisions occur provided that,

$b^2 < R_s^2 + \frac{4 Gm R_s}{\sigma^2}$ .

In terms of the escape velocity from the point of contact, $v_{\rm esc}^2 = 4 Gm / R_s$ , we have,

$b^2 < R_s^2 (1 + v_{\rm esc}^2 / \sigma^2)$ .

Equivalently, the cross-section for collisions is,

$A = \pi R_s^2 ( 1 + v_{\rm esc}^2 / \sigma^2)$ .

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 $M_p$ , orbiting the star at distance $a$. A nearby planetesimal has a radial separation from the planet of distance $d$ . If the planetesimal orbits the star, its angular velocity is,

$\Omega = \sqrt{GM_* / a^3}$ ,

whereas if it orbits the planet the angular velocity is,

$\Omega = \sqrt{GM_p / d^3}$ .

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,

$R_{\rm Hill} = \left( \frac{M_p}{3 M_*} \right)^{1/3} a$ ,

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, $R_{\rm Hill} / a$ , 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.

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Solar System Formation & Dynamics

Solutions midterm #1

Solutions, midterm 2

Writing assignment #4

Solutions to homework #3

Lecture #14: Stability of planetary orbits

Lecture #13: Keplerian orbits

Lecture #12: Giant planet formation

Lecture #11: Terrestrial planet formation

Lecture #10: Rates of planet growth

Lecture #9: Gravitational dynamics

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