Solar System Formation & Dynamics

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,

.

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,

.

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,

.

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.