Solar System Formation & Dynamics

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,

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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,

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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,

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

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

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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),

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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.