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The formation of exoplanets
The formation of exoplanets

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5 Conclusion

The focus of this course has been on how planets form around stars from the material in protoplanetary discs. These were some of the key learning points:

  1. Protoplanetary discs comprised of gas and solid material are believed to be the birthplaces of planets. In hydrostatic equilibrium, the density profile ρgas(z) of the gas in a disc as a function of vertical height z can be expressed as:

    rho sub gas of z equals rho sub zero times exp of negative z squared divided by two times cap h squared full stop
    Equation label: (Equation 5)

    Here, cap h equals c sub s solidus omega sub cap k (Equation 6) is the disc scale height, ρ0 is the density at the midplane (z = 0), c sub s equals left parenthesis cap p sub gas solidus rho sub gas right parenthesis super one solidus two (Equation 3) is the sound speed in the gas and omega sub cap k equals left parenthesis cap g times cap m sub asterisk operator solidus r cubed right parenthesis super one solidus two (Equation 2) is the Keplerian angular speed for an orbit at a distance r from a star of mass M*.

  2. In the radial direction, in addition to the gravitational force, there is also a force due to the pressure gradient of the gas dPgas/dr. Therefore, the orbital speed vorb(r) of the gas in the disc has two components: one due to the Keplerian speed, v sub cap k of r equals left parenthesis cap g times cap m sub asterisk operator solidus r right parenthesis super one solidus two (Equation 1), and one due to this extra pressure gradient, given by

    v sub orb squared of r equals cap g times cap m sub asterisk operator divided by r plus r divided by rho sub gas of r times d cap p sub gas of r divided by d r full stop
    Equation label: (Equation 9)

    Usually, dPgas/dr < 0, so the orbital speed is sub-Keplerian, vorb(r) < vK(r). The difference between the Keplerian speed and the orbital speed is normal cap delta times v equals v sub cap k minus v sub orb and is typically ~ 100 m s-1 at 1 au from a 1 M☉ star.

  3. The core-accretion scenario predicts that planets form by accumulation of initially sub-micron-sized dust grains to form metre-sized rocks, then kilometre-sized planetesimals and Mercury-sized planetary embryos, and eventually planetary cores up to several times the size of the Earth.

  4. The relation between the orbital speed and Keplerian speed of particles in a protoplanetary disc can be expressed as v sub orb equals v sub cap k times left parenthesis one minus eta right parenthesis super one solidus two (Equation 15) where eta equals n times left parenthesis cap h solidus r right parenthesis squared with n a numerical constant. Particles in the disc experience a radial drift inwards with a speed:

    v sub rad equals negative v sub cap k times eta divided by tau sub cap s plus tau sub cap s super negative one comma

    where tau sub cap s equals tau sub stop times omega sub cap k (Equation 12) is called the Stokes number. The Stokes number is related to the stopping time

    tau sub stop equals rho sub m divided by rho sub gas times s divided by c sub s
    Equation label: (Equation 14)

    where ρm is the material density of the particles and s is their radius. The maximum radial drift speed occurs when τS = 1 which corresponds to roughly metre-sized rocks. In this case, multirelation v sub rad of max equals negative eta times v sub cap k solidus two almost equals negative normal cap delta times v .

  5. Once planetesimals have formed, their mass Mp grows through collisions with other planetesimals at a rate:

    equation sequence part 1 d cap m sub p divided by d t equals part 2 pi times cap r sub p squared times omega sub cap k times cap sigma times left parenthesis one plus v sub esc squared divided by v sub rel squared right parenthesis equals part 3 pi times cap r sub p squared times omega sub cap k times cap sigma times cap f sub g comma
    Equation label: (Equation 17)

    where Rp is the planetesimal’s radius, vesc is its escape velocity, vrel is the relative velocity between the two impacting bodies, Σ is the surface density of the disc and Fg is the gravitational focusing.

  6. Planetary embryos continue growing into planetary cores by accreting leftover planetesimals within a feeding zone that extends a distance Δa either side of the core, such that normal cap delta times a equals cap c times cap r sub Hill . Here, C is a constant and RHill is the Hill radius that is defined as the distance from the planetary core at which its gravitational force dominates over the gravitational force of the star of mass M*, which it orbits at a distance a:

    cap r sub Hill equals left parenthesis cap m sub p divided by three times cap m sub asterisk operator right parenthesis super one solidus three times a full stop
    Equation label: (Equation 20)

  7. The total mass of material within the feeding zone is called the isolation mass and represents the final mass of the planetary core:

    cap m sub iso equals eight divided by Square root of three times left parenthesis pi times cap sigma times cap c right parenthesis super three solidus two times a cubed divided by cap m sub asterisk operator super one solidus two full stop
    Equation label: (Equation 21)

  8. Once the mass of the core reaches a few Earth masses, it starts to build up a gas envelope. This can lead to the formation of gas giant planets, ice giant planets or terrestrial planets, depending on the amount of gas accreted by the time the critical mass for hydrostatic equilibrium is reached. Many observed protoplanetary discs show gaps, bright rings, asymmetries, spirals and other structures where planets are forming within them.

  9. The disc-instability scenario provides an alternative way to form gas giants. In this model, a cold and/or massive disc fragments into clumps due to gravitational instabilities, and these clumps eventually evolve into gas giants. Two conditions need to be satisfied for disc fragmentation: the Toomre criterion:

    multirelation cap q equals omega sub cap k times c sub s divided by pi times cap g times cap sigma less than one comma
    Equation label: (Equation 22)

    where cs is the speed of sound, ωK is the Keplerian angular speed and Σ is the gas surface density; and the cooling criterion:

    tau sub cool less than or equivalent to one divided by three times omega sub cap k full stop
    Equation label: (Equation 24)

    The fact that both conditions need to be satisfied for fragmentation effectively limits the mass and semimajor axis values of the planets forming via the disc-instability scenario.

  10. The typical mass of a planet formed via fragmentation can be estimated from the Jeans mass, which may be expressed as

    cap m sub Jeans equals four times pi times cap m sub asterisk operator times left parenthesis cap h divided by r right parenthesis cubed full stop
    Equation label: (Equation 25)

    The Jeans mass is of order 1–2 times the mass of Jupiter for typical discs.

  11. Once formed, the planets interact with the disc and with each other, undergoing migration in some cases, until the system reaches its final configuration. Factors influencing the final composition and orbital configuration of planets can include interactions with the remaining gas in the disc, interactions with remaining planetesimals, planet–planet interactions and interactions with additional stellar companions.

  12. Neither the core-accretion nor disc-instability scenarios can explain all of the observed exoplanet population. Therefore, it is plausible that both scenarios play a role in planet formation, where different mechanisms are at play at different distances from the parent star.