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The formation of exoplanets
The formation of exoplanets

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2.3 The growth of planetary cores

Once kilometre-sized planetesimals have formed with a mass of ~ 1012 – 1013 kg, they are massive enough to interact significantly with their neighbours via gravity and modify their velocity, thus becoming prone to collisions.

In the same way as for the smaller particles, collisions of planetesimals with other planetesimals need to happen at sufficiently low speed to lead to accretion. Under this assumption, the rate at which a planetesimal of mass Mp and radius Rp grows with time t can be written as:

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 full stop
Equation label: (Equation 17)

Here, Σ is the surface density of the disc, vesc is the planetesimal’s escape velocity, vrel is the relative velocity between the two impacting planetesimals and Fg is the gravitational focusing, which is a dimensionless parameter that describes how the gravitational attraction between two bodies increases their collision probability.

  • How does dMp/dt change with the disc’s surface density Σ?

  • The growth rate scales linearly with the disc’s surface density, so the growth rate will be higher for discs with a higher mass in planetesimals.

  • And how does growth rate scale with the distance from the central star?

  • With everything else being equal, growth is slower at large distances where the Keplerian angular speed, ωK is smaller.

The following activity shows a quantitative example of the impact of the gravitational focusing on the growth rate, considering a planetesimal in an orbit similar to that of Jupiter.

Activity 4

  • a.Assuming that Fg is constant, starting from Equation 17 write an expression for the planetesimal’s radius growth rate dRp/dt as a function of its density ρp.

  • b.Estimate the value of Fg needed for a planetesimal at the same distance as Jupiter with ωK = 0.16 y-1 to grow to a radius of Rp = 1000 km in 105 years. Use a surface density of Σ = 100 kg m-2 and a planetesimal density of ρp = 3000 kg m-3.

Answer

  • a.Assuming the planetesimal to be spherical, its mass Mp can be expressed in terms of its radius Rp and density ρp as:

    cap m sub p equals four divided by three times pi times cap r sub p cubed times rho sub p comma
    Equation label: (Equation 18)

    then we note that

    d cap r sub p divided by d t equals d cap m sub p divided by d t division d cap m sub p divided by d cap r sub p full stop

    Since dMp/dt is given by Equation 17, and d cap m sub p postfix solidus d cap r sub p equals four times pi times cap r sub p squared times rho sub p (by differentiation of Equation 18), this means

    equation sequence part 1 d cap r sub p divided by normal d times normal t equals part 2 pi times cap r sub p squared times omega sub cap k times cap sigma times cap f sub g divided by four times pi times cap r sub p squared times rho sub p equals part 3 one divided by four times omega sub normal cap k times cap sigma divided by rho sub p times cap f sub g full stop
    Equation label: (Equation 19)

  • b.Using the values provided, we obtain:

    d cap r sub p divided by d t equals one divided by four multiplication 0.16 times y super negative one multiplication 100 times kg m super negative two divided by 3000 times kg m super negative three times cap f sub g
    d cap r sub p divided by d t equals 1.33 multiplication 10 super negative three times cap f sub g times m y super negative one full stop

    To reach a radius of 1000 km in 105 years, the desired growth rate must be dRp/dt= 106 m / 105 = 10 m y-1.

    Hence, the gravitational focusing needs to be approximately

    equation sequence part 1 cap f sub g almost equals part 2 10 m y super negative one divided by 1.33 multiplication 10 super negative three m y super negative one almost equals part 3 7500 full stop

The expression that involves Fg (Equation 17) depends on the relative velocities between planetesimals, the range of which is characterised by the velocity dispersion within the disc. Hence, velocity dispersion plays a crucial role in determining the accretion rates. It turns out that there are two regimes:

  1. In cases where gravitational focusing of planetesimals is initially very strong, runaway growth occurs. This is an accelerated phase in the growth of planetesimals such that the largest bodies get larger at a rapid and increasing rate, proportional to their mass. This phase is thought to be generally quite short, and ends when the velocity dispersion of the resulting planetary embryos increases to the point where gravitational focusing is ineffective.

  2. In cases where the largest planetary embryos grow quickly while the smallest grow slowly, oligarchic growth occurs. This leads to a bimodal mass distribution with a number of embryos comparable to the mass of the Moon, Mercury or Mars (~1023 kg) embedded in a large population of smaller planetesimals.