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# 2.4 The isolation mass

Once the oligarchic growth phase is over, the resulting embryos are relatively isolated and on initially circular orbits. They continue growing into planetary cores, by accreting the nearby leftover planetesimals within a feeding zone which extends a distance Δa either side of the planetary core. We may write the radius of this feeding zone as Δa = C RHill, where C is a constant and RHill is the Hill radius, defined as:

Equation label:(Equation 20)

Here, Mp is the mass of the planetary core, M* is the mass of the star and a is the radius of the orbit. The Hill radius is defined as the distance from the planetary core at which its gravitational force dominates over that of the star.

The growth of the cores continues until all the neighbouring planetesimals have been consumed. At this point, the mass of the core reaches the isolation mass Miso, defined as the total mass of the planetesimals within the feeding zone.

Equation label:(Equation 21)

The origin of Equation 21 is explored in the next activity.

## Activity 5

1. Write down an expression for the mass of planetesimals within the feeding zone in terms of the disc surface density Σ, distance to the central star a and feeding zone width Δa. Hence, derive Equation 21.

2. Use Equation 21 to evaluate Miso in the terrestrial planets region at a⊕ = 1.0 au and in the Jovian planets region at aJup = 5.2 au, for M* = 1 M☉, Σ = 100 kg m-2 and C = 2√3.

### Answer

1. The mass of planetesimals within the feeding zone is the area of the annulus with width 2Δa at a radius a, multiplied by the surface density. Hence,

   The width of the feeding zone is defined in terms of the Hill radius of the resulting planetary core (Equation 20) as Δa = C RHill. Therefore, once this mass is all contained within a single core, its mass is given by

   This simplifies to

   which may be rearranged to give the requested expression.

2. In the terrestrial planets region (where a = a⊕ = 1.0 au), this gives

   (This is about 0.066 times the mass of the Earth, or around the mass of Mercury.)

   Similarly, at the distance of Jovian planets (where a = aJup = 5.2 au), Equation 21 gives an isolation mass that is (5.2)3 larger. Hence,

   (This is about 9.3 times the mass of the Earth, which is around half the mass of Neptune.)

As shown in Activity 5, the fact that Miso ∝ a3 means that more massive cores up to several Earth masses can form at larger distances from the central star. However, at a ~ 10 au the speed of the planetesimals is too high for the collisions to lead to accretion, so it becomes increasingly hard to build massive planetary cores.

Planetary cores formed in this way will have sizes from around that of Mercury (with radius a few thousand kilometres) to several times that of the Earth (with radius a few tens of thousand kilometres).

## Activity 6

Summarise the typical sizes of objects involved in the various stages of the core-accretion scenario.

### Discussion

Initially, the particles are dust grains with a typical size of a micron or less which coagulate into millimetre-sized aggregates. These accumulate into rocks that are around one metre in size, which grow further into kilometre-sized planetesimals. Gravitational focusing helps these grow into planetary embryos with sizes and masses around that of the Moon, Mercury or Mars. These then become planetary cores by accreting leftover planetesimals within their feeding zone to reach a mass and size of a few times that of the Earth.

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