PX282 - L2 - core accretion

firstly, the young sun and the protoplanetary disc is formed by gravitational collapse
sub-micron sized dust particles are formed by condensation

image: NASA, CalTech, JPL, Leiden Observatory

growth of particles by pair-wise collisions*, up to $\sim 10$ km planetesimals
initially assisted by electrostatic forces, icy particles stick together readily, so they grow more rapidly beyond the snowline
there are two problems:
- particles become fragile as they grow and can be eroded by higher velocity collisions
- gas drag on particles in the disc, causing the larger particles to spiral in towards the sun
considering orbit with the gas disc:

\underset{keplerian orbits}{\underset{⏟}{\frac{v^{2}}{r} = \frac{G M_{⊙}}{r^{2}}}} + \underset{additional term}{\underset{⏟}{\frac{d P}{d r} \frac{1}{ρ}}}

the additional term is for support against gravity for gas only, assuming a negative pressure gradient in disc
$v_{g} ≃ 0.0995 v_{k e p} ⟹$ solid particles feel a headwind; at $1$ AU, $v_{w i n d} ≃ 100$ ms $^{- 1}$
there will be significant drag on the solids that leads to settling of solid particles into disc mid-plane and gradual spiral in towards the proto-sun
the drag force:

F_{D} = \frac{1}{2} C_{D} π R_{p}^{2} ρ_{g a s} (v_{k e p} - v_{g a s})^{2}

where, $C_{D}$ is the drag coefficient, $π R_{p}^{2}$ is the cross-section of the particle

\begin{matrix} τ_{D} ≃ \frac{m_{p} v_{k e p}}{| F_{D} |} \\ m_{p} = \frac{4}{3} π R_{p}^{3} ρ_{p} \\ τ_{D} ≃ \frac{8}{3} \frac{1}{C_{D}} \frac{ρ_{p}}{ρ_{g a s}} \frac{R_{p} v_{k e p}}{(v_{k e p} - 0.995 v_{k e p})^{2}} \\ ≃ \frac{8}{3} \frac{1}{C_{D}} \frac{ρ_{p}}{ρ_{g a s}} \frac{1}{{0.005}^{2}} \frac{R_{p}}{v_{k e p}} \end{matrix}

at 1 AU: $v_{k e p} = 3 \times 10^{4}$ ms $^{- 1}$ $⟹ v_{w i n d} ≃ 100$ ms $^{- 1}$
for a planet, $τ_{D} \sim 10^{8}$ yr $≫$ life time of a gas disc $(\sim 10^{6} - 10^{7})$
for a dust particle, $τ_{D} \sim$ seconds, so it is carried by gas
for a 1 m object, $τ_{D} \sim 1000$ years, so they will spiral into the sun and will be lost from the disc
there is a significant pattern for core-accretion scenario
possible solutions:
- structures in disc can trap particles in high pressure regions promoting rapid growth
- streaming instabilities can cause clumps of particles, shielding each other from the disc headwind
the planetesimals further grow into proto-planets $(\sim 0.1 M_{\oplus})$ assisted by mutual gravity
this stops at isolation mass, when the hill sphere of the planet has swept out an annuls of the disc
more massive cores form beyond the snowline, where ices can form
the surface density of disc:

\begin{matrix} Σ \propto r^{- 3 / 2} \\ ⟹ d m \propto 2 π r d r r^{- 3 / 2} \propto r^{- 1 / 2} d r \end{matrix}

where, $2 π r d r$ is the area of an annulus of disc, and $d m$ is the mass of solids in an annulus of thickness, $d r$

spacing of the planets $(d r)$ increases $\propto r ⟹ m_{c o r e} \propto r^{1 / 2}$
for jupiter:

M_{c o r e} (5 AU) ≃ M_{c o r e} (1 AU) \times 5^{1 / 2} \times 4 ≃ 9 M_{\oplus}

taking $M_{c o r e}$ at 1 AU to be $M_{\oplus}$ , and the factor of $4$ accounts for ice as well as rock
massive cores form rapidly beyond the snowline, capable of accreting gas
massive cores accrete gas from disc
gas fills the hill sphere of the planet, but it must radiate its gravitational potential energy to cool and settle on the planet
initially, this is slow as diffuse gas cools inefficiently (cooling rate $\propto ρ^{2}$ )
the gas accretion rate increases as gas mass increases, and the gas gets compressed, so it radiates more efficiently
eventually, it becomes a runaway process that rapidly accretes all the gas in disc annulus
this is how a gas giant is formed
the timescale is similar to disc lifetime
if the disc is dispersed during slow gas accretion phase, an ice giant is formed

image: Pollack, et al (1996); legend: $M_{z} :$ solids, $M_{X Y} :$ gases, $M_{p} :$ planet (combined)

\frac{3}{2} k T ≃ \frac{1}{2} m_{p} {\bar{v}}^{2}

where, $T$ is the temperature of the heated disc atmosphere $≃ 10000$ K

v_{e s c} = \sqrt{\frac{G M_{⊙}}{r}}

where, $r$ is the distance from the sun

r = \frac{2 G M_{⊙} m_{p}}{3 k T} ≃ 10 Au

beyond this $r$ , photoevaporation of the outer disc can be seen, shutting off the supply of gas to the inner disc
terrestrials assembled via giant collisions between protoplanets, after disc dispersal
the gravitational perturbation of orbits cause the eccentricity to be increased, causing crossing orbits and collisions
previously, orbits were circularised by interaction with the gas disc
evidence for giant collisions:
- presence and composition of earth's moon (same as earth's mantle)
- high density of mercury (missing mantle material, lost in collision)
- spin obliquity of uranus
from the radioisotope dating of moon rocks and meteorites, the moon can be dated back to $\sim 4.5$ Gyr
the solar system is $4.6$ Gyr old, and the assembly of the terrestrials took $\sim 10^{8}$ years