Astronomy 217

Prof. Andrew W. Steiner

Oct. 22, 2021

TA James Ternullo

Last Time

• Objects in the Solar System

Today

• Origin of the Solar System

Explaining our Solar System

• In developing a theory of the origins of our solar system, we have three sets of observations to help.
• The current solar system:
• Solar system is nearly coplanar with most of the angular momentum in planets.
• Inner, rocky, low mass, terrestrial planets are different than the outer, gaseous, massive Jovian planets.
• Existence of asteroid belt, Kuiper belt and comets.
• Extra-solar planets
• Young stars & Star-forming regions

Star-Forming Regions

• The close association of young stars with cold, dense clouds of interstellar gas and dust provides our first hint.
• A small compression of one of these dense clouds causes clumps in the cloud to gravitational collapse.
• These clumps can grow accreting more gas.
• Multiple clumps in a cloud usually form stars simultaneously.
• Once stars form, their radiation disperses the cloud.

Why Cold Dense Gas?

• For a gas cloud to collapse under gravity, gravity must overcome the thermal motions.
• For a gravitationally bound system with kinetic energy $K$ and gravitational potential energy $U$, $$E = K + U = U/2 \quad \mathrm{or} \quad 2K + U =0$$
• Substituting the thermal and gravitational energy of a cloud of $N$ atoms of radius $R$ and mass $M$ for $K$ and $U$ $$3 k T N < \frac{3 G M^2}{5 R}$$
• Re-espressing this in terms of density $$3 k T < \frac{4 \pi}{5} G \rho R^2$$
• Typically, $T_{\mathrm{ps}} \approx 10~\mathrm{K}$ and $\rho_{\mathrm{ps}} \approx 3 \times 10^{-15}~\mathrm{kg}/\mathrm{m}^{-3}$

Nebular Contraction

• We refer to the clump of gas and dust from which the sun formed as the pre-solar nebula.
• The formation process is called nebular contraction.
• Cloud of gas and dust contracts due to gravity but angular momentum means it spins faster and faster as it contracts, resulting in the formation of a disk.

Angular Momentum

• Conservation of angular momentum requires that product of radius and angular velocity must be constant as the pre-solar nebula collapses to form the Sun. $$L = m r^2 \omega = m v r$$
• During the collapse, the gas density increases from $\rho_{\mathrm{ps}} \approx 3 \times 10^{-15}~\mathrm{kg}~\mathrm{m}^{-3}$ to $\rho_{\odot} \approx 1400~\mathrm{kg}~\mathrm{m}^{-3}$
• Since mass is conserved, this requires a decrease in volume by a factor of $10^{18}$, thus $r_{\mathrm{ps}} \approx 10^6~R_{\odot}$
• Thus even small velocities in the pre-solar nebula produce significant angular momentum

Spinning a Disk

• In the absence of angular momentum, a clump would be rapidly converted to a sphere by gravity and gas pressure.
• In a rotating sphere, the centrifugal force opposes gravity at the equator but not along the pole, flattening the sphere.
• Shear forces and magnetic drag allow the central regions to transfer angular momentum outward, accentuating the disk.

Protoplanetary Disk

• Observations of young stars confirm that nebular contraction leaves a denser, hotter proto-star surrounded by a more rapidly spinning protoplanetary disk.
• Nebular contraction explains 3 key solar system observations:
• Planetary orbits all lie in (nearly) the same plane.
• The Sun has relatively little angular momentum
• Direction of orbital motion, and rotational motion of most planets & moons is the same as Sun’s rotation.

Planet Formation

• The theory for the transformation of the protoplanetary disk into planets, asteroids, comets, etc. must explain still more observations.
• Planets are relatively isolated in space
• Planetary orbits are nearly circular.
• Solar system is highly differentiated with small rocky planets in the interior and gas giants further out.
• Asteroids are very old, and not like either inner or outer planets.
• Explain icy Kuiper belt objects and comets.

T Tauri Star

• The initial collapse of the cloud takes ~ $10^5$ years, leaving a newborn T Tauri star surrounded by a cooling disk of gas and dust at 1000 K or less.
• T Tauri stars are in a highly active phase of their evolution and have strong solar winds.
• Accretion of material onto the star may continue for $10^7$ years but eventually, these winds sweep away the gas disk, leaving the planetesimals and gas giants (the debris disk).

Condensation

• Because of the increasing orbital size, the Sun provides significantly less energy to the outer planets.
• The flux suffers geometric dilution $$F(r) = \frac{L_{\odot}}{4 \pi r^2} = \frac{4 \pi R_{\odot}^2 \sigma_{\mathrm{SB}} T_{\odot}^4}{4 \pi r^2}$$
• A planet of radius $R$ in an orbit of radius $r$ will absorb $$W_p = \frac{L_{\odot}}{4 \pi r^2} ( \pi R^2) (1- A)$$
• $A$ is the albedo, the fraction of light reflected by the planet
• The albedo varies greatly depending on the planets composition, cloud cover, etc.

Planetary Temperature

• Before the gas is blown away by the wind of the T Tauri star, cool temperatures allow the dense gas in the disk to condense into solids and liquids.
• The local temperature limits what is able to condense.
• In the hotter inner regions of the disk, only refractory materials, those with high condensation temperatures, like metals and silicates may condense.
• In the cool outer regions, even volatile substances may condense.

Freezing in a Vaccum

• The terrestrial values for melting and freezing points don’t necessarily apply in the near vacuum of the protoplanetary disk.
• For example, the liquid phase of water disappears at ~0.01 atmosphere.
• The boiling point for water declines to a low of 200 K in a vacuum, but can rise to more that 600K under high pressure.

Planetesimals

• Solid or liquid condensations grow by extracting molecule after molecule from the surrounding gas, like snowflakes.
• When two condensations collide gently and are joined together by mechanical (electrostatic) forces, like snowballs, this is called accretion.
• Over a few million years, the combination of condensation and accretion build km sized planetesimals.

Coalescence

• With the gas expelled by the T Tauri wind, at least from the inner solar system, condensation and accretion cease.
• Planetesimals are large enough to drawn together and coalesce under the force of gravity.
• This coalescence builds gradually bigger planetesimals and ultimately protoplanets, averaging out the orbits.

Holding Onto Gas

• The gaseous atmospheres of the planets can not be held to the planet by mechanical forces, but only by gravity.
• As with the solar wind, high enough temperatures allow the gas to escape. The region of the atmosphere from which such escape it possible is the exosphere.

Holding Onto Gas II

• The chance of escape depends on the comparison of the thermal velocity to the escape velocity. $$v_{\mathrm{rms}} = \left( \frac{3 k T_{\mathrm{ex}}}{m} \right)^{1/2} \Leftrightarrow v_{\mathrm{esc}} = \left( \frac{2 G M}{R_{\mathrm{ex}}} \right)^{1/2}$$
• If $v_{\mathrm{rms}} \approx v_{\mathrm{esc}}$, the gas escapes rapidly, but even for $v_{\mathrm{rms}} < v_{\mathrm{esc}}$, the high-energy tail of the velocity distribution can escape

Keep Some, Lose Some

• In order to hold onto a particular gas for the billion year age of the solar system, $v_{\mathrm{rms}} < v_{\mathrm{esc}}/6$ $$\frac{3 k T_{\mathrm{ex}}}{m} < \frac{G M}{18 R_{\mathrm{ex}}} \quad \Rightarrow \quad T_{\mathrm{ex}}(m) < \frac{G M m}{54 k R_{\mathrm{ex}}}$$
• Thus holding onto gas of mass $m$ requires a temperature less than $T_{\mathrm{ex}}(m)$
• Alternatively, given $T_{\mathrm{ex}}$, one can find the gas molecules that a planet can retain: $$\mu > \frac{54 k T_{\mathrm{ex}} R_{\mathrm{ex}}} {G M m_p} \approx 7.1 \left( \frac{T_{\mathrm{ex}}} {1000~\mathrm{K}} \right) \left( \frac{R_{\mathrm{ex}}}{R_{\oplus}} \right) \left( \frac{M}{M_{\oplus}} \right)^{-1}$$
• For the Earth $\mu > 8$, for Mercury $\mu > 34$, for Jupiter $\mu > 0.4$

Differentiation

• Once a protoplanet has formed, its own self-gravity and continued interactions with planetesimals can alter its surface and interior structure.
• Chemical differentiation is the process by which denser elements sink toward the center, with lighter elements rising to the surface. This requires the planet to be fully molten. Heat can be provided by radioactivity as well as gravitational energy release due to the sinking of the denser material.
• Cratering resulting from impacting planetesimals alters the surface (until about 3.3 billion years ago) and trigger volcanism.
• Outgassing and chemical reactions can alter the atmosphere and surface.

Earth-Making

• The formation of the terrestrial planets with their rocky/metallic composition seems adequately explained by this theory.
• The thin atmosphere of Mercury, the presence of water on Earth and Mars fit with the temperatures of the respective condensates and their ability to hold onto these gases, as do their relatively small masses.
• The existence of silicate surfaces over possibly molten iron cores explains the densities and magnetic fields where present.

Jovian Formation

• The Jovian planets with their very large size and atmospheres rich in H and He seem also to be well explained by this theory.
• But a major question remains open.
• Retaining Hydrogen and Helium requires massive planets ( > 15 $M_{\oplus}$ )
• How did the Jovian planets grow large enough, fast enough to capture H and He before these were driven from the protoplanetary disk by the solar wind?

• Perhaps condensation, accretion and coalesce can work fast enough?
• Or perhaps hydrodynamic instabilities in the outer, cool regions of the nebula can grow large planets directly?
• Or perhaps the Jovian planets formed farther from the Sun, where the H and He persisted longer, and then “migrated” inward.
• Detailed information about the cores of Jovian planets could help distinguish between the possibilities.
• Extra-solar planets may tell the answer.

Asteroid

• The persistence and aging of planetesimals provides a natural explanation for asteroids, as fragments left over from the initial formation of the solar system.
• However, the existence of the over-dense Asteroid belt is not a direct consequence of this condensation theory.
• Its orbit between Mars and Jupiter is a clue.
• Perturbations by Jupiter’s gravity kept them from coalescing into a planet.
• But Jupiter’s gravity also shepherded the asteroids, keeping them being captured by an existing planet.

Interplanetary Debris

• The persistence of icy planetesimals far from the Sun, where the density was too small for planets to form also seems a natural consequence of the condensation theory.
• However, the existence of the over dense Kuiper belt requires explanation.
• Icy planetesimals far from the Sun were ejected into distant orbits by gravitational interaction with the Jovian planets, into the Kuiper belt and the Oort cloud.
• Some were left with extremely eccentric orbits and appear in the inner solar system as comets.

Building a Solar System

• The combination of nebular contraction and planetary condensation theory, seems to work well to explain the many features of our solar system.
• Solar system is evidently not a random assemblage, but has a single common origin.
• Observation of other solar systems are beginning to have a big impact on our theory of solar system formation.
• Some of these, like “Hot Jupiters”, are challenging our long held ideas.
• Even in our solar system there are also irregularities (Uranus’s axial tilt, Venus’s retrograde rotation, etc.) that must be understood in more detail.