Chapter 7: The Formation of Planetary Systems

Overview of Planetary Systems and their Formation

• Definition of a Planet: Large, round, isolated bodies that orbit a star.

• Planetary System: A comprehensive system consisting of a star, planets, moons, and various smaller bodies.

• Solar System Scope: Includes the Sun and every object orbitally bound to it, such as classical planets, moons, asteroids, and comets visible from Earth.

• Fundamental Observations:

  • All planets orbit in the same direction.

  • All planets orbit in the same flat plane.

  • These observations deduce that planets formed from a singular flat, rotating disk of material.

The Nebular Hypothesis

• Historical Origins:

  • First proposed in $1755$ by the German philosopher Immanuel Kant.

  • Arrived at independently in $1796$ by the French astronomer Pierre-Simon Laplace.

• Etymology: "Nebula" is the Latin word for "cloud."

• Core Concepts:

  • The hypothesis posits that a rotating cloud of interstellar gas gradually collapses and flattens under gravity to form a disk.

  • The Sun forms at the dense center of this cloud.

  • Planets form within the surrounding disk.

Physics of Cloud Collapse and Stability

• Hydrostatic Equilibrium: A state of balance in a stable object where inward and outward forces are equal.

  • Self-gravity: The mutual attraction between the parts of an object that pulls outer layers toward the center.

  • Opposing Forces: Self-gravity is resisted by structural strength, gas pressure, or radiation pressure.

  • Contraction Condition: If self-gravity exceeds the outward force, the object contracts.

• Balloon Thought Experiment:

  • Cold Environment: Placing a balloon in a freezer causes its circumference to shrink because the drop in temperature decreases internal pressure, allowing self-gravity (or external pressure) to dominate.

  • Warm Environment: Placing it in warm water causes expansion as pressure increases.

  • Application: This illustrates the maintenance of hydrostatic equilibrium ensuring stability in stars and planets. For star formation to begin, gravity must initially overcome all outward pressure forces.

Evidence for the Nebular Hypothesis

• Stellar Observations: Stellar astronomers observe young stars surrounded by rotating disks of gas and dust, often referred to as the "child" phase of stellar evolution.

• JWST Imagery: Recent images from the James Webb Space Telescope show protoplanetary disks edge-on, featuring conical outflows of gas/dust and narrow jets blasting into space.

• Meteorites: These space rocks are often mixtures of smaller pebbles and rocks, suggesting they formed through a process of aggregation. This indicates the early Solar System was a swirling disk containing both gas and solid material.

• Multidisciplinary Convergence:

  • Mathematicians/Theorists: Proposed the collapsing rotating cloud model.

  • Stellar Astronomers: Observed disks of gas and dust around young stars.

  • Planetary Scientists: Found evidence of aggregation in meteorites.

Angular Momentum and Disk Dynamics

• Definitions of Motion:

  • Tangential Speed ($v$): Linear speed along a circular path, directed tangent to the circumference.

  • Rotational Speed ($\omega$): Number of rotations or revolutions per unit of time (e.g., RPM).

  • Relationship: $v = r\omega$.

• Rotational Inertia: Also known as the moment of inertia, this is the resistance of an object to changes in its rotational state. It depends on mass ($m$) and the distribution of that mass relative to the axis of rotation.

• Angular Momentum ($L$):

  • Depends on velocity ($v$), mass ($m$), and size/radius ($r$).

  • Conservation of Angular Momentum: Angular momentum remains constant unless acted upon by an external force.

  • Example (Figure Skater): As a skater pulls their arms in (decreasing $r$), they rotate faster (increasing $\omega$) to keep $L$ constant.

  • Example (Pizza Dough): Spinning dough expands in diameter, causing the rotational speed to decrease to conserve angular momentum. A spherical nebula turns into a flat disk similar to how a ball of dough becomes a flat crust.

• The Paradox of Solar Rotation: A cloud $1\,\text{light-year}$ across rotating once per million years should rotate once every $0.6\,\text{seconds}$ when collapsed to the Sun's size. However, the Sun rotates roughly once every $25\,\text{days}$. Much of the original angular momentum was transferred to the orbiting planets.

Formation Mechanics of the Protoplanetary Disk

• Accretion Disks: Rapidly rotating, flattened structures of gas and dust spiraling into a central body. In young stars, these are called protoplanetary disks.

• Protostar: A large ball of gas at the center of the disk containing approximately $99\%$ of the system's mass; it is not yet hot enough to be a true star. The disk contains the remaining $1\%$.

• Collapse Process: Motion parallel to the rotation axis cancels out during collapse, while motion perpendicular to the axis is preserved, resulting in a flat disk.

• Traffic Circle Analogy: Material flowing into the disk is like cars entering a rotary with no exits, leading to a build-up of material circulating the center.

Growth of Particles and Planetesimals

• Aggregation: Small particles in the disk collide and stick together due to static electricity and gas motions.

• Planetesimals: Larger particles reaching approximately $1\,km$ in size. At this threshold, gravity becomes the dominant force, pulling in nearby objects.

• Protoplanets: Formed through the gravitational combination of planetesimals.

• Remnants: Today's asteroids and comets are the leftover planetesimals that did not incorporate into planets.

Temperature and Composition of the Disk

• Energy Conversion: Conservation of energy dictates that as the cloud shrinks, gravitational potential energy converts into kinetic, radiative, and thermal energy ($PE_{grav} \rightarrow KE + \text{Radiative} + \text{Thermal}$).

• Heat Distribution: Thermal energy is significantly higher in the inner portion of the disk because it is closer to the protostar and more gravitational energy is converted to heat there.

• Material Types:

  • Refractory Materials: Substances that do not melt at high temperatures (rocks and metals). Found in the inner disk.

  • Volatile Materials: Substances that melt or evaporate at moderate temperatures (e.g., water $H_2O$, ammonia $NH_3$). Found in the outer disk.

  • Organic Materials: Contain carbon-hydrogen bonds (e.g., methane $CH_4$, carbon monoxide $CO$).

• Planet Migration: Chaotic gravitational encounters can change a planet's orbit. Uranus and Neptune may have formed closer to Jupiter but were driven outward. A loss of angular momentum causes a planet to spiral inward.

Planetary Atmospheres

• Primary Atmosphere: The initial gas gathered from the protoplanetary disk, consisting mostly of low-mass Hydrogen ($H$) and Helium ($He$).

  • Core Accretion-Gas Capture: The process of gathering this initial atmosphere.

  • Loss: Young stars emit solar winds and radiation that drive out remaining gas. Low-mass planets have low escape velocities and cannot retain these light gases.

• Secondary Atmosphere: Formed after the loss of the primary atmosphere on low-mass planets.

  • Volcanism: Outgassing of heavy gases from a planet's interior.

  • Comet Impacts: Delivery of volatiles like water that evaporate and contribute to the atmosphere.

Models of Giant Planet Formation

• Core Accretion-Gas Capture Model: Large solid cores (planetesimals) grow big enough to gravitationally capture nearby gas. Mini-accretion disks form around the cores, funneling material and creating large moons. Jupiter had only about $10\,\text{million years}$ to form before the Solar wind cleared the disk's gas.

• Disk Instability (Gravitational Instability) Model: A "top-down" model where the disk fragments into massive clumps (planet-sized) that collapse under their own gravity. This is similar to how stars form and is more applicable to giant planets.

Specific Solar System Objects

• Terrestrial Planets: Mercury, Venus, Earth, and Mars. Formed within $5\,AU$ where only rock and metal were solid.

  • Mercury has no significant secondary atmosphere.

  • Venus, Earth, and Mars have substantial secondary atmospheres.

• Gas Giants: Jupiter, Saturn, Uranus, and Neptune. Formed in the outer disk where volatiles and organics could combine with refractory materials.

• Dwarf Planets: Round objects that orbit the Sun but have not cleared their orbital paths of smaller bodies.

• Asteroids: Small bodies found primarily inside Jupiter's orbit.

• Comet Nuclei: Icy planetesimals from the outer Solar System.

• Cataclysmic Impacts: Common in the early system.

  • Formed the Moon (Earth impact).

  • Caused Uranus to rotate on its side (perpendicular to its orbital plane).

  • Created heavy cratering on Mercury and the Moon.

Exoplanets: Definitions and Naming

• Exoplanet: A body with mass less than $13\,M_{Jupiter}$ orbiting a star other than the Sun.

• Brown Dwarf: A "failed star" with mass between $13$ and $80\,M_{Jupiter}$ ($0.08\,M_{Sun}$). They cannot fuse Hydrogen.

• Stars: Objects with mass greater than $0.08\,M_{Sun}$.

• Naming Convention:

  • The star takes the primary name (e.g., $51\,\text{Pegasus}$).

  • Planets are designated with lowercase letters starting with b for the first discovered (e.g., $51\,\text{Pegasus b}$).

  • Binary stars are labeled A (primary) and B (secondary).

  • Exomoons use Roman numerals (e.g., $I, II, III$).

  • Example: $16\,\text{Cygni Bb}$ is a planet orbiting the secondary star of the $16\,\text{Cygni}$ system.

Exoplanet Detection Methods

• Radial Velocity Method: Detects the "wobble" of a star caused by the gravitational pull of an orbiting planet. Uses Doppler shifts (blueshift toward us, redshift away). Effective for massive planets close to their stars.

• Transit Method: Measures the drop in a star's brightness as a planet passes in front.

  • Allows for the calculation of the planet's radius ($R$).

  • Missions like Kepler ($2009\text{--}2018$) and TESS (launched $2018$) use this.

• Direct Imaging: Taking an actual photograph of a planet. Difficult due to the star's glare. Uses tools like:

  • Coronagraph: Internal occulting shields in a telescope.

  • Starshade: External spacecraft flown to block starlight.

• Microlensing: A planet's gravity acts as a lens, temporarily brightening a background star. Can find planets far from their host stars or around dim stars.

• Astrometry: Precisely measuring the change in a star's position in the sky relative to other stars.

Unique Populations of Exoplanets

• Hot Jupiters: Jupiter-sized gas giants in extremely close orbits around their stars. Their presence suggests significant planetary migration.

• Super-Earths: Rocky planets more massive than Earth (> 1\,M_{Earth}).

• Mini-Neptunes: Gas planets with $2$ to $10\,M_{Earth}$.

• Rogue Planets (FFPs): Free-floating planets not bound to any star; they may be ejected from systems or form autonomously in the interstellar medium.

• Hycean Worlds: Hypothetical planets with hydrogen-rich atmospheres and liquid water oceans.

• Habitable Zone: The region around a star where the temperature allows liquid water to exist.

Working It Out Calculations

7.1: Angular Momentum Comparisons

• Formula: $L = m \times v \times r$

• Jupiter's Orbital Angular Momentum:

  • $m = 1.90 \times 10^{27}\,kg$

  • $v = 1.31 \times 10^4\,m/s$

  • $r = 7.79 \times 10^{11}\,m$

  • $L_{orbital} = 1.90 \times 10^{27}\,kg \times 1.31 \times 10^4\,m/s \times 7.79 \times 10^{11}\,m = 1.94 \times 10^{43}\,kg\,m^2/s$

• Sun's Spin Angular Momentum:

  • Formula for uniform sphere: $L_{spin} = \frac{4\pi m R^2}{5P}$

  • $R = 6.96 \times 10^8\,m$

  • $m = 1.99 \times 10^{30}\,kg$

  • $P = 24.5\,\text{days} = 2.12 \times 10^6\,s$

  • $L_{spin} = \frac{4 \times \pi \times 1.99 \times 10^{30}\,kg \times (6.96 \times 10^8\,m)^2}{5 \times (2.12 \times 10^6\,s)} = 1.14 \times 10^{42}\,kg\,m^2/s$

• Result: L_{orbital} > L_{spin}, proving most angular momentum resides in the planets' orbits.

7.2: Estimating Exoplanet Orbit Size

• Formula (Newton's Kepler's Third Law): $A^3 = \frac{G}{4\pi^2} \times M \times P^2$

• Data for HD70642b:

  • $P = 5.7\,yr = 1.8 \times 10^8\,s$

  • $M = 2 \times 10^{30}\,kg$

  • $A^3 = \frac{6.67 \times 10^{-20}\,km^3\,kg^{-1}\,s^{-2}}{4\pi^2} \times 2 \times 10^{30}\,kg \times (1.8 \times 10^8\,s)^2 = 1.1 \times 10^{26}\,km^3$

  • $A = 4.8 \times 10^8\,km = 3.2\,AU$

7.3: Estimating Exoplanet Radius

• Formula: $\text{fractional reduction in light} = \frac{R_{planet}^2}{R_{star}^2}$

• Data for Kepler-11c:

  • $R_{star} = 7.7 \times 10^5\,km$

  • Reduction = $0.077\% = 0.00077$

  • $0.00077 = \frac{R_{Kepler-11c}^2}{(7.7 \times 10^5\,km)^2}$

  • $R_{Kepler-11c}^2 = 4.6 \times 10^8\,km^2$

  • $R_{Kepler-11c} = 2.1 \times 10^4\,km = 3.3\,R_{Earth}$

Questions & Discussion

• Class Question 1: How does the nebular hypothesis explain that all planets orbit the Sun in the same direction?

  • Answer: All planets formed from a swirling, flat disk of gas orbiting the Sun. All planets orbit in the same direction as the disk from which they formed.

• Class Question 2: What will happen as a giant gas cloud contracts if it initially rotates slowly?

  • Answer: Its rotation will speed up (Conservation of Angular Momentum).

• Class Question 3: If Earth's mass were to suddenly double, what would happen to Earth's orbital speed?

  • Answer: It would become half as slow (to maintain constant $L = mvr$).

• Class Question 4: What physical process is responsible for starting cloud collapse in nurseries like the Carina Nebula?

  • Answer: A wind sweeping through the cloud and compressing parts.

• Class Question 5: Which planet lost its primary atmosphere and now has a substantial secondary atmosphere?

  • Answer: Venus.

• Class Question 6: For Jupiter to move closer to the Sun, what property would have to change?

  • Answer: Jupiter would need to lose angular momentum.

• Class Question 7: Why are there two types of planets in our Solar System?

  • Answer: The temperature in the inner disk was so hot that only rocks and metals were solid, while in the outer disk, ice bodies could form and accrete gas.

• Class Question 8: Which method of exoplanet discovery can measure the radius of a planet?

  • Answer: Transit.

• Check Your Understanding (CYU) Highlights:

  • Evidence for nebular hypothesis: Same orbital direction, flat system, disks observed around other stars.

  • Angular momentum location: Most goes into the orbital angular momentum of planets.

  • Rocky inner planets: Due to warm temperatures in the inner disk allowing only rock/metal solids.

  • Rocky planet beyond Neptune origin: Most likely formed close to the Sun and migrated out.

  • Most common method for finding Earth-mass planets: Transit method.