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Lecture 6-8: Solar nebula theory

Lecture 6-8: Solar nebula theory. Topics to be covered: Solar nebula theory. Solar nebula theory. Five steps to planet formation in Solar nebula theory. Collapse Heating via conversion of PE to KE. Spinning Spinning up of material to conserve AM. Flattening

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Lecture 6-8: Solar nebula theory

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  1. Lecture 6-8: Solar nebula theory • Topics to be covered: • Solar nebula theory PY4A01 Solar System Science

  2. Solar nebula theory • Five steps to planet formation in Solar nebula theory. • Collapse • Heating via conversion of PE to KE. • Spinning • Spinning up of material to conserve AM. • Flattening • Sphere to disk due to rotation. • Condensation • Gas to liquid and solid particles due to cooling. • Accretion • Solid particles ‘stick’ due to electrostatic and gravitational forces. • See extensive review by Lissauer: (http://adsabs.harvard.edu/abs/1993ARA%26A..31..129L) PY4A01 Solar System Science

  3. 1. Collapse - Solar Nebula Theory • For a cloud with M > MJwhere MJ (T3/)1/2=> cloud collapses. • Localised density enhancement (/) may cause collapse. • Gravity pulls material in, gas pressure opposes. • When pressure >> gravity => oscillations. • When pressure << gravity => collapse. • Cooling lowers pressure, can trigger collapse. PY4A01 Solar System Science

  4. 1. Collapse - Solar Nebula Theory • Consider N molecules of mass m in a volume of size L at a temperature T. • Gravitational energy: where M = Nm ~ L3. • Thermal energy: ET ~ NkT • Ratio is: where LJis the Jeans’ length: • Gravity wins when L > LJ. PY4A01 Solar System Science

  5. 1. Collapse - Solar Nebula Theory • Collapse timescale can be also be estimated from gravitational acceleration: • Time to collapse: => denser regions collapse faster. • Ignoring gravity, pressure waves travel at sound speed: • Sound crossing times is therefore, => Small hot regions oscillate more rapidly. PY4A01 Solar System Science

  6. 1. Collapse - Solar Nebula Theory • Ratio of the collapse time (tG) to the sound crossing time (ts) is: where LJis again the Jeans length: • Compare with form given in Jeans formation theory. PY4A01 Solar System Science

  7. 1. Collapse - Solar Nebula Theory • As gas cloud collapses, temperatures rise as potential energy converted to kinetic via: E = KE + U = const • From Conservation of Energy, KE increases as U decreases. • Temperature therefore rises as 1/2mv2 = 3/2 kT or T = 1/3k (1/2 m v2). • Some energy is radiated away thermally. The solar nebula becomes hottest near its center, where much of the mass collect to form the protosun. • Protosun eventually becomes so hot that nuclear fusion ignited in its core. PY4A01 Solar System Science

  8. 1. Collapse - Solar Nebular Theory • Initially, gas and dust with low AM fall to core of cloud. Material with high AM cannot due to centrifugal forces. • As gas and dust fall to equatorial plane, it collides with material falling in other direction => energy of this motion is dissipated as heat. • Consider a parcel of gas which falls from infinity to a circular orbit r. If half of gravitaional energy is converted to orbital kinetic energy via the remainder is available for heat. • For Mprotostar = 1 Msun, vc = 30 km/s at 1 AU => T ~ 7 x 104 K. This temperature is never reached as the time-scale for radiaitve cooling << heating time. PY4A01 Solar System Science

  9. 2. Spinning - Solar Nebula Theory • Solar nebula “spins-up” as it collapses to conserve AM. Lf = Li m vf rf = m vi ri m rf2i = m ri2f => f = i (ri / rf )2 • As ri > rf => f >> i. Cloud spins up rapidly as it contract. • Need a breaking mechanism. • Rotation ensures not all of material collapses onto the protosun: the greater the AM of a rotating cloud, the more spread out it will be along its equator. PY4A01 Solar System Science

  10. 3. Flattening - Solar Nebula Theory ar a • g = GM/r2is directed radially to centre. • a = r2is perpendicular to rotation axis. Radial component is ar= r2sin. • Net radial acceleration is • At pole ( = 0) => a(r) = GM/r2 • At equator ( =90) => a(r) = g - r2 • In disk, there’s a distance where g = r2=> this is point wherecontraction stops. g  g a PY4A01 Solar System Science

  11. 3. Flattening - Properties of the disk • Basic properties of disk depend on how gas behaves in a gravity field. • How does the disk shape determine? • What is the velocity of disk? • What is the density distribution of disk? • How does rotation and gas pressure effect shape? PY4A01 Solar System Science

  12. 3. Flattening - Properties of the disk • In hydrostatic equilibrium • z-component of gravity is • Equation of state for gas is P = cs2  • Density profile is thus: where the scale-height (ie thickness) is:  z PY4A01 Solar System Science

  13. 3. Flattening - Properties of the disk • Disk aspect ratio is defined as: • Condition for disk flaring is: (1 - q ) / 2 > 0 => q < 1 • For typical disks, (ie q = 1/2) • In general cases (eg. Galaxies) • Note, disk shape only depends on temperature, not density. PY4A01 Solar System Science

  14. 3. Flattening - Properties of the disk • Radial force balance where  = (r) is the surface density distribution. • Angular velocity of gas is not strictly Keplarian:    (1 - ) where  is the viscosity (0<  <1). • Typically,  ~ 0.001-0.01 => disk rotates slightly slower that would expect from Kepler’s laws. PY4A01 Solar System Science

  15. Summary of planet formation in disk 4. Solid planets formation 1. Disk formation 2. Dust sedimentation 5. Gaseous planets formation 3. Planetesimal formation 6. Disk dissipation PY4A01 Solar System Science

  16. Step 4: Condensation • Formation of planets requires “seeds” - chunks of matter that gravity can eventually draw together. Understanding these seeds and clumping is key to explaining the differing compositions of planets. • The process by which seeds were sown is condensation, when solid or liquid particles condense out of a gas. • Condensation is temperature dependent. When the temperature is low enough atoms/molecules solidify. PY4A01 Solar System Science

  17. Step 4: Condensation • Approximate equation for the temperature variation in Solar Nebula is T(r) 631 / r0.77 where r is in AU. “Ice line” where T = 273 K is located at ~3 AU from Sun. • T < 2,000 K, compounds of silicates (rock) and nickel-iron form. • T < 270 K, carbon compounds, silicates and ices form. • Planetary interiors to Mars • Nebula temperature > 400 K • Made of silicates and metals • Planets beyond Mars • Nebula temperature < 300 K • Made of silicates and ices PY4A01 Solar System Science

  18. Step 4: Condensation • Metals include iron, nickel, aluminum. Most metals condense into solid at temperatures of 1000-1600 K. Metals made up <0.2% of the solar nebula's mass. • Rocks are common on Earth’s surface, primarily silicon-based minerals (silicates). Rocks are solid at temperatures and pressures on Earth but melt or vaporize at temperatures of 500-1300 K depending on type. Rocky materials made up ~0.4% of the nebula by mass. • Hydrogen compounds are molecules such as methane (CH4), ammonia (NH3), and water (H2O) that solidify into ices below about 150 K. These were significantly more abundant than rocks and metals, making up ~1.4% of nebula's mass. • Light gases (H and He) never condense under solar nebula conditions. These gases made up the remaining 98% of the nebula's mass. • Note: Order of condensation scales with density. PY4A01 Solar System Science

  19. Step 4: Condensation • Terrestrial planets are made from materials that constituted ~0.6% of the nebula. • Jovian planets were formed in region where ~2% of material condensed. They also captured gas (98%). PY4A01 Solar System Science

  20. Step 4: Condensation • T~1500-2000K at the present-day orbit of Mercury • About Mercury metals can begin to aggregate together • Further out, rocky materials condense. • Most metals/rocks condensing around the present-day orbit of Mars (T~500K). • Hence inner planets have high metal/rock content and few volatile materials. PY4A01 Solar System Science

  21. Step 4: Condensation • Size and composition of planetesimals depends on temperature and distance from Sun. • Inner solar system • Within frost line, only rock and metals can condense. • Planetesimals therefore made of rock and metals. • Constitute ~ 0.6% of available material by mass. • Inner planetismals therefore grew more slowly. • Inner planets are therefore smaller. • Outer solar system • Beyond frost line, rock, metals and ices condensed. • Planetesmals therefore contain these materials. • Constitute ~ 2% of available material by mass. • Outer planetismals therefore grew more quickly. • Outer planetesmals are therefore larger. • These process resulted in elementary planetary cores. PY4A01 Solar System Science

  22. Step 4: Condensation • Densities and distances of objects in solar system supports this condensation theory: • Terrestrial planets: 3-6 g cm-3 => mainly rocks and metals. • Jovian planets: 1-2 g cm-3 => more ice and captured gas. • Inner Asteroids: contain metalic grains in rocky materials. • Outer Asteroids: less metals, and significantly more ice. PY4A01 Solar System Science

  23. Motion of dust in early disk • Dust particles experience drag force: also referred to as Epstien drag force. • In equilibrium, there is balance between drag and gravity: Ffric = mg • Using g = -2z, • Using  = 0exp(-z2/H2), the settling time is therefore: => Dust particles settle into the central plane. PY4A01 Solar System Science

  24. Step 5: Accretion • After condensation, growth of solid particles occurs due to collisions. • Accretion is growth of grains through collisions - the real planet building process. • Larger particles formed from both tiny chondrules about 1 mm in size, and from porous molecular aggregates held together by Van der Waals forces. • Accretion proceeds in two ways: • Collisions due to the geometric cross section - direct impacts on ‘seed’ grain. • Collisions due to gravitational attraction - sweeping-up of material from a region much larger than grain diameter. PY4A01 Solar System Science

  25. Step 5: Accretion - geometric • Consider spherical grain of radius r and geometric cross section s = p r2. If number of grains m-3 is ng, and relative velocity of the grains is vrelthen volume (V) swept out in time t is V = s vrel t, (or V = s vrelfor 1 second). • The number of particles (N) encountered in t is N = V ng = s vrel t ng = p r2 vrel t ng • In a given period, seed particle’s mass grows as m/t = m0 + N m0 = m0 (1 + pr2 vrel ng) where m0is the grain mass. • Mass of the seed particle therefore increases as r2 for geometrical collisions. s = p r2 vrelt PY4A01 Solar System Science

  26. Step 5: Geometric and gravitational accretion • Objects formed by geometric accretion are called planetesimals: act as seeds for planet formation. • At first, planetesimals were closely packed. • Then coalesced into larger objects, forming clumps few km across in few million years. • Once planetesimals had grown to few km, collisions became destructive, making further growth more difficult. • Gravitational accretion then begins to dominate. This then accretes planetesimals to form protoplanets. PY4A01 Solar System Science

  27. Step 5: Accretion - gravitational • When gravity important, grains accrete from larger volume than during geometric growth phase. • Consider “test” grain with velocity vi at a vertical distance s from a “seed” grain. Suppose “test” grain encounters “seed” grain with a final velocity vf.What is value of s such that the seed grain can capture the “test” grain? • Using conservation of angular momentum: mvf r = mvi sEqn. 1 and conservation of energy: where m is mass of “test” grain and M is mass of “seed” grain. • Eliminating vffrom Eqns. 1 and 2 gives: • As M ~ r3 =>s2~ r4. Eqn. 2 vi test grain s vf r s seed grain PY4A01 Solar System Science

  28. Step 5: Accretion - gravitational • The growth rate of the seed particle per unit time is therefore: m / t = m0 (1+ p s2 vrel ng) • As s2 ~ r4 => m / t ~ r4 =>runawayaccretion. • Once grains are large enough that gravity is important, accretion rate increases dramatically. • If critical size is achieved, a planetesimal will grow rapidly. Less massive objects grow at a much smaller rate. • Model calculation suggest that the first large size objects to form are planetesimals with sizes ~ few tens of km. PY4A01 Solar System Science

  29. 5. Accretion • These processes result in planetesimals of tens of kilometers in size in less than a million years or so. • Timescale for grows can be calculated from: • Not only do bigger planetesimal grow the fastest, but smaller planetesimals are quickly destroyed by fast collisions and turned into smaller fragments => typically one object will dominate a region. m/t ~ r4 m/t ~ r2 PY4A01 Solar System Science

  30. 5. Accretion - planet formation • Accretion therefore progresses according to: • Once planetesimals are formed, the following can occur: • The final stages in the growth of a Terrestrial planet are dramatic and violent. • Large Mars-sized protoplanets collide to produce objects such as the Earth and Venus (MEarth ~ 9 Mmars). Geometric Accretion Gravitational Accretion Planetesimals Planetesimals Protoplanets Planets PY4A01 Solar System Science

  31. 5. Accretion - the planets • Inner Planets • Formed slowly due to small amount of metals and rocks in early solar nebula. • Geometric accretion rate and gravitational accretion rate small. • By time inner planetesimals were formed and had significant gravitational fields, the nebula had been cleared out by the solar wind. • Then no nebular gas then present to capture an elementary atmosphere. • Outer Planets • Formed less violently. • Great quantities of ice at >3 AU resulted in large rock/ice cores forming. • Reason for rapid core growth is that ices have large cross-sectional area. PY4A01 Solar System Science

  32. 5. Accretion - The planetesimal graveyard • Asteroid belt is ‘resting ground’ for collision-evolved planetesimals that were not incorporated into a planet. • Total mass of asteroid belt ~5 x 1021 kg (which is about 1/3rd the mass of Pluto or 1/15th the mass of the Moon). • Ceres the largest asteroid has a diameter of 940 km and a mass of ~1021 kg. • A planet probably did not form in this region because of the rapid formation, and resulting large mass of Jupiter. PY4A01 Solar System Science

  33. Formation of giants: numerical simulation • Three stages for gas giant formation: • Core accretion. • Accretion of gas until Mgas ~ Mcore • Runnaway accretion of gas => Mgas >> Mcore • See Pollack et al. (1996). Stage 3 Stage 2 Stage 1 PY4A01 Solar System Science

  34. Summary of planet formation in solar nebula theory 4. Solid planets formation 1. Disk formation 2. Dust sedimentation 5. Gaseous planets formation 3. Planetesimal formation 6. Disk dissipation PY4A01 Solar System Science

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