PY4A01 Solar System Science Lecture 6-8: Solar nebula theory oTopics to be covered: oSolar nebula theory

PY4A01 Solar System Science Solar nebula theory oFive steps to planet formation in Solar nebula theory. 1.Collapse oHeating via conversion of PE to KE. 2.Spinning oSpinning up of material to conserve AM. 3.Flattening oSphere to disk due to rotation. 4.Condensation oGas to liquid and solid particles due to cooling. 5.Accretion oSolid particles ‘stick’ due to electrostatic and gravitational forces. oSee extensive review by Lissauer: (

PY4A01 Solar System Science 1. Collapse - Solar Nebula Theory oFor a cloud with M > M J where M J  (T 3 /  ) 1/2 => cloud collapses. oLocalised density enhancement (  /  ) may cause collapse. oGravity pulls material in, gas pressure opposes. oWhen pressure >> gravity => oscillations. oWhen pressure collapse. oCooling lowers pressure, can trigger collapse.

PY4A01 Solar System Science 1. Collapse - Solar Nebula Theory oConsider N molecules of mass m in a volume of size L at a temperature T. oGravitational energy:where M = Nm ~ L 3 . oThermal energy: E T ~ NkT oRatio is: where L J is the Jeans’ length: oGravity wins when L > L J.

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

PY4A01 Solar System Science 1. Collapse - Solar Nebula Theory oRatio of the collapse time (t G ) to the sound crossing time (t s ) is: where L J is again the Jeans length: oCompare with form given in Jeans formation theory.

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

PY4A01 Solar System Science 1. Collapse - Solar Nebular Theory oInitially, gas and dust with low AM fall to core of cloud. Material with high AM cannot due to centrifugal forces. oAs gas and dust fall to equatorial plane, it collides with material falling in other direction => energy of this motion is dissipated as heat. oConsider 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. oFor M protostar = 1 M sun, v c = 30 km/s at 1 AU => T ~ 7 x 10 4 K. This temperature is never reached as the time-scale for radiaitve cooling << heating time.

PY4A01 Solar System Science 2. Spinning - Solar Nebula Theory oSolar nebula “spins-up” as it collapses to conserve AM. L f = L i m v f r f = m v i r i m r f 2  i = m r i 2  f =>  f =  i (r i / r f ) 2 oAs r i > r f =>  f >>  i. Cloud spins up rapidly as it contract. oNeed a breaking mechanism. oRotation 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 3. Flattening - Solar Nebula Theory og = GM/r 2 is directed radially to centre. oa = r  2 is perpendicular to rotation axis. Radial component is a r = r  2 sin . oNet radial acceleration is oAt pole (  = 0) => a(r) = GM/r 2 oAt equator (  =90) => a(r) = g - r  2 oIn disk, there’s a distance where g = r  2 => this is point where contraction stops. g a a g  arar

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

PY4A01 Solar System Science 3. Flattening - Properties of the disk oIn hydrostatic equilibrium oz-component of gravity is oEquation of state for gas is P = c s 2  oDensity profile is thus: where the scale-height (ie thickness) is:  z

PY4A01 Solar System Science 3. Flattening - Properties of the disk oDisk aspect ratio is defined as: oCondition for disk flaring is: (1 - q ) / 2 > 0 => q < 1 oFor typical disks, (ie q = 1/2) oIn general cases (eg. Galaxies) oNote, disk shape only depends on temperature, not density.

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

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

PY4A01 Solar System Science Step 4: Condensation oFormation 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. oThe process by which seeds were sown is condensation, when solid or liquid particles condense out of a gas. oCondensation is temperature dependent. When the temperature is low enough atoms/molecules solidify.

PY4A01 Solar System Science Step 4: Condensation oApproximate equation for the temperature variation in Solar Nebula is T(r)  631 / r 0.77 where r is in AU. “Ice line” where T = 273 K is located at ~3 AU from Sun. oT < 2,000 K, compounds of silicates (rock) and nickel-iron form. oT < 270 K, carbon compounds, silicates and ices form. oPlanetary interiors to Mars oNebula temperature > 400 K oMade of silicates and metals oPlanets beyond Mars oNebula temperature < 300 K oMade of silicates and ices

PY4A01 Solar System Science Step 4: Condensation oMetals include iron, nickel, aluminum. Most metals condense into solid at temperatures of K. Metals made up <0.2% of the solar nebula's mass. oRocks 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 K depending on type. Rocky materials made up ~0.4% of the nebula by mass. oHydrogen compounds are molecules such as methane (CH 4 ), ammonia (NH 3 ), and water (H 2 O) 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. oLight gases (H and He) never condense under solar nebula conditions. These gases made up the remaining 98% of the nebula's mass. oNote: Order of condensation scales with density.

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

PY4A01 Solar System Science Step 4: Condensation oT~ K at the present-day orbit of Mercury oAbout Mercury metals can begin to aggregate together oFurther out, rocky materials condense. oMost metals/rocks condensing around the present-day orbit of Mars (T~500K). oHence inner planets have high metal/rock content and few volatile materials.

PY4A01 Solar System Science Step 4: Condensation oSize and composition of planetesimals depends on temperature and distance from Sun. oInner solar system oWithin frost line, only rock and metals can condense. oPlanetesimals therefore made of rock and metals. oConstitute ~ 0.6% of available material by mass. oInner planetismals therefore grew more slowly. oInner planets are therefore smaller. oOuter solar system oBeyond frost line, rock, metals and ices condensed. oPlanetesmals therefore contain these materials. oConstitute ~ 2% of available material by mass. oOuter planetismals therefore grew more quickly. oOuter planetesmals are therefore larger. oThese process resulted in elementary planetary cores.

PY4A01 Solar System Science Step 4: Condensation oDensities and distances of objects in solar system supports this condensation theory: oTerrestrial planets: 3-6 g cm -3 => mainly rocks and metals. oJovian planets: 1-2 g cm -3 => more ice and captured gas. oInner Asteroids: contain metalic grains in rocky materials. oOuter Asteroids: less metals, and significantly more ice.

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

PY4A01 Solar System Science Step 5: Accretion oAfter condensation, growth of solid particles occurs due to collisions. oAccretion is growth of grains through collisions - the real planet building process. oLarger particles formed from both tiny chondrules about 1 mm in size, and from porous molecular aggregates held together by Van der Waals forces. oAccretion proceeds in two ways: 1.Collisions due to the geometric cross section - direct impacts on ‘seed’ grain. 2.Collisions due to gravitational attraction - sweeping-up of material from a region much larger than grain diameter.

PY4A01 Solar System Science Step 5: Accretion - geometric oConsider spherical grain of radius r and geometric cross section s = p r 2. If number of grains m -3 is n g, and relative velocity of the grains is v rel then volume (V) swept out in time t is V = s v rel t, (or V = s v rel for 1 second). oThe number of particles (N) encountered in t is N = V n g = s v rel t n g = p r 2 v rel t n g oIn a given period, seed particle’s mass grows as  m/  t = m 0 + N m 0 = m 0 (1 + pr 2 v rel n g ) where m 0 is the grain mass. oMass of the seed particle therefore increases as r 2 for geometrical collisions.  =  r 2 v rel t

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

PY4A01 Solar System Science Step 5: Accretion - gravitational oWhen gravity important, grains accrete from larger volume than during geometric growth phase. oConsider “test” grain with velocity v i at a vertical distance s from a “seed” grain. Suppose “test” grain encounters “seed” grain with a final velocity v f.What is value of s such that the seed grain can capture the “test” grain? oUsing conservation of angular momentum: mv f r = mv i s Eqn. 1 and conservation of energy: where m is mass of “test” grain and M is mass of “seed” grain. oEliminating v f from Eqns. 1 and 2 gives: oAs M ~ r 3 => s 2 ~ r 4. s vivi r vfvf seed grain test grain  Eqn. 2

PY4A01 Solar System Science Step 5: Accretion - gravitational oThe growth rate of the seed particle per unit time is therefore:  m /  t = m 0 (1 + p s 2 v rel n g ) oAs s 2 ~ r 4 =>  m /  t ~ r 4 =>runaway accretion. oOnce grains are large enough that gravity is important, accretion rate increases dramatically. oIf critical size is achieved, a planetesimal will grow rapidly. Less massive objects grow at a much smaller rate. oModel calculation suggest that the first large size objects to form are planetesimals with sizes ~ few tens of km.

PY4A01 Solar System Science 5. Accretion oThese processes result in planetesimals of tens of kilometers in size in less than a million years or so. oTimescale for grows can be calculated from: oNot 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 ~ r 2  m/  t ~ r 4

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

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

PY4A01 Solar System Science 5. Accretion - The planetesimal graveyard oAsteroid belt is ‘resting ground’ for collision- evolved planetesimals that were not incorporated into a planet. oTotal mass of asteroid belt ~5 x kg (which is about 1/3rd the mass of Pluto or 1/15th the mass of the Moon). oCeres the largest asteroid has a diameter of 940 km and a mass of ~10 21 kg. oA planet probably did not form in this region because of the rapid formation, and resulting large mass of Jupiter.

PY4A01 Solar System Science Formation of giants: numerical simulation oThree stages for gas giant formation: 1.Core accretion. 2.Accretion of gas until M gas ~ M core 3.Runnaway accretion of gas => M gas >> M core oSee Pollack et al. (1996). Stage 1 Stage 2 Stage 3

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