1. PY4A01 Solar System Science Lecture 5-6 - Solar system formation theories oTopics to be covered: oLaplace nebula theory oJeans’ tidal theory oSolar nebula theory

2. PY4A01 Solar System Science Planet formation models oThree basic models have been proposed: oTidal theory: Planets formed from condensed gasses ‘ripped’ from an all ready formed Sun. oCapture theory: During a close stellar encounter, Sun captures material out of which planets form. oNebula theory: Planets formed at the same time as the Sun in the same gas cloud.

3. PY4A01 Solar System Science Laplace nebula theory oProposed by Pierre Laplace in Exposition du Système du Monde (1796). oConsists of 5 stages: 1.Slowly rotating, collapsing gas and dust sphere. 2.An oblate spheroid, flattened along the spin axis. 3.The critical lenticular form - material in equatorial region is in free orbit. 4.Rings left behind in equatorial plane due to further collapse. “Spasmodic” process leads to annular rings. 5.One planet condenses in each ring with Sun at centre. 1.2. 3.4.5.

4. PY4A01 Solar System Science Laplace nebula theory: Difficulties 1.Strongest criticism related to the distribution of AM. oThere is no mechanism for the partitioning of mass and AM. oMass and AM concentrated on the central star. 2.While still a student at Cambridge, Maxwell suggested that differential rotation between inner and outer parts of rings would prevent material from condensing. oGravitational attraction between objects in the rings would not be sufficient to overcome inertial forces. oRings would require much more mass than the planets they formed to overcome this effect.

5. PY4A01 Solar System Science Jeans’ tidal theory oLaplace theory was a monistic theory - same body of material in a single process gave rise to both the Sun and the planets. oJames Jeans (1917) proposed a dualistic theory that separated formation of Sun from formation of planets. oJeans’ Theory involved interaction between Sun and a very massive star in three stages: 1.Massive star passes within Roche Limit of Sun, pulling out material in the form of a filament. 2.Filament is gravitationally unstable, and breaks into series of blobs of masses greater than the Jeans’ critical mass, and so collapse to form proto-planets. 3.Planets were left in orbit about the Sun. 1. 2. 3.

6. PY4A01 Solar System Science Roche Limit oRoche limit is distance at which a satellite begins to be tidally torn apart. oConsider M with 2 satellites of mass m and radius r orbiting at distance R. Roche limit is reached when m is more attracted to M then to m. Occurs when F tidal ≥ F binding oThe binding force is: oForce of attraction between mass M and nearer satellite is: GMm/(R - r) 2. Force on more distant satellite is GMm/(R + r) 2 oTidal force experienced is thus, 2r R R-r R+r M mm Eqn. 4 Eqn. 5

7. PY4A01 Solar System Science Roche Limit (cont.) oWe can therefore rewrite the inequality F tidal ≥ F binding as: oRearranging then gives oAs M = 4/3  R 3  M and m = 4/3  r 3  m : oObjects which pass within this are torn apart. oThe Earth's Roche limit is ~18,470 km. Approximate Roche Limit

8. PY4A01 Solar System Science Jeans’ tidal theory: Proto-planet formation oJeans showed that filament would be unstable, and break into series of proto-planets. 1.Small density excess at A. 2.Gravitational attraction causes material in B and B’ to move towards A. 3.Material at C and C’ now experience an outward force and produce high-density regions at D and D’. oJeans treated as a wave-like problem, finding average distance between proto-planets to be: where  is ratio of the specific heats and l is effectively the wavelength. A A ABCD BB’ C’D

9. PY4A01 Solar System Science Jeans’ tidal theory: Difficulties 1.Very massive stars are rare and distant. oProbability of massive star coming close to another star is therefore very low. oSun’s nearest companion is Proxima Centauri (d =1.3 pc => R sun= /d ~2x10 -8 ). 2.Rotational period of Sun and Jupiter should be similar if Jupiter’s material was from Sun. oNot the case (P sun ~ 26 days and P jupiter ~ 10 hours). 3.In 1935, Henry Russell argued that it is not possible for the material from the Sun to acquire enough AM to explain Mercury, let alone the other planets. 4.Spitzer (1939) noted that material with solar densities and temperatures would give a minimum mass for collapse of ~100 times that of Jupiter.

10. PY4A01 Solar System Science Capture theory oModified version of Jeans’ theory, proposed in 1964 by M. Woolfson. oSun interacts with nearby protostar, dragging filament from protostar. oLow rotation speed of Sun is explained as due to formation before planets. oTerrestrial planets due to collisions between protoplanets close to Sun and giant planets. oPlanetary satellites due to condensation in drawn out filaments. Sun Proto-star moves on a hyperbolic orbit Material captured by Sun Depleted proto-star Tidally distorted proto-star

11. PY4A01 Solar System Science Capture theory: Difficulties oSpace between local stars too large for 9 planets and 60 moons to be caught by Sun. oMillions would have to pass, in order for one to be caught. oPlanets would tend to spiral into Sun, not begin encircling it. oMoons would not begin orbiting around planets; they would crash into Sun or into planets. oCannot explain why Sun and Planets have the same apparent age (4.5 Gyrs).

12. PY4A01 Solar System Science Solar nebula theory oCloud collapses according to the Jeans’ criterion. Collapse may be triggered by cataclysmic event such as shock wave from a supernova, or a spiral density wave. oOnce started, would continue to collapsing under Newton’s Laws. oF ~ 1/r^2 => Cloud “shrinks” by factor of 2, force increases by factor of 4. oAs collapse continues, temperature, density and rotation rate increase rapidly. oProceeds along five steps: 1.Heating 2.Spinning 3.Flattening 4.Condensation 5.Accretion

13. PY4A01 Solar System Science 1. Heating - 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.

14. PY4A01 Solar System Science 2. Spinning - Solar Nebula Theory oSolar nebula “spins-up” as it collapses to conserve AM. L f = L f 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 oCloud therefore spins up rapidly as it contract. oRotation also 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.

15. 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

16. PY4A01 Solar System Science 3. Flattening - Solar Nebula Theory oCollisions further flatten the disk. oGas moves in random directions at random speeds. Different clumps collide and merge, giving new clumps the average of their differing velocities. oOriginal cloud thus become more orderly as cloud collapses, changing the cloud's original lumpy shape into a rotating, flattened disk. oSimilarly, collisions between clumps of material in highly elliptical orbits reduce their ellipticities, making their orbits more circular.

17. 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.

18. 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

19. PY4A01 Solar System Science Step 4: Condensation oMetals 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. 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 500-1300 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.

20. 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%).

21. PY4A01 Solar System Science Step 4: Condensation oT~1500-2000K 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.

22. 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.

23. 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.

24. 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.

25. PY4A01 Solar System Science Step 5: Accretion - geometric oConsider spherical grain of radius r and geometric cross section  =  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 =  v rel t, (or V =  v rel for 1 second). oThe number of particles (N) encountered in t is N = V n g =  v rel t n g =  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 +  r 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

26. 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.

27. 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

28. 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 +  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.

29. 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. oThe bigger an object, the more able gravity is to attract and accrete nearby objects. oNot only will the rich get richer (i.e., the biggest planetesimal will grow the fastest), but the smaller planetesimals are quickly destroyed by fast collisions and turned into smaller fragments. oSo typically one object will dominate a region.  m/  t ~ r 2  m/  t ~ r 4

30. 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

31. 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

32. 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 10 21 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.