PY4A01 Solar System Science Lecture 5-6 - Solar system formation theories oTopics to be covered:...

PY4A01 Solar System Science

Lecture 5-6 - Solar system formation theoriesLecture 5-6 - Solar system formation theories

o Topics to be covered:

o Laplace nebula theory

o Jeans’ tidal theory

o Solar nebula theory

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Planet formation modelsPlanet formation models

o Three basic models have been proposed:

o Tidal theory: Planets formed from condensed gasses ‘ripped’ from an all ready formed Sun.

o Capture theory: During a close stellar encounter, Sun captures material out of which planets form.

o Nebula theory: Planets formed at the same time as the Sun in the same gas cloud.


Laplace nebula theoryLaplace nebula theory

o Proposed by Pierre Laplace in Exposition du Système du Monde (1796).

o Consists 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.


Laplace nebula theory: Difficulties Laplace nebula theory: Difficulties

1. Strongest criticism related to the distribution of AM.

o There is no mechanism for the partitioning of mass and AM.

o Mass 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.

o Gravitational attraction between objects in the rings would not be sufficient to overcome inertial forces.

o Rings would require much more mass than the planets they formed to overcome this effect.


Jeans’ tidal theoryJeans’ tidal theory

o Laplace theory was a monistic theory - same body of material in a single process gave rise to both the Sun and the planets.

o James Jeans (1917) proposed a dualistic theory that separated formation of Sun from formation of planets.

o Jeans’ 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.


Roche LimitRoche Limit

o Roche limit is distance at which a satellite begins to be tidally torn apart.

o Consider 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 Ftidal ≥ Fbinding

o The binding force is:

o Force of attraction between mass M and nearer satellite is: GMm/(R - r)2. Force on more distant satellite is GMm/(R + r)2

o Tidal force experienced is thus,

€

Fbinding =Gm2

(2r)2

2r

R

R-r R+r

M mm

€

Ftidal = GMm1

(R − r)2 −1

(R + r)2

⎛

⎝ ⎜

⎞

⎠ ⎟

=GMm

R2(1− r /R)−2 − (1+ r /R)−2

( )

=GMm

R2((1+ 2r /R) − (1− 2r /R)) => Ftidal =

4GMmr

R3

Eqn. 4

Eqn. 5


Roche Limit (cont.)Roche Limit (cont.)

o We can therefore rewrite the inequality Ftidal ≥ Fbinding as:

o Rearranging then gives

o As M = 4/3 R3M and m = 4/3 r3m:

o Objects which pass within this are torn apart.

o The Earth's Roche limit is ~18,470 km.

€

4GMmr

R3≥

Gm2

4r2

€

RRoche =16Mr3

m

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

€

RRoche = 2.520Rρ M

ρ m

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

Approximate Roche Limit


Jeans’ tidal theory: Proto-planet formationJeans’ tidal theory: Proto-planet formation

o Jeans 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’.

1. Jeans 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

A B C D

BB’

B’C’D

€

l =π

γGρ


Jeans’ tidal theory: DifficultiesJeans’ tidal theory: Difficulties

1. Very massive stars are rare and distant.

o Probability of massive star coming close to another star is therefore very low.

o Sun’s nearest companion is Proxima Centauri (d =1.3 pc => Rsun=/d ~2x10-8).

2. Rotational period of Sun and Jupiter should be similar if Jupiter’s material was from Sun.

o Not the case (Psun ~ 26 days and Pjupiter ~ 10 hours).

o 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.

o Spitzer (1939) noted that material with solar densities and temperatures would give a minimum mass for collapse of ~100 times that of Jupiter.


Capture theoryCapture theory

o Modified version of Jeans’ theory, proposed in 1964 by M. Woolfson.

o Sun interacts with nearby protostar, dragging filament from protostar.

o Low rotation speed of Sun is explained as due to formation before planets.

o Terrestrial planets due to collisions between protoplanets close to Sun and giant planets.

o Planetary satellites due to condensation in drawn out filaments.

Sun

Proto-star moves ona hyperbolic orbit

Material captured by Sun

Depleted proto-star

Tidally distortedproto-star


Capture theory: DifficultiesCapture theory: Difficulties

o Space between local stars too large for 9 planets and 60 moons to be caught by Sun.

o Millions would have to pass, in order for one to be caught.

o Planets would tend to spiral into Sun, not begin encircling it.

o Moons would not begin orbiting around planets; they would crash into Sun or into planets.

o Cannot explain why Sun and Planets have the same apparent age (4.5 Gyrs).


Solar nebula theorySolar nebula theory

o Cloud 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.

o Once started, would continue to collapsing under Newton’s Laws.o F ~ 1/r^2 => Cloud “shrinks” by factor of 2, force increases by factor of 4.

o As collapse continues, temperature, density and rotation rate increase rapidly.

o Proceeds along five steps:1. Heating2. Spinning3. Flattening4. Condensation5. Accretion


1. Heating - Solar Nebula Theory 1. Heating - Solar Nebula Theory

o As gas cloud collapses, temperatures rise as potential energy converted to kinetic via:

E = KE + U = const

o From Conservation of Energy, KE increases as U decreases.

o Temperature therefore rises as 1/2mv2 = 3/2 kT or T = 1/3k (1/2 m v2).

o Some energy is radiated away thermally. The solar nebula becomes hottest near its center, where much of the mass collect to form the protosun.

o Protosun eventually becomes so hot that nuclear fusion ignited in its core.

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2. Spinning - Solar Nebula Theory2. Spinning - Solar Nebula Theory

o Solar nebula “spins-up” as it collapses to conserve AM.

Lf = Lf

m vf rf = m vi ri

m rf2 i = m ri

2 f

=> f = i (ri / rf )2

o Cloud therefore spins up rapidly as it contract.

o Rotation 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.




3. Flattening - Solar Nebula 3. Flattening - Solar Nebula TheoryTheory

o g = GM/r2 is directed radially to centre.

o a = r2 is perpendicular to rotation axis.

Radial component is ar = r2 sin.

o Net radial acceleration is

o At pole ( = 0) => a(r) = GM/r2

o At equator ( =90) => a(r) = g - r2

o In disk, there’s a distance where g = r2=> this is point where contraction stops.

ga

ag

€

a(r) = g + ar

a(r) =GM

r2− rω2 sinφ

ar


3. Flattening - Solar Nebula Theory3. Flattening - Solar Nebula Theory

o Collisions further flatten the disk.

o Gas moves in random directions at random speeds. Different clumps collide and merge, giving new clumps the average of their differing velocities.

o Original cloud thus become more orderly as cloud collapses, changing the cloud's original lumpy shape into a rotating, flattened disk.

o Similarly, collisions between clumps of material in highly elliptical orbits reduce their ellipticities, making their orbits more circular.


Step 4: CondensationStep 4: Condensation

o 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.

o The process by which seeds were sown is condensation, when solid or liquid particles condense out of a gas.

o Condensation is temperature dependent. When the temperature is low enough atoms/molecules solidify.





o Approximate equation for the temperature variation in Solar Nebula isT(R) 631 / R0.77

where R is in AU. “Ice line” where T = 273 K is located at ~3 AU from Sun.

o T < 2,000 K, compounds of silicates (rock) and nickel-iron form.o T < 270 K, carbon compounds, silicates and

ices form.

o Planetary interiors to Mars o Nebula temperature > 400 Ko Made of silicates and metals

o Planets beyond Mars o Nebula temperature < 300 Ko Made of silicates and ices



o 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.

o 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.

o 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.

o Light gases (H and He) never condense under solar nebula conditions. These gases made up the remaining 98% of the nebula's mass.

o Note: Order of condensation scales with density.



o Terrestrial planets are made from materials that constituted ~0.6% of the nebula. o Jovian planets were formed in region where ~2% of material condensed. They also captured gas (98%).



o T~1500-2000K at the present-day orbit of Mercury

o About Mercury metals can begin to aggregate together

o Further out, rocky materials condense.

o Most metals/rocks condensing around the present-day orbit of Mars (T~500K).

o Hence inner planets have high metal/rock content and few volatile materials.



o Size and composition of planetesimals depends on temperature and distance from Sun.

o Inner solar systemo Within frost line, only rock and metals can

condense.o Planetesimals therefore made of rock and metals.o Constitute ~ 0.6% of available material by mass.o Inner planetismals therefore grew more slowly.o Inner planets are therefore smaller.

o Outer solar systemo Beyond frost line, rock, metals and ices condensed.o Planetesmals therefore contain these materials.o Constitute ~ 2% of available material by mass.o Outer planetismals therefore grew more quickly.o Outer planetesmals are therefore larger.

o These process resulted in elementary planetary cores.



o Densities and distances of objects in solar system supports this condensation theory:

o Terrestrial planets: 3-6 g cm-3

=> mainly rocks and metals.

o Jovian planets: 1-2 g cm-3

=> more ice and captured gas.

o Inner Asteroids: contain metalic grains in

rocky materials.

o Outer Asteroids: less metals, and

significantly more ice.


Step 5: AccretionStep 5: Accretion

o After condensation, growth of solid particles occurs due to collisions.

o Accretion is growth of grains through collisions - the real planet building process.

o Larger particles formed from both tiny chondrules about 1 mm in size, and from porous molecular aggregates held together by Van der Waals forces.

o Accretion 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.


Step 5: Accretion - geometricStep 5: Accretion - geometric

o Consider spherical grain of radius r and geometric cross section = r2. If number of grains m-3 is ng, and relative velocity of the grains is vrel then volume (V) swept out in time t is V = vrel t, (or V = vrel for 1 second).

o The number of particles (N) encountered in t is N = V ng

= vrel t ng = r2 vrel t ng

o In a given period, seed particle’s mass grows as

m/t = m0 + N m0 = m0 (1 + r2 vrel ng)

where m0 is the grain mass.

o Mass of the seed particle therefore increases as r2 for geometrical collisions.

= r2

vrelt


Step 5: Geometric and gravitational accretionStep 5: Geometric and gravitational accretion

o Objects formed by geometric accretion are called planetesimals: act as seeds for planet formation.

o At first, planetesimals were closely packed.

o Then coalesced into larger objects, forming clumps few km across in few million years.

o Once planetesimals had grown to few km, collisions became destructive, making further growth more difficult.

o Gravitational accretion then begins to dominate. This then accretes planetesimals to form protoplanets.

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Step 5: Accretion - gravitationalStep 5: Accretion - gravitational

o When gravity important, grains accrete from larger volume than during geometric growth phase.

o 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?

o Using conservation of angular momentum: mvf r = mvi s Eqn. 1

and conservation of energy:

where m is mass of “test” grain and M is mass of “seed” grain.

o Eliminating vf from Eqns. 1 and 2 gives:

o As M ~ r3 => s2 ~ r4.

s

vi

rvf

seed grain

testgrain

€

1/2mv i2 =1/2mv f

2 −GMm

rEqn. 2

€

s2 = r2 +2GMr

v i2


Step 5: Accretion - gravitationalStep 5: Accretion - gravitational

o The growth rate of the seed particle per unit time is therefore:

m / t = m0 (1 + s2 vrel ng)

o As s2 ~ r4 => m / t ~ r4 =>runaway accretion.

o Once grains are large enough that gravity is important, accretion rate increases dramatically.

o If critical size is achieved, a planetesimal will grow rapidly. Less massive objects grow at a much smaller rate.

o Model calculation suggest that the first large size objects to form are planetesimals with sizes ~ few tens of km.


5. Accretion5. Accretion

o These processes result in planetesimals of tens of kilometers in size in less than a million years or so.

o The bigger an object, the more able gravity is to attract and accrete nearby objects.

o Not 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.

o So typically one object will dominate a region.

m/t ~ r2

m/t ~ r4


5. Accretion - planet formation5. Accretion - planet formation

o Accretion therefore progresses according to:

o Once planetesimals are formed, the following can occur:

o The final stages in the growth of a Terrestrial planet are dramatic and violent.

o 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


o Inner Planetso Formed slowly due to small amount of

metals and rocks in early solar nebula. o Geometric accretion rate and

gravitational accretion rate small.o By time inner planetesimals were formed

and had significant gravitational fields, the nebula had been cleared out by the solar wind.

o Then no nebular gas then present to capture an elementary atmosphere.

o Outer Planetso Formed less violently.o Great quantities of ice at >3 AU resulted

in large rock/ice cores forming. o Reason for rapid core growth is that ices

have large cross-sectional area.

5. Accretion - the planets5. Accretion - the planets


5. Accretion - The planetesimal graveyard5. Accretion - The planetesimal graveyard

o Asteroid belt is ‘resting ground’ for collision-evolved planetesimals that were not incorporated into a planet.

o 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).

o Ceres the largest asteroid has a diameter of 940 km and a mass of ~1021 kg.

o A planet probably did not form in this region because of the rapid formation, and resulting large mass of Jupiter.

PY4A01 Solar System Science Lecture 5-6 - Solar system formation theories oTopics to be covered:...

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