ast 301, lecture 6 - stony brook university · james lattimer ast 301, lecture 6 the peak \late...

$: AST 301, Lecture 6 - Stony Brook University · James Lattimer AST 301, Lecture 6 The peak \Late Heavy Bombardment" was from about 4.0-3.8 Gyrs ago. The Earth’s high temperature$
AST 301, Lecture 6

James Lattimer

Department of Physics & Astronomy449 ESS Bldg.

Stony Brook University

February 13, 2015

Cosmic Catastrophes (AKA Collisions)[email protected]

James Lattimer AST 301, Lecture 6

Star Formation

• Dense cores of molecularclouds collapse into hotplasma which eventuallytriggers nuclear reactions.

• Conversion of gravitationalenergy both heats thematerial and releases infraredradiation. csep10.phys.utk.edu/astr161/lect/solarsys/nebular.html

• Dense clumps form in protostellar disc that eventuallythemselves collapse into planets.

• Conservation of angular momentum requires spin rate toincrease during collapse and disc formation.

• Emissions of protostar hidden by cooler infalling matter;eventually breaks out at poles (least resistance), producingjets.


• Sufficiently rapidlyrotating collapsesform double ormultiple stars.

• Young star’semissions sweep outgas and dust fromplanetary system.

C. R. O’Dell and S. K. Wong (Rice U.), WFPC2, HST, NASA

B. Reipurth (Univ. of Col.), NASA


The Jean’s Mass

The critical mass for gravitational stability against collapse iscalled the Jean’s mass MJ . This can be estimated where themagnitudes of the potential and kinetic energies are equal, sothe total energy is 0.Potential energy: Ω = −GM2/R .Internal (kinetic) energy: U = Mv 2

s /2Sound speed vs =

√2N0kBT with T the temperature.

Mass M = 4πρR3/3.

−Ω = U =⇒ MJ ≈v 22

2GRJ =

(5v 2

s

6G

)3/2(3

4πρ

)1/2

.

For T = 100K , vs = 1 km/s.For ρ = 10−24 g/cm3, MJ ≈ 2 · 104M and RJ ≈ 170 pc.Molecular clouds must fragment to form stars.


The Virial TheoremEquation of Hydrostatic Equilibrium is the balance betweenpressure force (due to a pressure gradient) and gravity:

dP

dr= −Gmρ

r 2,

dm

dr= 3πρr 2dr

Volume V = 4πr 3/3.

VdP =−1

3

Gm

rdm =

1

3dΩ∫

VdP = PV

∣∣∣∣R0

−∫

PdV =1

3Ω

PV = 0 at r = 0, and also at r = R (surface) if P = 0. Thus

Ω = −3

∫PdV .

For a perfect non-relativistic gas, P = 2ε/3. E is total energy.

Ω = −2

∫εdV = −2U , E = Ω + U =

Ω

2= −U

Half of Ω goes into heat (U) and half must be radiated awayduring collapse.


Virial Theorem Applied to an IS Cloud

Assume the cloud is bounded by the ISM pressure P0 > 0.Now Virial Theorem is∫ M

0

Gm

rdm = 3

∫ M

0

P

ρdm − 4πR3P0.

For an isothermal gas, P/ρ = N0kBT and so m ∝ r if P0 = 0.

GM2

R' 3N0kBTM − 4πR3P0, P0 '

3N0kBTM

4πR3− GM2

4πR4.

When R is small, P0 < 0. When R is large, P0 > 0 but tendsto 0 for R →∞.There must be a maximum of P0, at RJ , which is an instabilitylocation. For R < RJ , P0 is lowered and ISM forces collapse.This doesn’t happen if R > RJ . A new estimate for MJ is(

∂P0

∂R

)MJ

= 0 =⇒ RJ =4GMJ

9N0kbT, MJ =

9N0kBT

4GRJ .


Stabilizing Molecular Clouds

Clouds exceeding MJ are stabilized by magnetic fields if

B2

8π≥ 3GM2

4πR4

If B ' 30 µG,

M ≤ BR2/√

6G ' 50(R/pc)2M,

so 1M must originate from a region larger than about 0.14pc.

Rotation also provides support. Galactic rotation rate is2 · 10−16 s−1. Since Keplerian frequency is ΩK =

√GM/R3,

clouds rotating with the galactic rate are supported if

M ≥ Ω2R3/G = 0.8(R/pc)3M.

or R ≤ 0.93 pc for 1M.Thus, we have 0.14 pc < R < 0.93 pc for 1M.


Gravitational Collapse

Initially, collapse is isothermal because photons freely escapelow-density matter. Because MJ ∝ RJ , we haveMJ ∝ T 3/2/

√ρ and the Jean’s mass decreases as the cloud

begins to collapse. It fragments as smaller masses destabilize.

The gravitational free-fall timescale is τff =√

3/(8πG ρ). Thegravitational potential energy, half of which is radiatedaccording to the Virial Theorem, is |Ω| ≈ GM2/R , giving riseto a luminosity

L ≈ |Ω|2τff≈ GM2

2R

√8πG ρ

3=√

2G 3/2

(M

R

)5/2

.

This can’t be greater than the blackbody luminosityLbb = 4πR2σT 4

eff , giving

M ≤(

8π2R9σ2T 8eff

G 3

)1/5

.


The End of Fragmentation

Fragmentation stops when

R < RJ =4GMJ

9N0kBTeff

or

Mlim =81N2

0k2B

32G

(9N0kBTeff

2π2σ2G 2

)1/4

≈ 0.005T1/4eff M ≈ 0.03M

for Teff ≈ 100 K. This is greater than planetary masses, butsmaller than stellar masses which must be at least 0.1–0.2 M.

We can estimate the timescale for this stage of stellar collapse:

τff =

√3

8πG ρ=

√R3

2GM=

4√

2

27

GM

(N0kBTeff )3/2≈ 1000 yr

for M = 1M and Teff = 100 K.James Lattimer AST 301, Lecture 6

Planetary Disc Formation

Infall is radial near the spin axis of the cloud, but centrifugalforces become appreciable for large-enough r . Angularmomentum conservation implies that the angular velocityΩ = Ω0(r0/r)2 of a mass element with radius r increasesduring collapse (0 represents initial values). Radial infall holdsuntil Ω = ΩK =

√GM/r 3 (gravity far from the center

depends on the total mass M):

r = Rdisc =

(GM

Ω2

)1/3

=Ω2

0r40

GM≈ 20 AU

for M = 1M, Ω0 = 2 · 10−16 s−1 and r0 = 0.33 pc. Theformation of a solar system-sized disc is inevitable. A protostarforms at the center with a central density that increases in arunaway slowed when the star becomes optically thick.


The Protostar

The abrupt halt of gravitational collapse due to radiationtrapping unleashes a shock that traverses the infalling matterand ionizes it. The Virial Theorem with complete ionization,so that the internal energy is the ionization energy I , is:

U = −Ω

2≈ GM2

2R= I = MN0(χHX + χHeY )

where χH(χHe) = 15.8(19.8) ev are the ionization potentialsand X (Y ) = 3/4(1/4) are the abundances of H(He). ThusI/(MN0) = χ ≈ 17 eV. Thus the initial protostellar radius is

Rps ≈GM

2N0χ≈ 0.23 AU = 50R

for M = 1M. And if the internal energy isU = 3MN0KBT/(2µ), where µ ≈ 0.6 is the mean molecularweight, we find its mean temperature to be T ≈ 9 · 104 K.


Protostar Evolution

Although the mean temperature is of order 105 K, and rises asthe protostar contracts, the surface temperature cannotincrease beyond about 3500 K, the ionization temperature forH. This is because the opacity of matter increases abruptly atthis temperature (compare to the era of last scattering duringthe Big Bang).

The protostar contracts slowly compared to free-fall:

τcont ≈ −U

L≈ GM2

4πσR4T 4eff

= 1800 yr, τff =

√3GM

8πR3≈ 0.01 yr.

The contraction rate slows as collapse continues. Masscontinues to accrete onto the protostar during this stage,perhaps doubling its mass.

Contraction finally ends when central temperatures greaterthan 106 K are reached and deuterium nuclear reactions ignite.


Star Formation in the Hertzsprung-Russel Diagram


Star Forming Regions

H.Yang

(UIU

C)

&J.

Hester

(ASU

),N

ASA

C.R. O’Dell (Rice U.), NASA

NGC 604

M42

M8


Solar System Formation – History• Georges Buffon suggested (1745) that the planets formed

when a massive object (a comet, he thought) collided with theSun and splashed out debris that coalesced into planets.• Immanuel Kant proposed (1755) that the solar system formed

from the gravitational collapse of an interstellar gas cloud.• Pierre-Simon Laplace (1790) independently proposed Kant’s

idea and suggested planets formed in successive rings of gas.• Kant and Laplace idea is called the nebular hypothesis.• Laplace’s ring idea was eventually replaced by condensation of

solids into planetismals and gravitational accretion of solidsand gases into planets.• Modified nebular hypothesis agrees well with observations of

compositions of different objects in the solar system.• Chance close encounters are too rare.• New observations of extra-solar planets are forcing

modifications to explain surprising planetary orbits.James Lattimer AST 301, Lecture 6

Solar System Formation – Major Stages• Contraction: of a diffuse interstellar gas cloud due to gravity.

• Conservation of angular momentum: leads to increase inrotation rate and flattening. Thermodynamic laws result inheating as the central density rises.

• Condensation: Heavier elements condense into solid dust grains– metals and rocks (inner), ices like H2O and CO2 (outer).

• Accretion: Dust grains collide and stick.

• Continued accretion leads to formation of rocks; gravity helpsform planetismals and planets. Gases retained on outer planets.

• Clearing: Solar heating repels gaseous matter that is not boundto planets. Most dust is incorporated into planets.

• Comets can be ejected far beyond planets by gravitationalencounters with planets, which also causes planetary migrations.Asteroids are either material which never formed a planet(Jupiter’s influence?), or debris from collisions of planetismals.


Addiso

n-W

esley


Addison-W

esley


Nick

D.K

im,nearin

gzero

.net

Addison-Wesley

Cartoons in Science


Planetismals Form

www.astro.virginia.edu/class/skrutskie/astr121/notes/sscond.html

HH-30


Solar System Formation – Timeline


NASA/JPL-Caltech/R. Hurt


Wikipedia


Kuiper Belt

Wikipedia

Green = Kuiper belt objectOrange = Scattered disc object or CentaurMagenta = Trojan of JupiterYellow = Trojan of Neptune

Eris


Oort Cloud

herschel.jpl.nasa.gov/solarSystem.shtml


Formation of the Earth

• Earth formed 4.5 Gyrs ago.

• From a small accumulationof gas and dust, it grew byaccretion and bombardmentof rocky material.

• Impacts and radioactiveenergy release kept the earlyEarth very hot and molten.

• A large impact about 4.4 Gyrago resulted in the formationof the Moon.

The moon is the largest natural satellite of the Earth; theother moons being Cruithne , asteroid 2003 YN107, 1998 UP1and 2000 PH5.


• The peak “Late HeavyBombardment” was fromabout 4.0-3.8 Gyrs ago.

• The Earth’s high temperaturepromoted differentiation,forming the core and mantleas heavier elements (Fe, Ni,Co, Mn and S) sank andlighter elements (Si, Mg andO) floated.

www.physci.wsc.ma.edu/young/hgeol/geoinfo/timeline/formation/formation.html

6400 kmconve

ctiv

e

Lithosphere

• The molten character of the Earth destroyed most traces, butsome 4.4 Gyr-old zircons have survived.

• The Earth’s crust solidified and oceans formed about 4 Gyrago.

• Life seems to have formed on the Earth about the same time.


ast 301, lecture 6 - stony brook university · james lattimer ast 301, lecture 6 the peak \late...

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