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AST 301, Lecture 6 AST 301, Lecture 6 James Lattimer Department of Physics & Astronomy 449 ESS Bldg. Stony Brook University February 7, 2019 Cosmic Catastrophes (AKA Collisions) [email protected] James Lattimer AST 301, Lecture 6 Star Formation • Dense cores of molecular clouds collapse into hot plasma which eventually triggers nuclear reactions. • Conversion of gravitational energy both heats the material and releases infrared radiation. csep10.phys.utk.edu/astr161/lect/solarsys/nebular.html • Dense clumps form in protostellar disc that eventually themselves collapse into planets. • Conservation of angular momentum requires spin rate to increase during collapse and disc formation. • Emissions of protostar hidden by cooler infalling matter; eventually breaks out at poles (least resistance), producing jets. James Lattimer AST 301, Lecture 6 • Sufficiently rapidly rotating collapses form double or multiple stars. • Young star's emissions sweep out B. Reipurth (Univ. of Col.), NASA gas and dust from planetary system. C. R. O'Dell and S. K. Wong (Rice U.), WFPC2, HST, NASA James Lattimer AST 301, Lecture 6 The Jean's Mass The critical mass for gravitational stability against collapse is called the Jean's mass MJ . This can be estimated where the magnitudes of the potential and kinetic energies are equal, so the total energy is 0. Potential energy: Ω = −GM2=R. 2 Internal (kinetic) energy:p U = Mvs =2. Sound speed vs = 2N0kB T with T the temperature. Mass M = 4πρ¯R3=3. v 2 N k T 5v 2 3=2 3 1=2 −Ω = U =) M ≈ s R = 0 B R = s : J 2G J G J 6G 4πρ¯ For T = 100K, vs = 1 km/s. −24 3 4 Forρ ¯ = 10 g/cm , MJ ≈ 2 · 10 M and RJ ≈ 170 pc. Molecular clouds must fragment to form stars. James Lattimer AST 301, Lecture 6 Hydrostatic Equilibrium Fg = dFP dm = ρAdr dP GMr dm = − Ar 2 dP GMr ρ dr = − r 2 James Lattimer AST 301, Lecture 6 The Virial Theorem Equation of Hydrostatic Equilibrium is the balance between pressure force (due to a pressure gradient) and gravity: dP Gmρ dm = − ; = 4πρr 2dr dr r 2 dr Volume V = 4πr 3=3. Gm 1 VdP = − dm = dΩ 3r 3 Z R Z 1 VdP = PV − PdV = Ω 0 3 PV = 0 at r = 0, and also at rZ= R (surface) if P = 0. Thus Ω = −3 PdV : For a perfect non-relativistic gas, P = 2"=3. E is total energy. Z Ω Ω = −2 "dV = −2U; E = Ω + U = = −U 2 Half of Ω goes into heatJames Lattimer (U) andAST half 301, must Lecture 6 be radiated away during collapse. Virial Theorem Applied to an IS Cloud Assume the cloud is bounded by the ISM pressure P0 > 0. Now Virial Theorem is Z M Z M Gm P 3 dm = 3 dm − 4πR P0: 0 r 0 ρ For an isothermal gas, P/ρ = N0kB T and so m / r if P0 = 0. GM2 3N k TM GM2 ' 3N k TM − 4πR3P ; P ' 0 B − : R 0 B 0 0 4πR3 4πR4 When R is small, P0 < 0. When R is large, P0 > 0 but tends to 0 for R ! 1. There must be a maximum of P0, at RJ , which is an instability location. For R < RJ , P0 is lowered and ISM forces collapse. This doesn't happen if R > RJ . A new estimate for MJ is @P 4GM 9N k T 0 = 0 =) R = J ; M = 0 B R : @R J 9N k T J 4G J MJ 0 b James Lattimer AST 301, Lecture 6 Stabilizing Molecular Clouds Clouds exceeding MJ are stabilized by magnetic fields if B2 3GM2 ≥ 8π 4πR4 If B ' 30 µG, p 2 2 M ≤ BR = 6G ' 50(R=pc) M ; so 1M must originate from a region larger than about 0.14 pc. Rotation also provides support. Galactic rotation rate is −16 −1 p 3 2 · 10 s . Since Keplerian frequency is ΩK = GM=R , clouds rotating with the galactic rate are supported if 2 3 3 M ≥ Ω R =G = 0:8(R=pc) M : or R ≤ 0:93 pc for 1M . Thus, we have 0.14 pc < R < 0.93 pc for 1M . James Lattimer AST 301, Lecture 6 Gravitational Collapse Initially, collapse is isothermal because photons freely escape low-density matter. Because MJ / RJ , we have 3=2 p MJ / T = ρ¯ and the Jean's mass decreases as the cloud begins to collapse. It fragments as smaller masses destabilize. p The gravitational free-fall timescale is τff = 3=(8πGρ¯). The gravitational potential energy, half of which is radiated according to the Virial Theorem, is jΩj ≈ GM2=R, giving rise to a luminosity r 5=2 jΩj GM2 8πGρ¯ p M L ≈ ≈ = 2G 3=2 : 2τff 2R 3 R This can't be greater than the blackbody luminosity 2 4 Lbb = 4πR σTeff , giving 8π2R9σ2T 8 1=5 M ≤ eff : G 3 James Lattimer AST 301, Lecture 6 The End of Fragmentation Fragmentation stops when 4GMJ R < RJ = 9N0kB Teff or 81N2k2 9N k T 1=4 M = 0 B 0 B eff ≈ 0:005T 1=4M ≈ 0:03M lim 32G 2π2σ2G 2 eff for Teff ≈ 100 K. This is greater than planetary masses, but smaller than stellar masses which must be at least 0.1{0.2 M . We can estimate the timescale for this stage of stellar collapse: r p r 3 R3 4 2 GM τff = = = 3=2 ≈ 1000 yr 8πGρ¯ 2GM 27 (N0kB Teff ) 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 centrifugal forces become appreciable for large-enough r. Angular momentum conservation implies that the angular velocity 2 Ω = Ω0(r0=r) of a mass element with radius r increases during collapse (0 represents initial values). Radial infall holds p 3 until Ω = ΩK = GM=r (gravity far from the center depends on the total mass M): GM 1=3 Ω2r 4 r = R = = 0 0 ≈ 20 AU disc Ω2 GM −16 −1 for M = 1M ,Ω0 = 2 · 10 s and r0 = 0:33 pc. The formation of a solar system-sized disc is inevitable. A protostar forms at the center with a central density that increases in a runaway slowed when the star becomes optically thick. James Lattimer AST 301, Lecture 6 The Protostar The abrupt halt of gravitational collapse due to radiation trapping unleashes a shock that traverses the infalling matter and ionizes it. The Virial Theorem with complete ionization, so that the internal energy is the ionization energy I , is: Ω GM2 U = − ≈ = I = MN (χ X + χ Y ) 2 2R 0 H He where χH (χHe ) = 15:8(19:8) ev are the ionization potentials and X (Y ) = 3=4(1=4) are the abundances of H(He). Thus I =(MN0) = χ ≈ 17 eV. Thus the initial protostellar radius is GM Rps ≈ ≈ 0:23 AU = 50R 2N0χ for M = 1M . And if the internal energy is ¯ U = 3MN0KB T =(2µ), where µ ≈ 0:6 is the mean molecular weight, we find its mean temperature to be T¯ ≈ 9 · 104 K. James Lattimer AST 301, Lecture 6 Protostar Evolution Although the mean temperature is of order 105 K, and rises as the protostar contracts, the surface temperature cannot increase beyond about 3500 K, the ionization temperature for H. This is because the opacity of matter increases abruptly at this temperature (compare to the era of last scattering during the Big Bang). The protostar contracts slowly compared to free-fall: r U GM2 3GM τcont ≈ − ≈ 4 4 = 1800 yr; τff = 3 ≈ 0:01 yr: L 4πσR Teff 8πR The contraction rate slows as collapse continues. Mass continues to accrete onto the protostar during this stage, perhaps doubling its mass. Contraction finally ends when central temperatures greater than 106 K are reached and deuterium nuclear reactions ignite. James Lattimer AST 301, Lecture 6 Star Formation in the Hertzsprung-Russel Diagram James Lattimer AST 301, Lecture 6 Star Forming Regions M42 C.R. O'Dell (Rice U.), NASA NGC 604 M8 H. Yang (UIUC) & J. Hester (ASU), NASA James Lattimer AST 301, Lecture 6 Solar System Formation { History • Georges Buffon suggested (1745) that the planets formed when a massive object (a comet, he thought) collided with the Sun 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 solids and 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 in rotation rate and flattening. Thermodynamic laws result in heating 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 helps form planetismals and planets. Gases retained on outer planets. • Clearing: Solar heating repels gaseous matter that is not bound to planets. Most dust is incorporated into planets. • Comets can be ejected far beyond planets by gravitational encounters with planets, which also causes planetary migrations.
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