Protostars 1. First core and main accretion phase Stahler & Palla: Chapter 11.1 & 8.4.1 & Appendices F & G Early growth and collapse In a magnetized cloud undergoing contraction, the density gradually increases via ambipolar diffusion until the central Σ / B has surpassed the critical value. 1 2 The contracting deep interior effectively separates from the more slowly evolving outer portion of the cloud.
The structure that arises from the contraction is not yet a protostar but a temporary configuration known as the first core. To describe its growth and rapid demise, let s neglect (for now) the important element of rotation and magnetic support. The isothermal approximation breaks down! As its density climbs, the central lump becomes opaque to its own cooling radiation, and further compression causes its internal temperature to rise steadily. The enhanced pressure decelerates material drifting inward, which gently settles onto the hydrostatic structure. The settling gas radiates, removing energy from the outer skin and further enhancing compression. The core eventually stops expanding and begins to shrink. M ~ 5x10-2 M!, R ~ 5 AU " ρ ~ 10-10 g cm -3 3 The first core mostly consists of H 2 molecules and this ensures its early collapse 2T + 2U + W +M = 0 The object builds up from the least rotationally and magnetically supported portion of the cloud, so we can ignore these terms for now in the virial equilibrium equation. T µ 3R W GM 2 R ; U = 3 2 RT µ M GM R = 850 K % M (% R ( ' *' * & 5 10 2 M sun )& 5 AU) 1
With the addition of mass and shrinking of the radius, T soon surpasses 2000 K and collisional dissociation of H 2 begins " T starts to level off: the number of H 2 molecules in the core is XM/2m H (X = 0.70); the thermal energy per molecule is 3k B T/X = 0.74 ev (< 4.48 ev) @ T=2000 K. During the transition epoch, even a modest rise in the fraction of dissociated hydrogen absorbs most of the compressional work of gravity, without a large increase in temperature. H 2 dissociation begins As the density of the first core keeps climbing (whereas the T rise is damped by the dissociation process), the region containing atomic H spreads outwards from the center and increase the mass until the entire configuration becomes unstable and collapses: recall the isothermal Bonnor-Ebert sphere becomes unstable when the center to edge density ratio is ~14. This marks the end of the first core. The collapse of the partially dissociated gas takes the central region to much higher density and temperature " collisional ionization of the hydrogen. The true protostar is born. With a radius of several R!, a protostar of 0.1 M! has T > 10 5 K and density ~10-2 g cm -3. The gas approaching the protostellar surface now travels at free-fall velocities >> local sound speed. The steady rise in the protostellar mass gradually inflates this supersonic infall region and the cloud collapse proceeds inside-out " main accretion phase. Accretion Luminosity L acc GM * M $ M ' $ = 61 L sun & )& R * % 10 5 M sun yr -1 (% M * 1M sun ' $ )& (% R * 5R sun This is the energy per unit time released by infalling gas that converts all its kinetic energy into radiation as it lands on the stellar surface. Throughout the main accretion phase, L acc is very nearly equal to L rad, the average luminosity escaping. ' ) ( 1 6
The accretion luminosity is mostly generated close to the protostar s surface. The contribution from additional radiated energy (nuclear fusion + quasi-static contraction) are typically minor compared to L acc. The protostar is gaining mass with its luminosity coming mainly from external accretion. This radiation is able to escape the cloud because it is gradually degraded into the infrared regime as it travels outward. Observationally, then, protostars are optically invisible objects that should appear as compact sources at longer wavelengths. Most of the radiation is generated at the accretion shock, which constitutes the protostar s outer boundary (matter further inside is settling with relatively low velocity). 7 The gas raining down on the protostar originates in the outer envelope, where gas cooling is efficient. As the infalling gas continues to be compressed, the radiation eventually becomes trapped. The temperature quickly rises inside the dust photosphere (R phot ~ 10 14 cm), the effective radiative surface of the protostar. The dust envelope is the region bounded by R phot that is opaque to the protostar s radiation. Once the temperature climbs past about 1500 K, the grains vaporize. Inside the dust destruction front (R d ~ 10 13 cm), the opacity is greatly reduced (also the gas, which dissociates above 2000 K, is nearly transparent to the radiation field): opacity gap. Immediately outside the accretion shock itself, the gas is collisionally ionized and the opacity rises (radiative precursor).
Gas approaches R * with speeds close to the surface free-fall value V ff : " V ff = 2GM % * $ ' & R * 1/ 2 " M = 280 km s -1 * % $ ' 1 M sun & 1/ 2 " R * % $ ' 5 R sun & 1/ 2 Using the Rankine-Hugoniot jump conditions (equivalent to the conservation of mass, momentum and energy in J-type shocks; see Appendix F in S&P) in a perfect gas with γ = 5/3 and M 1 ( (u 1 -u 0 ) / a 1 ) >> 1, the immediate postshock temperature T 2 10 6 K. In velocity frame of the shock front: ρ2 v2 = ρ1 v1 P2 + ρ2 v2 2 = P1 + ρ1 v1 2 upstream v1 ρ1 P1 T1 downstream v2 ρ2 P2 T2 0.5v2 2 + ε2 + P2/ρ2 = 0.5v1 2 + ε1 + P1/ρ1 ε is internal energy per unit mass Post-shock temperature (strong, adiabatic shock): T2 = 3 µ mh Vshock 2 / (16 kb) = 2.9 x 10 5 K ( Vshock / 100 km/s) 2 (assumes µ=1.3) [see S&P eq. 8.50 & Appendix F] Such a hot gas emits photons in the extreme UV and soft X-ray regimes (λ hc/(k B T 2 ) 100 Å), mainly from highly ionized metallic species (e.g. Fe IX), but the material in the postshock settling region and the radiative precursor is opaque.
The protostar radiates into the opacity gap almost if it were a blackbody surface, with effective temperature: + L acc GM * M 4πR 2 * σ B T 4 eff L acc $ R * T eff M & 7300 K % ( $ 10-5 M sun yr -1 ' 1/ 4 M & % ( $ 1 M sun ' 1/ 4 R * & % ( $ 5 R sun ' 3 / 4 T eff roughly characterizes the spectral energy distribution of the radiation field. In the opacity gap, the characteristic temperature of the radiation and the gas temperature do not vary markedly. 11 Radiative Diffusion and the temperature of the envelope [see S&P Appendix G] In the dust envelope, the infalling matter is highly opaque to optical radiation. To determine the temperature, we need to consider the radiative diffusion equation. In a very optically thick medium, the specific intensity I ν B ν (T), with T = T kin (n tot >> n crit ). But I ν cannot be precisely B ν (T), otherwise F ν would vanishḟ I ν νµdω Suppose the net flux is locally in the z-direction and consider the variation of I ν (z,θ) w.r.t. θ. 12
The radiative transfer equation becomes: µ I ν (z,µ) z = ρκ ν I ν + j ν Assuming thermal emission (Kirchoff s Law applies): I ν (z,µ) B ν (T) µ ρκ ν Using this I ν (z,µ) to determine F ν : db ν dt dt dz Δs = Δz /µ µ = cosθ F ν = +1 I ν µdω = 2π dµµi ν = 4π 3ρκ ν db ν dt 1 dt dz 13 The frequency-integrated flux F rad : F rad = 0 F ν dν = 4π dt 3ρ dz 1 0 κ ν db ν dt dν Defining the Rosseland mean opacity κ: 1 κ 0 κ ν 1 db ν /dt dν 0 db ν /dt dν Radiative diffusion equation F rad = 16σ B T 3 3ρκ dt dz u ν 1 c I ν dω u ν dν = 4σ B T 4 /c 0
..back to the temperature of the envelope: T T 3 r = 3 ρ κ L acc 64πσ B r 2 F rad = L acc /4π r 2 $ T ' κ κ 0 & ) % 300 K( κ 0 = 4.8 cm 2 g -1 ; α = 0.8 α In the temperature regime of interest (100-600 K). Dimensional analysis tell us that: T(r) r γ γ 5 2(4 α) 15 The steady temperature decline continues until the gas becomes transparent to the IR radiation. The transition occurs when the mean free path of the average photon (1/ρκ) becomes comparable to the radial distance from the star. At this point, the entire dust envelope emanates as a blackbody of radius R phot and temperature T phot : Dust envelope temperature M = 1 M! ρκr phot =1 2 4 L acc = 4πR phot σ B T phot M r 3 / 2 ρ = 4π 2GM * L acc GM * M R * ( T + κ κ 0 * - ) 300 K, α $ For M = 10-5 M sun yr -1 and M * =1 M sun : R phot = 2.1 10 14 cm and T phot = 300 K Warning: crude approximations!
It is best to visualize R phot as the radius where a photon carrying the mean energy of the spectral distribution escapes the cloud. The wavelength of this photon is typically: λ hc /k B T phot = 49 µm From detailed modeling, the envelope becomes transparent to outgoing radiation once its temperature falls below several 100 K. In this regime, the dust temperature T d follows from a simple energy argument and it is found that: T d r 1/ 3 This optically thin profile is generally useful for modeling the observed emission at far-infrared and millimeter wavelengths from any dust cloud with an embedded star. Collapsing, rotating cloud Fluid streamlines Isodensity contours Temperature contours 450 K L rad = 21 L! M * = 0.5 M! dm/dt = 5x10-6 M! yr -1 Ω 0 = 1.35x10-14 s -1 1050 K 18
But realistic geometries are more complicated... Zhang & Tan, in prep.