Class 18: Protostellar evolution ASTR 4008 / 8008, Semester 2, 2020 Mark Krumholz Outline • Models and methods • Timescale hierarchy • Evolution equations • Boundary conditions • The role of deuterium • Qualitative results: major evolutionary phases • Evolution of protostars on the HR diagram • The birthline • The Hayashi and Heyney tracks Timescale hierarchy Similarities and differences to main sequence stars • Protostellar evolution governed by three basic timescales: • Mechanical equilibration: tmech ~ (R / cs) ~ (R3 / GM)1/2 ~ few hours • Energy equilibration (Kelvin-Helmholtz): tKH = GM2 / RL ~ 1 Myr • Accretion: tacc = M / (dM / dt) ~ 0.1 Myr −5 −1 • Numerical values for M = M⊙, R = 3 R⊙, L = 10 L⊙, dM / dt = 10 M⊙ yr • Implication: protostars are always in mechanical equilibrium, but are not generally in energy equilibrium while they are forming • By contrast, main sequence stars in energy equilibrium because tKH ≪ lifetime • This hierarchy applies to low-mass stars; somewhat different for high mass Equations of stellar structure For protostars • Equations describing mass conservation, hydrostatic balance, and energy transport are exactly the same as for main sequence stars: <latexit sha1_base64="zQBTVeYWRUq5mFjggJdG88FmPGo=">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</latexit> @r 1 = Mass conservation @M 4⇡r2⇢ @P GM = Hydrostatic balance @M −4⇡r4 @T 3RL Radiative flux (or equivalent equation for convective flux in Nuclear energy = 2 3 4 @M −256⇡ σSBT r convective regions) generation rate <latexit sha1_base64="y0iqzn1jiVV2d3dsV9NZcAy+Dh0=">AAACHXicdVDLSgMxFM34rPVVdekmWBRXQ8a+7EIounGhUMHWQqeUTJqxoZnMkGSEMsyPuPFX3LhQxIUb8W9MH+IDPRA4nHMuN/d4EWdKI/RuzczOzS8sZpayyyura+u5jc2mCmNJaIOEPJQtDyvKmaANzTSnrUhSHHicXnmDk5F/dUOlYqG41MOIdgJ8LZjPCNZG6uaKri8xSdwIS80wh2fpFz9P4R48gi6NFOMmnLgygCImaTeXR7ZTqJbLFYjsKkLFctGQSrlUcBzo2GiMPJii3s29ur2QxAEVmnCsVNtBke4koz2E0zTrxopGmAzwNW0bKnBAVScZX5fCXaP0oB9K84SGY/X7RIIDpYaBZ5IB1n312xuJf3ntWPuHnYSJKNZUkMkiP+ZQh3BUFewxSYnmQ0Mwkcz8FZI+NnVpU2jWlPB5KfyfNA9sp2Sji2K+djytIwO2wQ7YBw6ogBo4BXXQAATcgnvwCJ6sO+vBerZeJtEZazqzBX7AevsA26KiaQ==</latexit> @L Major difference is in energy equation: in a MS star, this would be = ✏nuc • @M • However, this is only true if the stars is in energy equilibrium; we cannot assume that this is the case for a protostar The non-equilibrium energy equation How protostars differ from main sequence stars • In a protostar, there is an additional potential source of energy: a mass shell can lose energy, which is added to the energy change across that shell • To include this use fundamental thermodynamic relation dU = P dV + T dS: change in energy equals work done + temperature times change in entropy <latexit sha1_base64="IhFtnEg+uriliZMdOCBk3ESXPeE=">AAACPXicdVDNS8MwHE3n15xfVY9egkPw4mhn1XkQRC8eFBT2BWsZaZZuwTQtSSqM0n/Mi/+DN29ePCji1avpnKhDHwQe7/d7Sd7zY0alsqwHozA1PTM7V5wvLSwuLa+Yq2tNGSUCkwaOWCTaPpKEUU4aiipG2rEgKPQZafnXp/m8dUOEpBGvq2FMvBD1OQ0oRkpLXbPuBgLh1I2RUBQxeJ5984sMHkGXxJIyvZq6IoQ8wRncgXU4YZM/bCrrmmWrYu/t79q70KrUDu09p6rJge1UHQfaFWuEMhjjsmveu70IJyHhCjMkZce2YuWl+YWYkazkJpLECF+jPuloylFIpJeO0mdwSys9GERCH67gSP3pSFEo5TD09WaI1EBOznLxr1knUUHNSymPE0U4/nwoSHTACOZVwh4VBCs21ARhQfVfIR4g3YvShZd0CV9J4f+kWdVFVawrp3x8Mq6jCDbAJtgGNjgAx+AMXIIGwOAWPIJn8GLcGU/Gq/H2uVowxp518AvG+wc5Ua/d</latexit> @L @s = ✏ T • No change in volume (dV = 0), so energy equation changes to @M nuc − @t • Computational procedure in practice: (1) given starting state r(M), T(M), solve flux equation to get L(M); (2) use L(M) in energy equation to find ds/dt in every shell; (3) update entropy of every shell; (4) solve equation of hydrostatic equilibrium to find new r(M), T(M), fixing entropy in each shell; (5) repeat Boundary conditions for protostars The second difference • System of equations for r(M), T(M), P(M), L(M) needs four boundary conditions • Two at inner boundary are same as for main sequence stars: r(0) = 0, L(0) = 0 • If protostar is not accreting, outer boundary conditions are the same as for main sequence stars as well: P(Mtot) = 0 (or something small if we are being more sophisticated), L(Mtot) = 4�r2(Mtot)�SBT4(Mtot) • BCs for accreting star are different because (1) accretion flow potentially provides finite pressure at outer radius, (2) accretion flow restricts escape of energy from star Spherical accretion flows Also known as “hot accretion” <latexit sha1_base64="Scwv1IKbI+lz1+7Rjm4HPORKe30=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXw8x0auuu6MZlBfvAdiyZNNOGZpIhyRTK0L9w40IRt/6NO//G9CGo6IELh3Pu5d57woRRpR3nw1pZXVvf2Mxt5bd3dvf2CweHTSVSiUkDCyZkO0SKMMpJQ1PNSDuRBMUhI61wdDXzW2MiFRX8Vk8SEsRowGlEMdJGuuvKoehROL73eoWiY7vl85Jbgo5dvXDLvmdIxfU934eu7cxRBEvUe4X3bl/gNCZcY4aU6rhOooMMSU0xI9N8N1UkQXiEBqRjKEcxUUE2v3gKT43Sh5GQpriGc/X7RIZipSZxaDpjpIfqtzcT//I6qY6qQUZ5kmrC8WJRlDKoBZy9D/tUEqzZxBCEJTW3QjxEEmFtQsqbEL4+hf+TpmeCsp0bv1i7XMaRA8fgBJwBF1RADVyDOmgADDh4AE/g2VLWo/VivS5aV6zlzBH4AevtE1/ikLs=</latexit> 2 • If accretion flow is spherical, ram pressure of infall is ⇢iv <latexit sha1_base64="N7WQhlrFh1gQZJCeXXj4YUSN/X8=">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</latexit> ˙ 2 • Density determined by mass accretion rate: M =4⇡r ⇢iv <latexit sha1_base64="HvCs3R/W6ZqfkapaNXqYm+Bei/c=">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</latexit> M˙ 2GM If material arrives at free-fall speed, bounding pressure is P (M)= • 4⇡ R5 <latexit sha1_base64="SFFYSzfMMv2ByIGJtM2GdxLhpv4=">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</latexit> r ˙ 2 4 GMM • Luminosity at surface takes form L(M)=4⇡R σSBT + fin R • First term is usual blackbody radiation, second is mechanical energy provided by accretion flow — but only a fraction fin of this is advected inward at accretion shock, with rest escaping immediately as radiation • Evaluation of shock properties suggests fin ≈ 1/4 Magnetically-channeled accretion flows Also known as “cold accretion” • Accretion flow onto star may cover only a small portion of surface due to channeling by stellar magnetic field • In this case, material being added to the star provides no confining pressure, and radiates away all its energy (and thus is added to star “cold”, with some specified, usually much lower, entropy) — then BCs are the same as for main sequence stars • This generally produces much smaller predicted radii for protostars, because material being added to star has far lower entropy • Reality probably somewhere between hot and cold extremes, but still debated The importance of deuterium The first thing that burns • By definition, protostellar phase is phase before star ignites H and settles onto main sequence • However, BBN produced ≈ 2 × 10−5 D/H, and D burns more easily than H because there is no need to wait for the weak nuclear force to convert protons to neutrons; ignites at ~106 K (compared to ~107 K for H) • Basic reaction is 2H + 1H → 3He; releases 5.5 MeV / D burned • Timescales: energy release comparable to H burning (7 MeV / nucleon), but fuel supply smaller by factor of 2 × 10−5; since H lasts ~1010 yr, expect D to last ~few × 105 yr — comparable to accretion time protostellar evolution 275 830 HOSOKAWA & OMUKAI Vol. 691 Figure 17.1: Kippenhahn and composi- tion diagrams for a protostar accreting 5 1 at 10− M yr− . In the top panel, Basic outline of the thick curve shows the protostellar radius as a function of mass, and gray and white bands show convective and evolution radiative regions, respectively. Hatched areas show regions of D and H burning, as indicated. Thin dotted lines show the radii containing 0.1, 0.3, 1, 3, and 10 M , as indicated. Shaded regions show four evolutionary phases: (I) convec- tion, (II) swelling, (III) KH-contraction, and (IV) the main sequence.