Stellar Objects: Star Formation and Protostars 1 Star Formation and Protostars 1 Preliminaries Objects on the way to become stars, but extract energy primarily from grav- itational contraction are called Young stellar objects here. Young stellar objects (YSO): the entire stellar system throughout all pre- main sequence (MS) evolutionary phases. • Protostar: the optically thick stellar core that forms during the adi- abatic contraction phase and grows during the accretion phase. The YSO is fully convective and evolves along the so-called Hayash track in the HR diagram. • PMS (pre-main sequence) star: the premature star that becomes visible once the natal envelope has been fully accreted and which still contracts towards the main sequence (MS), where the hydrogen burning domi- nates. In this evolutionary stage, the YSO has formed a radiative core, though still growing with time. Theoretically, the formation of a star may be divided into four stages (as illustrated in Fig. 2.1: (a) proto-star core formation; (b) the proto-star star builds up from the inside out, forming a disk around (core still contracts and is optically thick); (c) bipolar outflows; (d) the surrounding nebula swept away. The energy source is gravitational potential energy. While the total luminosity is comparable to the solar value, the proto-star stages have the 7 2 −1 −1 KH time scale [∼ 2 × 10 (M/M⊙) (L/L⊙) (R/R⊙) ]. They are also fully convective and is thus homogeneous chemically. Observationally, the proto-stars are classified into classes 0, 1, and 2, ac- cording to the ratio of infrared to optical, amount of molecular gas around, inflow/outflow, etc. Class 0 proto-stars are highly obscured and have short time scales (corresponding to the stage b?); a few are known. The class 1 and 2 proto-stars are already living partly on nuclear energy (c and d); but the total luminosity is still dominated by gravitational energy. The low-mass Stellar Objects: Star Formation and Protostars 2 YSO prototype is T Tauri. We still know little about high-mass YSOs, which evolve very fast and interact strongly with their environments. Signatures of YSOs: • variability on hours and days due to temperature irregularities on both the stellar surface and disk • emission lines from the disk and/or outflow • more inferred luminosity due to dust emission • high level of magnetic field triggered activities (flares, spots, corona ejection, etc) due to fast rotation and convection • Strong X-ray emission from hot corona. 2 The birth of stars From the virial theorem, 2E = −Ω, we have 3kT M M GM = r dM (1) µm r r A Z0 for the hydrostatic equilibrium of a gas sphere with a total mass M. Assum- ing that the density is constant, the right side of the equation is 3/5(GM 2/R). If the left side is smaller than the right side, the cloud would collapse. For the given chemical composition, ρ and T , this criterion gives the minimum mass (called Jeans mass) of the cloud to undergo a gravitational collapse: 3 1/2 5kT 3/2 M > M ≡ . (2) J 4πρ Gµm A For typical temperatures and densities of large molecular clouds, MJ ∼ 5 −1/2 10 M⊙ with a collapse time scale of tff ≈ (Gρ) . Such mass clouds may be formed in spiral density waves and other density perturbations (e.g., caused by the expansion of a supernova remnant or superbubble). Stellar Objects: Star Formation and Protostars 3 What exactly happens during the collapse depends very much on the temper- ature evolution of the cloud. Initially, the cooling processes (due to molecular and dust radiation) are very efficient. If the cooling time scale tcool is much −1/2 shorter than tff , the collapse is approximately isothermal. As MJ ∝ ρ decreases, inhomogeneities with mass larger than the actual MJ will collapse by themselves with their local tff , different from the initial tff of the whole cloud. This fragmentation process will continue as long as the local tcol is shorter than the local tff , producing increasingly smaller collapsing subunits. Eventually the density of subunits becomes so large that they become opti- cally thick and the evolution becomes adiabatic (i.e., T ∝ ρ2/3 for an ideal 1/2 gas), then MJ ∝ ρ . As the density has to increase, the evolution will al- ways reach a point when M = MJ , when we assume that a star is born. From this moment on the cloud would start to evolve in hydrostatic equilibrium. This way a giant molecular cloud can form a group of stars with their mass distribution being determined by the fragmentation process. The process depends on the physical and chemical properties of the cloud (ambient pres- sure, magnetic field, rotation, composition, dust fraction, stellar feedback, etc.). Much of the process is yet to be understood. We cannot yet theoret- ically determine the distribution of the initial mass function (IMF) of stars. The classic expression for the IMF, determined empirically is the so-called Salpeter’s law dn/dM = CM −x (3) where x = 2.35 for M/M⊙ ≥ 0.5 and x = 1.3 for 0.1 ≤ M/M⊙ ≤ 0.5 in the solar neighborhood. 3 Evolution of YSOs Still incomplete!!! The actually formation and evolution of a YSO is much more complicated, but roughly can be divided into the following stages: During the collapse the density increases inwards. The optically thick phase is reached first in the central region, which leads to the formation of a more-or- less hydrostatic core with free falling gas surrounding it. The energy released Stellar Objects: Star Formation and Protostars 4 by the core (now obeying the virial theorem) is absorbed by the envelope and radiated away as infrared radiation. Because of the heavy obscuration by the surrounding dusty gas, stars in this stage cannot be directly observed in optical and probably in near-IR. When the accretion is essentially completed, the YSO increases the temper- ature as the core continue to contract. The steady increase of the central temperature causes the dissociation of the H2, then the ionization of H, and the first and second ionization of He. The sum of the energy involved in all these processes has to be at most equal to the energy available to the star through the virial theorem. The simple estimate gives the maximum initial i radius Rmax of a YSO has to be Ri M max ≈ 50 (4) R⊙ M⊙ Therefore, the luminosity can be very large. 3.1 Evolution along the Hayashi Track Starting from the surface inward, we have encountered the simple atmosphere model in the heating transfer section. The temperature and pressure at the g ≈ 2 s 2 photosphere are T (τP )= Teff and P (τP ) 3 , where gs = GM/R is the κP local gravity and κP is the opacity κ at the photosphere. Note that we have assumed that convection plays no role in hear transport between the true and photosphere surface, consistent with our notion of a radiating, static, and visible surface (although this is not true for the sun). Next we consider the stellar “envelope”, which consists of the portion of a star that starts at the photosphere, and continues inward, but contains negligible mass, has no thermonuclear or gravitational energy sources, and is in hydrostatic equilibrium. We want to see how deep the radiative envelope may be until the convection takes over for heat transfer. In such a radiative envelope, ∇ = ∇rad with dlnT 3κL P ∇ = = . (5) dlnP 16πacGM T 4 Stellar Objects: Star Formation and Protostars 5 Have the opacity written in the interpolation form for an idea gas, κ = n −n−s κgP T , the above equation can be re-written as 16πacGM P ndP = T n+s+3dT. (6) 3κgL n+1 n+1 ′ n+1 n+s+4 n+s+4 P − PP =(K ) (T − TP ) (7) where 1 16πacGM 1/(n+1) K′ = (8) 1+ n 3κ L eff g where neff =(s + 3)/(n + 1). It is easy to prove that this can be written as 1 T n+s+4 1 ∇(r)= + eff ∇ − (9) 1+ n T (r) P 1+ n eff eff where ∇P is ∇ evaluated at the photosphere: 3L P ∇ P P = 4 = 1/8 (10) 16πacGMκP TP 2 2 4 where Pp = 2gs/3κP , gs = GM/R , and L = 4πR σReff are used. − For H opacity, n = 1/2 and s = −9, we have neff = −4 and 1 11 T (r) 9/2 ∇(r)= − + (11) 3 24 T eff Note that since temperature increases with depth, so does ∇. Eventually, when ∇ > ∇ad, the stellar material becomes convective. For simplicity, we assume that the convection is adiabatic. Thus at depths deeper than the critical depth, ∇ = ∇ad = 0.4 for ideal gas. The transition to convection 2/9 occurs at Tf = (8/5) Teff = 1.11Teff . The corresponding pressure Pf at the top of the convective interiors can be found P n+1 1 1 T (r) n+s+4 =1+ − 1 (12) P 1+ n ∇ T P eff P " eff # 2/3 which yield Pf = 2 PP . Stellar Objects: Star Formation and Protostars 6 For the assumed convection, the implying polytrope of index is 3/2 and ′ 5/2 P = Kn=3/2T (13) (K′ here is different from the previously defined??). ′ For a completely convective star, Kn=3/2 can be related to the the mass and radius of the star as defined by the E-solution polytrope 1 K′ ∝ (14) n=3/2 µ5/2M 1/2R3/2 ′ 5/2 2 4 Using Pf = Kn=3/2Tf and L = 4πσR Teff to replace R, we get 13/51 7/51 1/102 Teff ∝ µ (M/M⊙) (L/L⊙) (15) Therefore, such a completely convective star has a nearly constant effective temperature, independent of the luminosity.