Lecture 6: Protostellar Accretion Richard Alexander, 23rd March 2009 The angular momentum problem When we try to understand how young stars form, the obvious question to ask is how do young stars accrete their mass?. Naively we assume that they are able to form by simple gravitational collapse, but we must also consider the effects of rotation. Cores in star-forming molecular clouds are observed to have angular velocities of order Ω c 0 4 0 3 s, and we can thus compute the angular momentum of a core by appealing to the Jeans length and Jeans mass R J M J c s Gρ () c 3 s G 3/2. (2) ρ/2 (Here we have neglected order-of-unity constants for clarity.) Molecular clouds have temperatures T 0K, yielding sound speeds of order c s 0.2km/s. We therefore see that forming a star of solar mass by Jeans collapse requires densities that exceed ρ 0 9 g/cm 3, and requires that material fall inwards from distances of order R J 0.pc. The specific angular momentum of the collapasing core is thus j c Ω c R 2 J 0 2 0 22 cm 2 /s. (3) By contrast, the break-up velocity of a star (the maximum velocity at which it can rotate) can be computed by equating centrifugal acceleration with gravity thus Ω 2 b R = GM Ω b = R 2 (4) GM R 3. (5) A star like the Sun therefore has a break-up velocity Ω b 0 3 s (corresponding to a few hundred km/s), and a break-up specific angular momentum (assuming solid-body rotation) of j b Ω b R 2 08 0 9 cm 2 /s. (6) Thus j b j c, and most stars in fact rotate well below break-up. We see therefore that young stars have much lower angular momenta than the gas clouds from which they form, and how this angular momentum is lost is the so-called angular momentum problem of star formation. If we consider the solar system, we see that while the Sun contains more than 99% of the mass in the solar system, more than 99% of the angular momentum resides in the planets (primarily Jupiter). This fact, combined with the fact that all of the major planets lie in the same plane, led the likes of Kant and Laplace to suggest that the solar system formed from a disc, and we can appeal to accretion discs as a solution to the angular momentum problem in star formation. If we assume that discs around young stars are in Keplerian rotation, we can estimate their typical size from the angular momentum of the system. In a Keplerian orbit the specific angular momentum of a mass orbiting a star of mass M at radius R is j K = GM R (7) and if we set j K = j c we find that protostellar discs around solar-mass stars should have typical sizes of R = j2 c 0 3 0 4 AU. (8) GM
Star formation is in fact much more dynamic than the process we have described here, and interactions between protostars and their surroundings can redistribute some of the excess angular momentum. Nevertheless, when we observe young stars we see resolved discs with sizes of 00 000AU, and numerical simulations of star formation typically produce protostellar discs of similar sizes. Thus, disc accretion is crucial to the star formation process, and protostellar discs are an inevitable consequence of star formation. 2 Accretion in protostellar discs Having established that protostellar accretion primarily happens through discs, we now move on to discuss how such accretion occurs and how it can be detected. In this section I will focus on the socalled T Tauri stars, which are well-studied and relatively well understood. T Tauri stars are young ( Myr) stars of approximately solar mass, which are often surrounded by circumstellar discs but have already accreted most of their protostellar envelopes and are thus visible at optical wavelengths. They were originally identified on the basis of unusually strong emission lines which, as we will see below, are now know to be due to accretion. Significant accretion also occurs during earlier evolutionary phases, but these phases are much harder to study observationally (because the protostars are still heavily obscured), so for clarity we will not discuss them in detail. In principle there are two ways to drive accretion in protostellar discs: the outward transport of angular momentum through disc instabilities; and the removal of angular momentum by winds and/or jets. It is still not entirely clear which dominates, but in recent years the consensus has moved away from jets as a global sink of angular momentum. The launching footprint of jets has been observed to be no larger than a few AU, so it seems unlikely that jets can remove angular momentum from large radii in the disc. Jets may well be an important angular momentum sink at small radii, but we will neglect them for the remainder of this discussion. 2. Accretion disc theory Lecture 3 discussed classical accretion disc theory in some detail; here I merely review the points that are relevant for protostellar accretion discs. Our starting point is the diffusion equation for the evolution of a viscous disc Σ t = 3 [ R /2 ( νσr /2)]. (9) R R R Here Σ(R, t) is the disc surface density, R is cylindrical radius, and ν is the kinematic viscosity. This equation is quite general, and relies on only three assumptions: conservation of mass and angular momentum, and that the potential is Keplerian. Everything we don t understand about angular momentum transport is packaged up into the form of the viscosity, and we can infer a great deal about the qualitative behaviour of accretion discs by considering a few simple cases. Of particular relevance to protostellar discs is the solution derived by Lynden-Bell & Pringle (974). They showed, using a Green s function approach, that for the case of an isolated disc (i.e. no infall) where the viscosity is time-independent and a power-law in radius (ν R γ ), there is a family of self-similar solutions that satisfy Equation 9. Moreover, it was subsequently shown that under these conditions any arbitrary initial surface density profile will evolve towards the similarity solution over time. The solution was derived for any value of the power-law index γ, but for clarity we will consider only the case where γ = (i.e., ν R). In this case the surface density evolves as: Σ(R,t) = M d(0) 2πR 2 s R R s τ 3/2 exp ( R/R s τ ). (0) Here M d (0) is the initial disc mass, and R s is a scale radius that defines the initial disc size. The dimensionless time τ is given by τ = t t ν +, () 2
where t ν is the viscous time-scale at R = R s t ν = R2 s 3ν(R s ). (2) In this solution the disc surface density declines with time as mass is accreted, and the disc expands to conserve angular momentum. The asymptotic behaviour is that all of the mass is accreted, and all of the angular momentum is transported to R = (by an infinitesimal mass). At late times (t t ν ) the disc mass and accretion rate (at R 0) decline as power-laws in time: the disc mass as t /2, and the accretion rate as t 3/2. This analysis is heavily simplified, but it still serves to give a good qualitative understanding of accretion disc evolution. Moreover, as we will see, simple models of this type still provide a good fit to current data on protostellar discs. In order to compare to data, however, we must first try to understand how we observe accretion in the discs around young stars. 2.2 Magnetospheric accretion Observations of T Tauri stars reveal strong Zeeman broadening of some photospheric spectral lines, and the inferred surface magnetic fields are B kg in strength. Such fields are strong enough to disrupt an accretion disc at small radii, and as a result we expect accretion on to the stellar surface to be channelled through the stellar magnetosphere. Magnetospheric accretion has already been covered in the context of neutron stars, but the same basic theory can be re-scaled and applied to T Tauri stars as follows. The basic principle is that at some radius r m, magnetic pressure becomes large enough to disrupt the inward flow of material. This happens where the energy density in the magnetic field is comparable to the ram pressure of material in free-fall towards the star, so B 2 (r m ) 8π 2 ρ ffv 2 ff. (3) For a central star of mass M, the free-fall velocity at radius r is v 2 ff = 2GM /r. If the (spherical) mass accretion rate is Ṁ, mass continuity gives Ṁ = 4πr 2 v ff ρ ff, so we can substitute into equation 3 to find B 2 (r m ) Ṁ 2GM 8π 8πr 5/2. (4) If we make the simplifying assumption that the stellar magnetic field is dipolar, then B(r) = µr 3, where µ = B R 3 is the dipole moment. We can therefore re-arrange terms to find the magnetospheric radius r m µ 4/7 Ṁ 2/7 (GM ) /7, (5) where we have again neglected factors of order unity. Re-scaling to parameters typical of T Tauri stars gives ( ) 4/7 ( ) ( ) 2/7 2/7 ( ) B R Ṁ /7 M r m 0R 0.5kG 2R 0 8 M yr. (6) 0.5M In practice T Tauri magnetic fields are complex, and certainly not dipolar. However, the dipole component dominates the field at large radii, and we therefore expect the accretion flow around T Tauri stars to be channelled by the stellar magnetosphere inside 5 0 stellar radii. 2.3 Observations of disc accretion This simple model of magnetospheric accretion provides us with several means of observing accretion in T Tauri stars. The large fractional change in radius between r m and R means that the accreting gas reaches the stellar surface at close to its free-fall velocity (> 200km/s). This results in a so-called accretion shock on the stellar surface, and we can therefore observe accretion in the following ways: 3
The accretion shock is typically much hotter than the rest of the stellar photosphere, with temperatures of 0,000 5,000K. This gives rise to copious emission in the UV ( 2000 4000Å), and this UV excess is readily observed. Essentially this measures the accretion luminosity directly, and is therefore the most robust method of measuring accretion rates in T Tauri stars. The accretion shock provides excess continuum emission in the optical, where the stellar luminosity is significant. The excess emission fills in the standard photospheric absorption lines, and by comparing to template spectra it is possible to use this veiling to measure the accretion luminosity. This is fraught with systematics, but if done carefully can provide accurate measurements of accretion rates from optical spectra (which are often easier to obtain than observations in the UV). The high infall velocities in the accretion flow give rise to very broad emission lines, such as Hα. Unfortunately such lines are also emitted in the stellar chromosphere (because T Tauri stars are highly magnetically active), but it is possible to use the line profiles to determine accretion rates. These rates are somewhat model-dependent, but offer the fastest means of determining accretion rates in a large sample of objects. Accretion rates for T Tauri stars span the range 0 7 0 0 M yr, and show clear evidence for decline on a time-scale of a few Myr. This is consistent with observations of dust in discs (IR excess, etc.), which show that dust discs around young stars have a median lifetime of a few Myr. There is significant scatter (at least a factor of 0) in the observed accretion rates as a function of age, but the general trend is consistent with the similarity solution if α 0.0. 2.4 Disc thermodynamics Much of modern accretion disc theory discusses angular momentum transport and the magnitude of α, but the alpha-prescription also requires an understanding of the disc s thermal structure, in particular the radial temperature distribution. We saw in the previous lecture that accretion (viscous) heating is the dominant source of heating in discs around neutron stars and white dwarfs. However, in general this is not true for protostellar discs, where the effects of stellar irradiation are also important. The rate of viscous heating is proportional to the accretion rate through the disc, so there exists a critical accretion rate below which viscous heating is negligible. A flat disc intercepts /4 of the stellar luminosity, L, and we can therefore estimate this critical accretion rate by equating the energy released by accretion with the energy of irradiation thus 4 L = GM Ṁcrit 2R. (7) If we assume solar values, we find that Ṁ crit 0 8 M yr. Real T Tauri discs are flared, and thus intercept a higher fraction of the stellar luminosity that we have assumed here, and real T Tauri stars can often be much more luminous than the sun. We therefore see that accretion heating is probably not significant except at very early times in protostellar disc evolution, and this provides an a posteriori justification for a disc thermal structure (and therefore viscosity) that is more or less independent of the disc accretion rate. Much more detailed study is required to model the spectral energy distributions of protostellar discs in detail, but that is beyond the scope of this course. 3 Angular momentum transport We now turn to the thorny issue of angular momentum transport, which we have happily swept under the carpet until now. Observations of T Tauri stars suggest values of α 0.0, but the origin of this viscosity in protoplanetary discs remains unclear. It seems likely that some form of MHD turbulence (such as the magneto-rotational instability) drives the accretion, but this requires that the disc be 4
sufficiently ionized to couple to the magnetic field. We can appeal to cosmic rays and stellar X-rays as sources of ionization, and it seems likely that these can ionize some or all of the disc at large radii and/or late times (see the discussion of so-called dead zones from Lecture 4). However, at earlier times (especially in the embedded phase) it is not clear that the disc will remain sufficiently ionized, and we must instead turn to other instabilities as a source of angular momentum transport. Various instabilities have been proposed as means of transporting angular momentum in discs, but the most promising (apart from the MRI) is instability due to the disc s own gravity. 3. Gravitational instabilities in discs Consider a patch of a disc of size l. The mass of the patch is M Σl 2, and the gravitational potential energy of the patch (due to its own gravity) is The thermal energy is U G G M2 l GΣ 2 l 3. (8) U T 2 Mc2 s 2 Σl2 c 2 s, (9) and the rotational kinetic energy is U R 2 MΩ2 l 2 2 ΣΩ2 l 4. (20) If the disc is to become unstable we require that gravity overcome pressure rotational support, so we require that U G + U R + U T < 0. (2) Substituting, we find that 2GΣ 2 l 3 + ΣΩ 2 l 4 + Σl 2 c 2 s = Σl 2 ( 2GΣl + Ω 2 l 2 + c 2 s) < 0. (22) The term in brackets is quadratic in l, and has a minimum at l = GΣ/Ω 2. The condition for instability is met if the left-hand side is negative at this minimum, and therefore if c s Ω GΣ <. (23) This is approximately the famous Toomre (964) condition for gravitational instability in a disc. More careful analysis finds that the disc is unstable to axisymmetric perturbations if Q = c sω <. (24) πgσ The term on the left is usually referred to as the Toomre Q parameter, and lower values of Q give rise to instability. This form intuitively makes sense: increasing the temperature or the rotational rate increases Q and stabilises the disc, while increasing the disc mass lowers Q and makes the disc more unstable. We can use the Toomre criterion to estimate whether or not protostellar discs are likely to be gravitationally unstable. The disc thickness H c s /Ω, Ω 2 = GM /R 3, and the disc mass M d πσr 2. We can therefore re-arrange the Toomre criterion to find that the disc will be unstable if H R M d M. (25) Unlike the razor-thin discs found around compact objects, protostellar discs typically have H/R 0., and we therefore require that disc masses be 0% of the stellar mass in order to be unstable. Disc masses are measured from optically thin dust continuum emission, and these measurements suggests that most T Tauri discs are not gravitationally unstable: the median value is M d /M 0.0. However, the most massive protostellar discs (which are generally the youngest such objects) do appear to be massive enough to be unstable, so we now seek to understand how such unstable discs behave. 5
3.2 Thermodynamics and fragmentation We first note that, formally, the Toomre criterion tells us when the disc will become unstable to axisymmetric perturbations. However, a shearing disc becomes unstable to non-axisymmetric perturbations at slightly larger values of Q (Q.5 2), so we expect the initial development of the gravitational instability (GI) to be in the form of spiral density waves in the disc. These spiral density waves induce Reynolds and gravitational stresses in the disc, and these stresses can transport angular momentum and drive accretion. (It is left as an exercise for the enthusiastic student to show that only trailing spiral waves transport angular momentum outwards.) Spiral density waves can transport angular momentum, but if we consider their behaviour in detail it becomes clear that the development of GIs depends critically on the disc thermodynamics. From the Toomre criterion we see immediately that colder discs are more unstable. However, the instability results in heating, initially through adiabatic contraction of the unstable gas and and later through the weak shocks induced by the spiral density waves. This in turn drives the disc back towards stability, so the disc can only remain unstable if it is able to cool efficiently. We therefore expect the disc to be able self-regulate towards a state where heating from GIs balances cooling, and Q. Numerical simulations have shown that discs can attain such a self-regulated state in certain circumstances, and that discs in this state can drive quasi-steady angular momentum transport over long timescales (at least hundreds of orbital periods). If we assume that the angular momentum transport and heating from GIs both occur locally (i.e., that the GI behaves like an alpha-disc, with no wave-like transport of energy), and that the disc self-regulates to thermal equilibrium, we can derive an expression for the disc s cooling time-scale. The thermal energy per unit area of the disc can be written as U th = γ(γ ) Σc2 s, (26) where γ is the ratio of specific heats of the gas. (γ = 5/3 for a monatomic gas, or 7/5 for a diatomic gas.) Viscous heating (per unit area) occurs at a rate du dt = 9 4 νσω2 = 9 4 α gc 2 sσω, (27) where α g refers to the effective alpha of the GI-induced turbulence. If heating balances cooling, we therefore require the disc to cool on a time-scale t cool = U th du/dt = 4 9γ(γ ) Alternatively, one can parametrize the effective transport induced by GIs as α g Ω. (28) 4 α g = 9γ(γ ) t cool Ω. (29) In principle, this equation suggests that GIs can give rise to arbitrarily large values of α g, and therefore transport angular momentum at very large rates. This is not the case, however, as we can see by once again considering our unstable patch of disc. If the disc cooling balances the compressional heating, the patch is quasi-stable, but if the cooling rate becomes large the gas must contract very rapidly in order to maintain thermal equilibrium. For sufficiently rapid cooling pressure can no longer support the collapsing gas against its own gravity, and the result is that the disc fragments. Numerical simulations find that the fragmentation boundary occurs at t cool Ω 3 5, which in turn suggests that the maximum efficiency of angular momentum transport by GIs (in the local limit) is α g 0.. Note, however, that for sufficiently massive discs the local approximation breaks down, and the instability is instead dominated by low-order spiral modes (two- and three-armed spirals). The evolution of these modes is highly transient, and in this case a quasi-steady self-regulated state is not achievable. Instead the instability drives accretion that is highly time-variable, and not yet fully understood. 6
4 References and further reading In addition to the course textbook (Frank, King & Raine), there are a number of useful references that describe the physics of protostellar accretion and protoplanetary discs: Accretion Processes in Star Formation, by Lee Hartmann. (Cambridge University Press.) First published in the late 990s, this very readable textbook is now considered to be a classic in the field. A heavily updated second edition was published in 2008. Astrophysical Flows, by Jim Pringle & Andrew King. (Cambridge University Press.) This textbook was developed from a Masters-level course on astrophysical fluid dynamics, and gives a fairly mathematical, but readable, description of a number of interesting fluid instabilities. In particular, both the MRI and gravitational instabilities are discussed in some detail. Self-gravitating accretion discs, by Giuseppe Lodato. arxiv.080.3848 Recent review article on the physics of GIs in discs, with a strong focus on angular momentum transport and accretion. Lecture notes on the formation and early evolution of planetary systems, by Phil Armitage. astro-ph/070485 Lecture notes from a graduate-level course on planet formation. Much of the material concerns planet formation rather than accretion physics, but the section on protoplanetary discs is very instructive. In addition, a number of excellent review articles can be found in the Protostars & Planets series. The most recent meeting (PPV) was held in late 2005, and the book was published in early 2007; most of the articles are on astro-ph. PPIV was published in 2000: there is a copy in the Kaiser Lounge, and it is (currently) available online also. For PPIII or earlier, however, you will have to look in a library. See http://www.uapress.arizona.edu/onlinebks/ppiv/contents.php. 7