Abstract
A newborn star is encircled by a remnant disc of gas and dust. A fraction of the disc coalesces into planets. Another fraction spirals inward and accretes onto the star1. Accreting gas not only produces observed ultraviolet radiation, but also drags along embedded planets, helping to explain otherwise mysterious features of observed extrasolar systems. What drives disc accretion has remained uncertain. The magneto-rotational instability (MRI), driven by coupling between magnetic fields and disc rotation, supplies a powerful means of transport2, but protoplanetary disc gas might be too poorly ionized to couple to magnetic fields1,2,3,4,5,6. Here we show that the MRI explains the observed accretion rates of newly discovered transitional discs7,8, which are swept clean of dust inside rim radii of ∼10 AU. Stellar coronal X-rays ionize the disc rim, activating the MRI there. Gas flows steadily from the rim to the star, at a rate set by the depth to which X-rays ionize the rim wall. Blown out by radiation pressure, dust largely fails to accrete with gas. Our picture supplies one concrete setting for theories of how planets grow and have their orbits shaped by disc gas9, and when combined with photo-evaporative disc winds10 provides a framework for understanding how discs dissipate.
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Main
A typical transitional system contains a ∼106-yr-old star of mass M*≈1 solar mass (M⊙) orbited by a disc with a rim radius arim of the order of 10 AU as measured from its infrared spectrum7,8. Figure 1 illustrates the situation. Outside arim, optically thick dust abounds. Inside arim, only trace amounts of optically thin dust are observed. Transitional discs may bridge the evolutionary gap between conventional T Tauri discs that do not have large inner clearings and debris discs that are entirely optically thin.
Though nearly devoid of dust, the region interior to arim typically contains accreting gas, because near-ultraviolet emissions from the star imply that gas flows at a rate Ṁ≈10−9M⊙ yr−1 onto its surface7,8,11,12. According to our theory, such gas is leached from the rim by the MRI, a linear instability that amplifies magnetic fields in shearing discs and drives turbulence2. The inner clearing grows as the MRI eats its way out, and the accretion rate interior to the ever-expanding rim is entirely set by the accretion rate established at the rim.
For the MRI to be viable, gas must be sufficiently ionized to satisfy two conditions. First, magnetic fields must be frozen into whatever plasma exists13. Second, neutral hydrogen molecules, constituting the bulk of disc matter, must be dragged inward by accreting plasma14,15,16. We show below that for the problem at hand the second criterion of ambipolar diffusion, ignored in many studies3,4,5,6 of protoplanetary discs, supersedes the first criterion of ohmic dissipation.
Ionization is maintained by X-rays, emitted with luminosity LX≈1029 erg s−1 at energies EX≈3 keV from the central star4,17,18,19,20, most likely from its hyperactive corona20. Galactic cosmic rays, which tend to control ionization on much larger (for example, interstellar) scales, are negligible at the stellocentric distance of the rim. Stellar X-rays penetrate the exposed rim wall to a hydrogen column density N, measured radially. A limited column N* will satisfy the two conditions mentioned above. The MRI-active rim contains mass Mrim≈4πN*arimh μ, where h≈cs/Ω is the vertical density scale height, cs≈(k T*/μ)1/2 is the gas sound speed, Ω is the Kepler orbital frequency, T* is the temperature of MRI-active gas at the rim, k is Boltzmann’s constant and μ≈3×10−24 g is the gas mean molecular weight. This mass flows from arim to ∼arim/2 over the diffusion time tdiff≈arim2/ν, where ν=α csh is the turbulent diffusivity:
where G is the gravitational constant and the extra factor of three follows from a more accurate derivation. As measured by numerical simulations of the MRI13, which remain in their infancy, the transport parameter α might be ∼0.03–0.3, depending on the seed magnetic field. Notice how Ṁ in equation (1) does not depend explicitly on the surface-density profile of the disc. Given observations of arim and M*, we need only compute N* and T*, which we do below.
X-ray-driven MRI is not a new idea4, but our study is the first to apply it to transitional discs and to identify the correct criterion for where the MRI can operate under these conditions. We show below how the entire region interior to the rim, including the mid-plane, is MRI active. This contrasts with previous studies of X-ray-4,6 or cosmic-ray-3 driven MRI in which accretion is confined to the surface layers of a disc whose mid-plane properties cannot be calculated from first principles. Moreover, such surface accretion is unsteady3. The accretion rates that we derive are steady and can be directly compared to observations. Thus, transitional discs not only provide insight into the late stages of disc evolution, but also offer the first clean application of ideas—namely, MRI triggered by non-thermal ionization processes3,4—pioneered for more complex systems.
Determining N* requires that we calculate the degree of ionization as a function of N. The X-ray ionization rate per H2 molecule is
where σX≈4×10−24(EX/3 keV)−2.81 cm2 is the photo-ionization cross-section and η≈81 accounts for the number of secondary ionizations produced per absorbed 3 keV photon4. Freshly ionized H2+ rapidly converts to molecular ions such as HCO+. Most such molecules dissociate in collisions with electrons, but some transfer their charge to gas-phase atomic metals such as magnesium5,21. The latter channel is important because ionized metals tend to keep their charge, neutralizing slowly either by radiative recombination with electrons or collisions with negatively charged dust grains21,22. In the Methods section we show how these considerations yield a quartic equation for xe=ne/n, the number density of free electrons to the number density of H2 molecules, in terms of ζ/n, recombination rate coefficients and the fractional abundance xmet,tot of all heavy metals in the gas phase. To relate n to the column N penetrated, we spread all 4πN arimh molecules comprising the X-ray-penetrated rim into the volume 2πarim2h interior to the rim to estimate that n≈2N/arim. Rate coefficients depend on temperature T, which we estimate using a simple thermal balance. Gas is heated by fast photoelectrons at a rate per H2 of LXσXexp(−N σX)f/(4πarim2), where f≈0.5 is the fraction of X-ray energy deposited as heat23. Gas cools by ro-vibrational transitions of CO, whose abundance is assumed to be cosmic and whose level populations are assumed to be thermal23. The resultant solution xe of the quartic is shown against N in Fig. 2 for parameters appropriate to the transitional system GM Aur7,17, for various choices of xmet,tot. Results for other systems (TW Hyd8,18, DM Tau7,19) are similar to within factors of two.
For neutral hydrogen to accrete by the MRI, a given H2 molecule must collide with enough ions within the e-folding time of the instability14,15,16, 1/Ω:
where βin≈1.9×10−9 cm3 s−1 is the rate coefficient for ions to share their momentum with neutrals24, and the critical ambipolar diffusion number Am* is measured in numerical simulations15 to be roughly 100. Linear growth rates for the MRI drop dramatically with the degree to which (3) is not satisfied16.
Figure 2 plots Am against N for GM Aur. Roughly speaking, Am>Am* for N=N*≈5×1023 cm−2. This value of N*, which we adopt henceforth, is only an order-of-magnitude estimate, given uncertainties in Am* and in how sharp criterion (3) is in determining the nonlinear, saturated state of the instability. In any case, the corresponding ionization fractions, xe≈10−8–10−7, are so large that the magnetic Reynolds numbers, Re≈107–108, far exceed the critical values, Re*≈102–104, seemingly required for magnetic flux freezing13. Accretion by the MRI in transitional discs is limited by ambipolar diffusion, not by ohmic dissipation. For N=N*, we find from our thermal balance model that T=T*≈230 K. Results for TW Hyd and DM Tau are similar.
Armed with T* and N*, we use equation (1) to plot Ṁ against arim in Fig. 3, adjusting α as necessary to reproduce the observations for GM Aur, TW Hyd and DM Tau. Best-fit α-values are 0.007–0.035, of the order of those seen in current simulations of the MRI13. The transitional system CoKu Tau/4 is not detected in X-rays25 and has an unmeasurably small26,27 accretion rate (Ṁ<10−10M⊙ yr−1). Possibly CoKu Tau/4 has a softer X-ray spectrum, a prediction that is subject to test. If, say, Am<10 at its rim, then its 3 keV luminosity LX<1028(10−8/xmet,tot) erg s−1.
We expect little accreting material outside arim. The X-ray heated rim has a vertical thickness greater than that of material immediately outside. Thus, as depicted in Fig. 1, some fraction of the disc beyond the rim will dwell in the rim’s X-ray shadow and be magnetically inert. How long the MRI takes to eat its way out to arim depends on the unknown surface density Σ of the original disc. The clearing time tclear=Σ arim2/Ṁ≈106 yr if Σ≈102 g cm−2, comparable to that of the minimum-mass solar nebula in the vicinity of 10 AU.
Once dislodged from the rim, matter must still travel up to 3 decades in radius, from ∼10 to ∼0.01 AU, to reach the stellar surface. We now show that the MRI continues to provide transport at a≪arim. Observed spectra demand that inside the rim the disc contain so few grains as to be optically thin at mid-infrared wavelengths7,8. We can understand this (see Methods section) as a consequence both of the limited number of grains contained within the MRI-active mass Mrim and of stellar radiation pressure28, which blows out a large fraction of submicrometre-sized grains. Having lost its primary source of continuum opacity, the gas heats not by incident starlight, but by accretional (ohmic) dissipation. Cooling proceeds through gas line transitions, at a rate that is difficult to estimate because it requires solving simultaneously for the thermal, chemical and excitation state of the gas. Here we find it adequate to normalize the temperature of mid-plane gas at a≪arim to the minimum value, obtained by equating the energy flux from accretion with that emitted by a black body:
where . Mass continuity then implies a vertical hydrogen column density of
to the mid-plane. Observations29 suggest . To estimate Am, we take the ionization rate ζ from radiative transfer simulations of X-ray-irradiated discs4, scaled to as given in (4), and insert these rates into the aforementioned quartic for xe. For a 3 keV thermal plasma emitting LX=1029 erg s−1, these simulations4, which account for multiple Compton scattering of X-rays, give ζ=10−16 s−1 at 1 AU and ζ=2.5×10−16 s−1 at 0.1 AU, at the mid-plane. Because dust is largely absent, metal ions can only recombine radiatively with electrons, and the quartic gives Am≈90 at 1 AU and Am≈120 at 0.1 AU, for xmet,tot=10−6. The MRI-active vertical columns at 0.1–1 AU, of the order of , are larger than the MRI-active radial column at the rim, N*≈5×1023 cm−2. This follows from the Am criterion (3), which assigns importance to the total (ion) density xen, not the fractional (electron) density xe; total densities increase dramatically from the rim to the star. At ∼0.01 AU, thermal ionization is sufficient to sustain the MRI3,5,6. Thus, the MRI plausibly operates everywhere interior to the rim, carrying steadily inward all of the mass drawn from the rim.
Our investigation can be extended in several directions. Detailed considerations of thermal balance will enable construction of models that smoothly span all radii a<arim. The extent to which dust can be entrained in the gas flow inside arim can then be calculated and compared with observation7,8. For preliminary calculations in this regard, see our Methods. Finally, our theory is designed to explain only systems with large rims. Most discs do not show such inner clearings, yet their host stars still accrete1 at rates up to ∼10−7M⊙ yr−1. Whether our ideas can be expanded to apply to conventional T Tauri stars at earlier stages of their evolution is an outstanding issue, related to the unsolved problem of the origin of transitional discs.
Disc properties inside the rim are insensitive to those outside, because the MRI can only draw a radial column of N*≈5×1023 cm−2 from the rim at any time. Our picture therefore provides a robust setting for theories of how planets form and how their orbits evolve9 within transitional discs. A protoplanet lying interior to the rim will interact with gas whose density, temperature and transport properties are definite and decoupled from uncertain initial conditions. Our study also supplies part of the answer to how discs dissipate. Except for matter that gets locked into planets, the inner disc drains from the inside out by the MRI, whereas material beyond the rim photoevaporates by stellar ultraviolet radiation10.
Methods
Ionization balance
Molecular ions produced by X-ray ionization dissociate collisionally with electrons or transfer their charge to gas-phase atomic metals. In equilibrium, the fractional number abundance xmol+ of molecular ions relative to hydrogen molecules is therefore given by
where βdiss≈3×10−7(T/230 K)−1/2 cm3 s−1 and βt≈10−9 cm3 s−1 are rate coefficients for dissociation and charge transfer21,30, and xe and xmet are the fractional number densities of electrons and free neutral metals, respectively. Ionized metals neutralize by either radiative recombination with electrons or collisions with negatively charged dust grains21,22,30:
where xmet+ is the fractional abundance of metal ions, βrec≈4×10−12(T/230 K)−1/2 cm3 s−1 is the radiative recombination coefficient and βgr is the recombination coefficient, measured per H2 molecule, for grains. We assume that the total grain surface area available for recombination is dominated by grains of radius s containing a fraction Zs of the total mass. It follows that βgr≈3×10−20(μm s−1)(Zs/10−4)(T/230 K)1/2 cm3 s−1. Though unknown, the factor Zs is likely to be considerably less than the maximal value permitted by solar abundance gas, Z⊙≈10−2, because of grain growth. Fortunately, our results are insensitive to βgr because radiative recombination is usually more efficient than recombination onto grains (see the end of this section). A constraint on Zs/s from observations is contained in the next section on radiation blow-out.
We combine equations (5) and (6) with relations for charge and number conservation
to derive, under approximations specified below, a quartic equation for xe. Our final equation is a quartic and not a cubic21 because we allow for the possibility that nearly all gas-phase atomic metals might be ionized, via equation (8).
Equations (5)–(7) yield21
The first term on the left-hand side dominates the second term when
This is always the case for Am>Am*≈100, as is evident in Fig. 2. Therefore we drop the second term in (9) to write
Combining equations (5), (6) and (8), we have
which simplifies, according to the same approximation embodied in (10), to
We solve (12) for xmet and substitute into (11) to write
It is safe to ignore βrec in comparison to βt, so
We solve the quartic (13) numerically, taking ζ either from (2) when evaluating conditions at the rim, or from Monte Carlo simulations4 when studying the mid-plane at a≪arim. The solutions are insensitive to the uncertain parameter βgr∝Zs/s. For example, in Fig. 2, increasing the grain recombination efficiency by a factor of 100 above our nominal value decreases the peak value of Am from 140 to 40.
Radiation blow-out of grains
Submicrometre-sized grains feel an outward stellar radiation force that just exceeds stellar gravity. Then the time for such grains to travel from a to ∼2a is
where tstop≈ρps/(csμ n)<1/Ω is the time for grains of radius s and internal density ρp≈1 g cm−3 to attain terminal velocity according to the Epstein gas drag law28,31. Radiation blow-out is faster than aerodynamic drift (the latter caused by radial pressure gradients in gas10) by (a/h)rim2∼102.
We compare tblow to tdiff≈a2/ν, the time for gas to diffuse from a to ∼a/2. At the rim of the disc of GM Aur, we find the timescales match by coincidence: tblow,rim≈tdiff,rim≈1×105 yr. This indicates that about half of the dust in Mrim is expelled, leaving the other half entrained with accreting gas and spread between arim and ∼arim/2. The geometric optical depth of entrained dust, measured perpendicular to the mid-plane, is
Because radiation at a wavelength of 10 μm originates from grains with temperatures of ∼300 K, observed 10 μm spectra of GM Aur constrain the optical depth in grains at ∼1 AU but not near the rim at 24 AU, where grain temperatures are much lower. Therefore, we cannot directly compare our calculated τrim with 10 μm observations for GM Aur. For the case of TW Hyd, however, we can more easily make this comparison, because its rim is located at ∼4 AU. Scaling to the parameters of that system, we find tblow,rim≈5×103 yr and tdiff,rim≈2×104 yr, which suggests that more than half the dust in Mrim is expelled. Further reducing our estimate in (14) by a factor of two, and accounting for the smaller aspect ratio (h/a)rim, we estimate τrim≈0.1 for TW Hyd. Observationally8, the vertical optical depth of dust interior to the rim of TW Hydra’s disc is τ10≈0.05 at a wavelength of 10 μm. This is essentially the same as the geometric optical depth for micrometre-sized grains because of the silicate resonance band at 10 μm wavelength. Thus, our crude estimate of τrim≈0.1, on the basis of assumed values for Zs=10−4 and s=1 μm, is within a factor of two of the observed optical depth, agreement that we consider acceptable.
To satisfy the observation that the disc remains optically thin at a≪arim, whatever dust still lies between arim and arim/2 must fail to penetrate within arim/2. Radiation pressure ensures that dust does not continue to fall in, provided that tblow/tdiff∝n T3/2a5/2 decreases with decreasing a. This seems likely to obtain because T should drop sharply just inside arim/2 once mid-plane gas becomes too dense to be heated effectively by X-rays.
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Acknowledgements
We thank A. Glassgold, D. Hollenbach, U. Gorti, E. Feigelson, M. Pessah, C. Espaillat, J. Brown and C. Salyk for discussions. S. Balbus and an anonymous referee provided criticisms. This work was motivated by the Hebrew University Winter School for Theoretical Physics, and supported by grants from the National Science Foundation and the Alfred P. Sloan Foundation.
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Chiang, E., Murray-Clay, R. Inside-out evacuation of transitional protoplanetary discs by the magneto-rotational instability. Nature Phys 3, 604–608 (2007). https://doi.org/10.1038/nphys661
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DOI: https://doi.org/10.1038/nphys661