Draft version September 3, 2011 Preprint typeset using LATEX style emulateapj v. 11/10/09 ELEMENT ABUNDANCE OF GAS AROUND β PICTORIS Ji-Wei Xie1;3, Alexis Brandeker2, Yanqin Wu1 1Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada; 2Department of Astronomy, Stockholm University, SE-106 91 Stockholm, Sweden and 3Department of Astronomy & Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, 210093, China Draft version September 3, 2011 ABSTRACT Recent observations have shown that carbon in the gas around β Pictoris is > 100 times over- abundant with respect to the solar abundance. Although it is thought that such an overabundant in carbon is crucial to retain the metal elements in the disk, its origin is however unclear. In this paper, we establish a simple analytical model to study gas removing process and thus calculate the abundance of various elements in the gas disk around β Pictoris. Gas removing rate is controlled by the inward flow-viscosity accretion-and the outward flow-radiation drift. If the disk viscosity (using classical α disk model) is low, thus radiation drift dominates the gas loss, then carbon can become highly overabundant. In order to produce the observed overabundance of carbon, a low viscosity of α < 10−3 and a gas production with solar abundance are preferred. 1. INTRODUCTION In this work, we are going to study such a process for A debris disk is a circumstellar disk of dust and de- various elements with a simple analytical model, focusing bris in orbit around a star. The dust in debris disk on the relative abundance evolution of various elements. is generally considered being produced by cascade col- This paper is organized as the following. We describe our lisions which grind down large particles (planetesimals) analytical model in section 2, with some related deriva- to the observed small grains (Wyatt 2008; Krivov 2010). tions in detail can be found in the Appendix A and B. Although most detected debris disks are lacking of gas, Our results and comparison to observations are discussed people did find significant gas in the debris disk around in section 3. Finally, we conclude in section 4. β Pictoris (Hobbs et al. 1985), which is the subject of 2. MODEL this work. Some metallic gas components, such as neutral sodium 2.1. Collision VS. Radiation Drift (Na I), are found to be in stable orbits with small radial A particle (ion or neutral atom) with β > 0:5 is accel- velocities (Olofsson et al. 2001; Brandeker et al. 2004). erated to drift outward due to the radiation force, and This was not expected as Na I should be subject to very at the same time it is decelerated through collisions with strong radiation pressure (the ratio of radiation force and other braking particles that are not subjected to signifi- gravitational force being β ∼ 360). For such a reason, cant radiation force. The dominant braking particles are some agent is needed to brake these metallic gas down to carbon ions (C II). The competing process between radi- low observed radial velocity. As suggested by Fern´andez ation acceleration and collision deceleration leads to an et al. (2006), if the gas is overabundant in carbon by a equilibrium drift velocity. factor ≥ 10, then then the gas can be self-braking. In- For a neutral tracer particle, its equilibrium drift ve- deed, such a carbon rich gas has already been discovered locity is (see the Appendix A for detailed derivation) (Roberge et al. 2006), and it could be produced through 1=2 −1 collisional vaporization of the dust (Czechowski & Mann mx mxmc NCII vneu 1 ∼ 0:8β −3 2007) or by photo desorption (Chen et al. 2007). mc mp(mx + mc) 100 cm Furthermore, the latest observations (Brandeker et al. r −2 M P ol −1=2 2011) shows that the C and O are ∼ 400 times over- star ms−1; (1) 3 abundant relative to cosmic (solar) abundance (in solar 100 AU M 10 A˚ abundance, ratio of C and Fe density is ∼ 10) . Such a large overabundant in C and O cannot be reproduced di- where mp, mx are the mass of a proton and a atom rectly by collisional vaporization of the dust. Instead, it of element \x", respectively, r is the radial distance to the center star of mass M , N is the number den- might indicate that the gas disk has underwent some cer- star CII tain evolution processes which increase the abundances sity of CII , and P ol is the polarizability of neutral atom of C and O to current high values. One possibility is \x". On the other hand, there is another limiting velocity that various elements experience various radiation pres- (vneu 2) that the particle can reach before it is ionized, sure, and thus are removed at different rates. Since C GM 1 v ∼ β star and O do not experience any significant radiation pres- neu 2 r2 Γ sure, they would stay behind and thus become enhanced Γ relative to other elements. ∼ 10(β − 0:5) AU ms−1; (2) 10−7 s−1 [email protected] where G is the gravity constant, ΓAU is the ionization 2 Xie et al. h i rate, Γ, of the neutral atom \x" at r = 1 AU. Following Nc 2.3. Abundance Ratio: N Brandeker (2011) 1, the expected velocity of the neutral x tracer is 2.3.1. equilibrium case γ vneu 2 vneu = (3) Once the equilibrium is realized, gas production rate γ + 1 should be equating to the gas removing rate. For those , where γ = vneu 1=vneu 2. elements that are subject to little radiation force, such For an ionized tracer particle, its equilibrium drift ve- as carbon, their removing rates are only governed by the locity due to collisions with carbon ions can be derived viscosity accretion inflow, while for elements that are as (see the appendix B for detailed derivation) subject to significant radiation force, such as sodium, their removing rates are governed by both the viscosity 2 1=2 −1=2 accretion inflow and the radiation drift outflow. Hence, −2 mx NCII v ∼ 3:7 × 10 β 2 ion m (m + m ) 100 cm−3 the equilibrium equations (zero-order estimate ) for the c c x field gas particle, carbon, and for the tracer particle, \x", r −1 M 1=2 T can be written as star disk ms−1 (4) 100 AU M 100 K Σc − + Sc ∼ 0 (9) Therefore, the total effective drift velocity of tracer \x", tv which is driven by radiation force, is Σ Σ − x − x + S ∼ 0 (10) t t x ux = fxvneu + (1 − fx)vion; (5) r v where S and S denote the gas production rate of carbon where fx is the neutral fraction of element \x", which can c x be calculated from the equation of ionization equilibrium and element \x" (see the Appendix C for details about (see Appendix C). the gas production). The equilibrium surface densities then can be solved as Figure 1 shows vneu 1; vneu 2; vneu and vion for all the elements with β > 0:5. Σc ∼ Sctv • (1) From the top panel of figure 1, we see that most tvtr elements' neutral drift velocities v are close to Σx ∼ Sx ; (11) neu tv + tr vneu 2 (ionization limit), except for Be, B, Na, Mg, P, S, and Co for which vneu 2 (neutral-ion collision and the equilibrium abundance ratio between carbon and limit) begins to affect vneu. the tracer is N Σ m S m t • (2) Form the bottom panel of figure 1, we see that c = ce x = c x v + 1 ; neutral dirft (vneu ∗ fx dominates the final effective Nx e Σxe mc Sx mc tr outward drift for most elements, except for Ca. Λ t = c v + 1 ; (12) 2.2. Viscosity Accretion VS. Radiation Drift Λx tr Beside the outward drift driven by radiation, gas par- where mc is the atomic mass of carbon, and Λc,Λx are ticles are also subject to viscosity accretion toward the the abundance of carbon and tracer \X" in the source central star. The typical timescale for viscosity accretion material (see table-1 and the appendix). The time for (t ) and for radiation drift (t ) can be estimated as the v r carbon (tce ) and the tracer (txe ) to reach their equilibri- follows ums are respectively, r2 t ∼ tce = tv; v ν tvtr −1 5 α r txe = : (13) ∼ 1:4 × 10 yr (6) tv + tr 0:1 100 AU As t < t , thus the whole system reach its equilibrium r xe ce tr ∼ in ux t = t = t ; (14) u −1 r e ce v ∼ 4:7 × 105 x yr: (7) 1 m s−1 100 AU which denpends only on the viscosity accretion. Equating tv and tr, then the critical α, at which the 2.3.2. non-equilibrium case: if tage < tv inflow driven by viscosity accretion is comparable to the outflow driven by radiation force, can be derived as The system would not reach its equilibrium tage < te ≡ tv, where tage is the age of the system. Depending on the u α ∼ 0:33 x : (8) value of tr, such case can be divided into two sub-cases cr 1 m s−1 as the following. 1 Note that the different names of some variables as compared 2 For such an zero-order estimate, variables which depend on to those in Brandeker (2011). The two neutral limiting velocities, radial distance r, such as Σc,Σx, tr and tv, are not treated as a vneu 1 and vneu 2, are corresponding to the vdrift and vion in distribution of r but a characteristic scalar of the gas disk, namely Brandeker (2011), while vion in this paper denotes the limiting the corresponding value at r = Rdisk, where Rdisk (typically 100 velocity of ionized tracer particles.