ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: Disks The lack of correlation in Figure 9 implies a decoupling of the mm grains and gas. We can test this further by assuming instead that there is no decoupling, and by comparing the inferred gas content of the inner disk with the accretion rate. Assuming a standard ISM gas-to-dust ratio, we compare the dust mass and accretion rate in Figure 12 and overplot the resulting timescale for the gas and dust content of the inner disk to accrete onto the star. Similar to Figure 9 , no correlation appears to be present ( r corr = − 0.24 ± 0.26), which is again consistent with decoupling of the mm grains and accreting gas. Second, we find that the lifetimes of most inner disks are relatively short compared to the lifetime of the disk, typically less than 10 4 yr, assuming an ISM gas-to-dust ratio of 100. Figure 12. Inner disk dust mass vs. accretion rate. 0.5 dex of error is assumed for the accretion rate. An ISM-like gas-to-dust ratio of 100 is assumed for the inner disk, the validity of which is discussed in Section 4.5 . No correlation appears to be present ( r corr = − 0.24 ± 0.26), which is again consistent with decoupling of the mm grains and accreting gas. Assuming a gas-to-dust ratio of 100, the lifetime of the inner disks is typically less than 10 4 yr, well below the typical lifetime of a protoplanetary disk. Given the high occurrence rate of inner disks in our sample, a short lifetime would require that the inner disk is regularly replenished by episodes of enhanced accretion from the outer disk. However, depletion of dust in the inner disk due to radial drift such as suggested in the previous section may cause the gas-to-dust ratio of the inner disk to be several orders of magnitude larger (Pinilla et al. 2012 ), implying proportional longer lifetimes. While we do not have a direct measurement of gas content of the inner disk to assess this scenario, viscous accretion disk theory predicts a link between the gas surface density, accretion rate, and viscosity, which we can use to estimate the gas content. We assume the following relations:with ν the disk viscosity, α the viscosity parameter, c s the sound speed, H the vertical height, Ω( r ) the angular velocity, k B the Boltzmann constant, μ ≈ 2 the mean molecular weight, m p the proton mass, G the gravitational constant, and M * the stellar mass. The third equation assumes that the disk is vertically isothermal. These equations result in a relation between the gas surface density and the accretion rate (Manara et al. 2014 ):The gas surface density profile is derived from the dust profile where ξ inner and ξ outer are the gas-to-dust ratios for inner and outer disks, respectively:while for the temperature profile, we use Equation ( 4 ). This means that we can express the gas-to-dust ratio ξ inner :withand equivalently, we can express viscosity α as We note that both quantities α and ξ inner are independent of r and can be evaluated at any radius, and we can compute either quantity while