Inner dusty regions of protoplanetary discs - I. High-resolution temperature structure

Abstract

Our current understanding of the physical conditions in the inner regions of protoplanetary discs is being increasingly challenged by more detailed observational and theoretical explorations. The calculation of the dust temperature is one of the key features that we strive to understand and this is a necessary step in image and flux reconstruction. Here, we explore the coexistence of small (0.1‐μm radius) and large (2‐μm radius) dust grains, which can coexist at distances from the star where small grains would not survive without large grains shielding them from the direct starlight. Our study required a high‐resolution radiative transfer calculation, which is capable of resolving the large temperature gradients and disc‐surface curvatures caused by dust sublimation. This method of calculation was also capable of resolving the temperature inversion effect in large grains, where the maximum dust temperature is at a visual optical depth of τV∼ 1.5. We also show disc images and spectra, with disentangled contributions from small and large grains. Large grains dominate the near‐infrared flux, mainly because of the bright hot inner disc rim. Small grains populate almost the entire interior of the inner disc, but they appear at the disc’s surface at distances 2.2 times larger than the closest distance of the large grains from the star. Nevertheless, small grains can contribute to the image surface brightness at smaller radii because they are visible below the optically thin surface defined by stellar heating. Our calculations demonstrate that the sublimation temperature does not provide a unique boundary condition for radiative transfer models of optically thick discs. The source of this problem is the temperature inversion effect, which allows the survival of optically thin configurations of large grains closer to the star than the inner radius of the optically thick disc. Future attempts to derive more realistic multigrain inner disc models will need the numerical resolution shown in our study, especially if the dust dynamics is considered where grains can travel through zones of local temperature maxima.

Constraints on the height of the inner disk rim in pre-main-sequence stars

“…Isella & Natta 2005;Tannirkulam, Harries & Monnier 2007), because big dust grains can survive in the optically thin zone closer to the star than R in (see examples by Kama, Min &Dominik (2009) andVinković (2012)). Hence, in this estimate we ignored optically thin dust within R in because we have no information on its density structure (Vinković 2012).…”

“…This is also the place of local temperature maximum equal to local values of T sub (see Figure 3 in Kama, Min & Dominik 2009). High-resolution radiative transfer calculation shows that the entire inner rim experiences temperatures close to T sub , with dominantly radial temperature variation close to the rim (Vinković 2012). Hence, a vertically isothermal disk is a good approximation as long as we consider only the rim properties.…”

“…Even the most resilient dust grains sublimate in these conditions, but the exact sublimation distance from the star depends on the grain size and chemistry and on the local gas pressure. Removal of dust grains creates large steps in the opacity gradient, which complicates modeling the dusty disk structure (Kama, Min & Dominik 2009;Vinković 2012). Modeling efforts have therefore been devoted to approximate methods, motivated by the peculiar near-infrared bump in the spectral energy distribution of many pre-main-sequence stars.…”

“…Small grains have often been included into the dust opacity of the inner rim, but now we know that the shape of inner disk rim is dictated solely by big grains ( 1µm) because they are more resilient to sublimation than smaller grains. Hence, big grains populate the inner disk surface and shield smaller grains in the disk interior from direct stellar radiation (Kama, Min & Dominik 2009;Vinković 2012). An unexpected consequence of radiative transfer in big grains is that the location of the inner disk rim becomes a nontrivial problem, with the temperature inversion effect producing a local temperature maximum within the dust cloud (see the appendix).…”

Two-dimensional Monte Carlo radiative transfer modelling of the disc-shaped secondary of Epsilon Aurigae

“…The disc is considered here as to have formed by the accretion process and hence we use a standard flared accretion density structure (Lynden‐Bell & Pringle 1974) described as where ω is the radial coordinate of the disc mid‐plane and the scale height increases with radius as h = h 0 (ω/ R star ) β . For our models we adopt a flaring parameter β= 1.25 which was based on the accretion disc models at hydrostatic equilibrium (D’Alessio et al 1999) and was used to describe the structure of the accretion disc (Wood et al 2001; Thi, Woitke & Kamp 2011; Vinkovic 2012) and the value of α= 3(β− 0.5) = 2.25 (Shakura & Sunyaev 1973). We take h 0 = 0.05 R star such that the disc will have a thickness of 0.76 au at its outer edge of radius 3.8 au as constrained by the IR interferometric observations of Kloppenborg et al (2010).…”