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Appendix A: Synthetic position velocity diagrams for a Keplerian disk

	Fig. A.1 Synthetic position-velocity diagrams for the 1 mm methanol line (gray contours). The contour levels and axis ranges are the same as in Fig. 4. The red points with error bars show the line centroid along the cut. The diagrams have been computed assuming that the emission originates in a Keplerian disk with different masses of the central protostar (see text).
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To test the Keplerian disk scenario, we have computed synthetic P.V. diagrams for the 1 mm methanol line. We have assumed that the emission follows a Gaussian distribution, with a FWHM of 0.44′′ (as determined from the visibility fit). The line-of-sight velocity is assumed to be given by (see, e.g., Guilloteau et al. 2012) (A.1)where r and θ are the cylindrical coordinates in the disk plane, i is the inclination of the disk rotation axis with respect to the line of sight (i.e., i = 90° for an edge-on disk), v_LSR is the source

velocity in the local standard of rest, and is the Keplerian velocity (A.2)with G the gravitational constant and M_∗ the mass of the central protostar. The line is assumed to have a Gaussian shape, with a local linewidth () equal to , in agreement with the value determined in DM Tau by Guilloteau et al. (2012). Although this value is uncertain, it cannot exceed because a disk with a larger would be unstable. We assume that the disk is rotating about an axis with i = 60°. This is consistent with the disk rotating about the outflow axis, which has i ≥ 45° (Codella et al. 2014). Finally, we assume a v_LSR of 6.5 km s^-1, a source distance of 235 pc, and M_∗ is left as a free parameter. The disk is not truncated in radius; we merely assume that its line emission decreases as a Gaussian, as mentioned above. We compute a synthetic data cube with a pixel size of 0.1″, and a channel width of 0.5 km s^-1. The intensity of the emission as a function of channel is computed assuming that the line-of-sight velocity varies as a function of r and θ following Eq. (A.1). This data cube is then convolved with the synthesized beam, assumed to be a Gaussian with a FWHM of 0.8′′. The convolved data cube is then scaled so that the peak intensity matches the observations (453  mJy/beam). Finally, Gaussian noise is added in each channel so that the signal-to-noise ratio also corresponds to the observations (~21).

Figure A.1 shows the synthetic position velocity diagrams we obtain for different values of M_∗. We find that our observations are inconsistent with a Keplerian disk, regardless of the mass of the central object. For M_∗ = 0.05  M_⊙ and 0.1  M_⊙, the synthetic P.V. diagrams have two peaks, while the observed P.V. diagram has one peak (see Fig. 4). For M_∗ = 0.01  M_⊙, the synthetic P.V. diagram is single peaked, and the predicted first-order moment along the cut is constant, within the error bars. However, the predicted linewidth is much smaller than observed (compare the size of the P.V. contours along the vertical axis in Figs. 4 and A.1). A broader linewidth would require a larger value of . However, we find that even for a very turbulent disk with , the predicted linewidth is 2–3 times smaller than observed. We conclude that the observed methanol emission can not arise from a Keplerian disk, and must originate in the infalling and perhaps slowly rotating inner envelope. A more detailed modeling of the methanol line emission is needed to determine the precise velocity field of the inner region of the protostar envelope.

© ESO, 2014