From our analysis, the peak locations move in radius as a function of azimuth. We introduce a spiral model to explain the features. This method was used in the disks of SAO 206462 (Muto et al., 2012), V1247 Orionis (Kraus et al., 2017) and MWC 758 (Benisty et al., 2015) to explain the spiral features.
The model is based on the spiral density wave theory, where the planet embedded in the disk is known to launch spiral waves (Lin & Shu, 1964; Ogilvie & Lubow, 2002).
The model was first predicted by Rafikov (2002), using WKB approximation.The spiral pattern can be described by the equation.
Figure 3.3: Results of NW(upper panel), SW(middle panel) and SE(bottom panel) spiral fitting. For left panels, color scale is the deprojected continuum emission with SU weight-ing. The red circles and the black squares are the peak locations. The blue line is the density wave launched by the predicted planet with 95% confident level in magenta arcs.
The diamonds are the predicted planet location at (rc, θ) = (0.′′45± 0.′′05, 253.1°± 24.6°), (0.′′35± 0.′′01, 312°± 12°) and (0.′′42± 0.′′01, 154.7°± 11.5°) for NW, SW and SE respec-tively. The disk aspect ratio (hc) of three features are 0.03±0.02 for NW, 0.018±0.003 for SW and 0.027±0.003 for SE. For right panels, the axes are distance and azimuth. The red circles are the selected peak locations to fit with the spiral feature model. The blue curve is the best-fit result of the spiral feature. The diamond is the predicted planet locations.
θ(r) = θ0+ sgn(r− rc)
This equation has five parameters. The rcand θ0 is the launching point, the predicted planet location in polar coordinates. α is related to the disk’s rotation, Ω(r) ∝ r−α. β is related to the sound speed (i.e., temperature), c(r) ∝ r−β. hc is the disk aspect ratio.
Disk aspect ratio is the ratio of the scale height to the radius at the location rc, hc= c(rc) / rc× Ω(rc) (Muto et al., 2012; Benisty et al., 2015). We fix α=1.5 assuming the disk is in Keplerian rotation and β=0.45 following Andrews et al. (2011).
We fit three features with the model. Fig. 3.3 shows the result of NW(upper panel), SW(middle panel) and SE(bottom panel) fitted with the spiral feature model. Diamonds are the predicted planet location at (rc, θ0) = (0.′′45± 0.′′05, 253.1°± 24.6°), (0.′′35± 0.′′01, 312°± 12°) and (0.′′42± 0.′′01, 154.7°± 11.5°) for NW, SW and SE, respectively. The disk aspect ratio (hc) of three features are 0.03±0.02 for NW, 0.018±0.003 for SW and 0.027±0.003 for SE.
Chapter 4 Discussion
4.1 Spirals
In order to check the degeneracy of the spiral model, I test my fitting with different bound values. With a smaller disk aspect ratio, the planet location tends to be closer to the spiral, while the pitch angle is larger around the planet. With setting a lower bound of hc
= 0, the spiral fittings result in small disk aspect ratios∼0.03 of our features, with planets close to the spirals.
The disk aspect ratio of a rotationally flatted disk is typically∼0.1. The disk aspect ratio is found to range from 0.05-0.25 obtained by a model which resolved the disks using scatter light data (Andrews, 2015). A low value indicates a colder disk, which is difficult to launch a spiral. The typical lower limit is hc∼0.01-0.03 (Muto et al., 2012). Our fitting results have small values of hcwhich approach the lower limit to detect spirals in the colder disk.
The disk aspect ratio is the ratio of the scale height to the radius. The scale height is the ratio of the sound speed to the Keplarian angular speed. We calculate the disk aspect ratio by the definition directly. The sound speed is obtained by the kinetic theory of gases with the temperature profile from Boehler et al. (2018). With the central star of 2 solar mass, we can calculate the Keplarian angular speed. The disk aspect ratio is∼0.07-0.09 over the disk.
The hc obtained from the spiral model result is too low for the disk. One possible
reason is that the spiral model is only suitable for linear and weakly non-linear regions.
However, with the uncertainties of the parameters obtained from the best-fit, the hcof 0.03 is within the expected disk aspect ratio of∼0.07.
I also find second-best fit results with higher limit bounded hcto the arm 1 and arm 2 fittings. These results have smaller adjust r-square than low hc cases. The hc and the predicted planet location (rc, θ0) are not independent. As the upper bound value of hs increases, these fitting have favored in the largest hc, up to the bounds. The higher the hc is, rcwould be smaller and closer to the phase center(the stellar location). However, larger hcwould produce too much infrared flux for the SED fitting (Andrews et al., 2011). The hcin upper limit bounded cases is not well constrained.
With the physical value range of disk aspect ratio (0.03 < hc < 0.2 (Andrews et al., 2011)), upper limit bounded is also a possible result. The disk aspect ratio obtained from the spiral model to the spirals seen in the NIR polarized image is 0.18 (Grady et al., 2013) and 0.2 (Benisty et al., 2015). The planet locations of our arm 1 and arm 2 suggested by the upper limit of hcare similar to the one determined by Benisty et al. (2015) with the same disk aspect ratio.
Figure 4.1: (a): Polarized intensity image (color scale) with peak locations (marked as red and cyan crosses) from the two Gaussian fitting. Both the NIR image and peak locations are projected. (b): Triple rings (marked as white solid, dotted and dashed curves) identified by Dong et al. (2018) superimposed with the deprojected image and the peak locations (marked as red dots and black squares) in the radius and azimuth coordinates.
The SE spiral at∼ 250° clearly deviates from the elliptical inner ring identified by Dong et al. (2018). (c): Analysis method by Dong et al. (2018) applied to our ALMA image.
The blue dots are the intensities averaged in azimuth ranging from 250° to 300° every 0.′′04 in radius. A wider inner ring at∼0.′′35 is seen, which we identify as the arm 2 spiral using our two Gaussian analyses.