Source properties
R Doradus is an M-type AGB star that belongs to the class of semi-regular pulsators. On a timescale of about 1,000 days, it switches between two pulsation modes with periods of 362 and 175 days. The distance to R Doradus has been determined to be 55 ± 3 pc using revised Hipparcos measurements6. There is no usable Gaia parallax. From CO observations, it was determined that R Doradus has a relatively low mass loss rate (about 10−7 M⊙ yr−1) and wind expansion velocity (about 5.7 km s−1)30. Previous ALMA observations also indicated that R Doradus rotates fast for a giant star, with a rotation velocity at the surface of about 1.0 ± 0.1 km s−1 compared to a rotation velocity of a few tens of metres per second expected for solitary AGB stars31. It has been suggested that the apparent rotation could be the result of a chance alignment of convective cells20. However, the rotation has been observed in several molecular lines at four different epochs that, including the observations presented here, span more than 6 years (for example, refs. 12,31). This is much longer than the convective timescales found in our analysis, and hence a chance alignment of convective cells can be ruled out.
In our comparison with the convective theory, we adopt the values for the effective temperature of Teff = 2,710 ± 70 K. For the surface gravity, we use log[g] = −0.6 ± 0.1, based on models that indicate that the initial mass was 1–1.25 M⊙ and that the current mass is 0.7–1.0 M⊙, combined with interferometric measurements that yield a stellar diameter in the infrared of DIR = 51.18 ± 2.24 mas (ref. 28). It is expected that this diameter, which corresponds to a radius of RIR = 1.4 ± 0.1 au = 298 ± 21 R⊙, indicates the size of the stellar photosphere. We can compare this with the (τ = 1) size of the star determined with ALMA at 338 GHz obtained by visibility fitting. Using a combination of the last three epochs, we fit a nearly circular stellar disk with F338GHz = 521 ± 18 mJy, D338GHz = 59.8 ± 0.4 mas and an axis ratio of 0.99 ± 0.01. This means a brightness temperature Tb = 2,270 ± 130 K and, taking into account the uncertainty on the distance, a radius R338GHz = 1.64 ± 0.09 au = 353 ± 19 R⊙ = 1.18 ± 0.11 RIR.
Observations, data reduction and imaging
The AGB star R Doradus was observed in ALMA bands 6 and 7 as part of the ALMA project 2022.1.01071.S (principal investigator: T.K.). The band 7 observations were taken between 5 July and 2 August 2023 using four spectral windows centred at 331.2, 333.0, 342.1 and 345.1 GHz. Each spectral window had a bandwidth of 1.875 GHz and 1,920 channels. The integration time of the individual visibilities was set to 2.02 s. The observations were taken in the largest ALMA configurations (C-9 and C-10) with the quasars J0519-4546 and J0516-6207 as bandpass–amplitude and phase calibrator, respectively. Details of the observations are presented in Extended Data Table 1. The calibration of the last three epochs was carried out using the ALMA pipeline in CASA v6.4.1.1232. The first two epochs were labelled as semi-pass in the ALMA quality assurance and for these the calibration was carried out manually by staff from the Nordic ALMA Regional Center node using CASA v6.5.4.9. In the first epoch, there was an issue with the bandpass calibration that needed to be solved manually. In the second epoch, one of the antennas needed to be flagged, resulting in a loss of some of the longest baselines. For both epochs, the requested angular resolution and sensitivity were not reached. After the initial calibration of each epoch, molecular lines were identified and flagged before the data were averaged to an integration time of 6.06 s and to 50 channels per spectral window. Subsequently, two steps of phase-only self-calibration were carried out on the stellar continuum. The self-calibration improved the signal-to-noise ratio by a factor of about 2.5 on the continuum. Finally, images, using all four spectral windows, were produced for the five epochs using superuniform visibility weighting33. This method increases the relative weight of the visibilities at the longer baselines, which minimizes the beam size at the expense of signal-to-noise ratio. The superuniform beam characteristics and continuum root mean square (r.m.s.) noise are also given in Extended Data Table 1. The increase of r.m.s. noise compared to more regular Briggs weighting (with a robust parameter of 0.5) depends on the telescope distribution and was a factor of about 2, 1.5, 1.5, 3 and 7 for the five epochs, respectively. The improvement in angular resolution between superuniform and uniform weighting ranged from about 2% in the final epoch to about 15% in the third epoch. The superuniform-weighted images of the final three epochs are presented in Fig. 1 and those of the first two epochs are shown in Extended Data Fig. 1. The images of the highest-angular-resolution epochs were used to derive the angular radial profile presented in Fig. 3. We verified that reducing the angular resolution to match that of the epoch with the largest beam smooths out the observed structures and the derived velocities and thus use the highest-angular-resolution results. The fits of the stellar disk and the spatial PSD analysis were carried out on the calibrated visibilities. Although we focus our analysis on the higher-resolution band 7 observations, we also include the continuum result from the band 6 observations in the first section of the Methods. The observational details for these observations are also included in Extended Data Table 1, and the calibration, self-calibration and imaging steps were identical to those carried out for the band 7 observations. The four spectral windows are centred at 218.9, 220.8, 230.0 and 232.9 GHz, and the increase in the continuum r.m.s. noise between Briggs weighting and superuniform weighting is a factor of 1.5.
Spatial PSD analysis
The spatial PSD is regularly used to derive information about, for example, the turbulent structure of the interstellar medium (for example, refs. 34,35) as well as the convective structure of the solar photosphere (for example, ref. 36). The spatial PSD is given by the two-dimensional Fourier transform of an image. However, considering interferometric images are themselves the Fourier transform of the interferometric visibilities, the spatial PSD is equal to the modulus squared of the complex visibilities34. We can thus calculate the PSD directly from our interferometric visibilities without introducing potential artefacts during the imaging process. As the PSD would be dominated by the power at the scales of the stellar disk, we first use the uv-fitting code uvmultifit37 to fit the stellar disk in uv-coordinates (for a discussion on the disk profile, see the first section of the Methods). Subsequently, we subtract the disk from the visibilities, after which, we calculate, for a phase centre towards the centre of the star, the PSD using visibilities annularly averaged in equally spaced bins of uv−distance (in units of kλ). From the inverse of the uv−distance, we can directly obtain the angular scale x in milliarcseconds. In addition, we also determine the PSD for a position offset from the star. We chose an offset of 7 as to avoid a contribution to the off-source PSD from the source signal. As the first two epochs have worse spatial resolution, we produce the PSD only for the final three epochs. The results are shown in Fig. 2. By comparing the on- and off-source PSD, we can determine which structures are significant and which are possibly due to correlated noise in the visibilities.
Stellar disk profile
To check how well a top-hat-shaped stellar disk profile fits the observations, we have also investigated the stellar disk profile in the image plane. We produced a radially averaged profile of R Doradus based on the combined data for the final three epochs. We then compared this profile with a top-hat-shaped stellar disk model convolved with the interferometric beam. The results of this comparison are shown in Extended Data Fig. 2. The stellar disk model can accurately describe the observations with residuals at a level of about 2% of the peak emission. This means that the 338 GHz optical depth τ338GHz increases steeply over only a small change in the radius. The radial motions we observe thus reflect the physical motion of the 338-GHz optical depth surface. As the optical depth is a strong function of the density11, the motions closely reflect the motion of the shocks induced by the convection.
Band 6 observations
The ALMA band 6 (225 GHz) observations and the radius as a function of position angle are shown in Extended Data Figs. 3 and 4, respectively. Fitting the interferometric visibilities yields a completely circular stellar disk with F225GHz = 221.4 ± 0.1 mJy and D225GHz = 61.8 ± 0.1 mas. This means a brightness temperature Tb = 2,006 ± 8 K and, including the distance uncertainty, a radius R225GHz = 1.70 ± 0.09 au = 365 ± 20 R⊙ = 1.22 ± 0.11 RIR. In a comparison between the radii of the 225-GHz observations and the last epochs of the 338-GHz observations in Fig. 4, it is clear that there is a very good correspondence. As the different observations are completely independent, this shows that the pattern seen in the radii is intrinsic to the source at the time of the observations.