Main

Himawari-8 is a Japanese geostationary meteorological satellite orbiting 35,786 km above the equator at 140.7° E (ref. 10). Since 7 July 2015, Himawari-8 has taken images of the entire disk of the Earth once every 10 min using its optical and infrared imager, the Advanced Himawari Imager (AHI; Extended Data Fig. 1). The Himawari-8 satellite also observes the region of outer space around the edge of the Earth’s disk during every scan; this motivated us to develop a new concept: using meteorological satellites as ‘space telescopes’ for astronomy (Fig. 1).

Fig. 1: Betelgeuse observed with the Himawari-8 meteorological satellite near the edge of the Earth’s disk.
figure 1

a, False-colour image taken by the Himawari-8 satellite on 19 January 2020. Betelgeuse is located at the top right corner of the image. b,c, Zoomed-in images of band 7. d, The response functions of Himawari-8; see the basic properties of the Himawari-8 data in Extended Data Fig. 1. e, The observed SED after averaging all observations with a model spectrum fitted to it.

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One of the unique aspects of this concept of using Himawari-8 for astronomy is that it enables us to obtain high-cadence time series of mid-infrared images, which are hard to acquire with the usual astronomical instruments. Small ground-based telescopes can acquire time-series measurements12, but the observations only cover wavelengths within the atmospheric window, and they are interrupted by the Sun for several months in most cases13,14. In contrast, astronomical satellites and aircraft-carrying telescopes can observe stars in most wavelength ranges15,16,17, but the observations require higher costs than ground-based telescopes. Survey telescopes orbiting the Earth for non-astronomical purposes—such as the Himawari-8 meteorological satellite—have the potential to overcome these problems. In this work, we demonstrate that the mystery of the Great Dimming of Betelgeuse can be solved with the aid of the time-series observations from Himawari-8, especially those in the mid-infrared, with which the amount of dust around Betelgeuse can be determined.

Examining the images taken by the Himawari-8 satellite, we found that several bright stars, including Betelgeuse, sometimes appear in the images (Extended Data Fig. 2). We therefore measured the amount of light coming from Betelgeuse, typically once per 1.72 d, between January 2017 and June 2021, and we made a 4.5 yr catalogue of light curves in 16 bands covering 0.45–13.5 μm (Extended Data Fig. 3a). Comparing the light curves, that is, the time series of spectral energy distributions (SEDs), with a model SED for Betelgeuse, we determined the time variation of the main parameters of Betelgeuse, namely the radius R, effective temperature Teff and extinction A(V) in the optical range. In addition to these three parameters, which can be determined from optical or near-infrared observation8,18, we also succeeded in determining the dust optical depth τ10 at 10 μm using our mid-infrared light curves. With this τ10, we will examine the dust-production history around Betelgeuse.

Figure 2 shows the main results of this paper, namely, the time variations of the stellar parameters R, Teff, A(V) and τ10 together with the AB magnitudes for band 3 at 640 nm, which is located around the central part of the Johnson R band. From the time variations of R, Teff and A(V), we found that the Great Dimming of Betelgeuse, the drop of ~1.2 mag in the V magnitude4, is likely to be caused by a cooling of ~140 K in Teff, which results in a dimming of the V magnitude by ~0.6 mag, and an increase of ~0.6 mag in A(V), which, by definition, darken the V magnitude by ~0.6 mag. This result supports a previously suggested scenario in which the Dimming was caused by a combination of a decrease in Teff and an increase in A(V) (refs. 1,8), although several shortcomings in our model SED should be kept in mind; some of the shortcomings are examined in Methods and Extended Data Fig. 4.

Fig. 2: Time variation of the stellar parameters of Betelgeuse.
figure 2

The black dots with error bars represent the measured values and their s.e.m., and the blue lines and blue shading show versions smoothed by averaging five points to guide the eye and their s.e.m. a, The light curve of Betelgeuse at band 3, centred at 640 nm, in AB magnitudes. b–e, The results of the SED fitting. b, R. c, Teff. d, A(V). e, τ10. The time of the Great Dimming is indicated by the red vertical line.

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However, the cause of the enhanced A(V) during the Great Dimming is still under debate. Several works have proposed that occultation by a newly formed circumstellar dust cloud might be a promising cause1,8. In contrast to this dust-formation scenario, several other works have argued that the dust formation is unnecessary to explain the Dimming. For example, three-dimensional (3D) effects7 or molecular opacity19,20, rather than dust extinction, can also explain the Dimming.

To examine whether or not the enhanced A(V) during the Dimming is due to the circumstellar dust, we focus on the time variation of τ10. Since τ10 was determined using the mid-infrared light curves of bands 12–14 at around 10 μm, where O-rich dust emissions dominate the flux21,22, it reflects the amount of circumstellar dust that emits the infrared light. Therefore, using our infrared τ10 measurements, in contrast to optical diagnostics, we can directly trace the amount of circumstellar dust. Our measurements in Fig. 2e show an enhancement of τ10 during the Great Dimming, though there is a slight time delay between the peaks of A(V) and τ10. This enhancement of τ10 indicates that a clump of gas did in fact produce dust that shaded the photosphere of Betelgeuse and contributed to the Great Dimming.

Two potential dust reservoirs have been found so far for red supergiants. The first is a diffuse and spread-out dust layer located a few to several tens of stellar radii from the photosphere21,22. In addition, instruments with very high spatial resolution have recently been used to find circumstellar dust clouds around red supergiants just a few tenths of a stellar radius above the photosphere23,24. The distribution and amount of dust in the spread-out layer remain mostly unchanged with time22,25, and thus it is unlikely that dust condensation in this layer is related to the large time variation of A(V) and τ10 during the Dimming that we found. In contrast, the latter, lukewarm region26 coexists with the warm chromosphere, the temperature of which9,27 is sufficiently high for dust sublimation28. Therefore, a dust clump that condenses near the photosphere may soon be sublimated by chromospheric heating, or, if the radiative acceleration is insufficient, it may fall back to the photosphere to be sublimated29. Considering these points, we conclude that the enhancements of A(V) and τ10 during the Dimming may have occurred very close to the photosphere. This hypothesis is also supported by the fact that the polarized brightness of this circumstellar envelope changed substantially around the Dimming30,31.

To further examine the dust-condensation process, we focus here on the time variation of the amount of molecular gas around Betelgeuse, in which the dust condensation occurred, and in particular on H2O molecules. In addition to the dust emission seen at wavelengths around 10 μm, the number of molecules, particularly H2O, can also be traced using Himawari-8 photometric data. Since H2O molecules in the Earth’s atmosphere absorb starlight32 in the wavelength ranges where the H2O molecular features from stars exist33, the observation of extraterrestrial H2O molecules usually requires space telescopes15. One of the missions of the Himawari-8 satellite is to observe the distribution of water vapour in the Earth’s upper atmosphere10, and thus Himawari-8 has the wavelength bands where H2O features exist. These bands, bands 8–10 at around 6–8 μm, are therefore useful for time-series observations of H2O molecules around Betelgeuse. We focus here on the flux variation of band 8, where the signal-to-noise ratio of the H2O feature is maximized, thanks to the strength of this feature33,34 and the sensitivity of the Himawari-8 satellite (Extended Data Fig. 1). We note that the 6–8 μm wavelength range is a complicated region in which other molecular bands are present, for example, SiO molecules. Thus, we should keep in mind the possibility of contamination by these molecules, and further spectroscopic observations are required to fully disentangle the problem. Nevertheless, band 8 is the least contaminated by other molecules15,20,34.

Figure 3 shows the time variation of the flux excess in band 8, defined as the ratio of the observed flux to predicted photospheric flux minus 1, which is likely to trace the amount of H2O molecular gas around Betelgeuse, maybe in the MOLsphere15,34. When gas clumps outside the photosphere are present that emit or absorb H2O molecular features, the excess is positive or negative, respectively. In the first half of the observing period, that is, before September 2018, to within the error bars the flux excess was almost consistent with zero, which indicates that there was no extra detectable H2O gas around the photosphere. In contrast, most of the data points after September 2018 show negative flux excesses. This indicates the emergence of a clump and/or a layer of cool gas containing H2O molecules, in which the dust that dimmed Betelgeuse may have formed. Considering the clumpy nature of the dust cloud that dimmed Betelgeuse1 and the existence of the warm gas clump found from ultraviolet observations of the Mg ii h and k lines around October 20199, the H2O gas that we found is also likely to be a clump.

Fig. 3: Molecular H2O emission and absorption.
figure 3

The flux excess in band 8, where molecular H2O features exist. The blue line and blue shading show a smoothed version and s.e.m. by averaging five points of the original black dots with error bars (measured values and s.e.m.) to guide the eye. We found a rapid transition in the flux excess from negative (absorption) to positive (emission) in early April 2019 (green vertical line), which occurs 10 months before the Great Dimming in early February 2020 (red vertical line).

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Finally, and more interestingly, we tentatively identified a rapid transition in the H2O feature from absorption to emission in early April 2019 (the green vertical line in Fig. 3). After visual inspection of the unbinned light curve for band 8, we found that the timescale of this transition is likely to be about a week or less. This timescale is much shorter than either the theoretical hydrodynamic timescale of a few weeks to a few months35 or previously observed timescales of changes in the H2O features seen in evolved red stars36,37. Because the gas clump cannot move outside the line of sight to the photosphere on such a short timescale, this transition indicates the occurrence of an episodic bursty event. One may speculate that the cause of this unusual event is related to another rare event, the Great Dimming; if so, it would be easy to imagine that such an unusual event could occur. One conceivable scenario involves a shock wave passing through the clump. It has been suggested that such a shock emerged from the bottom of the photosphere in January 2019 and propagated from there to the outer envelope of Betelgeuse19. Together with the fact that A(V) began to increase at the time of this transition (Fig. 2d), this shock might be related to the triggering process of the Great Dimming. Future theoretical investigations and time series of spatially resolved infrared observations would enable us to tackle this further mystery.

Methods

We first aimed to establish and test a new concept, using meteorological satellites as astronomical space telescopes, to obtain multiband, long-term and continuous light curves uninterrupted by the Sun. Second, we applied this technique to Betelgeuse to investigate its mysterious Great Dimming.

Images of celestial objects, particularly the Moon, obtained by meteorological satellites have long been used only to calibrate and ensure the quality of Earth observations38,39,40. In contrast, we attempted to use these images to investigate stellar astrophysics. Here, we thoroughly describe the data properties and the photometric methods and their validation.

Basic properties of the Himawari-8 scanning observation

Geostationary meteorological satellites, such as Himawari-8, observe the entire disk of the Earth by sequential scans with several rows, each of which is termed a swath10,41. Each swath is scanned with a one-dimensional (1D)-like detector array from west to east. Due to this scanning procedure, the properties of images obtained by Himawari-8 have several differences compared with those obtained by typical astronomical observations using two-dimensional devices. Nevertheless, we can use the reduced and distributed data, termed the L1B product, as ordinary astronomical images, provided that several caveats are taken into consideration.

First, pixel navigation and data sampling are well controlled and regular, and their errors are usually smaller than 0.5 pix for Himawari-842. Thus, we can precisely transform the position in the image onto its location in the sky.

Second, the fill factor of the pixels in the north–south direction is almost 100%, and the east–west sampling interval is smaller than the pixel size41. As such, we do not need to be concerned with undersampling due to gaps between pixels. Moreover, after the resampling process during the reduction described below, each image is transformed to the fixed square grid, which is very easy to use.

Third, because each row of unresampled images is observed by a single detector, inaccurate calibration of the detector occasionally introduces a stripe pattern into the obtained images. The strength of such a stripe has been shown to be small for observations of Earth scenes43, but it has not yet been closely examined for faint sources such as stars. Therefore, it should be kept in mind that flux measurements of stars are possibly influenced by this relatively large systematic error.

Finally, other possibly minor effects include quantization noise, nonlinearity of the detector response, and frequency of the calibration. We will further examine some causes of systematic and/or statistical errors and validate the photometric results in Supplementary Discussion.

In this work, we limit our analyses to the images observed with the AHI on board the Himawari-8 satellite, which is a ‘third-generation meteorological satellite’ characterized by access to as many as 16 bands10. There also are three other third-generation satellites equipped with 16-band imagers: GOES-16 and GOES-1744, which are operated by the National Aeronautics and Space Administration and the National Oceanic and Atmospheric Administration, and GEO-KOMPSAT-2A45, which is operated by the Korea Meteorological Administration. Although they have been operational since November 2016, March 2018 and July 2019, respectively, flux values outside the Earth’s disk are found to be masked and unavailable in their published standard L1B products, which makes these data unusable for our analyses. Combined analyses with these satellites would improve the precision of our results, if their images of the space around the Earth’s disk were to be made publicly available.

Reduction of the AHI images is performed by the Japan Meteorological Agency41,46. This involves geometric regridding into a fixed coordinate system using a Lanczos-like resampling; a linear and global flux calibration using deep space as the background, with a solar diffuser for the optical and a blackbody for the infrared as standard flux sources; and nonlinear flux calibrations based on prelaunch ground tests. No corrections are made for atmospheric effects, nor is masking for cosmic rays performed.

Photometry of the Himawari-8 image

About 6 yr of 16-band images have been obtained by the Himawari-8 satellite at a frequency of once every 10 min. We restricted the scope of our analyses to data observed from January 2017 to June 2021 because systematic biases in the flux calibration and in the positioning were identified in observations from July 2015 to November 2016 (Supplementary Discussion). From this large number of images, we tried to find and measure the light from Betelgeuse, which is moving with the speed of the Earth’s rotation, 15° h−1, and which is found to be located in the field of view about once every two days. First, we calculated the times when Betelgeuse is in the direction of the Earth using the Astropy package47, and we collected all the images in which Betelgeuse could be detected. Then we searched for the location of the Betelgeuse signal in each of these images in bands 1–10, where the signal-to-noise ratio (SNR) is high enough to detect the star. For each of these high-SNR bands, we searched for the location of the star by choosing the pixel with the largest counts among those pixels where Betelgeuse is expected to be located. In contrast, we inferred the locations in the other bands (bands 11–16) by shifting from the position of Betelgeuse in band 7, where the SNR is the highest, by around 30 arcsec depending upon the differences in the times of observation for the two bands.

The pixel scale of the published images in each band is larger than the actual radius of the point-spread function estimated from the radius of the telescope. Thus, the light from a point source occupies only 1 × 1 pix2 of the unreduced images, and it spreads out to ~2 × 2 pix2 during the reduction process (Fig. 1a–c). Due to this small point-spread function radius, point-spread function photometry does not work well, so instead we used the aperture-photometry technique with a 7 × 7 pix2 square aperture. The background flux in the aperture was estimated and subtracted by fitting the flux within a square annulus around the star using a two-dimensional polynomial function. We set the inner width of the annulus to be 7 pix and the outer width to be 43, 21 and 21 pix for the bands with spatial resolutions of 0.5, 1.0 and 2.0 km, respectively. We selected these widths to minimize the impact of stripe-shaped fluctuations of the background. We selected the order of the polynomial to be in the range 0–4 for each measurement so that the Akaike information criterion calculated from the fitting residuals is minimized. Then, we estimated the stellar flux by summing the background-subtracted spectral irradiances within the 7 × 7 pix2 aperture.

Since the background shapes are not guaranteed to be continuously connected between adjacent swaths, we excluded from our analyses cases in which the 7 × 7 pix2 aperture spans multiple swaths. Furthermore, we trimmed the square annulus used for background estimation by requiring that the annulus does not span multiple swaths. Moreover, to avoid contamination from terrestrial light, we introduced distance thresholds from the edge of the Earth’s disk (Extended Data Fig. 1).

Validation of the photometry

Here, we performed a wide variety of validation steps, not only with Betelgeuse but also with other bright stars located in the field of view of the Himawari-8 satellite (Extended Data Fig. 2). From the validation described in Supplementary Discussion and Supplementary Figs. 1–6, we found no evidence of any serious systematic bias in our photometry, and we thus conclude that the measurements are ready for further analyses. Nevertheless, it would be important, in a coming decade, to examine whether or not there are overlooked systematic effects in our method.

SED fitting to the Himawari-8 light curves

Figure 1d,e shows the SED averaged in time together with a model SED having the typical atmospheric parameters of Betelgeuse. The photometric bands with wavelengths of ≲8 μm (bands 1–10) are almost unaffected by the circumstellar dust emission, and thus they can be used to determine the atmospheric parameters of Betelgeuse. In contrast, the other bands receive radiation from silicate-dust emission at around 10 μm, and they can thus be used to measure the amount of dust formed around Betelgeuse. We therefore first fitted the SED for bands 1–10 with a model photosphere for Betelgeuse to determine the photospheric parameters—R, Teff and A(V)—and we then determined τ10 using bands 12–14, which have the highest SNR for the dust emission. Five examples of the fitting results are shown in Extended Data Fig. 3b–f.

Our final time-series photometric catalogue of Betelgeuse is an incomplete one; the catalogue lacks fluxes in several bands on several observations. This is especially serious for longer-wavelength bands, where the sky-background emission is stronger, and thus a larger number of photometric measurements was removed from the catalogue when the star was located near the edge of the Earth’s disk. Moreover, the signal-to-noise ratios in the mid-infrared bands are typically small, ~1. To deal with these difficulties, we binned each light curve over 5–12 observations, requiring that all the binned data points for bands 1–10 include five or more points. Note that the number of days in each bin is automatically determined by the observation frequency, which depends on the observation epoch. For example, there is a clump of observations in May–June 2018 with a frequency of 0.85 d, which leads to the smaller number of days in the bins. Finally, we calculated the resulting median and its standard error for each bin.

Model SED for the photosphere

We fitted each point of this ‘complete’ SED curve for bands 1–10 with a photospheric model SED using the LMFIT package48. To obtain the model spectrum for the photosphere, we interpolated a grid of BT-NextGen model spectra49 with solar metallicity and the surface gravity log g = 0.0, assuming that this 1D model can be used to model the spectra of Betelgeuse in any phase. We used Cardelli’s reddening law50 with the selective-to-total extinction ratio RV = 3.1 (ref. 51), assuming that this law can be extrapolated to longer wavelengths. We also used the parallax of 6.55 mas measured with the Hipparcos satellite52. Although the accuracy of the parallax of Betelgeuse is controversial53,54, the choice of parallax value only changes the fitted radius in a relative way and does not affect the interpretation of the results. For the same reason, we do not take into account the error in the parallax value. The model spectra are then convolved with the response function of the AHI (September 2013 version retrieved from https://www.data.jma.go.jp/mscweb/en/himawari89/space_segment/spsg_ahi.html) to calculate the model SEDs for Betelgeuse, and we fitted the observed SED with the model.

Dependence on the model selection

For the grid of model spectra, we first tried the MARCS grid55 with 15 M⊙ (refs. 54,56,57,58,59), which is usually used to model the spectra of red supergiants51,60. We found that the MARCS grid gives a smaller Akaike information criterion than the BT-NextGen grid, probably because the wavelengths and the strengths of the molecular bands in the MARCS grid are more accurate. Moreover, considering the fact that several previous works51 succeeded in reproducing observed optical spectra of red supergiants dominated by molecular TiO bands with MARCS models, MARCS models appear to be an appropriate grid for the synthesis of TiO bands in red supergiants. However, we found that the results of the fits obtained using the MARCS grid were physically unreasonable for most cases that we considered, for example, A(V) < 0 mag, although the reason, possibly related to 3D effects and/or circumstellar gas, is unclear. Thus, instead of the MARCS grid, we used the BT-NextGen grid to compute the model SED. Testing both the MARCS and NextGen grids, we found that the model parameters we obtained using the BT-NextGen grid vary with time in the same way as those obtained using the MARCS grid; that is, the differences between the time variations obtained using the two grids are roughly constant (Extended Data Fig. 4a–c). Therefore, the results of this paper using the NextGen grid are not affected quantitatively by the choice of model grid, as long as the time variations are interpreted in a relative way.

We have also tested whether or not other model assumptions affect the results. For example, a change in log g from 0.0 to −0.5 makes Teff systematically ~80 K warmer and A(V) systematically ~0.3 mag larger. Similarly, changing RV from 3.1 to 4.4 (ref. 51) decreases Teff by ≲10 K and A(V) by ≲0.05 mag. From these tests, we concluded that, although the absolute values of R, Teff and A(V) depend on the model assumptions, their relative variations appear to have been determined to an accuracy of ~10 K in Teff and ~0.05 dex in A(V), both of which are much smaller than the fitting errors, ~40 K and ~0.2 dex, respectively.

3D effects in the photosphere

The temperatures that previous works8,19,61,62 and we have both derived assumed a 1D geometry for the model atmosphere. Thus these temperatures are affected by 3D effects in the optical spectrum of the real star7. In fact, the SED of a model with a given Teff, especially the strength of the TiO molecular absorption bands that appear in the optical spectra of red supergiants8, has a variety of shapes due to 3D effects35. Therefore, when we fit the observed SED of the real 3D photosphere of Betelgeuse with a 1D model SED, the fitted values of Teff and A(V) depend not only on Teff and A(V) themselves but also on the 3D effects35. In particular, a time variation of the patchiness of the surface63,64 could affect the time variations of Teff and A(V) (ref. 7).

As a simplified test of the 3D effect on our fitting result, inspired by the optical image of Betelgeuse obtained during the Dimming1,25, we considered a model photosphere composed of two regions with different Teff, which is similar to the model used in a previous work7. In this model, we assume that the cooler part of the photosphere has a varying temperature and a varying area and that the warmer part has a fixed temperature of 3,650 K. In contrast to the previous model7, we are able to consider two additional free parameters, A(V) and R; without these two parameters we could not obtain a satisfactory fit to our multiwavelength data. Altogether, we considered four free parameters (R, A(V) and the temperature and area fraction of the cooler part); for example, when the entire area of the photosphere is occupied by the cooler part, the fraction is 1.0. With this four-parameter model, due to the strong degeneracy between A(V) and the area, sometimes the fitting failed and gave unphysical results. To remove these cases, we excluded fitted results with areas greater than 1.0 or less than 0.0 as well as results for which the temperatures reached the boundaries of the acceptable range (we assumed 2,800 and 4,000 K for the lower and upper boundaries, respectively).

Extended Data Fig. 4d–g compare the fitted results obtained with the three-parameter model adopted in the main text and the four-parameter model described here. We found that with the four-parameter model the contribution of the increased A(V) to the Dimming decreases but that it is still larger than zero (~0.4 mag). This result supports the robustness of our results in a qualitative sense. Since the four-parameter model examined here is too simple to take 3D effects fully into account7, future theoretical modelling with self-consistent hydrodynamic models65 will be necessary.

Comparison with previous radial-velocity (RV) measurements

To check our SED fitting results, we compared the radius determined herein with previous RV measurements. Because the radius is given by the integration of the RV, we can validate the radius curve using literature RV curves.

First, we found time lags of up to ~80 d between the times of maximum brightness and minimum radius (Fig. 2). Although such a large lag was not expected in a simple 1D hydrodynamic model54, a lag between light and RV curves has been observationally found for Betelgeuse9,19,66. Moreover, we found clockwise trajectories between the R/Teff and band 3 flux for most observation periods (Extended Data Fig. 3h–o), which has been predicted theoretically54,67. The exception, the relation between R and band 3 flux between 27 October 2017 and 24 December 2018 shown in Extended Data Fig. 3i, could be attributed to effects other than the fundamental-mode pulsation, such as the long secondary period68; however, the actual cause remains unclear.

Second, we estimated the RV curve by calculating the differential of our radius curve, and compared it with literature RV curves19,66. This comparison reveals an overall good agreement between our data and the results cited above, supporting the validity of our radius measurement, and hence other parameters. However, minor differences of up to ~5 km s−1 may exist, particularly in September 2018–November 2018 and September 2019–December 2019. Plausible causes of these differences are (1) our imperfect modelling of Himawari-8’s SEDs with a 1D model atmosphere, as discussed in the previous subsection, and (2) differences in the atmospheric layers that were observed in the present and previous studies, and hence the real, physical difference in RVs19.

Model SED for the dust emission

After fitting the SED curve for bands 1–10, we calculated the expected photospheric flux curves for bands 11–16 and subtracted them from the observed flux curves. These residual fluxes are likely to originate from the circumstellar dust emission, and they are useful for estimating the variation of the amount of circumstellar dust.

For the model spectra of the circumstellar dust emission, we used the radiative-transfer code DUSTY V269. For the settings of the DUSTY calculation, we basically followed the procedure described in a previous work70, except for several modifications. First, we only fitted τ10 and fixed the other parameters. We used the ‘warm silicates’ dust composition71, a constant grain size of 0.3 μm and a constant dust temperature of 1,300 K at the inner boundary, which roughly corresponds to the sublimation temperature of silicate dust28. Although inner-boundary temperatures less than 1,300 K are sometimes found using observations of red supergiants70, we adopted this value since a different inner-boundary temperature only results in a systematic bias in τ10 as long as the temperature is independent of time. We also tried the assumption of a constant radius at the inner boundary, and we found that the results do not change in a qualitative sense. For the stellar radiation that illuminates the dust, we used the time variation of Teff determined in the previous section to calculate the natal photospheric SED, after applying spline smoothing to reduce the noise in each measurement. Then, we calculated the expected silicate-dust emission in bands 12–14 on the τ10 grid and determined the τ10 value that best matches the observed flux-residual curve for each of bands 12–14. Finally, we calculated the mean of the three τ10 values determined from bands 12–14 to obtain the final estimate of τ10. We note that the τ10 that we determined is in fact not the actual optical depth at 10 μm; it is rather a model parameter that we use to match the observed and model fluxes of bands 12–14, and we use it as an indicator of the amount of circumstellar silicate around Betelgeuse.