# The Photospheric Temperatures of Betelgeuse during the Great Dimming of
2019/2020: No New Dust Required

Graham M. Harper Center for Astrophysics and Space Astronomy
University of Colorado Boulder
389 UCB
Boulder, CO 80309
USA  Edward F. Guinan  Astrophysics and Planetary Science Department
Villanova University
Villanova, PA 19085
USA  Richard Wasatonic  Astrophysics and Planetary Science Department
Villanova University
Villanova, PA 19085
USA  Nils Ryde  Lund Observatory
Department of Astronomy and Theoretical Physics
Lund University
Box 43, SE-221 00 Lund
Sweden

(Received August 27th 2020; Revised September 15th 2020; Accepted October 13th
2020)

###### Abstract

The processes that shape the extended atmospheres of red supergiants, heat
their chromospheres, create molecular reservoirs, drive mass loss, and create
dust remain poorly understood. Betelgeuse’s $V$-band ”Great Dimming” event of
2019 September /2020 February and its subsequent rapid brightening provides a
rare opportunity to study these phenomena. Two different explanations have
emerged to explain the dimming; new dust appeared in our line of sight
attenuating the photospheric light, or a large portion of the photosphere had
cooled. Here we present five years of Wing three-filter ($A$, $B$, and $C$
band) TiO and near-IR photometry obtained at the Wasatonic Observatory. These
reveal that parts of the photosphere had a mean effective temperature $(T_{\rm
eff}$) significantly lower than that found by Levesque & Massey (2020).
Synthetic photometry from MARCS -model photospheres and spectra reveal that
the $V$ band, TiO index, and $C$-band photometry, and previously reported
4000-6800Å spectra can be quantitatively reproduced if there are multiple
photospheric components, as hinted at by Very Large Telescope (VLT)-SPHERE
images Montargès et al. (2020). If the cooler component has $\Delta T_{\rm
eff}\geq 250$ K cooler than 3650 K, then no new dust is required to explain
the available empirical constraints. A coincidence of the dominant short-
($\sim 430$ day) and long-period ($\sim 5.8$ yr) $V$-band variations occurred
near the time of deep minimum (Guinan et al., 2019a). This is in tandem with
the strong correlation of $V$ mag and photospheric radial velocities, recently
reported by Dupree et al. (2020b). These suggest that the cooling of a large
fraction of the visible star has a dynamic origin related to the photospheric
motions, perhaps arising from pulsation or large-scale convective motions.

stars: individual: $\alpha$ Ori – stars: circumstellar matter – stars: mass
loss –lines: profiles

††journal: ApJ††facilities: (Wasatonic Observatory)

##  1 Introduction

In 2020 February Betelgeuse ($\alpha$ Orionis; $\langle{V}\rangle\sim 0.45$;
M1-2 Ia-Iab) became the faintest it has been on record in the visual/$V$ band
($V=+1.61$) — the Great Dimming of 2019/2020. At this time the bright red
supergiant (RSG) thus attracted much attention with the public and in the
amateur and professional astronomy communities. Betelgeuse is a semi-regular
variable that has been known to vary, at least in the modern era, since 1839
(Herschel, 1840). It typically shows two dominant periods at $V$ band and in
its photospheric radial velocity. Since 1985 the shorter period has been near
420 days, while the longer period prior to 1933 was $\simeq 5.87$ years
((Goldberg, 1984, and references therein), Dupree et al. (1987) and Smith et
al. (1989)). Analysis of visual observations made by AAVSO observers from 1918
to 2006 (Kiss et al., 2006) yields similar, but slightly different, results
with periods of $388\pm 30$ days and $5.61\pm 1.1$ years, respectively. Both
of these periods vary in length and amplitude111We estimate a short period of
$430\pm 10$ day from our 25 years of multi-epoch photometry at $V$. The
variations are typically $\Delta V\simeq 0.5$ mag peak-to-peak but during the
Great Dimming $\Delta V\simeq 1.1$ mag and the fading was visible to the naked
eye when viewing the constellation Orion. Observations were made at many
observatories, e.g., Kashyap et al. (2020) reported a non-detection with the
Chandra X-ray telescope, Harper et al. (2020) reported that SOFIA-EXES
observation showed the circumstellar emission line-to-continuum flux ratio was
unchanged with respect to the normal state, and Gehrz et al. (2020) reported
that multi-band infrared photometry between 1.2 and 8.8$\mu$m was essentially
unchanged from previous observations. No exotic high-energy phenomena or new
dust emission was detected.

The origin of the dominant short and long periods observed on red supergiants
(RSGs) is not clearly established. It has been proposed that the shorter
period is a fundamental, or low-order overtone radial pulsation, e.g., Chatys
et al. (2019), Soraisam et al. (2020), and for Betelgeuse Joyce et al. (2020),
conclude that the short period is a fundamental mode driven by the
$\kappa$-mechanism. The long secondary period may be related to flow
timescales of giant convection cells (Stothers, 2010). Of particular interest
for Betelgeuse is that the minimum of the relatively well-defined short period
appeared to coincide with the long-period minimum during 2020 February (Guinan
et al., 2019b). In addition to causing the star to be dimmer overall by about
$0.4$ mag, it is possible that this coincidence may have a bearing on the deep
minimum observed, which is greater than a simple addition of amplitudes,
perhaps reflecting a nonlinear response.

VLT-SPHERE observations222https://www.eso.org/public/images/eso2003c/, made
with the H$\alpha$-continuum filter, in 2019 January and December show that
the brightness across the southern photosphere had dimmed markedly in December
(Montargès et al. 2020). The spatial scale of the dimmed region is compatible
with large convection cells that are expected and observed to be a significant
fraction of the stellar radius (e.g., Wilson et al., 1997; Freytag et al.,
2002; Montargès et al., 2016; López Ariste et al., 2018). These photospheric
phenomena are expected to be related to the mechanisms that ultimately heat
RSG chromospheres, perhaps through acoustic or magnetic shocks, fuel the
quasi-static molecular reservoirs that sit between the upper photosphere and
chromosphere (O’Gorman et al., 2020), drive mass loss, and create dust, all of
which are poorly understood processes.

The $V$ band is dominated by molecular absorption from titanium oxide (TiO)
molecules, and is sensitive to the temperature of the stellar atmosphere. The
reduction in $V$-band brightness might be expected to correspond to a
significant reduction in photospheric temperature across a large fraction of
the visible photosphere. However, Levesque & Massey (2020) used the strength
of TiO absorption bands in the 4000–6800Å spectral region to derive an
effective temperature $T_{\rm eff}=3600\pm 25$ K for 2020 February 15. This is
near the time when the star was faintest at $V=+1.61$ mag. This is only a
small decrease from $T_{\rm eff}=3650\pm 25$ K when $V\simeq 0.5$ (Levesque &
Massey, 2020). They invoke large dust grains to explain the large decrease in
$V$-band brightness. Dharmawardena et al. (2020) reported that 450 and 850
$\mu$m sub-mm fluxes declined by $\sim 20$% during the dimming period.
Spatially resolved ALMA observations show that the 870$\mu$m flux originates
from the extended upper photosphere or lower chromosphere, near $1.3R_{\ast}$
(O’Gorman et al., 2017). This wavelength region is on the Rayleigh-Jeans tail
of the Planck function, indicating a reduction in the mean local gas
temperature also by 20% if the angular size of the region remained constant.
Dharmawardena et al. (2020) also point out that a large region of lower
$T_{\rm eff}$ can qualitatively reproduce their observations and the Levesque
& Massey (2020) spectra. More recently Dupree et al. (2020b) discussed results
from spatially resolved Hubble Space Telescope near-ultraviolet (NUV)
spectroscopy. They report on the enhancement of chromospheric Mg II 2800Å h &
k emission during 2019 October, 3 to 4 months prior to the Great Dimming.
Their study linked the Mg II emission to mass ejection, perhaps leading to
dust formation.

While Betelgeuse appeared close to the Sun, as seen from Earth, in 2020
June/July, the STEREO Solar Mission obtained images of that region of the sky
(Dupree et al., 2020a). The star decreased in brightness by 30% from its 2020
late-April maximum, declining to $V\simeq+0.80$ mag by the end of July. Our
photometry, along with photometry from D. Carona (2020, private
communication), define a local light minimum in mid-August ($V=+0.85$ mag) and
slowly rising to $V=+0.68$ mag by early-October 2020.

The origin of the unprecedented large dimming of Betelgeuse during 2019/2020
is the subject of this paper. Here we report Wing three-filter and $V$-band
photometry of Betelgeuse for the last 5 years, including good coverage of the
Great Dimming event. We analyze the data with synthetic photometry from MARCS
models to shed light on the origin of the reduction in $V$-band brightness
during 2019/2020. In §2 the Wing three-filter near-IR observations and
synthetic photometry are described, and the results are presented in §3. The
discussion of the findings is given in §4, and conclusions are given in §5.

##  2 Observations

Photometric observations of Betelgeuse have been conducted at Wasatonic
Observatory (Allentown, PA) since 1996 September. The observations were
carried out with a $f/10$ 20-cm (8 inch) Celestron Schmidt-Cassegrain
Telescope (SCT) for the first five observing seasons, followed by using a
$f/10$ 28-cm (11 inch) SCT for the remaining 19 seasons. An uncooled Optec
SSP-3 photometer was employed. The SSP-3 has a red-near-IR sensitive PIN-
photodiode detector with a broad spectral range from 3000 to 11000Å.

$V$-band photometry was conducted using a wide-band V filter (5550Å) that
closely matches the $V$ bandpass of the Johnson $UBV$ system. In addition,
three intermediate band Wing red/near-IR filters were employed that form a
subset of Wing’s eight-color system (White & Wing, 1978; Wing, 1992; Wasatonic
et al., 2015). The Wing $A$ filter is centered on the strong $\gamma(0,0)$ TiO
7190Å band head, while the central wavelengths of the Wing $B$ (7500Å) and $C$
(10240Å) filters are centered on near-IR continuum regions that are free of
strong absorption lines in M stars, see Table 1. Additional observing details
can be found in the Appendix.

Table 1: Characteristics of the Four Photometric Filters Filter | Spectral Region | Central-$\lambda$ | Bandpass (FWHM)
---|---|---|---
|  | (Å) | (Å)
$V$ |  | 5550 | 910
$A$ | TiO $\gamma(0,0)$ | 7190 | 115
$B$ | Continuum | 7540 | 110
$C$ | Continuum | 10240 | 445

###  2.1 TiO Photometry

Data from the first (1996/97) season were reported by Morgan et al. (1997),
but here we report on the most recent 5 years of photometry pertinent to the
2019/2020 Great Dimming. To estimate a typical nightly magnitude uncertainty
for the Wing photometry, we note the strong linear correlation between 5 years
of $V$ and Wing $A$-band magnitudes (Pearson Linear Coeff. =0.973, $N=209$).
This is not surprising given the presence of strong TiO absorption in both
bands. Thus, for evolved M stars like Betelgeuse, changes in $V$ magnitude are
more of a measure of $T_{\rm eff}$ than they are of stellar luminosity.
Similar but slightly weaker correlations are found for $B$ and $C$ magnitudes.

Hipparcos photometry (ESA, 1997) shows that Betelgeuse varies smoothly on
time-scales of days, while the observed V mags show a mean standard deviation
of $\sigma(V)=0.025$ mag from smooth curves through seasonal data. This
includes a typical on-source photometric uncertainty of 0.015 mag and
uncertainties from extinction corrections and calibration-star observations.
For each Wing-filter magnitude we assume a linear relation with $V$ mag, and
from the scatter in the $V$ versus Wing magnitudes we infer nightly
representative standard deviations: $\sigma\left(A\right)=0.068$,
$\sigma\left(B\right)=0.051$, and $\sigma(C)=0.037$.

###  2.2 TiO Index

The TiO index was calculated from the standardized Wing $A$, $B$, and $C$
filter magnitudes from Equation (1) (Wing, 1992):

${\rm TiO\>Index}=A-B-\left[0.13\left(B-C\right)\right].$ (1)

The near-IR ($B-C$) color term is applied to account for the decrease in the
star’s continuum, and thus between the $B$ and $A$ filters. Because the
($B-C$) color typically varies between $-0.05$ and $+0.13$, this photometric
correction is small.

The TiO absorption bands increase in strength with later spectral type and
decreasing $T_{\rm eff}$. The calibration of the Wing TiO index with spectral
type and $T_{\rm eff}$ are given by Wasatonic et al. (2015), and are based on
Levesque et al. (2005). The spectral types of the Wing calibration stars
extend from $\sim$K5 to M7.5 III stars (in which the TiO-index changes from
$\sim 0.25$ to 2.0). The $T_{\rm eff}$-TiO index relation is given, following
Wasatonic et al. (2015)333 I.e., the Wasatonic et al. (2015) relation with
slightly updated coefficients, see Appendix., by

$T_{\rm eff}^{Was}=3902.4-421.62\times{\rm TiO}+63.931\times{\rm TiO}^{2}.$
(2)

This empirical calibration is based on evolved stars that are less luminous
and less variable than Betelgeuse, but as MacConnell et al. (1992) point out,
the TiO strength is nearly independent of luminosity. For stars later (cooler)
than $\sim$M7, vanadium oxide (VO) bands and other absorption lines start to
contaminate the Wing $B$ and $C$ bands, but this does not affect the present
observations of Betelgeuse, which display mean spectral types between M2 and
M4.

A small calibration was applied to our $C$ filter (10240Å) to transform it to
the Wing standard system where the original $C$ filter is centered on 10400Å.
Details of the $C$ band 10240Å color correction and conversion to Wing 10400Å
magnitudes are given by Wasatonic et al. (2015). As discussed by them, the $C$
bandpass is centered on a spectral region with no strong absorption lines and
also is near the maximum spectral energy distribution of early M-stars like
Betelgeuse. Because of this, the $C$ magnitude can serve as a proxy for the
apparent bolometric magnitude ($m_{\rm bol}$). This permits the luminosity to
be computed if the distance is known, and if not, the relative change in
luminosity (see Wasatonic et al., 2015).

The individual uncertainties of the Wing filter magnitudes propagate into the
TiO index through the use of Equation (2), and give
$\sigma\left(TiO\right)=0.089$, and $\sigma\left(T_{\rm eff}\right)\simeq 29$
K per night. These relatively large individual nightly uncertainties appear to
arise from the small aperture of the telescopes, relative high airmass of the
stars, and corresponding large extinction corrections. The nightly
uncertainties are reduced using weekly mean values (seven-day bins), which
mostly include between 1 and 4 nights. The large number of observations
clearly reveals the strong trends in the photometry.

###  2.3 Synthetic Wing Three-filter and $V$-band Photometry

With the improvements in atomic and molecular data and stellar photospheric
modeling, synthetic photometry is a powerful technique to explore the
observational consequences of changes in the properties of a stellar
atmosphere. Here we are exploring the differential photometric behavior of
Betelgeuse where all the atmospheric and spectral parameters are kept constant
except for $T_{\rm eff}$. We generated a grid of spherical MARCS photospheric
models (Gustafsson et al., 2008) with $\log{g_{*}}=0.0$,
$M_{\ast}=15M_{\odot}$, $5$ ${\rm km\,s}^{-1}$ microturbulence, and with
$T_{\rm eff}\left(K\right)=[3200,3300,3400,3450,3500,3550,3600,3650,3700]$.
This $T_{\rm eff}$ sampling captures the smooth changes in synthetic
magnitudes. These models are one-dimensional, assume hydrostatic equlibrium,
are steady-state, and are computed in local thermodynamic equilibrium. The
MARCS-model photospheres have 153,910 opacity samples between 910Å and 20
$\mu$m. These photospheres were used to create synthetic spectra for the $V$
band, and the Wing $A$, $B$, and $C$ bands, using a wavelength sampling of
$1.5$ ${\rm km\,s}^{-1}$. For the carbon, nitrogen, and oxygen (CNO)
abundances we adopt the values from Ryde et al. (2006), who scaled the CNO
values from Lambert et al. (1984) (who adopted $T_{\rm eff}=3800$ K,) to
$T_{\rm eff}=3600$ K. Specifically we adopt $A_{\rm C}=8.29$, $A_{\rm
N}=8.37$, and $A_{\rm O}=8.52$, for the models and the synthetic
spectra.444The abundance $A_{\rm X}$ is given through the following
definition: $A_{\rm X}\equiv\log{n_{\rm X}}-\log{n_{\rm H}}+12$, where $n_{\rm
X}$ is the number density. We adopt the carbon isotopic ratio 12C/13C$=7$ from
Harris & Lambert (1984). The linelists used include metals and CH, ${\rm
C}_{2}$, CN, NH, OH, MgH, AlH, SiH, CaH, CrH, TiO, ZrO, VO, H2O, and FeH (see
Gustafsson et al., 2008, and references therein).

The synthetic spectra were then reddened with the same interstellar parameters
adopted by Levesque et al. (2005), namely $A_{V}=0.62$, $R_{V}=3.1$, with the
reddening prescription of Cardelli et al. (1989). Following Levesque et al.
(2005), this reddening also includes any potential contribution from the
amorphous silicates in the low optical depth circumstellar envelope (see e.g.,
Rowan-Robinson et al., 1986). We adopt a constant photospheric angular
diameter of 44 mas (see for example, Montargès et al., 2016). Synthetic Wing
three-filter and $V$-band photometry was performed using the filter
transmissions and detector response of the Optec SSP-3. The zero-point
magnitude offsets were derived from the Vega model spectrum of Castelli &
Kurucz (1994), using the calibrations given by Wing (1992, Table 2). Table 2
gives the resultant synthetic reddened magnitudes for MARCS-model spectra with
different effective temperatures.

Table 2: Temperature Sensitivity of Betelgeuse Synthetic Photometry to $T_{\rm eff}$ Band | $C_{0}$ | $C_{1}$ | $C_{2}$
---|---|---|---
$V$ | 142.05 | $-7.414\times 10^{-2}$ | $+9.683\times 10^{-6}$
$\sigma\left(C_{V}\right)$ | $3.84$ | $0.223\times 10^{-2}$ | $0.322\times 10^{-6}$
$A$ | 116.62 | $-6.134\times 10^{-2}$ | $+7.899\times 10^{-6}$
$\sigma\left(C_{A}\right)$ | $3.87$ | $0.224\times 10^{-2}$ | $0.325\times 10^{-6}$
$B$ | 28.81 | $-1.576\times 10^{-2}$ | $+1.96\times 10^{-6}$
$\sigma\left(C_{B}\right)$ | $2.82$ | $0.164\times 10^{-2}$ | $0.237\times 10^{-6}$
$C$ | 2.46 | $-1.421\times 10^{-3}$ |  $\cdots$
$\sigma\left(C_{C}\right)$ | $0.08$ | $0.023\times 10^{-3}$ |  $\cdots$

These dedicated Betelgeuse models also permit the determination of a model-
dependent $T_{\rm eff}$–TiO calibration. We find that for a given TiO index
the inferred $T_{\rm eff}$ is between 10 K (near 3300 K) and 70 K (near 3600
K) lower than that given by Equation (2). There are several possible causes
for this small difference aside from different stellar parameters and
abundances, e.g., van Belle et al. (2009) find empirically a non-negligible
variation in $T_{\rm eff}$ within an individual RSG spectral-type class,
namely $\pm\geq 100$ K. It is also known that 1-D photospheric models are not
perfect representations of real (physical) M supergiants, especially in the
outer layers where strong molecular lines are formed (see, for example Arroyo-
Torres et al., 2015; Chiavassa et al., 2011). Equation (2) is also based on
less luminous cool evolved stars. Also, Davies et al. (2013) have pointed out
that $T_{\rm eff}$ based on TiO band-head fits to 1D MARCS models are
systematically lower, by over 100 K, than those based on spectral energy
distributions.

In §4 we will use the MARCS synthetic photometry to model the photometric
behavior of Betelgeuse when the surface is covered with regions with different
values of $T_{\rm eff}$, and then compare these to the observed photometry.

##  3 Results

Figure 1: Five years of photometry, described in §2.1, and the calculated
$T^{Was}_{\rm eff}$ (gold) derived from the TiO index using Equation (2). The
$T_{\rm eff}$ and $C$ magnitudes have been averaged into 7 day bins. Also
shown are the $V$ mag (green, unbinned) and $C$ mag (blue) data. The $1\sigma$
error bars are also shown, but for $V$ they are similar to the symbol size.
The three red bulls-eye symbols are the TiO-based $T_{\rm eff}$ previously
reported in Guinan et al. (2019b) and Guinan & Wasatonic (2020), and that
occur on 2019 September 15, December 7, and 2020 January 31, or
HJD-2450000=8742, 8825, 8895, respectively. The gray circle and error bar are
the mean $T_{\rm eff}$derived from the optical spectrum of Levesque & Massey
(2020). The TiO-based $T_{\rm eff}$s are significantly cooler than derived
from the optical spectrum. The TiO-based $T_{\rm eff}$s are clearly correlated
with $V$ over the past 5 seasons, including during the recent large dimming
event. The correlation with $C$ is also present, but the magnitude range is
smaller.

Figure 1 shows the $T_{\rm eff}$ averaged on a grid of 7 day bins for the past
5 years, with the propagated uncertainties reflecting the number of nights
included in each bin. During the Great Dimming the discrepancy between the
TiO-based $T_{\rm eff}$ and that based on the strength of TiO band heads in
the 4000–6800Å spectral region by Levesque & Massey (2020) is immediately
apparent. In the most recent observing cycle, the TiO-based $T_{\rm eff}$
declined from $T_{\rm eff}\simeq 3645\pm 15$ K (2019 September 21/HJD=2458748;
7 day bin) to $\simeq 3520\pm 25$ K (for individual 2020 February
15/HJD=2458895, 22, and 29 7 day bins), i.e., $\Delta T_{\rm eff}\simeq-125$
K. (For this discussion we omit the 7-day bin at $3491\pm 17$ K). The
difference between September and February of $\Delta T_{\rm eff}\simeq-125$ K
corresponds to a change of spectral type from M2 to M4 in six months. White &
Wing (1978) noted that changes of spectral subtype by 0.5 in RSGs seldom
occurred over such a short interval. Figure 1 also shows the clear correlation
between $V$ and TiO-based $T_{\rm eff}$ over the last 5 years of photometry,
including the large dimming event where no sudden change in trend is apparent
as $V$ increased beyond its typical range. This correlation can be understood
as a result of the strong temperature sensitivity of TiO molecular absorption;
lower gas temperatures lead to increased TiO formation and deeper absorption.
During the recent dimming event $V$ increased by about $+1.1$ mag from 2019
September to 2020 mid-February (a reduction in flux by a factor of 2.75 in
flux). The $C$ mag, which is a measure of the $1.02\mu$m continuum flux, also
follows the same $T_{\rm eff}$ trend, but it only increased by $\Delta
C\simeq+0.25$ mag (a reduction in flux by a factor of 1.25), the largest on
record.

The most interesting result from the TiO photometry is the significant
difference in $T_{\rm eff}$ at minimum $V$-band brightness, i.e., $\leq 3520$
K and the $T_{\rm eff}=3600\pm 25$ K derived by Levesque & Massey (2020) from
the strength of TiO band heads between 4500 and 6800Å. Given that both
temperature estimates are based on TiO bands from low-lying energy levels it
is unlikely that such a difference would occur because of different formation
heights during a dynamic event. If the star had uniformly reduced its $T_{\rm
eff}$ from $3650$ to $3600$ K while maintaining the same angular size, then
$V$ would increase by $+0.2$ and the flux in the $V$ band would only decrease
by $\simeq 17$%. If the $T_{\rm eff}$ had decreased to $3520$ K $V$ would
increase by 0.6 and the flux decrease by 43%. However, the observed increase
in $V$ was $+1.1$.

Figure 2: The logarithm of the flux density ratio of $T_{\rm eff}=3600$ K to
$T_{\rm eff}=3300$ K spectra computed from the MARCS models with all other
stellar parameters kept constant. The spectral ratio has been binned to $\sim
6$Å. Relative filter responses have also been plotted with arbitrary offsets
for clarity. The blue curve is the $V$-band response, the red curves are the
Wing $A$ and $B$ bands, and the green curve the VLT-SPHERE H$\alpha$-continuum
filter. As can be seen, the flux for $T_{\rm eff}=3600$ K ranges between a
factor of 3 to 12 brighter within the $V$ band, with a filter mean of $\sim
7$. The flux ratio is $\sim 8$ in the Wing $A$ filter, while in the continuum-
dominated spectral region of the Wing $B$ filter at 7540Å, the ratio is only
$\sim 1.7$ (0.22 dex).

Importantly, below 4500Å the flux ratio declines from the high values in $V$
band.

Clues to the origin of these apparent discrepancies are the two VLT-SPHERE
H$\alpha$-continuum images ($\lambda$6449Å, $\Delta\lambda=41$Å) obtained in
2019 January and December (Montargès et al. 2020). These reveal that in
December the southern hemisphere was dimmer than the north, showing brightness
variations across the star. Once it is recognized that regions of different
$T_{\rm eff}$ may exist across the visible hemisphere, the different empirical
constraints can be reconciled.

Synthetic photometry shows that the $V$ magnitude is, as expected, very
sensitive to $T_{\rm eff}$ (see Table 2), and by illustration Figure 2 shows
the flux ratio of the MARCS spectrum for $T_{\rm eff}=3600$ K to that for
$T_{\rm eff}=3300$ K. The continuum flux ratio in the Wing $B$ filter (and
$C$, not shown) is $\sim 1.7$, being much less sensitive to $T_{\rm eff}$ than
the TiO-dominated $V$ and Wing $A$ (7190Å) filter bands, the latter includes
the strong $\gamma(0,0)$ band. Here the mean flux ratios are $\sim 7-8$.

Consider the case of a photospheric region covering an area of $50\%$ of the
star’s visible hemisphere with $T^{cool}_{\rm eff}\sim 3300$ K and the rest of
the visible hemisphere having $T_{\rm eff}=3600$ K. The cool component would
have very low emission across most of the $V$\- and $A$-band spectral region,
as shown in Figure 2, where the TiO bands dominate. The resulting combined
spectrum (3600 and 3300 K) would then yield a $T_{\rm eff}^{optical}\leq 3600$
K, but geometrically diluted, i.e, the total flux is reduced by the fraction
of the star covered in cooler material. The TiO index, on the other hand, is
determined primarily by the $A$ and $B$ filters, which will have different
relative contributions to the combined spectrum. The flux in $A$ band, from
the cool component, like $V$, would contribute very little to the total
spectrum. However, $B$ will contribute much more flux than $A$ because it is
much less sensitive to $T_{\rm eff}$. The combined spectrum, would appear to
have a deeper $A$-filter absorption than the 3600 K component alone, lowering
the $T_{eff}$ from 3600 K to a value between that of the two components, which
would explain the TiO-based $T_{\rm eff}\simeq 3520$ K. In this scenario
$T_{\rm eff}^{cool}<T_{\rm eff}^{TiO}<T_{\rm eff}^{optical}\leq T(3600{\rm
K})$.

In this example the $V$ brightness would be $\simeq 55\%$ of that without the
cool component. During the Great Dimming the reduction in $V$ brightness was
greater than this, which is easily accommodated with a fractional area of the
cool component of $>50$%.

For this geometric dilution interpretation the difference between the normal-
and dimmed-state spectra would be mostly independent of wavelength between
4500 and 6700Å. However, below 4500Å the ratio of hot- to cool-component
fluxes decreases, and the differences between the combined spectra at these
wavelengths also decreases, as pointed out by Dharmawardena et al. (2020).
This model would also explain the much smaller decrease in the combined
continuum flux at $C$ (1.02 $\mu$m). The concept of a mean $T_{\rm eff}$ is no
longer useful in this case. Figure 3 shows a simple two-component model that
illustrates these points: the cool component has $T_{\rm eff}=3300$ K but with
a fractional area filling factor of 60%. This model shows the spectra in the
same fashion as Levesque & Massey (2020, Figure 1) to aid comparison. Between
4700 and 6500Å the overall flux decrease between the 3650 K and two-component
model matches the difference ($\simeq 0.3$ dex) between the Levesque & Massey
(2020) 2004 and 2020 spectra. Furthermore, the combined spectrum is not that
different from a diluted $T_{\rm eff}\simeq 3600$ K which may explain the
$T_{\rm eff}$ found by Levesque & Massey (2020).

Different combinations of $T_{\rm eff}$ and area fraction are possible; the
cooler the cool component, the closer the total combined spectrum resembles
that of a diluted $T_{\rm eff}=3600$ K spectrum. The 10240Å region (not shown)
is reduced by only 0.1 dex (or $\Delta C=+0.25$ mag) as observed. Furthermore,
since the $C$ mag is a measure of the luminosity of the visible hemisphere,
and $L\propto T_{\rm eff}^{4}$, then for the model above the relative
luminosity $L(2-{\rm comp})/L({\rm 3650\>K})\simeq
0.4+0.6(3300/3650)^{4}\simeq 0.80$, which matches $\Delta C=+0.25$ and
$L(2-{\rm comp})/L({\rm 3650\>K})\simeq 2.512^{-0.25}\simeq 0.79$, as
expected.

Figure 3: A two-component photospheric model illustrating the dimmed state on
2020 February 15. The top orange curve is for $T_{\rm eff}=3650$ K
representative of its more typical state, and the gray curve represents 60% of
the visible hemisphere covered by $T_{\rm eff}=3300$ and 40% covered with
$T_{\rm eff}=3600$ K. For clarity relative filter responses have also been
plotted with arbitrary offsets. The blue curve is the $V$-band response, the
red curves are the Wing $A$ and $B$ bands, and the green curve the VLT-SPHERE
H$\alpha$-continuum filter. The 4500- 6700Å spectral region (indicated by
black bar at top) has been binned to match Levesque & Massey (2020, Figure 1).
The two-component gray curve is a factor of $\sim 2$ (0.3 dex) less than the
3650 K curve, a similar difference to the 2004 and 2020 observations of
Levesque & Massey (2020). The gold lower curve shows the cool component’s
contribution, which illustrates how the $V$-band contribution is almost
negligible, and why the combined spectrum in this region resembles a
geometrically diluted $T_{\rm eff}=3600$ K spectrum. However, below 4500Å, the
cooler spectrum is not reduced as much, and the differences become less
significant.

We can exploit the Wing three-filter and $V$-band photometry to further
examine the temporal behavior during the last two seasons including the
dimming episode. We construct a simple model where the star is composed of two
different components: one with a typical $T_{\rm eff}=3650$ K with an area
fraction $(1-A)$, where $A$ is the variable fractional area of the visible
hemisphere, and one component with $T_{\rm var}$ and area $A$. This extra
component may mimic hot spots, or cool regions, as observed by previous
interferometry studies. The total flux from the system is then

$F_{\rm total}=\left(1-A\right)F_{\rm 3650\,K}+AF_{T_{\rm var}}.$ (3)

This model assumes that the angular size of the photosphere is constant and
uses steady-state MARCS models. It also assumes that each component includes
both limb and disk emission contributions, i.e., each component is sector-
shaped. Using the $T_{\rm eff}$-magnitude relations given in Table 2, we
compute the apparent magnitudes of the $A_{\rm mod}$, $B_{\rm mod}$, $C_{\rm
mod}$, and $V_{\rm mod}$ magnitudes for a fine two-dimensional grid of $A$
($0\to 1$) and $T_{\rm var}$ ($3000\to 3750$ K) and then find the least-
squares minimum for each 7 day bin, namely we minimize

$\left({V_{\rm obs}-V_{\rm
mod}\over{\sigma\left(V\right)}}\right)^{2}+\left({A_{\rm obs}-A_{\rm
mod}\over{\sigma\left(A\right)}}\right)^{2}+\left({B_{\rm obs}-B_{\rm
mod}\over{\sigma\left(B\right)}}\right)^{2}+\left({C_{\rm obs}-C_{\rm
mod}\over{\sigma\left(C\right)}}\right)^{2}.$ (4)

The $\sigma$s are the individual uncertainties for each 7 day bin, and because
the $V$ magnitudes have the smallest uncertainties they have the greatest
weight in the solution. The TiO-dominated $A$ and $V$ magnitudes are strongly
correlated, and the continuum-dominated $B$ and $C$ magnitudes are also
correlated.

Figure 4: Results of the two-component model, Equation (3), for each 7 day
bin. Left: the model $T_{\rm eff}$ for the whole star, $T_{\rm mod}^{\rm
Was}$, calculated from Equations (1) and (2) with the Wing and $V$ magnitudes
from each best-fit solution. The corresponding $V$ and $C$ magnitudes are
shown below. Note that if the internal TiO-$T_{\rm eff}$ calibration from the
MARCS models had been used the computed $T_{\rm eff}$ would be lower by 10–70
K (see §2.3). Top right Top: the values of $T_{\rm var}$ corresponding to the
best-fit solutions for the season of the Great Dimming. Bottom right: the
corresponding area fractions of the photosphere covered with $T_{\rm var}$,
with a simple polynomial trend curve (red). Both parameters of the two-
component model show significant scatter, but the general trends of decreasing
$T_{\rm eff}$ and increasing $A$ are seen with increasing date, until the deep
$V$-band minimum when $T_{\rm var}$ begins to increase.

The best-fit solutions for the magnitudes were also converted to $T^{\rm
Was}_{\rm mod}$ using Equations (1) and (2), and the results for the dimming
cycle are shown in Figure 4. (Recall that $T_{\rm var}$ is the $T_{\rm eff}$
of the input MARCS models.) There is a tension between $T_{\rm var}$ and $A$,
for example a reduction in stellar flux can be caused by a reduction in
$T_{\rm eff}$ for a given fractional area, or from a smaller reduction in
$T_{\rm var}$ but with a great area fraction. There is considerable scatter in
the solutions which is not real. Significant changes in the brightness and/or
the areas of large features in hydrodynamic simulations occur on timescales
greater than 7 days (Chiavassa et al., 2009). The scatter probably reflects
the simple, i.e., nonphysical nature of the model, and noise in the
observational data. The assumption that each temperature component be sector-
shaped is unlikely to be good, especially when a convective feature emerges
within the stellar disk.

During the 2018/2019 season (not shown), $T_{\rm var}$ has a high area factor
and starts near $3650$ K, declining to $\simeq 3550$ K by mid-season and then
increasing again to $\sim 3650$ K. Figure 4 shows the 2019/2020 season, and
September starts with $T_{\rm var}\simeq 3550-3600$ K with $A\sim 0.3-0.4$,
slightly cooler than the typical $T_{\rm eff}$, but during the Great Dimming
$T_{\rm var}$ steadily declines while $A$ increases. By 2019 December 28 the
model suggests $T_{\rm var}\sim 3485^{+15}_{-165}$ K, with $A\sim
0.85^{+0.15}_{-0.25}$, cooling further to $T_{\rm var}=3360^{+65}_{-45}$ K on
2020 February 15 with a similar $A$-range. During the subsequent increase in
$V$ brightness in 2020 March, $T_{\rm var}$ increased while the fractional
area remained the same. The VLT-SPHERE 2019 December image suggests an area
factor on the lower end of that suggested by the model. Because the photometry
is spatially unresolved this model does not tell us where the cool region is
or whether it is comprised of multiple cool components. This very simple model
is probably the best that can be justified with the available four photometric
bands. The two-component example shown in Figure 3 included 60% of the area
covered by $T_{\rm eff}=3300$ K, although material at $3200$ K leading to more
distinct band heads could not be ruled out. Clearly, in reality the star
probably has a distribution of $T_{\rm eff}\left(A\right)$. The addition of
multi-epoch optical spectra to the analysis during this dimming event would
improve the constraints on the geometric dilution term $A$, and hence $T_{\rm
var}$.

##  4 Discussion

We have found that the Wing three-filter near-IR and $V$-band photometry can
be explained by an inhomogeneous photosphere, with one component having a
large fractional area that is significantly cooler than the typical $T_{\rm
eff}=3650$ K expected for an M2 Iab RSG. The $T_{\rm eff}$ of cool component
is less than the mean $T_{\rm eff}$ based on the TiO index, i.e., $<3520$ K.
Dharmawardena et al. (2020) have also reached a similar conclusion based on
their observations of reduced sub-mm fluxes555 Given that the sub-mm emission
is from an optically thick region above the photosphere (O’Gorman et al.,
2017), the 20% sub-mm flux reduction implies a $\sim 20$% reduction in local
gas temperature if the angular size of the emitting region is unchanged,
larger than the 5% adopted in that paper. during the dimming event, and an
examination of model multi-component optical spectra. The appearance of new
dust is not required to interpret the Great Dimming.

Levesque & Massey (2020) derived a $T_{\rm eff}=3600\pm 25$ K for the minimum
of the Great Dimming, which is only slightly cooler than expected for a M2 Iab
spectral type. To explain the factor of two flux reduction (0.3 dex) observed
between their 2004 and 2020 spectra they proposed that the dimming was
predominantly caused by attenuation from new dust in the line of sight. The
grains were assumed to be large because of the relatively wavelength-
independent flux reduction between 4500 and 6700Å. However, if large dust
grains were injected into our line of sight then the Wing $A$ (7190Å) and $B$
(7540Å) spectral regions would be similarly affected, and the TiO-based
$T_{\rm eff}$ values would be expected to be near $3600$ K, which they are
not.

If dust did attenuate the $V$-band flux then some thermal dust-emission
signatures might be expected. However, Gehrz et al. (2020) reported infrared
photometry at 1.2, 2.2, 3.6, 4.9, 7.9, and 8.8 $\mu$m, that does not reveal
evidence of an increase in emission compared to previous observations reported
50 years prior (Gehrz & Woolf, 1971). The 2020 values are consistent within
the error bars longward of 4.8 $\mu$m with those reported in 1971, and they
are 0.2–0.3 mag fainter than those obtained between 1991 April 25 and May 3
(Fouque et al., 1992) when the nearest $V$ measurement epoch, 3 to 4 weeks
prior, had $V\sim 0.55$ (Krisciunas, 1992). The decrease in $J$ band
brightness is similar to what we find for the $C$ band (1.024 $\mu$m).
Dharmawardena et al. (2020) presented a schematic dust model to mimic the
dimming event, adopting alumina and Mg-Fe silicate dust grains. The addition
of new dust leads to an increase in flux longward of 1.25 $\mu$m rising to 60%
at 8 $\mu$m. Because these simulations are idealized (e.g., a spherical shell
rather than a localized dust cloud that might reflect starlight into a
different direction) the infrared evidence cannot conclusively rule out the
formation of any new dust; however, it does not provide support for the
presence of dust to attenuate the $V$-band fluxes.

From SOFIA-EXES spectra, Harper et al. (2020) found that the ratio of
circumstellar emission-line fluxes of [Fe II] at 25.99 $\mu$m and [S I] at
25.25 $\mu$m to the local continuum fluxes was not significantly different
between the dimming episode and two previous epochs when the star was in its
brighter normal state. The last two observations were made with the same
aperture. The continuum captured in the EXES apertures is mostly stellar with
a contribution from extended circumstellar silicate dust emission (Harper et
al., 2001). Given that the collisionally excited emission lines and the
photospheric and dust continuum are formed by different physical processes in
different spatial regions, the constancy of the EXES line-to-continuum flux
ratios suggests that the stellar continuum near 26$\mu$m has not significantly
changed. It would be unlikely that the changes in continuum and CSE emission
would change in the same direction by the same amount. The Dharmawardena et
al. (2020) schematic dust model predicts a continuum increase by a factor of
1.87 at 26 $\mu$m (T. Dharmawardena 2020, private communication) which is not
evident in the SOFIA-EXES spectra.

The observed enhancement in Mg II h & k emission and NUV continuum in 2019
October, prior to the Great Dimming, could have resulted in an ejection of hot
dense plasma potentially leading to dust formation (Dupree et al., 2020b).
However, the dust (if formed) may not necessarily be in our line of sight.

While some dust-related event may have been associated with the Great Dimming,
the evidence suggests that the dimming in $V$ (as well as in the Wing three-
filter TiO near-IR photometry) was due to a significant fraction of the
visible stellar surface having a $T_{\rm eff}$ $\leq 250$ K cooler than
normal.

###  4.1 Polarimetry: Changes in Illumination and/or Dust?

There have been reports of polarimetry obtained at epochs near the Great
Dimming, and changes in dust have been described as a potential cause of the
changes in polarization (Cotton et al., 2020a, b; Safonov et al., 2020). Here
we briefly discuss this point.

The observations of Betelgeuse by Hayes (1984) found that both the degree of
linear polarization and position angle in the $B$ band changed significantly,
and in an orderly fashion, during four consecutive observing seasons. The non-
repeating polarimetric pattern had a timescale of $\sim$ 0.5–1 year, and it
was suggested that the origin may be related to the rotation of large
convective cells. This time-scale is consistent with convective (but non-
giant) cells (Schwarzschild, 1975), and also for $\sim 0.1R_{\ast}$ cells in
hydrodynamic simulations that have time-scales of a few months to one year
(Chiavassa et al., 2009). Linear continuum polarization may occur from
atmospheric Rayleigh scattering and/or Mie scattering off circumstellar dust.
The observed dependence of linear polarization with wavelength suggests
atmospheric Rayleigh scattering of radiation from a non-axisymmetric
illumination source at, or near, the stellar surface. Aurière et al. (2016)
have confirmed this interpretation with observations of Betelgeuse’s line de-
polarization spectrum, which cannot be created with dust scattering
alone.666The polarized spectrum from the photosphere may subsequently be
polarized by circumstellar dust scattering.

Cotton et al. (2020a, b) found that the 4250Å and 4800Å polarization decreased
during the $V$ brightness minimum, which could result from a decrease in
asymmetric illumination. However, Safonov et al. (2020) employing differential
speckle polarimetry found that the net polarization brightness did not change
until 2020 February, when it subsequently increased as the $V$ brightness
increased. They suggest the initial constancy supports the idea of a dust
cloud along the line of sight because the extended envelope would still be
illuminated by the rest of the star. However, that does not easily explain the
subsequent increase in polarization brightness that occurred when the star
returned towards its normal brightness. Dust is known to exist close to the
photosphere, Kervella et al. (2016) detected dust at $\sim 3R_{\ast}$, and
Haubois et al. (2019) detected dust near $\sim 1.5R_{\ast}$. Changes in
polarization during the dimming may instead be related to changes in
illumination from the stellar surface and/or changes in the distribution of
existing dust.

###  4.2 Radial Velocity - Light Variations

Dupree et al. (2020b) presented photospheric radial velocities from the last
two short-period ($\sim 430$ days) cycles obtained with the STELLA robotic
observatory (Strassmeier et al., 2004). Figure 5 reproduces these data777Using
WebPlotDigitizer: https://automeris.io/WebPlotDigitizer together with our $V$
magnitudes.

During these cycles the maximum (positive) radial velocity (i.e., the maximum
contraction velocity of the photosphere) occurs about 35 days ($\simeq 0.08$
Period (P)) after the time when the star is faintest and coolest (see Figures
1 and 5). In addition, the maximum stellar expansion velocity occurs during
the star’s, only partially covered, broad maximum brightness. This phasing is
similar to the behavior of pulsating yellow supergiant Cepheid variables. For
example, in the case of the well observed classical Cepheid prototype,
$\delta$ Cep (F5 Ib – G1 Ib; $P=5.366$ days), minimum brightness of the star
also occurs $\sim$0.085 P prior to the star’s maximum contraction radial
velocity (see Engle et al., 2017, and references therein). This behavior
indicates that the observed $V$, temperature, and radial-velocity variations
of Betelgeuse (observed over the star’s dominant $\sim$430 day period) may
arise from radial pulsations. But unlike Cepheids, which have repeatable light
and radial-velocity variations, Betelgeuse has multiple periods and complex,
semi-regular light variations.

It is interesting that the range of radial-velocity variation ($\Delta v=10$
${\rm km\,s}^{-1}$) during the dimming cycle was higher than the previous
cycle ($\Delta v=6$ ${\rm km\,s}^{-1}$), and the former is typical of the
larger variations in radial velocity previously recorded (Goldberg, 1984, and
references therein), suggesting a particularly dynamic photospheric event.
This could arise from a more vigorous pulsation episode or a larger than usual
giant convection cell, or possibly both. The coherent nature of the last two
seasons of photospheric radial velocities and the mean $T_{\rm eff}$, and the
coincidence of the minimum of the short- and long-period variations suggests
that the Great Dimming was a stronger pulsation or shock event, leading to
enhanced photospheric cooling. During the dimming event the measured
photospheric radial velocities are weighted toward the hotter diminishing
photospheric component, and therefore because of the smooth changes in radial
velocity with time, the larger cooling component presumably shares similar
infall velocities.

Figure 5: Radial velocities as reported by Dupree et al. (2020b) fit with a
smooth curve and our $V$ magnitudes. For ease of comparison the $V$ mags are
plotted with largest values (minimum brightness) at the top. The center-of-
mass (CoM) stellar radial velocity of $20.9\pm 0.3(1\sigma)$ ${\rm
km\,s}^{-1}$ is taken from Harper et al. (2017), is based on a combination of
estimators, and is consistent with the ALMA spatially-resolved molecular
diagnostics (Kervella et al., 2018). During these epochs the mean photospheric
radial velocity is smaller than the CoM velocity. The mean values for each
cycle are shown (average of leading minimum and maximum). The offsets in mean
radial velocities with the CoM velocity could be caused by the radial velocity
contribution arising from the long ($\sim 5.6$ – 6.0 yr) period. Sanford
(1933, and references therein) find peak-to-peak radial velocity ranges of
between 4.1 and 6.1 ${\rm km\,s}^{-1}$.

Previous observations show similar, but not identical, behavior. Dupree &
Stefanik (2013) have discussed the relationship between radial velocity and
$V$-band observations for a 16 year span, including the radial-velocity
measurements of Smith et al. (1989). Even though the coverage is typically not
uniform, they find that there are times when the opposite correlation of
maximum radial velocity and maximum brightness (hence temperature) is
observed, while at another time the opposite occurs. Gray (2008) presented
radial-velocity and temperature measurements from the neutral lines of V I at
6251.83Å, Fe I at 6252.57Å, and Ti I at 6261.11Å, between epochs 2002.5 and
2007.5. These data suggest a general pattern between $T_{\rm eff}$ -$v_{rad}$,
where the temperature increased before the expansion phase (blue shifted
lines), and cooling occurred before the contraction phase, similar to that
shown in Figure 5. However, unlike the Gray (2008) results, during the last
two seasons (with more dense coverage) Betelgeuse shows a clear correlation
between the size of the ranges of $v_{\rm rad}$ and the $T_{\rm eff}$.

Kravchenko et al. (2019) have studied the temporal relation of $v_{\rm rad}$
and $T_{\rm eff}$ in the more luminous RSG $\mu$ Cep (M2- Ia; (Keenan &
McNeil, 1989)) and in radiation-hydrodynamic (RHD) simulations. Following the
ideas of Gray (2008), Kravchenko et al. (2019) developed a time-dependent
scenario to interpret the quasi-periodic variations as a photospheric shock,
driven by activity in the deeper hidden convection zone. However, a noticeable
difference in these studies is that the phase relation between $v_{rad}$ and
$V$ is not as coherent as that shown in Figure 5 for Betelgeuse. We note that
sparse temporal coverage makes it hard to draw firm conclusions from previous
observational studies, and also existing RHD models for RSGs do not show such
coherent variations.

Perhaps the key point to take away from the high precision STELLA radial-
velocity data and our photometry is that the correlation seen in the last two
cycles between $V$ and $v_{\rm rad}$, namely the phase offset and amplitudes,
shows no sudden deviation or discontinuity. This suggests that a new
phenomenon has not occurred, e.g., dust formation in our line of sight that
changes the $V$ brightness discontinuously during the dimming. The $V$ and
radial velocities measurements, along with Wing photometry, are closely
correlated in the last two seasons where we have observations to show it.

##  5 Conclusions

The Wing three-filter and $V$-band photometry indicate a significantly cooler
mean $T_{\rm eff}$ during the 2019/2020 Great Dimming (see Figure 1) than that
inferred from the strength of the TiO band heads in the 4000–6800Å spectra by
Levesque & Massey (2020). These findings can most simply be reconciled by the
presence of a large area, $\geq 50$%, of even cooler photospheric material.
The observed $V$-band spectral region is then a geometrically diluted spectrum
of the normal $T_{\rm eff}$ component. This picture also explains why the
differences in the 2004 and 2020 spectra below 4500Å are smaller than in the
rest of the spectral region, as pointed out by Dharmawardena et al. (2020).
This is a consequence of the $T_{\rm eff}$-sensitivity of $V$-band spectral
region, as shown in Figure 2. No new dust is required to explain these
observations. Independent information from observations made during the Great
Dimming, namely IR photometry (Gehrz et al., 2020) and SOFIA-EXES spectra
(Harper et al., 2020) also do not reveal any signature of dust emission.
Changes in linear polarization are not so easily interpreted, but they may be
related to changes in photospheric illumination of pre-existing dust, known to
be present close to the photosphere.

The photometric data support the hypothesis that the Great Dimming was a
greatly enhanced (amplified) continuation of regular changes in mean $T_{\rm
eff}$, $V$, and radial velocity, over the short 430-day period cycle. The deep
minimum occurred close to the time predicted. The increased range in radial
velocity during the dimming cycle, see Figure 5, may reflect a dynamic origin
of the cooling of large areas of the photosphere.

It is hoped that by combining all the observations obtained before, during,
and after the Great Dimming a more complete understanding of RSG variability
can be achieved.

Differential photometry was carried out in the usual pattern of sky-
comparison-variable-comparison-(check star)-sky that generally repeated 3-4
times for each filter. Typical integration times per measurement were 30-50
s.. These individual measures were averaged to form nightly means. The star
was generally observed for 50-60 min per night. The primary comparison star
was $\phi^{2}$ Ori (40 Ori, G9 III-IV; $V=+4.09$ mag, $B-V=0.96$). $\phi^{2}$
Ori is an older inactive high-velocity star; no significant variability ($<3$
mmag) is indicated from Hipparcos photometry (ESA, 1997). The check star,
$\delta$ Ori, (Bellatrix, HD 35468, B2 V, $V=+1.64$, $B-V=-0.22$.) was
observed differentially with respect to the comparison star on most nights in
the V band. Bellatrix does seem to show possible small light variations ($\pm
0.015$ mag). Wing standard stars were also observed and used as alternate
check stars. Differential atmospheric extinction corrections were applied, and
the mid-times of the observation converted to Heliocentric Julian Dates.
Equation (1) is based on a larger set of calibration observations than used in
Wasatonic et al. (2015). All calibration observations obtained between 1996
and 2009 were used to revise the atmospheric extinction and Wing-system
transformation coefficients. Equation (2) differs from the 2015 version by
less than 15 K between 3200 and 3800 K.

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Financial support for G.M.H. was provided by NASA through a SOFIA Betelgeuse
Flash Archival Research award #07-0073 issued by USRA. N.R. acknowledges
support from the Royal Physiographic Society in Lund through the Stiftelse
Walter Gyllenbergs fund and Märta and Erik Holmbergs donation. The Wasatonic
Observatory (Allentown, PA) is supported by Villanova University. We thank T.
Dharmawardena for providing numerical details of their dust model, and we also
thank RWH for invaluable assistance. We acknowledge the reviewers for their
helpful comments which improved the presentation of this paper. This research
has made very extensive use of NASA’s Astrophysics Data System Bibliographic
Services. This research has also made use of the VizieR catalogue access tool,
CDS, Strasbourg, France (Ochsenbein et al., 2000), including the 5th Edition
of the Catalog of Infrared Observations (Gezari et al., 1999).