# Informed systematic method to identify variable mid and late-T dwarfs

Natalia Oliveros-Gomez Departamento de Astronomía, Universidad de Guanajuato.
Callejón de Jalisco, S/N, 36023, Guanajuato, GTO, México  Elena Manjavacas
AURA for the European Space Agency (ESA), ESA Office, Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD, 21218 USA  Department of
Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Afra Ashraf Department of Physics & Astronomy, Barnard College 3009 Broadway,
New York, NY 10027, USA  Department of Astrophysics, American Museum of
Natural History 200 Central Park West, New York, NY 10024, USA  Daniella C.
Bardalez-Gagliuffi Department of Physics & Astronomy, Amherst College, 25 East
Drive, Amherst, MA 01003, USA  Department of Astrophysics, American Museum of
Natural History 200 Central Park West, New York, NY 10024, USA  Johanna M. Vos
Department of Astrophysics, American Museum of Natural History 200 Central
Park West, New York, NY 10024, USA  Jacqueline K. Faherty Department of
Astrophysics, American Museum of Natural History 200 Central Park West, New
York, NY 10024, USA  Theodora Karalidi Department of Physics, University of
Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816, USA  Daniel
Apai Steward Observatory, The University of Arizona, 933 N. Cherry Avenue,
Tucson, AZ 85721, USA  Lunar and Planetary Laboratory, The University of
Arizona, 1629 E. University Blvd., Tucson, AZ 85721, USA

###### Abstract

The majority of brown dwarfs show some level of photometric or spectro-
photometric variability in different wavelength ranges. This variability allow
us to trace the 3D atmospheric structures of variable brown dwarfs and
directly-imaged exoplanets with radiative-transfer models and mapping codes.
Nevertheless, to date, we do not have an informed method to pre-select the
brown dwarfs that might show a higher variability amplitude for a thorough
variability study. In this work, we designed and tested near-infrared spectral
indices to pre-select the most likely variable mid- and late-T dwarfs, which
overlap in effective temperatures with directly-imaged exoplanets. We used
time-resolved near-infrared Hubble Space Telescope Wide Field Camera 3 spectra
of a T6.5 dwarf, 2MASS J22282889–431026, to design our novel spectral indices.
We tested these spectral indices on 26 T5.5–T7.5 near-infrared SpeX/IRTF
spectra, and we provided eight new mid- and late-T variable candidates. We
estimated the variability fraction of our sample in $\mathrm{38^{+4}_{-30}}$%,
which agrees with the variability fractions provided by Metchev et al. (2015)
for mid-to late-T dwarfs. In addition, two of the three previously known
variables in our SpeX spectra sample are flagged as variable candidates by our
indices. Similarly, all seven known non-variable in our sample are flagged as
non-variable objects by our indices. These results suggest that our spectral
indices might be used to find variable mid- and late-T brown dwarf variables.
These indices may be crucial in the future to select cool directly-imaged
exoplanets for variability studies.

Variability — Brown dwarfs — T-dwarfs – Cool directly-imaged exoplanets –
Spectral indices

##  1 Introduction

Many brown dwarfs and planetary-mass objects show photometric and spectro-
photometric variability (e. g, Apai et al. 2013, Radigan et al. 2014, Metchev
et al. 2015, Zhou et al. 2018, Biller et al. 2018, Vos et al. 2019, Manjavacas
et al. 2019a, Vos et al. 2020, Vos et al. 2022). Although other scenarios are
possible (Tremblin et al., 2015, 2016), the most likely reason to explain
brown dwarf variability is the existence of heterogeneous clouds coverage
(Apai et al., 2013) in their atmospheres at various pressure levels.

Variability studies are a powerful tool for the understanding of the dynamics
of substellar atmospheres (e.g. Yang et al. 2016, Apai et al. 2017, Manjavacas
et al. 2021), which overlap in effective temperatures and colors to directly-
imaged exoplanets (Faherty et al., 2016). Time-resolved spectrophotometric
observations allow us to detect variability at different pressure-levels of
the atmospheres of brown dwarfs (e.g. Yang et al. 2016). Using radiative-
transfer models, we can provide an interpretation of the heterogeneous cloud
layers that are introducing the variability measured, at each pressure level
(Saumon & Marley, 2008). If we have a long enough monitoring baseline covering
at least few rotations of the object, mapping codes like Stratos (Apai et al.,
2013), Aeolus (Karalidi et al., 2015), Starry (Luger et al., 2019), or mapping
exoplanets (Cowan & Fujii, 2017) can be used to derive 2D or 3D brightness
distributions that are consistent with the data and may provide exploratory
maps of the atmospheres. Such a thorough analysis can be only performed in
variable brown dwarfs. However, to determine if a specific brown dwarf is
variable or not, several hours of photometric or spectrophotometric monitoring
need to be spent to obtain a light curve with enough time coverage to
determine the variability of the object. So far, the brown dwarfs that have
been monitored for variability surveys have been selected blindly, many of
them resulting in no detecting any significant variability after expending
considerable telescope time.

In the past years, spectral indices in the near-infrared have been used to
identify binary L+T or M+T brown dwarf candidates (Burgasser et al., 2006,
2010; Bardalez Gagliuffi et al., 2014). These spectral indices have been found
to detect some highly variable brown dwarfs, like 2MASS J21392676+0220226
(Burgasser et al., 2006) or 2MASS J13243553+6358281 (Looper et al., 2007) that
did not show any signs of binarity, after high resolution imaging follow-up
(Gagliuffi et al. 2015, Burgasser et al. 2016). Manjavacas et al. (2019b)
applied these spectral indices to search for binary candidates in a L, T, and
Y-dwarf spectral library of near-infrared HST/WFC3 data, obtaining eight
spectral binary candidates. Manjavacas et al. (2019b) found that five out of
the eight binary candidates were also flagged as known variables in the
literature, suggesting that these spectral indices might be also helpful to
search for brown dwarf variables. With this aim, Ashraf et al. (2022) selected
which indices to target brown dwarf binaries would be best at differentiating
L/T transition (L7-T3) variables from the rest of the population. Some of the
variability candidates proposed by Ashraf et al. (2022) were recently
confirmed as variables (Vos et al., 2022).

In this paper, we designed a novel set of spectral indices to find variable
candidates with mid- and late-T (T5.5-T7.5) spectral types using Hubble Space
Telescope time-resolved near-infrared spectroscopy of 2MASS J22282889–431026
(2M2228) obtained with the Wide Field Camera 3 instrument. 2M2228 is a T6.5
brown dwarf (Burgasser et al., 2003), at a distance of d = 10.64$\pm$0.79 pc
(Faherty et al., 2012). Buenzli et al. (2012) estimated a $\mathrm{T_{eff}}$ =
900 K, and a log g = 4.5-5.0, typical of T-dwarf. 2M2228 is variable with a
period of 1.43 hr and a peak to peak amplitude of $1.85\pm 0.07\%$ in the
[4.5] Spitzer channel, $1.85\pm 0.07\%$ in the $J$-band, $2.74\pm 0.11\%$ in
the $H$-band and $5.3\pm 0.6\%$ in $\mathrm{H_{2}O}$-band (Buenzli et al.,
2012). Metchev et al. (2015) measured its variability amplitude in 4.6 % in
the Spitzer [3.6] band, and an amplitude of 1.6 % in the [4.5] channel.
Buenzli et al. (2012) measured a phase shift of 89.38$\pm$13.18o between the
$J$\- and $H$-band HST/WFC3 light curves, which was interpreted as evidence of
large-scale longitudinal-vertical structures. The variability amplitudes and
phase-shifts were confirmed by Yang et al. (2016) in data obtained four years
later, which indicates the light curve of 2M2228 is relatively stable in long
time scales.

It is particularly important to design spectral indices for mid and late-T-
type objects, since they are generally faint. Thus, variability surveys are
more challenging to perform than for L and L/T transition brown dwarfs. The
spectral indices presented in this paper might also help to identify variable
cool directly-imaged exoplanets in the advent of higher resolution and
wavelength coverage James Webb Space Telescope near-infrared spectra.

This paper is structured as follows: In Section 2 we provide the details of
the HST/WFC3 observations for 2M2228. In Section 3, we explain the development
of the near-infrared spectral indices for detecting variability in T-type
dwarfs. In Section 4, we compare our results with previous studies. In Section
5 we test the robustness of our spectral indices. In Section 6, we discuss the
variability fraction obtained for our sample, and the physics behind the
spectral indices. Finally, in Section 7, we summarize our conclusions.

##  2 OBSERVATIONS

The data used for our analysis were obtained on July 7, 2011 with the infrared
channel of the Wide Field Camera 3 (WFC3) on-board the Hubble Space Telescope
(HST) and its G141 grism, providing spectra covering a wavelength range
between 1.10 and 1.70 $\mu$m (GO-12314, GO-70170; PI: D. Apai). These spectra
were published in Buenzli et al. (2012).

The observations were performed using a 256 x 256 pixels subarray of the WFC3
infrared channel, with a field of view of approximately 30×30′′. A SPARS25 was
used 1024 × 1024 pixels readout mode with 11 reads per exposure. Each sub-
readout has an exposure time of 22.34 s, after an initial first short read at
0.27 s. Allowing for a maximum exposure number of 27,000 counts which is below
the average detector capacity. The spectra were acquired over six consecutive
HST orbits, spanning 8.57 hr, obtaining a total of 54 spectra. In each orbit
we obtained 9 spectra with a cadence of 241.99 s. The resolution of the
spectra is $\sim$100, and the signal-to-noise ratio was $\sim$124.

##  3 Index Method

###  3.1 Designing novel spectral indexes

Spectral indices are parameters that allow us to compare the flux in different
spectral ranges within the same spectrum of one object. In this study, we
designed spectral indices using near-infrared HST/WFC3 spectra to find
candidate variable mid- and late-T dwarfs using any single-epoch near-infrared
spectra, with the aim of facilitating the search for photometric and spectro-
photometric variability in mid- and late-T dwarfs.

To design our spectral indices, we used the time-resolved near-infrared
HST/WFC3 spectra of 2M2228 published in Buenzli et al. (2012). The near-
infrared light curve of 2M2228 presented in Buenzli et al. (2012) showed a
variability amplitude between 1.45% and 5.3%, depending on the wavelength
range of the 2M2228 used to create the light curve (Fig. 1). In the near-
infrared light curve, each data point is a near-infrared time-resolved
spectrum itself (see Fig. 1). To design our spectral indices, we explored
which wavelength ranges of the HST/WFC3 spectra of 2M2228 vary within the
object’s rotation due to its heterogeneous cloud-coverage.

Figure 1: 2M2228 near-infrared light curve to identify the regions of
variability based on visual inspection. The least variable spectra are those
corresponding to the center of the light curve (flux $\sim$ 1, black line).
The most variable spectra are those corresponding to the peaks for the light
curve (dashed orange lines). Figure 2: Top panel: template for 2M2228 after
median-combining the 40 HST/WFC3 spectra (black), and example of one variable
spectrum for 2M2228. Bottom panel: residuals after subtracting the variable
spectrum of 2M2228 of the 2M2228 template. We highlight in red the most
variable region (the peaks of the the $J$-band and $H$-band), and in blue the
least variable regions of the spectrum according to the residuals shown below.
Figure 3: Ranges of wavelengths used to calculate spectral indices on a sample
brown dwarf spectrum. The red corresponds to the numerator, and the blue
corresponds to the denominator in equation 1.

We obtained the most and least variable wavelength ranges of the 2M2228
spectrum. For that, we subtracted each of the maximum flux spectra of 2M2228
from a spectral template (Fig. 2), created by median combining all 54 HST/WFC3
2M2228 spectra. Then, we identified the corresponding wavelength ranges in
which the difference between the most variable 2M2228 spectrum and the
template spectrum is maximum and minimum (in absolute value). The 2M2228
template spectrum would be similar to a single-epoch spectrum of 2M2228 in
which several minutes of integration are needed to obtain a high signal-to-
noise spectrum. The most variable wavelength ranges are those which have
higher absolute difference values, and the least variable wavelength ranges
are those that have absolute differences (subtraction) close to zero. We found
that the most and least variable regions of each spectrum are similar for all
spectra. In Fig. 2, we highlighted in blue the most variable wavelength ranges
of the spectra, and in red the least variable wavelength ranges. The
wavelength ranges are defined moving average the entire spectrum with 0.03 mm
wide windows that are shifted every 0.01 mm. After obtaining the mean values
in each window, we selected two regions as the most and least variable. The
most variable region show a variability amplitude 1.3 times larger than the
overall variability amplitude of the near-infrared light curve. The least
variable wavelength range has 0.4 times the variability amplitude of the near-
infrared light curve. The least and most variable wavelength ranges are shown
in Table 1 and Fig. 3.

Following this process, we consolidated the spectral ranges to create two
spectral indices: one to measure the variability in the $H$-band and the other
the $J$-band, which are calculated with equation 1.

$\textup{index}=\frac{\int_{\lambda_{1}}^{\lambda_{2}}F_{1}(\lambda)d\lambda}{\int_{\lambda_{3}}^{\lambda_{4}}F_{2}(\lambda)d\lambda}$
(1)

In equation 1, the numerator is one of the most variable region of the
spectrum, and the denominator is a adjacent least variable region. Both cover
the same wavelength width. In addition, we created two other indices,
comparing the $J$\- and $H$-bands with each other, one corresponding to the
index color, and the other is a ratio between the $H$\- and $J$-band, to
compare the two most variable bands in our spectrum. Finally, we supplemented
our indices with two other indices published in Bardalez-Gagliuffi et al.
(2014) were used $J$-slope and $J$-curve, focusing on the band that has the
highest variability, according to the residuals in Fig. 2. The six indices
examined are described in Table 1 and shown in Fig. 3.

We measured the six spectral indices for the 54 HST/WFC3 NIR spectra of
2M2228. We compared these measurements indices in index-index plots (see Fig.
4). As observed in Fig. 4, we see the values of the indices for the most
variable and least variable spectra clustering in specific areas of the index-
index plots. Among the initial index-index plots, we selected 12 of them that
segregated better the most and least variable spectra of 2M2228, and did not
resemble any of the other plots. To identify the regions of the index-index
plots in which the variable mid- and T-dwarfs would be located, we color coded
the points with higher (red) to lower variability amplitudes (blue) spectra,
as shown in Fig. 4.

Figure 4: Index-index plots comparing the designed spectral indexes to find
mid- and late-T brown dwarf variables. The blue points correspond to the
spectra that show the least variability amplitude (close to the 1.0 of the
light curve in Fig. 1), and the red correspond to the most variable spectra
(close to the 1.007 or 0.992 of the light curve in Fig. 1). The black star in
the center corresponds to the indexes calculated for the template spectrum
with no variability. We show the provisional variable regions (in yellow)
using the indices calculated only with observed spectra.

###  3.2 Areas of high variability likelihood in index-index plots

The points in Fig. 4 corresponding to the most variable spectra (red points)
coincide with the the maximum (or minimum) of the combined near infrared light
curve of 2M2228, and the least variable spectra are those corresponding to the
points of the light curve located near a value of 1.0, as shown in Figure 1.
From this analysis, we were able to visually identify the regions where most
of the variable spectra are in the index-index plots, as shown in the Fig. 4
(yellow areas). The variability regions presented in Fig. 4, are limited to
the 54 spectra available for 2M2228. Thus, to provide robustly regions of high
likelihood of variability in the index-index plots, we generated synthetic
spectra, similar to the HST/WFC3 near-infrared spectra, using a Monte Carlo
simulation. For that purpose, we redefined each point of the spectrum using a
Gaussian random number generator. The mean value of the Gaussian is the
original flux in the spectrum, and the standard deviation is the uncertainty
of each point of the sample spectrum.

We generated two different sets of synthetic spectra: the first set similar to
the most variable spectrum in the 2M2228 light curve (in the light curve in
Fig. 1 at 0.3 hr, normalized flux $\approx$ 0.99024), and second set from the
least variable spectrum in the object’s light curve. The least variable
spectrum of 2M2228 is the one with relative flux value closest to 1.0 in the
target’s light curve (in Fig. 1 at 3.4 hr, normalized flux = 0.99997). We
generated 100 new synthetic spectra using the most variable spectrum of
2M2228, and 100 new synthetic spectra using the least variable spectrum of
2M2228. With these 200 new synthetic spectra, the six indices are calculated
for each spectrum and plotted alongside the initial 54 spectra of 2M2228 in
the index-index comparison plots (Fig. 5). Using the newly generated synthetic
spectra, we refine the initial variable areas in the index-index plots. The
new regions of variability are shown in Fig. 5 and the values for the
polygons’ limits are described in Table 2.

Figure 5: Index-index plot as in Fig. 4 including the value of the indexes
calculated for the synthetic spectra created using the most variable spectra
of 2M2228 (purple stars), and the least variable 2M2228 spectra (green stars).
We show the updated new variable regions (in grey) using the synthetic
spectra.

###  3.3 Candidate selection

Using the areas defined in Section 3.2, it is possible to denote variability
candidacy based on the number of times a spectrum is found within the regions
of variability. Figure 6 shows two histograms with the frequency in which
spectra are identified as variable or non-variable in a given number of index
plots. We found that 70% of the synthetic spectra generated using the most
variable 2M2228 spectrum are within the variable regions in at least 11 of the
12 index-index plots (upper panel). For the case of the synthetic spectra
generated using the least variable 2M2228 spectrum, we found that 100% of the
spectra fall in the variable region in less than 9 of the index-index plots
(lower panel). For this reason, we identify as variable candidates those found
at least in 11 regions in our index-index comparison plots.

Figure 6: Upper plot: Histogram showing how many synthetic variable spectra fall in variable areas of index-index plots. The histogram shows that most of the variable synthetic spectra fall at least in 11 variable areas in the index-index plots. Bottom plot: Histogram showing how many synthetic non-variable spectra fall in variable areas. The histogram shows that most of the non-variable synthetic spectra fall in less than 10 variable areas. Thus, we mark the threshold on 11 plots to catalog an object as a variable candidate.  Spectral | Numerator Range111Color index case, numerator is the first value. | Denominator Range222Color index case, denominator is the second value. | Feature | Reference
---|---|---|---|---
index | ($\mu$m) | ($\mu$m) |  |
H/J | 1,51-1,62 | 1,205-1,315 | Comparison of H-band and J-band | (1)
J-H | 1,205-1,315 | 1,51-1,62 | J-H Color index | (1)
H | 1,59-1,62 | 1,64-1,67 | Difference variability in H-band | (1)
J | 1,22-1,25 | 1,15-1,18 | Difference variability in J-band | (1)
$J$-slope | 1.27-1.30 | 1.30-1.33 | 1.28 $\mu$m flux peak shape | (2)
$J$-curve | 1.26-1.29 | 1.14-1.17 | Curvature across J-band | (2)
Table 1: Boundary ranges used for the creation of the spectral indices. References: (1) This paper, (2) Bardalez-Gagliuffi et al. (2014) x-axis | y-axis | Coordinates
---|---|---
H | J | (0.445,0.215), (0.46,0.215), (0.463,0.2045), (0.475,0.19), (0.445,0.19)
H | H/J | (0.445,0.4185), (0.457,0.4185), (0.464,0.4146), (0.46,0.410), (0.445,0.410)
J-H | H/J | (0.04375,0.422), (0.0465,0.422), (0.0465,0.410), (0.0459,0.410),(0.045,0.4146)
J-H | H | (0.0465,0.485), (0.0465,0.445), (0.044,0.445)
J-H | J | (0.0447,0.215), (0.04615,0.215), (0.04615,0.19), (0.045,0.2045)
J-slope | J | (0.612,0.215), (0.6275,0.215), (0.632,0.2042), (0.6275,0.1915), (0.612,0.1915)
J-slope | H | (0.612,0.47), (0.632,0.463), (0.646,0.448), (0.612,0.448)
J-slope | H/J | (0.612,0.4186), (0.6275,0.4186), (0.632,0.4146), (0.6275,0.4105), (0.612,0.4105)
J-slope | J-H | (0.612,0.046), (0.65,0.046), (0.65,0.0447), (0.612,0.04515)
J-slope | J-curve | (0.613,0.165), (0.653,0.142), (0.615,0.142)
H | J-curve | (0.447,0.164), (0.481,0.142), (0.447,0.142)
J-H | J-curve | (0.0462,0.164), (0.0462,0.142), (0.0438,0.142)
Table 2: Boundaries for the variable areas within the index-index plots in
Fig. 4.

##  4 IDENTIFICATION OF CANDIDATE VARIABLES

Once robust areas were defined within the index-index plots, we are able to
use our spectral indices to identify variable mid- to late-T brown dwarf
candidates.

We used all available near-infrared (0.7–2.5 $\mu$m) spectra from the
SpeX/IRTF spectral
library333http://pono.ucsd.edu/~adam/browndwarfs/spexprism/html/tdwarf.html
(R$\sim$120) with spectral types between T5.5 and T7.5. We end up with 26
brown dwarfs with a single epoch spectrum. We test our spectral indices with
brown dwarfs with spectral types between T5.5 and T7.5 since they are within
one spectral type of 2M2228 (T6.5). In this sample, we included the SpeX
spectrum of 2M2228 itself to confirm the validity with our method with spectra
taken with other instruments different from WFC3.

In Table 3, we show the details of the T5.5-T7.5 sample that we used to test
our method. We calculated the indices for the 26 brown dwarfs, and created the
index-index plots, concluding that 10 of the 26 T5.5-T7.5 dwarfs were selected
by the indices as variable candidates. This represents a $\sim$38% variability
occurrence for the T5.5 - T7.5 spectral sample considered in this study.

Figure 7: Histogram of the number of times each of the mid- to late-T dwarfs
in Table 3 fall in the variability regions in each of the 12 index-index
plots.  Table 3: T dwarfs with spectral type between T5.5-7.5 available at the
SpeX Spectral Library. Variability references: (1) Koen et al. (2004), (2)
Buenzli et al. (2014), (3) Radigan et al. (2014), (4) Allers et al. (2020),
(5) Dupuy & Liu (2012), (6) Metchev et al. (2015), (7) Clarke et al. (2008),
(8) Vos et al. (2018), (9) Buenzli et al. (2012)

Name | RA | DEC | SpeX NIR |  $J$-mag | Our variable | N° | Variability | Reference
---|---|---|---|---|---|---|---|---
|  |  | SpT |  | candidate | regions | literature | variability studies
2MASSI J0243137–245329 | 02 43 13.71 | -24 53 29.83 | T6.0 | $15,381\pm 0,05$ | No | 04/12 | Non-variable | (1)(2)(3)
2MASSI J0727182+171001 | 07 27 18.24 | 17 10 01.20 | T7.0 | $15.600\pm 0.061$ | Yes | 11/12 |  |
2MASSI J0937347+293142 | 09 37 34.87 | 29 31 40.99 | T6.0 | $14.648\pm 0.036$ | No | 05/12 |  |
2MASSI J1047538+212423 | 10 47 53.85 | 21 24 23.46 | T6.5 | $15.819\pm 0.059$ | No | 10/12 | Variable | (4)
2MASSI J1217110–031113 | 12 17 11.10 | -03 11 13.17 | T7.5 | $15.860\pm 0.061$ | Yes | 11/12 |  |
2MASSI J1534498–295227 | 15 34 49.84 | -29 52 27.41 | T5.5 | $14.900\pm 0.054$ | No | 03/12 |  |
2MASSI J1553022+153236 | 15 53 02.28 | 15 32 36.91 | T7.0 | $15.825\pm 0.071$ | Yes | 11/12 | Binary | (5)
2MASSI J2356547–155310 | 23 56 54.77 | -15 53 11.13 | T5.5 | $15.824\pm 0.057$ | No | 03/12 |  |
2MASS J003451+052305 | 00 34 51.57 | 05 23 05.09 | T6.5 | $15.140\pm 0.004$ | Yes | 11/12 |  |
2MASS J051609–044549 | 05 16 09.45 | -04 45 49.93 | T5.5 | $15.984\pm 0.079$ | No | 04/12 | Non-variable | (6)
2MASS J111451–261823 | 11 14 51.33 | -26 18 23.56 | T7.5 | $15.858\pm 0.083$ | No | 10/12 |  |
2MASS J122554–273946 | 12 25 54.32 | -27 39 46.67 | T6.0 | $15.260\pm 0.047$ | No | 08/12 |  |
2MASS J123147+084733 | 12 31 47.53 | 08 47 33.11 | T5.5 | $15.570\pm 0.072$ | No | 04/12 |  |
2MASS J123739+652614 | 12 37 39.19 | 65 26 14.80 | T6.5 | $16.053\pm 0.087$ | No | 08/12 |  |
2MASS J154627–332511 | 15 46 27.18 | -33 25 11.18 | T5.5 | $15.631\pm 0.051$ | No | 03/12 | Non-variable | (3)(7)
2MASS J161504+134007 | 16 15 04.13 | 13 40 07.92 | T6.0 | $16.350\pm 0.092$ | Yes | 12/12 |  |
2MASS J182835–484904 | 18 28 35.72 | -48 49 04.63 | T5.5 | $15.175\pm 0.056$ | No | 06/12 | Non-variable | (3)(7)
Gliese 570D | 14 57 14.96 | -21 21 47.75 | T7.5 | $15.324\pm 0.048$ | Yes | 11/12 |  |
HD 3651B | 00 39 18.91 | 21 15 16.80 | T7.5 | $16.31\pm 0.03$ | No | 10/12 |  |
SDSS J175805+463311 | 17 58 05.45 | 46 33 09.91 | T6.5 | $16.152\pm 0.088$ | Yes | 11/12 |  |
SDSSp J111010+011613 | 11 10 10.01 | 01 16 13.08 | T5.5 | $16.161\pm 0.008$ | No | 07/12 | Non-variable | (8)
SDSSp J134646–003150 | 13 46 46.34 | -00 31 50.13 | T6.5 | $16.000\pm 0.102$ | Yes | 11/12 |  |
SDSSp J162414+002915 | 16 24 14.36 | 00 29 15.81 | T6 | $15.494\pm 0.054$ | No | 05/12 | Non-variable | (1)(3)
2MASS J005019–332240 | 00 50 19.94 | -33 22 40.27 | T7.0 | $15.928\pm 0.070$ | Yes | 11/12 | Variable | (3) (6)
ULAS J141623+134836 B | 14 16 23.94 | 13 48 36.32 | T7.5 | $17.259\pm 0.017$ | No | 10/12 | Non-variable | (6)
2MASS J222828–431026 | 22 28 28.89 | -43 10 26.27 | T6.5 | $15.662\pm 0.073$ | Yes | 12/12 | Variable | (6)(9)

From the SpeX sample, ten objects have been monitored for variability in
previous works (Radigan et al. 2014, Metchev et al. 2015). The only known
variables, apart from 2M2228 are J0050–3322 and J1047+2124. There are seven
others of the T5.5-T7.5 sample that have been confirmed as non-variable or
with amplitudes below noise level in previous works (Clarke et al., 2008;
Radigan et al., 2014; Buenzli et al., 2014; Metchev et al., 2015).

Since our spectral indices are designed in the $J$\- and $H$-band, there are
expected to be applicable to brown dwarfs with similar colors according to the
main sequence of mid- and late-T dwarfs. In Fig. 8, we show the 26 mid- to
late-T dwarfs used to test our indices.

Figure 8: Color–magnitude diagrams in the MKO system showing all ultracool
dwarfs from Table 3 in Dupuy & Liu (2012) with errors $<0.5$ mag. Symbol
colors indicate spectral types: M6–L2 (brown), L2.5–L9 (red), L9.5–T4
(purple), $>$T4.5 (blue), and the brown dwarfs with IRTF/SpeX spectra used in
this work (black). J1416B is one of the objects included in our sample of weak
variables that have a blue color.

In the following section, we list the objects selected as variable and non-
variable candidates by our indices. The full list of variable candidates
selected by our indices can be found in Table 3.

###  4.1 Variables candidates

The following objects were selected by our indices as variable candidates.

####  4.1.1 2MASS J22282889–431026

2M2228 was classified as a T6.5 spectral type (Burgasser et al., 2003).
Buenzli et al. (2012) measured a variability amplitude of 1.85$\pm$0.07 % in
the $J$-band, and a 2.74$\pm$0.11% in the $H$-band using near-infrared
HST/WFC3 spectra. The $J$-, $H$-band and $\mathrm{H_{2}O}$ band at 1.4 $\mu$m
light curves show phase shifts. Buenzli et al. (2012) found that the phase lag
increases with decreasing pressure level, or higher altitude, probing the
heterogeneity in the atmosphere of 2M2228 in both horizontal and vertical
directions.

To test our spectral indices, we used a single-epoch SpeX/IRTF spectrum of
2M2228 from Burgasser et al. (2004). 2M2228 was found in the variable areas in
12 out of 12 our index-index plots, validating the effectiveness of our
indices.

####  4.1.2 2MASS J00501994–3322402

J0050–3322 is classified as a T7.0 brown dwarf (Kirkpatrick et al., 2011).
Using a forward modelling analysis, Zhang et al. (2021) measured an effective
temperature of $\mathrm{T_{eff}}=947^{+23}_{-24}K$, surface gravity log g =
$4.06^{+0.26}_{-0.26}$, and metallicity of, Z = -0.11${}^{+0.15}_{-0.15}$.
J0050-3322 was monitored by Metchev et al. (2015) with Spitzer, finding no
significant variability in the [3.6] channel, and a variability amplitude of
$1.07\pm 0.11$% in the [4.5] channel. J0050–3322 have a regular light curve,
with a period of $P=1.55\pm 0.02$ hr, being one of the fastest brown dwarf
rotators. J0050–3322 was found in the variable areas in 11 out of 12 our
index-index plots.

###  4.2 Non-variables candidates

The following objects were selected as non-variable candidates by our spectral
indices.

####  4.2.1 2MASSI J1047538+212423

J1047+2124 is a T6.5 dwarf (Burgasser et al., 2000). Filippazzo et al. (2015)
provided physical parameters for this object: $\mathrm{T_{eff}}=880\pm 76$ K,
$log(g)=4.96\pm 0.49$ dex, age around $0.5-10$ Gyr, M= $41.61\pm
26.03~{}\mathrm{M_{jup}}$ and R = $4.96\pm 0.49~{}\mathrm{R_{jup}}$. Allers et
al. (2020) used the Infrared Array Camera on the Spitzer Space Telescope used
the Infrared Array Camera on the Spitzer Space Telescope, on its [4.5]
channel, for two days, during 7 hr and 14 hr each. Allers et al. (2020) found
sinusoidal variability of $0.5\%$, and adopt a period of $1.741\pm 0.007$ hr.

In addition, Allers et al. (2020) measured the wind speed for this object, in
radio observation taken with Karl G. Jansky Very Large Array (VLA), which
generates an (unknown) atmospheric inhomogeneity that dominates the
photometric variability, with similar period, $1.758\pm 0.0012$ hr. J1047+2124
was found in the variable areas in 10 out of 12 our index-index plots,
indicating that it is non-variable candidate. Since J1047+2124 is a false
negative, in section 5, we analyze in detail this particular object.

####  4.2.2 ULAS J141623.94+134836.3 B

J1416+1348 is a resolved L+T subdwarf system, separated by 9” (Burgasser et
al., 2010). The primary dwarf is ULAS J141623.94+134836.3 (J1416+1348A) with a
L7 spectral type. On the other hand, the secondary object J1416+1348B, is a
T7.5 dwarf. Burgasser et al. (2000) noted the unusually blue $H-W2$ and
$W1-W2$ colors of this object and connected these properties to its metal-poor
atmosphere. The very blue $Y-K$ color of this companion suggests high gravity
and low metallicity, with $logg=5.21^{+0.26}_{-0.25}$ dex and metallicity
$Z=-0.39^{+0.14}_{-0.11}$. Zhang et al. (2021) derived an effective
temperature of $\mathrm{T_{eff}}=695\pm 33$ K.

Variability studies have been performed on both objects. J1416+1348B was
initially studied in Khandrika et al. (2013) and marginal evidence of $J$-band
variability was found using the Gemini infrared camera on the 3 m Shane
telescope at Lick Observatory. Radigan et al. (2014) and Metchev et al. (2015)
found no significant variability. Metchev et al. (2015) provided an upper
levels of variability of A[3.6] $<$0.91 % and A[4.5] $<$0.59 %. J1416+1348B
was found in the variable areas in 10 out of 12 our index-index plots,
indicating that it is a non-variable candidate.

####  4.2.3 2MASS J05160945–0445499

2MJ0516–0445 is a T5.5 dwarf, and it is at a parallactic distance of $12\pm 2$
pc. Faherty et al. (2012) measured its effective temperature
$\mathrm{T_{eff}}=1211\pm 137$ K, $log(g)=5.02\pm 0.48$ dex, and reported an
age between $0.5-10$ Gyr. Faherty et al. (2012) estimated a mass of M=
$47.18\pm 27.53~{}\mathrm{M_{jup}}$ and R = $0.97\pm 0.17~{}\mathrm{R_{jup}}$.
Metchev et al. (2015) measured its variability with the Spitzer telescope,
finding no significant variability, but providing upper limits on its
variability of A[3.6] $<$ 0.83 % and A[4.5] $<$ 0.81 %. J0516–04454 was found
in the variable areas in 4 out of 12 our index-index plots, indicating that it
is a non-variable candidate.

####  4.2.4 SDSSp J162414.37+002915.6

J1624+0029 is a T6.0 dwarf (Burgasser et al., 2002). Smart et al. (2013)
provided the physical parameters for this object: $\mathrm{T_{eff}}=936\pm 78$
K, $log(g)=4.98\pm 0.48$ dex, age between $0.5-10$ Gyr, M = $42.84\pm
26.06~{}\mathrm{M_{jup}}$ and R = $0.94\pm 0.15~{}\mathrm{R_{jup}}$. Koen et
al. (2004) and Radigan et al. (2014) did not find any significant variability
for this object with ground-based near-infrared monitoring. On the other hand,
Buenzli et al. (2014) found a tentative variability for 2M1624+0029 with
HST/WFC3 data in the water band. J1624+0029 was found in the variable areas in
5 out of 12 our index-index plots, indicating that it is a non-variable
candidate.

####  4.2.5 2MASS J15462718–3325111

J1546–3325 is a T5.5 dwarf (Burgasser et al., 2002). Smart et al. (2013)
derived some physical parameters for this object: $\mathrm{T_{eff}}=1002\pm
84$ K, $log(g)=5.00\pm 0.48$ dex, age between $0.5-10$ Gyr, M = $44.13\pm
26.34~{}\mathrm{M_{jup}}$ y R = $0.94\pm 0.16~{}\mathrm{R_{jup}}$. Koen et al.
(2004) found no variability in the near-infrared after 4.4 hr of photometric
monitoring. Similarly, Radigan et al. (2014) found no significant variability
for J1516-3325 after 3.8 hr of $J$-band monitoring. J1546–3325 was found in
the variable areas in 3 out of 12 our index-index plots, indicating that it is
a non-variable candidate.

####  4.2.6 2MASSI J0243 137–245329

J0243–2453 is a T6 dwarf (Burgasser et al., 2002). Filippazzo et al. (2015)
provided physical parameters for this object: $\mathrm{T_{eff}}=972\pm 83$ K,
$log(g)=4.99\pm 0.47$ dex, age around $0.5-10$ Gyr, M= $43.59\pm
26.41~{}\mathrm{M_{jup}}$ and R = $0.94\pm 0.16~{}\mathrm{R_{jup}}$. Buenzli
et al. (2014) found no variability $\sim 40$ minutes near-infrared monitoring.
Similarly, Radigan et al. (2014) did not find any significant variability in
the $J$-band after 3.7 hr of monitoring. J0243–24532 was found in the variable
areas in 4 out of 12 our index-index plots, indicating that it is a non-
variable candidate.

####  4.2.7 2MASS J18283572-4849046

J1828–4849 is a T5.5 brown dwarf (Burgasser et al., 2000). Faherty et al.
(2012) provided the most relevant physical parameters for this object:
$\mathrm{T_{eff}}=1060\pm 103$ K, $log(g)=5.01\pm 0.48$ dex, age between
$0.5-10$ Gyr, M= $45.21\pm 27.07~{}\mathrm{M_{jup}}$ and R $=0.95\pm
0.16~{}\mathrm{R_{jup}}$. Our spectral indices did not select J1828–4849 as a
variable candidate, since it appeared only in 7 out of 12 variable areas.
Nevertheless, Radigan et al. (2014) found significant variability for the
object, with a variability amplitude of $0.9\pm 0.1\%$, with a 5 hr rotational
period. However, they found that the J1828–4849 light curve followed the same
trend of a calibration star. Thus, it is possible that the variability
measured for J1828–4849 is not intrinsic. Clarke et al. (2008) found an
attenuation of 0.04 mag for the J1828–4849 light curve. Nevertheless, they
found a strong correlation with the airmass, suggesting that the attenuation
measured could be contamination from the control star they used. J1828–4849
was found in the variable areas in 6 out of 12 our index-index plots,
indicating that it is a non-variable candidate.

####  4.2.8 SDSSp J111010+011613

J1110+0116 (Geballe et al., 2002) is a T5.5 brown dwarf, member of the 150 Myr
old AB-Doradus moving group (Gagné et al., 2015). Vos et al. (2018) monitored
it for 9 hr of observation with Spitzer in the [4.5] channel, detecting no
variability. For periods $<$18 hr, they place an upper limit of 1.25% on the
peak-to-peak variability amplitude of this object. J1110+0116 was found in the
variable areas in 7 out of 12 our index-index plots, indicating that it is a
non-variable candidate.

###  4.3 New Variable Candidates

The following targets are selected as new variable candidates by our spectral
indices.

####  4.3.1 2MASSI J0727182+171001

J0727+1710 was classified as a T7.0 dwarf by (Burgasser et al., 2002). This
object was photometrically monitored in a search for variability with the 1.6
m telescope of the Observatoire du Mont-Megantic by Girardin et al. (2013),
finding variability Peak-to-Peak Amplitudes $<13$ mag. Pineda et al. (2016)
conducted an optical survey with the Keck telescopes looking for $H\alpha$
emission at 6563 $\text{\AA}$, which is possibly indicative of magnetic or
auroral activity in brown dwarf atmospheres sometimes resulting in rotational
variability. However, J0727+1710 did not demonstrate $H\alpha$ emission, with
values of $log(L_{H\alpha}/L_{bol})<-5.9$. J0727+1710 was found in the
variable areas in 11 out of 12 our index-index plots, indicating is a variable
candidate.

####  4.3.2 2MASSI J1217110–031113

J1217–0311 is a T7.5 brown dwarf (Burgasser et al., 2000). Radigan et al.
(2014) did not detect significant photometric variability during 3.18 hr of
monitoring. This object does not show $H\alpha$ emission (Pineda et al.,
2016), with values of $log(L_{H\alpha}/L_{bol})<-5.9$. J1217–031 was found in
the variable areas in 11 out of 12 our index-index plots, indicating that it
is a variable candidate.

####  4.3.3 2MASSI J1553022+153236

J1553+1532 was discovered by Burgasser et al. (2002), and classified as T6.0
by Burgasser et al. (2010). Dupuy & Liu (2012) discovered that it is actually
is a 0.349” binary pair of T7 and T6.5 brown dwarfs resolved with HST with an
estimated mass ratio of $0.9\pm 0.02$ and a bolometric luminosity ratio of
$0.31\pm 0.12$ (Burgasser et al., 2006). J1553+1532 was found in the variable
areas in 11 out of 12 our index-index plots, indicating that it is a variable
candidate.

####  4.3.4 2MASS J00345157+0523050

J0034+0523 is a T6.5 brown dwarf discovered by Burgasser et al. (2004). Its
spectral analysis revealed that this object has a depressed K-band peak,
indicating potentially a metal-poor atmosphere. Its spectrum is similar to the
unusual T6 brown dwarf, J0937+2931 (Burgasser et al., 2002), since both
objects have spectral indicators of a high pressure photosphere, which can
either be indicative of high surface gravity or a metal poor atmosphere.
J0034+0523 has a very blue near-infrared color $J-K=-0.93\pm 0.03$, and it is
one of the fastest rotating brown dwarfs found to date with a maximum rotation
period of 1.4 hours and $vsini=90\pm 2kms^{-1}$ (Hsu et al., 2021). J0034+0523
was found in the variable areas in 11 out of 12 our index-index plots,
indicating that it is a variable candidate.

####  4.3.5 2MASS J16150413+1340079

J1615+1340 was not monitored for variability studies in previous works.
J1615+1340 was discovered by Looper et al. (2007), who classified it as a T7.0
spectral type brown dwarf. J1615+1340 was found in the variable areas in 12
out of 12 our index-index plots, indicating that it is a variable candidate.

####  4.3.6 SDSS J175805.46+463311.9

J1758+4633 was not monitored for variability studies in previous works. It was
discovered by Geballe et al. (2002), and classified it as a T7.0 brown dwarf.
J1758+4633 shows also signs of low surface gravity (Burgasser et al., 2006).
J1758+4633 was found in the variable areas in 11 out of 12 our index-index
plots, indicating that it might is variable candidate.

####  4.3.7 SDSSp J134646.45–003150.4

J1346–0031 was not monitored for variability studies in previous works.
J1346–0031 was discovered by Burgasser et al. (2000) and classified it as a
T6.0 spectral type brown dwarf. This object does not have any noticeable
magnetic activity, with H$\alpha$ emission values of
$log(L_{H\alpha}/L_{bol})<-5.2$. J1346–0031 was found in the variable areas in
11 out of 12 our index-index plots, indicating that it is a variable
candidate.

##  5 Robustness of the Index Method

Since we have a limited sample of mid and late-T with variability information,
we aim at testing the method for a larger sample of spectra. To verify the
robustness of our method, we generated synthetic spectra using a Monte Carlo
simulation for the known variables and non-variables dwarfs that were selected
as such by our spectral indices (see Sections 4.1 and 4.2). Using these 8
objects (2 variables and 6 non-variables), we performed a similar Monte Carlo
simulation as the one described in Section 3 using their IRTF/SpeX near-
infrared spectra. 100 synthetic spectra were generated for each object using
the Monte Carlo simulation. We calculated the six spectral indices mentioned
in Table 1. Then, these spectral indices are compared with each other in the
index-index plots as in Fig. 4. Finally, we calculated how many of the
synthetically generated spectra are found in more than 11 variable regions
within the index-index plots for the variable and the non-variable spectra. In
Figure 9, we present the histogram showing in how many variability regions the
synthetic spectra created using the SpeX spectra of the two variable objects
(2M2228 and 2M0050–3322) are found. For J0050–3322, 100% of the synthetic
spectra fall the variable regions in more than 11 index-index plots. For
J2228, 89% of the synthetic spectra are found in more than 11 variable regions
in the index-index plots.

Figure 9: Histogram of the number of variable zones in which the variable
synthetic spectra generated with the SpeX spectra are found. Most of the
synthetic spectra generated from the SpeX spectra of 2M2228 and J0050–3322 are
in more than 11 variable zones in the index plots.

On the other hand, in Figure 10, we show in how many variable regions the
synthetic spectra generated using the SpeX spectra of the known non-variable
objects are found. The majority of the objects fulfill the non-variable
condition since 100% of their spectra are found in less than 11 variable
regions.

Figure 10: Histogram of the number of times in which non-variable synthetic
spectra generated using the SpeX spectra fall. Only J1110+0116 have a few
spectra in more than 11 out of 12 regions.

As mentioned in Section 4.2.2, J1416+1348B is found in 10 index-index plots,
falling in the non-variable category. J1416+1348B is a blue object (Burgasser
et al. 2010, see Fig. 8), thus its spectrum and colors differ from the
standard mid- to late-T spectral sequence. Thus, J1416+1348B might be a
contaminant in our poll of variable/non-variable candidates. J1416+1348B has
an upper limit on variability (Metchev et al., 2015).

We consider if objects that found in 10/12 areas should be considered also
variable candidates. To test how robust this limit is, using a Monte Carlo
simulation as in Section 5, we generated 100 synthetic spectra for each of the
four objects that found in 10/12 areas (J1114, HD3651, J1416B and J1047), and
we estimated how many of the newly generated 100 synthetic spectra found in
more than 11 out of 12 areas. As shown in Fig. 11, with the exception of
J1114–2618 (found only 12/100 as candidate variable), most of the
synthetically generated spectra from the other three objects found in more
than 11 of 12 areas. HD3651 76/100 variable spectra, J1416+1348B 73/100
variable spectra, J1047+2124 100/100 variable spectra.

In conclusion, to acknowledge this finding, the objects that found in 10 out
of 12 variability areas might be considered “weak variable candidates”.

Figure 11: Histogram of the number of times the synthetic spectra of brown
dwarfs that are in the 10/12 limit of our variability regions.

##  6 Discussion

###  6.1 Recovery Rates

In Table 3, we find that 10 objects in the sample of 26 T5.5-T7.5 dwarfs were
monitored for variability studies in the literature. Three of them were found
to be variable (2M2228, J0050-3322 and J1047+2124), while the other seven are
not variable. The index method presented in Section 3 was able to recover two
of the three known variable brown dwarfs (recovery rate of 67%). For the known
non-variable brown dwarfs, our index method was able to recover all seven
known non-variables (recovery rate of 100%), we find single false negative,
J1047+2124, which would suggest a false-negative rate of 12.5% (1/8). As
mentioned in Section 4.2.1, J1047+2124 is a borderline variable candidate,
selected as variable by 10/12 regions, and we decided to consider it as a
”weak variable candidate”. Further variability monitoring campaigns to observe
the new variable candidates in Section 4.3 are needed to confirm the recovery
rates of our index method.

###  6.2 Variability Fraction

Using the index method presented in this paper (Section 3) over a sample of 26
mid and late-T dwarfs SpeX near-infrared spectra with spectral types between
T5.5 and T7.5, and based on the assumption that the variable candidates are
indeed variable, we aim at estimating the variability fraction given by our
indices, and compare it with other estimations in the literature.

We found that 10 of the 26 objects are variable candidates. This suggests a
variability fraction of $\mathrm{38^{+4}_{-30}}$% for this spectral range.
From the 10 variable candidates, only 2 are known variables (2M2228 and
J0050–3322). We followed the same method as in Burgasser et al. (2003) to
estimate the uncertainties of the variability fraction using a Poisson
distribution. Considering that the total number of objects in our sample is
26, the mean of the distribution is 10 (variable candidates found by our
indices), and the minimum variable fraction of our sample is 2/26 $\sim$ 8%.
We integrated the probability of the distribution until reaching 0.68,
equivalent to 1-$\sigma$ Gaussian limits. We estimated the variability
fraction of our sample in $38^{+4}_{-30}$% according to our indices. However,
taking into account the false negative rate of the indices, the true fraction
of variability could be slightly higher.

We compared this variability fraction with the results of ground-based
(Radigan et al., 2014) and space-based (Buenzli et al. 2014, Metchev et al.
2015) surveys to understand if the variability fraction reported in this work
is consistent with those. Radigan et al. (2014) monitored a sample of 57 L,
L/T and T-dwarfs in the $J$-band using the Du Pont 2.5 m telescope at Las
Campanas Observatory and the Canada–France–Hawaii Telescope on Mauna Kea.
Radigan et al. (2014) had 15 dwarfs with a spectral type later than T5.5, and
reported significant variability for three of them, implying a variability
fraction of $\sim$20%. Because of the limited stability and sensitivity of the
ground-based data, this 20% should probably be considered as a lower limit
rather than an intrinsic variability fraction. Similarly, Buenzli et al.
(2014) monitored a sample of 22 L5 to T6 dwarfs with the Hubble Space
Telescope/Wide Field Camera 3 in the $J$\- and $H$-bands, finding six brown
dwarfs with confident variability and five brown dwarfs with tentative
variability. This leads to a variability fraction between 27–50%. In addition,
Metchev et al. (2015) monitored a sample of 44 L3 to T8 dwarfs using the
Spitzer Space Telescope in the [3.6] and [4.5] channels. Metchev et al. (2015)
monitored five T-dwarfs with spectral types later than T5.5, finding
significant variability for two of them ($\sim$40%). Metchev et al. (2015)
concluded that most of the T-dwarfs in their sample show low level variability
($>$0.2%). The difference in variability fractions is most likely due to three
factors: the difference in sensitivity between ground and space-based
telescopes, the wavelength range at which each sample was observed, and the
different time-scales: the Spitzer observations were usually much longer than
what is possible from the ground. The $J$-band traces deeper pressure levels
of the mid to late-T atmospheres ($\sim$5 bar, Yang et al. 2016, see their
Fig. 19) than the [3.6] and [4.5] Spitzer bands ($\sim$0.2 bar and $\sim$0.6
bar, respectively, Yang et al. 2016). Given that most of the cloud deck is
expected to be at pressures above $\sim$1 bar (Yang et al., 2016), in
principle, we do not expect a high variability in the Spitzer channels. A
visual representation of the atmospheric structure of a similar spectral type
object, 2M0050-3322, can be found in Manjavacas et al. (2022), their Fig. 19.
Nevertheless, the sensitivity of the Spitzer Space Telescope is higher than
ground-based observatories. The variability fraction we report on this work,
38${}^{+4}_{-30}$%, is consistent to the Spitzer variability fraction reported
by Metchev et al. (2015).

###  6.3 Physics behind the index method

The spectral indices presented in this work compare the wavelength ranges of
the $J$\- and $H$-bands of the HST/WFC3 spectra of 2M2228 that vary the most
with the wavelength ranges that vary the least as the object rotates. The most
variable ranges of the $J$\- and $H$-bands are usually located around the
maximum of the bands (see Fig. 3), and the least variable are usually located
at the wings of the bands (see Fig. 3). In addition, two of our indices
compare the $J$\- and $H$-band flux between them. To understand the reason why
certain wavelength ranges show higher flux variations than others, we use the
vertical structure predicted for an object of similar spectral type and
surface gravity to 2M2228 by radiative-trasfer models (Saumon & Marley, 2008)
in Yang et al. (2016). In Fig. 12 we show the pressure levels probed by
HST/WFC3 and the G141 grism, similar to the contribution functions presented
in Yang et al. (2016), their Fig. 18. As seen in Fig. 12, the peaks of the
$J$\- and $H$-bands around 1.25 $\mu$m and 1.58 $\mu$m (shaded in red),
respectively, probe the deepest layers of the atmosphere of a mid to late-T
dwarf that we are able to trace. The peak of $J$-band probes the 30–40 bar
level, and the $H$-band peak probes the 10–20 bar level. In the meantime, the
wings trace upper layers: the $\sim$10 bar level is traced with wavelengths
around 1.17 $\mu$m and 1.32 $\mu$m for the $J$-band, and the $\sim$6 bar level
is traced with wavelengths around 1.65 $\mu$m in the $H$-band (see Fig. 12,
shaded in blue). As discussed in Section 3, these the wavelength ranges
covered by the wings barely change during the rotational period of 2M2228.

In addition, the condensate mixing ratio (mole fraction) of the clouds present
in 2M2228’s atmosphere ($\mathrm{Al_{2}O_{3}}$, Fe, and
$\mathrm{Mg_{2}SiO_{4}}$ clouds) is predicted to decrease with decreasing
pressure (see Yang et al. 2016, their Fig. 19). Thus, as we probe higher
levels in the atmosphere of 2M2228, we expect to have less dense clouds, and
therefore potentially less variability too. This scenario would explain why
the variability observed in the wings of the $J$\- and $H$-band is smaller
than in their respective peaks. Finally, the high signal-to-noise of the
HST/WFC3 near-infrared spectra is enough to quantify the differences in
variability amplitude, and thus to design the spectral indices that allowed us
to identify new $>$T5.5 variable candidates (see Section 4) with a single-
epoch near-infrared spectrum.

Figure 12: Pressure levels probed by the HST/WFC3 + G141 wavelength range for
T-dwarfs with temperatures of 900 K (red line) and 600 K (blue line), covering
the T5-T8 spectral types according to radiative-transfer models. We shadowed
in red the most variable wavelength ranges of the spectra, which were used as
numerator for J and H indexes (see Fig. 3). We shadowed in blue the least
variable wavelength ranges of the spectra, which were used as denominator for
indixes J, H and J-slope

.

###  6.4 Top-of-the-atmosphere features

In contrast to early-T dwarfs, mid- and late-T dwarfs show in general
relatively low variability amplitudes (Radigan et al., 2014; Metchev et al.,
2015). Variability surveys from the ground and space-based facilities have not
found variability amplitudes above the 2% for mid- and late-T dwarfs, with the
only exception of Ross 458 c (Manjavacas et al., 2019a). Nevertheless, Metchev
et al. (2015) showed that most mid- and late-T dwarfs do show low-level
variability ($>$0.4%). Even though few mid- and late-T dwarfs have been
monitored for spectrophotometric variability with space-based facilities that
provide high signal-to-noise data, they are expected to show bands and spots
as Solar System planets do, according to General Circulation models (GCMs)
(Tan & Showman, 2021). GCMs provide instantaneous model maps for the top-of-
the atmosphere thermal flux, and predict the most likely light curve for such
an object. The variability amplitude predicted for a T-dwarf of a similar
spectral type to 2M2228 (2M0050–3322, Manjavacas et al. 2022) is about
$\sim$1% for an edge-on object. The predicted variability amplitude decreases
to about 0.2% for a pole-on object. GCMs predict a band near the equator
driven by the radiative feedback of MnS and $\mathrm{Na_{2}S}$ clouds. A more
extended sample of mid- and late-T dwarfs with high signal-to-noise
spectrophotometric monitoring data from space-based facilities would help to
constrain the properties and dynamics of mid- and late-T dwarfs, and would
allow to confirm the prediction of GCMs.

###  6.5 Analogs to Cool Directly-Imaged Exoplanets

Brown dwarfs are analogs to directly-imaged exoplanets, as they share
effective temperatures, colors, and sometimes surface gravities and masses
(e.g. Faherty et al. 2016). This is the case of the 51 Eridani b directly-
imaged exoplanet (600–750 K, Macintosh et al. 2015), HD 19467 b (1040$\pm$40
K, Wood et al. 2019), Gl 758 b (600–750 K, Bowler et al. 2018), or HD 4113C
(500–600 K, Cheetham et al. 2018), which show significant variability as in
the case of HR8799bc Biller et al. (2021), among others. Thus, a comprehensive
atmospheric characterization of the technically easier to observe isolated
T-dwarfs might shed light on the atmospheric structures of cool directly-
imaged exoplanets (Apai et al., 2017). High signal-to-noise spectro-
photometric time-resolved data provide information about the variability
introduced at different pressure levels of the atmospheres of these objects,
which allow us to understand at which pressure levels most of the
heterogeneous clouds introducing the measured variability are found, using
radiative-transfer models (e.g. Yang et al. 2016). Continued monitoring of few
rotational periods of these objects allow us to produce a map of its surface
and reconstruct which atmospheric structures (band and/or spots) have most
likely produced its light curve at different pressure levels.

Mapping Codes like Stratos (Apai et al., 2013), Aeolus (Karalidi et al., 2015)
or Starry (Luger et al., 2019) allow us to generate 2D maps of a given light
curve expanding over 2-3 rotational periods. Such a detailed atmospheric
characterization is yet technically challenging for directly-imaged exoplanets
(Apai et al., 2016). Nevertheless, with the advent of the first scientific
data from the just launched James Webb Space Telescope, we will be able to
characterize more cool directly-imaged exoplanets. The spectral indices
presented in this paper will be of a special importance, since they will allow
the community to pre-select the most likely variable T-dwarf type-like
directly imaged exoplanets to be monitored for spectro-photometric
variability. Since until now no systematic method to pre-identify variable
T-dwarfs exists, these spectral indices will very likely save hours of
precious telescope time.

##  7 Conclusions

  1. 1.

We designed a set of novel near-infrared spectral indices to find variable
candidate mid- and late-T ($>$T5.5) dwarfs using a single-epoch near infrared
spectrum. To design these novel spectral indices, we used HST/WFC3 near-
infrared, high signal-to-noise, time-resolved spectra of the T6.5 variable
2M2228-4310 published in Buenzli et al. (2012).

  2. 2.

We designed six new spectral indices to find potential variable mid- and
late-T dwarfs. With those six indices, we created twelve index-index plots
which allow us to segregate the variable and non-variable spectra of 2M2228,
and to define areas within those plots where we expect to find variable mid-
and late-T dwarfs.

  3. 3.

Using a Monte Carlo simulation of the most variable and least variable spectra
of 2M2228, we refined the areas in which variable mid- and late-T dwarfs
should be find within the index-index plots.

  4. 4.

Using the synthetically generated spectra of 2M2228, we flagged the objects
that were found in more than 11 out of 12 variable areas as variable
candidates. Objects that were found in 10 out of 12 variable areas might be
flagged as weak candidate variables, since some of them have been found to be
marginally variable.

  5. 5.

Using the newly-generated near-infrared spectral indices, we identified
potential mid- and late-T variable brown dwarfs using their single-epoch
spectra. We used all the mid- and late-T dwarf near-infrared spectra available
at the SpeX/IRTF spectral library with spectral types between T5.5 and T7.5.
We reported a variability fraction of $38^{+4}_{-30}$% for our late-T dwarf
sample assuming that the variable candidates selected are indeed variable. The
variability fraction reported agrees with the variability fraction provided by
Metchev et al. (2015) for the same range of spectral types.

  6. 6.

From the 10 variable candidates selected by the spectral indices, three were
known variables (2M0050–3322, 2M2228-4310 and J1047+2124). Of the objects
selected as non-variable, seven are known non-variables (J1416+1348B,
J0501–0445, J1624+0029, 2MASS J1546-3325, J1024–2453, J1828-4849, and
J1110+0116). Using the index method we are able to recover two of the three
known variables (recovery rate of 67%), and all the known non-variables
(recovery rate 100%). J1047+2124 is a false negative, which would suggest a
false negative rate of 12.5%.

  7. 7.

We reported eight new mid- and late-T variable candidates, which will need to
be confirmed with near-infrared photometric monitoring follow-up.

  8. 8.

To test the robustness of our indices, we produced 100 synthetic spectra using
the SpeX/IRTF spectra of the known variable and non-variable mid- and late-T
dwarfs with a Monte Carlo simulation. We calculated their spectral indices and
place them in the index-index plots. We concluded that the great majority of
the synthetic spectra created for the known variables were also classified as
variables by our indices. Similarly, the synthetic spectra created for the
known non-variables were classified as non-variable by our indices, with the
exception of J1416+1348B.

  9. 9.

The indices presented in this work measure the difference in variability at
the peak of the $J$\- and $H$-bands, and at their sides of the bands. The peak
of the $J$\- and $H$-bands trace the deeper layers of the atmosphere of mid-
and late-T dwarfs (30-40 bar, and 10–20 bar, respectively), and the sides of
the bands trace upper levels ($\sim$10 bar and $\sim$6 bar, respectively).
Thus, if a difference in heterogeneous cloud coverage exist at these different
pressure levels, the variability should change among those, affecting slightly
the flux at the peaks of the $J$\- and $H$-bands, and at their sides.

  10. 10.

Mid- and late-T dwarfs are analogs to cool directly-imaged exoplanets in terms
of temperatures and colors. Thus, the spectral indices presented in this work
could help to pre-select the most-likely variable exoplanet candidates in the
near future, to be monitored in spectro-photometric variability studies with
the James Webb Space Telescope or the 40 m class telescopes scheduled to start
observations during this decade.

##  8 Acknowledgments

We thank the RECA444https://www.astroreca.org/en/ (Red Estudiantes Colombianos
en Astronomía) Summer Internship Program for providing an avenue to initiate
the discussions that led to this work.

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