HUBBLE SPACE TELESCOPE SPECTROSCOPY OF BROWN DWARFS DISCOVERED WITH THE WIDE-FIELD INFRARED SURVEY EXPLORER

3.1.1. WFC3/HST

3.1.2. IRAC/Spitzer

HST spectroscopic observations were carried out using Wide Field Camera 3 (WFC3) as part of Cycle 18 program 12330 (PI: Kirkpatrick), Cycle 19 program 12544 (PI: Cushing), and Cycle 20 programs 12970 (PI: Cushing) and 13178 (PI: Kirkpatrick) and are summarized in Table 3. The G141 (1.1–1.7 μm, $R\equiv \lambda /{\Delta }\lambda \approx 130$) and G102 (0.9–1.1 μm, $R\approx 210$ ) spectroscopic reductions were performed using the methods described in Cushing et al. (2011), with the exception of one final step, described in the following paragraphs.

Because the G141 and G102 grism modes are slitless, source spectra are occasionally influenced by photons from a nearby source or uneven background fluctuations. For this reason, we developed a custom source extraction routine that allows us to define source apertures and background regions on the individual stamp images of a spectrum produced with the AXEDRIZZLE routine (e.g., Figure 7). Once an aperture is defined, we use the aperture corrections of Kuntschner et al. (2011) and the appropriate grism sensitivity curves¹² to flux-calibrate the final spectrum. There is also a 2% absolute flux calibration uncertainty (Kuntschner et al. 2011), which we account for in all synthetic photometry calculations (Section 5.3) and model fits (Section 5.4). An example of the functionality of our extraction technique is shown in Figure 7, where the effects of uneven background fluctuations are mitigated.

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Several sources had multiple HST visits (i.e., spectroscopic observations occurring on different dates). In principle, direct and spectroscopic images from multiple visits can be combined using standard AstroDrizzle routines to produce final images with improved S/N. Yet, because our sources are nearby brown dwarfs, they tend to have significant proper motions, even on timescales as short as a month. For HST grism observations, identical positions are critical for the placement of extraction apertures and wavelength zero points. For sources with multiple visits, AXEDRIZZLE stamp images from each visit are median combined to produce a final spectroscopic image. Spectra are then extracted using the method described above. Objects with both G141 and G102 data are stitched together at 1.1 μm with no scaling done between the two spectra. The complete spectroscopic sample is shown in Figures 8 and 9. We note here that the depressed H-band peak of WISE 0647–6232 seen in Kirkpatrick et al. (2013) does not appear in our reduced spectrum. This is likely due to either the added signal of our spectrum or our improved spectral extraction technique. WISE 0535–7500 is located in an extremely crowded field, and thus retrieving a clean G102 spectrum was particularly difficult, even with multiple observations at multiple roll angles. While we did manage to extract a G102 spectrum for this object, we urge caution in its interpretation, as it is still likely contaminated. We also note that a different roll angle allowed us to extract a cleaner J-band spectrum than that found in Kirkpatrick et al. (2012), though the zeroth-order light from a nearby object fell directly in the H-band portion of WISE 0535–7500's spectrum. For this reason, we do not use the H-band portion of this object's spectrum.

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Upon inspection of their spectra, two objects that met the selection criteria were found to not be brown dwarfs. The spectra for both of these objects were flat and devoid of spectroscopic features and are likely extragalactic in nature. The AllWISE positions and photometry for these contaminants are provided in Table 1.

Spectral types for the six new brown dwarf discoveries (WISE 0325–5044, WISE 0404–6420, WISE 0825+2805, WISE 1206+8401, WISE 2212–6931, and WISE 2354+0240) were determined following the methods of Cushing et al. (2011) and Kirkpatrick et al. (2012). These discoveries are classified via by-eye comparisons of the width of the J-band peak to the spectral standards defined in Burgasser et al. (2006); Cushing et al. (2011), and Kirkpatrick et al. (2013). Comparisons are shown in Figures 8 and 9, while the spectral types are provided in Table 1.

The spectrum of WISE 0325–5044 is an excellent match to the T8 spectral standard 2MASSI J0415195–093506 (Burgasser et al. 2006); hence, we classify it as a T8. Similarly, WISE 0404–6420 and WISE 2212–6931 show excellent agreement with the T9 spectral standard UGPS J072227.51–054031.2 (Cushing et al. 2011) and are classified as such. WISE 1206+8401 is an almost perfect match to the Y0 spectral standard WISE J173835.53+273259.0 (Cushing et al. 2011); therefore, we are confident in assigning it a Y0 spectral classification. Even though the spectrum of WISE 2354+0240 is somewhat noisy, it shows a J-band peak significantly narrower than the Y0 spectral standard and matches well with the Y1 standard WISE J035000.32–565830.2 (Kirkpatrick et al. 2012). Therefore, we assign a spectral type of Y1 for this object, making it the fourth such object to receive such a designation, along with WISE J035000.32–565830.2 (Kirkpatrick et al. 2012), WISE J064723.23–623235.5 (Kirkpatrick et al. 2013), and WISE J154151.65–225024.9 (see Section 4.2). The most ambiguous of all classifications was that of WISE 0825+2805, which is shown separately in Figure 10. A comparison of WISE 0825+2805 with the Y0 and Y1 spectral standards shows that the width of the J-band peak for this object is intermediate between the two. For this reason we assign WISE 0825+2805 a spectral type of Y0.5.

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While all of these discoveries are part of an extensive parallax program to measure their distances, we provide preliminary estimates here based solely on W2 photometry and the polynomial fits from Dupuy & Liu (2012) and Kirkpatrick et al. (2012). Distances estimates are provide in Table 4. Uncertainties are based solely on the photometric uncertainty of the W2 measurement.

Figures 8 and 9 also show HST spectroscopy of five previously known brown dwarfs, including WISE J0335+4310 (T9–Mace et al. 2013), WISE J07344–7157 (Y0–Kirkpatrick et al. 2012), WISE J1541–2250 ("Y0.5?"–Kirkpatrick et al. 2012), WISE J1639–6847 (Y0–Y0.5–Tinney et al. 2012), and WISE J2220–3628 (Y0–Kirkpatrick et al. 2012). Spectral types were reexamined using the methods described in Section 4.1. Spectral types determined in this way agree well with the spectral types determined in Mace et al. (2013); Kirkpatrick et al. (2012), and Tinney et al. (2012). With the improved S/N from HST, we reclassify W1541–2250 as Y1 because the width of its J-band peak is a much better match to the Y1 standard than that of the Y0 standard. This determination supersedes the "Y0.5?" suggested in Kirkpatrick et al. (2012). The J-band peak of WISE 1639–6847 matches well with the Y0 standard, in agreement with the Y0–Y0.5 spectral type determined in Tinney et al. (2012). However, the Y-band peak and $Y-J$ color of this object are unusual compared to other Y0 dwarfs (see Section 5.1 and Figure 13); therefore, we classify WISE 1639–6847 as Y0pec.

In Figure 11 we show the entire combined HST G102/G141 near-infrared spectra for a complete sequence of late-type brown dwarfs from T8 to Y1. There is a distinct trend for the Y-, J-, and H-band peaks as temperatures decrease. The Y-band peak appears more symmetrical going from T8 to Y1. This occurs because the blue wing of the Y-band peak becomes more suppressed for later spectral types. The wavelength at which the peak reaches a maximum also occurs at a different location as temperature decreases, gradually becoming bluer with spectral type. The J-band peak becomes narrower as temperatures decrease, while the H-band peak appears to become more symmetrical for later spectral types. The peaks of both the J and H bands occur at the same position throughout the sequence.

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G102 observations were made for five of the six new brown dwarfs presented in Section 4.1 (the exception being WISE 2354+0240), all five dwarfs presented in Section 4.2, and five additional brown dwarfs, including WISE 0535–7500 (≥Y1–Kirkpatrick et al. 2012), WISE 0647–6232 (Y1–Kirkpatrick et al. 2013), WISE 1405+5534 (Y0pec–Cushing et al. 2011), WISE 1542+2230 (T9.5–Mace et al. 2013), and WISE $2209+2711$ (Y0:–Cushing et al. 2014a). The G102 spectra are presented in Figure 12.

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WISE 1639–6847 shows an increase in the relative height of the Y-band peak compared to other Y0 type dwarfs. The peak is also shifted blueward from the peak of the T9 standard UGPS J072227.51–054031.2. This is a surprising result considering that we find that the J-band peak matches well with the Y0 standard (see Figure 9). Because the Y-band shape of this object is so unusual, we classify it as a Y0pec. We note that a similarly large Y-band peak is seen in the FIRE spectrum of WISE 1639–6847 presented in Tinney et al. (2012) (see their Figure 7).

The Y0pec dwarf WISE 1405+5534 (which is categorized as "pec" due to a wavelength shift of its H-band peak) shows a Y-band peak slightly different from that of the T9 and Y0 dwarfs. Though noisy, the Y-band peak of WISE 1405+5534 is more symmetrical about its maximum than the normal T9-Y0 dwarfs. We also note here that the J-band peak appears slightly narrower than that of the Y0 standard (see Figure 9). For this reason, we reclassify WISE 1405+5534 as Y0.5. The near-infrared colors of WISE 1405+5534 support this change (see Section 5.3).

WISE 2209+2711 (Y0:–Cushing et al. 2014a), while noisy, shows a very peaked appearance, more similar to that of the Y1s than the Y0s. This may be providing a hint that this object is possibly Y1 or intermediate between Y0 and Y1.

The appearance of ammonia (NH₃) in the near-infrared spectra of cool brown dwarfs has been suggested for some time as the harbinger of the Y spectral class (Burrows et al. 2003; Leggett et al. 2007; Kirkpatrick 2008). Models predict that NH₃ absorption bands could become prominent in brown dwarfs with temperatures ≲450 K. Cushing et al. (2011) show a possible detection of NH₃ on the blue wing of the H-band peak of WISE 1738+2732, but this NH₃ feature is itself confused with a strong H₂O band. An additional NH₃ absorption component is predicted to emerge that will affect the shape of the Y-band spectrum of extremely cool dwarfs, eventually causing a bifurcation of the peak around 300 K. So far, a firm detection of NH₃ absorption at 1.03 μm has been elusive (Leggett et al. 2013). Leggett et al. (2013) have suggested that vertical mixing can explain the lack of NH₃ in the spectrum of the Y0 dwarf WISE 2056+1459. Vertical mixing keeps the atmospheric chemistry from reaching equilibrium, thus enhancing the N₂ abundance and decreasing the NH₃ abundance (Saumon et al. 2003; Hubeny & Burrows 2007; Morley et al. 2014). Now, with an increased sample size of late-type brown dwarfs for which Y-band spectra are available, we can investigate the effects of NH₃ in these atmospheres. If the lack of an NH₃ detection in the spectrum of the Y0 dwarf WISE 2056+1459 is due to vertical mixing, as suggested by Leggett et al. (2013), then such a phenomenon may be ubiquitous in early Y dwarfs. We find no obvious evidence of the presence of NH₃ in the Y-band spectra for any of our late-type dwarfs.

We synthesized $Y-$, $J-$, and H–band photometry using the MKO-NIR photometric system for each brown dwarf in our sample. We also synthesized HST F105W, F125W, and F140W photometry using available system throughput tables.¹³ Uncertainties are computed using a Monte Carlo approach. For each spectrum, a value is selected from a normal distribution using the flux density at each wavelength as the mean and the flux density uncertainty as the standard deviation. Synthetic photometry is performed on the resulting spectrum, and the process is repeated 1000 times. We take the mean photometric value as the true value and the standard deviation of the distribution as the uncertainty. The resulting magnitudes and uncertainties are listed in Table 5. To investigate trends as a function of spectral type, we also synthesize photometry on the entire sample of T dwarfs available from the Spex Prism Spectral Library¹⁴ (as of June, 2014) using the same method described above. Colors as a function of spectral type are shown in Figure 13.

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Our synthetic YJH photometry generally agrees well with published photometry from Cushing et al. (2011); Kirkpatrick et al. (2012); Morley et al. (2012); Dupuy & Kraus (2013); Leggett et al. (2013); Beichman et al. (2014); Cushing et al. (2014a); Kirkpatrick et al. (2014), and Leggett et al. (2014), with only a few exceptions (56 out of 62 [87%] measurements have differences that are within 3σ of zero). Our measured H magnitude for WISE 0410+1502 and J magnitude for WISE 1405+5534 differ significantly from the measured values in Cushing et al. (2011) and Kirkpatrick et al. (2012), though they do agree with the values quoted in Leggett et al. (2013). Our J- and H-band magnitudes for WISE 1541–2250 disagree with the values given in Morley et al. (2012) and Leggett et al. (2014), respectively. Using our measured magnitudes, WISE 1541–2250 does not stand out prominently from the other Y1 dwarfs in color–color and color versus spectral type diagrams. Our measured J magnitude for WISE 1738+2732 disagrees with the value quoted in Leggett et al. (2013), though it does agree with the measurements from Cushing et al. (2011) and Kirkpatrick et al. (2012). Lastly, our measured J and H magnitudes for WISE 2056+1459 agree well with the measured values from Cushing et al. (2011) and Kirkpatrick et al. (2012), though they do not agree with the values from Leggett et al. (2013). While we cannot rule out extreme variability as the source of magnitude differences, it is more likely that some measurements are erroneous. Since our photometry comes from our slitless grism spectroscopy, one potential source of error in our measurements is contamination from nearby bright objects or uneven background fluctuations, though our reduction process (Section 3.2) should minimize such effects.

Previous investigations of color/spectral type relations show distinct changes at the T/Y boundary (Lodieu et al. 2013; Leggett et al. 2013, 2014). We see similar changes in Figure 13, namely, the $Y-J$ (and $F105W-F125W$) color plummets for spectral types later than T8, and the $J-H$ color plateaus at ∼T7 and slowly turns redder for the Y dwarfs. As noted by Leggett et al. (2010), the blueward trend in the $Y-J$ colors of late-type dwarfs may be due to K I condensing into KCl as temperatures decrease. This condensing weakens the broad 0.77 μm K I doublet, resulting in a brighter Y band. A similar trend is seen in the $z-J$ colors of Y dwarfs in Lodieu et al. (2013). The redward trend of the $J-H$ color for the latest Y dwarfs occurs because the J-band peak for these objects continues to narrow, while the H-band peak does not (see Figure 11).

Figure 14 shows the $Y-J$ (when available) versus $J-H$ and $J-H$ versus $J-W2$ colors for our complete sample of brown dwarfs along with synthesized colors from the model spectra of Saumon et al. (2012); Morley et al. (2012), and Morley et al. (2014). While neither the cloudy nor cloud-free models reproduce the trends seen for our measured dwarfs, the cloudy models do a better job approximating the colors of our late-type dwarf sample than the cloud-free models. Morley et al. (2012) show that the inclusion of clouds is an efficient way to redden the $J-H$ colors of late-type brown dwarfs. Figure 14 also shows that models with lower surface gravities and lower f_sed values do not fit the observed color trends as well.

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The effective temperatures of brown dwarfs are typically estimated in two ways. If the distance to a brown dwarf is known and a significant fraction of its emergent flux can be measured, then its bolometric luminosity can be computed. Since brown dwarfs all have similar radii due to the equation of state of their partial degenerate interiors, an effective temperature can then be computed using the Stefan Boltzmann law (e.g., Golimowski et al. 2004; Vrba et al. 2004; Dupuy & Kraus 2013). Alternatively, model spectra can be compared directly to observed spectra and/or photometry to infer not only the effective temperatures of brown dwarfs but also their surface gravity and metallicity (e.g., Cushing et al. 2008; Witte et al. 2011; Leggett et al. 2013). Because accurate distances to many of the dwarfs in our sample are not known, we have taken the second approach and compared our near-infrared spectra and mid-infrared photometry to atmospheric models in order to estimate their atmospheric parameters.

In particular, we compare are spectra and photometry to several different sets of solar-metallicity models, including the cloud-free models from Saumon et al. (2012), the cloudy models of Morley et al. (2012), and the cloudy models of Morley et al. (2014). The latter set is a new suite of models that, in addition to including sulfide and chloride condensates, also includes water-ice clouds for objects with T$_{eff}\;\leqslant \;450$ K. Some models from Morley et al. (2014) are also "partly cloudy", calculated with 50% cloudy and 50% hole surface coverage. The models are arranged in a grid of T_eff, log g, and f_sed values. The f_sed parameter describes the efficiency of cloud sedimentation in the models, where larger f_sed values correspond to larger grain sizes, which lead to an increased sedimentation efficiency and hence thinner clouds. We adopt the f_sed = "nc" or "no clouds" notation of Cushing et al. (2008) to refer to the cloud-free models. Table 6 summarizes the T_eff, log g, and f_sed combinations that were used for model fitting.

In order to compare the models to our observed spectra, each model spectrum was resampled to be uniform in ln(λ) space, smoothed with a Gaussian kernel to a resolving power of HST WFC3, and resampled onto the same wavelength grid as the HST WFC3 spectrum. Synthetic flux density values in the [3.6] and [4.5] Spitzer bands were computed from the models as described in Cushing et al. (2006). Following Cushing et al. (2008), we evaluate a goodness-of-fit parameter for each model spectrum, defined as

$$\begin{eqnarray}&&{{G}_{k}}=\mathop{\sum }\limits_{i=1}^{n}{{\left( \frac{{{f}_{i}}-{{C}_{k}}{{{\mathcal{F}}}_{k,i}}}{{{\sigma }_{i}}} \right)}^{2}},\end{eqnarray} \tag{ 1 }$$

where n corresponds to each data pixel, f_i is the flux density of the data, ${{{\mathcal{F}}}_{k,i}}$ is the flux density of the model k, and ${{\sigma }_{i}}$ are the errors for each observed flux density. The C_k parameter is a multiplicative constant given by

$$\begin{eqnarray}&&{{C}_{k}}=\frac{\sum \;{{f}_{i}}{{{\mathcal{F}}}_{k,i}}/\sigma _{i}^{2}}{\sum {{{\mathcal{F}}}_{k,i}}/\sigma _{i}^{2}}.\end{eqnarray} \tag{ 2 }$$

C_k is equivalent to $({{R}^{2}}/{{d}^{2}})$, where R is the stellar radius and d is the distance. For each model we compute a C_k value and corresponding G_k value and identify the best-fitting model as that model with the lowest overall G_k value. Because in most instances the difference between the best-fitting models for each dwarf is marginal, we proved a range of best-fit model parameters from the three best-fitting models for each brown dwarf in Table 7. The (R²/d²) and ${{\chi }^{2}}$ values in Table 7 refer to the model with the best fit.

Table 7.  Model-derived Physical Characteristics^a

AllWISE Name	Spec.	Teff	log g	fsed	% Cloudy	(R/d)	${{\chi }^{2}}$/d.o.f	Ageb
	Type	(K)	(cm s−1)		(%)	(Rjup/pc)		(Gyr)
WISEA J032504.52–504403.0	T8	550–600	4.0	3–4	100	2.468 × 10−2	5213/211	0.08–0.3
WISEA J033515.07+431044.7	T9	500–550	4.0–4.5	3–5	100	1.973 × 10−2	5276/211	0.2–1.5
WISEA J035000.31–565830.5	Y1	300–350	5.0	..., 5	0,50	4.704 × 10−2	2873/128	U
WISEA J035934.07–540154.8	Y0	400	4.0–5.0	2, 3, 5	100	3.641 × 10−2	2494/128	1.5–8
WISEA J040443.50–642030.0	T9	550–600	4.0	3–4	100	1.861 × 10−2	4458/211	0.08–0.3
WISEA J041022.75+150247.9	Y0	400	4.0–4.5	2, 3, 4	100	1.007 × 10−1	5172/128	1.5–8
WISEA J053516.87–750024.6	≥Y1	450–500	5.0	..., 5	0,50	8.397 × 10−3	2744/179	4–U
WISEA J064723.24–623235.4	Y1	350–400	5.0	..., 5	0,50	2.609 × 10−2	2621/211	8–U
WISEA J073444.03–715743.8	Y0	450	4.0–4.5	3–5	100	2.410 × 10−2	3109/211	0.4–2
WISEA J082507.37+280548.2	Y0.5	400	4.5–5.0	..., 4, 5	0,100	1.808 × 10−2	3313/211	3–U
WISEA J094306.00+360723.3	T9.5	450–500	4.0–4.5	2, 3, 5	100	4.167 × 10−2	4506/128	0.3–2
WISEA J120604.25+840110.5	Y0	400–450	4.0–4.5	3, 5	100	4.040 × 10−2	3475/211	0.4–3
WISEA J140518.32+553421.3	Y0.5	350–400	5.0–5.5	..., 5	0,100	3.218 × 10−2	4117/211	U
WISE J154151.65–225024.9	Y1	400	4.0–4.5	...	0	2.072 × 10−2	4102/211	0.6–3
WISEA J154214.00+223005.2	T9.5	450–500	4.0–4.5	3–5	100	2.127 × 10−2	4051/211	0.3–2
WISEA J163940.84–684739.4	Y0pec	400	5.0	..., 5	0,100	3.186 × 10−2	10859/211	8–U
WISEA J173835.52+273258.8	Y0	400	4.0–4.5	2, 4, 5	100	6.445 × 10−2	6598/128	0.6–3
WISEA J205628.88+145953.6	Y0	400–450	4.0–4.5	2, 4	100	5.541 × 10−2	10770/128	0.4–3
WISEA J220905.75+271143.6	Y0:	500–550	4.0–4.5	4, 5	100	6.682 × 10−3	3106/211	0.2–1.5
WISEA J221216.27–693121.6	T9	500–600	4.0	3, 5	100	1.541 × 10−2	5389/211	0.08–0.4
WISEA J222055.34–362817.5	Y0	400–450	4.0–5.0	2, 3, 5	100	3.204 × 10−2	4574/211	1–6
WISEA J235402.79+024014.1	Y1	300–400	4.0–5.0	2, 5	50,100	5.961 × 10−2	10663/128	1.5–U

Notes:

^aModel-derived parameters are generally unreliable for reasons explained in Section 5.4. ^bAges are derived for the T_eff and log g ranges in columns 3 and 4 using the Saumon & Marley (2008) cloudless evolutionary models. Some combinations of T_eff and log g derived from the atmospheric fits are unphysical in that brown dwarfs cannot evolve to have such values, so the corresponding ages are denoted as "U" for unphysical.

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The derived effective temperatures are generally correlated with spectral type in that the effective temperatures decrease with increasing spectral type: WISE 0325–5044 (the lone T8) has an effective temperature range of 550–600 K, the T9 dwarfs have temperatures between 500 and 600 K, the T9.5 dwarfs have temperatures between 450 and 500 K, the Y0 dwarfs have temperatures between 400 and 450 K, the Y1 dwarfs have temperatures between 300 and 400 K. Exceptions include the Y0: dwarf WISE 2209+2711 (T_eff = 500–550 K) and the Y1 dwarf WISE 0535–7500, which is located in an extremely crowded field and thus has contamination issues (see Section 5.1).

A comparison with the effective temperatures derived from bolometric luminosities in Dupuy & Kraus (2013) finds some agreement, though there are several discrepancies. The main discrepancies appear for the objects with very uncertain parallaxes in Dupuy & Kraus (2013) (WISE 0359–5401, WISE 0535–7500, WISE 1541–2250, and WISE 1639–6847). For these objects with uncertain parallaxes, two (WISE 0359–5401 and WISE 1639–6847) are found to have T_eff values much colder in Dupuy & Kraus (2013) than our model-fitting derived values, while the other two (WISE 0535–7500 and WISE 1541–2250) are found to be much warmer. It is unclear whether these discrepancies are solely due to parallax uncertainties or other systematic errors (see below).

Although atmospheric models can be computed for any given T_eff/log g values, not all combinations are physical because theoretical evolutionary models limit the T_eff /log g values that brown dwarfs can obtain. We therefore compared our derived T_eff/log g values to the low-mass, solar-metallicity, cloud-free evolutionary models of Saumon & Marley (2008) in order to confirm that our best-fit models are all physical (Figure 15). We note that the atmospheric models used in our analysis were not used as boundary conditions for the evolutionary models and thus our comparisons are not technically self-consistent. Inspection of Figure 15 suggests that several of the derived surface gravities are not physical (those objects with log g values of 5 and T_eff estimates ≤400 K). Evolutionary models can also be used to estimate the ages of brown dwarfs. Most of our sources have ages of several Gyr, as expected for objects near to the Sun, though some objects were found to have younger (a few hundred Myr) or older (>5 Gyr) ages than expected. Indeed, a trend is apparent whereby the earlier spectral types in our sample (T8 and T9) are preferentially fit with lower log g values and hence younger ages, while the latest spectral types (Y0.5 and Y1) are preferentially fit with larger log g values (some of which are unphysical). Although it is possible that this trend is real, it is more likely a result of systematic errors in the model fits. Fits with unphysical gravities are indicated as such in the age column of Table 7.

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While the derived temperatures are generally consistent with previous work, the fits to the data are, in many cases, rather poor. The discrepancies between the data and models generally fall into two categories. Because the HST spectra contain over 200 data points, the two IRAC points carry little weight in the fitting process, and thus the models "match" the near-infrared data reasonably well but often miss the Spitzer/IRAC points badly. Figure 16 shows the best fits to the T8 to Y1 spectral sequence from Figure 12. In some instances (e.g., WISE 0325–5044, WISE 1541–2250), the Spitzer model fluxes can be off by factors of 0.2 to 4, respectively. Since much of the energy of cold brown dwarfs emerges at 5 μm, the bolometric luminosity of the best-fitting models disagrees with the observations by similar factors. A factor of 2 to 5 in L_bol can result in an effective temperature difference of up to 200 K (see Figure 15).

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Closer inspection also reveals large differences between the models and observations in the shapes of the near-infrared bands. In particular, the heights of the model H-band peaks are significantly different from the observations, with the observed H-band peak consistently being higher than that of the best-fitting model. The height, width, and position of the J-band peak are generally well matched to the models. For the two latest spectral types, the J-band peak of the observations does not reach the same heights as those from the best-fitting models. The Y-band peaks for the two T dwarfs match well with the models, while the fits to the Y-band peaks of the Y dwarfs do not. The main discrepancy in the Y-band spectra of the Y dwarfs is the absence in the observations of the NH₃ absorption feature expected to occur for temperatures $\lt 500$ K. The problems with these particular model fits are nicely summarized by the color–color diagrams shown in Figure 14.

Some of the issues in the near-infrared are likely a result of the fact that the models do not account for non-equilibrium NH₃ chemistry due to vertical mixing in the atmosphere (see Section 5.2). Figure 17 shows the near-infrared spectrum of WISE 1541–2250 (Y1) along with the cross-section spectrum of NH₃. Ammonia absorption shapes the entire near-infrared spectrum—from the blue wing of the H-band peak, to the width of the J-band peak, to the shape of the Y-band peak. Morley et al. (2014) show that when all else is equal, the decrease in the NH₃ abundance broadens the J- and H-band peaks and removes the strong NH₃ absorption feature in the Y band. The inclusion of non-equilibrium chemistry, which would result in significant changes to the model spectra, would no doubt result in different effective temperature and surface gravity estimates. Based on the H band alone, the change would be 300 K at a fixed gravity. Given the significant mismatches identified between these model spectra and data, we conclude that we cannot reliably derive reliable atmospheric parameters by fitting these model spectra to the observed spectra and photometry of our sample.

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Leggett et al. (2014) state that the Y, H, and ch1 model fluxes are too low by about a factor of two compared to photometric measurements for late-type brown dwarfs. By extension, this suggests that the J and $W2$ magnitudes are not affected and thus the $J-W2$ color may be able to provide a reasonable effective temperature estimate when compared with colors from model spectra. Figure 18 shows the $J-W2$ color as a function of effective temperature for three different models—a cloud-free model from Saumon et al. (2012) (log g = 5), a 50% cloud coverage model from Morley et al. (2014) (log g = 5, f_sed = 5, with H₂O), and a 100% cloud coverage model from Morley et al. (2012) (log g = 5, f_sed = 3, without H₂O). Because our grid of models is, as of yet, incomplete (see Table 6), we show the only 50% cloud coverage model (with H₂O) for which the entire 300–1000 K temperature range is available, and the 100% cloud coverage model does not extend to temperatures <400 K. Even when limited to these three models, Figure 18 shows that effective temperature estimates for $J-W2$ values less than ∼7 typically have ranges greater than 100 K. For $J-W2$ values greater than ∼7, varying percentages of cloud cover result in minimal differences in effective temperature, though the 100% cloudy model does not extend to this range. The range of temperatures for a single $J-W2$ value will only increase with the inclusion of additional models with varying log g and f_sed values.

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Furthermore, as noted previously, the $J-$ band flux will also be affected by the reduced amount of NH₃ present in the atmosphere due to vertical mixing (not only Y and H). Figure 12 of Morley et al. (2014) displays equilibrium and disequilibrium models at 450, 300, and 200 K. For the 450 K models, including disequilibrium chemistry will change the Y–, J–, and $H-$ band magnitudes by ∼0.2, ∼0.3, and ∼0.6, respectively, implying that these near-infrared bands are affected by similar amounts when vertical mixing is included. Because of the large temperature ranges for individual $J-W2$ values, uncertain temperature ranges for $J-W2$ colors due to our incomplete model grid, and the similar influence of vertical mixing on the $J-$ band magnitude when compared to the Y and H bands, we conclude that estimating temperatures based solely on model $J-W2$ colors is unsound. Since our observations are limited to near- and mid-infrared wavelengths and the current models do not include non-equilibrium chemistry, we lack a method to determine accurate effective temperatures for the objects in our sample.

Given the limitations of these particular models, determining effective temperatures by computing bolometric luminosities may therefore be preferable (e.g., Dupuy & Kraus 2013). However, determining luminosities in this way currently relies on models to estimate the flux levels for up to 50% of the total luminosity (Dupuy & Kraus 2013). Dupuy & Kraus (2013) note that all current atmospheric models provide similar bolometric corrections in the wavelength ranges where flux has not yet been directly measured. Nevertheless, minimizing the dependence of the derived bolometric luminosities on the models should be a high priority, and therefore determining accurate effective temperatures for the lowest-mass brown dwarfs should focus on obtaining flux measurements at wavelengths longer than ∼5 μm. The near-infrared (NIRSpec and NIRCam) instruments and in particular the mid-infrared instrument (MIRI) aboard the forthcoming James Webb Space Telescope could prove invaluable in providing the additional flux coverage needed to determine accurate effective temperatures for the lowest-mass brown dwarfs.