---
abstract: 'We examine the hypothesis that the red near-infrared colors of some L dwarfs could be explained by a "dust haze" of small particles in their upper atmospheres. This dust haze would exist in conjunction with the clouds found in dwarfs with more typical colors. We developed a model which uses Mie theory and the Hansen particle size distributions to reproduce the extinction due to the proposed dust haze. We apply our method to 23 young L dwarfs and 23 red field L dwarfs. We constrain the properties of the dust haze including particle size distribution and column density using Markov-Chain Monte Carlo methods. We find that sub-micron range silicate grains reproduce the observed reddening. Current brown dwarf atmosphere models include large grain (1–100 $\mu m$) dust clouds but not submicron dust grains. Our results provide a strong proof of concept and motivate a combination of large and small dust grains in brown dwarf atmosphere models.'
author:
- 
- 
- 
- 
- 
title: 'Exploring the Role of Sub-micron Sized Dust Grains in the Atmospheres of Red L0 – L6 Dwarfs'
---

Introduction and Problem
========================

Brown dwarfs are substellar objects with intermediate masses bridging the stellar and planetary regimes. Brown dwarfs do not have high enough mass to sustain hydrogen fusion in their cores, so they keep cooling with time. As brown dwarfs age, they also contract and their surface gravities increase. The cool temperatures of brown dwarfs allow condensates to form in their atmospheres, which shape their emergent spectra [@Burrows01].

There is a wide spread in $J - K_s$ colors of L dwarfs of a given spectral type. So-called 'red' L dwarfs have redder-than-average near-infrared (NIR) colors, and have notably redder spectral slopes through the NIR than typical L dwarfs [@faherty13]. Low-gravity objects tend to be systematically redder but some field-aged L dwarfs also have redder colors. Red, low-gravity L dwarfs can be considered 'exoplanet analogs' and are of particular interest [@Kirkpatrick08; @cruz09; @Faherty09; @faherty13]. While the range of NIR colors of L dwarfs has not been fully explained, it is commonly attributed to variation in metallicity, gravity, and/or cloud properties [@saumon08; @Marley12]. Brown dwarfs are thought to have refractory particles typically ranging 1–100 $\mu m$ in size in their atmospheres and these particles shape their emergent spectra [@AM01; @Cushing08]. Current brown dwarf atmosphere models either use these large dust grains that are organized into discrete cloud layers, or small submicron grains that are mixed throughout the atmosphere [@Allard01]. These atmosphere models reproduce the general trend of L dwarf spectra but fail to give a reasonable explanation for the observed reddening in the NIR. Model spectral energy distribution (SED) fitting sometimes gives unrealistically low gravities and small $f_{\rm sed}$ (cloud sedimentation efficiency parameter) values [@AM01; @Cushing08] and radii too small compared to evolutionary models [@Liu13] for red L dwarfs. A better dust treatment is needed that can account for the observed reddening in L dwarf spectra. In efforts to understand the red objects empirically, it has been found that dust described by the interstellar reddening law can de-redden red L dwarf spectra to look like standard objects [@Looper10; @Marocco14]. Interstellar reddening is the extinction of starlight caused by the interstellar dust. The interstellar grains, which have radii less than 1 $\mu m$, suppress blue light more effectively than red light. As a result, distant stars look redder than they actually are. The interstellar reddening law (extinction law) is an empirical relationship between the extinction at any wavelength $A(\lambda)$ and the visual extinction $A(V)$ [@CCM89]. We do not expect significant interstellar reddening in brown dwarfs since they are so close to the Sun on a galactic scale, so the use of the interstellar reddening law to de-redden brown dwarf spectra is not physically motivated. However, ISM-like grains ($< 1~\micron$) in the atmospheres of red L dwarfs might produce results similar to the interstellar reddening law and explain the observed reddening.

@Looper10 de-redden the optical spectra of TWA 30 (young M5 star) using the interstellar reddening law described by @CCM89. Although the reddening seen towards TWA 30 may be due to dust in the disk within the line of sight, @Looper10 demonstrated that the interstellar reddening law can be successfully used in this case and it can be useful for dealing with reddening due to small grains other than the interstellar medium (ISM). @Marocco14 de-redden the spectra of several L dwarfs including ULAS J222711$-$004547 (L7pec) using two interstellar extinction curves [@CCM89; @Fitz99] and found that sub-micron size dust grains can explain the reddening effect. Furthermore, @Cushing06 ascribe a flattening in the spectra of mid-type L dwarfs seen at 9–11 $\micron$ to a population of small silicate grains above the main cloud deck.

These three results imply that brown dwarfs with red spectral energy distributions may have small grains like the ISM which are smaller than 1 $\micron$ in addition to larger grains ranging from 1–100 $\micron$ currently included in the models in their atmospheres that scatter and absorb the emergent light.

In this work, we develop a prescription for a dust haze in L dwarf atmospheres and test whether it can account for the characteristics of red L dwarf spectra. By constraining the nature of this dust haze we aim to better understand the physical cause of the reddening in brown dwarfs. Independently, two previous studies explored a similar dust haze analysis, @Marocco14 for brown dwarfs and @Bonn15 for directly imaged exoplanets. @Marocco14, @Bonn15, and this work are all motivated by the success of the interstellar reddening law in de-reddening unusually red L dwarf spectra and aim to explain the observed reddening by introducing a layer of small dust grains in the upper atmospheres. Despite the similar ideas and concepts, the method we describe here is distinct from those previous studies as described in $\S\ref{sec: Model}$ and $\S\ref{subsec: Comparison}$. The sample of L dwarfs studied in this analysis is presented in $\S\ref{sec: Data}$. In $\S\ref{sec: reddening}$, we present our method of using spectral observations to estimate the reddening. We explain our dust haze model in detail in $\S\ref{sec: Model}$, and model fitting method in $\S\ref{sec: Method}$. Finally, we present our results in $\S\ref{sec: Results}$ and conclusion in $\S\ref{sec: Conclusion}$.

Sample and Spectral Observations {#sec: Data}
================================

In order to study the observed reddening, we compiled a sample of low-resolution NIR spectra of 23 L dwarfs with low gravity features in the optical [@cruz09] and 23 red field L dwarfs [@Kirkpatrick10].

Our red field objects have spectral features indicative of field gravity with $J-K$ colors redder than the spectral standards [@Kirkpatrick10]. This definition of 'red' is specific to the purpose of this analysis. For example, @faherty13 defined 'red' as having a redder $J-K$ color than the mean $J-K$ of normal objects in the spectral type as opposed to comparing to the spectral standard. Therefore, red objects in our sample may be different from objects that are defined red by @faherty13 or other papers. The objects in our sample are listed in Table 1. Our sample includes 18 new spectra of red L dwarfs obtained with the SpeX spectrograph on the Infrared Telescope Facility (IRTF) [@Rayner03]. Observations were obtained over 28 nights during 2003–2011. The targets and observation dates are listed in Table \[table: data\]. All objects were observed in clear and dry conditions. The targets were observed dithered pairs (ABBA) to enable pair-wise subtraction. We used the 0.5$\arcsec$ slit and prism-dispersed mode to obtain $\lambda/\Delta\lambda \thickapprox $ 120 spectra covering 0.7-2.5 $\mu$m. The data were reduced using the SpeXtool package [@Cushing04], nearby A0 V stars were observed for flux calibration and telluric correction [@Vacca03], and internal flat field and Ar arc lamp exposures were obtained for pixel response and wavelength calibration.

Estimating the Observed Reddening {#sec: reddening}
=================================

In order to estimate the observed reddening, we compared red L dwarfs (including young and field) to the field spectral standards. The observed reddening was obtained by dividing the spectrum of the field standard by the spectrum of the red L dwarf. The top panel of Figure $\ref{fig: spec}$ shows spectra of a red L dwarf and a spectral standard, illustrating the redder spectral slope of the red object. The bottom pane is the ratio of the two spectra and visualizes the estimated observed reddening of the red object. The small-scale features seen in the observed reddening are due to gravity-sensitive spectral features such as FeH, VO, and the triangular-shaped H band [@Kirkpatrick06; @Allers13]. We assume these features are not caused by reddening, so we treat the overall shape of the observed reddening as a smooth curve.

Modeling the Observed Reddening {#sec: Model}
===============================

In this paper, we develop a prescription for a hypothesized dust haze of small particles in the atmospheres of the so-called 'red' L dwarfs to explain the observed reddening in their spectral energy distributions (SEDs). In order to model the observed reddening, we used Mie theory to calculate the 'raw' extinction coefficients due to forsterite grains. Then we averaged the raw extinction coefficients over various particle size distributions to calculate the 'effective' extinction coefficients and to generate a model grid to compare with the observed reddening.

Dust Haze Prescription
----------------------

In Figure $\ref{fig: model}$, we show an illustration of the proposed dust haze in the upper atmospheres of red L dwarfs. Our model prescribes that the dust haze must be high in the atmospheres so that it is too cool (since the temperature decreases as the altitude increases) to radiate significantly in the NIR. The prescribed dust haze lies above the main cloud deck, so the dust grains in the haze affect the emergent spectra. Our prescription does not have thickness or height, therefore we are not constraining the position or dimensions of the dust haze any further than lying above the main cloud deck. The dust haze grains were modeled by forsterite grains. As shown by @Lodders06, forsterite ($\rm Mg_{2}SiO_4$) is thought to exist in L dwarf atmospheres among other dust species such as corundum ($\rm Al_{2}O_3$), enstatite ($\rm MgSiO_3$), and iron. Corundum condenses at higher temperatures in the atmospheres of late M dwarfs. Liquid iron and silicates condense in early L dwarfs between 1600 – 1840 K. Since iron grains form deeper in the atmosphere, the dust is most likely silicate. The silicate grains in the L dwarf atmospheres are thought to be a mixture of forsterite and enstatite. Since the extinction curves of forsterite and enstatite have similar shapes, forsterite was used in our analysis.

Extinction is the sum of absorption and scattering, and is the fraction of incoming light that gets affected by interactions with particles. Reddening is a type of extinction, which occurs when extinction is more effective at shorter (bluer) wavelengths than at longer (redder) wavelengths, and has the effect of making the spectral slope redder. We used Mie theory to model the reddening effect of the proposed dust haze on the emergent spectra of red L dwarfs. Mie theory applies when the scattering particle is spherical and its size is similar to the wavelength of the scattered light, which is appropriate for sub-micron size grains in the NIR. For larger particles, Mie scattering is independent of wavelength while for smaller particles, it is wavelength dependent. Mie scattering reduces to strongly wavelength dependent Rayleigh scattering when particle sizes are much smaller than wavelength. We employed a Mie code, described in , and the refractive indices of forsterite (G. Sloan, pers. comm.) to compute the 'raw' extinction coefficients, $Q_{\rm ext}(r, \lambda)$, for particles of radii between 0.01 and 10 $\micron$.

Particle Size Distributions {#subsec: distribution}
---------------------------

In order to calculate effective extinction coefficients from the 'raw' extinction coefficients directly computed from the Mie code, we need to choose a particle size distribution $n(r)$. We considered three different particle size distributions commonly used to model grains, clouds and hazes in various settings: power law, Gaussian, and Hansen distributions. Figure $\ref{fig: dist}$, compares the shape of the three distributions. We describe the motivation to use these particle size distributions and results below.

We considered a power law particle size distribution $n(r) \propto r^{p}$ with $p = -3$ and particle radius $r$ ranging between 0.01 – 10 $\micron$ to model theoretical extinction due to the proposed small dust grains. Power law particle size distributions with $p \approx -3.5$ are typically used to characterize interstellar dust and grains in the circumstellar disks around young brown dwarfs [@Mathis77; @Draine06; @Luhman05; @BF14]. Figure $\ref{fig: dist}$ shows two different power law particle size distributions with $p =  -3$ and -3.5. We also considered a Gaussian particle size distribution $n(r) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ with the mean radius $\mu = 0.5 \micron$ and a width $\sqrt{2}\sigma = 0.1\times \mu$. This particle size distribution is adopted from @Marocco14, who de-reddened ULAS J222711$-$004547 using corundum and enstatite, and other red L dwarfs using corundum with Gaussian particle size distributions and found $\mu \thicksim 0.5 {\mu}m$.

We also considered the Hansen particle size distribution, which is a variation of the gamma distribution and is expressed as follows. $$n(r) = r^{\frac{1-3b}{b}}e^{-\frac{r}{ab}}
\label{eq: eq1}$$ where $a$ is the mean effective radius and $b$ is the effective variance. Following @Hansen, $a$ and $b$ are defined as $$a = \frac{\int_{0}^{\infty}\! r{\pi}r^2 n(r)dr}{\int_{0}^{\infty}\! {\pi}r^2 n(r)dr}$$ $$b = \frac{\int_{0}^{\infty}\! (r - a)^2 {\pi}r^2 n(r)dr}{a^2 \int_{0}^{\infty}\! {\pi}r^2 n(r)dr}$$ respectively.

The Hansen particle size distribution successfully reproduces the observed particle size distributions of different types of water clouds in Earth's atmosphere (fair weather cumulus, altostratus, and stratus clouds) as shown in Figure 1 of @Hansen. Figure $\ref{fig: dist}$ shows two Hansen particle size distributions with the mean effective radius $a = 0.2~\micron$ for effective variances $b$ = 0.1 and 0.5. The Hansen particle size distribution with a large variance (dashed green, $b = 0.5$) is similar to power law particle size distributions in the regime of small particle size ($\lesssim 0.1~\micron$).

Computing Modeled Extinction Curves and Comparing to Observations {#subsec: Qeff}
-----------------------------------------------------------------

In order to account for a range of particle sizes and to smooth over the small-scale interference patterns in forsterite extinction coefficients, we computed effective extinction coefficients by averaging the raw extinction coefficients over a particle size distribution. Effective extinction coefficients are defined as $$Q_{\rm ext}(\lambda) = \frac{\int_{r_{\rm min}}^{r_{\rm max}}\! {\pi}r^2Q_{\rm ext}( r, \lambda)n(r)dr}{\int_{r_{\rm min}}^{r_{\rm max}}\! {\pi}r^2n(r)dr}
\label{eq: eq4}$$ where $n(r)dr$ is the number of particles per unit volume with radius between $r$ and $r+dr$. The integration limits we employed are 0.01$-10~\micron$. Particles smaller than this range would be too small (a few atoms) to scatter light and particles that exceed $10~\micron$ tend to be grey at all wavelengths (i.e. the extinction they cause will be independent of wavelength).

In order to compare the observed reddening to the modeled extinction, we assume that the observed flux, $I$, from a red L dwarf can be modeled as $$I(\lambda) = fI_{0}(\lambda)e^{-\tau(\lambda)}$$ where $I_{0}(\lambda)$ is the flux of the field standard L dwarf, $f$ is a scaling factor, and $\tau(\lambda)$ is the optical depth of the dust haze in the red L dwarf atmosphere, assuming the red L dwarf can be described by the field L dwarf surrounded by the dust haze as in Figure 2. The scaling factor $f$ is determined by the distances and sizes of the objects. $$f = \frac{{d_0}^2}{d^2}\frac{R^2}{{R_0}^2}$$ where $d_0$ and $d$ are the distances to the field and the red L dwarfs, and $R$ and $R_0$ are the radii of the red and the field L dwarfs, respectively.

Solving Equation $\ref{eq: eq1}$ for the optical depth we get $$\tau(\lambda)  = \ln{f} + \ln{\frac{I_0(\lambda)}{I(\lambda)}}.$$ The optical depth $\tau(\lambda)$ is related to the Mie extinction coefficient $Q_{\rm ext}(\lambda)$ as $$\tau(\lambda) = N{\pi}a^{2}Q_{\rm ext}(\lambda)$$ where $N$ is the column density of the haze layer and $a$ is the effective radius of the scattering grains which we assume are composed of forsterite. $Q_{\rm ext}(\lambda)$ is the wavelength dependent forsterite grain extinction coefficient which we calculated using the Mie code . We averaged the forsterite coefficients over a particle size distribution $n(r)$ as described in Equation \[eq: eq4\]. Combining Equations 7 and 8, we get $$\ln{\frac{I_0(\lambda)}{I(\lambda)}}  = N{\pi}a^{2}Q_{\rm ext}(\lambda) + C
\label{eq: eq9}$$ where $N$, $a$, and $Q_{\rm ext}(\lambda)$ are the column density of the dust haze, the mean effective radius found from the particle size distribution $n(r)$, and the effective forsterite extinction coefficients of small grains from Equation \[eq: eq4\]. The constant term $C$ accounts for the scaling factor $f$ and any differences in grey atmospheric opacity between the L dwarfs. Since the reddening we observe is wavelength dependent, we relegate any grey component to the $C$ term. $C$ is independent of wavelength because $Q_{\rm ext}(\lambda)$ approaches a constant for $a >> \lambda$. The RHS of Equation \[eq: eq9\] is the modeled extinction and the LHS is the observed reddening. As explained later in $\S\ref{sec: Method}$, we used MCMC methods to constrain the parameters in the modeled extinction ($N$, $a$) that best reproduce the observed reddening.

### Evaluation of Particle Size Distributions and Creating a Model Grid

Figure $\ref{fig: residual}$ shows model fits to the observed extinction using the three different particle size distributions. The top panel shows the observed reddening (black, same as the bottom panel in Figure $\ref{fig: spec}$), theoretical extinction curves using the Hansen, power law, and Gaussian particle size distributions. $\chi^2$ value and the degrees of freedom for each fit are reported in the legend. The bottom panel shows the residual between the observed reddening and the theoretical extinction curve as a percentage of the observed flux. Figure $\ref{fig: residual}$ demonstrates that all three modeled extinction curves fit the observed reddening reasonably. The Hansen and power law particle size distributions reproduce the smooth shape of the observed reddening, while the Gaussian particle size distribution results in a less smooth extinction curve.

For the remainder of our analysis, we adopt the Hansen particle size distribution because of the favorable behavior shown in Figure $\ref{fig: residual}$, the fact that it reflects the microphysics and structure of Earth's clouds, and because it has various helpful properties that are useful for algebraic manipulation [@Hansen].

In Figure $\ref{fig: Qext}$, we show our model grid of forsterite extinction coefficients for various Hansen particle size distributions. The effective forsterite extinction curves for small mean particle sizes ($a \leq 0.4\,\micron$) more closely resemble the observed reddening than those with a larger mean particle size. For these small mean particle sizes, extinction coefficients are wavelength dependent and resemble the curved shapes of the observed reddening (Figure $\ref{fig: spec}$). For particles larger than 0.4 $\micron$, extinction coefficients are less wavelength dependent and the resulting extinction shapes are flat ('grey'), which do not fit the observed reddening. The curves for $a = 1.0 \micron$ are shown as representatives of the curves with $a > 0.4 \micron$.

The range of effective variance $b$ was determined based on the shape of the Hansen particle size distribution. Small effective variances ($b <  0.1$) result in a particle size distribution concentrated at the mean effective radius, and large effective variances ($b > 1$) make the particle size distribution wide and resemble a power law particle size distribution. Thus, we decided to use Hansen particle distribution with a parameter grid of mean effective radius $a$ between 0.05 and 0.4 $\micron$, and effective variance $b$ between 0.1 and 1.0 as priors for the rest of our analysis.

Methods: Fitting the Models to the Observed Reddening {#sec: Method}
=====================================================

We use Markov-chain Monte Carlo (MCMC) fitting to estimate the best-fit parameters and their uncertainties. MCMC is a Bayesian inference method which provides a sampling approximation of the posterior probability distribution function (PDF). An MCMC run produces a chain of positions in parameter space, and a histogram of these positions provides the approximation of the posterior PDF. MCMC allows for more in-depth probabilistic data analysis than $\chi^2$ minimization, for example, it efficiently approximates the full posterior PDF, which in turn provides uncertainties on and illustrates covariances between model parameters. The Goodman-Weare (G-W) algorithm improves upon the Metropolis-Hastings (M-H) algorithm by changing the method for choosing trial positions [@goodman2010]. The G-W algorithm deploys an ensemble of chains, known as "walkers", instead of a single chain. The trial position for each walker is chosen from the ensemble's location in parameter space, with some probability for choosing a position outside the occupied region. This method does not require hand-tuning the step size for each parameter, and the selection of trial positions can be parallelized. The G-W algorithm more efficient than the M-H algorithm in both human working hours and computation time. [@goodman2010; @foreman-mackey2013]

We use the open-source python implementation of the G-W algorithm, <span>*emcee*</span> [@foreman-mackey2013], to fit the observed reddening with our model grid described in $\S\ref{subsec: Qeff}$. The modeled extinction curves are parameterized by the mean particle size $a$ and effective variance $b$ for the Hansen distribution and the column density $N$ of forsterite grains. We assume an unnormalized flat prior probability distribution for each parameter. The effective extinction coefficients are linearly interpolated and the modeled extinction at each wavelength point is calculated as Equation 9.

We also model the vertical offset between the observed reddening and the extinction curve with a constant $C$, and include a tolerance parameter $s$. The tolerance $s$ estimates the uncertainty in the model as a single value across the extinction curve; it accounts for the fact that the photon-noise uncertainties are smaller than the typical difference between each observed reddening point and the corresponding point on the extinction curve. If we denote observed reddening points as $r = \{r_i\}$ and the corresponding uncertainties as $\sigma_0 = \{\sigma_{0,i}\}$, we compute the natural logarithm of the likelihood function as $${\rm{ln}}\mathcal{L}( r|a,b,N,C,s,\sigma_0) = -\frac{1}{2} \sum_i \left[
\frac{(r_i - (N\pi a^2 Q_i^2 + C)}{(\sigma_{0,i}^2 + s^2)}
+ \ln(2\pi\ (\sigma_{0,i}^2 + s^2))
\right]$$ The natural logarithm of the posterior PDF is given by $$\ln({\rm {PDF}})(a,b,N,C,s|r,\sigma_0) = {\rm{ln}}\mathcal{L}(r|a,b,N,C,s,\sigma_0) + {\rm{ln}}\mathcal{P}(a,b,N,C,s,\sigma_0)$$ We assume an unnormalized flat prior on each parameter, so $\mathcal{P}(a,b,N,C,s,\sigma_0)=1$ and ${\rm{ln}}\mathcal{P}(a,b,N,C,s,\sigma_0)=0$. We pass a function for $\ln(\rm PDF)$ to <span>*emcee*</span>, which uses that function to determine acceptance of each step in parameter space. We typically use 100 walkers. After we iterate for 200 steps to generate a new set of initial positions for the walkers, we reset the walkers and restart from the new initial positions. We iterate for 2000 steps after a burn-in period of 200 steps.

Results and Discussion {#sec: Results}
======================

Fitting Dust Haze Parameters {#subsec: fitresult}
----------------------------

We used the MCMC method described in $\S\ref{sec: Method}$ to fit dust haze extinction models to the observed reddening curves for each object and constrain the physical properties of the proposed dust haze. We plot the 1D and 2D marginalized posterior PDFs for all parameters in Figure Set $\ref{fig: triangle}$. Models corresponding to 100 randomly-drawn parameter sets from the posterior PDF are shown with the data in Figure Set $\ref{fig: fit}$. The constrained properties of the dust haze include mean effective radius, effective variance of the Hansen distribution ($\S\ref{subsec: distribution}$), and column density of the dust haze. In Figure Set 6, we show the posterior distributions for each parameter. Each figure has 1-D distributions for the parameters and 2-D contours for each combination of parameters. Gaussian-like 1-D distributions and round 2-D contours indicate no covariances. Quantiles (16, 50, 84 $\%$) are shown with dashed lines and are used to report the uncertainties on the parameter fits. In many objects, the PDFs for the mean effective radius $a$, column density $N$, tolerance parameter $\log s$, and vertical offset constant $C$ have clear peaks and therefore are well constrained. The variance $b$, on the other hand, is not well constrained in most objects.

The column density values $N$ are comparable to the value of typical brown dwarf atmospheres ($\sim 10^8\,\rm cm^{-2}$), which indicates that our results are reasonably realistic. However, there is a correlation between parameters $a$ and $N$ as seen in the 2-D contours. The relationship between $a$ and $N$ is shown in Equation \[eq: eq9\]. In order to compute the optical depth, we multiply the column density $N$, scattering cross section $\pi a^2$, and effective extinction coefficient $Q_{\rm ext}(\lambda)$. The effective extinction curves are similar over a small range of grain radii, so we are constraining the product $Na^2$. Therefore, $a^2$ and $N$ are inversely proportional and this relationship appears in the posterior distributions. In Figure Set 7, we show the resulting model fits to the observed reddening. The black line is the observed reddening ($\S\ref{sec: reddening}$) and the green lines are 100 models randomly drawn from the posterior distributions. The models reproduce the overall shape of the observed reddening.

In Figure Set 8, we compare a de-reddened spectrum of a red L dwarf, the spectrum of the field standard L dwarf, and the original red L dwarf spectrum. The de-reddened spectrum is the spectrum of a red L dwarf corrected by the best-fit forsterite extinction curve determined by the MCMC analysis. $\chi^2$ values between the red and standard spectra, and between the de-reddened and standard spectra are reported. The $\chi^2$ values are used simply to quantitatively demonstrate that the de-reddened spectrum is a better fit to the standard spectrum than the original red L dwarf spectrum. The tolerance parameter $s$ is not included in the calculation because it is not necessary for this purpose. In all cases the de-reddened spectrum fits the standard spectrum better than the original red L dwarf spectrum, as reflected by the significant improvement of $\chi^2$ value after the de-reddening.

In Figures $\ref{fig: combo1}$ and $\ref{fig: combo2}$, we show an example of the de-reddened spectra (Figure Set 8) and model fits (Figure Set 7) for each spectral type. In all spectral types, the de-reddened spectra looks much closer to the standard spectra than the original red L dwarf spectra. These results show that the submicron-sized dust haze prescription can successfully account for the red SED and $J - K_s$ colors of L dwarfs.

In Figure $\ref{fig: dchisq}$, we show improvement in $\chi^2$ due to the proposed dust haze prescription. The ratio of $\chi^2$ before de-reddening to $\chi^2$ after de-reddening is plotted against $\Delta (J - K_s)$ color, which is the difference in $J - K_s$ color between the red L dwarf and the field standard L dwarf. We use $\Delta (J - K_s)$ because we compare the spectra of red L dwarfs to the standards to isolate the observed reddening. The value of $\chi^2$ is improved for all objects, which shows that the de-reddened spectra fit the field standards better, and in most cases substantially better ($> 10$x), than the original red L dwarf spectra.

Correlation with Gravity {#subsec: correlation}
------------------------

It has been widely noted that low-gravity L dwarfs have redder NIR spectral energy distributions compared to the field-gravity spectral standards. A discussion of how clouds behave differently at low and moderate gravity is given in @Marley12. In addition to the cloud height differences discussed by @Marley12, one might expect to see a correlation between dust haze properties and low-gravity spectral features. We hypothesized that the proposed dust haze might dissipate over time due to grain growth by condensation. Large condensed particles are expected to fall out of the dust haze as a result of the sedimentation rate exceeding the remixing rate by eddy turbulence [@Marley99]. Reduced convective velocities and perhaps less vigorous gravity wave excitation with age could also contribute to more efficient dust settling. Thus, young, low gravity L dwarfs might have optically thicker dust hazes which may explain their red NIR colors. The proposed dust haze could also explain the reddening within field L dwarfs and the properties of the dust haze might be correlated with age. In Figures $\ref{fig: ascat}$ and $\ref{fig: nscat}$, we show scatter plots of mean effective radius $a$ and column density $N$ versus $\Delta(J-K)$ color, respectively. Green symbols denote low-gravity objects and magenta denote field-gravity. Circles denote objects with PDFs with strong constraints, while squares and diamonds denote objects with weak constraints for some of the parameters. There are visually noticeable differences between the distributions of the low-gravity and field-gravity objects in both Figure $\ref{fig: ascat}$ and Figure $\ref{fig: nscat}$. In Figure $\ref{fig: ascat}$, we show that there is no correlation between mean effective radius $a$ and $\Delta(J-K)$ color for the low-gravity objects, while the distribution of the field-gravity objects show a correlation. In Figure $\ref{fig: nscat}$, the distribution of the low-gravity objects span a wider range of column densities, while the field-gravity objects tend to have lower column densities. Even though there are visible differences between the distributions of the low-gravity and field-gravity objects, the two distributions partially overlap and the correlations are not strong. In order to quantitatively determine if the distributions of the low-gravity objects and the field-gravity objects are indeed different, we performed a two-dimensional K-S test on the two distributions in both Figure $\ref{fig: ascat}$ and Figure $\ref{fig: nscat}$. The probability of the two distributions in Figure $\ref{fig: ascat}$ drawn from the same parent distribution is $2.32\cdot 10^{-4}$, and the probability of the two distributions in Figure $\ref{fig: nscat}$ drawn from the same parent distribution is $5.11\cdot 10^{-6}$. These results show that the low-gravity and the field-gravity objects are most likely drawn from different distributions. The lack of strong correlations between the dust haze properties and gravity might be due to a model grid not spanning a wide enough range of effective variance, $b$. We consider 0.1 $< b <$ 1.0 because Hansen particle size distributions for large $b$ look like a power law size distribution (Figure $\ref{fig: dist}$; Equation 1). This makes the results for these objects somewhat unreliable. In most of those objects, $b$ tends to be close to the upper limit and therefore is not well constrained (Figure Set \[fig: triangle\]). Further, our initial assumptions about low-gravity and field L dwarfs may be unrealistic. We assumed that low-gravity and field L dwarfs with the same base spectral type (e.g. L2 and L2$\gamma$) have the same effective temperatures and the low-gravity L dwarf has the hypothetical dust haze of small grains in the upper atmosphere. However, recent evidence suggests that low-gravity L dwarfs do not necessarily share the same physical properties with field L dwarfs just because they share the same base spectral type [@Luhman12; @Filippazzo15]. Young, low gravity L dwarfs might have cooler effective temperatures than field L dwarfs in the same spectral classification [@faherty13]. This suggests that an earlier type standard would be a better comparison. We could be comparing objects with different effective temperatures and thus, not setting accurate estimates of the dust haze properties. Since the overall shape of a spectrum is very sensitive to the effective temperature, comparing objects with the same effective temperatures might be more useful for future analysis. Finally, the overlapping distributions of $a$ and $N$ may indicate that the dust hazes of low-gravity and red field L dwarfs are not different from from one another. It might be the case that gravity does not play a major role in determining the properties of the dust haze and the same distribution of dust haze properties exist in both low-gravity and red field L dwarfs. Some red field L dwarfs might have a dust haze for reasons other than gravity, such as differing rotation rates, composition, or evolutionary history.

Comparison to @Marocco14 {#subsec: Comparison}
------------------------

Independently, @Marocco14 (hereafter, M14) present results from a similar analysis. @Bonn15 closely followed the approach of @Marocco14 so we do not consider it further here. Like us, they are motivated by the utility of the interstellar extinction law in de-reddening red L dwarf spectra, use Mie theory to characterize a high-altitude population of sub-micron dust grains, isolate the observed reddening by comparing red L dwarf spectra to spectral standards.

However, M14 consider corundum ($\rm Al_{2}O_3$), enstatite ($\rm MgSiO_3$), and iron while we use forsterite ($\rm Mg_{2}SiO_4$). We use forsterite as a test dust particle because extinction curves for forsterite, enstatite –both of which are silicates– and corundum all behave similarly in the near infrared. We use silicate instead of iron because silicate grains form higher in the atmosphere than iron. Actual dust grains in the brown dwarfs atmospheres are most likely a mixture of these species. Our model includes more parameters than M14. They have two parameters: characteristic grain radius $\it r$ and normalization of the extinction hamptonpamela@example.net $\mu$m while our model includes mean effective grain radius $a$, effective grain size variance $b$, column density $N$, vertical offset $C$, and tolerance factor $s$ as described in $\S \ref{sec: Model}$. The $N$ and $C$ parameters account for normalization as shown in Equation 9. M14 perform $\chi^2$ minimization to select the best fit parameters and therefore do not have uncertainties, while we use MCMC to determine best fit parameters and their uncertainties.

The range of the characteristic grain radius M14 found for corundum and enstatite are slightly larger than what we found. They obtained $r =$ 0.4 – 0.6 $\micron$ for corundum and enstatite, while our results for the mean effective radius for forsterite are generally smaller ($0.15 - 0.35 \micron$). Their results for the maximum radius of iron are 0.15 – 0.3 $\micron$, closer to the values we found for forsterite grains.

M14 applied their method to five red L dwarfs, and we applied our method to 46 red L dwarfs including low-gravity and field-gravity objects. They used objects with later spectral types (L5 – L7), so there is only one common object between their sample and ours. 2MASS 0355+1133 is used in both studies but compared to different spectral standards. M14 used SDSS J0835+1953 as the L5 standard while we used 2MASS 1507-1627. They found the characteristic grain radius for 2MASS 0355+1133 to be 0.4 $\micron$ and we found the mean effective radius to be $0.3\substack{+0.03 \\ -0.02}~\micron$.

Regardless of the different methods, these two studies show similar results. The reproducibility of the results demonstrates the viability of sub-micron size dust grains, and warrants further study and inclusion of small dust grains in future atmosphere models.

Conclusion and Summary {#sec: Conclusion}
======================

The success of the interstellar reddening law in de-reddening L dwarf spectra led us to investigate the possibility of a population of small submicron-sized dust grains in red L dwarf atmospheres that can possibly explain their red NIR colors.

In order to isolate and characterize the reddening, we compare spectra of red L dwarfs to field spectral standards. The observed reddening is treated as smooth power-law shaped curves. We use Mie theory to model the dust haze of small particles with theoretical extinction curves. We use a Markov-Chain Monte Carlo algorithm to fit the theoretical extinction curves to the observed reddening in order to find the best-fit values and uncertainties for the properties of the dust haze. We apply this method to 23 L dwarfs with low-gravity spectral features and 23 field L dwarfs with redder $J-K$ colors than the spectral standards. We find that small forsterite grains ($< 0.5~\micron$) reproduce the observed reddening. There are differences in grain properties between low-gravity and field L dwarfs, which indicates that dust haze properties may be correlated with surface gravity. The best-fit column densities of forsterite grains are reasonable compared to typical brown dwarf atmospheres. These results suggest that a dust haze of small particles with a Hansen particle size distribution can account for the observed red NIR spectral energy distributions of brown dwarfs. However, to rigorously explore this hypothesis other particle species and particle size distributions should be explored as well.

This work is a proof of concept and it provides a strong motivation for including small dust grains in future atmosphere models of brown dwarfs. Future work on the role of small grains in brown dwarf atmospheres will include other grain species such as corundum and enstatite. The dust haze analysis can be applied to other studies including variability in brown dwarfs, exoplanet atmospheres, and the interstellar/intergalactic medium.

We thank our anonymous referee for thorough and helpful comments. This material is based upon work supported by the National Science Foundation under Grant No. AST-1313278. Support for this project was provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

<span> </span>

Ackerman, A. S., and Marley, M. S. 2001, Astrophysical Journal, 556, 872 Allard, F., Hauschildt, P. H., Alexander, D. R., Tamanai, A., and Schweitzer, A. 2001, Astrophysical Journal, 556, 357 Allers, K. N., Liu, M. C., Dupuy, T. J., and Cushing, M. C. 2010, Astrophysical Journal, 715, 561 Allers, K. N. and Liu, M. C. 2013, Astrophysical Journal, 772, 79 Bardalez Gagliuffi, D. C., Burgasser, A. J., Gelino, C. R., Looper, D. L., Nicholls, C. P., Schmidt, S. J., Cruz, K. L., West, A. A., Gizis, J. E., and Metchev, S. 2014, Astrophysical Journal, 794, 143 Bonnefoy, M., Zurlo, A., Baudino, J. L., Lucas, P., Mesa, D., Maire, A.-L., Vigan, A., Galicher, R., Homeier, D., Marocco, F., Gratton, R., Chauvin, G., Allard, F., Desidera, S., Kasper, M., Moutou, C., Lagrange, A.-M., Baruffolo, A., Baudrand, J., Beuzit, J.-L., Boccaletti, A., Cantalloube, F., Carbillet, M., Charton, J., Claudi, R. U., Costille, A., Dohlen, K., Dominik, C., Fantinel, D., Feautrier, P., Feldt, M., Fusco, T., Gigan, P., Girard, J. H., Gluck, L., Gry, C., Henning, T., Janson, M., Langlois, M., Madec, F., Magnard, Y., Maurel, D., Mawet, D., Meyer, M. R., Milli, J., Moeller-Nilsson, O., Mouillet, D., Pavlov, A., Perret, D., Pujet, P., Quanz, S. P., Rochat, S., Rousset, G., Roux, A., Salasnich, B., Salter, G., Sauvage, J.-F., Schmid, H. M., Sevin, A., Soenke, C., Stadler, E., Turatto, M., Udry, S.,, Vakili, F., Wahhaj, Z., and Wildi, F. 2015arXiv151104082B Broekhoven-Fiene, H., Matthews, B., Duchene, G., Di Francesco, J., Scholz, Al., Chrysostomou, A., Jayawardhana, R. 2014, Astrophysical Journal, 789, 155 Bouy, H, Brandner, W, Mart'n, E. L., Delfosse, X., Allard, F., and Basri, G. 2003, Astronomical Journal, 126, 1526 Burgasser, A. J., and McElwain, M. W. Astronomical Journal, 131, 1007 Burgasser, A. J. 2007, Astrophysical Journal, 659, 655 Burgasser, A. J. 2008, PhT, 61, 70 Burgasser, A. J., Liu, M. C., Ireland, M. J., Cruz, K. L., and Dupuy, T. J. 2008, Astrophysical Journal, 681, 579 Burgasser, A. J., Cruz, K. L., Cushing, M. C., Gelino, C. R., Looper, D. L., Faherty, J. K., Kirkpatrick, J. D., and Reid, I. N. 2010 Astrophysical Journal, 710, 1142 Burgasser, A. J., Simcoe, R. A., Bochanski, J. J., Saumon, D., Mamajek, E. E., Cushing, M. C., Marley, M. S., McMurtry, C., Pipher, J. L., and Forrest, W. J. 2010, Astrophysical Journal, 725, 1405 Burgasser, A. J., Bardalez-Gagliuffi, D. C., and Gizis, J. E. 2011, Astronomical Journal, 141, 70 Burrows, A., Hubbard, W. B., Lunine, J. I., and Liebert, J. Reviews of Modern Physics, 73, 719 Cardelli, J. A., Clayton, G. C., and Mathis, J. S. 1989, Astrophysical Journal, 345, 245 Chiu, K., Fan, X., Leggett, S. K., Golimowski, D. A., Zheng, W., Geballe, T. R., Schneider, D. P., and Brinkmann, J. 2006, Astronomical Journal, 131, 2722 Cruz, K. L., Reid, I. N., Liebert, J., Kirkpatrick, J. D., and Lowrance, P. J. Astronomical Journal, 126, 2421 Cruz, K. L., Burgasser, A. J., Reid, I. N., and Liebert, J. 2004, Astrophysical Journal, 604, 61 Cruz, K. L., Reid, I. N., Kirkpatrick, J. D., Burgasser, A. J., Liebert, J., Solomon, A. R., Schmidt, S. J., Allen, P. R., Hawley, S. L., and Covey, K. R. 2007, Astronomical Journal, 133, 439 Cruz, K. L., Kirkpatrick, J. D., and Burgasser, A. J. 2009, Astronomical Journal, 137, 3345 Cushing, M. C., Vacca, W. D., $\&$ Rayner, J. T. 2004, Publications of the Astronomical Society of the Pacific, 116, 362 Cushing, M. C., Roellig, T. L., Marley, M. S., Saumon, D., Leggett, S. K., Kirkpatrick, J. D., Wilson, J. C., Sloan, G. C., Mainzer, A. K., Van Cleve, J. E., and Houck, J. R. 2006, Astrophysical Journal, 648, 614 Cushing, M. C., Marley, M. S., Saumon, D., Kelly, B. C., Vacca, W. D., Rayner, J. T., Freedman, R. S., Lodders, K., and Roelling, T. L. 2008, Astrophysical Journal, 678, 1372 Dahn, C. C., Harris, H. C., Vrba, F. J., Guetter, H. H., Canzian, B., Henden, A. A., Levine, S. E., Luginbuhl, C. B., Monet, A. K. B., Monet, D. G., Pier, J. R., Stone, R. C., Walker, R. L., Burgasser, A. J., Gizis, J. E., Kirkpatrick, J. D., Liebert, J.,, and Reid, I. N. 2002, Astronomical Journal, 124, 1170 Delfosse, X., Tinney, C. G., Forveille, T., Epchtein, N., Bertin, E., Borsenberger, J., Copet, E., de Batz, B., Fouque, P., Kimeswenger, S., Le Bertre, T., Lacombe, F., Rouan, D., and Tiphene, D. 1997, Astronomy $\&$ Astrophysics, 327, 25 Draine, B. T. 2006, Astrophysical Journal, 636, 1114 Faherty, J. K., Burgasser, A. J., Cruz, K. L., Shara, M., Walter, F. M., and Gelino, C. R. 2009, Astrophysical Journal, 137, 1 Faherty, J. K., Rice, E. L., Crus, K. L., Mamajek, E. E., and Nunez, A. 2013, Astronomical Journal, 145, 2 Fan, X., Knapp, G. R., Strauss, M. A., Gunn, J. E., Lupton, R. H., Ivezic, Z., Rockosi, C. M., Yanny, B., Kent, S., Schneider, D. P., Kirkpatrick, J. D., Annis, J., Bastian, S., Berman, E., Brinkmann, J., Csabai, I., Federwitz, G. R., Fukugita, M., Gurbani, V. K., Hennessy, G. S., Hindsley, R. B., Ichikawa, T., Lamb, D. Q., Lindenmeyer, C., Mantsch, P. M., McKay, T. A., Munn, J. A., Nash, T., Okamura, S., Pauls, A. G., Pier, J. R., Rechenmacher, R., Rivetta, C. H., Sergey, G., Stoughton, C., Szalay, A. S., Szokoly, G. P., Tucker, D. L., York, D. G., SDSS Collaboration, 2000, Astronomical Journal, 119, 928 Filippazzo, J. C., Rice, E. L., Faherty, J. K., Cruz, K. L., Van Gordon, M. M., and Looper, D. L. 2015, arXiv:1508.01767 Fitzpatrick, E. L., 1999, Publications of the Astronomical Society of the Pacific, 111, 63 Ford, E. B. 2005, Astronomical Journal, 129, 1726 Foreman-Mackey, D. et al. 2013, Publications of the Astronomical Society of the Pacific, 125, 306 Gagne, J., Lafreniere, D., Doyon, R., Artigau, E., Malo, L., Robert, J., and Nadeau, D. 2014, Astrophysical Journal, 792, 17 Gagne, J., Faherty, J. K., Cruz, K. L., Lafreniere, D., Doyon, R., Malo, L., Burgasser, A. J., Naud, M.-E., Artigau, E., Bouchard, S., Gizis, J. E., Albert, L. 2015, 219, 33 Geballe, T. R., Knapp, G. R., Leggett, S. K., Fan, X., Golimowski, D. A., Anderson, S., Brinkmann, J., Csabai, I., Gunn, J. E., Hawley, S. L., Hennessy, G., Henry, T. J., Hill, G. J., Hindsley, R. B., Ivezic, Z., Lupton, R. H., McDaniel, A., Munn, J. A., Narayanan, V. K., Peng, E., Pier, J. R., Rockosi, C. M., Schneider, D. P., Smith, J. Allyn, Strauss, M. A., Tsvetanov, Z. I., Uomoto, A., York, D. G., Zheng, W. 2002, Astrophysical Journal, 564, 466 Gizis, J. E., Monet, D. G., Reid, I. N., Kirkpatrick, J. D., Liebert, J., and Williams, R. J. 2000, Astronomical Journal, 120, 1085 Gizis, J. E., Kirkpatrick, J. D., and Wilson, J. C. 2001, Astronomical Journal, 121, 2185 Gizis, J. E., Reid, I. N., Knapp, G. R., Liebert, J., Kirkpatrick, J. D., Koerner, D. W., and Burgasser, A. J. 2003, Astronomical Journal, 125, 3302 Goldman, B. et al. 1999, Astronomy $\&$ Astrophysics, 351, 5 Goodman, J. and Weare, J. 2010, Comm. App. Math. Comp. Sci., 5, 65 Hansen, James E. 1971, Journal of the Atmospheric Sciences, 28, 1400 Kirkpatrick, J. D., Beichman, C. A., and Skrutskie, M. F. 1997, Astrophysical Journal, 476, 311 Kirkpatrick, J. D., Reid, I. N., Liebert, J. Cutri, R. M., Nelson, B., Beichman, C. A., Dahn, C. C., Monet, D. G., Gizis, J. E., Skrutskie, M. F. 1999, ApJ, 519, 802 Kirkpatrick, J. D., Reid, I. N., Liebert, J., Gizis, J. E., Burgasser, A. J., and Brown, M. E. 2000, Astronomical Journal, 120, 473 Kirkpatrick, J. D., Barman, T. S., Burgasser, A. J., McGovern, M. R., McLean, I. S., Tinney, C. G., and Lowrance, P. J. 2006, Astrophysical Journal, 639, 1120 Kirkpatrick, J. D., Cruz, K. L., Barman, T. S., Burgasser, A. J., Looper, D. L., Tinney, C. G., Gelino, C. R., Lowrance, P. J., Liebert, J., and Carpenter, J. M. 2008, Astrophysical Journal, 689, 1295 Kirkpatrick, J. D., Looper, D. L., Burgasser, A. J., Schurr, S. D., Cutri, R. M., Cushing, M, C., Cruz, K. L., Sweet, A. C., Knapp, G. R., and Barman, T. S. 2010, Astrophysical Journal, 190, 100 Knapp, G. R., Leggett, S. K., Fan, X., Marley, M. S., Geballe, T. R., Golimowski, D. A., Finkbeiner, D., Gunn, J. E., Hennawi, J., Ivezic, Z., Lupton, R. H., Schlegel, D. J., Strauss, M. A., Tsvetanov, Z. I., Chiu, K., Hoversten, E. A., Glazebrook, K., Zheng, W., Hendrickson, M., Williams, C. C., Uomoto, A., Vrba, F. J., Henden, A. A., Luginbuhl, C. B., Guetter, H. H., Munn, J. A., Canzian, B., Schneider, Donald P., Brinkmann, J. 2004, Astronomical Journal, 127, 3553 Liu, M. C., Magnier, E. A., Deacon, N. R., Allers, K. N., Dupuy, T. J., Koston, M. C., Aller, K. M., Burgett, W. S., Chambers, K. C., and Draper, P. W. 2013, Astrophysical Journal, 777, 20 Lodders, K. and Fegley, B., Jr. 2006, Astrophysics Update 2, Springer Praxis Books Looper, D. L., Kirkpatrick, J. D., Cutri, R. M., Barman, T., Burgasser, A. J., Cushing, M. C., Roellig, T., McGovern, M. R., McLean, I. S., Rice, E. L., Swift, B. J., and Schurr, S. D. 2008, Astrophysical Journal, 686, 528 Looper, D. L., Mohanty, S., Bochanski, J. J., Burgasser, A. J., Mamajek, E. E., Herczeg, G. J., West, A. A., Faherty, J. K., Rayner, J., Pitts, M. A., and Kirkpatrick, J. D. 2010, Astrophysical Journal, 714, 45 Luhman, K. L., D'Alessio, P., Calvet, N., Allen, L. E., Hartmann, L., Forrest, W. J., Watson, D. M., Luhman, K. L., Uchida, K. I., Green, J. D., Sargent, B., Najita, J., Sloan, G. C., Keller, L. D., and Herter, T. L. 2005, Astrophysical Journal, 620, 51 Luhman, K. L. 2012, Astronomy $\&$ Astrophysics, 50, 65 Marley, M. S., Gelino, C., Stephens, D., Lunine, J. I., $\&$ Freedman, R. 1999, Astrophysical Journal, 513, 879 Marley, M. S., Saumon, D., Cushing, M. C., Ackerman, A. S., Fortney, J. J., Freedman, R. 2012, Astrophysical Journal, 754, 135 Mathis, J. S., Rumpl, W., Nordsieck, K. H. 1977, Astrophysical Journal, 217, 425 Marocco, F., Day-Jones, A. C., Lucas, P. W., Jones, H. R. A., Smart, R. L., Zhang, Z. H., Gomes, J. I., Burningham, B., Pinfield, D. J., Raddi, R., and Smith, L. 2014, Monthly Notices of the Royal Astronomical Society, 439, 372 Rayner, J. T., Toomey, D. W., Onaka, P. M., Denault, A. J., Stahlberger, W. E., Vacca, W. D., Cushing, M. C., and Wang, S. 2003, PASP, 115, 362 Reid, I. N., Kirkpatrick, J. D., Gizis, J. E., Dahn, C. C., Monet, D. G., Williams, R. J., Liebert, J., and Burgasser, A. J. 2000, Astronomical Journal, 119, 369 Reid, I. N., Lewitus, E., Burgasser, A. J., and Cruz, K. L. 2006, Astrophysical Journal, 639, 1114 Reid, I. N., Cruz, K. L., Kirkpatrick, J. D., Allen, P. R., Mungall, F., Liebert, J., Lowrance, P., and Sweet, A. 2008, Astronomical Journal, 136, 1290 Ruiz, M. T., Leggett, S. K., and Allard, F. 1997, Astrophysical Journal, 491, 107 Saumon, D. and Marley, M. S. 2008, Astrophysical Journal, 689, 1327 Siegler, N., Close, L. M., Burgasser, A. J., Cruz, K. L., Marois, C., Macintosh, B., and Barman, T. 2007, Astronomical Journal, 133, 2320 Toon, O. B. and Ackerman, T. P. 1981, Appl. Optics, 20, 20 Vacca, W. D., Cushing, M. C., $\&$ Rayner, J. T. 2003, Publications of the Astronomical Society of the Pacific, 115, 389 Wilson, J. C., Miller, N. A., Gizis, J. E., Skrutskie, M. F., Houck, J. R., Kirkpatrick, J. D., Burgasser, A. J., and Monet, D. G. 2003, Proceedings of IAU Symposium, 211, 197 Zhang, Z. H., Pokorny, R. S., Jones, H. R. A., Pinfield, D. J., Chen, P. S., Han, Z., Chen, D., G‡lvez-Ortiz, M. C., and Burningham, B. 2009, Astronomy and Astrophysics, 497, 619

<span><span>**Fig. Set**</span> </span><span><span>**6.**</span></span> <span> <span>**Posterior distributions for MCMC parameters**</span> </span>

<span><span>**Fig. Set**</span> </span><span><span>**7.**</span></span> <span> <span>**Model fits**</span> </span>

<span><span>**Fig. Set**</span> </span><span><span>**8.**</span></span> <span> <span>**De-reddened spectra**</span> </span>

![An example of de-reddened spectra (equivalent to Figure Set 8) and model fits (equivalent to Figure Set 7) for spectral types L0–L3. The left column shows the spectral standard (black), the red L dwarf (red), and the de-reddened (green) spectra. The right column shows the best-fit model (green) and the observed reddening (black).<span data-label="fig: combo1"></span>](f9_1.pdf){width="\textwidth" height="\textheight"}

![Same as Figure \[fig: combo1\] for spectral types L4–L6 including 0355-type. The left column shows the spectral standard (black), the red L dwarf (red), and the de-reddened (green) spectra. The right column shows the best-fit model (green) and the observed reddening (black).<span data-label="fig: combo2"></span>](f9_2.pdf){width="\textwidth" height="\textheight"}

<span>llllllll</span> 0345432+254023 & L0 & 1.33 & 0 & @Kirkpatrick97 &@Kirkpatrick99 & @BM06\
2130446-084520 & L1 & 1.33 & 0 & @Kirkpatrick08 & @Kirkpatrick08 & @Kirkpatrick10\
13054019-2541059& L2 & 1.67 & 0 & @Ruiz97 & @Kirkpatrick99& @Burg07\
1506544+132106 & L3 & 1.63 & 0 & @Gizis00&@Gizis00 & @Burg07\
21580457-1550098 & L4 & 1.86 & 0 & @Kirkpatrick08&@Kirkpatrick08 & @Kirkpatrick10\
1507476-162738 & L5 & 1.46 & 0 & @Reid00 & @Kirkpatrick00& @Burg07\
10101480-0406499 & L6 & 1.89 & 0 & @Cruz03 & @Cruz03 & @Reid06\
01415823-4633574 & L0$\gamma$ & 1.74 & 0.410 & @Kirkpatrick06 & @cruz09 & @Kirkpatrick06\
00325584-4405058 & L0$\gamma$ & 1.51 & 0.178 & @Gold99 & @cruz09 & @BG14\
02103857-3015313 & L0$\gamma$ & 1.57 & 0.236 & Cruz et al. in prep & Cruz et al. in prep & This Paper & 2003 Sep 04\
0241115-032658 & L0$\gamma$ & 1.76 & 0.434 &@Cruz07 & @cruz09 & This Paper & 2006 Aug 21\
03231002-4631237& L0$\gamma$ & 1.69 & 0.357 &@Reid08 & @Cruz07 & This Paper & 2007 Nov 13\
2213449-213607 & L0$\gamma$ & 1.62 & 0.290 & @Cruz07 & @cruz09 & @BG14\
23153135+0617146 & L0$\gamma$ & 1.80 & 0.466 & Cruz et al. in prep & Cruz et al. in prep & This Paper & 2007 Nov 14\
1711135+232633 & L0$\gamma$ & 1.44 & 0.113 & @Cruz07 & Cruz et al. in prep & This Paper & 2008 Jul 13\
15525906+2948485 & L0$\beta$ & 1.46 & 0.126 & @Wilson03 & @cruz09 & @BG14\
03572695-4417305 & L0$\beta$ & 1.46 & 0.127 & @Bouy03 & @cruz09 & @BG14\
0117474-340325 & L1$\gamma$ & 1.69 & 0.358 & @Cruz03 & @Gagne15 & This Paper & 2006 Aug 21\
0518461-275645& L1$\gamma$ & 1.65 & 0.317 & @Cruz07 & @Gagne15 & @BG14\
00550564+0134365 & L2$\gamma$ & 2.00 & 0.328 & Cruz et al. in prep & Cruz et al. in prep & This Paper & 2003 Sep 04\
0536199-192039 & L2$\gamma$ & 1.91 & 0.244 & @Cruz07 & @Gagne15 & This Paper & 2003 Sep 03\
15515237+0941148 & L3:$\gamma$ & 2.01 & 0.149 & @Reid08 & @Gagne15 & This Paper & 2008 Jul 13\
1726000+153819 & L3.5$\gamma$ & 1.81 & 0.380 & @Kirkpatrick00 & @Allers13 & @BG14\
05012406-0010452 & L4$\gamma$ & 2.02 & 0.159 & @Reid08 & @cruz09& @Filippazzo15\
22495345+0044046 & L4$\gamma$ & 2.22 & 0.369 & @Geba02 & @Gagne15 & @Allers10\
05120636-2949540 & L5$\gamma$ & 2.18 & 0.718 & @Cruz03 & @Gagne15& @BG14\
03264225-21020572 & L5$\beta/\gamma$ & 2.21 & 0.752 & @Gizis03 & @Gagne15 & This Paper & 2007 Nov 13\
21543454-1055308 & L5$\beta/\gamma$ & 2.24 & 0.381 & @Gagne14 & @Gagne15 & This Paper & 2003 Aug 11\
03552337+1133437 & 0355-type & 2.52 & 1.06 & @Reid08 &@Gagne15 & @faherty13\
1615425+495321& 0355-type & 2.48 & 0.623 & @Cruz07 &@Gagne15& This Paper & 2005 Mar 23\
02355993-2331205 & L1 & 1.48 & 0.154 & @Burg08b & @Gizis01 & @Burg08b\
05431887+6422528 & L1 & 1.52 & 0.187 & @Reid08 & @Reid08 &@BG14\
06022216+6336391 & L1: & 1.58 & 0.263 & @Reid08& @Reid08 &@BG14\
00165953-4056541 & L3 & 1.88 & 0.254 &@Kirkpatrick08 & @Kirkpatrick08 &@Burg10c\
23392527+3507165 & L3.5 & 1.77 & 0.144 & @Reid08 & @Reid08 & This Paper & 2003 Sep 04\
11000965+4957470 & L3.5 & 1.81 & 0.178 & @Reid08 & @Reid08 & This Paper& 2004 Nov 08\
00511078-1544169 & L3.5 & 1.81 & 0.181 & @Kirkpatrick00 & @Kirkpatrick00 & @Burg10a\
23174712-4838501 & L4pec & 1.97 & 0.109 & @Reid08 & @Reid08 & @Kirkpatrick10\
0337036-175807 & L4.5 & 2.04 & 0.180 & @Kirkpatrick00 & @Kirkpatrick00 & @BG14\
02082363+2737400 & L5 & 1.84 & 0.382 & @Kirkpatrick00& @Kirkpatrick00 & @Burg10a\
0835425-081923 & L5 & 2.03 & 0.573 & @Cruz03 & @Cruz03 & This Paper & 2004 Nov 08\
03582255-4116060 & L5 & 2.01 & 0.548 & @Reid08 & @Reid08 & This Paper & 2007 Nov 14\
09054654+5623117 & L5 & 1.67 & 0.205 & @Reid08 & @Reid08 & @Burg10a\
12281523-1547342 & L5 & 1.61 & 0.151 & @Delf97 & @Burg10c& @Burg10a\
12392727+5515371 & L5 & 1.92 & 0.459 & @Kirkpatrick00 & @Burg10a & @Burg10a\
03101401-2756452 & L5 & 1.84 & 0.376 & @Cruz07 & @Cruz07 & @BG14\
06244595-4521548 & L5 & 1.89 & 0.425 & @Reid08 & @Reid08 & @BG14\
0652307+471034 & L5 & 1.82 & 0.357 & @Cruz03 & @Cruz03 & This Paper & 2004 Nov 07\
14383259+5722168 & L5 & 1.59 & 0.129 & @Zhang09 & @Zhang09 & @BG14\
1326298-003831 & L5 & 1.90 & 0.435 & @Fan00 & @Cruz03 & @Geba02\
22120703+3430351& L5: & 1.95 & 0.486 & @Reid08 & @Reid08 & This Paper & 2004 Sep 05\
21481633+4003594 & L6 & 2.38 & 0.49 & @Looper08 & @Looper08 & @Looper08\
22443167+2043433 & L6.5 & 2.45 & 0.56 & @Dahn02 & @Kirkpatrick08 & @Looper08\

\[table: data\]

{'source': 'arXiv_out/1606'}