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The first wave of fundamental parameters to be determined includes effective temperature, metallicity, surface gravity, and alpha enhancement, constrained in that order. Our technique utilizes an iterative least squares method in which the general procedure is that for every model a “reduced square residual” (henceforth “RSR”) is calculated for every color (e.g.,

V −R, V −I, ..., R −I, R −J, ..., etc.) by comparing the model and observed colors: (model color_{observed color error}−observed color)2_{. In theory the best-fitting model spectrum is the one with the lowest}

RSR across all color combinations. However, due to occasional local minima and some model parameters being poorly constrained by our filters and/or modeled with insufficient accuracy or precision, the actual procedure turns out to be not so straightforward.

Temperature and metallicity appear to be the two parameters best probed by the com- bination of our V RIJ HKW1W2W3 photometry and the BT-Settl model grid. Varying surface gravity appears to have negligible effects on the overall RSR values, in line with find-

ings by Leggett et al. (1998, 2000), and alpha enhancement has a maximum of three options in this version of the grid (fewer in some sections). Through trial and error, the most accurate (see§5.2) means of determining the four parameters appears to be to let alpha enhancement float and set two of the three remaining parameters as constant for a given iteration as the third parameter is calculated — e.g., when determining temperature, metallicity and surface gravity are set constant so that the best temperature can be calculated independently of the other two parameters. Because surface gravity is the remaining parameter least probed by our method, and temperature has a far broader range of options (1500-5000 K at 50 K intervals vs. −0.5 to 0.5 dex at 0.5 dex intervals) we constrain temperature, metallicity, surface gravity, and alpha enhancement in that order, as detailed carefully next. Bear in mind that the grid is discrete, and our calculated values are not necessarily equal to these discrete values, as explained below.

Step 1 Calculate a temperature, T1, with metallicity = 0 and surface gravity = 5.0.

Step 2 Calculate a metallicity, M1, with temperature set to the grid value closest to T1 and surface gravity = 5.0.

Step 3 Repeat Steps 1 (for T2) and 2 (for M2), using the grid values closest to the most recent calculation (i.e. M1, and then T2). This is to account for the possibility that the starting temperature and/or metallicity are far from the best fit.

Step 4 Calculate surface gravity, G1, with temperature and metallicity set to the grid values closest to the just-calculated T2 and M2.

Step 5 Repeat steps 1-4 using the grid values closest to the calculated values, to account for a possibly inaccurate starting surface gravity.

Step 6 Repeat steps 1-3 to account for possible changes introduced by our new (and final) surface gravity value.

Step 7 Calculate the best alpha enhancement, which may simply be zero if that is the only option given the other parameters.

Summary The order of calculations is as follows: (T1 ⇒ M1⇒ T2 ⇒ M2⇒ G1) ⇒ (T3 ⇒M3 ⇒ T4 ⇒ M4 ⇒ G2) ⇒ (T5 ⇒ M5 ⇒ T6 ⇒ M6 ⇒ A1), with alpha enhancement floating until the last step, and one parameter being constrained at a time, while the remaining two are set to either their initial value or the closest matching grid value to their most recently calculated value. The resulting boldfaced values — T6, M6, G2, and A1 — comprises the final adopted set of stellar parameters.

For example, the first three calculation results may be as follows, with settings for the calculation in parentheses and setting values carried over from previous steps in italics:

T1 = 3257 K (M = 0 dex, G = 5.0, A floating) M1 = +0.27 dex (T = 3300 K, G = 5.0, A floating) T2 = 3278 K (M = +0.5 dex, G = 5.0, A floating)

T1 = 3257 K ⇒ M1 = +0.27 dex ⇒ T2 = 3278 ⇒ M2 = +0.32 dex ⇒ G1 = 4.71 ⇒ T3 = 3310 K ⇒ ... ⇒ G2 = 4.68 ⇒ ... ⇒ T6 = 3314 K ⇒ M6 = +0.36 dex ⇒ A1

= 0.17 dex.

The word “calculate” has a slightly different meaning than its usual connotation in this context — the temperature, metallicity, surface gravity, and alpha enhancement are calcu- lated to be the means of the results for each color combination, and the standard deviations are then assigned as errors. To clarify, each color combination (e.g., V −R) yields a best- fitting spectrum in the grid corresponding to the lowest RSR. The corresponding RSR value indicates how closely that best-fitting spectrum matches observations. However, it is unlikely that the properties of any given star are an exact match to any of the spectra in the grid (e.g., exactly 3000 K, metallicity = 0.0, log g = 5.0, and alpha enhancement = 0.0). It is far more likely that the true values are somewhere between the values of the grid spectra. Also, because the other three parameters, especially surface gravity, aren’t as well-probed as temperature by the model spectra and/or our filters, it is important to sample each of the four parameters individually in order to decipher values between the grid spectra. Be- cause our chosen parameter ranges have 41 effective temperature options, but only 3, 3, and 2 metallicity, surface gravity, and alpha enhancement options, respectively, as described next we employ a different procedure to determine effective temperature than for the other three properties. Henceforth, we are calculating one property at a time by applying these procedures to one color combination at a time.

the true temperature of the star; color is generally taken to be a proxy for temperature, so it makes sense that there would be stark mismatches when comparing observed colors to the colors exhibited by a model star that is hundreds of Kelvin warmer or cooler. As shown in Figure 4.5 for one star, 2MA1645-1319, for a sufficient number of model grid points, e.g., our maximum of 41 temperature options, plots of RSR for each of the 14 colors used for this cool star vs. effective temperature usually yield roughly parabolic-shaped curves. We show temperatures within 500 K of the temperature producing the lowest RSR to probe the minimum (marked by a red X in each panel of Figure 4.5). This range is chosen in an effort to (1) avoid some of the starkly different RSRs that occasionally show up at larger temperature offsets from the minimum, (2) include enough points (usually around 15) to generate a fit of sufficient quality, (3) avoid the dominance of the warmer side of the curve, smoother due to improved model quality, on the fit, and (4) avoid a lopsided curve sampled much more extensively on the warm or cool side for cooler and warmer stars respectively. Our 1500 to 5000 K range allows this wiggle room on either side of the true effective temperature, expected to be about 3900 to 1900 K for M0-L2.5 dwarfs (Pecaut & Mamajek 2013; Filippazzo et al. 2015).5 Note that because the models are not perfect, especially for the lowest mass stars, and because the color mismatch between observed and model temperature may not be exactly quadratic with temperature, we opt to use a third order polynomial for these fits. This polynomial fit is applied to a 1 K interval array spanning 500 K on either side of the initial minimum (the grid temperature with the lowest RSR) using IDL’s POLY_FIT routine. The minimum is calculated to be at the point where the derivative of the polynomial

5

is zero and the concavity >0.

As can be seen in Figure 4.5, this procedure is performed on all colors assigned to a given star (9 for warm stars and 14 for cool stars). At the end of this procedure, the mean value and standard deviation are recorded as the “final” temperature for that particular iteration through the temperature procedure. As the code moves on to the next steps (metallicity, surface gravity, and/or alpha enhancement), the grid value closest to this result (e.g., 2300 K for a final value of 2284 K) is used until the next iteration of the temperature procedure. When the entire iterative process is over, the final mean value and standard deviation are the final results for the star. As discussed in §6.2.1, our effective temperature results match established methods to determine low mass star effective temperatures, and compare well with previous efforts, while providing values for many more, and broader range of stars.

For the other three parameters there are not enough value options to fit a polynomial as was done for temperature — a maximum of only three values for metallicity = -0.5, 0.0, 0.5; three values for gravity = 4.5, 5.0, 5.5; and two values for alpha enhancement 0.0, 0.2. As we move from color to color, we still determine a best value for a given color, but it is prudent to give the extracted parameter values with high RSRs less weight than those with superior fits, rather than just taking the mean. Therefore, for each color (e.g. V −K), the metallicity/surface gravity/alpha enhancement is determined according to the weighted mean of all N grid options, i (and j), for that iteration, with the weights set to be the inverse

Figure 4.5: Example 3rd order polynomial fits to RSR vs. effective temperature for the cool star 2MA1645-1319 (V −K = 9.455), using the procedure described in §4.3.3. The y-axis is intentionally cut off at RSR = 1500 for consistency between plots. The red triangles mark the 5 temperatures with the lowest RSRs, and the X indicates the minimum of the fitted curve, which corresponds the final effective temperature and RSR for each color. The effective temperature of the star is taken to be the mean of these temperatures, and their standard deviation is assigned to be the error in the temperature determination.

of the RSR, e.g., for metallicity Z:

Zcolor = N ∑ i=1 Zi RSR−color,i1 N ∑ j=1 RSR−_color,j1 (4.6)

is taken to be the final result for the star. The closest grid value is used as the fixed value for calculations of the remaining three parameters, as is done with effective temperature.

Although we do not find much consistency between metallicity reported from our pro- cedure and what one would expect for young stars and cool subdwarfs, this is somewhat expected because we are using broadband filter measurements that are relatively insensitive to the oftentimes subtle features of low-mass star spectra for at least the milder versions of young red dwarfs and cool subdwarfs. In fact, as explained next, our best-fitting spectrum will need some adjustment to match the photometry observations used to determine the luminosities of the stars.