DETECTION OF STARS WITHIN ∼0.8 in OF KeplerOBJECTS OF INTEREST

Once the spectrum is normalized and on a constant ${\rm \ \Delta\ }\lambda /\lambda$ wavelength scale, we search through the SpecMatch library of stellar spectra to find the best fit for the primary star. The SpecMatch library consists of Keck-HIRES spectra of 640 FGKM stars scattered throughout the H-R diagram, with a concentration to the main sequence and subgiants. These spectra were obtained as part of the California Planet Search. Each spectrum has S/N = 100–200 per pixel between 3800 and 8000 Å with the same spectral resolution as the bulk of the spectra analyzed here, R = 60,000.

All of the SpecMatch library spectra are available online at the Keck Observatory Archive. Apart from M dwarfs, each spectrum was analyzed with an advanced version of the spectroscopic analysis package, SME, as described in Valenti & Fischer (2005). SME was later modified based on the available parallax information, for a revised, improved version of SME (Brewer et al. 2014, in preparation). This analysis yielded the values of ${{T}_{{\rm eff}}}$, logg, and [Fe/G] for all FGK stars in the library. Stellar parameters for M dwarfs were adopted from Rojas-Ayala et al. (2012).

SpecMatch stellar parameter values range in ${{T}_{{\rm eff}}}$ = [3250,7260], logg = [1.46,5.00], and [Fe/H] = [−1.475,0.558]. We assume that the primary star lies on main sequence and exclude the subgiants from the library as possible best-fit candidates. This choice was made due to the preferential choice of subgiants as the best-fitting stars, even when the observed primary star was on main sequence. Excluding the giants improves the quality of the best fit and allows for a better subtraction of the primary spectrum, which is explained in Section 3.3.1. This decision was made with our efforts in mind; since our primary goal is to search for the secondary lines in the spectrum, we sacrifice some of the accuracy of the primary star parameters in order to achieve a better fit to the primary star absorption lines. Nevertheless, if the spectral type of the primary star is known, the algorithm can also be modified to include the subgiants in the fitting process.

We look for the library spectrum that produces the lowest ${{\chi }^{2}}$ value when the absorption lines of the library spectrum are aligned with the studied spectrum. We vary the rotational broadening of each library spectrum and the depth of the absorption lines by diluting the spectrum. The latter is to correct for the finite grid spacing of the metallicity in our library, and is implemented to ensure as complete subtraction of the primary star absorption lines as possible. Due to the inclusion of the constant dilution factor, we are unable to determine the logg of the primary star. However, as mentioned, the determination of the parameters for the primary star is beyond the scope of this paper, but it can be performed using a different method if so desired.

Taking the above into account, the ${{\chi }^{2}}$ is computed using

$$\begin{eqnarray}&&{{\chi }^{2}}=\mathop{\sum }\limits_{i}{{(S_{i}^{{\rm lib}}-d\cdot (S_{i+p}^{{\rm obs}}(v{\rm sin} i)-1)+1)}^{2}},\end{eqnarray} \tag{ 1 }$$

where ${{S}^{{\rm lib}}}$ is the SpecMatch library spectrum, ${{S}^{{\rm obs}}}(v{\rm sin} i)$ is the rotationally broadened observed spectrum, d is the dilution factor, and p is the Doppler shift of the primary star in pixels.

Discrepancies between each library star and the actual spectrum are both due to Poisson noise and intrinsic spectral differences. This is mostly due to the finite grid spacing in the parameter space of our library. Consequently, we cannot establish a good uncertainty estimate for each point along the spectrum, and the actual value of ${{\chi }^{2}}$ bears no significance; rather, it is the relative ${{\chi }^{2}}$ among all the library spectra that we are concerned with.

In order to prevent outliers from being identified as best-fitting stars, we first identify the approximate temperature of the primary star by finding the minimum of the ${{\chi }^{2}}$ versus library star temperature distribution, as shown in Figure 2. The vertical scatter of the points at the same ${{T}_{{\rm eff}}}$ is due to discrepancies in other parameters, such as logg and metallicity. Note that we could also plot ${{\chi }^{2}}$ as a function of other parameters as well; we choose the temperature as the distribution that has the most easily identifiable minimum.

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Based on the polynomial fit to the ${{\chi }^{2}}$ distribution we eliminate the outliers that differ from the polynomial fit for more than 3σ at any value of the effective temperature. We then repeat the polynomial fitting, and choose the lowest ${{\chi }^{2}}$ value within the ± 200 K range about the minimum of the polynomial fit as our best-fitting library spectrum. While restricting the range of possible ${{T}_{{\rm eff}}}$ to ± 200 K around the polynomial minimum could introduce larger uncertainty in the determined ${{T}_{{\rm eff}}}$ of the primary star, it does ensure that the chosen library spectrum is the best achievable fit to the observed spectrum and has approximately correct effective temperature. The restricted range excludes any outliers that have not yet been eliminated at the edges of the effective temperature range ($\gt 6100\;{\rm K}$ and $\lt 3400$) due to the edge effects on the shape of the polynomial.

Because the main goal of the primary star fitting is the subtraction of its absorption lines from the spectrum, we do not report any of the best-fit parameters for the primary star. The accuracy of these results is compromised due to several manipulations of the library spectra to account for the finite grid of our library parameter space. Nevertheless, all the described steps do ensure that the primary star absorption lines are fitted almost perfectly and subtracted almost completely. If the parameters for the primary star are known, however, any or all of the values in the parameter space we search over (logg, $[{\rm Fe}/{\rm H}]$, rotational broadening, ${{T}_{{\rm eff}}}$) can be constrained to a known range or fixed to a known value.

Once the best-fit library spectrum is identified and its lines are both broadened and diluted to match the primary starʼs absorption lines, we subtract that best-fit library spectrum from the original spectrum, leaving residuals as shown in Figure 4.

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We re-normalize the residuals described above back to unity, and search for the secondary set of absorption lines. Our algorithm is focused mostly on faint secondary stars, so the spectral differences among two stars with ${\rm \ \Delta\ }{{T}_{{\rm eff}}}\lt 100$ K are almost indiscernible. To enhance efficiency, we create 26 median spectra with ${{T}_{{\rm eff}}}$ intervals of 100 K starting at 3300 K and up to 6100 K, rather than utilizing each individual library spectrum.

We omit ${{T}_{{\rm eff}}}$ where there are less than three spectra per 100 K temperature interval. Consequently, not all of the 100 K temperature intervals are represented. In particular, we do not have a representative median library spectrum at effective temperatures 3800 and 3900 K.

We then calculate ${{\chi }^{2}}$ as a function of Doppler shift between the residuals and each median library spectrum, and normalize it such that the median is 1. A significant ${{\chi }^{2}}$ minimum occurring at the same ${\rm \ \Delta\ RV}$ for several neighboring effective temperatures of the median library spectrum indicates a possible second star in the spectrum. An example of a residual ${{\chi }^{2}}$ function is shown in Figure 5.

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Unless the primary and the secondary spectrum are sufficiently different, we cannot detect secondary stars with ${\rm \ \Delta\ RV}\lt 10\;{\rm km}\;{{{\rm s}}^{-1}}$ relative to the primary star, due to the normal widths of the absorption lines. None of the library spectra are the exact match to the studied ones, so there will always be some imperfections in the best-fitting library star, resulting in larger residuals at the locations of primary starʼs absorption lines. These residuals may cause either small peak or a small dip at ${\rm \ \Delta\ RV}\;$= 0 ${\rm km}\;{{{\rm s}}^{-1}}$. Thus, if both stars have exactly the same radial velocity, we cannot differentiate between the minimum at ${\rm \ \Delta\ RV}$ = 0 ${\rm km}\;{{{\rm s}}^{-1}}$ that is due to the imperfect subtraction of the primary, or that due to an actual second star. Furthermore, for secondary sets of absorption lines with slightly different radial velocities than the primary star, their spectral lines are still blended with the primary starʼs absorption lines. This occurs if the RV separation is less than $10\;{\rm km}\;{{{\rm s}}^{-1}}$, which is the typical width of an absorption line. This blending causes a fraction of secondary absorption lines to be subtracted away together with the primary star, making a faint second star appear even fainter, and in most cases impossible to detect.

Due to the limitations described, we mask out the central region with ${\rm \ \Delta\ RV}\;\lt$ 10 ${\rm km}\;{{{\rm s}}^{-1}}$, as shown in Figure 6. The only exception is the case where the two spectral types are sufficiently different that the primary star spectrum does not interfere with the secondary, regardless of their RV separation. This is the case of a G-type primary and an M dwarf secondary star, as further discussed in Section 4.1.4.

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As mentioned in Section 3.3.2, we obtain 26 distinct residual ${{\chi }^{2}}$ functions, differing in the effective temperature of the library spectrum superimposed with the studied residual spectrum. We expect to find a unique ${\rm \ \Delta\ RV}$ location of the secondary star, if one is detectable. Because a secondary set of absorption lines would cause a minimum in ${{\chi }^{2}}$ over a range of library spectral types that are close in ${{T}_{{\rm eff}}}$ to the actual temperature, we assume the secondary star ${\rm \ \Delta\ RV}$ to be the mode of the ${{\chi }^{2}}$ minimum locations for a set of all 29 residual ${{\chi }^{2}}$ functions. The exception here is the case where the minimum occurs only for ${{T}_{{\rm eff}}}$ ∼ 3500 K. Due to a significantly different M dwarf spectrum, it is possible that the secondary star is detected only in a few residual ${{\chi }^{2}}$ functions with ${{T}_{{\rm eff}}}\;\lt \;\sim$3800 K. Those cases are examined and assessed on an individual basis.

Using the mode minimum ${\rm \ \Delta\ RV}$ location, we plot the ${{\chi }^{2}}$ values at those locations versus the temperature of the median library spectrum used to construct each respective ${{\chi }^{2}}$ function. For most residual ${{\chi }^{2}}$ functions, this ${{\chi }^{2}}$ is indeed the ${{\chi }^{2}}$ minimum. Ideally, the depth of the minimum should only depend on how well the library spectrum correlates with the spectrum of the secondary star, producing the lowest ${{\chi }^{2}}$ minimum when the library spectrum temperature is closest to the effective temperature of the secondary star. However, mostly due to the low S/N of the residuals, we noticed a dependence of the residual ${{\chi }^{2}}$ minimum on the temperature of the library spectrum that was independent of the actual ${{T}_{{\rm eff}}}$ of the secondary star. In order to correct for this, we divide the ${{\chi }^{2}}$ minima plot by the calibration function, a construction of which is described in more detail in Section 4.1.2.

We then plot the calibrated ${{\chi }^{2}}$ minimum values versus the temperature of the median library spectrum, as shown in Figure 9 (left). In order to determine approximate temperature of the secondary star, we follow the same procedure as for the primary star, as outlined in Section 3.1.1. In brief, we fit the points with a polynomial, find a minimum, remove the outliers that differ from the fit by more than 3 σ, re-fit the remaining points, and restrict the possible range of secondary star ${{T}_{{\rm eff}}}$ to ± 200 K around the polynomial minimum. Within that range, we then compute the mean ${{T}_{{\rm eff}}}$ of the three median library spectra corresponding to the three lowest ${{\chi }^{2}}$ minima. We adopt this mean as the estimated temperature of the secondary star. Taking the mean of the three lowest ${{T}_{{\rm eff}}}$ values corrects for the missing median templates, because not every 100 K ${{T}_{{\rm eff}}}$ is represented due to the varying parameter populations of SpecMatch library.

We cannot extrapolate the behavior of the ${{\chi }^{2}}$ minima function to values of ${{T}_{{\rm eff}}}$ higher than 6100 K or lower than 3300 K. Thus, any secondary stars whose ${{T}_{{\rm eff}}}$ is identified at ∼6000 K, is not guaranteed to be at that temperature; rather this serves as the lower temperature limit, as the secondary star can be hotter than ∼6000 K but not cooler. Similarly, secondary stars whose estimated ${{T}_{{\rm eff}}}$ is 3300 K can be cooler than this value, but not hotter. This is discussed more in detail in Section 6.2.

After determining ${{T}_{{\rm eff}}}$ of the second star, we use the corresponding residual ${{\chi }^{2}}$ function for further analysis. We aim to estimate the brightness of the secondary star relative to the primary star based on the depth of that residual ${{\chi }^{2}}$ function minimum.

We do so by synthesizing the effect of secondary spectra. We inject another spectrum into the original studied spectrum. We scale down the injected spectrum such that it contributes either 3% or 1% of the total flux, and shift its absorption lines to a known relative radial velocity. The injected spectrum has ${{T}_{{\rm eff}}}$ close to that already estimated for the secondary star temperature, assuring that the superimposed library spectrum fits both the actual secondary star and the injected spectrum equally well. The properties of all the possible injected spectra are listed in Table 1.

Table 1. Stellar Parameters for Injected SpecMatch Library Spectra

Name	${{T}_{{\rm eff}}}$ (K)	logg (mag)	M (${{M}_{\odot }}$)	R (${{R}_{\odot }}$)	$[{\rm Fe}/{\rm H}]$ (mag)	Source
GL 273	3293	$\cdots$	0.29	0.31	−0.17	Rojas-Ayala et al. (2012)
GL 687	3395	$\cdots$	0.40	0.40	−0.09	Rojas-Ayala et al. (2012)
GL 408	3526	$\cdots$	0.38	0.38	−0.09	Rojas-Ayala et al. (2012)
GL 250B	3569	$\cdots$	0.45	0.43	0.01	Rojas-Ayala et al. (2012)
GL 686	3693	$\cdots$	0.45	0.43	−0.28	Rojas-Ayala et al. (2012)
HIP 24284	3992	4.88	0.41	0.36	−0.49	SME analysis
HIP 41689	4088	4.83	0.47	0.39	−0.45	SME analysis
HD 217357	4200	4.74	0.55	0.46	−0.17	SME analysis
HIP 105341	4298	4.72	0.58	0.55	−0.05	SME analysis
HIP 99205	4397	4.70	0.63	0.59	−0.18	SME analysis
HIP 36551	4501	4.69	0.65	0.60	−0.30	SME analysis
HIP 103650	4602	4.68	0.69	0.63	−0.04	SME analysis
HIP 63762	4701	4.62	0.74	0.69	0.09	SME analysis
HD 220221	4797	4.58	0.79	0.75	0.18	SME analysis
HD 51866	4906	4.59	0.80	0.75	0.13	SME analysis
HD 23356	4988	4.60	0.78	0.73	−0.04	SME analysis
HD 216520	5097	4.55	0.78	0.77	−0.19	SME analysis
HD 205855	5204	4.53	0.86	0.83	0.05	SME analysis
HD 75732	5295	4.49	0.97	0.92	0.39	SME analysis
HD 58727	5399	4.52	0.97	0.90	0.24	SME analysis
HD 147750	5496	4.52	0.90	0.85	−0.11	SME analysis
HD 20619	5600	4.42	0.87	0.94	−0.25	SME analysis
HD 12661	5699	4.38	1.09	1.11	0.35	SME analysis
HD 148284	5799	4.39	1.11	1.10	0.29	SME analysis
HD 205351	5896	4.29	1.11	1.23	0.09	SME analysis
HD 27859	5997	4.41	1.14	1.09	0.14	SME analysis
HD 48682	6104	4.36	1.18	1.18	0.11	SME analysis

Note. No precise logg values are available for M dwarfs.

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We then treat this original spectrum, with its synthetic injected secondary star spectrum, as the new studied spectrum, and repeat the whole procedure outlined in Sections 3.1.1 through 3.3.2. We record the ${{\chi }^{2}}$ value at the ${\rm \ \Delta\ RV}$ location of the injected spectrum, as annotated in Figure 7. At that particular ${\rm \ \Delta\ RV}$, the difference between the injected spectrum ${{\chi }^{2}}$ and the value of the original ${{\chi }^{2}}$, both shown in Figure 7, represents the minimum depth that would be caused by a secondary star of the injected brightness.

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To obtain a more statistically useful sample of possible secondary spectra, we repeat the injection for several different values ${\rm \ \Delta\ RV}$s, and record the difference between the value of the ${{\chi }^{2}}$ minimum at the location of the injected secondary spectrum and the original residual ${{\chi }^{2}}$ value at that location. Using those differences, we then calculate the median ${{\chi }^{2}}$ minimum depth relative to the original residual ${{\chi }^{2}}$ caused by a secondary star of that particular brightness.

Subtracting the median depth for both the 3% and 1% injected secondary spectrum from the original residual ${{\chi }^{2}}$ without the injected secondary spectrum, we obtain the plot shown in Figure 8. At each ${\rm \ \Delta\ RV}$, the two colored curves represent the value that the ${{\chi }^{2}}$ function would have if there was a secondary star of the specified brightness located at that particular ${\rm \ \Delta\ RV}$. This allows for a visual comparison of the actual residual ${{\chi }^{2}}$ minimum to the characteristic ${{\chi }^{2}}$ minima depths for 3% and 1% secondary stars.

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To estimate the relative brightness of the actual secondary star, we extrapolate the median ${{\chi }^{2}}$ minimum caused by the 3% and 1% injected secondary spectrum to the actual ${{\chi }^{2}}$ minimum for the residual function. We use a least-squares linear fit, with a restriction that 0% secondary star (none present) causes a ${{\chi }^{2}}$ minimum of depth 0. We report both the ${\rm \ \Delta\ RV}$ of the secondary star, as well as the estimated relative brightness as a percentage of total flux, on the plot.

Since our wavelength domain encompasses approximately the V and R broadbands of the classical photometry, the relative brightness of the two stars can be used to compute the dilution factor of the Kepler light curves. Thus, if the primary star has any planet candidates, this dilution factor can help determine a more accurate radius of the planet.

Figure 9 has two panels, with the left showing the calibrated residual ${{\chi }^{2}}$ minimum as a function of ${{T}_{{\rm eff}}}$ and the right showing ${{\chi }^{2}}$ versus ΔRV. This includes the minima curves for 1% and 3% injected secondaries, shown in red and blue, respectively (see Section 3.3.4).

This final Figure 9 is intended to summarize our knowledge about the secondary star: estimated ${{T}_{{\rm eff}}}$, estimated percentage contribution to the total flux spectrum in the V and R bands, and the radial velocity of the secondary relative to the primary star. Any additional parameters for the secondary star cannot be determined accurately enough to be published.

On the final secondary star plot, we annotate any minimum that contributes at least 0.5% of the total flux. Depending on the S/N of the original spectrum (typically 50–200 per pixel), the spectral types of the two stars, and their ${\rm \ \Delta\ RV}$, secondaries with fluxes between 0.5% and 1% relative to the primary star can be detected with our algorithm.

We visually inspect each diagnostic plot to assess the probability that the identified minimum is indeed due to a secondary star. When there is no clear ${{\chi }^{2}}$ minimum, or when the ${{\chi }^{2}}$ minimum is due to fluctuations, we establish the threshold limits on the secondary star as follows. Any undetected secondary must contribute less to the total flux than would have been revealed above the fluctuations in the ${{\chi }^{2}}$ functions. If this percentage is not marked on the plot, the brightness limit is 0.5% relative to the primary star. Otherwise, the appropriate percentage is marked on the plot. M dwarf secondaries contributing more than 0.5% of the total flux can typically be detected in our spectra if present, or ruled out if not present.

We used the synthetic binary spectra with added noise to determine the uncertainties in the deduced parameters for the secondary star. We investigated each temperature combination and each flux ratio separately. We examined the recovery rate by counting all the cases where the secondary star was identified, disregarding the accuracy of the estimated parameters. Results are summarized in Table 2. The table can be read as follows: ${{T}_{{\rm eff}}}$ values running horizontally denote the effective temperature of the secondary star, and the values running vertically down the first column denote ${{T}_{{\rm eff}}}$ of the primary star. For example, the 60% rate quoted in the last column of the first data row means that the secondary star in the binary spectrum consisting of 99% of a 3500 K star and 1% 3500 K star was successfully identified in 6 out of 10 cases.

Table 2. Percentage Recovery Rates for Injection-recovery Experiment

Primary ${{T}_{{\rm eff}}}$	Secondary ${{T}_{{\rm eff}}}$ (K)
	3500	4000	4500	5000	5500	6000
	1% Secondary
3500	50%	40%	40%	60%	60%	60%
4000	70%	70%	70%	70%	70%	70%
4500	80%	70%	80%	80%	70%	70%
5000	80%	80%	70%	70%	80%	70%
5500	100%	90%	90%	100%	100%	80%
6000	80%	70%	80%	70%	80%	80%
	3% Secondary
3500	100%	90%	100%	100%	100%	100%
4000	90%	90%	90%	90%	90%	90%
4500	100%	100%	100%	100%	100%	90%
5000	90%	100%	100%	100%	90%	90%
5500	100%	100%	100%	100%	100%	100%
6000	90%	100%	100%	100%	90%	90%
	10% Secondary
3500	100%	100%	100%	100%	100%	100%
4000	100%	100%	100%	100%	100%	100%
4500	100%	100%	100%	100%	100%	100%
5000	100%	100%	100%	100%	100%	100%
5500	100%	100%	100%	100%	100%	100%
6000	100%	100%	100%	100%	100%	100%

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For all synthetic spectra, we added a Poisson noise at the level of 2%. Since we subtract the primary star absorption lines, a 1% secondary implies an S/N for the residual spectrum of only 0.5, 3% has an S/N of 1.5, and a 10% secondary yields a residual spectrum with an S/N of 5. Of course, such low S/N values per pixel are overcome by the thousands of absorption lines, detected in thousands of pixels, each contributing to the detectability of the secondary star. Thus the effective S/N of the secondary star is much higher than the per-pixel S/N.

We isolated the synthetic binary cases where the binarity of the spectrum was established, and examined the predicted secondary star temperature as opposed to the actual values. As mentioned in Section 3.3.3, we noticed a dependence of the depth of the residual ${{\chi }^{2}}$ minimum on the temperature of the library spectrum, which was superimposed with the studied spectrum to construct the residual ${{\chi }^{2}}$. Except when the secondary star was an M dwarf, the residual ${{\chi }^{2}}$ minimum was always the deepest for ${{T}_{{\rm eff}}}\;\sim$ 4500 K, regardless of whether or not that was the actual ${{T}_{{\rm eff}}}$ of the secondary star. We suspect that this bias is caused by the nature of the 4500 K spectrum itself, as it contains an abundance of deep and sharp absorption lines from neutral metals that dominate FGK spectra. With the primary star subtracted from the spectrum, the residuals are dominated by Poisson noise at a 2% level, especially when the secondary star contributes only a few percent to the total flux. These neutral metal absorption lines tend to accidentally align with the noise much more frequently due to their deep, sharp features. Such accidental alignments cause a deeper ${{\chi }^{2}}$ minimum for the 4500 K library star with the studied spectrum of any ${{T}_{{\rm eff}}}$, particularly for the residuals with S/N values near unity.

This bias was strong enough to influence the estimated temperature of the second star and make our deduced raw derived secondary star parameters inaccurate. In order to correct for this temperature bias, we calculated the average residual ${{\chi }^{2}}$ minimum distribution for all the test cases of a particular flux ratio, excluding non-detections. If the distribution was unbiased against library spectrumʼs ${{T}_{{\rm eff}}}$, the mean distribution should be approximately constant with respect to temperature, as we ensured that there were an equal number of cases at each secondary star ${{T}_{{\rm eff}}}$.

We found, however, that this was not the case; the mean distribution peaked at 4500 K, as expected. Since the shape of any true ${{\chi }^{2}}$ minimum distribution will be affected by this peak, we use the mean distribution, normalized such that the maximum occurs at ${{\chi }^{2}}=1$, as our calibration function. Calibration functions for a 1%, 3%, and 10% secondary star are shown in Figure 10.

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As we can see from Figure 10, calibration functions are almost identical among the different relative brightnesses of the two stars. Therefore, there will be negligible errors resulting from dividing a residual ${{\chi }^{2}}$ minimum distribution by these calibration functions for cases where the relative brightness of the studied star is not 1%, 3%, or 10% exactly.

Using calibrated ${{\chi }^{2}}$ minimum distribution, we estimated the second star temperature and relative brightness, as described in Sections 3.3.3 and 3.3.4. We then compared the deduced values to the known parameters, and the discrepancies are shown in Tables 3 and 4, respectively.

Table 3. Systematic Errors and Uncertainties for the Estimated Secondary Star Temperature

Primary ${{T}_{{\rm eff}}}$ (K)	Secondary ${{T}_{{\rm eff}}}$ (K)
	3500	4000	4500	5000	5500	6000
	1% Secondary
3500	75 ± 70	565 ± 200	30 ± 695	−245 ± 415	45 ± 465	500 ± 465
4000	−45 ± 155	175 ± 440	−55 ± 530	−470 ± 440	−230 ± 320	180 ± 330
4500	0 ± 125	195 ± 455	80 ± 220	−285 ± 260	−355 ± 70	115 ± 180
5000	−85 ± 155	500 ± 125	230 ± 135	−205 ± 230	−14- ± 390	50 ± 20
5500	5 ± 155	230 ± 440	140 ± 235	−260 ± 235	−375 ± 95	60 ± 25
6000	−85 ± 155	500 ± 125	230 ± 125	−205 ± 230	−140 ± 390	50 ± 20
	3% Secondary
3500	−60 ± 185	470 ± 340	200 ± 355	−85 ± 410	−180 ± 360	85 ± 65
4000	−5 ± 130	380 ± 420	195 ± 325	−80 ± 480	−255 ± 440	205 ± 450
4500	25 ± 135	385 ± 370	250 ± 350	−195 ± 295	−355 ± 145	50 ± 150
5000	0 ± 135	80 ± 370	250 ± 250	−210 ± 385	225 ± 435	330 ± 495
5500	−5 ± 160	375 ± 255	225 ± 255	−220 ± 320	−375 ± 105	50 ± 150
6000	0 ± 135	80 ± 375	250 ± 250	−210 ± 370	205 ± 435	330 ± 495
	10% Secondary
3500	−15 ± 145	295 ± 170	22 ± 185	70 ± 245	−155 ± 345	−50 ± 50
4000	−5 ± 130	250 ± 120	−50 ± 220	−15 ± 275	−230 ± 280	−50 ± 50
4500	−10 ± 130	205 ± 135	100 ± 50	−20 ± 315	−195 ± 320	100 ± 14
5000	0 ± 150	205 ± 85	25 ± 150	30 ± 255	−145 ± 245	−20 ± 50
5500	−5 ± 150	190 ± 160	130 ± 65	−95 ± 345	−320 ± 125	−50 ± 20
6000	0 ± 150	300 ± 80	150 ± 100	30 ± 255	−145 ± 245	−40 ± 50

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Table 4. Systematic Errors and Uncertainties for the Estimated Percentage Flux of the Secondary Star

Primary ${{T}_{{\rm eff}}}$ (K)	Secondary ${{T}_{{\rm eff}}}$ (K)
	3500	4000	4500	5000	5500	6000
	1% Secondary
3500	0.25 ± 0.10	−0.28 ± 0.89	0.03 ± 0.33	−0.30 ± 0.24	−0.21 ± 0.29	−0.15 ± 0.34
4000	0.21 ± 0.11	0.20 ± 0.26	− 0.01 ± 0.11	−0.18 ± 0.24	−0.10 ± 0.18	−0.04 ± 0.21
4500	0.10 ± 0.26	0.11 ± 0.26	−0.07 ± 0.12	−0.25 ± 0.34	−0.16 ± 0.11	−0.14 ± 0.16
5000	0.32 ± 0.19	0.32 ± 0.19	0.03 ± 0.19	−0.04 ± 0.25	−0.02 ± 0.17	−0.05 ± 0.22
5500	0.15 ± 0.18	0.11 ± 0.26	−0.11 ± 0.17	−0.12 ± 0.20	−0.23 ± 0.31	−0.07 ± 0.14
6000	0.32 ± 0.18	0.32 ± 0.19	0.03 ± 0.19	−0.04 ± 0.25	−0.02 ± 0.17	−0.05 ± 0.22
	3% Secondary
3500	0.34 ± 0.90	0.67 ± 0.45	0.00 ± 0.40	−0.21 ± 0.66	−0.34 ± 0.70	−0.35 ± 0.74
4000	0.32 ± 0.43	0.71 ± 0.65	−0.08 ± 0.32	−0.23 ± 0.57	−0.24 ± 0.78	0.19 ± 0.69
4500	0.38 ± 0.43	0.62 ± 0.50	0.07 ± 0.69	−0.25 ± 0.50	−0.31 ± 0.52	−0.02 ± 0.34
5000	0.31 ± 0.86	0.11 ± 0.79	−0.12 ± 0.23	−0.18 ± 0.18	−0.11 ± 0.14	−0.49 ± 0.16
5500	0.25 ± 0.49	0.67 ± 0.49	−0.21 ± 0.38	−0.18 ± 0.42	−0.45 ± 0.53	0.04 ± 0.29
6000	0.18 ± 0.87	0.11 ± 0.79	−0.12 ± 0.23	0.18 ± 0.18	0.11 ± 0.14	0.69 ± 0.16
	10% Secondary
3500	1.17 ± 1.34	3.09 ± 1.21	0.35 ± 1.85	0.06 ± 2.25	−2.63 ± 2.50	−1.17 ± 1.42
4000	1.26 ± 1.40	3.43 ± 1.09	0.23 ± 1.40	−1.92 ± 1.44	−1.84 ± 0.77	−0.68 ± 1.44
4500	1.32 ± 1.55	2.95 ± 0.77	−0.12 ± 1.05	−1.26 ± 0.83	−1.97 ± 1.08	−1.07 ± 1.97
5000	−0.07 ± 0.15	2.23 ± 1.43	0.29 ± 0.70	−1.25 ± 0.93	−1.76 ± 0.83	−1.55 ± 1.53
5500	0.55 ± 1.56	2.00 ± 1.00	−0.84 ± 0.78	−1.71 ± 0.41	−1.71 ± 0.53	−0.47 ± 0.50
6000	−0.07 ± 0.16	2.23 ± 1.43	0.29 ± 0.70	−0.42 ± 0.58	−1.76 ± 0.83	−1.55 ± 1.53

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Table 3 lists the discrepancies in the SpecMatch ${{T}_{{\rm eff}}}$ value and that derived from our algorithm, (${{T}_{{\rm SpecMatch}}}\;-\;{{T}_{{\rm deduced}}}.$) Table 4 lists the discrepancies in the set secondary star percentage contribution to the total flux and the value derived by our algorithm. We know the actual flux contribution precisely, because we artificially scaled down the secondary spectra when constructing synthetic binaries. The values are listed in terms of the percentage of the total flux, (${{\%}_{{\rm known}}}\;-{{\%}_{{\rm deduced}}}$). Each primary–secondary temperature combination consisted of 10 trial cases, with different realizations of added Poisson noise.

We report both the systematic difference (the mean discrepancy) and the associated standard deviation. Since the systematic differences are not constant over all temperature combinations or relative brightnesses, we cannot apply a single correction to obtain more accurate values. As such, Tables 3 and 4 serve more as illustrations of the injection-recovery results.

We noticed larger ${{T}_{{\rm eff}}}$ systematic differences for 4000 K secondary stars. This is mostly due to the scarce SpecMatch library populations in that region, with some 100 K ${{T}_{{\rm eff}}}$ intervals missing altogether. On the other hand, the algorithm did consistently correctly identify the M dwarf secondaries, owing their detectability to their distinct spectral features. As expected, the systematic differences in ${{T}_{{\rm eff}}}$ and the standard deviations decreased for brighter secondary stars, as the secondary spectrum becomes brighter than the 2% noise.

On the other hand, absolute errors in percentage increased when the secondary star was brighter. This was expected because the errors in the linear extrapolations of ${{\chi }^{2}}$ minimum depth become larger for brighter stars. Nevertheless, the errors were still mostly only around 1%, and the relative errors rarely exceed 50%. We also noticed a pattern that the relative brightness of hotter secondary stars was mostly overestimated, while cooler stars were underestimated. These trends can be useful when establishing limits on brightnesses of any real secondaries.

We combine the systematic error and the uncertainty to form more conservative uncertainties to be used with the secondary star parameters deduced in the analysis of actual spectra. These are shown in Table 5. Columns two through seven show the absolute uncertainty in the effective temperature of the secondary star for several possible primary–secondary star pairs; columns eight through 13 show the relative uncertainty in the flux ratio. Based on the effective temperatures of the primary–secondary pair and their flux ratio, appropriate uncertainty for both the effective temperature of the secondary star and its relative brightness can be read from Table 5.

Table 5. Uncertainties in the Effective Temperature of the Secondary Star

Primary ${{T}_{{\rm eff}}}$ (K)	Secondary ${{T}_{{\rm eff}}}$ (K)
	3500	4000	4500	5000	5500	6000	3500	4000	4500	5000	5500	6000
	Absolute ${{T}_{{\rm eff}}}$ Uncertainty, ${{\sigma }_{{{T}_{{\rm eff}}}}}$ (K)						Relative Flux Ratio Uncertainty a , ${{\sigma }_{f}}/f$
	1% Secondary						1% Secondary
3500	150	750	750	650	500	950	0.35	1.15	0.35	0.55	0.50	0.50
4000	200	600	600	900	550	500	0.30	0.45	0.10	0.40	0.30	0.25
4500	150	650	300	550	450	300	0.35	0.35	0.20	0.60	0.25	0.30
5000	250	650	350	450	400	100	0.50	0.50	0.20	0.30	0.20	0.30
5500	150	650	400	500	450	100	0.35	0.35	0.30	0.30	0.55	0.20
6000	250	650	350	450	550	100	0.50	0.50	0.20	0.30	0.20	0.25
	3% Secondary						3% Secondary
3500	250	800	550	500	550	150	0.40	0.35	0.15	0.30	0.35	0.35
4000	150	800	500	550	700	650	0.25	0.45	0.15	0.25	0.35	0.30
4500	150	755	600	500	500	200	0.25	0.35	0.25	0.25	0.25	0.10
5000	150	450	500	600	650	850	0.40	0.30	0.10	0.10	0.10	0.20
5500	150	650	500	550	500	200	0.25	0.40	0.20	0.20	0.30	0.10
6000	150	450	500	600	650	850	0.35	0.30	0.10	0.10	0.10	0.30
	10% Secondary						10% Secondary
3500	150	450	200	300	550	100	0.25	0.45	0.20	0.25	0.50	0.25
4000	150	350	250	350	500	100	0.25	0.45	0.15	0.35	0.25	0.20
4500	150	350	150	350	500	100	0.30	0.35	0.10	0.20	0.30	0.30
5000	150	300	200	300	400	100	0.10	0.35	0.10	0.20	0.25	0.30
5500	150	350	200	450	450	100	0.20	0.30	0.15	0.20	0.20	0.10
6000	150	400	250	300	400	100	0.05	0.35	0.10	0.10	0.25	0.30

Note.

^a The flux ratio of the secondary and primary star is denoted by f, where $f={{F}_{B}}/{{F}_{A}}$.

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As a part of the injection-recovery experiment, we also more closely examined the case of a G-type primary star with a temperature of 5500 K and an M dwarf secondary at 3500 K. Due to its low temperature, the M dwarf spectrum contains many molecular lines in the visible spectrum, which isquite distinct from the spectrum of FGK-type stars. As such, it is much easier to detect an M dwarf secondary spectra, even when the ${\rm \ \Delta\ RV}$ for the two stars is low.

We considered a situation where the two stars are separated by only 5 ${\rm km}\;{{{\rm s}}^{-1}}$, placing the secondary set of absorption lines into the masked-out region on our residual ${{\chi }^{2}}$ versus ${\rm \ \Delta\ RV}$ plot. We constructed three sets of 20 different synthetic binary stars with a 5500 K primary and a 3500 K secondary star, with the secondary contributing either 1%, 3%, or 5% of the total flux. We then executed the algorithm on each spectrum, determined the recovery rate, and calculated the discrepancies in the deduced ${{T}_{{\rm eff}}}$ and percentage flux. Results are shown in Table 6. Each companion percentage contribution sample consisted of 20 test cases.

Table 6. Injection-recovery Experiment: Recovery Rates and Parameter Uncertainties for a G-type Primary Star and an M dwarf Secondary at +5 ${\rm km}\;{{{\rm s}}^{-1}}\;{\rm \ \Delta\ RV}$

Secondary Star	Recovery	${{T}_{{\rm eff}}}$ (K)	Relative Flux (%)
Brightness	Rate	${{T}_{{\rm actual}}}\;-\;{{T}_{{\rm deduced}}}$	${{\%}_{{\rm actual}}}\;-\;{{\%}_{{\rm deduced}}}$
1%	40%	35 ± 170	0.01 ± 0.22
3%	90%	10 ± 130	1.18 ± 0.56
5%	90%	−5 ± 140	2.05 ± 0.70

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These results quantify the ability of our algorithm to detect M dwarf secondary stars even when the two sets of absorption lines overlap. Some of the M dwarf absorption lines still get subtracted away together with the primary star, thus the relative brightness of the secondary was consistently underestimated for all cases. This suggests an important limitation on our algorithm; when the secondary stars are detected in the ${\rm \ \Delta\ RV}\;\lt$ 10 ${\rm km}\;{{{\rm s}}^{-1}}$ region, the estimated relative brightness of the second star is more of a lower limit rather than an actual value.

Besides considering M dwarf secondary stars with low ${\rm \ \Delta\ RV}$ separation, we also explored the detectability limit in terms of the relative brightness of the two stars. We constructed synthetic binaries where the M dwarf secondary contributed 0.05%, 0.1%, and 0.5% of the total flux. The recovery rate for the 0.5% M dwarf secondary star was 40%, whereas none of the fainter companions were detected. Since all of the spectra had a 2% percent noise, this places the M dwarf detectability to an S/N ∼0.25.

KIC 10319590 is a precessing eclipsing binary, with time-varying eclipse depths (Slawson 2011; Rappaport et al. 2013). There is very little evidence of the secondary star in the ${{\chi }^{2}}$ function for the primary star shown in Figure 11, indicating that the secondary is faint.

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The residual ${{\chi }^{2}}$ function shown in Figure 12 shows evidence of a secondary set of absorption lines at ΔRV = $+44{\rm km}\;{{{\rm s}}^{-1}}$. Based on the residual ${{\chi }^{2}}$ minimum distribution, we estimate the secondary star ${{T}_{{\rm eff}}}$ = 4300 ± 500 K, with uncertainty based on Table 5.

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Deduced percentage of the total flux for the secondary star is 3.54%, yielding the flux ratio ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.036 ± 0.07. The comparison of our results to known parameters is limited by the lack of the available information.

KIC 5473556 is another example of a SB2 eclipsing binary (R. Angus 2013, private communication). For this system, the companion is sufficiently bright to cause a secondary minimum in the ${{\chi }^{2}}$ function for the primary star, as shown in Figure 13.

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There is a clear detection shown in the residual ${{\chi }^{2}}$ (Figure 14) at ΔRV = +89 ${\rm km}\;{{{\rm s}}^{-1}}$. We estimate the secondary star ${{T}_{{\rm eff}}}$ at 6000 ± 100 K. At the relative flux of ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.22 ± 0.07, this could either be a bound companion or a background star, as the primary star ${{T}_{{\rm eff}}}$ is similar to the estimated secondary star value.

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In order to resolve the ambiguity, we executed our algorithm using a different observation taken at a different time. If the two stars orbit each other, a different orbital phase should yield a different ${\rm \ \Delta\ RV}$ value. This was indeed the case, as another observation shows a companion at ${\rm \ \Delta\ RV}\;\approx \;-115$ ${\rm km}\;{{{\rm s}}^{-1}}$.

The flux ratio of 0.22 ± 0.07 and approximately the same ${{T}_{{\rm eff}}}$ for the two stars indicates that our deduced companion temperature might be an overestimate, and the actual companion ${{T}_{{\rm eff}}}$ most likely lies at the lower uncertainty limit, ${{T}_{{\rm eff}}}\;\sim$ 5900 K. Moreover, it is possible that only a fraction of the flux from the companion actually entered the slit during the observation. Since the parameters for the companion of KIC 5473556 are not known, we cannot compare our derived flux ratio, ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.22 ± 0.07, to a known value.

KIC 8572936 is an SB2 binary system that is also known for its circumbinary planet, Kepler-34 (Welsh et al. 2012). The two stars are both Sun-like stars, and their fluxes are comparable; ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.8475 ± 0.005 (Welsh et al. 2012). The effective temperatures of the two stars are similar, ${{T}_{{\rm eff},{\rm A}}}$ = 5932 ± 130 K and ${{T}_{{\rm eff},{\rm B}}}$ = 5867 ± 130 K.

This is an interesting case due to the similarity between the two stars, both in their spectral types and in the fractional contribution to the total flux. The main problem with such cases is that the two sets of absorption lines interfere with the best-fit for the primary star, thus we inevitably subtract some of the secondary lines together with the primary star. This leads to a somewhat larger uncertainty in the derived flux ratio for the stars using our method; however, the detection of both stars is clear and unambiguous, as seen in Figures 15 and 16.

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The estimated temperature of the companion of ${{T}_{{\rm eff}}}$ = 6000 ± 100 K, is in agreement with ${{T}_{{\rm eff}}}$ = 5812 ± 150 K quoted by Welsh et al. (2012). We calculated the flux ratio of ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.82 ± 0.25, which, in the uncertainty limit, also matches the Welsh et al. (2012) value of 0.8475 ± 0.005.

The case of KIC 9837578 is similar to KIC 8572936. It is also an SB2 binary system, with the two stars of similar ${{T}_{{\rm eff}}}$, but the flux ratio in the Kepler bandpass is somewhat lower, ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.3941 (Welsh et al. 2012).

We find a secondary star at ΔRV = −81 ${\rm km}\;{{{\rm s}}^{-1}}$, with the estimated effective temperature of 5600 ± 400 K (Figures 17 and 18). Taking into account the uncertainty, ${{T}_{{\rm eff}}}$ agrees with the Welsh et al. (2012) value of 5202 ± 100 K. The estimated flux ratio for the two stars of 0.56 ± 0.14 is larger than the Welsh et al. (2012) value by 7%. Some of this discrepancy can be attributed to a slightly different wavelength bandpass of our spectra as compared to Welsh et al. (2012), and some due to the mixing of the two sets of absorption lines, as both stars are of relatively similar spectral types with the secondary star contributing a significant amount of the total flux.

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HD 61994 is a double-lined spectroscopic binary with a period of 552.8 days (Strassmeier et al. 2012). When the spectrum was taken, the relative radial velocity for the two stars was only +18 ${\rm km}\;{{{\rm s}}^{-1}}$, which is an interesting example for cases where the relative radial velocity is low and the secondary star is something other than an M dwarf. Furthermore, the companion is faint—the visible brightness ratio for the two stars ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ is only 0.069 (Strassmeier et al. 2012).

The primary ${{\chi }^{2}}$ deviates slightly from the characteristic single-star ${{\chi }^{2}}$ function at ΔRV ≈ $-20\;{\rm km}\;{{{\rm s}}^{-1}}$, shown in Figure 19. However, more convincing evidence for the presence of a secondary star comes from the residual ${{\chi }^{2}}$ function shown in Figure 9, indicating a secondary star at ΔRV =$\;-16\;{\rm km}\;{{{\rm s}}^{-1}}$. Its residual ${{\chi }^{2}}$ minimum distribution function sets the secondary star ${{T}_{{\rm eff}}}$ at 4200 ± 650 K. Within the uncertainty limit, this agrees with the Strassmeier et al. (2012) value of ${{T}_{{\rm eff}}}$ = 4775 ± 150 K. The calculated flux ratio, ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.055 ± 0.022, also matches the flux ratio of ${{{\rm F}}_{B}}/{{{\rm F}}_{A}}$ = 0.069, quoted by Strassmeier et al. (2012).

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KOI-54 is a non-eclipsing, SB2 binary system. The two stars are in an almost face-on, highly eccentric orbit (Welsh et al. 2011).

This example shows the ability to detect the presence of both stars even when their effective temperatures are outside the range of our library and above the highest SpecMatch library spectrum temperature used to construct the residual ${{\chi }^{2}}$ panels. The primary star has a temperature ${{T}_{{\rm eff}}}$= 8800 ± 200 K, and the companion lags behind this value only slightly with ${{T}_{{\rm eff}}}$ = 8500 ± 200 K (Welsh et al. 2011). This is over 2000 K higher than any of our library spectra.

Nonetheless, we are able to see the minimum in the ${{\chi }^{2}}$ function shown in Figure 20 due to each of the two stars, as well as detect a clear minimum indicating the presence of the companion after the primary star was subtracted, as seen in Figure 21. We do not state the parameters obtained for KOI-54 system, because they are incorrect due to the large discrepancy between the stellar spectra for the KOI-54 stars and the available library spectra; more on this issue is explained in Section 6.2.

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HD 16702 is identified as a binary star in Díaz et al. (2012), with the mass ratio for the companion and the primary star being ${{{\rm M}}_{B}}/{{{\rm M}}_{A}}$ = 0.35 − 0.41 (Díaz et al. 2012). We detect a secondary star at ${\rm \ \Delta\ RV}$ = −5 ${\rm km}\;{{{\rm s}}^{-1}}$, with an estimated ${{T}_{{\rm eff}}}$ = 3500 ± 250 K and contributing 1.06% of the total flux, as shown in Figures 22 and 23.

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Since the only known companion parameter is its mass, we performed some analysis to confirm that our detection agrees with Díaz et al. (2012) value of 0.40 ± 0.05 ${{M}_{\odot }}$. The spectrum was taken when the velocity of both stars was almost exclusively along the line of sight, so we can predict the mass of the companion based on their radial velocity separation. Using nine precise RV measurements (Marcy & Butler 1992), we determine that the primary star velocity relative to the center of mass of the binary system is at its maximum at the time of observation, v_p = −2.2 ${\rm km}\;{{{\rm s}}^{-1}}$. Using the primary mass of 0.98 ${{M}_{\odot }}$ (Díaz et al. 2012), and ${\rm \ \Delta\ RV}$ of −5 ${\rm km}\;{{{\rm s}}^{-1}}$, as determined by our algorithm, we predict the mass ratio ${{{\rm M}}_{B}}/{{{\rm M}}_{A}}$= 0.30. This indicates a companion mass of ∼0.3 ${\rm km}\;{{{\rm s}}^{-1}}$, which is only slightly below the Díaz et al. (2012) value. Using known calibrations, both the flux ratio of ∼0.01 and the companion temperature of ∼3500 K are consistent with a star whose mass is approximately 0.4 ${{M}_{\odot }}$.

KOI-1361—Also known as Kepler-61, it has one confirmed planet (Ballard et al. 2013). We determined the primary star temperature to be ${{T}_{{\rm eff}}}$ = 4265 ± 145, which is ∼100 K larger than Ballard et al. (2013). We identified an even cooler companion, with ${{T}_{{\rm eff}}}$ = 3600 ± 130 K, at a radial velocity of +40 ${\rm km}\;{{{\rm s}}^{-1}}$ relative to the primary star. Ballard et al. (2013) reports a companion $0.$94 in North of the primary star, detected using adaptive optics. The magnitude difference between the primary star and the companion, as reported by Ballard et al. (2013), is 2.98 mag in Kepler bandpass. This corresponds to the flux ratio of ${{{\rm F}}_{A}}/{{{\rm F}}_{B}}$ = 0.064, which is only slightly above the value deduced by our algorithm, ${{{\rm F}}_{A}}/{{{\rm F}}_{B}}$ = 0.022 ± 0.003. Some of this difference can be attributed to the difference in the Kepler bandpass regions as compared to the regions used in our algorithm. It is also possible that the spectrometer slit was not necessarily aligned to capture all of the light from the companion, thus the flux ratio is correspondingly lower.

KOI-5—The observed RV trend indicates the possibility of a stellar companion (Wang et al. 2014). Low ${\rm \ \Delta\ RV}$ of +11 ${\rm km}\;{{{\rm s}}^{-1}}$ might have caused some of the secondary absorption lines to be subtracted away with the primary star, thus the quoted flux ratio of 0.066 ± 0.020 is a lower limit rather than an actual value. We analyzed several different observations of the same star, and the RV separation varied with observation time. This indicates that the secondary and the primary star form a bound system. The final companion plot is shown in Figure A1.

KOI-354—Detection of the secondary star to the KOI-354 is marginal, thus we do not list it in Table 9. The system consists of a 5936 ± 102 primary star (cfop.ipac.caltech.edu) and an M dwarf secondary with ${{T}_{{\rm eff}}}$ = 3500 ± 250 K. The secondary star has a low relative radial velocity of +11 ${\rm km}\;{{{\rm s}}^{-1}}$, and thus the flux ratio of ${{{\rm F}}_{A}}/{{{\rm F}}_{B}}$ = 0.005 ± 001 is a lower limit. The secondary star plot for KOI-354 is shown in Figure A4.

KOI-1613—Due to a low RV separation of +10 ${\rm km}\;{{{\rm s}}^{-1}}$, the flux ratio of 0.044 ± 0.013 is a lower limit. Law et al. (2013) detected a secondary star at a separation of $0.$22 in, which is 1.3 mag fainter than the primary star. A 1.3 magnitude difference indicates a flux ratio of 0.30, which is much higher than our lower limit. Since they observed the system in a different bandpass (roughly 7000–8000$\overset{\circ}{A}$), this flux discrepancy could be due to a different flux ratios in different wavelength regions. However, we suspect that most of the discrepancy can be attributed to the low ${\rm \ \Delta\ RV}$ separation of the two stars of similar spectral types. At 10 ${\rm \ \Delta\ RV}$ = ${\rm km}\;{{{\rm s}}^{-1}}$, the two sets of absorption lines are barely resolved, and most likely a large fraction of the secondary star was subtracted away together with the primary, leaving only a weak signal in the residuals. The final secondary star plot is shown in Figure A18.

KOI-2059—M dwarf secondary star has a small RV separation of only +5 ${\rm km}\;{{{\rm s}}^{-1}}$, making the flux ratio of 0.016 ± 0.008 a lower limit. Nevertheless, due to a significantly different secondary spectral type of the secondary compared to the primary with ${{T}_{{\rm eff}}}$ of 5996 ± 103 K (cfop.ipac.caltech.edu), we do not expect the actual flux ratio to deviate significantly from this lower limit. Using AO, a companion was detected at a separation of $0.$38 in, 1.1 mag fainter (Law et al. 2013). A magnitude difference of 1.1 predicts the flux ratio of about 0.36, which is again larger than the ratio predicted by our algorithm. Again, since the Law et al. (2013) observation was made in the near-infrared region, where the flux of the M dwarfs is considerably larger, we do expect that the flux ratio of an M dwarf and a ∼5000 K primary star is higher as compared to the ratio in the V and R bandpasses. Therefore, our flux ratio is still consistent with the Law et al. (2013) value, especially when accounting for the fact that the ratio predicted by our algorithm as a lower limit. The final secondary star plot is shown in Figure A23.

Here we provide notes about selected KOIs that have one planet candidate, and an indication from our spectroscopic analysis of a secondary set of absorption lines in the spectrum. A complete listing of all such detections is in Table 9.

KOI-151—Due to a low ${\rm \ \Delta\ RV}$ separation of +14 ${\rm km}\;{{{\rm s}}^{-1}}$, the flux ratio of 0.012 ± 0.006 represents a lower limit for KOI-151 and its secondary star. The final secondary star plot is shown in Figure A2.

KOI-219—Both the flux ratio of 0.330 ± 0.033 as well as the secondary star ${{T}_{{\rm eff}}}$ of 6000 ± 100 K represent lower limits due to a restricted ${{T}_{{\rm eff}}}$ library range and low RV separation, respectively. The final secondary star plot is shown in Figure A3.

KOI-652—KOI-652 is at least a triple star system, with a possible marginal detection of the fourth star. All three companions are M dwarfs, with each contributing only a few percent to the total flux. The final companion plot is shown in Figure A5. In Table 9, we do not list the faintest companion at −44 ${\rm km}\;{{{\rm s}}^{-1}}$, because its detection is only marginal.

KOI-698—This system is analyzed in Santerne (2012). Assuming a main sequence companion, our ${{T}_{{\rm eff}}}$ estimate for the secondary star of 4800 ± 600 K indicates a secondary mass of 0.7–0.8 ${{M}_{\odot }}$, which is below the predicted value of 1.1 ± 0.1 ${{M}_{\odot }}$ by Santerne (2012). The final secondary star plot is shown in Figure A6.

KOI-716—The detection of KOI-716 is marginal, and we do not include it in the Table 9. The KOI-716 secondary star is extremely faint and at the threshold of detection, with a flux ratio of ${{{\rm F}}_{A}}/{{{\rm F}}_{B}}$ = 0.004 ± 0.001. The primary star is a ${{T}_{{\rm eff}}}$ = 6096 ± 150 K (cfop.ipac.caltech.edu), and the secondary star is an M dwarf with ${{T}_{{\rm eff}}}$ = 3500 ± 250 K. Due to a relatively low RV separation of $+18\;{\rm km}\;{{{\rm s}}^{-1}}$, the flux ratio might be underestimated, yet we do not expect the secondary star contribution to be larger than 1% of the total flux. KOI-716 has one planetary candidate, and the secondary star plot is shown in Figure A7.

KOI-984—AO imaging of KOI-984 revealed a companion of approximately equal magnitude 1.5 arcsec apart (Law et al. 2013). Our algorithm, however, detects a marginal secondary star at ${\rm \ \Delta\ RV}$ = $+33\;{\rm km}\;{{{\rm s}}^{-1}}$ with ${{T}_{{\rm eff}}}$ = 3500 ± 150 K, contributing less than 1% to the total flux. This suggests that only a small fraction of the light from a secondary star star actually entered the slit, and while we can confirm the possibility of the companion, its estimated parameters are highly inaccurate. The final secondary star plot is shown in Figure A9. Since the detection is marginal, we do not list it in Table 9.

KOI-1020—The existence of the companion already speculated by (Hirano et al. 2012). The final secondary star plot is shown in Figure A10.

KOI-1137—Despite low RV separation of +11 ${\rm km}\;{{{\rm s}}^{-1}}$, the two stars are of sufficiently distinct spectral types that we do not expect the flux ratio to be significantly higher than the calculated lower limit of 0.020 ± 0.004. The final secondary star plot is shown in Figure A12.

KOI-1452—The primary star has ${{T}_{{\rm eff}}}$ = 7172 ± 211 K (cfop.ipac.caltech.edu), which is outside our library range. While the binarity of the spectrum is clear, the parameters for the secondary star cannot be determined accurately enough to be published. This is mostly due to a high rotational broadening of the secondary absorption lines, indicating a secondary star with a spectral type outside the range of our SpecMatch library. The final secondary star plot is shown in Figure A17.

KOI-1784—Due to a low RV separation of −13 ${\rm km}\;{{{\rm s}}^{-1}}$, the flux ratio of 0.192 ± 0.058 represents a lower limit. The final secondary star plot is shown in Figure A21.

KOI-2075—The secondary star is faint, contributing less than 1% to the total flux. While the flux ratio of 0.008 ± 0.004 is a lower limit due to a low RV separation of −13 ${\rm km}\;{{{\rm s}}^{-1}}$, we do not expect that a large fraction of the secondary absorption lines got subtracted away together with the primary star, as the two spectral types differ significantly from each other. The final secondary star plot is shown in Figure A24.

KOI-2215—This is a triple-lined spectrum, with both secondary and tertiary stars clearly detectable. Both additional stars are treated separately, but the uncertainties on the parameters might be slightly higher due to the blending of spectral lines as the spectral types of all three stars are relatively similar. The flux ratio for the third star of 0.155 ± 0.047 is a lower limit due to a low RV separation of −15 ${\rm km}\;{{{\rm s}}^{-1}}$. The final secondary star plot is shown in Figure A25.

KOI-2457—The primary star effective temperature of 6728 ± 155 (cfop.ipac.caltech.edu), exceeds our temperature range of the SpecMatch library. This hinders our ability to determine accurate parameters for the secondary star at a low ${\rm \ \Delta\ RV}$ separation of +14 ${\rm km}\;{{{\rm s}}^{-1}}$. Nevertheless, our analysis indicates a secondary star with a lower limit flux ratio of 0.076 ± 0.023. The final secondary star plot is shown in Figure A27.

KOI-3471—The spectrum of KOI-3471 contains four stars, all but one of comparable brightness. We provide estimates for the parameters for each of the constituent stars, but expect larger errors due to significant blending of their spectral lines. Since all of the stars are contained within less than 80 ${\rm km}\;{{{\rm s}}^{-1}}$ ${\rm \ \Delta\ RV}$, we expect that the temperatures and the flux ratios for additional stars are affected by the presence of the other stars. The final plot is shown in Figure A41.

KOI-3573—Low RV separation of $-15\;{\rm km}\;{{{\rm s}}^{-1}}$, together with the similar spectral types of the two stars, suggest that the actual flux ratio is probably higher than the calculated lower limit of 0.109 $|pm$ 0.033. As the two sets of absorption lines overlap slightly, it is likely that a fraction of the secondary absorption lines got subtracted away together with the primary star, thus making the secondary star appear fainter. The final secondary star plot is shown in Figure A47.

KOI-3583—The secondary star ${{\chi }^{2}}$ minimum is significantly broadened, indicating rotational broadening of secondary star absorption lines. Thus, despite the RV separation of −23 ${\rm km}\;{{{\rm s}}^{-1}}$, there is a possibility for the blending of the two sets of absorption lines, and the calculated flux ratio of 0.109 ± 0.033 is a lower limit. The final secondary star plot is shown in Figure A48.

KOI-3606—Both the secondary and the primary ${{\chi }^{2}}$ minimum are broadened, indicating a significant rotational broadening of the spectral lines. As this is characteristic of hot stars, we assume that the actual ${{T}_{{\rm eff}}}$ for the secondary star is above the quoted lower limit of 6000 ± 100 K. The final secondary star plot is shown in Figure A51.

KOI-3721—The spectrum of KOI-3721 contained three sets of absorption lines, with the faintest star contributing only ∼1% to the total flux. The brighter secondary star ${{\chi }^{2}}$ is broad, indicating significant rotational broadening. This is consistent with the estimate of the secondary star ${{T}_{{\rm eff}}}$ above 6000 ± 100 K. The final plot is shown in Figure A52.

KOI-3782—The calculated flux ratio is a lower limit due to a low RV separation of $-7\;{\rm km}\;{{{\rm s}}^{-1}}$, but we do not expect a significant deviation from the stated flux ratio due to significantly different spectral types of the two stars. The final secondary star plot is shown in Figure A53.

KOI-4713—Due to similar spectral types of the two stars, as well as the small RV separation of −15${\rm km}\;{{{\rm s}}^{-1}}$, the calculated flux ratio of 0.344 ± 103 is a lower limit. The final secondary star plot is shown in Figure A60.

KOI-4871—Despite the RV separation of −23 ${\rm km}\;{{{\rm s}}^{-1}}$, we state the flux ratio of 0.012 ± 0.003 as the lower limit due to the similarity of the primary and secondary star spectral type. This similarity allows for a chance of overlap of some of the absorption lines, and some of the secondary absorption lines might have been subtracted away with the primary star. The final secondary star plot is shown in Figure A61.

KOI-1152—KOI-1152 has a known companion that was previously detected using AO (Law et al. 2013). The planet candidate was identified as a false positive (cfop.ipac.caltech.edu). We analyzed several different observations of the same star, and the RV separation varied with different observation times. This indicates that the secondary and the primary star form a bound system. The final secondary star plot is shown in Figure A13.

KOI-3000—The flux ratio of 0.055 ± 0.019 is a lower limit rather than an actual value due to a low RV separation of +10 ${\rm km}\;{{{\rm s}}^{-1}}$. Nevertheless, the two stars have sufficiently different spectral types that we do not expect a significant departure from the stated flux ratio. The final secondary plot is shown in Figure A32.

KOI-3035—Despite moderate RV separation of +20 ${\rm km}\;{{{\rm s}}^{-1}}$, both sets of spectral lines are rotationally broadened, causing the two sets of absorption lines to overlap slightly. Therefore, the flux ratio of 0.212 ± 0.064 is a lower limit rather than an actual value. The final secondary star plot is shown in Figure A34.

KOI-3162—The flux ratio of 0.553 ± 0.166 is a lower limit due to a low RV separation of −13 ${\rm km}\;{{{\rm s}}^{-1}}$. The final secondary star plot is shown in Figure A36.

KOI-3216—Flux ratio of 0.326 ± 0.033 is a lower limit due to a low RV separation of +13 ${\rm km}\;{{{\rm s}}^{-1}}$. The final secondary star plot is shown in Figure A37.

KOI-4355—The secondary star to KOI-4355 exhibited a broad minimum, indicating significant rotational broadening. Due to shallower spectral lines, the uncertainties on the secondary star parameters might be somewhat larger. The final secondary star plot is shown in Figure A58.

KOI-4457—The calculated flux ratio of 0.012 ± 0.003 is a lower limit due to a low RV separation of −15 ${\rm km}\;{{{\rm s}}^{-1}}$, but we do not expect significant deviation from the stated value due to the significantly different spectral types of the two stars. The final secondary star plot is shown in Figure A59.