APOGEE/Kepler Overlap Yields Orbital Solutions for a Variety of Eclipsing Binaries

TL;DR

This paper presents a new methodology combining Kepler light curves and APOGEE spectra to derive orbital solutions for eclipsing binaries, revealing detailed system parameters and evolutionary insights.

Contribution

The study introduces a simultaneous analysis technique for Kepler and APOGEE data, improving radial velocity precision and enabling the study of tertiary companions in eclipsing binaries.

Findings

Precise orbital parameters for seven eclipsing binaries.

Identification of tertiary companions in three systems.

Estimation of stellar ages and evolutionary histories.

Abstract

Spectroscopic Eclipsing Binaries (SEBs) are fundamental benchmarks in stellar astrophysics and today are observed in breathtaking detail by missions like TESS, Kepler, and APOGEE. We develop a methodology for simultaneous analysis of high precision Kepler light curves and high resolution near-IR spectra from APOGEE and present orbital solutions and evolutionary histories for a subset of SEBs within this overlap. Radial velocities extracted from APOGEE spectra using the Broadening Function technique are combined with Kepler light curves and to yield binary orbital solutions. The Broadening Function approach yields more precise radial velocities than the standard Cross-Correlation Function, which in turn yields more precise orbital parameters and enables the identification of tertiary stars. The orbital periods of these seven SEBs range from 4 to 40 days. Four of the systems (KIC 5285607, KIC 6864859, KIC 6778289, and KIC 4285087) are well-detached binaries. The remaining three systems have apparent tertiary companions, but each exhibits two eclipses along with at least one spectroscopically varying component (KIC 6449358, KIC 6131659, and KIC 6781535). Gaia distances are available for four targets which we use to estimate temperatures of both members of these SEBs. We explore evolutionary histories in H-R diagram space and estimate ages for this subset of our sample. Finally, we consider the implications for the formation pathways of close binary systems via interactions with tertiary companions. Our methodology combined with the era of big data and observation overlap opens up the possibility of discovering and analyzing large numbers of diverse SEBs, including those with high flux ratios and those in triple systems.

Tables

Table 1. Table 2: Promising SEBs observed by APOGEE and Kepler, sorted by Kepler magnitude ( K p subscript 𝐾 𝑝 K_{p} )

KIC	APOGEE ID	Visits	$K_p$	$P_{\text{orb}}$ (day)	SE Depth (frac)	Morphology	Reference	Notes
9246715	2M20034832+4536148	2	10.08	171.28	0.1124	0.11	Rawls et al. (2016)
2708156	2M19302686+4318185	3	10.67	1.89	0.0625	0.57		Only 3 visits
3120320	2M19291007+3817041	3	11.28	10.27	0.0127	0.14		Kepler APOGEE/EB WG
4851217	2M19432016+3957081	6	11.32	2.47	0.1815	0.58		Low S/N ratio
3439031	2M19203184+3830492	3	11.50	5.95	0.4156	0.33		Kepler APOGEE/EB WG
5285607	2M19390532+4027346	6	11.69	3.90	0.0403	0.36	This work
6449358	2M19353513+4149543	25	11.72	5.78	0.0120	0.31	This work
10206340	2M19245882+4714573	3	11.78	4.56	0.2431	0.61		Only 3 visits
6864859	2M19292405+4223363	25	11.93	40.88	0.2426	0.06	This work
4931073	2M19351913+4001522	6	12.18	26.95	0.0564	0.08		Kepler APOGEE/EB WG
3127817	2M19355993+3813561	6	12.24	4.33	0.0512	0.48		Kepler APOGEE/EB WG
3335816	2M19184759+3824238	3	12.40	7.42	0.0106	0.16		Kepler APOGEE/EB WG
5284133	2M19373173+4027078	6	12.50	8.78	0.0492	0.15		Future work
3542573	2M19232622+3838017	3	12.61	6.94	0.0837	0.25		Only 3 visits
2711114	2M19240341+3758109	3	12.63	2.86	0.0022	0.29		Only 3 visits
4281895	2M19441242+3923418	6	12.76	9.54	0.0652	0.13		Only 3 visits
4660997	2M19340328+3942410	6	12.78	0.56	0.2527	0.62		Ellipsoidal variations
4473933	2M19363898+3933105	6	12.87	103.59	0.0126	0.25		Low S/N ratio
2305543	2M19280644+3736023	3	12.97	1.36	0.1052	0.50		Only 3 visits
3241619	2M19322278+3821405	3	13.06	1.70	0.1625	0.44		Only 3 visits
4285087	2M19463571+3919069	6	13.19	4.49	0.2408	0.31	This work
2576692	2M19263432+3748513	3	13.19	87.88	0.2588	0.04		Kepler APOGEE/EB WG
6131659	2M19370697+4126128	27	13.20	17.53	0.1036	0.09	This work
4847832	2M19401839+3957298	6	13.20	30.96	0.3200	0.08		Kepler APOGEE/EB WG
5025294	2M19414825+4010323	6	13.27	5.46	0.0010	0.18		Future work
6778289	2M19282456+4215080	25	13.31	30.13	0.1619	0.11	This work
3248332	2M19383951+3819588	6	13.37	7.36	0.0974	0.20		Kepler APOGEE/EB WG
6610219	2M19320615+4200049	25	13.58	11.30	0.2899	0.20		Low S/N ratio
6781535	2M19321788+4216489	25	14.14	9.12	0.0305	0.12	This work
4077442	2M19452193+3908260	6	14.35	0.69	0.0703	0.59		Kepler APOGEE/EB WG
3247294	2M19374558+3822510	6	14.35	67.42	0.1032	0.02		Kepler APOGEE/EB WG
3848919	2M19241352+3858278	3	14.48	1.05	0.3418	0.57		Low S/N ratio
5460835	2M19411125+4039416	6	14.72	21.54	0.0228	0.06		Low S/N ratio
4075064	2M19432862+3908535	6	15.71	61.42	0.0821	0.00		Low S/N ratio

Table 2. Table 3: Binary Orbital and Stellar Parameters

	KIC 5285607aaBroad Gaussian prior on flux ratio used based on BF results ($\sigma =20$% BF value).	KIC 6864859	KIC 6778289	KIC 6449358	KIC 4285087	KIC 6131659	KIC 6781535aaBroad Gaussian prior on flux ratio used based on BF results ($\sigma =20$% BF value).
$f_M$bb$f_M$ is the mass function as defined in §2.4; instead of fitting directly for $M_1$ and $M_2$, we used $f_M$ for KIC 6449358 because it is a single-lined spectroscopic binary.	$\cdots$	$\cdots$	$\cdots$	${0.126}_{-0.002}^{+0.002}$	$\cdots$	$\cdots$	$\cdots$
$M_1\left(M_{\odot }\right)$	${1.557}_{-0.035}^{+0.038}$	${1.411}_{-0.028}^{+0.028}$	${1.512}_{-0.022}^{+0.022}$	$\cdots$	${1.135}_{-0.014}^{+0.013}$	${0.942}_{-0.010}^{+0.010}$	${1.003}_{-0.038}^{+0.039}$
$M_2\left(M_{\odot }\right)$	${1.346}_{-0.033}^{+0.033}$	${1.354}_{-0.028}^{+0.028}$	${1.092}_{-0.018}^{+0.019}$	$\cdots$	${1.101}_{-0.014}^{+0.013}$	${0.703}_{-0.008}^{+0.008}$	${1.034}_{-0.040}^{+0.040}$
$R_1\left(R_{\odot }\right)$	${2.003}_{-0.054}^{+0.062}$	${1.655}_{-0.013}^{+0.012}$	${1.748}_{-0.009}^{+0.009}$	${2.1254}_{-0.0006}^{+0.0007}$	${1.033}_{-0.012}^{+0.010}$	${0.908}_{-0.003}^{+0.003}$	${1.199}_{-0.065}^{+0.113}$
$R_2\left(R_{\odot }\right)$	${1.679}_{-0.087}^{+0.063}$	${1.455}_{-0.012}^{+0.012}$	${0.998}_{-0.005}^{+0.005}$	${0.6977}_{-0.0004}^{+0.0005}$	${1.026}_{-0.011}^{+0.011}$	${0.616}_{-0.003}^{+0.003}$	${1.434}_{-0.114}^{+0.066}$
$P_{\text{orb}}\left(\text{d}\right)$	${3.89940111}_{-4.5e-8}^{+4.4e-8}$	${40.8778425}_{-3.0e-7}^{+2.9e-7}$	${30.1301385}_{-3.5e-7}^{+3.5e-7}$	${5.77679431}_{-4.9e-8}^{+5.0e-8}$	${4.48603142}_{-5.4e-8}^{+5.1e-8}$	${17.52782741}_{-2.3e-7}^{+2.2e-7}$	${9.12208640}_{-2.4e-7}^{+2.5e-7}$
$t_{pe}\left(\text{d}\right)$ccThe fit zeropoint for the time of primary eclipse is in units of BKJD (BJD - 2454833). The primary eclipse is defined here as the deeper of the two. This differs from the KEBC (Kirk et al., 2016) primary eclipse definition for KIC 6864859 only because the two eclipses have very similar depths.	${126.577647}_{-9.7e-6}^{+1.1e-5}$	${158.3189733}_{-7.6e-6}^{+9.2e-6}$	${122.100819}_{-1.0e-5}^{+1.0e-5}$	${122.0739252}_{-7.2e-6}^{+7.1e-6}$	${133.450881}_{-1.1e-5}^{+1.1e-5}$	${127.042190}_{-1.1e-5}^{+1.1e-5}$	${138.838193}_{-6.6e-5}^{+4.8e-5}$
$i\left(\text{rad}\right)$	${1.3656}_{-0.0009}^{+0.0015}$	${1.5415}_{-0.0001}^{+0.0001}$	${1.55876}_{-0.00001}^{+0.00001}$	${1.5327}_{-0.0001}^{+0.0001}$	${1.5210}_{-0.0001}^{+0.0001}$	${1.5558}_{-0.0001}^{+0.0001}$	${1.4749}_{-0.0004}^{+0.0007}$
$e\sin \omega$	$1.3e-6_{-1.2e-4}^{+9.0e-4}$	$-{0.0254}_{-0.0005}^{+0.0006}$	$-{0.2337}_{-0.0004}^{+0.0004}$	${0.00060}_{-0.00040}^{+0.00046}$	$1.8e-4_{-3.0e-4}^{+1.4e-3}$	$-{0.0127}_{-0.0005}^{+0.0004}$	${0.0215}_{-0.0026}^{+0.0034}$
$e\cos \omega$	$-8.6e-7_{-4.1e-6}^{+3.5e-6}$	$-{0.634115}_{-0.00001}^{+0.00001}$	${0.051090}_{-0.000005}^{+0.000005}$	$-{0.00023}_{-1.1e-5}^{+1.0e-5}$	$-5.9e-5_{-3.8e-6}^{+3.7e-6}$	$7.1e-5_{-1.8e-6}^{+1.7e-6}$	${0.25005}_{-0.00001}^{+0.00001}$
$F_2/F_1$	${0.6579}_{-0.1025}^{+0.0959}$	${0.7256}_{-0.0124}^{+0.0139}$	${0.19155}_{-0.00002}^{+0.00002}$	${0.010752}_{-6.5e-6}^{+6.5e-6}$	${0.9455}_{-0.0356}^{+0.0396}$	${0.1480}_{-0.0010}^{+0.0010}$	${1.0654}_{-0.2384}^{+0.2195}$
$q_{1,1}$	${0.3690}_{-0.0157}^{+0.0158}$	${0.4313}_{-0.0176}^{+0.0158}$	${0.2930}_{-0.0146}^{+0.0146}$	${0.3570}_{-0.0099}^{+0.0082}$	${0.4632}_{-0.0413}^{+0.0369}$	${0.1961}_{-0.0039}^{+0.0042}$	${0.3786}_{-0.0366}^{+0.0389}$
$q_{1,2}$	${0.0335}_{-0.0242}^{+0.0407}$	${0.1113}_{-0.0112}^{+0.0121}$	${0.3112}_{-0.0169}^{+0.0179}$	${0.2961}_{-0.0068}^{+0.0085}$	${0.3046}_{-0.0460}^{+0.0518}$	${0.9959}_{-0.0068}^{+0.0031}$	${0.7549}_{-0.1037}^{+0.0938}$
$q_{2,1}$	${0.4027}_{-0.0710}^{+0.0249}$	${0.3662}_{-0.0259}^{+0.0269}$	${0.3991}_{-0.0260}^{+0.0265}$	${0.3436}_{-0.1270}^{+0.1530}$	${0.5470}_{-0.0509}^{+0.0680}$	${0.0424}_{-0.0070}^{+0.0076}$	${4056}_{-0.0553}^{+0.0711}$
$q_{2,2}$	${0.0905}_{-0.0579}^{+0.1343}$	${0.2377}_{-0.0213}^{+0.0226}$	${0.3220}_{-0.0306}^{+0.0329}$	${0.3189}_{-0.2208}^{+0.3426}$	${0.2275}_{-0.0381}^{+0.0427}$	${0.9380}_{-0.0923}^{+0.0459}$	${0.2689}_{-0.1750}^{+0.2181}$
$M_1+M_2\left(M_{\odot }\right)$	${2.903}_{-0.064}^{+0.068}$	${2.765}_{-0.052}^{+0.052}$	${2.604}_{-0.038}^{+0.038}$	$\cdots$	${2.236}_{-0.026}^{+0.025}$	${1.645}_{-0.017}^{+0.017}$	${2.040}_{-0.074}^{+0.073}$
$M_2/M_1$	${0.863}_{-0.013}^{+0.014}$	${0.960}_{-0.015}^{+0.015}$	${0.723}_{-0.009}^{+0.009}$	$\cdots$	${0.969}_{-0.008}^{+0.007}$	${0.746}_{-0.005}^{+0.005}$	${1.030}_{-0.025}^{+0.026}$
$R_1+R_2\left(R_{\odot }\right)$	${3.679}_{-0.033}^{+0.031}$	${3.110}_{-0.020}^{+0.019}$	${2.746}_{-0.013}^{+0.013}$	${2.8231}_{-0.0009}^{+0.0010}$	${2.060}_{-0.008}^{+0.008}$	${1.525}_{-0.005}^{+0.005}$	${2.632}_{-0.033}^{+0.034}$
$R_2/R_1$	${0.839}_{-0.067}^{+0.055}$	${0.879}_{-0.007}^{+0.008}$	${0.5708}_{-0.0003}^{+0.0003}$	${0.3283}_{-0.0002}^{+0.0002}$	${0.993}_{-0.018}^{+0.020}$	${0.679}_{-0.003}^{+0.003}$	${1.222}_{-0.153}^{+0.113}$
$b$ddThe impact parameter $b$ is defined as $a\cos i/R_1$, where $q_1^1,q_2^1$ and $q_1^2,q_2^2$ are triangularized quadratic limb darkening coefficients for star 1 and star 2 (see Section 2.4 for details). We use natural log for the systematic LC and RV error terms for fitting flexibility.	${1.5146}_{-0.0576}^{+0.0473}$	${1.2407}_{-0.0075}^{+0.0081}$	${0.3860}_{-0.0005}^{+0.0005}$	${0.3413}_{-0.0008}^{+0.0009}$	${0.7215}_{-0.0076}^{+0.0079}$	${0.5533}_{-0.0042}^{+0.0041}$	${1.9034}_{-0.1341}^{+0.0960}$
$e$	$6.7e-5_{-6.2e-5}^{+9.6e-4}$	${0.63462}_{-0.00001}^{+0.00001}$	${0.2392}_{-0.0004}^{+0.0004}$	${0.00064}_{-0.00033}^{+0.00045}$	$3.4e-4_{-2.6e-4}^{+1.3e-3}$	${0.0127}_{-0.0004}^{+0.0005}$	${0.2510}_{-0.0002}^{+0.0005}$
$\ln \left({\sigma }_{\text{LC, sys}}\right)$	$-{8.76}_{-0.01}^{+0.01}$	$-{8.84}_{-0.01}^{+0.01}$	$-{8.48}_{-0.01}^{+0.01}$	$-{8.419}_{-0.006}^{+0.006}$	$-{6.81}_{-0.01}^{+0.01}$	$-{7.54}_{-0.01}^{+0.01}$	$-{8.12}_{-0.01}^{+0.01}$
$k_0\left({\text{m s}}^{-1}\right)$	${61668.8}_{-488.9}^{+532.1}$	${93945.5}_{-142.6}^{+139.6}$	${72332.2}_{-212.4}^{+213.8}$	${68694.7}_{-147.0}^{+144.0}$	${91275.4}_{-256.1}^{+248.5}$	${89867.7}_{-125.2}^{+127.1}$	${44547.4}_{-627.5}^{+633.7}$
$\ln \left({\sigma }_{\text{RV, sys}}\left({\text{m s}}^{-1}\right)\right)$	${7.40}_{-0.23}^{+0.26}$	${6.54}_{-0.13}^{+0.14}$	${7.04}_{-0.11}^{+0.12}$	${6.27}_{-0.20}^{+0.23}$	${6.65}_{-0.23}^{+0.29}$	${6.82}_{-0.10}^{+0.10}$	${8.28}_{-0.11}^{+0.12}$

Table 3. Table 4: Parameters related to RV extraction and temperature estimates

	KIC 5285607	KIC 6864859	KIC 6778289	KIC 6449358	KIC 4285087	KIC 6131659	KIC 6781535
BF Flux Ratio ($F_2/F_1$)	$0.620\pm 0.027$	$0.811\pm 0.028$	$0.462\pm 0.029$	$0.392\pm 0.014$	$0.997\pm 0.023$	$0.648\pm 0.028$	$1.253\pm 0.115$
ASPCAP $T_{\text{eff}}$ (K)aaDR14 (Pérez et al., 2016)	$6495\pm 156$	$6417\pm 159$	$6572\pm 162$	$6237\pm 179$	$5664\pm 146$	$4845\pm 98$	$5749\pm 125$
Gaia parallax (mas)bbBailer-Jones et al. (2018)	$1.254\pm 0.0216$	$1.4897\pm 0.0241$	$0.9093\pm 0.0222$	$1.1974\pm 0.0264$	$1.619\pm 0.0312$	$-0.5117\pm 1.0713$	$\cdots$
Gaia distance (pc)bbBailer-Jones et al. (2018)	$799\pm 14$	$671\pm 11$	$1100\pm 27$	$835\pm 18$	$617\pm 12$	$\cdots$	$\cdots$
$\log g_1$ (cgs)ccComputed directly from $M$ and $R$ as reported in Table 3.	$4.028\pm 0.013$	$4.150\pm 0.010$	$4.133\pm 0.008$	$\cdots$	$4.454\pm 0.006$	$4.496\pm 0.005$	$4.161\pm 0.019$
$\log g_2$ (cgs)ccComputed directly from $M$ and $R$ as reported in Table 3.	$4.118\pm 0.015$	$4.244\pm 0.012$	$4.479\pm 0.010$	$\cdots$	$4.478\pm 0.007$	$4.705\pm 0.006$	$4.253\pm 0.021$
$T_{\text{eff}}$ offset (K)ddEl-Badry et al. (2017)	$350$	$80$	$250$	$\cdots$	$20$	$350$	$100$
Adopted $T_{\text{eff},1}$ (K)	$6845\pm 328$	$6497\pm 159$	$6822\pm 162$	$6737\pm 178$	$5689\pm 146$	$5195\pm 98$	$5849\pm 125$
Adopted $T_{\text{eff},2}$ (K)	$6716\pm 293$	$6541\pm 283$	$7265\pm 440$	$8788\pm 658$	$5735\pm 105$	$\cdots$	$\cdots$

Table 4. Table 5: Measured radial velocities from APOGEE spectra

Time (BJD)	Orbital Phase	$v_1$ ($\mathrm{km}{s}^{-1}$)	$v_2$ ($\mathrm{km}{s}^{-1}$)
KIC 5285607
2455813.69967	0.040	$41.404\pm 0.101$	$86.618\pm 0.173$
2455823.72647	0.611	$118.406\pm 0.061$	$-3.961\pm 0.108$
2455840.66112	0.954	$88.218\pm 0.067$	$31.481\pm 0.109$
2455849.57835	0.241	$-24.927\pm 0.060$	$162.515\pm 0.105$
2455851.64874	0.772	$148.087\pm 0.060$	$\cdots$
2455866.56945	0.598	$113.109\pm 0.063$	$2.025\pm 0.114$
KIC 6864859
2456557.73263	0.194	$107.492\pm 0.058$	$\cdots$
2456559.72256	0.243	$110.939\pm 0.079$	$78.670\pm 0.069$
2456560.72029	0.267	$112.129\pm 0.062$	$77.683\pm 0.056$
2456584.63147	0.852	$74.820\pm 0.060$	$117.933\pm 0.063$
2456585.62998	0.877	$57.225\pm 0.064$	$133.366\pm 0.052$
2456760.90512	0.164	$106.432\pm 0.072$	$80.051\pm 0.077$
2456761.87222	0.188	$108.904\pm 0.073$	$78.499\pm 0.079$
2456762.86801	0.213	$109.800\pm 0.057$	$77.234\pm 0.048$
2456763.88053	0.237	$111.281\pm 0.053$	$76.175\pm 0.054$
2456787.80872	0.823	$81.571\pm 0.078$	$104.978\pm 0.091$
2456788.84246	0.848	$76.011\pm 0.053$	$116.930\pm 0.062$
2456812.75131	0.433	$114.944\pm 0.053$	$73.779\pm 0.051$
2456814.75480	0.482	$114.257\pm 0.049$	$73.825\pm 0.050$
2456815.78485	0.507	$113.825\pm 0.053$	$73.899\pm 0.049$
2456816.76560	0.531	$113.034\pm 0.061$	$73.480\pm 0.061$
2456817.76131	0.555	$112.440\pm 0.056$	$74.591\pm 0.048$
2456818.76390	0.580	$111.523\pm 0.056$	$75.498\pm 0.054$
2456819.76154	0.604	$110.927\pm 0.058$	$75.851\pm 0.053$
2456820.75533	0.629	$109.049\pm 0.059$	$75.654\pm 0.066$
KIC 6778289
2456557.73261	0.635	$47.229\pm 0.045$	$107.444\pm 0.045$
2456559.72254	0.701	$38.941\pm 0.049$	$117.686\pm 0.054$
2456560.72027	0.734	$36.535\pm 0.046$	$122.226\pm 0.053$
2456584.63145	0.528	$60.403\pm 0.059$	$87.443\pm 0.058$
2456585.62996	0.561	$57.486\pm 0.047$	$94.149\pm 0.053$
2456757.89237	0.278	$101.533\pm 0.048$	$31.407\pm 0.053$
2456760.90514	0.378	$85.993\pm 0.051$	$54.705\pm 0.056$
2456762.86803	0.443	$72.743\pm 0.049$	$\cdots$
2456763.88055	0.477	$72.718\pm 0.052$	$\cdots$
2456783.83502	0.139	$115.038\pm 0.049$	$12.937\pm 0.055$
2456784.82136	0.172	$113.771\pm 0.046$	$16.022\pm 0.055$
2456785.82484	0.205	$111.263\pm 0.044$	$18.581\pm 0.051$
2456786.79785	0.237	$108.112\pm 0.045$	$22.629\pm 0.048$
2456787.80874	0.271	$102.199\pm 0.047$	$29.240\pm 0.050$
2456788.84248	0.305	$98.364\pm 0.046$	$36.995\pm 0.044$
2456814.75483	0.165	$113.882\pm 0.048$	$15.109\pm 0.053$
2456815.78487	0.200	$111.327\pm 0.047$	$17.508\pm 0.052$
2456816.76563	0.232	$107.622\pm 0.050$	$21.517\pm 0.052$
2456818.76392	0.298	$97.639\pm 0.048$	$35.617\pm 0.053$
2456819.76156	0.331	$93.260\pm 0.046$	$42.669\pm 0.050$
2456820.75535	0.364	$87.120\pm 0.048$	$50.657\pm 0.054$
KIC 6449358
2456557.73275	0.431	$47.972\pm 0.092$	$\cdots$
2456559.72268	0.775	$116.332\pm 0.066$	$\cdots$
2456584.63158	0.087	$44.828\pm 0.067$	$\cdots$
2456585.63008	0.260	$20.640\pm 0.056$	$\cdots$
2456757.89224	0.080	$46.407\pm 0.069$	$\cdots$
2456760.90501	0.601	$96.766\pm 0.090$	$\cdots$
2456761.87212	0.769	$116.275\pm 0.066$	$\cdots$
2456763.88043	0.116	$36.021\pm 0.056$	$\cdots$
2456784.82126	0.741	$115.987\pm 0.057$	$\cdots$
2456787.80865	0.258	$20.638\pm 0.057$	$\cdots$
2456815.78483	0.101	$40.477\pm 0.069$	$\cdots$
2456816.76558	0.271	$21.538\pm 0.066$	$\cdots$
2456818.76389	0.617	$101.418\pm 0.065$	$\cdots$
2456819.76152	0.790	$114.484\pm 0.060$	$\cdots$
KIC 4285087
2455813.69984	0.864	$154.531\pm 0.062$	$25.294\pm 0.060$
2455823.72663	0.099	$43.398\pm 0.060$	$140.891\pm 0.056$
2455840.66127	0.874	$150.887\pm 0.057$	$31.327\pm 0.058$
2455849.57849	0.862	$154.496\pm 0.056$	$25.174\pm 0.057$
2455851.64888	0.323	$17.397\pm 0.057$	$167.332\pm 0.062$
2455866.56955	0.649	$158.478\pm 0.060$	$22.212\pm 0.057$
KIC 6131659
2456368.99876	0.384	$62.544\pm 0.048$	$125.604\pm 0.091$
2456411.91961	0.833	$125.989\pm 0.045$	$41.382\pm 0.070$
2456557.73279	0.152	$55.739\pm 0.042$	$134.879\pm 0.069$
2456559.72271	0.265	$49.076\pm 0.041$	$145.012\pm 0.076$
2456560.72045	0.322	$52.988\pm 0.043$	$140.191\pm 0.075$
2456584.63160	0.686	$128.199\pm 0.045$	$39.531\pm 0.064$
2456585.63010	0.743	$131.456\pm 0.045$	$35.183\pm 0.075$
2456757.89221	0.571	$105.771\pm 0.047$	$67.116\pm 0.077$
2456758.90157	0.629	$120.004\pm 0.052$	$48.763\pm 0.084$
2456760.90499	0.743	$131.105\pm 0.044$	$35.406\pm 0.092$
2456761.87209	0.798	$129.464\pm 0.048$	$37.476\pm 0.080$
2456762.86788	0.855	$122.805\pm 0.042$	$45.003\pm 0.093$
2456763.88040	0.913	$111.491\pm 0.099$	$61.304\pm 0.105$
2456783.83490	0.051	$78.027\pm 0.051$	$105.677\pm 0.086$
2456784.82125	0.108	$63.945\pm 0.053$	$124.186\pm 0.111$
2456785.82473	0.165	$54.481\pm 0.042$	$137.809\pm 0.089$
2456786.79774	0.220	$49.620\pm 0.044$	$144.068\pm 0.089$
2456787.80864	0.278	$49.600\pm 0.040$	$144.493\pm 0.078$
2456788.84238	0.337	$54.269\pm 0.041$	$137.917\pm 0.082$
2456812.75129	0.701	$129.389\pm 0.044$	$36.993\pm 0.089$
2456814.75479	0.815	$127.626\pm 0.044$	$38.646\pm 0.085$
2456815.78484	0.874	$119.797\pm 0.047$	$50.101\pm 0.083$
2456816.76560	0.930	$105.694\pm 0.045$	$68.143\pm 0.089$
2456817.76131	0.987	$91.467\pm 0.048$	$\cdots$
2456818.76390	0.044	$79.392\pm 0.046$	$\cdots$
2456819.76154	0.101	$65.338\pm 0.047$	$122.387\pm 0.086$
2456820.75533	0.158	$55.054\pm 0.040$	$136.794\pm 0.088$
KIC 6781535
2456557.73097	0.853	$128.940\pm 0.057$	$-37.069\pm 0.049$
2456559.72097	0.071	$29.463\pm 0.109$	$69.329\pm 0.204$
2456560.71874	0.180	$1.983\pm 0.057$	$85.464\pm 0.045$
2456584.63091	0.801	$122.687\pm 0.060$	$-30.426\pm 0.052$
2456585.62946	0.911	$109.784\pm 0.057$	$-20.436\pm 0.047$
2456757.89316	0.795	$119.307\pm 0.060$	$-29.253\pm 0.052$
2456760.90580	0.125	$12.366\pm 0.076$	$75.156\pm 0.060$
2456762.86860	0.340	$-4.835\pm 0.064$	$92.365\pm 0.049$
2456763.88108	0.451	$4.541\pm 0.060$	$82.610\pm 0.049$
2456783.83465	0.639	$\cdots$	$43.811\pm 0.041$
2456784.82095	0.747	$100.401\pm 0.057$	$-9.574\pm 0.049$
2456785.82438	0.857	$128.143\pm 0.053$	$-37.060\pm 0.046$
2456786.79735	0.964	$82.332\pm 0.064$	$7.094\pm 0.053$
2456787.80820	0.074	$23.165\pm 0.158$	$62.731\pm 0.119$
2456788.84189	0.188	$1.170\pm 0.047$	$86.509\pm 0.045$
2456812.74399	0.809	$124.321\pm 0.056$	$-33.597\pm 0.049$
2456814.75319	0.028	$\cdots$	$43.896\pm 0.046$
2456815.78320	0.141	$8.643\pm 0.068$	$79.192\pm 0.050$
2456817.75959	0.358	$-3.732\pm 0.059$	$92.410\pm 0.045$
2456818.76215	0.468	$7.142\pm 0.064$	$80.126\pm 0.049$
2456819.75975	0.577	$20.702\pm 0.354$	$57.626\pm 0.204$
2456820.75351	0.686	$73.353\pm 0.070$	$15.883\pm 0.061$

Equations

$$f_M=\frac{M_2^3{\sin }^{3}i}{(M_1+M_2{)}^2}.$$

$$F_{\text{sum}}=\frac{\sigma T_{\text{avg}}^4(R_1^2+R_2^2)}{d^2},$$

$$F_1=\frac{F_{\text{sum}}}{1+F_{\text{ratio}}}$$

$$F_2=F_{\text{sum}}-F_1.$$

$$T_{\text{avg}}=T_2\frac{F_{\text{ratio}}^{-1}{T}_{\text{ratio}}^{-1}+1}{F_{\text{ratio}}^{-1}+1},$$

$$T_1=\sqrt[4]{\frac{F_1{d}^2}{\sigma R_1^2}}$$

$$T_2=\sqrt[4]{\frac{F_2{d}^2}{\sigma R_2^2}}.$$

Full text

††software: Astropy (Astropy Collaboration et al., 2013; Price-Whelan et al., 2018), PyAstronomy (Czesla, 2018), Matplotlib (Hunter, 2007), Apogee (Bovy, 2016), Makecite (Price-Whelan et al., 2018), Numpy (Van Der Walt et al., 2011), Pandas (McKinney, 2010), Emcee (Foreman-Mackey et al., 2013), Gaussfitter (Ginsburg, 2017), Scipy (Jones et al., 2001)

1 Introduction

The Apache Point Observatory Galactic Evolution Experiment (APOGEE) is studying our Galaxy in fantastic detail by providing high resolution spectra for some 150,000 stars (Majewski et al., 2015). Some of these belong to double-lined spectroscopic eclipsing binaries (SEBs), and a further subset have been observed by the Kepler spacecraft (Borucki et al., 2010) and appear in the Kepler Eclipsing Binary Catalog (Kirk et al., 2016). These APOGEE/Kepler SEBs which have several APOGEE spectra at different epochs give a unique opportunity to combine the spectra with the Kepler light curve to model the binary orbit and directly measure fundamental stellar parameters, including mass and radius. They can then be used to explore and constrain stellar evolution, stellar populations, and orbital kinematics.

While much work has gone into exploring Kepler eclipsing binaries (EBs) as a population, fewer studies have maximally utilized complementary spectra to fully characterize these stellar systems. A notable exception is Matson et al. (2017), which found the radial velocities of 40 Kepler binaries, 35 of them double-lined and the remainder single-lined. Their work used medium resolution ground-based spectra, but the authors note that high resolution spectra is more optimal. In another example, Torres et al. (2018) used K2 light curves of the Ruprecht 147 cluster together with high resolution spectra and the cluster’s well-modeled metallicity to constrain the orbital parameters extracted from spectroscopic binary cluster members. In addition, Lehmann et al. (2012) analyzed the quadruple system KIC 4247791 by combining Kepler light curves and moderate resolution spectra.

Many studies have used APOGEE and Kepler data together, such as the APOKASC catalog (Pinsonneault et al., 2018, 2014) which combines APOGEE stellar parameters with Kepler asteroseismology. However, such works tend to ignore stellar multiplicity. The SEB overlap between APOGEE and Kepler in particular remains relatively unexplored. There is also synergy with the Kepler planet survey which identifies candidate planet systems, some of which are found to be eclipsing binaries, with or without tertiaries, with follow-up radial-velocity observations with APOGEE (Fleming et al., 2015). The frequency of binaries, with and without tertiary companions, is a necessary component of computing transiting exoplanet astrophysical false-alarm probabilities (e.g. Morton et al., 2016).

EBs have long been used as fundamental benchmarks for stellar astrophysics (e.g., Torres et al., 2010), including more recently as benchmarks for exoplanet properties (e.g., Stassun et al., 2017), to test asteroseismic inferences of stellar parameters (e.g., Gaulme et al., 2016), and even for assessing trigonometric parallaxes (Stassun & Torres, 2018). In addition, as they are often observed as SEBs, EBs are useful for assembling reliable statistics on the occurrence of higher order multiples (e.g., tertiary companions) and on the relationship of companion properties to the properties of the EB. For example, Tokovinin (1997) found the incidence of wide tertiaries to be strongly linked to the orbital period of the inner binary. Additional well-studied EBs can help to further test these relationships.

In this work, we identify 33 promising APOGEE/Kepler SEBs and compute full orbital solutions with a suite of stellar parameters for seven of them. In §2, we detail our sample selection, data processing, and modeling methodology. We further show how the Broadening Function technique is a superior method to extract multiple velocity components from APOGEE spectra. Subsequently §3 discusses each of the seven modeled systems in turn and presents orbital solutions. Finally, §4 places the SEBs in the context of each star’s stellar evolutionary history and explores the relationship of the EB orbital properties to the presence of tertiary companions.

2 Data and Methods

2.1 Sample Selection

We use the following criteria and filters to arrive at a candidate sample of promising SEBs in the APOGEE/Kepler overlap. We begin with the Kepler EB catalog compiled by Kirk et al. (2016). From this catalog we select targets which have both their primary and secondary eclipses observed by Kepler; this limits our selection to binaries with inclinations close to 90 degrees. We further require the light curve to be semi- or well-detached, with the morphology parameter significantly less than 1. Next, a luminosity limit of $H<14$ magnitudes was imposed, as fainter targets are unlikely to have $H$-band APOGEE spectra with a sufficiently high signal-to-noise ratio. We also require the targets to have multiple cross-correlation function (CCF) peaks from the APOGEE pipeline (Nidever et al., 2015) visible by eye in one epoch. Finally, the binaries must have been observed by APOGEE at least three times, and thus have at least three apVisit spectra, with no quality flags present.

Taken together, these criteria result in 33 candidates, which are listed in Table 2, plus one additional candidate which has already been analyzed (Rawls et al., 2016). Of these, we perform a detailed analysis of seven. Notes in Table 2 indicate why we choose to exclude the other systems at this time. Several are being investigated by the Kepler APOGEE EB Working Group, some have only three APOGEE visits which would make a good RV curve solution challenging without additional spectra, some have low signal-to-noise (S/N) ratios, one shows significant ellipsoidal variations which are not included in our photometric model, and two remain good candidates for future analyses.

2.2 Radial Velocities (RVs) from APOGEE

Spectra

The standard observing mode for APOGEE spectra has a total exposure time of roughly three hours, which is usually collected over a series of visits on different days. The visits are then combined into one spectrum per target (an apStar spectrum). We instead utilize individual visit spectra (apVisit), which are identified with their plate ID, date (MJD), and fiber ID. These may be retrieved from the SDSS Science Archive Server search tool with a simple search by APOGEE ID. We continuum normalize the visit spectra and then “de-spike” them to remove erroneous spectral features due to tellurics. De-spiking consists of identifying outliers above or below the continuum by 0.7 or 3 times the standard deviation of the normalized flux, respectively. The “below continuum” factor is larger to avoid unintentionally removing real absorption line features. Around each outlier spike, a $\pm 6$ Å window is also flagged for removal. The python scripts used to retrieve, continuum normalize, and de-spike apVisit spectra are publicly available on GitHub111https://github.com/mrawls/apVisitproc. They rely heavily on the apogee python package on GitHub described in Bovy (2016).

In the main APOGEE reduction pipeline (Nidever et al., 2015), RVs are measured using the CCF. In this approach, a template spectrum and a series of visit spectra for a given target are cross-correlated, giving the RV of the target star relative to the template.

The CCF method works because an observed stellar spectrum can be represented as a convolution of two functions: that of the astrophysical target (which includes “natural broadening” components such as thermal broadening, microturbulence effects, and instrumental broadening) and another function called the Broadening Function (BF) that contains the important RV information. The BF is formally presented in Rucinski (1992, 1999, 2002, 2004). Although the CCF method is very close to the real convolution that occurs in an APOGEE spectrum, cross-correlating a template spectrum with an observed stellar spectrum yields a function which inherits the natural broadening components present in both spectra. In this way the CCF is essentially a non-linear proxy of the BF. Therefore, instead of using the CCF, in this work we measure the BF directly.

To extract BFs from our target spectra, we use a modified version of the BF software suite from Rawls et al. (2016) which is based on the method introduced by Rucinski (1992). A PHOENIX BT-Settl model atmosphere spectrum (Husser et al., 2013) is selected to match the target’s approximate spectral parameters as reported by APOGEE. The match cannot be exact as the two stars in a binary may not have identical spectral types and the model grid has a finite sampling in stellar parameters. In general, a mismatch in spectral type between template and target causes the BF to change in its intensity scale and quality, but the amplitudes of the RV components remain unchanged (Lu et al., 2001).

We examine the BF peaks by eye to identify their approximate locations on the radial velocity axis and use a least-squares fitting procedure to fit one or more Gaussians to the BF. The location of each Gaussian’s mean is the RV, which we then correct with the barycentric velocity provided with each apVisit spectrum. Our reported RV uncertainties come from the error in fitting a Gaussian to each BF peak using least-squares. Much like in the APOGEE CCF pipeline (Nidever et al., 2015), it is ultimately the uncertainty in the measurement of the BF peak, which depends partially on its semi-arbitrary width, that determines the reported uncertainties for the RVs. This systematically underestimates the uncertainty of each RV measurement. The software used to extract RVs as described here is publicly available on GitHub.

In Figures 1 and 2, we demonstrate how the BF method produces significantly better separated peaks for APOGEE double-lined SEBs than the CCFs generated by the APOGEE pipeline. Due in part to the BF method having less of a “peak-pulling” effect, this more defined separation dramatically improves our ability to measure the RV of each component. We present the BF and measured RVs for each of our seven targets in Appendix A.

We also measure flux ratios in the APOGEE $H$ band with our BF peaks, as the ratio of the peak areas is directly proportional to the flux ratio of the binary (Bayless & Orosz, 2006; Stassun et al., 2007). These BF flux ratios will generally differ from Kepler-derived flux ratios because of the difference in wavelength; APOGEE is an $H$-band spectrograph and Kepler has a broad visible light bandpass. The BF flux ratios are discussed further in Section 3.8 alongside other variables for RV extraction and temperature estimation.

2.3 Kepler Light Curve Processing

We use minimally-processed Kepler long-cadence simple aperture photometry (SAP) to construct EB light curves for each target. Each light curve, and its uncertainties, is normalized by the median raw flux value of all available quarters. We de-weight data of poor quality by selecting times with Kepler quality flags > 0, and inflating the normalized flux uncertainties at these times by a factor of 10.

2.4 Orbital and Mass Solutions with KEBLAT

With RVs in hand, we turn to the photometric modeling of the Kepler light curves. We utilize a modular Python tool dubbed “KEBLAT” which is capable of separate or simultaneous modeling of the binary light curve, spectral energy distribution (SED), and RV time series (Windemuth et al., 2018). Here, we simultaneously model Kepler light curves and APOGEE RVs of each EB in our sample to determine orbital solutions ($P,\ e,\ \omega,\ i,\ t_{PE}$), stellar parameters ($m_{1},\ m_{2},\ r_{1},\ r_{2},\ \frac{F_{2}}{F_{1}}$), quadratic limb darkening coefficients under triangular reparameterization ($q_{1,1},\ q_{1,2},\ q_{2,1},\ q_{2,2}$; Kipping 2013), and systemic radial velocity $k_{0}$. For parameter sampling purposes, we transform individual mass and radius parameters to sums and ratios, and parameterize $e$ and $\omega$ as $e\cos\omega$ and $e\sin\omega$.

Given a system’s total mass, period, eccentricity, argument of periastron, inclination, and time of primary eclipse, KEBLAT uses a Keplerian solver to compute the instantaneous positions and velocities of each stellar component. The positions, along with specified sizes and relative flux of the stars are then used to determine the instantaneous light contribution during eclipse via a quadratic limb-darkening (Mandel & Agol, 2002) model for spherical bodies.222The assumption of spherical stars requires that the stars be sufficiently detached to avoid tidal and rotational distortions. We account for finite sampling effects (Kipping, 2010) on the light curve by down-sampling 1-minute eclipse profiles to the Kepler long cadence $\left(dt=0.0204\ \textrm{d}\right)$. Stellar and instrumental noise is marginalized by fitting the lowest non-linear order quadratic polynomial around each eclipse. We apply quarterly crowding values from Kepler to model third light contamination. To account for underestimated observational uncertainties and additional noise, we fit for a systematic light curve error $\sigma_{\mathrm{sys,LC}}$, which we add in quadrature to the observed errors.

The z-component of the velocity, as solved by Kepler’s equation, is used to model the extracted RVs. For double-line eclipsing binary systems, where the RVs of both components are detected, the amplitudes of the primary and secondary RV are related to the masses of the secondary and primary, respectively. For single-line EBs, where only the RV of the brightest component is detected, only the “mass function” $f_{M}$ of the system can be constrained, where

[TABLE]

As with the light curve data, we fit for a systematic radial velocity error parameter $\sigma_{\mathrm{sys,RV}}$ to account for underestimated noise.

We combine RVs with Kepler light curve information to model the system and find a best fit solution. We first determine the light curve and RV solutions separately, and then fit RV and light curve simultaneously. The simultaneous RV+LC model has 17 free parameters in total. The model is optimized via a least-squares algorithm lmfit (Newville et al., 2016), and then uses the best-fit solution to seed Monte Carlo Markov Chain (MCMC) simulations with emcee (Foreman-Mackey et al., 2013), in order to sample the posterior distributions of each parameter. We use broad, uniform priors and run the Markov chains with 128 walkers for $\sim$100,000 iterations, visually inspecting trace plots for convergence. We report the 50%, 16%, and 86% values, i.e., the mean and $1\sigma$ uncertainties for each parameter. For more details on the KEBLAT model, including parameter bounds, see Windemuth et al. (2018).

2.5 Radius ratio—flux ratio—inclination Degeneracy

For light curves with partially or grazing eclipsing geometries, there exists a degeneracy between radius ratio and flux ratio when eclipses are observed in a single photometric band. For this reason, we use additional constraints on the Kepler light curve flux ratios with spectroscopic $H$-band flux ratios obtained from the BF for SEBs exhibiting shallow eclipses (KIC 5285607 and KIC 6781535). For these two systems, we place a Gaussian prior on the RV+LC solution with $\mu$ centered around the BF-derived flux ratio and $\sigma=0.2$.

2.6 Temperatures from Flux Ratios and Radii

Obtaining effective temperatures of the stars in these SEBs requires additional analysis. The KEBLAT model does not provide a measure of stellar temperatures directly, but only indirectly via the flux ratio in the Kepler bandpass. In addition, the APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP) processing reports only a single “combined light” effective temperature for each system (Pérez et al., 2016). This ASPCAP temperature is likely to be biased due to the contamination of the brighter star’s spectrum by the fainter star. Here we address both of these issues to estimate individual stellar effective temperatures.

The orbital solutions described in §2.4 yield sums and ratios of radii. In addition, the light curve analyses yield flux ratios ($F_{\textrm{ratio}}\equiv\frac{F_{2}}{F_{1}}$) in the Kepler bandpass which are primarily constrained by the observed eclipse depths.

By assuming the ASPCAP effective temperature is the flux weighted average of the system, $T_{\textrm{avg}}=(F_{1}T_{1}+F_{2}T_{2})/(F_{1}+F_{2})$, and defining the primary star as the one that provides the majority of the light, we can use the following relationships between the flux ratio $F_{\textrm{ratio}}$, $T_{\textrm{avg}}$, stellar radii $R_{1}$ and $R_{2}$, and distance $d$ to find flux and temperature estimates for the individual binary components separately. First, we solve for the binary’s flux sum using Gaia distance estimates from Bailer-Jones et al. (2018):

[TABLE]

from which we can compute the individual fluxes as

[TABLE]

and

[TABLE]

Then, we use the relationship between $T_{\textrm{ave}}$, ${F_{\textrm{ratio}}}$, and the individual stellar fluxes to solve for the temperature of each star:

[TABLE]

which yields

[TABLE]

and

[TABLE]

However, these resulting temperature estimates are likely systematically underestimated. In the ASPCAP pipeline, APOGEE spectra are compared to a synthetic spectral model to resolve quantities like effective temperature. When a detached binary signature is present in stellar spectra, the additional component can cause the spectrum to be fit by a cooler synthetic template. This leads to a systematic underestimation in the binaries’ effective temperatures of roughly 300 K (El-Badry et al., 2017). The systematic underestimation is a function of the effective temperature of the primary and the mass ratio of the system. We follow their method for each system to correct for this effect.

3 Results

The joint light curve and RV analysis for each of the seven systems is presented in detail in the following subsections. We present orbital and mass solutions together in Table 3.

In all systems, we define the primary eclipse ($\phi=0$) as the deeper eclipse. This corresponds to when the primary star is eclipsed by the secondary star. Usually, the primary star is the brighter of the two, and the light curve definition of primary and the RV definition of primary agree. However, we note the secondary star in KIC 6781535 is brighter in the APOGEE $H$-band than the primary. In the figures that follow, we color code the RV of the primary star in red and the RV of the secondary star in orange.

{rotatetable*}

3.1 KIC 5285607

KIC 5285607 is a grazing ($i=79^{\circ}$) 3.9 d eclipsing binary with similar mass stellar components ($M_{1}=1.56\leavevmode\nobreak\ M_{\odot},M_{2}=1.35\leavevmode\nobreak\ M_{\odot}$). The stars are in a circular orbit, as exhibited by the sinusoidal shape of the RV and occurrence of secondary eclipse at a phase of $\sim 0.5$ as seen in Figure 3.

Because the eclipses in the light curves are shallow (4% loss of light), the impact parameter is highly degenerate with the flux and radius ratio. That is, a solution with similar flux contributions from both components in a more inclined system yields the same shallow eclipses as a solution with a much brighter primary component in a more edge-on system. This degeneracy can be ameliorated with additional information from spectra. Therefore, we place a Gaussian prior on the flux ratio parameter based on the BF fits, with a 0.2 1$\sigma$ width. With this flux ratio constraint, we find the secondary star is about 80% the size of the primary, with absolute dimensions of $2.0\leavevmode\nobreak\ R_{\odot}$ and $1.7\leavevmode\nobreak\ R_{\odot}$ for the primary and secondary components, respectively.

3.2 KIC 6864859

KIC 6864859 is a highly eccentric ($e\sim 0.6$), slightly inclined ($i=88.32^{\circ}$) 40.9 d eclipsing binary with components of similar mass ($M_{1}=1.41\ M_{\odot}$, $M_{2}=1.35\ M_{\odot}$) and radii ($R_{1}=1.66\ R_{\odot}$, $R_{2}=1.46\ R_{\odot}$). The best-fit model is shown in Figure 4. The system’s highly eccentric orbit gives rise to irregularly shaped RVs and a secondary eclipse near phase $\sim$ 0.125.

Figure 5 shows clear brightening events of the system between primary and secondary eclipses, with maximal amplitudes $\sim$0.3 ppt at phase $\sim$0.065, the predicted phase of periastron passage from KEBLAT. This behaviour is consistent with tidal distortions in an eccentric orbit near periastron, and is symptomatic of a class of objects known as “heartbeat” stars (Thompson et al., 2012). Both primary and secondary eclipse residuals exhibit small amplitude ($\sim$0.5 ppt) coherent structures; these are likely due to the non-spherical shape of the stars which is not explicitly modeled in KEBLAT.

We note that in our reduction process, some apVisit spectra were eliminated due to a very low signal-to-noise ratio that persisted after being run through our de-spike program. The majority of the remaining visits for this target resulted in well-separated peaks from the BF.

3.3 KIC 6778289

KIC 6778289 is a 30.1 d eclipsing binary with stellar components $M_{1}=1.51\ M_{\odot}andM_{2}=1.09\ M_{\odot}$ in an eccentric ($e=0.2$), nearly edge-on ($i=89.3^{\circ}$) orbit. The simultaneous RV+LC fit is shown in Figure 6. The radii are $R_{1}=1.75\ R_{\odot}$ and $R_{2}=1.0R_{\odot}$. This difference in radius gives rise to the difference in primary and secondary eclipse shape. The flat-bottom secondary eclipse indicates a total eclipse of the secondary component, while the primary eclipse is more V-shaped, e.g., more grazing. The larger residuals during secondary eclipse is consistent with starspot modulation. The system has low ($\approx 1\%$) third light contamination in the Kepler light curve which does not appear in the BF.

3.4 KIC 6449358

KIC 6449358 is a 5.8 d circular EB which may be a gravitationally bound to a distant tertiary companion.

The BF for this object exhibits two clear peaks, however, the second-brightest peak is relatively stationary in RV while the brighter peak varies by $\sim$60 km/s over one orbit. This is shown in Figure 7 as well as in Appendix A Figure 19. If the $\sim$stationary BF peak corresponded to a stellar binary component, it would require a system with extremely large mass ratio $M_{2}/M_{1}>10$, which would be consistent with a white dwarf. However, the light curve constrains the radius ratio to be $R_{2}/R_{1}\sim 0.3$, which makes this scenario physically implausible.

A more likely explanation for the $\sim$stationary RV component is that it belongs to a tertiary star, and that the true secondary stellar component of the EB is too faint to be robustly detected by APOGEE. Indeed, the flux ratio in the Kepler bandpass is $F_{2}/F_{1}\sim 0.01$. Thus, we effectively treat KIC 6449358 as a single-lined SEB in our model, and as a result we are only able to constrain the mass function $f_{M}$ of the binary. Specifically, to reproduce the observed RV amplitude, our sum and ratio of masses solutions are degenerate, tending toward two extremes: high total mass ($\sim 4\ M_{\odot}$) with a low mass ratio ($q\sim 0.3$), or low total mass ($1\ M_{\odot}$) with a higher mass ratio ($q\sim 0.67$).

We note that some APOGEE visits do suggest a small, third BF peak (see Appendix A Figure 20 for details). These marginal BF peaks have large radial velocity variations from visit to visit, consistent with a low-mass star. This supports the scenario with total mass $\sim 2.3M_{\odot}$ and mass ratio $q\sim 0.45$.

Figure 7 shows the best KEBLAT model fit to the light curve and radial velocities obtained using mass function $f_{M}=\frac{M_{2}^{3}\sin i}{M_{1}+M_{2})^{2}}$ her than $M_{1}$ and $M_{2}$. The $\sim$stationary RV points, which are not fit, correspond to the putative third star, either a line-of-sight coincident or a tertiary companion in a hierarchical triple system. We favour the latter scenario, as the eclipses show timing variations consistent with perturbations by a bound, tertiary component. These eclipse timing variations (ETVs) have been used to identify and characterize many Kepler EBs (e.g. Borkovits et al., 2016), and we fit these ETVs using a simple linear ephemeris based on the observed times of primary eclipse. We then compute the observed minus computed $(O-C)$ eclipse times as a function of time. The result is shown in Figure 8. There is a clear parabolic or sinusoidal trend in the ETVs with an amplitude of $\sim 0.0006$ d; the ETV timescale indicates that the perturbing tertiary body has a minimum period $\sim 1450$ d. The architecture of this type of hierarchical triple – short, circular inner binary orbited by a distance tertiary companion – is consistent with dynamical processing via the Kozai (1962) mechanism.

3.5 KIC 4285087

KIC 4285087 is an equal mass binary ($M_{1}\approx M_{2}\approx 1.1M_{\odot}$) in a circular, slightly inclined ($i=87.3^{\circ}$) 4.5 d orbit. We show the best-fit solution in Figure 9. The components are main-sequence dwarfs with similar radii ($R_{1}\sim R_{2}\sim 1R_{\odot}$). The eclipses are similar depth ($\sim 30\%$), duration ($\sim 0.2$ d), and shape (V), consistent with equal mass dwarfs orbiting each other.

3.6 KIC 6131659

KIC 6131659 is a mass ratio of 0.75 binary ($M_{1}=0.9\ M_{\odot},M_{2}=0.7\ M_{\odot}$) in a 17.5 d, close-to-circular orbit. Figure 10 shows the simultaneous RV+LC fit to the data. The primary and secondary eclipses are relatively deep, with 35% and 10% loss of total system light, respectively. The residuals to the light curve fit show correlated structure, which may be due to poor limb darkening modeling and/or a varying third light component which deviates from Kepler crowding values.

There is a third light component readily visible in the BF (see Appendix A Figure 22), but it is not RV-variable. This suggests it may be a line-of-sight contamination source or a gravitationally bound body in a hierarchical triple with an orbital period much longer than 17.5 d. The light curve does not show apparent eclipse timing variations, but this does not preclude the presence of a gravitationally bound tertiary.

3.7 KIC 6781535

KIC 6781535 is an eccentric ($e=0.25$), grazing ($i=84^{\circ}$), 9.1 d binary. The best-fit solution (see Figure 11) yields binary components of similar mass $(M_{2}/M_{1}\approx 1.0$) but slightly different radii ($R_{2}/R_{1}\approx 1.2\pm 0.1$), which suggests a slightly evolved system. The shallow eclipses poorly constrain the system’s impact parameter, flux ratio, and radius ratio. As a result of this degeneracy, we apply a Gaussian prior on the light curve flux ratio parameter from spectra, following the same method as used for KIC 5285607 (see §3.1).

Similarly to KIC 6131659, there is a third light component visible in the BF (see Figure 23) that is not RV variable, indicating either a line-of-sight coincident third star or gravitationally bound tertiary companion. There are symmetric structures in the light curve residuals, most noticeably during primary eclipse, consistent with variable third light contribution, changes to the binary orbit due to additional bodies, or starspot modulations. Because the system exhibits shallow, grazing eclipses, it was not conducive to an ETV analysis.

3.8 Supplemental Physical Parameters

In addition to the main results in Table 3, we report some additional physical parameters in Table 4. As discussed in Section 2.6, we can use the BF peak area ratios to measure the $H$-band flux ratio of each system. We can also combine Gaia parallax information with our measured fluxes, radii, and the ASPCAP $T_{\textrm{eff}}$ to estimate individual stellar temperatures. These parameters, along with adopted values from external sources, are summarized in Table 4.

4 Discussion

4.1 Evolutionary Histories

With such well-characterized stars, we can investigate each binary’s age and evolutionary history with two different approaches. In the following we assume normal Milky Way metallicities $-0.5<$[M/H]$<0$.

First we explore the H-R diagram in $\log g$ vs. $\log T_{\rm eff}$ after first correcting our temperature estimates following El-Badry et al. (2017). We calculate $\log g$ for each star directly from the KEBLAT mass and radius. We also use the KEBLAT masses and radii to determine the system ages in the mass-radius space directly, which avoids any dependence on distance or on our disentangling of the individual component temperatures.

In both approaches, we use Dartmouth evolutionary tracks (Dotter et al., 2008) and consider only the portion of the track with $\log g\geq 4.1$. This effectively only includes the main sequence. For consistency, we adhere to the KEBLAT definition of star 1 (primary) and star 2 (secondary) in which star 1 is the member of the SEB being eclipsed during the primary eclipsing event, and star 2 (secondary) as the member eclipsed during the secondary eclipsing event.

Figure 12 shows all of the SEBs in the $\log g$ vs. $\log T_{\rm eff}$ diagram, and Figure 13 shows each SEB system individually. In general, the systems are broadly consistent with ages ranging from about 0.8 to about 3 Gyr, and the two components of each system appear to be consistent with a common age. Figure 14 represents these systems in the mass-radius diagram, where again all six systems modeled appear consistent with coevality for the same range of ages as above.

4.2 Mass-Luminosity Relationships

In order to verify that our targets are on the main sequence, we create a mass-luminosity plot (Figure 15) using the stellar masses from Table 3 and calculated $H$-band luminosities explained below. These luminosities are independent of the ASPCAP temperature estimates and corrections from El-Badry et al. (2017) used to derive Figures 13 and 14. The results represent a comparison to theoretical models that complements the mass-radius relationship presented in Figure 14, and is less reliant on light curve modeling, which may have degenerate radius solutions in grazing geometries.

To calculate $H$-band luminosities, we use distances derived from Gaia parallaxes (Bailer-Jones et al., 2018) to convert apparent $H$-band magnitudes from 2MASS (Skrutskie et al., 2006) to absolute magnitudes. We check that the $H$-band magnitudes were not taken during eclipse by cross referencing the time of 2MASS observations to the EB ephemeris. We then compute the system $H$-band luminosities from absolute magnitudes using the sun as a reference, with $H$-band magnitude of $3.32$ from Cohen et al. (2003). We disentangle the separate luminosities for each stellar component in the system using the observed APOGEE $H$-band flux ratios (see Table 3). In Figure 15, we show each system using the same plotting convention as Figures 12 to 14, where solid and open circles correspond to primary and secondary components, respectively. We over-plot for comparison theoretical masses and $H$-band magnitudes from Dartmouth isochrones (Dotter et al., 2008) at sub-solar and solar metallicity and a range of ages (0.8–5 Gyr). In general, as previously concluded, members of the same binary system follow the same evolutionary track, i.e., are coeval.

We exclude KIC 6449398 from this analysis, because it is a single-lined SEB. We also exclude KICs 6131659 and 6781535, which have negative parallax values from Gaia. For this reason we could not include these targets in the spectroscopic H-R diagram for our systems with Gaia distances. We did not include reddening corrections to the distance modulus calculations; however, reddening effects should be minimal in the near-IR (2MASS and APOGEE $H$-band) compared to the visible (Kepler).

4.3 Tertiary Companions

In our subset of SEBs, we identified three candidate triple systems. The binaries with possible tertiary companions are KIC 6131659, KIC 6781535, and KIC 6449358 (see Sections 3.6, 3.7, and 3.4, respectively). The first two of these systems exhibit clear third BF component peaks in nearly all RV visits. We fit these third peaks with Gaussians, similar to the Gaussian fits for the primary and secondary BF components. In both cases, the third BF components do not have radial velocity variations above the noise, which suggest these third members are either line-of-sight contamination sources or gravitationally bound in hierarchical triples with orbital periods much longer than that of the observed SEB.

While statistics on multiplicity are not complete, numerous studies have found tertiary systems composed of a tight binary orbited by a distant third member. Statistics from Tokovinin et al. (2006) indicate that roughly 63% of spectroscopic binaries have tertiary companions in a wide orbit. In binaries with shorter periods (less than 3 days) this percentage rises to 96%, but in longer period binaries (12 or more days) this percentage is only 34%.

The incidence of triples among Kepler close eclipsing binaries (as are the ones in our analysis) is at least $\sim$20% (Rappaport et al., 2013; Conroy et al., 2014), and likely higher for tertiaries with longer periods beyond Kepler’s finite observing time.

Evidence from both spectroscopic and photometric observations indicate KIC 6449358 belongs to a hierarchical triple system (see §3.4). The BF for this system shows a stationary tertiary peak in a few of the APOGEE visits. The mid times of eclipses in the Kepler light curve also exhibit sinusoidal variations in time; these eclipse timing variations indicate that the tertiary has a period $\gtrsim 1450$ d.

Interestingly, among our sample we do not detect tertiary companions among the shortest-period binaries. In particular, neither of the two EBs with $P_{\rm orb}<5$ d exhibits a clear tertiary in our data. It is not yet clear whether existing observations might already exclude the presence of tertiaries at very large separations that might not appear in our data; additional imaging observations might be required to identify such stars. At the same time, two of the four EBs with $P_{\rm orb}>9$ d are triples, which would appear to be an over-abundance of tertiaries among the longest-period EBs, albeit with a small sample. However, we note that one of these (KIC 6781535, $P_{\rm orb}\approx 9$ d) is a modestly evolved system (see Figure 14), and excluding that case yields an occurrence of 1/3 triples among our EBs with $P_{\rm orb}>12$ d), fully consistent with the results of Tokovinin et al. (2006).

5 Summary

We thoroughly characterize seven SEBs that have been observed by both Kepler and APOGEE. Our targets are selected from the Kepler EB catalog, and limited to bright, detached EB targets with both primary and secondary eclipses observed by Kepler, high inclination, and multiple APOGEE visits. We identify an additional 26 SEBs which may warrant similar studies. We demonstrate that the BF is a superior method for extracting RVs from APOGEE visit spectra compared to the CCF used in the present reduction pipeline. This is particularly true for systems with multiple RV variable components. While such an analysis is beyond the scope of this work, if the BF method were applied to the full data set of multiple-visit APOGEE targets, it would most likely reveal many previously unknown SEBs and other interesting RV-variable sources.

RVs are extracted from apVisit spectra using the BF method and the Kepler light curves are normalized, de-weighted and modeled using KEBLAT. The light curve and RV solutions are first determined individually, and then computed simultaneously. We use the resulting physical parameters to estimate stellar temperatures, investigate coevality, and explore candidate triple systems.

Using our analysis we find our target’s binary members are coeval with ages ranging from 1 to 3 Gyr, assuming normal Milky Way metallicity $(-0.5<[M/H]<0)$. The exception is KIC 6781535 which lies closer to a slightly metal poor $([M/H]\sim-1.0)$ 3 Gyr isochrone. Our systems being broadly consistent with coevality confirms a common assumption in star formation that members of multiples form at the same time, and also effectively calibrates stellar evolution modeling.

Overlap between large scale surveys like APOGEE, Kepler, and Gaia allows us to discover and analyze many diverse SEBs, including systems with very low flux ratios and those in higher order systems. The statistics on the triples within our subset with respect to the orbital period of inner binaries is broadly consistent with statistics from the field Tokovinin (1997), though there may be some tension with our sample in that the shortest-period EBs do not appear to be spectroscopic triples. This is in contrast to the expectation that shortest-period EBs are most likely to be hierarchical triples. It is possible that very wide tertiaries do exist in these systems but have yet to be identified via imaging.

We have shown that through tools like KEBLAT and the BF analysis of APOGEE spectra, it is possible to perform high quality analysis of large numbers of SEBs with a variety of properties. This opens up great promise for future SEBs identified in TESS and SDSS-V data.

We would like to thank Scott Fleming for critical guidance and discussion and Paul A. Mason for valuable brainstorming and advice. We recognize the SDSS Faculty And Student Team (FAST) initiative for supporting this work through a partnership with New Mexico State University. JMCC thanks the Fisk-Vanderbilt Masters-to-PhD Bridge Program, Amanda Cobb, Kelly Holley-Bockelmann, and Nancy Chanover for continued empowerment of a woman and new mother in STEM. MLR celebrates that this work has encompassed two births, one wedding, and multiple graduations among the lead authors.

Appendix A Stellar Radial Velocities

Here, we include one Broadening Function plot for each of the seen targets. These illustrate how we measured radial velocities for each component of the spectroscopic binaries from APOGEE visit spectra, as discussed in Section 2.2. Note each figure uses uncorrected RV on the abscissa, before barycentric corrections have been applied. The final corrected RVs with uncertainties are reported in Table 5 below.

\startlongtable