Figure 11. SED fits to the quadruple system TIC 121088959/60. The red curve is the model for star B (TIC 121088959), while the blue curve is for the composite of stars Ca, Cb, and A. The model SED for the individual stars Ca, Cb, and A are shown in green and light blue, respectively. The match to the measured fluxes is made by integrating the model fluxes over the appropriate filter bands.

Figure 12. Allowed orbits in the \( P _ { \mathrm { A C } } - e _ { \mathrm { A C } } \) plane. The analytic constraints are shown by the colored curves and are discussed in the text. Additionally, the orbits are required to produce a motion of star A with respect to an essentially motionless star B, yielding ∼8 km \( \mathbf { s } ^ { - 1 } \) in the radial direction and a relative PM of ∼5 mas \( \mathrm { y r } ^ { - 1 } \).

Powell et al. 2021, Borkovits et al. 2021, and Kostov et al. 2021 for details). Therefore, in addition to the six SED points for each stellar image, we make use of four other important constraints. From our fit of the EB lightcurve we take (i) the ratio of stellar radii in the EB to be \( R _ { \mathrm { C b } } / R _ { \mathrm { C a } } = 0 . 6 7 6 \pm 0 . 1 \); (ii) the sum of the scaled stellar radii to be \( ( R _ { \mathrm { C b } } + R _ { \mathrm { C a } } ) / a = 0 . 0 5 5 \pm 0 . 0 1 8 \); (iii) the eclipse depth ratio of primary to secondary to be 1.5 \( + \) 0.15; and (iv) the third light for the eccentric EB to be 89.9% \( ^ + \) 1.5%. These 16 total constrains, plus the assumption that the stars are on the ZAMS, then prove sufficient to fit uniquely for the four stellar masses4. The results are shown in Figure 11 and the fitted stellar parameters are given in Table 5.

We find that the masses of the stars in the eccentric EB are a \( M _ { C _ { \mathrm { { o } } } } \, \simeq \, 0 . 1 9 0 \, \mathrm { M } _ { \mathrm { { \odot } } } \) and \( M _ { C \mathrm { h } } \, \simeq \, 0 . 1 3 8 \, \mathrm { M } _ { \odot } \). The mass of star B (TIC 121088959) is \( M _ { \mathrm { R } } \simeq 0 . 3 8 6 \, \mathbf { M } _ { \odot } \), while the brighter member of TIC 121088960 (star A) is only slightly more massive at \( M _ { \Delta } \, \simeq 0 . 4 3 0 \, \mathrm { M } _ { \ G } \).

# 6.2 Constraining the Triple Orbit AC

Armed with reasonably good estimates for the masses of the four stars comprising this system, we can now place some constraints on the orbit of the eccentric EB around what we have called “star A”. Let us refer to this orbit as the ‘AC’ orbit or that of the ‘inner triple’. There are five constraints that we use to limit the range of orbital periods and other parameters for the AC system.

These include the facts that (i) the AC system must be dynami-cally stable; (ii) the ABC system (i.e., that of the quadruple), must also be dynamically stable; (iii) there is no observed apsidal motion of the binary C (2 \( - \, \sigma \) limit of \( \lesssim 1 0 ^ { \circ } \) between the two TESS ob-servations spaced by two years); (iv) the difference in RV between TIC 121088959 and TIC 121088960 (i.e., between stars A and B) is 8 \( ^ + \) 0.3 km \( \mathbf { s } ^ { - 1 } \) over an 11-day interval; and (v) there is a difference in PM between stars A and B of 5 \( \pm \) 0.2 mas \( \mathrm { y r } ^ { - 1 } \).

The last two of these constraints (the difference in RVs and PMs) were from different epochs separated by a few years. Nonetheless, for simplicity in the calculations, we take this difference in time to be considerably less than the orbital period of the AC binary, and there-fore these two constraints are applied locally around the orbits being examined. This becomes a good approximation for \( P _ { \mathrm { A C } } \gtrsim 1 0 \) years. Furthermore, with regard to these same two constraints, we assume that both the Gaia and ground-based observations are dominated by star A in the AC system (TIC 121088960). The expression for the minimum \( P _ { \mathrm { { A C } } } \) allowed before apsidal advance in the C binary would be observed is from Rappaport et al. (2017), Eqn. (13). The lower and upper bounds on \( P _ { \mathrm { A C } } \) for the dynamical stability of orbits AC and ABC, respectively, are given by Eqn. (16) of Rappaport et al. (2013)5 modified with a leading factor of \( ( 1 + e _ { \mathrm { i n n e r } } ) ^ { 3 / 2 } \) which we adopt from the Eggleton-Kiseleva stability requirement (Eggleton& Kiseleva 1995, Eqn. 1; Mikkola 2008, Eqn. 10). We take this to account for the fact that the more eccentric the inner binary is in a triple system, at a fixed binary orbital period, the more the two stars are separated at apastron. Finally, with regard to the constraint on the stability of the ABC orbit we estimate a limit on the outer orbital period by noting that the projected physical separation of AC and B on the sky is ' 320 AU. We could use this as a proxy for the semimajor axis of the outer orbit. However, to be somewhat more conservative, we take the semi-major axis of the outer orbit to be \( \lesssim \) 1000 AU. This does not set a rigorous upper limit on \( a _ { \mathrm { A R C } } \), but we believe it is reasonably conservative.

Our approach was to randomly select orbital parameters for the AC orbit uniformly in \( \log P _ { \mathrm { A C } } \), from 1 to \( 1 0 ^ { 7 } \) days; uniformly in eccentricity from 0 to unity; uniformly in argument of periastron, \( \omega \) from 0 to 2 \( \pi \); and orbital inclination according to \( \cos ^ { - 1 } ( \mathcal { R } ) \), where \( \mathcal { R } \) is a draw from uniform random number generator. We followed each of \( 1 0 ^ { 5 } \) trial orbits around a complete orbit. Constraints (i), (ii), and (iii) listed above were immediately applied to each trial orbit, and if any of those tests failed, that particular orbit was rejected. For constraints (iv) and (v) listed above, those conditions were checked everywhere around an entire orbit. If they were never satisfied, the orbit was also rejected.

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4 The approach of using SED fitting on multiple star systems has been demonstrated by numerous groups, but perhaps none more dramatically than for the sextuple star system TIC 168789840 (see Powell et al. 2021).

5 See Mardling & Aarseth (2001) and Mikkola (2008) for the original ex-pressions.

Table 5. Mass Estimates for the Four Stars in TIC 121088959/60

<table><tr><td>Parameter</td><td> Star Ca</td><td> Star Cb</td><td> Star A</td><td> Star B</td></tr><tr><td>Mass \( \mathbf { M } _ { \odot } \))</td><td>0.190 \( ^ + \) 0.009</td><td> 0.138 \( ^ + \) 0.005</td><td> 0.430 \( ^ { + } \) 0.008</td><td> 0.386 \( ^ + \) 0.008</td></tr><tr><td>Radius \( \mathbf { R } _ { \odot } \))</td><td>0.201 \( ^ + \) 0.007</td><td> 0.160 \( ^ + \) 0.005</td><td> 0.394 \( ^ + \) 0.005</td><td> 0.358 \( ^ + \) 0.007</td></tr><tr><td>\( T _ { \mathrm { e f f } } \)(K)</td><td>3228 \( ^ + \) 25</td><td> 3033 \( ^ + \) 30</td><td> 3512 \( + \) 10</td><td> 3468 \( ^ + \) 10</td></tr><tr><td>Luminosity \( \mathrm { L } _ { \odot } \))</td><td>0.0040 \( ^ + \) 0.0004</td><td> 0.0020 \( \mathrm { ^ { + } } \) 0.0002</td><td> 0.0213 \( ^ + \) 0.0007</td><td> 0.0167 \( + \) 0.0007</td></tr><tr><td>a \( \mathbf { R } _ { \odot } \))</td><td>6.08 \( + \) 0.07</td><td> 6.08 \( \mathrm { ~ \texttt ~ { ~ \, ~ \, ~ } ~ } \) 0.07</td><td>...</td><td>...</td></tr></table>

Figure 13. Allowed orbits in the \( P _ { \mathrm { A C } } - \omega _ { \mathrm { A C } } \) plane.

The results of acceptable AC orbits are shown in Figs. 12 and 13. In Figure 12 we show the orbits which satisfy all five constraints in the \( P _ { \mathrm { A C } } - e _ { \mathrm { A C } } \) plane. Since constraints (i)-(iii) are analytic functions of \( P _ { \mathrm { A C } } \) and \( e _ { \mathrm { A C } } , \) and a reasonable assumption about \( P _ { \mathrm { A B C } } \) (see above), we show those as solid limiting curves. The remainder of the constraints come from the RV and PM differences between stars A and B. It seems clear that inner orbital periods \( P _ { \mathrm { A C } } \) ranging from about a year to 1000 years are acceptable. All eccentricities seem possible. However, for nominal eccentricities in triple systems of∼0.2 − 0.7 (see Figure 9 of Borkovits et al. 2016) we more likely can expect to find \( P _ { \mathrm { A C } } \) in the range of 1 − 50 yr. In Figure 13 we show \( \omega _ { \mathrm { A C } } \) vs \( P _ { \mathrm { A C } } \). These two parameters are slightly correlated, with more allowed orbits having \( \omega _ { \Delta C } \simeq 9 0 ^ { \circ } \)◦and \( 2 7 0 ^ { \circ } \)◦and covering a more restricted range of \( P _ { \mathrm { A C } } \). The inclination angles \( i _ { \mathrm { A C } } \) (not shown in the plots) are uniformly distributed between a minimum of \( 6 8 ^ { \circ } \) and \( 9 0 ^ { \circ } \).

# 7 COMPARISON WITH OTHER LOW-MASS ECCENTRIC BINARIES

Similar to the case of the vast majority of close main-sequence bi-naries, it is clear that since the radii of the low-mass protostars were much larger than the present-day separation of its stars, the eccentric binary Cab cannot have been formed in its present orbital configu-ration (see, e.g. Kiseleva et al. 1998; Fabrycky & Tremaine 2007). Perhaps the most interesting question about this close pair is how it has managed to retain such a high eccentricity despite the requisite orbital shrinkage during its past history. Some possibilities are as follows: (i) the initial eccentricity of the originally wider orbit was much higher, and while there is ongoing tidal friction causing orbital shrinkage and circularization, the system is still sufficiently young that there has not been enough time to circularize the orbit; (ii) the shrinkage of the initial orbit, which was wide enough to accommo-date the protostars, was the result of some other mechanism(s) beside tidal dissipation such as, e.g., (a) escape of an additional stellar com-ponent or, (b) accretion-driven migration (see, e.g. Tokovinin & Moe 2020, and further references therein). Or, another possibility is, (iii) that the observed current high eccentricity is a consequence of on-going dynamical interactions with the more distant, third and fourth stellar components of the quadruple system.

We constructed a so-called “\( P - e \)” diagram showing the known cases of eccentric binaries in Figure 14 (upper panel). We plot-ted there the same datasets as in a previous paper by Zasche et al. (2021): small red dots are from the SB9 catalogue (Pourbaix et al. 2004); yellow dots show eclipsing binaries from the catalogue of eccentric binaries by Kim et al. (2018); blue dots show Kepler bina-ries by Kjurkchieva et al. (2017); black dots show those from ASAS published by Shivvers et al. (2014); cyan points are from Halbwachs et al. (2003); green points are from Triaud et al. (2017); and magenta points are from (Latham et al. 2002). All of these were studied as spectroscopic and/or eclipsing binaries. However, some of the most extreme points from the SB9 catalogue have very uncertain orbits and should not be considered as real eccentricities. Moreover, most of the data shown here represent much more massive stars, for which the circularization process is different due to their internal structure (as recently proven on real data, see e.g. Van Eylen et al. 2016).

For these reasons, we have tried to compare the comparable, i.e., plot our unique low-mass system with other stars having derived eccentricities and both components of the M spectral type. Specifi-cally, (i) the set of Kepler eclipsing binaries yielded 7 systems with GAIA photometric index (\( B _ { p } - R _ { p } ) \, > \, 1 . 8 \) mag; (ii) the spectro-scopic survey by Latham et al. (2002) contains only one system with effective temperature below 4000 K; and (iii) the survey Halbwachs et al. (2003) provided two binaries with masses below \( 0 . 6 \, M _ { \odot } . \) These are plotted in the bottom panel of Fig. 14. These data are then com-plemented with other M+M binaries taken from various dedicated studies (Stassun et al. 2006; Morales et al. 2009; Carter et al. 2011;Irwin et al. 2011; Gómez Maqueo Chew et al. 2012; Kraus et al. 2015; David et al. 2016; Gillen et al. 2017; Kraus et al. 2017; Lubin et al. 2017; Irwin et al. 2018; Murphy et al. 2020; Acton et al. 2020). The data for these low-mass binaries are plotted together in the bot-tom panel of Fig. 14 with curves representing the close pericenter approaches (i.e., \( 1 . 5 \times R _ { \star } \, = \, a \cdot ( 1 - e ) \)) when they likely collide with each other. The periastron separations are calculated for dif-ferent spectral types (from M0V to M6V) according to their typical radii and masses according to Pecaut & Mamajek (2013), assuming both components are similar to each other (same masses and radii). One can also ask whether some proximity effect near periastron pas-sage would also be visible on the light curve of the binary. However, we have calculated that such an effect is so small that it cannot be detected with the current precision of the available TESS data.