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.