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tokens: this correlation is presently unclear. Figure 8. Periastron distance (top panel; ) and the tidal force acting on the primary star at periastron divided by the star’s surface gravity (bottom panel; see Equation ( 4 )), both as a function of the full amplitude of the photometric HB signal at periastron. In both panels the markers’ radius is linear in the primary star’s (See Table 1 ). 4.5. Higher Multiplicity Systems Many of the HB stars examined here may be members of triple or higher multiplicity systems. Several works (e.g., Duquennoy Mayor 1991 ; Meibom Mathieu 2005 ; Tokovinin et al. 2006 ; Raghavan et al. 2010 ) have found that the fraction of higher multiplicity systems among short period and highly eccentric binaries is very high, exceeding 90% for binaries with days (Tokovinin et al. 2006 ). Indeed, tertiary bodies may excite the orbital eccentricity of HB progenitors via Kozai–Lidov oscillations, producing close periastron passages and allowing them to be detected as HB systems. Quadruple systems composed of two binaries may also be common among HB systems, and these types of systems have been predicted by Pejcha et al. ( 2013 ), who found that HB systems formation is greatly enhanced in quadruple systems relative to triple and binary systems. Despite the probable multiplicity of many HB systems, we find it unlikely that many of our Keplerian RV solutions have been greatly affected by orbital motion induced by a third body. A putative third body exciting the orbital eccentricity would likely have an orbital period, P 3 , more than several times longer than the HB period (depending on its mass, inclination, and eccentricity) in order to allow long term dynamical stability (Kiseleva et al. 1994 ). If a typical HB progenitor system was born with an orbital period days before having its orbital eccentricity excited, we expect for most third bodies, with much larger values of P 3 in most cases. Although the orbital RV amplitude induced by such a third body can be several km s −1 , we expect only a small RV change over the ∼100 days baseline of our observations, and therefore this motion can be safely included as a constant RV offset. We are currently pursuing a follow-up survey on many of these HB systems that will reveal whether any of them have tertiary companions with orbital periods . 4.6. Future Prospects The orbits measured here comprise a relatively large sample obtained with a single instrument, and they can facilitate several follow-up scientific studies in addition to that of tidal circularization and the shape of the e – P distribution (Section 4.3 ). Other examples include: 1. Stars in binary systems with highly eccentric orbits are expected to rotate near a “pseudo-synchronous” rotation period, where the stellar rotation is synchronized with the orbital motion close to periastron which is faster than the mean orbital motion along the entire orbit. The pseudo-synchronous rotation rate depends on eccentricity and orbital period