| Summary: | Heartbeat star systems have been reported to exhibit two distinct different orbital dynamic evolution processes: apsidal precession (e.g., KIC 4544587) and orbital decay (e.g., KIC 3766353). While experiencing similar dynamic tidal interactions, these binary systems display different dynamical behaviors, which is a puzzling phenomenon. In this work, we deduced a theoretical relation between the timescale of stellar pulsation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>pul</mi></msub></semantics></math></inline-formula> and orbital periods <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>orb</mi></msub></semantics></math></inline-formula> of heartbeat stars based on the resonance criteria representing the orbital local low-energy configuration. The theoretical relation shows that when the ratio of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>orb</mi></msub></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>pul</mi></msub></semantics></math></inline-formula> is an integer, the specific orbital period is captured in the resonance state, resulting in resonance locking. The resonance criteria are verified by periodograms of the pulsations and orbits of the two systems KIC 4544587 and KIC 3766353 from observations. KIC 4544587 is an apsidal precession heartbeat star with eight observed resonant frequencies available from observations and has an almost integer ratio of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mi>pul</mi></msub><mo>/</mo><msub><mi>P</mi><mi>orb</mi></msub><mo>=</mo><mn>67.968</mn></mrow></semantics></math></inline-formula>. On the contrary, KIC 3766353 is undergoing the process of orbital shrinkage with only three weak pulsation–orbital resonance frequencies available and shows a non-integer ratio of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mi>pul</mi></msub><mo>/</mo><msub><mi>P</mi><mi>orb</mi></msub><mo>=</mo><mn>83.163</mn></mrow></semantics></math></inline-formula>. Given the results, the theoretical relation is a potential proxy to distinguish between apsidal precession and orbital decay binary systems. Furthermore, we predict that the orbital period of KIC 3766353 will be reduced to 2.492 days, at which time it will be transformed into apsidal precession.
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