Fine-tuning of light-time effect in triple systems

Abstract

The sequence of eclipses of binary stars is subject to inequalities for various reasons. The presence of a third component in the system causes periodic motion of the binary's center of mass along the line of sight of an observer. The finite value of the light velocity implies that the epochs of eclipses periodically advance and delay with respect to the exact orbital period of the binary, a phenomenon termed the light-time effect (LITE). We aim to refine two aspects of the mathematical treatment of LITE. First, we provide both generalized and more accurate analytic formulation describing the light-travel time in the binary system itself presented in previous works. Second, we analytically estimate the so far neglected coupling of LITE with the dynamical interaction of the binary orbit with the motion of the third star. Our principal results are given in a simple analytical form, which is suitable for the analysis of photometric observations that require minimization over a multidimensional parameter space of the triple system. The leading correction to the traditional formulation of LITE due to the light-travel time in the binary system may be detectable for triple systems with a period ratio of $P_2/P_1\lesssim 20$, for which accurate photometric observations are available. On the other hand, the correction due to the dynamical coupling of the two orbits with $P_2$ periodicity is small, but may become relevant in the future.

Figures (1)

• Figure 1: Upper panels: LITE correction due to finite light travel time in the binary system: black triangles are from our analytical model (e.g., $\Delta t_\pm^{\rm in}$ and $\Delta t_\pm^{\rm out}$ from Eqs. \ref{['EC3']} and \ref{['EC5']} for primary eclipses), red and blue stars from direct numerical integration of light travel through the the system (primary and secondary eclipses). We used parameters of a well constrained multiple system $\xi$ Tauri nemr2016 that harbors inner triple system with a distant fourth component (for sake of simplicity the effect of the fourth star was neglected in this example). The triple system is near coplanar with inclination $i\simeq 87^\circ$, inner and outer periods $P_1=7.146$ d and $P_2=146$ d, eccentricities $e_1=0$ and $e_2=0.2$. The results are shown for three different values of the argument of pericenter of the outer orbit, $\omega_2=0$ (left panel), $\omega_2=\pi/2$ (middle panel), and $\omega_2=3\pi/2$ (right panel; the present value is $\simeq 10^\circ$ and precesses with a rate $\dot{\omega}_2\simeq 2^\circ$ yr$^{-1}$ due to interaction with the eclipsing binary). The masses are $m_0=2.23$ M$_\odot$, $m_1=2$ M$_\odot$ and $m_2=3.74$ M$_\odot$. Time at abscissa in days covers three revolutions of the outer orbit, LITE on the ordinate in seconds. The mean value of the correction $\simeq a_1/(2c)$ would combine with the second term on the right hand side of Eq. (\ref{['lite0']}) to be near zero. The $P_2$-periodic variation with an amplitude of $\simeq 15$ s is significant and exceeds uncertainty of the TESS photometry. Middle and lower panels: the residuals between the exact numerical simulation and our analytic formulation are shown for primary eclipses by red stars in the middle panel and secondary eclipses by blue stars in the lower panel. Their average value is zero and no signal above $\simeq 0.005$ seconds (namely $0.03$% of the LITE correction) is observed.