A comprehensive investigation of the variable overcontact system EH Cancri

Abstract

Precise multi-color CCD-derived photometric data were obtained from EH Cnc at two sites during 2010, 2014 and 2018 wherein each epoch used a different instrument. This has provided a unique opportunity to investigate parameter uncertainty following Roche modeling of light curves optimized by differential corrections using the Wilson–Devinney code. Furthermore, new radial velocity data from EH Cnc presented in this study for the first time has produced absolute physical and geometric parameters for this A-subtype W UMa-type variable. Analysis of eclipse timing data confirms the presence of sinusoidal-like excursions in the eclipse timing residuals. We address whether these are due to magnetic activity cycles, the so-called “Applegate effect”, or related to a light-time effect (LiTE) resulting from the presence of a third gravitationally bound cool low mass white dwarf. A model using the PAdova & TRieste Stellar Evolution Code (PARSEC) has provided valuable insight about the evolutionary history of EH Cnc as a trinary system.

Appendices

Appendices

1.1 Appendix A: RV error analyses

The error estimate for a given fitted quantity (such as $K_{1}$, $K_{2}$, or a derived quantity such as mass) was determined as follows: Let $Q = f (y_{1}, y_{2},\ldots , y_{n}$), where $Q = K_{1}, K_{2},\ldots , y_{i}$ are the individual RV points, and f is a function not known explicitly but recoverable from the Solver fit. If we increase quantity $y_{i}$ by its estimated error $\delta y_{i}$ and reapply Solver we obtain a new value, $Q_{i}^{\prime} = f (y_{1}, y_{2}, \ldots, y_{i} + \delta y_{i},\ldots , y_{n})$. The resulting difference, $\mid Q_{i}^{\prime} - Q\mid$ may be taken as $\delta Q_{i}$, a value due to error in point $y_{i}$. Then, assuming statistical independence, the error in Q is given by the standard formula:

$$\begin{aligned} \delta Q = \sqrt{\delta Q_{1}^2 + \delta Q_{2}^2 +\cdots +\delta Q_{n}^2}. \end{aligned}$$

(A1)

In this case, the number of data points was sufficiently small so that values for all the $\delta Q_{i}$ were computed. This procedure has been used in other circumstances where the number of points, n, was too many for all values $\delta Q_{i}$ to be evaluated. In this case, a random sample of points, $y_{i}$, was taken and results scaled up.

1.2 Appendix B: Derivation of phase-smearing correction (R. H. Nelson)

Let $I(\lambda ,t)$ equal the instantaneous spectrum, and compute the RVs by a broadening function routine such that

$$\begin{aligned} V(t)=B\cdot [I(\lambda ,t)], \end{aligned}$$

(B1)

where B is some undefined function. (If the procedures and parameters of the broadening function process are fixed, B will return a unique value for a given input, and is therefore a function.) But, since instantaneous spectra cannot be taken, we need to integrate over some time interval:

$$\begin{aligned} \Delta t = t_{2}-t_{1}=T. \end{aligned}$$

(B2)

At the mid-time, $t_{m}$, we compute

$$\begin{aligned} \tilde{V}(t_{m}) = B\cdot \left[ \frac{1}{T}\int _{t_{m}-\frac{T}{2}}^{t_{m} +\frac{T}{2}}I(\lambda ,t)\,{\mathrm{d}}t \right] . \end{aligned}$$

(B3)

It follows that since function B is linear (Ruciński 1992; Nelson 2010c),

$$\begin{aligned} \tilde{V}(t_{m}) = \frac{1}{T}\int _{t_{m}-\frac{T}{2}}^{t_{m}+\frac{T}{2}} B\cdot [I(\lambda ,t)]\,{\mathrm{d}}t = \frac{1}{T}\int _{t_{m}-\frac{T}{2}}^{t_{m}+\frac{T}{2}} V(t)\,{\mathrm{d}}t. \end{aligned}$$

(B4)

If the orbit is circular, and outside of eclipse then

$$\begin{aligned} V(t)= K\sin (\omega t), \end{aligned}$$

(B5)

where $\omega =2\pi /P$, K is the amplitude, and P is the period. Then

$$\begin{aligned} \tilde{V}(t_{m}) = \frac{-K}{\omega T}\left[ \cos \left( \omega t_{m}+\frac{\omega T}{2}\right) -\cos \left( \omega t_{m}-\frac{\omega T}{2}\right) \right] =\frac{2K}{\omega T}\sin \left( \frac{\omega T}{2}\right) \cdot \sin (\omega t_{m}). \end{aligned}$$

(B6)

Set

$$\begin{aligned} x=\frac{\omega T}{2}=\frac{\pi T}{P}=\frac{\pi (t_{2}-t_{1})}{P}. \end{aligned}$$

(B7)

Then

$$\begin{aligned} V(t) =\frac{\sin (x)}{x}\cdot K\sin (\omega t_{m})= f\cdot V_{m}, \end{aligned}$$

(B8)

where $f=\frac{\sin (x)}{x}$. Typical values of f are 0.95, 0.97, etc. Note that this correction affects only the scale of the RV plots, and cannot change the computed mass ratio. Therefore, the “true” value

$$\begin{aligned} V_{m}=\frac{\tilde{V}(t_{m})}{f}. \end{aligned}$$

(B9)