Theoretical RR Lyrae Lightcurve

Preparing model

Blazhko modulation

Each Blazhko period modulates the amplitude, phase, and mean light of the light curve.

\[ \begin{array}{r@{\quad}r@{\;}c@{\;}l} \eqlabel{Pulsation phase} & \phi_0(t) & = & \operatorname{frac}\!\left(\frac{t-\Delta t_{\mathrm{LTT}}}{P_0}\right) \\ \eqlabel{Blazhko phase} & \psi_j(t) & = & \operatorname{frac}\!\left(\frac{t-\Delta t_{\mathrm{LTT}}}{\cpb{P_{B,j}}} + \coffset{\psi_{0,j}}\right) \\ \eqlabel{Amplitude scale} & A(t) & = & 1 + \sum_j \camp{a_j} \sin(2\pi\psi_j) \\ \eqlabel{Phase shift} & \Delta\phi_B(t) & = & \sum_j \cphase{d\phi_j} \cos(2\pi\psi_j) \\ \eqlabel{Mean shift} & \Delta M(t) & = & \sum_j \cmean{dM_j} \sin(2\pi\psi_j) \\ \eqlabel{Shifted template} & \widetilde{M}_0(t) & = & M_0\!\left(\phi_0-\Delta\phi_B\right) \\ \eqlabel{Magnitude} & M(t) & = & \bar{M} + A(t)\left[\widetilde{M}_0(t)-\bar{M}\right] + \Delta M(t) \end{array} \]

Default parameters fitted from OGLE-BLG-RRLYR-07497.

Light travel time effect

The binary companion shifts the apparent pulsation phase by the line-of-sight orbital delay.

\[ \begin{array}{r@{\quad}r@{\;}c@{\;}l} \eqlabel{Periastron epoch} & T_{\mathrm{peri}} & = & T_{\mathrm{ref}} + \cperi{\eta_{\mathrm{peri}}}\,\corb{P_{\mathrm{orb}}} \\ \eqlabel{Mean anomaly} & \mathcal{M} & = & \frac{2\pi(t-T_{\mathrm{peri}})}{\corb{P_{\mathrm{orb}}}} \\ \eqlabel{Kepler equation} & E - \cecc{e}\sin E & = & \mathcal{M} \\ \eqlabel{Orbit coordinate} & y_E & = & \sqrt{1-\cecc{e}^{2}}\sin E \\ \eqlabel{Orbit coordinate} & x_E & = & \cos E-\cecc{e} \\ \eqlabel{True anomaly} & \nu & = & \operatorname{atan2}\!\left(y_E,\;x_E\right) \\ \eqlabel{Time delay} & \Delta t_{\mathrm{LTT}} & = & \frac{\casini{a\sin i}}{c}\, \frac{(1-\cecc{e}^2)\sin(\nu+\comega{\omega})}{1+\cecc{e}\cos\nu} \\ \eqlabel{Observed phase} & \phi_{\mathrm{obs}}(t) & = & \operatorname{frac}\!\left(\frac{t-\Delta t_{\mathrm{LTT}}}{P_0}\right) \end{array} \]

Default orbital parameters fitted from OGLE-BLG-RRLYR-06498.