Secondary minima for eccentric orbit

[pavolgaj]

Secondary minimum is not at E=0.5 for eccentric orbit. Adding offset of epoch manually or calculate it..

[pavolgaj]

Solved on paper;) Waiting for implementation - in the calculation of epoch (all classes) + new window in GUI.

[pavolgaj]

General solution:

1. calculate epoch for secondary minima (i.e. the difference in epochs between primary and secondary minima) $\Delta E$ from eccentricity $e$ and argument of pericenter $\omega$:
   1.0 some general relations for true anomalies of primary ( $\nu_0$ ) and secondary minima ( $\nu_1$ ):
   $$\Delta\nu\equiv\nu_1-\nu_0=\pi$$ $$\nu_0=\frac{\pi}{2}-\omega$$
   1.1 difference in eccentric anomalies between minima: $$\Delta\epsilon = -2\arctan \frac{\sqrt{1-e^2}}{e\cos\omega} $$
   1.2 sum of eccentric anomalies in both minima: $$\epsilon_0+\epsilon_1 = -2\arctan \left(\sqrt{1-e^2}\tan\omega\right) $$
   1.3 difference in mean anomalies: $$\Delta M = \Delta\epsilon-2e\sin\frac{\Delta\epsilon}{2}\cos\frac{\epsilon_0+\epsilon_1}{2}$$ or $$\Delta M = \Delta\epsilon+\frac{2e\sqrt{1-e^2}\cos\omega}{1-e^2\sin^2\omega}$$
   1.4 difference in epochs: $$\Delta E = \frac{\Delta M}{2\pi}$$

2. determine the type of minima from its time $t$ using reference time $T_0$, orbital period $P$ and calculated epoch difference $\Delta E$:
   2.1 calculate "observed epoch": $$E_{obs}=\frac{t-T_0}{P}$$
   2.2 calculate observed phase: $$f_{obs}=E_{obs}-round(E_{obs})$$
   2.3 type of minima is:

• primary ( $type=0$ ) if $|f_{obs}| < \min\left( |f_{obs}-\Delta E|, |f_{obs}-\Delta E+1| \right)$
• secondary ( $type=1$ ) otherwise
  2.4 real epoch is $$E=round(E_{obs}-type \Delta E) + type \Delta E $$

[pavolgaj]

Some figures and examples - epoch/phase of secondary minima as a function of the argument of pericenter (eccentricity) for different values of eccentricity (argument of pericenter).

[pavolgaj]

Example of wrong calculation of epochs for secondary minima and O-Cs for hypothetical EB with period 15 days and eccentricity 0.05 (argument of pericenter is 40 deg). In this system phase of secondary minima is 0.51 (no 0.5 as for circular orbit).

Neglecting eccentricity causes a shift between primary and secondary minima for about 4 hours!

If we assume eccentric orbit, there isn't any shift.

[pavolgaj]

Fix in dev branch - commit 2b6d743.