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I'm not quite sure how I find the orbital phase ($\phi$) of a transitting exoplanet. For example, I have this RV data in this article:

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First of all, I don't know how that phase or "cycles" is derived. But that's not the phase $\phi$ I'm looking for but this (in the same article):

enter image description here

As you see, that phase goes from 0 to 1, and I know it has to do with the time of passage from the periastron and the data time (the HJD column), but I'm not able to derive this number in the interval [0,1]. So, if we assume the orbital parameters are known, how can we derive that number?

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The phase as plotted on the graph is constrained in the range [0,1) to plot everything in one orbital cycle (this makes it easier to see the shape of the RV curve). The phase can be obtained by taking the fractional part of the total number of cycles. So the first row in the table with the cycles number 28.677 corresponds to phase 0.677.

Update: As to what the cycles number is, it is a measurement of time since a reference time (usually the transit time), counted in units of the orbital period of the planet. In the case of a transiting planet, the orbital period will be known from the transit interval, so it doesn't need to be derived from the RV curve.

In this case, the first row in Table 1 has time $t = \mathrm{HJD}\ 2455107.37937$, while in Table 2 the transit time is given as $E = \mathrm{HJD}\ 2454967.27571$ and the orbital period as $P = 4.885525\ \mathrm{days}$. So the number of cycles can be calculated as $\mathrm{cycles} = (t-E)/P$ giving 28.677 (to 3 decimal places).

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