I want to convert the list of Julian date values of observing brightness of a Cepheid variable into its phase.
The equation I found in the Internet is the following:

$$\mathrm{Phase}= \frac{\mathrm{HJD}-T_0}{P} - \mathrm{INT}\left[\frac{\mathrm{HJD}-T_0}{P}\right]$$ What is the $\mathrm{INT}$ in this equation? I only know that $T_0$ is the one reference point among the list of Julian date values and $P$ is the period.

Also, what kind of criteria exist in determining the reference point?


INT refers to the floor function. So $\mathrm{INT}(x)$ is the largest integer not exceeding $x$.

In this case, the function is used to get the fractional part of $\frac{\mathrm{HJD}-T_0}{P}$, i.e. the part after the decimal point. So if for example $\frac{\mathrm{HJD}-T_0}{P}=3.4$, then $\frac{\mathrm{HJD}-T_0}{P}-\mathrm{INT}\left(\frac{\mathrm{HJD}-T_0}{P}\right)=0.4$.


You are free to pick your reference point any way you want. You could pick a $\text{HJD}$ at which the star is brightest, or dimmest, or exactly average, or you could pick the $\text{HJD}$ of the first observation, or you could even set $T_0 = 0$. If two people pick different $T_0$, then the phases they find differ by a constant value (modulo 1).

It is convenient to set $T_0$ at or before the smallest $\text{HJD}$ in your observations, so that all $\text{HJD} - \text{T_0}$ are nonnegative, because the $\text{INT}$-like functions provided by calculators, computer languages, and software libraries don't all round negative numbers in the same direction. Many round all numbers towards zero, so that $+3.4$ yields $+3$ (as we want) and $-3.4$ yields $-3$, but here we want $-3.4$ to yield $-4$, otherwise you get negative phases before $T_0$ and positive phases after $T_0$.

If your observations of brightness are evenly spaced in time, then you can use a Fast Fourier Transform to find the approximate period and phase.


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