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Phase calculation of a variable with period variation

MMN
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Sat, 03/14/2020 - 17:35

Hello, I have collected a series of minima of an eclipsing binary. The trend shows a variation over time well resolved with an ephemeris with quadratic element as the formula:

JDmin = Epoch + P*E + a*E^2

Now, to build a light curve obtained over a period of years and to take into account the variation of the period, I think that it is not correct to use the simple formula: 

Phase = (HJD -E) / P

In fact using the above formula the light curve shows a shift over the phase.

What is the correct formula to use?
Thank you!

Massimiliano Martignoni (MMN)

Sat, 03/14/2020 - 20:15
If you substitute JDmin from

If you substitute JDmin from your first equation, for E in your second equation, for each observation, this will give you a useful phase for each observation, for making a phased light curve.

In other words, for each observation, compute E for the first equation, the cycle number (without decimals, just the integer whole number), and then compute the start time for that cycle.  That's what JDmin gives you, using your first equation.

Then use this in your second equation for E.  Note that in your second equation, E means something different:  it is the start time of the cycle, or the epoch for that cycle.

Limitations of this:

1) determining the cycle number is a bit tricky.  If you just compute  (HJDobs-Epoch)/Period, you get a number with a fractional component, i.e. numbers other than zero to the right of the decimal point.  If you always chop off the numbers to the right of the decimal, you'll get phases from 0 to 1.  You can also round up for fractions greater than 0.75, and then you'll get phases from -0.25 to +0.75, which sometimes makes a better plot.

2) a more serious limitation, is you will probably still still get "phase smear".  I.e. your first equation may do a very good job of fitting the the trend curve, but it won't be perfect.  You'll get a better light curve if you can limit the observations you are using to as short a span as possible, while still giving a well populated light curve.

I hope this helps.

Gary Billings

P.S.  now that you've fitted the parabolic ephemeris (your first equation), subtract that from your time of minima, and re-analyze the resulting "residuals" for additional details!  Often these "second order" effects are very interesting.