What happens if the period isn't quite right?

You go out and measure the brightness of a variable star on one night. This particular star varies rapidly, so that it undergoes several cycles during a single night. Based on your measurements last night, you decide

the period is P = 0.10000 days (how convenient!)
the star reached minimum light last night at exactly 11:30 PM, which corresponds to Julian Date 2,453,850.650

Over the next week, your colleague Dr. Binary is scheduled to use the telescope. You want to let him know when he can expect the star to reach minimum light. How can you tell him?

Exercise:

What time(s) tonight will the star be at minimum light? Express your answer(s) in local time.

Express these same times in Julian Date.

At what times TOMORROW night can Dr. Binary expect to see minima? Use Julian Dates again.

Can you come up with a compact, relatively simple way to describe the times of minimum light for any particular date in the future? In other words, an equation?

The usual form of the ephemeris for a variable star looks something like this:



          Time of min  =  T₀  +  N * Period

where T₀ is the time of one minimum -- any one will do -- and N is any integer. Below is an example from the literature; note that for some reason, astronomers have chosen to use the letter E to stand for "number of cycles since the given minimum."

Yes, this is W UMa ... but measured long ago. See the references at the end

The ephemeris shown above should provide accurate predictions for the time of minimum. However, there is always the possibility of an error creeping into our measurements. Let's look at three different types of error.

getting the time of one minimum wrong
miscalculating the period itself
assuming incorrectly that the period is constant

What if the time of one minimum is not quite right?

At one point during your single night of observing, you measured the star's light to decrease to a minimum at JD 2,453,850.650. Dr. Binary observes the star over the next couple of weeks and measures several times of minimum himself. He finds


   your prediction     his measurement
 -------------------------------------------------
       851.650             851.660

       852.750             852.760

       854.850             854.860

       857.850             857.860

       859.650             859.660

       865.750             865.760
--------------------------------------------------

When astronomers face a disagreement between predictions and observations, they often resort to graphs instead of tables or equations. One common tool is the O - C graph . To make it, you put

the date (JD) of an observation on the horizontal axis
the difference between Observed time and Computed time on the vertical axis

Plot these particular measurements on the graph paper provided.

Exercise:

What sort of pattern do you see in the graph?

What if the period is not quite right?

At one point during your single night of observing, you measured the star's light to decrease to a minimum at JD 2,453,850.650. You also determined a period of P = 0.10 days.

Suppose that (in some alternate universe) Dr. Binary observes the star over the next couple of weeks and finds a different set of results. He measures


   your prediction     his measurement
 -------------------------------------------------
       851.650             851.660

       852.750             852.771

       854.850             854.892

       857.850             857.922

       859.650             859.740

       865.750             865.901
--------------------------------------------------

Compute the (O-C) value for each of Dr. Binary's times of measured minimum light. Then plot the points on your graph -- use a different symbol than you used for the earlier dataset.

Exercise:

What sort of pattern do you see in the graph?
What is the period, based on Dr. Binary's measurements?

What if the period isn't really a period?

At one point during your single night of observing, you measured the star's light to decrease to a minimum at JD 2,453,850.650. You also determined a period of P = 0.10 days.

Suppose that (in some second alternate universe) Dr. Binary observes the star over the next couple of weeks and finds a different set of results. He measures


   your prediction     his measurement
 -------------------------------------------------
       851.650             851.685

       852.750             852.812

       854.850             854.930

       857.850             857.879

       859.650             859.605

       865.750             865.214
--------------------------------------------------

Compute the (O-C) value for each of Dr. Binary's times of measured minimum light. Then plot the points on your graph -- use a different symbol than you used for the earlier dataset.

Exercise:

What sort of pattern do you see in the graph?
What is the period, based on the first pair of Dr. Binary's measurements in the table? (How many cycles fall between them?)
What is the period, based on the final pair of Dr. Binary's measurements in the table? (How many cycles fall between them?)

Over very long periods of time, astronomers do see some stars change their period slightly. This is one of the few ways in which we can actually see (a very few) stars evolve (at least a little tiny bit) within a single human lifetime.

For more information

The Variable Star of the Month for August, 2002: XZ Cygni is an easy-to-read article describing long-term changes in a pulsating star. Thanks to the AAVSO for writing it.
Determining the Ephemerides of Shortperiod Eclipsing Binaries: V566 Ophiuchi is a nice example of a student project, showing step-by-step how one can analyze the light curve of a variable star.
Four-colour UVBY observations of W UMa by Linnaluoto and Piirola provided some of the examples in this lecture.