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 = T_{0}+ N * Period

where **T _{0}** is the time of one minimum -- any one
will do -- and

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

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?

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?

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.

- 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.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.