MadSci Network: Astronomy Query:

### Re: What is a variable star Observed minus Calculated (O-C) diagram?

Date: Wed Feb 9 13:01:06 2000
Posted By: Jeff Brown, Faculty, Astronomy
Area of science: Astronomy
ID: 948953297.As
Message:

An "O - C" diagram is a tool used by astronomers to assess what is going on with a variable star. (There are a few variable stars bright enough to see with the unaided eye; these include eclipsing binaries (see here and here, among other places), and various kinds of pulsating variables of which Cepheids are the most famous type.)

Several classes of variable stars are periodic, that is, they are doing something over and over with a more or less regular cycle. An obvious example is two stars orbiting each other; some kinds of pulsating variable stars show impressively regular periods as well. One of the things an astrophysicist does in trying to understand a variable star is predict its behavior. As an example, for an eclipsing binary, this means determining the orbital parameters of the binary system and from those predicting when the eclipses of the system will occur. (We use the word "predict", which commonly implies future events, but the same word is used for computing estimates for events in the past, which have already been observed. If variable star has a short period, like a few days, then accumulating new data may be quickly done, but there are systems with periods of years, and often an astronomer's career is not long enough to hope to accumulate enough observations to check a model in one's lifetime.)

Once one has got a set of predictions for a star, then one should compare those predictions to real observations, and one way of presenting this comparison is the use of an O - C diagram. (There's an O - C diagram for the eclipsing binary star Beta Lyrae here, which is Figure 8 of a technical paper on that star.) The horizontal axis of the diagram is easiest to understand: it is simply time, usually expressed in days, most often the Julian Day number (JD) of the observation. The vertical axis is the "O - C" part which gives the diagram its name and its interpretive power. For each observed event (an eclipse, to continue the example) one takes the observed time of the event (that's the "O" part) and subtracts the time predicted from the model for the event (that's the "C", calculated, part). The difference, Observed minus Calculated or "O - C", is plotted on the vertical axis of the graph.

The pattern that shows up in the O - C diagram can tell you not only whether your model is any good, but also whether there are things going on in the real star that aren't included in your model. If your model is perfect and you have got the right value for the period of the variable star, then the points in the O - C diagram will be a straight and level line across the graph: if the difference between the observations and model is zero for all time, you can easily imagine that your graph then will look like a bunch of points, all with O - C at or near zero, for a long time. If you have the right idea for what's going on but your period is slightly off, then your model will get the event times right for a short while, but the discrepancy will accumulate as times goes on. The O - C diagram in that case will look like a slanted straight line. A more interesting possibility is if the O-C diagram turns out to be curved; that is what results if the period of the system is changing slowly. A period change like that means that something else that you haven't included is going on in the system, and that something else may be novel astrophysics.

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