The observation of variable stars (often called "variables") can be undertaken by amateurs for simple pleasure, as an educational exercise, as a contribution to serious scientific research, or all of these.

However, be warned. Many people over time and across the world have found the observation of variable stars to be addictive. Of course, the careful study of individual stars is not for everyone, but one of the great things about astronomy is the breadth of activities that an amateur can choose from. If you find that using you eyes to study variables is not attractive, you may consider the measurement of these stars using photoelectric instruments. These may be CCD cameras, or photoelectric photometers with photodiode or photomultiplier tube detectors such as those manufactured by Optec Inc. in the United States of America.

Binoculars are easy to use, there is almost no setup time, and of course you can only observe the brighter variables, but it is therefore easier to find your targets. Most observers use 7x50 or 10x50 instruments (7x or 10x magnification, with objective lenses - the big ones at the front of the binoculars - having a diameter of 50mm). Some observers use more powerful instruments for fainter targets, but binoculars which have a higher magnifying power are larger, and it will therefore be more difficult to hold the field of view steady, unless you use some form of mount.

Finding Variable Stars with "Go To" Computerised Telescopes

Many modern telescopes are computerised, with a "go to" function, allowing the user to key in the identity of the target, and then to sit back and watch as the telescope swings toward it. If you wish to find variable stars this way, I recommend that you create a chart with a reasonably bright "finder star" near the variable, and that you programme the telescope to go to that bright star first. Unless you have a very expensive setup, your target may not be in the centre of the field of view, and if that is the case, having a bright, easily recognizable target makes life easier.

Variable Stars and Comparison Stars

The principle is to compare the variable with two comparison stars, one slightly fainter (preferably no more than a few tenths of a magnitude) and one slightly brighter than the variable. Then, try to estimate if the brightness of the variable is about half way between those of the comparison stars, or if it seems to be closer in brightness to one comparison star than the other. Figure 1. below is a cartoon depicting three "stars". A is the brighter comparison star, B the variable, and C the fainter comparison star. The diameters of the black circles represents the relative brightness of the stars.

Figure 1. Cartoon to illustrate the comparative visual brightness of a variable star (B) with a brighter comparison star (A) and a fainter comparison star (C).

B looks fainter than A, but brighter than C. The thing is, how much fainter than A, and how much brighter than C? To answer this question, it may be useful to imagine the brighter comparison star, the variable and the fainter comparison star on a straight line. Then imagine that the differences in magnitude between the bright comparison star and the variable, and the variable and the faint comparison star can be expressed as one or more steps.

For example, suppose that we think that the following is the situation:

COMP1 (1 STEP) V (1 STEP) COMP2

That is, we think that the difference in brightness between COMP1 and the variable is the same as the difference in brightness between the variable and COMP2.

This is better illustrated with an example, in which the magnitude of COMP1 is 5.1, and the magnitude of COMP2 is 5.5. Application of the above expression yields:

5.1 (1 STEP) V (1 STEP) 5.5

But the usual way to express this is:

5.1 (1) V (1) 5.5

We calculate the magnitude of the variable as follows:

There are two brightness "steps" between COMP1 and COMP2. The ratio of the steps is 1:1, and the brightness of the variable is ½ of the way (1 step) between the brightnesses of COMP1 and COMP2 (which are 2 steps apart). So, with the COMP1 magnitude being 5.1 and the COMP2 magnitude 5.5, the difference is 5.5 - 5.1 = 0.4 magnitude. The magnitude of the variable is ½ of that difference fainter than COMP1 (remember that the higher the magnitude figure, the fainter the star - a 5.5 magnitude star is fainter than a 5.1 magnitude star). The calculation is therefore as follows:

Variable = 5.1 + (1/2x0.4) = 5.1 + 0.2 = 5.3

This is probably easier to understand on a visual analogue scale, where 5.1 is the magnitude of COMP1, 5.3 is the magnitude of the variable V, and 5.5 is the magnitude of COMP2:

5.1 5.2 5.3 5.4 5.5

This example can also be used to illustrate the principle that each "step" can be of any magnitude range, and therefore can be defined however you wish. So, the "step" from 5.1 to 5.3 can be thought of as 2 steps, each of 0.1 magnitude. Similarly, 5.3 to 5.5 can be 2 steps of 0.1 magnitude. When magnitude differences are small, using 0.1 magnitude as a step can sometimes be convenient.

Lets look at another example.

5.1 (2) V (1) 5.8

Here there are 2 + 1 = 3 steps between COMP1 and COMP2.  The ratio between the two steps is 2:1.  The variable is thus estimated to be 2/3 (i.e., 2 steps over 2+1=3 steps) of the way between the brightness of COMP1 and the brightness of COMP2.  Magnitude of COMP2 - magnitude of COMP1 = 5.8 - 5.1 = 0.7 2/3 of 0.7 = 0.46.  Magnitude of variable = 5.1 + 0.46 = 5.56.  After rounding up, the magnitude of the variable is reported as 5.6, since it is usual practice to report a visual estimate of magnitude to the nearest 0.1 magnitude.

If the result of a calculation yields a figure such as 6.55, simply round off, and report the magnitude as 6.5.

In practice, if the magnitude differences between the variable and the comparison stars are quite small, I simply conjure up a mental image of a visual analogue scale, and estimate the magnitude of the variable directly.

Observational Techniques

It is obvious that you need to look at the variable star, and look at each of the comparison stars that are relevant to a particular estimate. Start by comparing the variable quickly with nearby comparison stars. You will soon realize which of the comparison stars are brighter, and which are fainter than the variable.

Don't try to make any quantitative estimates at this point, as the aim is to select two comparison stars, one which you can see is just brighter and one which you can see is just fainter than the variable. You may at this point find that the variable star is indistinguishable in brightness from one of the comparison stars. If that is the case, find two more comparison stars, one fainter and one brighter than the first one, just to check that the variable is indeed between these two. But if you are confident that the variable looks identical to one of the comparison stars, then the magnitude of that comparison star becomes your estimate of the magnitude of the variable.

However, you will often have to study the variable and two comparison stars, one brighter and one fainter than the variable, and at this point you must now do this carefully. If the variable and the comparison stars are in the same field of view, you can, if you wish, hold the binoculars or telescope steady, and move your eye(s) from one star to another. If you do this, it is recommended that you draw an imaginary line between the two stars that you are comparing at any one moment, and make that line parallel with a line joining your two eyes. This procedure will result in some gymnastics if you try to carry it out properly, so be careful if you need to use a ladder to reach the eyepiece of your telescope! Start with one of the comparison stars and the variable, and flick your eyes from one to the other and back again, looking at each star for only 1 to 2 seconds. You will gain an impression of the difference between the two stars. Now, do the same thing with the other comparison star and the variable. After doing this, you may be able to judge if the variable is closer in brightness to one of the comparison stars than the other. You may have to repeat the observing sequence until you are confident about your observations. I have personally found this method to be awkward to apply rigidly, because of the necessity to twist my head around to line up the stars as described. Therefore, I usually use the another method - which is to bring each star into the centre of the field in turn to observe it.

When I first read about the latter method, I thought that it would be very difficult to carry out. In fact, I found that it allowed me more easily to distinguish between stars that differed by only a small part of a magnitude (e.g., 0.2 magnitude). What I do is this. Bring the variable into the centre of your field of view, look at it for 1 to 2 seconds, look away to the dark sky for a second, look at the variable for 1 to 2 second, and repeat this sequence several times until you feel that you have a "visual memory" of the brightness of the star. Now bring a comparison star into the centre of the field and repeat the procedure of alternately looking at the star and at the nearby sky. You should be able to detect a difference. After repeating this with the variable and the other comparison star, you may be able to judge if the variable is closer in brightness to one of the comparison stars than the other. You may have to repeat the observing sequence until you are confident about your observations.

My personal experience is that it requires some persistence and practice to feel reasonably confident with your estimates. Don't be discouraged if you don't make perfect estimates on your first night out. Keep trying, and I think that you will find that your skill will develop.

Optimizing the Accuracy of Your Estimates

First of all, avoid errors. Ensure that you identify the variable and the comparison stars correctly, that your judgment of the relative brightness of the variable and the comparison stars is as good as you can make it, and that you take care with calculations of the type we have just described above.

Accuracy of estimates will be improved if the magnitude difference between the relevant comparison stars is small, ideally only a few tenths of a magnitude. If the difference is greater than one magnitude, it will be more difficult for most observers to make accurate estimates. It is not always possible to achieve the ideal, particularly in parts of the sky away from the plane of the Milky Way, where the numbers of stars of any magnitude that you could see in the field of view of your instrument may be quite small.

Be aware of the colours of the stars that you observe. The Johnson-Morgan B-V colour index is most widely used. Blue stars have negative indices (e.g., Rigel has a colour index of -0.03), and orange to red stars have a much higher colour index (e.g., 2.5 or higher). It is ideal if the colour of your comparison stars is close to the colour of the variable. This is possible for some variables, such as eclipsing binaries, but it is not possible, for example, for Mira-type long period variable. By their very nature, Mira variable are red, and most very red stars are variable! Therefore, comparison stars for most long period variables will be bluer than the variable.. Why is this important? The response of the human eye to the red end of the visual spectrum differs considerably among visual observers. Two observers with different red responses may differ in their estimates of the same star by several tenths of a magnitude. In my personal experience of observing V Crucis, another observer and I differed by up to one magnitude.

How accurate can you be? With practice, your estimates may differ by no more than 0.2 magnitude from the actual figure. How can you determine this? A good way is to choose some star fields, select stars that are NOT variable, make observations of the sort described above, then go to your planetarium programme and note the measured magnitudes of the stars. The magnitudes in the Bright Star Catalogue (stars down to 6th magnitude), and the Tycho-2 Catalogue (for stars down to 10th or 11th magnitude) are generally good enough for checking visual estimates, and for assigning magnitudes to comparison stars.