Stars are like people. Just as there are, in a sense, billions of types of people, there are tens of thousands of types of variable stars. However hard we try to classify variables, we always run into exceptions, and when we create a new category for the exceptions to the old category, that usually turns out to have exceptions too. Although it is an exaggeration to say that every variable star in the sky is in a class by itself, it is useful to think of variables as individuals capable of surprises.

There is a different aspect, however, of the philosophy of classification. Just as we can find strength and beauty by looking at the diversity of language, behavior, and culture in people, a look at the kinds of variation in stars will help to show just how extraordinary this field of study really is.

Some classifications are obvious. You would never want to confuse a stately, mature, slowly varying red giant like Mira with a little dwarf star that erupts every two months. An Orion variable, changing brightness for no apparent rhyme or reason, would not be confused with a slightly unpredictable semiregular. The eclipsing binaries, revolving together in clockwork fashion, are not the same as the intricate ‘breathing’ of the Cepheids. Astronomers recognize these broad divisions that help us both to understand the different patterns of variation and to plan our observing programs for them.

Variable star observing was something people had to appreciate and discover. As an observing discipline, it was forced on us time and again as bright exploding suns intruded in our otherwise placid sky. Opening and closing suddenly with each major nova, observers could watch these intruders with fear and with questions, but these stars did not appear to have any lasting effect on early civilization. Novae and comets shared the uncertain interpretation of being signs of something else, rather than being objects of intrinsic interest.

Ancient Chinese and native American records of supernovae are the earliest surviving variable star reports. The “guest star” of 1054, for example, appeared at a time when these people would be looking beyond, to new things, and would keep careful records of anything extraordinary.

The history of variable star observing closely parallels the sudden launching of Renaissance astronomical curiosity. The quiet, orderly sky of Ptolemy offered comfort with safe spheres, all fixed in content, of which the stellar was the most distant or purest.

In 1572 a bright new light punched a hole through that sphere, shattering the concept of order without change. Observations by Tycho recorded not only the appearance of this incredible sun but also its changing brightness over time.

There is something magical about a star that, in its final phase of life, announces its end to the entire Universe. On July 4, 1054, Chinese skywatchers were stunned by something new in the sky. The “guest star” they reported was the outburst of a supernova, triggered by a major instability and collapse within the star. Since then at least three other supernovae have been easily visible from Earth, in 1572 and 1604, a time when European civilization was just about ready to accept new thoughts on the stability and arrangement of stars in space, and in 1987, when our understanding of the process of a supernova was finally good enough to be tested.

In 1054, native American records consisting of paintings and carvings on rocks welcomed a bright new star in Taurus. More accurate Chinese records have given us information so that we can pinpoint the date of the event as July 4, 1054. Much brighter than nearby Aldebaran, the exploding star could be visible in daylight.

In cold weather, we draw inward as the frigid weather and short days beckon us away from the stars and towards the armchair. This is an unfortunate loss for northern hemisphere observers who forego the unparalleled richness and diversity of the sky at this time of year, a sky that dares us to defy the inside comforts and go outside and watch. This is a time of challenge.

With its stunning belt and sword regions, Orion is the first area we would look to for possible variables, and we will not be disappointed! Orion's Great Nebula harbors some of the most fascinating variables of the entire sky. Lurking within the nebula are some 50 variable stars, 10 of which are bright enough to be observed with a 15 cm (6 inch) telescope.

With telescope, winter offers a host of unparalleled, delightful stars that are infrequently observed because of the clouds and the cold. U Geminorum, which can rocket from 14.2 to 8.8 in a few hours, is a highlight of winter observing.

Observing hints for cold weather

A winter night can be a devastating experience, which under no circumstances should be taken lightly. During his search for trans-Neptunian planets, one quiet, wind-free night, Pluto discoverer Clyde Tombaugh opened the shutter of the 13 inch telescopic camera and began an exposure.

With growing experience you can follow several enticing stars throughout their entire range with standard 7 x 5 0 binoculars. These four stars are easy to find; just use the charts in Figs. 13.1, 13.2, and 13.3.

R Scuti

This most interesting bright variable ranges over about two magnitudes in almost five months. However, it shows many irregularities, and even its period, listed as 140 days, is not precise. Because the inspiring open cluster NGC6705 (M11) is nearby, R Scuti is as easy to find as it is fun to estimate. The range is large, it is well placed in the sky from May to September and is a fine example of an easily observed bright variable.

R Scuti is a special type of star, in the RV Tauri class. At first its light performance mimics Beta Lyrae, although the period is much, much longer and the maxima are sharper. But then the shallower of the two minima gradually deepens until it becomes the most pronounced low point, and the star now behaves more like a long period version of Delta Cephei. You should estimate R Scuti once each week.

X Herculis, g Herculis, and RR Coronae Borealis

These are semiregular variables that provide an interesting weekly project (Fig. 13.3). X Herculis is a semiregular red giant that seems to have been deliberately placed in the sky so that you would have an easy time finding and estimating it.

Someone may have once told you that astronomy calls for large, expensive equipment. You may even have flipped through the pages of an astronomy magazine in amazement at all the fantastic technology on display there. At some point in your development as an astronomer, you may feel that a telescope will build your interest and extend the power of your observations. For now, you are probably much better off with a simple pair of binoculars, and this is true for viewing some variable stars. Because binoculars are mass-produced and sold almost everywhere, they are far less expensive than are telescopes, even those of the same size. By taking advantage of both your eyes, binoculars present the sky in an efficient, almost three-dimensional way. In variable star observations binoculars are used widely, for many of the semiregular variables are bright enough to be followed with them.

The only problem with binoculars is that the two small telescopes that form their optical system must be precisely aligned. So many binoculars lose their adjustment with the bumps and insults of regular use. You should at least be sure to start with a good pair; test them before purchasing by pointing them to a distant building or mountain. Holding them securely, make certain that the image in one side is precisely the same as that in the other; landmarks should fall in the same place in both circular fields of view.

What is the first thing you notice about the stars? Quite likely, it is their differing brightnesses. Although this appears obvious, it is the single most important concept with which you should become familiar before you can be a variable star observer.

Why do stars differ in brightness? Is it because they are at different distances from us, so that the farther stars appear fainter, like the glow from lamps at the far end of the street? Or are the stars themselves of different brightnesses? As you have likely guessed, both are correct. Sirius, a blue star off the southeast corner of Orion the Hunter, is normally the brightest star in the night sky, not so much because it is actually large and bright, but because it is close. At a mere 8 light years away, Sirius is one of our nearest neighbors. However, there exists a star only a little farther away, Wolf 359, whose intrinsic brightness is so low that it cannot be seen with the unaided eye. Meanwhile, the brightest star in Cygnus the Swan, Deneb, is well over 1000 light years away from us. S Doradus, intrinsically one of the brightest stars of all, appears to us as a faint star shining dimly because it is so far away, actually in a neighboring galaxy.

To reconcile these two factors, astronomers have created two independent systems of describing brightness or “magnitude”.

I owe Rick Hill ten dollars.

It all started one evening when we were discussing the unpredictability of these strange stars we call recurrent novae. Suddenly Rik proposed a wager. Suppose, for example, T Corona Borealis erupts every 80 years. With its last outburst in, say, 1946, I could bet ten dollars that it would next erupt around 2026.

This would be quite a wager, with uncertainty so great that were Rik to bet on 2000 and me to say 2050, we would stand an equal chance of winning.

Since I didn't want to keep ten dollars in some distant bank that long, I suggested we try a dwarf nova, a star that erupts about every 50 days, not years. “No, SS Cyg is too regular,” Rik countered. “There are variable stars that make much better bets.”

Anyway, I talked him into a $10 wager on SS Cygni. A week later, at 11:30 p.m., in the midst of a dark southern Arizona night, the telephone rang ominously.

It was Rik. “Have you seen SS Cyg tonight?” SS Cygni is regular? Who says? It went into outburst seven days early! And I lost my bet!

Even at minimum this star shines at a not-too-faint twelfth magnitude, visible with a city-bound 20 cm (8 inch) telescope. The field is not easy to find, but once you succeed, it is easy to remember (Fig. 18.1).

Clyde Tombaugh, who discovered Pluto in 1930, began his career with astronomy on a Kansas farm. To brighten up one long day of farming he asked himself, “How many cubic inches are there in Betelgeuse?” His answer, with what we know today, would have been 10 to the 41st power! On the next clear night he looked skyward, with a twinkle in his eye, to the reddish chief of Orion. One of its secrets given away, Betelgeuse twinkled back.

Orion is master of the winter sky. From city sky or country, from almost any part of the world, the majestic figure of the Hunter dominates the sky with belt, sword, and club. Look to the southeast early in a January or February evening, or to the south in the March evening, and discover Orion. The keys to this constellation are the three stars that line up in a neat row. The westernmost one is called Mintaka, a delightful Arabic name meaning Belt. Using the belt as a beacon, Betelgeuse is one of the easiest stars in the sky to find. The three stars in a row are surrounded by a four-sided figure of four bright stars. The star in the northeast corner of the figure is Betelgeuse. The best time to see Betelgeuse is on January 29, when it is in the sky most of the night.

Z Ursae Majoris and the AAVSO method

A huge red giant, Z Ursae Majoris displays some unusual behavior in this late stage of its life. Although it usually ranges from magnitude 6.5 to 8.3, it occasionally surprises us. Once I watched it drop to 9.5 — more than a full magnitude below its normal minimum.

With the earlier starter stars we used a simple method where 1 represents the brightest and 5 the faintest. Now that we have some understanding of what the process means, we need not use it; now we are going to play the variable star game by the rules as outlined by the American Association of Variable Star Observers.

Look closely at the two charts (Figs. 8.1 and 8.2) for Z Ursae Majoris. Fig. 8.1 is designed for binoculars and has north up. Fig. 8.2 is meant for Newtonian viewing where the image is usually inverted. South is up, north down, and east and west are exchanged.

In Fig. 8.1 you recognize the Big Dipper, and near the top of the bowl is the circle-and-dot symbol for the variable star. Near other selected stars are numbers like 66, 54, and 64. These numbers represent magnitudes and tenths, but the decimal point is left out to avoid confusion with the other points that represent stars. Therefore, read “66” as “6.6” and so on. If you look at the area with binoculars, and are able to find Z and see that it is easily visible, then you can use Fig. 8.1.

As the number and quality of your observations grows, your need to record them accurately and efficiently also becomes apparent. Since the recording procedure aids the accuracy of your observations, I suggest that you record your observations in a way that is designed to prevent your being influenced by previous observations.

First, record your data in an observing log that contains the results for each night's observing session. This record may well contain information about all the observing you do, including the Moon and planets, double stars and galaxies, even the Sun. You would begin with the date, times, observing site, weather conditions, seeing, quality of the sky, and instrument used. Then you would list each star that you observed during the night, by name, time, and magnitude estimate. This should be a permanently bound notebook, not a spiral or ring binder that will fall apart before it is of use to someone else.

The second step begins after your session is over, when you transfer your data to a special file for each star. If you file all of your variables on index cards and if you keep them in the order of variable star designation, you will find the compilation of your monthly observing report an easy operation. Several generations of variable star observers have found index cards or sheets of paper appropriate, but now computer programs may handle these records more quickly.

Variation on the Sun! For many years the AAVSO has had a section for observation of the Sun, the logic being the star around which we revolve is a variable. In a stretched sense this may be true, but if we were to observe the Sun as we watch other stars, from light years away, we would find it shining at a constant brightness, without any indication whatsoever of its 11 year cycle of variation that we see manifested in sunspot activity.

Whether we worship it, plan our lives by its schedule, tan ourselves by its light, bask in its warmth, or study it, the Sun is a star whose importance cannot be overstated. And when we observe it through our telescope, we learn much about the the churning, changing nature of the star around which our planet turns.

An amateur astronomer and pharmacist of Dessau, Germany, Heinrich Schwabe, discovered the Sun's “variation” in 1843 through his long series of meticulous observations of its activity. After buying a small telescope he began to search for a planet inside Mercury's orbit, hoping to find it transiting the Sun's surface. This “Vulcan” idea still lives, and as late as 1982 infrared searches have attempted to find such a planet. It has not been found. Schwabe's serendipitous discovery was the 11 year sunspot cycle.

The most obvious solar feature is the sunspots, magnetic storms on the solar surface that appear dark because they are cooler than the rest of the surface.

Two of the most famous Mira variables in the sky, R Leporis and Chi Cygni, are challenging, but for different reasons; R Leporis is unusually red, and Chi Cygni lies in a rich field of stars.

R Leporis

Until you've seen R Leporis at maximum, you haven't seen red. Here is a star whose redness offers us a new interpretation of color, a transcendent presence of vivid hue from a great distance.

In my early years of stargazing, I was guided by an old book by J. B. Sidgwick called Introducing Astronomy. During hundreds of observing sessions it taught me faithfully, pointing out the constellations one by one, as well as the inspiring contents of each. I especially remember the description of M42 — as exciting to read about as the nebula was to look at. Then I'd turn the page for Lepus, the Rabbit, just to see what glories were hidden from me in the little constellation that couldn't quite hop above the treetops of my southern horizon. It was Introducing Astronomy that taught me about R Leporis, Hind's Crimson Star, that shone in the sky like a drop of blood. As much as I longed to see this star, I expected I never would until either the trees fell down or I moved to a better site.

Of all the heavenly bodies visible in the night from the Earth, the Moon is the most spectacular. It far outshines even the most brilliant planet, moves so quickly that you can see its motion against the stars, and provides a wealth of detail in the shadowy features of its disc. Yet its motion is the most difficult to predict and it is for that reason we have left it until last. It is of course in orbit about the Earth but the Sun and other members of the Solar System perturb that orbit to such an extent that many corrections are needed to calculate the Moon's position accurately.

In the next few sections we use a simple method to find the position of the Moon. The method takes account only of the principle perturbations to the orbit yet gives results which are accurate enough for most purposes. We also calculate the times of moonrise and moonset, the phases of the Moon, and the circumstances of both solar and lunar eclipses. The calculations are lengthy but the satisfaction you feel when you accurately predict, for example, the occurrence of a lunar eclipse, cannot be denied.

The Moon's orbit

To an Earth-bound observer, the Moon appears to be in orbit about the Earth, making one complete revolution with respect to the background of stars in 27.3217 days. This period is called the sidereal month. During this time the Earth moves on along its own orbit so that the Sun's position changes with respect to the stars.

Practical astronomy with your calculator has been written for those who wish to calculate the positions and visual aspects of the major heavenly bodies and important phenomena such as eclipses, either for practical purposes or simply because they enjoy making predictions. I have tried to cut a path through the complexities and difficult concepts of rigorous mathematics, taking account only of those factors which are essential to each calculation and ignoring corrections for this and that, necessary for the very precise predictions of astronomical phenomena. My simple methods are usually sufficient for all but the most exacting amateur astronomer, but they should not be used for navigational purposes. For example, the times of sunrise and sunset can be determined to within one minute and the position of the Moon to within one fifth of a degree.

The second edition included much more material in response to letters and requests from readers of the first edition. I also corrected many errors. This new third edition continues the same process. You will find four new sections on generalised coordinate transformations, nutation, aberration, and selenographic coordinates; I have improved the sunrise/set and moonrise/set Calculations so that they work properly everywhere in the world; the section on precession now contains a rigorous method of calculating precession as well as the approximate one of previous editions; I have taken account of the new J2000 astronomical system where appropriate, and have updated the material by assuming the epoch 1990 January 0.0 as the starting point for calculations; and I have corrected mistakes and clarified obscurities wherever I have been aware of them in the second edition.