Variable light output from stars can be caused by several phenomena. Basically there are three types of variable stars intrinsic variables, eruptive variable, and extrinsic variables. With some variable star projects, only timing of the period is important. Others require filter photometry and careful calibration of the system and comparison stars.

Intrinsic Variables
The intrinsic variables have their light variations caused by internal phenomena. The stars may pulsate, e.g., Cepheid variables. This pulsation may be a long period or short period pulsation, semi-regular, or irregular. Also, a relatively new group of stars, known as RS CVn variables are believed to have large star spots (similar to sun spots only much larger) that travel around on the surface of the star. These spots are cooler than the rest of the surface so as they migrate they cause the star's brightness to change.

These stars vary because of physical instabilities in their atmospheres, which give rise to stellar pulsations. Therefore, the variations are intrinsic to the stars themselves. As the stars expand and contract, both the luminosity and temperature change. Without discussing all classes of intrinsic variables, we call attention to three types and refer readers to the texts by Hoffmeister et al. (1985) and Petit (1987) for additional details.

One common type of intrinsic variable claims delta Cephei as its classical prototype. Hence, these stars are called Cepheid variables. The periods of Cepheids range from a few days to a few weeks. The pulsations are quite regular, and the pulsation period is related to the mean stellar luminosity. This relationship is of vital importance as it allows Cepheids to act as "standard candles" in calibrating distances to nearby galaxies.

Assuming the physical conditions in remote Cepheids are similar to those found in local examples, the only difference between them is the reduced intensity due to distance. Therefore, from a simple determination of the period of the Cepheid, a value for its distance can be found. Of course, observers with small telescopes are not going to measure light from individual stars in external galaxies.

The use of standard candles depends on the calibration of galactic counterparts that are within the range of modest instruments. Although the basic physical state of Cepheids is understood, new observations of selected systems are warranted because changes can arise in their light curves, indicating changes in the underlying atmospheric conditions.

A second type of intrinsic variable is exemplified by the prototype star RR Lyrae. Objects of this type are quite numerous in the galaxy, particularly within globular clusters.

Like Cepheids, the pulsation periods of RR Lyrae stars are rather stable, with nearly all stars having a period less than one day. However, some stars show systematic changes in their photometric period and need further monitoring to define their behavior. The stars can be used as standard candles, but only to a limited extent and in nearby galaxies, because RR Lyrae stars are intrinsically fainter than Cepheid variables.

The very broad class of stars known as long-period variables is a third type of intrinsic variable star. The most famous of these is Mira, or o Ceti. This star was the first to be recognized for having luminosity variations. The star cycles between V=3 and V=9 (roughly) with a period of slightly less than one year. The variation is much more pronounced in the visual bandpasses (i.e., >6 magnitude in B and V) than in the near-infrared (i.e., perhaps only 1-2 mag in R and I). Therefore, the change in color of Miras is very pronounced.

There is a long history of visual observations of Mira variables, with organizations such as the American Association of Variable Star Observers coordinating much of the effort. In addition to visual observations, the AAVSO also maintains a network of photometric observers.

Those interested in contributing to this research should write to AAVSO, 25 Birch Street, Cambridge, MA 02138. Wing and Hall (1983) have called attention to the important work that can be done on Miras using small telescopes, particularly at near-infrared wavelengths. Contact Dr. Douglas Hall, Dyer Observatory, Vanderbilt University, Nashville, TN 37235, for additional information.

Eruptive Variables
The eruptive variables are those that tend to produce sudden outbursts of energy. Examples are Flare stars, Nova and Supernova types.

Extrinsic Variables
The extrinsic variables have there light variations caused by external reasons, e.g., another object (star) passing in front of them. These are known as eclipsing binaries. While they may be eclipsing binaries, the systems may contain more than two stars and may have intrinsic and eruptive variables within the system. Much information about the star system can be learned from the light curves. By using filters, e.g., UBV filters, additional information about the system can be obtained.

The number of identified variable stars is quite substantial, with over 25,000 entries listed in the General Catalogue of Variable Stars. The vast majority of these objects have been little studied. Often, the only parameters known are the general type and range of variability and an estimate of the period. Amateur astronomers can fill a valuable scientific role by observing important stars using photoelectric photometry.

The question that remains is "What stars are appropriate targets?" To answer this, beginning photometrists are encouraged to contact active astronomers at a local college before heading to the telescope. Not only can they give advice regarding object selection (of course, biased by their own research), but also they should include the names of backyard scientists as co-authors of resulting professional publications.

In this section several types of projects are prresented that may be of interest to photometrists using small telescopes.

The appropriate project to pursue depends on the available equipment, time, and weather. Generally, stars with shorter periods need more frequent observing, either on a single night or several successive nights. This restriction is often troublesome if a complete light curve (i.e., covering at least one cycle) cannot be obtained during a reasonably short time interval. It becomes very frustrating if poor weather rolls in on the night(s) needed to complete photometric coverage at all phases.

Calculating Phase
Both eclipsing and intrinsic variable stars often have published periods. If a zero point in time is also given, then new observations can be directly compared to old ones by calculating the phase corresponding to the time of the measurements.

For an observation obtained at time T, the phase, f, is expressed as

f= mod((T-T_o)/P,1) for T > T_o
f = 1 - mod((To-T)/P,1) for T_o >T

where P is the period and T_o is a specified time that defines phase zero. Depending of the star, T_o may refer to one of several photometric phenomena. For example, T_o may be the time of mid-eclipse or minimum light in an eclipsing binary, or the value may refer to the time of maximum light in a long-period variable. The mod function is defined in the section on LST.

Therefore, use the above equations to calculate the expected phases of a given variable before going out to observe. Extend the calculations to all nights or portions thereof that may be available and only undertake projects that have a reasonable chance of completion. Of course, make sure that the object is well placed in the sky (i.e., at zenith distances <60 degrees) for all planned observing sessions. There are few things worse than partial photometric coverage!

Example:
If the published parameters for a star are P=3.456789 days and T_o= 2447654.321, what is the phase of an observation obtained at T= 448123.456?

Note:
T and T_o are Heliocentric Julian Dates (HJD)

f = mod((2448123.456 - 2447654.321) /3.456789,1)
f = mod(135.7141,1) = 0.7141

Therefore, since T_o the star has completed 135 cycles and is currently 0.7141 through another. The star is said to be at phase 0.7141. In general, Heliocentric Julian Dates are necessary to calculate f. An exception to this occurs if P is large enough so that the time correction becomes negligible. The first equation covers the special case when T_o is in the future and cycles are being counted in reverse order. Hence, phase -0.7 is the same as 0.3.

RS CVn Systems
The RS CVn systems, named after their bright prototype, are interesting versions of eclipsing binary systems. The observed light curves of RS CVn systems show a superposition of eclipses with other variations that are attributed to star spots on one of the components.

Due to stellar rotation (which is not synchronized with the orbital period - which is usually a few days or weeks), the signature of the star spots does not remain constant in terms of phase. Therefore, the composite light curve changes shape from one observing season (i.e., the continuous run of days that a star is well positioned for observing) to the next.

Dr. Douglas Hall has been instrumental in coordinating the efforts of amateur astronomers working on RS CVn systems. Contact him for additional information at douglas.s.hall@vanderbilt.edu.

From a photometric perspective, the advantages of studying these stars are that a complete light curve can be constructed over the course of several months and it need not be obtained in all bandpasses. Usually, only observations in V are needed because there is little variation with color.

Be Stars
These stars are early-type stars that show hydrogen emission lines in their spectra. The emission is thought to arise in an equatorial disk or ring that surrounds a rapidly rotating B star. As a class, these stars exhibit variability on many time scales, from hours to years. The variations are mostly irregular, but are correlated with spectral changes. Evidence for an underlying repetition has been found in some systems. The brightest Be star is Cas with V~2.5. Other bright examples are Tau and the Pleiades member Tau.

Observers interested in pursuing photometric work on Be stars should contact Dr. Petr Harmanec at Ceskoslovenska Akademie Ved, Astronomicky Ustav, 251 65 Observator Ondrejov, Czechoslovakia, and inquire about the international observing campaign on these objects. Similar to observations of RS CVn stars, it is not essential that daily monitoring be obtained. Instead, observations of Be stars are needed every few days or weeks over the course of the observing season.

Eclipsing Systems
These systems contain two or more stars in orbit around each other. The observed variation in intensity lies solely on the perspective from which they are viewed (i.e., on geometric considerations) rather than intrinsic properties of the stars. If the orbital plane on the two of the stars is sufficiently close to the system's line of sight, then light from one star is periodically eclipsed by its companion.

These systems usually are too far away and cannot be resolved with a telescope. Photoelectric photometry provides a means to gather information about these systems, see Figure 46.

Figure 46 An Eclipsing Binary Star System

Eclipsing Binary Star Systems
The importance of studying eclipsing binaries lies in the fact that they are governed by Kepler's Laws. Particularly, the third law that relates orbital period to the semi-major axis (i.e., recall that P²(m1 + m2)=a³ if P is measured in years, m₁ and m₂ in solar masses, and a in astronomical units). The range of periods is very extensive - from just a few hours (as seen in cataclysmic variables and low-mass X-ray binaries) to several decades (as demonstrated by Aurigae).

Although there are several subtypes of eclipsing binaries, the classical example is Algol, or Persei. This system has deep primary eclipses (1.28 mag in V) which last several hours and repeat every 2.87 days.

Systems that consist of a cool giant or supergiant star in orbit with a hot dwarf are particularly interesting. The periods of these systems are typically years, with eclipses that last a weeks or months.

The value of observing these systems lies in using the small hot star as a probe of the extended atmosphere that surrounds the large cool component. Although the duration of total eclipse is virtually independent of the bandpass used, partial phases last much longer for shorter wavelengths than for longer ones. As more data on individual long-period systems are gathered, it is possible to define variations in the size of cool-star atmospheres.

For persons who cannot observe on a continuous basis, these systems offer the advantage of isolating the most important observing time to distinct intervals near the eclipses. Only a limited amount of photometry is needed at other times.

The disadvantages of working on long-period systems are that often none of the bright systems is near eclipse at a convenient time to observe and that nearly all the "action" is concentrated on a few specific nights - which may be cloudy.

An astronomer with modest equipment can make significant contributions to astronomy by doing photoelectric photometry of these systems. These are unique in that they not only provide the observer with an observing project, but also allow the ambitious observer to do some analysis of the data. There is a great deal that can be derived from the data. When using filters information about the spectrum of the stars can be obtained along with the orbital periods, orbital inclination, size of the components, and masses of the components.

Suggested Long Period Eclipsing Binary Star Systems
Following is a list of some long-period eclipsing binary stars that are in need of observations. These star systems make up a class known as the Aurigae class of eclipsing binaries. These are all fairly bright and easy. They can be rather exciting when the star system goes into eclipse. They can all be seen from the northern hemisphere except V777 Sagittae which is at -26^o declination..

STAR SYSTEM	PERIOD
Epsilon Aurigae	9885 days
VV Cephei	7430.5 days
31 Cygni	3784 days
32 Cygni	1147.6 days
Zeta Aurigae	972.2 days
V777 Sagittae	936.07 days
22 Vulpeculae	249.1 days

The following are brief write-ups on each of these star systems. A summary of data follows these descriptions. These data summaries are the results of observations at the Hopkins Phoenix Observatory, research of papers, star atlases, and various star catalogues. Several years of effort went into compiling these summaries. If information is found to be incorrect or if data that are indicated as unknown become known, please contact the author and this information be included in future editions.

Epsilon Aurigae (Aur)
The Epsilon Aurigae system is perhaps the most interesting eclipsing star system. It has puzzled astronomers for over 150 years. It is a bright star (3rd magnitude) located about 3o southwest of Capella and eclipses once every 27.1 years. It is at the vertex of a triangular group of stars known as "The Kids". Zeta Aurigae, another interesting long-period eclipsing binary, makes up one of the other two stars.

What makes this star system so intriguing is not just its long period but the length of its eclipse and what happens during the eclipse. Typically the eclipse lasts about two years which with the 27.1 year period means the eclipsing body must be gigantic. There have been no satisfactory explanations for this. To make matters even more interesting, there seems to be a mid-eclipse brightening. How can this be? One explanation, according to James Kemp, is the eclipsing body is a giant cloud of gases enclosing two small stars in orbit around each other. These stars sweep out an area in the middle. It would be a bit like a giant donut. This donut must be tilted such that as it eclipses the primary star, the system's total light decreases until the "donut-hole" allows some of the primary star's light to sneak through at mid-eclipse, see Figure 47.

Figure 47 Epsilon Aurigae Star System

To try to unravel this system's mystery, a concentrated effort was undertaken during the 1982-1984 eclipse. Hundreds of astronomers, amateurs and professionals, from around the world, observed the eclipse. Space born satellites observed in the ultraviolet and infrared. Ground based observations were photometric, spectroscopic, and polarmetric. Photometric observations were made with UBV filters, narrow band filters, and at wavelengths into the far infrared. Despite the concentrated efforts, Epsilon Aurigae still remains a mystery. The secondary eclipse was due to occur around 1996/1997. Because the secondary eclipse light variation is on the order of the primary star's pulsations, separating the seconardy eclipse from the pulsations is difficult. An effort is underway to try to predict the pulsations through continuing observations. If these pulsations can be predicted, observation of the secondary eclipse may be possible.

For those astronomers still interested, the late summer of 2009 will be a good time to start a new campaign to observe the next eclipse. Surely, by the end of the next eclipse, astronomy will have unraveled the mystery of Epsilon Aurigae.

Reference: Stencel, R.E. , North American Workshop on the Recent Eclipse of EPSILON AURIGAE , January 16-17, 1985, NASA Publications.

VV Cephei (Cep)
The VV Cephei star system has the third longest period with a period of 20 years or 7430.5 days (note KQ Pup has a period of 9752 days). VV Cep is a 5th magnitude system. The total eclipse is less than Epsilon Aurigae by nearly half a year (totality for Epsilon Aurigae is 670 days and for VV Cephei it is 490 days). The last eclipse of VV Cephei was due in 1997. VV Cep is one of the most massive binaries known. Its estimated mass is 100 solar masses and the primary star has a diameter of 1621 times the Sun. If it were our star, its surface would extend to between the orbits of Saturn and Uranus

Reference: Van De Kamp, P. 1978, Sky & Telescope; The Distances of VV Cephei and Epsilon Aurigae, 397-399, November 1978.

31 Cygni (Cyg)
Several star systems (31 and 32 Cygni, Aurigae, 22 Vul, and VV Cephei) make up what is called the Aurigae type system. These star systems are unique because they allow study of the primary star's atmosphere. When the smaller blue star passes behind the primary star, the ingress and egress of the eclipse are times when the blue star's light is passing through the primary star's atmosphere. It is these parts of the eclipse that are of most interest as they yield information about the primary star's atmosphere.

31 Cygni is a 4th magnitude star and is easy to observe. It has a period of 10.4 years. As with the other stars of the Aurigae group, the eclipse of 31 Cygni is shallow (0.12 magnitudes) in the "V" band but very pronounced (0.5 and 1.7 magnitudes respectively) in the "B" and "U" bands.

Reference: Stencel, R.E., Hopkins, J.L., Hagen, W., Fried, R., Schmidtke, P.C., Kondo, Y., Chapman, R.D., 1984, The Astrophysical Journal; THE 1982 ECLIPSE OF 31 CYGNI, 281:751-759, June 15.

32 Cygni (Cyg)
This star system is similar to the 31 Cygni system and is also a member of the group of Aurigae systems. 32 Cygni is a 4th magnitude star and has a period of 3.15 years.

Zeta Aurigae (Aur)
The Aurigae star system is a 6th magnitude system with a period of 2.7 years. It consists of a very hot and luminous blue star plus an even more luminous orange super giant star. Because of the vast difference in sizes, only the primary eclipse where the orange super giant passes in front of the smaller blue star, is observable. Along with the other star systems listed here, the z Aurigae system allows observations of the chromospheric, or outermost layers, of the orange super giant star. By careful observations during the ingress and egress phase of the eclipse, information about the super giant's atmosphere can be gained.

As might be expected, the UBV observations of the eclipse will produce a small change in the "V" (0.1 to 0.2 magnitudes) but nearly 2 magnitudes of change in the "U" band.

Reference: Darling, D., 1983, Astronomy; The Curious Case of Zeta Aurigae, 66-70, March 1983

V777 Sagittae (Sgr)
Little data are available on this 2.6 year eclipsing binary star system. This would make an excellent project for an observer in the Southern Hemisphere.

22 Vulpeculae (Vul)
This star system is a new member of the Aurigae group. This 5th magnitude star system was discovered during the 1984 eclipse. The eclipse in the "V" band is only 0.05 magnitudes but increases to nearly 0.2 magnitudes for the "B" band and nearly 0.5 magnitudes in the "U" band.

Reference: Parsons, S.B., Ake, T.B., and Hopkins, J.L., 1985, Publications of the Astronomical Society of the Pacific ; THE AUGUST 1984 ECLIPSE OF 22 VULPECULAE, 97:725-730, August 1985.

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