Introduction

These stars vary in luminosity caused by the star pulsating. The two forces at play are radiation pressure pushing outward and gravity squeezing inward. When the star is at its maximum diameter, radiation pressure is at a minimum, the temperature is low and gaseous velocity is zero. Gravity then makes the star contract, the pressure then increases, and so the temperature and the gaseous velocity. At its minimum diameter, the pressure will make the star “inflate.” The pressure and temperature and pressure will diminish and when the maximum diameter is reached, the whole process starts again. The repetition period is often vary irregular.

RR Lyraea

Definition

RR Lyrae variables exist in large numbers. They were once known as “cluster variables” because the first of them were found in globular clusters. There are currently more than 6000 in the General Catalog.

The typical star is RR Lyrae, the brightest of this group. It varies from visual magnitude 7.06 to one of 8.12 with a period of 13h 36m or 0.56682326d. The light curve is highly asymmetric with the rise to the maximum taking 2h 35m and the fall taking about 11h. The luminosity variation accompanies a significant change in color: B-V changes from +0.14 at max to +0.45 at minimum. So the amplitude of 1.06 in visual magnitude corresponds to one of 1.37 in the B magnitude. The spectra show a large change from A8 to F7.

Description

There are three subtypes: RR, RRB, RRC

RR (such as RR Lyr) is characterized by an amplitude near to, or greater than, one magnitude, a period of about 0.5 day and by marked asymmetry. It is believed that the oscillations take place at the fundamental frequency. The shortest period 0.22d, the longest(modal value) is 0.55d. The maximum periods, although rare, exceed 1.3 d.

RRAB (such as RX Leo) is characterized by less amplitude than the first group (0.5-0.8 magnitude), the period is longer (-0.7 day), the maximum is more rounded and more symmetric. It is believed that the oscillations take place at the fundamental frequency.

RRC (such as RZ Cep) is characterized by an almost sinusoidal light curve with a rounded maximum, 0.5 amplitude, and 0.3 day period. It is believed that the oscillations take place at the first harmonic frequency. They are slightly bluer, and so hotter than the first two groups. The shortest period 0.20d, the longest (modal value) is 0.34d. The maximum periods are 0.55 d. Very few of this type compared to the two above.

Distribution

RR Lyrae stars are population II stars so there are large numbers in the galactic nucleus, in the halo, and in the globular clusters (although some have none). They are “old” less massive stars being on average 0.43-0.48 that of the Sun, their radii about 4-5 times larger than the Sun, resulting in low density. They form a quasi-spherical system, density decreasing slowly from the galactic plane (they are still found at 10kpc from it) and increasing sharply towards the center of the Galaxy. At 4kpc the density is four times greater than near the Sun (9.5kpc). Unlike the Cepheid’s, there is no correlation between period and galactic position.

The Cepheids

Definition

The typical star, Delta Cephei, has been known since antiquity. Hipparchus cataloged it in 126 Goodricke detected its variability in 1784. It is a supergiant with spectrum varying from F5 at maximum to G2 at minimum. The period is 5d 8h 47m 30s; maximum magnitude is 3.4 V amplitude of 0.83 (1.30 in B, 1.66 in U). The rising part of the cycle lasts 1d 14h 30m, the decline lasts 3d 18h 17m. Its absolute magnitude is -3.4. There is a 7.5 magnitude, spectral type AO V companion 41sec from and physically connected to the primary. There are 700 known in the galaxy.

Typical Cepheids, or those with small amplitudes, are often associated with galactic clusters. Some long term Cepheids belong to O associations, and are very young objects. The mass of Cepheids is not known with great precision, estimates put it at four solar masses for short period stars and ten solar masses for the longest period stars of this type. Like all supergiants Cephids are evolving rapidly and the Cephid state should last no longer than 10 million years.

Description

The Cepheids are in two classes, the Classical Cepheids are population I stars and Delta Cephei is the prototype star. The W Viriginis type Cephids are population II stars.

The Classical Cepheids themselves fall into two categories. In the first group the amplitudes of the variations are from 0.7 to 1.2 magnitudes (the highest of the groups and the asymmetry is well marked at 0.25 to 0.30. Those stars with periods of 7-9 days have a distinct bump in the descending part of the light curve. Those with short periods have a smooth light curve. For stars with periods of 10-12 days the amplitude is smaller than stars with 7-9 day amplitudes. The asymmetry is 0.4. For stars with a period of 15 days the bump appears at the beginning of the rise to maximum, disappears towards 20-30 days but a small nodule is then seen at the maximum.

The typical star of the W Virginis type was discovered by Schonfeld in 1866. There are now over 200 of this type known. As in the Classical Cepheids class, this class is made up of two groups, depending on their periods. The first group contains stars of very short periods of less than 2.5 days. They sometimes have a peculiar light curve with one bump on the rising part of the curve and another on the decline. Amplitudes are often greater than one and the asymmetry is pronounced.

The second group has periods longer than 10 days with most between 15 and 20 days. The amplitude often exceeds one magnitude. For periods up to 20-23 days, the asymmetry becomes less noticeable and there are bumps on the descending part of the curve which appear for the longer periods. It is not always easy to differentiate between the two groups.

It is very rare to find Cepheids with periods between 2.5 and 10 days.

Like all pulsating variables the Cepheids are subject to variation in their periods. These are often small but can be determined by the accumulated effect on the star’s period. The changes in period may be continuous, with a regular increase or decrease, or they may be abrupt. In some cases the period alternately increases and decreases. Finally, the variations can become very complex. The long-period variables often have large variations I period, particularly those belonging to population II stars where variations can be as much a 3% of the period, or about one half a day at a period of 17 days.

The Period-Luminosity Relationship

In 1912 H. Leavitt noticed that the Cepheid’s in the Small Megallanic Cloud had a strong correlation between their period and magnitude. H. Shapley found eleven bright Cepheids whose distance could be determined through parallax. From this he deduced the absolute magnitudes and used these as a calibration for the period-luminosity relationship. Since then it has been possible to use Cepheids to determine distances to other galaxies.

Soon a complication became obvious. There seemed to be two different versions of Cepheid variables, the other now called RR Lyrae stars after the archetype stars. Shapley’s calibration of zero was incorrect. In 1952 Baade found that the Cepheids in the Andromeda galaxy were 1.5 magnitudes fainter than they should have been. It was determined that the distances determined by the correlation must be multiplied by 2. This immediately made the observed galaxies two as large and twice as far away. Actually this meant that the universe itself was twice as large as thought; this cause a revision of the Hubble constant.

Distribution

The Classical Cepheids make up a typical population I; most of them are in the spiral arms and form a very flat system, 90% being less than 100 parsecs from the galactic plane and none at all at more than 300 parsecs. They are young stars, many, mainly long-period stars are associated with clouds of hydrogen in the galactic arms.

The W Virginis type Cepheids are distributed throughout the galactic halo and their numbers decrease only very slowly as the distance from the galactic plane increases. They are still found at several thousand parsecs from the galactic plan.

The average distance to the galactic plan is 90 parsecs for Classical Cepheids and 610 parsecs for W Virginis stars. The mean radial velocity is 15/km/s and 36 km/s, respectively.

Classical Cepheids are distributed differently according to their period. Those with the shortest periods occur in large numbers in the outer arms (the Cassiopia, Orion, Canis Major-Pupis region. Those with long periods often occur in arms nearer the galactic center (Carina-Centaurus, Vela, Sagittarius).

Population II Cepheids with the shortest periods are found mostly towards the galactic longitude of 180 degrees, or towards the anti-center, whereas CW stars located in the direction of the galactic center have short (often less than 5 days) periods.

Typical Cepheids, or those with small amplitudes, are often associated with galactic clusters. Some long term Cepheids belong to O associations, and are very young objects.

Long-period

Definition

There are a large number (+5000) of long-period variables. The typical star in the group is Omocron Ceti or Mira, the first variable star ever discovered (Fabricius 1596). The period was first determined by Bouillaud in 1667. He found a value for P of 333 days but also noted that this was not constant and that the brightness was not the same from one maximum to another. W. Herchel and then Argelander obtained more precise period measurements until eventually determining the period to be 331d 15h.

At some maximums Mira reaches a magnitude of 2 at others it barely reaches a magnitude of five. Minima also show irregularity, varying from magnitude 8.3 to 10.1, Maximum amplitude is 8.1 magnitudes, and average is 6.5 magnitudes. The period fluctuates from 310 to 370 days, with a mean value of 331.6 days.

The Mira light curve is further complicated by the fact that Mira is a double or possible a triple. The companion was discovered in 1923 and has a Beq spectrum and is itself a variable, from magnitude 9.6-12.0.

The (still uncertain) orbital period of the stars is 400 years, with a mean angular separation of 0.85 and eccentricity of 0.66. There could also be a third companion with a period of about 29 years.

Mira has a maximum visual magnitude of about -2 and the companion VZ Cet about +4.5. Measured by interferometer the diameter of Mira is 0.053”, or about 4 AU.

Description

Long-period variables have spectra of type M, C, or S, with M the most common type, all others making up less than 10%. Many long-period variables have an intense emission lines (Me), particularly of Hydrogen. Originating from giants they belong to luminosity classes II and III.

The 1356 M-type stars have periods from 80 to more than 700 days, and an asymmetry between 0.4 and 0.5. The modal value is 280-290 days. The 69 S-type stars have longer average periods, with a mode of 360 days. Stars with periods of less than 200 days have a much more rounded light curve. The 70 C-type stars have a maximum of 450 days, with an asymmetry of 0.5 (same rise and fall times).

M and S stars are low temperature, Omicron Ceti fluctuates between 2640 and 1920K, and for R Tri between 2400 and 1950K. Some are even cooler.

The spectrum changes completely in the course of a cycle. The emission lines, especially the Balmer series (of Si and FE), which are intense near maximum, decreasing to almost nothing near minimum, appearing again at the rising phase of the next cycle. Absorption bands (Titanium Oxide – M stars, Zincronium Oxide – Stars, Lanthanum Oxide) are weak at maximum but very strong at minimum. Traces of Technetium sometimes found in S-type stars. At maximum emission lines and absorption lines differ by 10-20km/s, at minimum they are almost identical.

Photoelectric measurements of these very red stars show intense spectra in the infrared regions, at maximum, the V-I index is very high: +5.0 for Omicron Ceti, +8.5 for T Cep and more than 10 in some cases. In the infrared Mira reaches magnitude -2 and Xi Cyg magnitude -5.

The C stars are completely different from the M and S stars. They are even redder than the M stars: for V Aql the B – V index is +4.0 and the U – B +6.5. So a star of visual magnitude 7 would have on of 11 in B and 17.5 in U! Interestingly they are hotter than the M or S stars. The spectrum of a C0 star is that of a G star ant to a K to M star for the latest C class stars. There is an anomaly in that C13 is abundant in C stars, being rare on the Earth and the Sun.

The shortest period long-period stars are the brightest, variables with a period of 175 days have a mean absolute magnitude of -2.7, -1.7 for periods of 250 days, -0.5 for 350 days, and +0.9 for periods of 500 days. For C and C stars the mean spectrum passes from M3 for periods of 150 days to M5 for periods of 270 days. All of these data contain significant scattering around the mean values.

The mechanism of the long-period variables begins with pulses arising in the deepest layers of the star and is propagated to the outer regions which are less dense as the surface is approached. In the less dense areas of the star the pulsation causes a shock wave that heats up and thus disassociates the molecules (titanium oxide, oxides of carbon, cyanogen and water) and dust particles (metallic silicates) in the upper atmosphere of the star. As the shock wave passes the temperature falls and the molecules recombine and the atmosphere become opaque again.

Distribution

The shorter-period stars of the long-period variables (less than 200 days) belong to Population II and so are fond in the galactic halo, in the direction of the galactic center, and in globular clusters. They have relatively high radial velocities, from 65 to more than 200km/s. Long-period variables with periods of more than 200 days are intermediate Population I stars. They occur 1000 – 1200 parsecs from the galactic plane and their radial velocities are from 15 – 50km/s, depending on the period.

The S stars are very young and form a flat system typical of Population I stars occurring less than 100 parsecs from the galactic plane.

For long-period variables, in general, in the direction of the galactic center the proportion of periods over 200 days is high, those above 300 days are rare. In the direction away from the center the period is 350 days with many periods greater than 400 days. Like RR Lyrae stars, long-period variables are not good indicators of distance.

Semi and irregular-period

Definition

Semiregular variables, which are giants or supergiants of intermediate and late spectral types showing noticeable periodicity in their light changes, accompanied or sometimes interrupted by various irregularities. Periods lie in the range from 20 to GT 2000 days, while the shapes of the light curves are rather different and variable, and the amplitudes may be from several hundredths to several magnitudes (usually 1-2 mag in V).

Description

Semi-regular variables is the appropriate name given to red pulsating stars which are distinguished from long-period variables by an often more complex light curve and by a smaller magnitude rarely exceeding 2 magnitudes. The term irregular is given to those stars whose fluctuations are too time-independent to define a cycle. There is no clear line separating semi-regular and irregular variables, some stars moving from one type to another and back again.

There are four subgroups: SRa, SRb, SRc, and SRd. Tpyes SRa and SRb include giants with M, S, or C spectra. Type SRc includes M supergiants. Type SRd contains supergiants with G or K spectra. Included on this category will be RV Tauri stars which are usually discussed separately.

RV Tauri Type

In 1912 H. Leavitt noticed that the Cepheid’s in the Small Megallanic Cloud had a strong correlation between their period and magnitude. H. Shapley found eleven bright Cepheids whose distance could be determined through parallax. From this he deduced the absolute magnitudes and used these as a calibration for the period-luminosity relationship. Since then it has been possible to use Cepheids to determine distances to other galaxies.

Soon a complication became obvious. There seemed to be two different versions of Cepheid variables, the other now called RR Lyrae stars after the archetype stars. Shapley’s calibration of zero was incorrect. In 1952 Baade found that the Cepheids in the Andromeda galaxy were 1.5 magnitudes fainter than they should have been. It was determined that the distances determined by the correlation must be multiplied by 2. This immediately made the observed galaxies two as large and twice as far away. Actually this meant that the universe itself was twice as large as thought; this cause a revision of the Hubble constant.

RV Tauri has a relatively regular cycle, but the maxima and the minima are not constant from one cycle to the next. The curious light curve consists of one maximum that is followed by a shallow minimum and the by a second maximum, which is then followed by a deeper minimum. Their period is defined between the two deep minimums. Superimposed on these fluctuations is a principle cycle of period P(b) separating the fluctuations of maximum amplitude. P(b) is actually a modulation period. For the typical RV Tarui, the normal period is 78.7 days and that of the principle cycle is 1224 days.

These stars are not common with only about 100 known in the galaxy. Periods run from 30 to more than 200 days with those of over 150 days being rare. These stars are supergiants; with spectral types from F to K an sometimes M, with intense emission lines when near maximum. They are very bright with absolute magnitude of -3 to -4. They are very large and of low density. Belonging to Population II they are found in globular clusters and certain galaxies.

In 1955 Tsesevitch theorize that the pulsation stars in the hotter layers, creates a shock wave that arrives at the very tenuous outer layers of the star and amplifies the variations in the radius. An exact model is illusive but it is known that they shock wave encourages the formation of emission lines near the maximum as is noted in observations.

SRa Type

Semiregular late-type (M, C, S or Me, Ce, Se) giants displaying persistent periodicity and usually small (LT 2.5 mag in V) light amplitudes (Z Aqr). Amplitudes and light-curve shapes generally vary and periods are in the range of 35-1200 days. Many of these stars differ from Miras only by showing smaller light amplitudes;

The light curves of these stars resemble long-period variables with smaller amplitude and greater symmetry, D approx. 0.5. While the periods are not very regular they are no more irregular than those of the long-period variables and so differentiating these two groups is arbitrary (greater than 2.5 magnitudes are classified as long-period variables. There are at least 1200 known in the galaxy. The spectral type may be M, with or without emission, type C, or rarely type S. Of the 1200 stars those 470 with known spectra include 385 M type, 68 C type, and 17 S type.

The periods range from 40-1000 days: form M type a modal value is between 150-200 days and a small secondary from 250-300 days. In contrast, the C and S stars have only one marked peak at 350 days.

SRb Type

Semiregular late-type (M, C, S or Me, Ce, Se) giants with poorly defined periodicity (mean cycles in the range of 20 to 2300 days) or with alternating intervals of periodic and slow irregular changes, and even with light constancy intervals (RR CrB, AF Cyg). Every star of this type may usually be assigned a certain mean period (cycle), which is the value given in the Catalogue. In a number of cases, the simultaneous presence of two or more periods of light variation is observed.

Several superposed cycles are the main attribute of these stars. For example, WCyg has at least two periods of 130.8 and 199.8 days, producing fluctuations that are complex even showing quasi-constant brightness at times. There is a modulation period 8-15 times that of the normal period (maximum amplitude): 900 days for g Her, 960d for AF Cyg, 2450d for W Ori, and 2270d for V Aql.

There are at least 700 variables of this type. There spectra include M, C, and S stars; of 429 stars with known spectra, 358 are type M, 61 are type C, and 10 are type S. Periods range from 30-1000 days, with a strong peak at100-125 days for M stars, and two peaks (200-250d and 350-400d) for C and S stars. However there are many M stars with periods of 30-40 days.

SRc Type

Semiregular late-type (M, C, S or Me, Ce, Se) supergiants (Mu Cep) with amplitudes of about 1 mag and periods of light variation from 30 days to several thousand days;

Semiregular variable giants and supergiants of F, G, or K spectral types, sometimes with emission lines in their spectra. Amplitudes of light variation are in the range from 0.1 to 4 mag, and the range of periods is from 30 to 1100 days (SX Her, SV UMa).

SRc are all extremely bright supergiants with absolute magnitudes of -5 to -6. The typical star is mu Cep, discovered by W. Herschel (1782) has am amplitude of 1.5 magnitudes and very complex variations. Over the years observations and statistical data methods have provided many different periods, superimposed. There is no clear agreement on the periods of these stars as a group.

There are very few of these stars, about 40 being known, some are very bright (Betelgeuse and Antares). They are typical Population I stars and are present in some open clusters, about ten in the Perseus double cluster. All of the double cluster variables are fairly bright , 6 –8 magnitude with an amplitude of one or more.

SRd Type

These stars have similarities to RV Tauri stars. The stars are yellow supergiant stars of type F5 – K, with radial velocities of 100km/s or more. The ten of these stars found in globular clusters are of about an absolute magnitude of -1.6.

There are very few of these stars about 50 being known. The amplitude of the variation is 2 magnitudes on average. Neither the period nor the light curve is regular. It is not easy to determine a period for these stars, even an average period, with a range of 30 – 200 days with a modal value between 80 and 100 days.

Irregular Variable Type

The irregular variables are of two types: Lb giants of spectral type M (sometimes K), S, or C: and Lc supergiants of M type. The Lb stars are fairly numerous with at least 1200 being known. Type M predominates with 637 stars, 18 S stars, and 125 C stars. There are only about 40 of the rare LC type stars. As these are irregular-period stars no generalizations cab be made. There are stars that have slow continuous variations and others whose variations are extremely abrupt. There are a number of bright stars of the irregular type.

The SRa and SRb stars and irregulars with C or S spectra have the same galactic distribution as long-period variables with the same spectral type. That is, they form a flat system of typical Population I stars. SRc stars also form a flat system located less that 100 parsecs from the galatci plane.

Misc. Pulsating Variables

Stars in this classification are not a homogeneous group, being of several groups none of which is very numerous. Their amplitude is often small so that relatively sophisticated equipment is needed for observations.

Dwarf Cepheids Stars

Dwarf Cepheids were included with the RR Lyrae stars. In 1955 H. Smith distinguished it as a separate group, having shorter periods and weaker absolute magnitudes than the RR Lyrae stars. They have kept their traditional RR designation. The typical star is AI Vel, with a magnitude of 6.41 – 7.13. The normal period is 0.1116 day (2h 40m). In fact there are several superposed periods that make the light curve very complex. A 0.3792 day modulation period has been demonstrated.

About 60 stars are known, with periods from 0.05 – 0.25 day and the most frequent period between 0.10 – 0.20 day. RRs stars are of the moderately late A or F spectral types with a appreciable variations between the maximum and minimum. Variations in radial velocity are also large (100km/s amplitude). The spectral changes have the same period as the photometric variations and are closely connected with the pulsation.

The dwarf Cepheids have always been considered to be Population II stars, as are the RR Lyrae stars. However, some indications such as the relative abundances of the elements in the spectra imply that they might belong to a much younger population.

Delta Scuti Stars

These stars were initially confused with the dwarf Cepheids, which are of the same spectral type. They are differentiated by their smaller amplitude, often less that 0.1 magnitude. A major search for this type of star between 1970 and 1975 revealed more than 100 stars. Periods are even shorter than the dwarf Cepheids (0.02 to 0.25 day) but without a marked modal value, although a significant portion fall between 0.05 and 0.15 day.

Superimposed periods have been detected in this type of star. In some a modulation period 20 – 50 times longer that the principle period has been observed. Spectral types range from A5 to F5. Appreciable variation is observed in the radial velocity, always with a period equal to the photometric period. The stars are slightly less bright than the dwarf Cepheids with absolute magnitudes from +2 to + 5.

The Delta Scuti stars are Population I. They are often present in relatively young galactic clusters: several are known in the Haydes cluster and 10 or so in the Praesepe cluster.

Many Delta Scuti stars are spectroscopic binaries with periods that may be short (as in KW Aur with P= 3.789 days) but may also be long (FM Ver, 38.324 days). There is undoubtedly a relationship between the binary nature of these stars and their amplitudes.

Beta Canis Majoris Stars

This group has a somewhat controversial designation. It was first called the Beta Cehpeid type as it was the first star of the group to be identified in the early 1900’s. The General Catalog uses the designation Beta Canis Majoris because it is the brightest of the group, this has been rejected by numerous individuals. There are still a number of authors and researchers that use the old name in preference to the current official designation.

These stars have short periods with a modal value between 0.115 and 0.20 day, although ES Vul has a period of 0.610 days. The always very small amplitude of these stars generally being less that 0.1 magnitude. The radial velocity varies with the same period as its brightness but an appreciable range, often more than 100km/s. These latter variations are also complex, often out of phase with the light curve. A theory for the observed characteristics is still incomplete at best.

More than half of these stars have two pulsation periods, but these very little from each other. The typical star B CMa has a period(1) = 0.25002246 day and period(2) = 0.25130003 day, so than the modulation is particularly long at 49.1236 days, which is nearly 200 times the principle period. Several stars have shown variations in period, such variations are rare and most periods have been steady over the last 50 years.

More than 50 stars of this type are known. They are giants or sub-giants with a remarkable uniformity of their spectral types, which range from B0 to only B3. Their absolute magnitude ranges from -3 to -5. They obey a period-luminosity relationship peculiar to this group, the longer periods corresponding to the brightest stars.

These are young stars found in O-B stellar associations, especially in Per I, and in the young galactic clusters. Several are spectroscopic binaries.

These stars are currently not well understood with no model for their behavior to be proffered.

Alpa2 Canem Venaticorum Stars

This peculiar group was only established in 1950. It is characterized by an A type spectrum (B9 – A5) which shows a completely abnormal abundance of certain heavy metals and, in contrast, a marked deficiency of elements such as oxygen and light metals such as calcium. Three groups can be distinguished from a spectral point of view: those which show many silicon lines, those in which manganese lines are predominant, and those showing chromium, strontium, and rare earths such as europium. The spectral types are denoted by Ap followed by the most abundant elements such as, A0p CR, EU for CS Vir.

These stars are at the same time photometric variables, with an amplitude that is always small and a period that is often long, spectroscopic variables, with time variations in line intensities and profiles, and magnetic variables, with magnetic fields that are strong (several thousand gauss) but also vary greatly. It is believed that the observed period is that of the rotation of the star about its own axis as all the variations: photometric, spectroscopic, and magnetic are the same.

There are several hundred stars of type Ap and over 100 with known variability. The periods vary widely with a small peak around 2-3 days but with a range between 0.5 and several hundred days(314 days for Delta Equ). The visual amplitude is a small 0.1 magnitude at most Measurements show an enormous amplitude variation with wavelength, for UU Com 0.02 magnitude in visual light and 0.80 in ultraviolet. Strangely each star has a wavelength at which there is NO variation in brightness. This phenomenon remains totally unexplained. Finally the long and short wavelengths are out of phase, if brightness is at a maximum in the ultraviolet, it is at a minimum in the red, and vice versa.

As the observed periods are equal to the rotational period of the respective star, technically these should not be considered pulsating variables. Pulsations of very short period were discovered in the 1980’s: 30 minutes for UU Com and just over 6 minutes for DO Eri. These are of very small magnitude and are independent of the “normal” photometric variations in the stars. The cause seems to be an overabundance of metals that are not uniform in the atmosphere but are localized at certain points by the magnetic field of the star. Rotation makes these variations in brightness perceptible. The localization appears as rings and segments not just as irregular patches.

Some of these stars are spectroscopic binaries with periods different from the periods of the light variations, in other words the period of revolution of the pair is different from that of the star’s rotation.

ZZ Ceti Stars

This is another special group. The typical star ZZ Ceti, is a white dwarf of type DA and 14.09 magnitude, showing periodic but not radial oscillations (variations not due to changes in radius). The oscillations are due to shock waves which traverse the atmosphere. ZZ Ceti has two periods of 213 and 274 seconds.

There are currently more than 20 known all DA type with absolute magnitudes between +11 and +13. Their periods range from 2 to 20 minutes, with an always small amplitude, 0.2 – 0.2 at most.

While only extremely well equipped amateurs could make measurements of this type of variable due to their dimness, minute amplitude variation, and complex light curves they are very important to astronomers interested in the atmospheres of white dwarfs. The oscillations often have superposed on them rapid variations or “flickering”. The variations are thus complex and difficult to analyze.

Variable B Supergiants Stars

These are hot supergiants of spectral type A or B, with absolute magnitudes of -7 or -8 and so high luminosities. Kappa Cas, type B1 Ia, has been one of the best observed with a maximum V = 4.16, a period of 7 days, and amplitude of 0.04 magnitude – although the variations are not strictly periodic. There are additional long variations in these stars that seem to be superposed in the semi-regular ones.

The typical star of the 15 or so known is Alpha Cyg with a spectrum of A2 Ia. The periods are all of the same order (5 – 10 days) and the amplitude variations are always les then 0.1 magnitude. They are thus unspectacular variables but important in the study of the internal structure of supergiants.