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1. Introduction 1. Introduction Three variable stars with short periods and high-amplitude, CY Aqr, BP Peg, and GP And, are selected for the study, and the characteristic of each variable star is analyzed from their light curves. These three variable stars are difference a little, CY Aqr is probability a binary system, BP Peg is a type of delta Scuti star with two stable periods (Rodriguez et al., 1992), and GP And is a simple delta Scuti star. 1.1 Delta Scuti stars Delta Scuti, the fourth bright star in Scutum at V magnitude, 4.71, stand out as the prototype of one of these. On the HR diagram or temperature-luminosity diagram, the kind of variable stars were located in intersects of main sequence with instability strip shown in Figure 1. Figure 1. Delta Scuti stars were in intersects of main sequence with instability strip. Delta Scuti stars is the group of the second most numerous of pulsators in the Galaxy, after the pulsating white dwarfs, and their spectrum belong to type A to early F. Most delta Scuti stars belong to Population I (Antonello, Broglia & Mantegazza, 1986), but a few variables show low metals and 6 high space velocities typical of Population II (Rodriguez E., Rolland A. & Lopez de coca P., 1990). The delta Scuti stars is divided into two types, variable stars with high-amplitude delta Scuti (HADS) and high-amplitude SX Phe (HASXP) (Breger, 1983；Andreasen, 1983；Frolov and Irkaev, 1984). Both of them have asymmetrical light curve in V with amplitudes > 0.25 magnitude and probably hydrogen-burning stars in the main sequence or post main sequence stage. Furthermore, many delta Scuti stars have multiple pulsation periods, for example, delta Scuti’s chief period is 4.65 hours, while the secondary is 4.48 hours. In some cases, the delta Scuti stars pulsate for only the fundamental periods, and others pulsate with periods of the fundamental and first-overtone. Both of them fall in the H-R diagram on the same region as the small-amplitude delta Scuti stars, and they are typically more luminous and are restricted to a narrower temperature domain. Radial pulsation and small rotational velocities are also used to classify apart from the low-amplitude delta Scuti variables. The delta Scuti stars are usually used as distance scale in terms of the period-luminosity (P-L) relation (such as Woltjer, 1956; Frolov, 1962, 1965; McNamara and Feltz, 1978; Eggen, 1964, 1994; Nemec et al., 1994; and McNamara, 1995). 1.2 SX Phoenicis stars In 1980s, SX Phe stars were hard to differentiate from dwarf Cepheids and delta Scuti stars, because both their periods were shorter than 0.25 days. However, the light-curves of these two types of the characteristics are very different. The amplitudes of light curve of dwarf Cepheids are typically about 0.5 magnitude, and that of delta Scuti stars are about 0.01. The delta Scuti stars of Population II or old disk population are also called the SX Phe variables, and they are discovered in the general field (Rodriguez & Breger, 2001) and globular clusters (Rodriguez & Lopez-Gonzalez, 2000). Due to the process of photoelectric photometry and CCD photometry, the times of maximum light of these stars can be determined much more precision than before. For some delta Scuti stars, the period is change, which should reflect the evolutionary changes in radius hence. The change of period of delta Scuti stars or SX Phe stars provide a good tool for deducing their evolutionary status and the evolutionary tracks across the H-R diagram. The period changes is predicted by the evolution models within the boundaries of the delta Scuti instability strip should always be in time positive sense, except in the zone of the overall contraction phase when the evolutionary periods decrease (Rodiiguez 2004). However, the prediction is not always borne out by the observations (Breger & Pamyatnykh 1998). On the other hand, Khokhuntod et al. (2007) supported the predictions after eliminating time light-travel time effect in the binary system for some case. 7 1.3 GP Andromedae o h m s o GP And (=BD +0 4900, α2000 = 00 55 18.10 , δ2000 = +23 09′ 49″) is an already well-known high-amplitude delta Scuti star. In 1956, Strohmeier et al. were first time to discover its variability. The spectral type and period of GP And were A3 (Lange, 1969, 1970) and 0.07868270d (Gieseking et al., 1979, respectively). The light curve seems to display a quite complex variability phenomenon (Gieseking et al., 1979; Burchi et al., 1993). The photographic, photoelectric, or CCD photometry with maximum light times has been published by Splittgerber (1976), Eggen (1978), Gieseking et al. (1979), RodrIguez et al. (1993), Burchi et al. (1993), Schmidt et al. (1995), Agerer & Hhilxscher(1998, 2002. 2003), Agerer et al. (1999, 2001), Hubscher (2005), Hubscher et al. (2005) and Szeidl et al. (2006). In these literature, the information are usually the time of maximum light, and only few value of period are derived (Table 1). Table 1. period of GP And year T0 period Name Note 1979 2433861.438 0.07868270 F. Gieseking et al. 1993 2441909.49167 0.078682737 E. Rodrigues et al. 2006 2447005.6146 0.078682760 Szeidl et al. Konkoly observatory 8 1.4 BP Pegasi h m s BP Peg (α2000 = 21 33 13.53 , δ2000 = +22° 44′ 24″ ) with the V =11.69-12.28 was discovered to be a variable star by Masani and Broglia (1954), and it was classified a delta Scuti star with two stable oscillation periods of 0.109543375 days and 0.084610 days (Rodriguez et al., 1992), corresponding to double-mode variables (Masani and Broglia, 1954; Figer, 1983; Rodriguez, 1989). The light amplitude (~0.5 in V) would place the star among the RRs stars or dwarf Cepheids in the classification scheme of earlier versions of the General catalog Variable Stars(GCVS). Until McNamara (1985), the delta scuti variable stars with large-amplitude, ΔV＞0.3m, was designed of dwarf Cepheid differ from the small-amplitude delta Scuti variable stars. At present, BP Peg is classified a delta Scuti variable star in the GCVS(internet, 2007). Broglia (1959) detected a variation of the period between the 1953 and 1958 and 1959 observations with a linear ephemeris with a period of 0.10954347d. Furthermore, Broglia (1959) found a modulation period on the order of 0.37 days with amplitude in the maxima of 0.45 mag in the visual. They were able to determine the same parameters previously established with their photometry. Table 2. Period of BP Peg on literature year T0 period Name note 1959 2443014.5768 0.10954347 Broglia(1959) 1983 2443014.5786 0.109543375(P0)Figer(1983) 0.084610(P1) 1989 0.10954357 Kim & Joner(1989) 1992 0.109543375(P0) Rodriguez et al.(1992) 0.084610(P1) The ratio of the two periods of BP Peg, P1/ P0 ≈ 0.77 is corresponding to the fundamental mode and to the first overtone (Rodriguez, 1989). Many Physical parameters like temperature, surface gravity, metal abundance, rotational velocity, and radial velocity have been undertaken by photometric and spectroscopic study to investigate. The photometric reddening value of E(b-y) equal to 0.067 mag had been determined by Kim et al. (1989). A superficial gravity value of 3.85 and a ratio [Fe/H] of −0.08 were obtained. Photometry and theoretical grids were determined an average temperature, Te of 7470 K. From the T e =7470 K and theoretical models, a bolometric magnitude of Mbol 1.4 mag, a mass of 1.85 M⊙ and an age of 1.3 109 yr were finally derived for new uvby-β and spectroscopic observations of this star. Rodriguez et al. (1992) also reported uvby-β of three large amplitude delta Scuti stars, BP Peg, among them. 9 1.5 CY Aquarii ° h m s ° The short-period pulsating star CY Aqr (=BD +0 4900, α2000=22 37 47.9 , δ2000=+01 32′ 4.0″, V=10.42-11.16) with a period of 0.061days (88min) and large amplitude (0.74mag in the V band) is a subtype of SX Phe variable star. It is a typical single–mode pulsator (McNamara, Powell, & Joner, 1996) in an early post-main-sequence stage of evolution with masses apparently consistent with their evolutionary state (Andreasen, 1983). Since it was discovered by Hoffmeister in 1934 (Jensch, 1934), and it had the shortest period until 1951 when the SX Phe with the 80 minutes of period was found (Eggen, 1952). Because of its very short period, many researchers had studied. The early work done on this star are found (Hardie 1961, Tolbert 1961, Mahdy 1980, Szeidl 1980, and Powell Joner, & McNamara, 1995; Fu and Sterken 2003; etc.) shown on Table 3. Table 3. The period of CY Aqr in literature year T0 (HJD) Period (day) authors note 1954 2432180.3862 0.061038484 Ashbrook, 1954 not include Fitch, 1960 and 1961 2431291.6653 0.061038502 Hardie & Tolbert, 1961 Hardie & Tolbert, 1961 2431291.6639 0.061038469 Hardie & Tolbert, 1961 Include Fitch, 1960 1966 2431291.6657 0.061038475 1968 2434283.7112 0.0610383405 Zissell, 1968 1975 2440894.6513 0.061038351 Percy, 1975 1980 2434308.4314 0.061038328 Mahdy & Szeidl, 1980 1980CoKon..74....1M Ewa Bohusz & Andrzej 1980 2434283.7082 0.0610383405 Udalski 1986 2440892.6367 0.061038318 Rolland et al., 1986 2003 2426159.4967 0.0610383716 Fu et al., 2 Percy (1975) reanalyzed the earlier observations and also studied the change of period, and he find a drastic change between 1952 and 1953, and then remained constant. Furthermore, the changes of period of CY Aqr are also noted by many researchers, such as Coates, et al.
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