Научная статья на тему 'A Study of Double-Mode High-Amplitude Delta Scuti Variable Stars Pulsating in the Fundamental and First Overtone Modes. I'

A Study of Double-Mode High-Amplitude Delta Scuti Variable Stars Pulsating in the Fundamental and First Overtone Modes. I Текст научной статьи по специальности «Физика»

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Peremennye Zvezdy
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variable stars / переменные звезды

Аннотация научной статьи по физике, автор научной работы — Khruslov A.V.

I present a detection of 118 new double-mode high-amplitude Delta Scuti variable stars, HADS(B) type, pulsating in the fundamental and first overtone modes. We analyzed all observations available for these stars in the ASAS-SN online public archive using the period-search software developed by Dr. V.P. Goranskij for Windows environment. Additionally, in individual cases, I used data of the SuperWASP, CSS, ASAS-3, and NSVS surveys. Light elements and parameters of the light curves of both oscillations were obtained.

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Текст научной работы на тему «A Study of Double-Mode High-Amplitude Delta Scuti Variable Stars Pulsating in the Fundamental and First Overtone Modes. I»

Peremennye Zvezdy ( Variable ¡Stars) 41, No. 8, 2021 Received 10 December; accepted 17 December.

DOI: 10.24412/2221-0474-41-59-72

A Study of Double-Mode High-Amplitude 6 Scuti Variable Stars Pulsating in the Fundamental and First Overtone Modes. I

A. V. Khruslov1'2

Sternberg Astronomical Institute, Moscow State University, Universitetskij pr. 13, Moscow 119992, Russia; khruslov@bk.ru

Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya str. 48, Moscow 119017, Russia

I present a detection of 118 new double-mode high-amplitude S Scuti variable stars, HADS(B) type, pulsating in the fundamental and first overtone modes. We analyzed all observations available for these stars in the ASAS-SN online public archive using the period-search software developed by Dr. V.P. Goranskij for Windows environment. Additionally, in individual cases, I used data of the SuperWASP, CSS, ASAS-3, and NSVS surveys. Light elements and parameters of the light curves of both oscillations were obtained.

1 Introduction

In the course of my study of high-amplitude 8 Scuti variable stars (HADS) included in the ASAS-SN Variable Stars Database (Jayasinghe et al. 2018, 2019a, 2019b, 2020, 2021), I discovered 118 new double-mode HADS variables, pulsating in the fundamental and first overtone modes. Searching for double periodicity, I mainly used data available in the photometric archive of the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappe et al. 2014, Kochanek et al. 2017). Additionally, I analyzed available data from other archives: the Wide Angle Search for Planets (SuperWASP, Butters et al. 2010), the Catalina Sky Surveys (Drake et al. 2009), the All Sky Automated Survey (ASAS-3, Pojmanski 2002), the Northern Sky Variability Survey (NSVS, Wozniak et al. 2004).

The SuperWASP observations are available as FITS tables, which were converted into ASCII tables using the OMC2ASCII1 program as described by Sokolovsky (2007).

Among HADS type stars of the ASAS-SN lists, I selected the stars with phased light curves showing a somewhat larger scattering. For double-mode candidates, I analyzed all observations using Deeming's method (Deeming 1975) implemented in the WinEfk code2 written by V.P. Goranskij. For several stars, I also detected 1 or 2 additional non-radial modes (see comments to Table 1).

For HADS(B) type stars, this paper supplements our previous publications (Khruslov 2009, 2011, 2014b, 2015c, 2018, 2020).

The light curves and data from the ASAS-SN surveys are available online in the html version of this paper as a zip-archive. The light curves are given in the format displayed in Fig. 1.

1 http://scan.sai.msu.ru/swasp_converter/

2 http://www.vgoranskij.net/software/

The radial pulsation modes were identified by the period ratio (Pshort/Plong)• The period ratio of pulsations in the fundamental and first-overtone modes typical of doublemode HADS stars (F/1O) is P1/P0 = 0.75 — 0.78 (Petersen & Christensen-Dalsgaard 1996).

The stars were identified in the GSC (Morrison et al. 2001) and USNO-BI.O (Monet et al. 2003) catalogs. The tabulated coordinates of the variables were drawn from the Gaia DR2 catalog (Gaia Collaboration, Brown et al. 2018).

2 Results

In the right ascension interval 00h — 12h, I detected 118 new HADS(B) stars pulsating in the fundamental and first-overtone modes. Their magnitudes are between 10m3 and 16m 1 in the V band of the ASAS-SN photometric system. The highest peak-to-peak amplitude of light variations is 0m9. The fundamental-mode periods are between 0d049 and 0d219.

The light curves for one of the stars (No. 35) are displayed in Fig. 1. Top panels present data folded with the fundamental-mode and first overtone periods. Bottom panels show the same curves after prewhitening the other oscillation (if /i+/o, fi—fo and other interaction frequencies or non-radial frequencies were excluded, it is also noted). Along with the light curves, we present power spectra of the double-mode variables, for the raw data and after subtraction of the dominant mode (fundamental or first-overtone oscillations). The structure of the power spectra shows that the secondary periods are real.

Information on these stars is presented in Tables 1 and 2. Table 1 contains equatorial coordinates (J2000); star numbers from the GSC and USN0-B1.0 catalogs; magnitudes at maximum and minimum in the V band of the ASAS-SN photometric system; number for comments on individual stars, which are given below the table.

Table 2 presents the light elements (period in days, and epoch of maximum, HJD), semi-amplitude and asymmetry parameter M — m for the fundamental mode (P0, E0, A0, and M — m) and first-overtone mode (Pi, E1, A1 and M — m) oscillations, and the period ratio P1/P0.

The information on detected interaction frequencies is contained in Table 3: periods and semi-amplitudes of the frequencies f1 + f0, f1 — f0 and f1 + 2/0; other modes for two cases, a and "b", are given below the table.

Two of the stars are contained in the General Catalog of Variable Stars (GCVS, Samus et al. 2017), No. 57 (V0807 Aur) and No. 88 (PP Cnc). Both stars were classified as 8 Scuti variables (DSCT type in the GCVS classification system).

The Petersen diagram for double-mode HADS variables, pulsating in the fundamental and first overtone modes, and studied in this paper is displayed in Fig. 2.

Table 1. Positions, identifications, and magnitudes

No. Coordinates (J2000) GSC USN0-B1.0 V, mag Comments

1 00 10 32.32 +32 23 48.1 02264-00469 1223-0003416 14.40- 14.73 1,2

2 00 19 59.60 +49 07 55.6 - 1391-0009181 14.75- 15.18 3

3 00 23 38.86 +28 37 45.1 01737-01308 1186-0005809 12.78- 13.21 1,2,3,4,5,6

4 00 44 23.08 +28 13 13.9 01744-00939 1182-0013918 13.25- 13.75 1,2,3,4,7

5 00 54 09.83 -53 57 50.2 08473-00232 0360-0006187 14.37- 14.91 3

6 01 06 06.76 -31 06 50.1 06999-00176 0588-0012017 14.31- 15.00 3,4,7

7 01 13 19.52 -32 31 10.8 07002-02069 0574-0014622 12.53- 12.88 8

8 01 22 42.45 +08 54 06.9 00614-01301 0989-0012023 12.14- 12.65 8,9,10

9 01 28 30.96 -35 30 37.2 07004-00789 0544-0012430 13.81- 14.22 3,4,7

10 01 37 21.66 +10 10 31.8 - 1001-0013950 15.24- 15.74 2

11 01 38 59.08 +43 33 22.4 02826-00354 1335-0031024 14.15- 14.67 2

12 02 03 28.96 -54 55 07.5 08482-00617 0350-0014997 14.28- 14.68 3

13 02 14 38.20 +52 54 43.9 - 1429-0074165 15.04- 15.82 -

14 02 38 00.27 +28 08 02.2 01779-01144 1181-0042094 13.25- 13.88 1,3,4,5,11

15 02 59 56.45 -22 49 08.2 06438-00986 0671-0039139 14.18- 14.81 1,3,7

16 03 05 41.29 +84 02 21.6 04616-01295 1740-0006756 13.84- 14.45 7

17 03 11 06.90 +34 29 09.6 - 1244-0044952 15.04- 15.62 1

18 03 22 28.34 -49 53 56.4 - 0401-0025899 15.18 -15.75 3

19 03 26 11.39 -59 21 25.3 08499-00895 0306-0021812 14.36- 15.08 3,7

20 03 31 51.68 -09 45 45.8 05296-00428 0802-0034347 14.45 -15.0 3,7

21 03 55 44.26 +44 15 25.7 02876-00685 1342-0085570 11.65- 12.17 1

22 03 57 05.32 +24 56 21.4 01817-00636 1149-0046971 12.70- 13.02 1,2,8

23 03 58 10.89 +19 35 13.9 01257-00447 1095-0043931 14.00- 14.59 2,5,12

24 03 58 31.38 -58 43 54.1 08507-01489 0312-0024539 13.08- 13.37 7,13

25 04 02 27.52 +55 20 33.6 - 1453-0119425 13.81- 14.25 1,2,7

26 04 03 20.09 -83 41 58.6 09492-02623 0063-0009008 11.70- 12.05 8,9,14

27 04 22 16.95 -63 18 11.3 08872-01146 0266-0027244 14.00- 14.32 7

28 04 54 25.87 +15 16 00.0 01280-01425 1052-0055729 15.00- 15.40 2

29 04 57 35.63 +68 09 38.2 04342-02642 1581-0114191 13.62- 14.05 2,7

30 04 58 01.28 -50 11 08.0 08083-01319 0398-0040619 12.56- 13.04 3,8,9

31 05 09 15.12 +44 55 01.0 - 1349-0130586 13.62- 14.25 1,2,4

32 05 21 07.82 +42 04 07.0 - 1320-0146648 14.04- 14.66 1,7

33 05 22 49.70 +56 52 55.6 03756-00466 1468-0165058 14.73- 15.15 1

34 05 27 39.16 -11 50 14.5 05340-01215 0781-0088861 13.95- 14.42 3,7,8,15

35 05 36 58.20 +38 44 37.6 02910-01193 1287-0123535 12.70- 13.16 1,2,7

36 05 42 19.20 +34 52 19.9 02413-00945 1248-0108043 11.87- 12.38 1,16

37 05 45 57.22 +04 51 55.5 00124-01857 0948-0067579 13.92- 14.26 2,7,17

38 05 46 59.00 -13 42 36.5 05359-00679 0762-0062688 13.53- 14.01 3,4,5,18

39 05 49 59.44 +57 45 10.8 03758-00811 1477-0201646 14.72- 15.10 2,19

40 05 52 15.04 +38 26 21.3 - 1284-0134768 14.10- 14.59 1,7

41 05 54 02.76 +31 55 00.3 - 1219-0113872 14.67- 15.11 1,2

42 06 04 55.57 +05 33 43.7 00138-01752 0955-0078740 14.57- 15.09 -

43 06 05 58.96 -32 06 17.5 07075-00372 0578-0105650 14.80- 15.35 3

44 06 11 07.91 -33 46 00.2 07080-00744 0562-0062360 12.96- 13.36 1,3,7,8

45 06 12 26.82 -26 54 53.8 06513-00659 0630-0111793 14.90- 15.30 1,3

46 06 13 34.34 -25 42 21.1 - 0642-0076933 15.45- 16.05 3

Figure 1. The light curves and power spectra of star No. 35 (GSC 02910-01193) from ASAS-SN data.

Figure 2. The Petersen diagram for the HADS(B) stars.

Table 1 (continued)

No. Coordinates (J2000) GSC USN0-B1.0 V, mag Comments

47 06 14 41.51 -27 46 48.4 06513-00255 0622-0106776 14.08- 14.62 1,3,7,8

48 06 16 57.22 -28 36 08.5 - 0613-0073887 15.55- 16.10 -

49 06 16 58.97 -21 33 18.9 05946-01171 0684-0088861 13.12- 13.75 7,8

50 06 22 30.32 +23 19 32.9 01878-01104 1133-0125277 12.89- 13.36 20

51 06 28 13.65 -12 37 12.3 05372-01751 0773-0113783 13.27- 13.65 8

52 06 29 08.67 -18 29 48.2 05951-02954 0715-0083260 14.30- 14.98 -

53 06 30 11.25 -25 55 05.9 - 0640-0084457 15.02- 15.37 -

54 06 32 52.61 +16 08 37.7 - 1061-0110555 14.40- 15.00 2,17

55 06 35 10.78 +56 30 27.9 03773-01053 1465-0179678 14.25- 14.75 1,2,3

56 06 36 07.77 +20 26 14.4 01337-01189 1104-0116641 12.56- 13.05 2

57 06 39 48.82 +46 57 14.9 - 1369-0180384 14.06- 14.58 3,4,5,21

58 06 41 46.61 -32 45 27.5 07091-00259 0572-0111428 13.74- 14.16 7

59 06 43 24.79 +21 27 36.0 - 1114-0135450 14.59- 15.05 1, 2, 3

60 06 46 34.46 -25 07 51.7 06525-01065 0648-0094430 12.94- 13.34 1,3,7

61 06 49 44.02 -45 16 41.2 08117-01123 0447-0065424 13.42- 13.83 1,3,7,22

62 06 52 16.92 +53 55 46.6 03766-02065 1439-0177038 14.76- 15.20 2,23

63 06 57 43.33 +23 37 12.9 01894-01205 1136-0131242 12.74- 13.08 1,2,3,8

64 06 59 38.96 -38 01 59.0 07628-02239 0519-0082879 13.67- 14.18 7,8

65 06 59 42.34 -44 41 56.0 07640-01727 0453-0068408 13.62- 14.00 7

66 07 03 42.34 +20 11 54.5 - 1101-0133503 15.2- 15.6 2

67 07 03 42.89 +25 21 57.6 01899-01601 1153-0134988 13.50- 13.91 1,3

68 07 04 48.34 +76 11 10.2 04526-01334 1661-0044981 13.25- 13.71 2,3,7

69 07 04 57.42 -50 59 43.3 - 0390-0068170 14.82- 15.76 7

70 07 05 19.26 +07 21 52.9 - 0973-0155337 14.03- 14.51 -

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71 07 18 48.94 +14 56 20.8 - 1049-0145251 14.50- 15.30 -

72 07 21 47.38 +28 27 41.3 - 1184-0151544 15.29- 15.88 2

73 07 25 31.29 +19 19 15.5 - 1093-0144508 14.50- 15.35 2,7

74 07 28 27.18 -02 07 59.7 04821-00218 0878-0212072 12.94- 13.42 7,8

75 07 30 31.64 -44 51 51.3 - 0451-0082290 14.83- 15.33 17,24,25

76 07 32 09.75 -52 09 41.9 08145-00183 0378-0110856 12.48- 12.72 7,8

77 07 36 28.00 -10 04 25.2 - 0799-0140796 13.94- 14.39 -

78 07 44 33.38 -09 52 47.6 - 0801-0149500 14.38- 14.95 3

79 07 50 28.42 -07 07 32.6 - 0828-0201846 14.63- 15.31 -

80 07 52 33.31 -10 01 42.9 - 0799-0154194 14.25- 14.55 -

81 08 10 49.17 -64 37 42.8 08932-00427 0253-0100471 13.25- 13.59 7,8

82 08 15 19.82 -51 18 29.2 08161-00028 0386-0095948 13.44- 13.70 7,8

83 08 21 12.85 -24 38 23.9 - 0653-0192122 14.06- 14.41 3

84 08 22 34.68 -10 07 00.8 - 0798-0171654 15.30- 15.80 3

85 08 23 57.42 +31 41 58.5 02470-00257 1216-0165249 13.36- 13.70 1,2,3,5,26

86 08 25 10.87 -66 46 29.3 08937-00164 0232-0169767 14.93- 15.44 7

87 08 27 43.65 -68 22 14.7 - 0216-0153143 15.00- 15.45 7

88 08 41 44.18 +25 30 30.6 01945-02080 1155-0159535 11.66- 12.35 1,9,27

89 08 42 57.83 +01 36 47.8 - 0916-0169855 15.13- 15.68 3

90 08 52 32.01 -29 11 48.8 - 0608-0206442 15.38- 16.13 -

91 08 55 52.22 -17 01 40.2 - 0729-0267352 14.89- 15.57 -

92 08 58 34.41 -30 27 02.2 - 0595-0209401 14.27- 14.98 7

Table 1 (continued)

No. Coordinates (J2000) GSC USN0-B1.0 V, mag Comments

93 09 18 01.10 -07 37 44.1 05458-00494 0823-0248424 13.58- 14.20 3,7,8

94 09 19 18.19 -35 39 04.9 07165-02056 0543-0195152 13.31- 13.93 1,7,8

95 09 33 52.79 +77 43 17.5 04544-01447 1677-0062769 14.18- 14.52 2,3

96 09 34 14.45 -08 00 45.2 05460-00148 0819-0239063 14.25- 14.65 4

97 09 42 42.46 +46 24 34.7 03426-01265 1364-0207401 14.10- 14.45 1

98 09 47 05.65 +32 01 55.0 02501-01015 1220-0200692 12.99- 13.44 1,2,3,5,28

99 09 50 57.71 -39 45 24.9 07698-01304 0502-0197972 14.61- 15.17 7

100 09 51 04.36 -77 51 51.6 09400-01546 0121-0051312 12.29- 12.75 7,8

101 09 51 05.75 +19 08 26.5 01414-00707 1091-0176888 14.30- 14.67 3

102 09 56 32.24 -36 15 23.8 07181-01296 0537-0216969 14.61- 15.09 3

103 10 10 04.41 -37 10 36.8 - 0528-0285690 15.26- 16.06 1,29

104 10 18 40.40 -21 44 49.7 06072-00342 0682-0271795 10.31- 10.73 1,7

105 10 34 38.26 +52 09 43.3 03448-00306 1421-0244083 12.27- 12.61 2,3

106 10 49 15.92 -48 39 08.0 08202-02421 0413-0222512 13.61- 14.18 3,8

107 10 54 13.49 -77 24 12.6 09414-00551 0125-0072422 13.81- 14.19 7,8

108 11 01 11.59 -51 13 45.2 - 0387-0239318 14.86- 15.66 7

109 11 04 42.10 -48 47 56.4 08208-00254 0412-0232165 12.03- 12.44 8,9,30

110 11 15 17.05 -42 07 10.5 07734-00267 0478-0297017 13.86- 14.28 1,3,8

111 11 15 51.70 -31 29 48.1 07202-00985 0585-0261637 14.20- 14.50 1

112 11 24 13.26 -08 54 54.3 05508-00686 0810-0228352 14.10- 14.80 2,3,31

113 11 34 19.96 -32 54 34.5 07220-00586 0570-0335039 14.12- 14.59 1,3,7

114 11 43 41.00 -06 33 48.6 04937-00474 0834-0234264 13.66- 14.16 3,7

115 11 45 01.59 -71 11 20.3 - 0188-0235100 13.94- 14.48 7,8,17

116 11 47 31.84 -19 30 09.7 06096-00315 0704-0249943 12.35- 12.74 3,8

117 11 51 05.89 -53 12 21.0 08631-01733 0367-0336708 14.02- 14.34 3,17

118 11 52 56.20 -20 38 50.0 06099-01115 0693-0265666 12.77- 13.26 1,3,7,8

Comments to Table 1.

1. Data from 1SWASP were used to improve the light elements.

2. The variable is contained in the ZTF catalog, type DSCT.

3. The star is contained in the Gaia DR2 catalog as a variable, type DSCT or SXPHE.

4. Data from the Catalina survey (CSS or SSS) were used to improve the light elements.

5. The variable is contained in the Catalina surveys periodic variable star catalog (Drake et al. 2014).

6. The variable is contained in the TAROT suspected variable star catalog (TSVSC1 TN-N322130012-234-67-2, Damerdji et al. 2007).

7. The variability was detected in the ASAS-SN survey.

8. Data from the ASAS-3 survey were used to improve the light elements.

9. The variability was detected in the ASAS-3 survey (Pojmanski 2002).

10. Additional non-radial pulsation, Pn = 0d04539784, An = 0.012.

11. Variability was detected by Wetterer and McGraw (2007), [WM2007] 42. According to V. Benishek (announcement of October 13, 2015 in the AAVSO Variable Star Index), the star is classified as a multiperiodic HADS variable, with two periods given, P1 = 0d0824245 (which is wrong) and P2 = 0d 1164448. The ASAS-SN catalog gives type HADS with a wrong period, P = 0d. 0824245 d.

12. Wrong type (EW) in the Catalina periodic variable star catalog (Drake et al. 2014).

13. Additional non-radial pulsation in the ASAS-SN catalog, Pn = 0"?07653493, An = 0.019.

14. Two additional non-radial pulsations. Pni = 0?04610757, An1 = 0.006; Pn2 = 0d04583652, An2 = 0.005.

15. Light elements improved only for f0.

16. The variability was detected by E. Conseil (announcement of February 14, 2012 in the AAVSO Variable Star Index, VSX J054219.2+345219). In addition to the two radial oscillations, I detected one non- radial mode, Pn = 0d 1180603, An = 0.018. Interaction mode fn + f0: Pn+0 = 0d06805571, An+0 = 0.009.

17. A close companion can somewhat influence the ASAS-SN photometry, amplitude is underestimated.

18. Additional non- radial pulsation, Pn = 0d0688260, An = 0.013. In the CSS data, fn is not detected.

19. The variability was discovered by Vitali Nevski (announcement of April 22, 2019 in the AAVSO Variable Star Index, NEV301).

20. P0 possibly varies, individual light elements for the ASAS-3 and 1SWASP data time interval are given.

For ASAS-3: f0: HJD(max) = 2454000.0100 + 0d 1397980 x E;

fi: HJD(max) = 2454000.0638 + 0d 1074006 x E;

P1/P0 = 0.7683;

For 1SWASP: f0: HJD(max) = 2454100.1023 + 0d 139820 x E;

f1: HJD(max) = 2454100.0510 + 0d 1074008 x E;

P1/P0 = 0.7681;

two interaction modes, f1 — f0 and f1 + f0, were detected.

21. GCVS variable (V0807 Aur), type DSCT, P = 0d 1043815. The variability was discovered by Monninger (2009, Object No.13). The new light elements were improved using Monninger's data. In the Catalina surveys periodic variable stars catalog (Drake et al. 2014), a wrong period P = 0d 116583 is given.

22. The period in the ASAS-SN catalog, P = 0d. 095300, is wrong. Additional nonradial oscillation: Pn = 0d05395933, An = 0.010.

23. Two periods P0 and P1 suggested in the ZTF as possibilities.

24. The star was included in the Gaia DR2 catalog as a variable, type RRAB.

25. Inaccurate coordinates are given in the ASAS-SN catalog.

26. The NSVS data were used for the light elements.

27. GCVS variable (PP Cnc), type DSCT, period P = 0d 1595357. In addition to the two radial oscillations, I detected one non- radial mode, Pn = 0d09466795, An = 0.018.

28. The variability was detected by A. Ditkovsky (announcement of July 10, 2012 in the AAVSO Variable Star Index).

29. The variability was discovered by Denis Denisenko (announcement of May 26, 2017 in the AAVSO Variable Star Index, DDE 128).

30. Additional non-radial oscillation, Pn = 0d 11628672, An = 0.008.

31. The variability was detected by Palaversa et al. (2013), LINEAR 1426326, type 5 Scuti.

Table 2. Light elements and parameters of light curves

No. Po Eo Ao M - m Pi Ei Ai M - m Pi/Po

1 0.0726208 0.0392 0.052 0.50 0.0562745 0.0035 0.026 0.48: 0.7749

2 0.0678568 0.0340 0.103 0.42 0.0528000 0.0300 0.045 0.50 0.7781

3 0.09468345 0.0830 0.127 0.40 0.07318673 0.0075 0.032 0.47 0.7730

4 0.06633370 0.0076 0.149 0.37 0.05137296 0.0185 0.039 0.43 0.7745

5 0.0730171 0.0663 0.125 0.47 0.0566927 0.0450 0.050 0.42 0.7764

6 0.05965608 0.0305 0.180 0.40 0.04668118 0.0156 0.047 0.45 0.7825

7 0.05697485 0.0562 0.131 0.38 0.04440837 0.0138 0.009 0.53: 0.7794

8 0.0866064 0.0050 0.033 0.50 0.06741625 0.0159 0.152 0.41 0.7784

9 0.05734314 0.0188 0.128 0.42 0.04438018 0.0307 0.027 0.43 0.7739

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10 0.0750599 0.0308 0.089 0.42 0.0580951 0.0325 0.033 0.46 0.7740

11 0.1314774 0.0535 0.076 0.44 0.1014740 0.0710 0.112 0.43 0.7718

12 0.0609297 0.0346 0.072 0.45 0.0474852 0.0062 0.058 0.42 0.7793

13 0.1090979 0.0163 0.174 0.40 0.0842605 0.0693 0.072 0.50: 0.7723

14 0.11644490 0.0555 0.077 0.41 0.08985065 0.0545 0.120 0.39 0.7716

15 0.05594349 0.0414 0.101 0.46 0.04367466 0.0340 0.110 0.45 0.7807

16 0.0923944 0.0506 0.153 0.34 0.0715963 0.0680 0.049 0.47 0.7749

17 0.07424527 0.0401 0.109 0.36 0.05753546 0.0359 0.073 0.42 0.7749

18 0.0488586 0.0115 0.137 0.40 0.0377969 0.0204 0.041 0.41 0.7736

19 0.0573055 0.0058 0.134 0.39 0.0447423 0.0202 0.094 0.41 0.7808

20 0.0647820 0.0145 0.119 0.41 0.0500718 0.0475 0.049 0.47 0.7729

21 0.1186905 0.0360 0.089 0.43 0.09171797 0.0315 0.092 0.37 0.7727

22 0.1631526 0.0469 0.086 0.44 0.1251222 0.0513 0.034 0.54 0.7669

23 0.1450347 0.0675 0.070 0.41 0.1117387 0.0602 0.102 0.41 0.7704

24 0.09511733 0.0477 0.072 0.43 0.07354215 0.0444 0.027 0.45 0.7732

25 0.09959518 0.0830 0.064 0.43 0.07701590 0.0687 0.095 0.38 0.7733

26 0.08565983 0.0032 0.133 0.43 0.06629983 0.0628 0.016 0.50 0.7740

27 0.0736922 0.0393 0.044 0.47 0.0569386 0.0465 0.045 0.48 0.7727

28 0.1143558 0.0193 0.116 0.43 0.0886835 0.0165 0.029 0.34 0.7755

29 0.1025558 0.0567 0.053 0.45 0.0793083 0.0448 0.074 0.46 0.7733

30 0.09396564 0.0241 0.196 0.32 0.07263767 0.0307 0.008 0.52 0.7730

31 0.09352360 0.0493 0.159 0.33 0.07253164 0.0386 0.050 0.45 0.7755

32 0.10091302 0.0260 0.187 0.35 0.07795395 0.0096 0.028 0.49 0.7725

33 0.08282542 0.0520 0.075 0.50 0.06487509 0.0348 0.053 0.46 0.7833

34 0.06509253 0.0311 0.144 0.39 0.0503301 0.0020 0.018 0.46 0.7732

35 0.07113790 0.0420 0.140 0.41 0.05506798 0.0065 0.036 0.45 0.7741

36 0.16067870 0.1043 0.165 0.33 0.12319683 0.0912 0.016 0.50 0.7667

37 0.0974633 0.0795 0.089 0.40 0.0753941 0.0508 0.017 0.62: 0.7736

38 0.08904091 0.0335 0.165 0.37 0.06870810 0.0183 0.020 0.41 0.7716

39 0.0767990 0.0650 0.087 0.42 0.0595345 0.0033 0.028 0.50 0.7752

40 0.2192864 0.0490 0.087 0.39 0.1724252 0.1680 0.086 0.37 0.7863

41 0.08777719 0.0827 0.080 0.41 0.06797523 0.0560 0.031 0.41 0.7744

42 0.1322361 0.112 0.068 0.50 0.1018154 0.0887 0.083 0.42 0.7700

43 0.0611311 0.0185 0.122 0.43 0.0475319 0.0445 0.036 0.43 0.7775

44 0.06275252 0.0582 0.123 0.43 0.04842434 0.0207 0.025 0.49 0.7717

45 0.1232893 0.0083 0.059 0.46 0.0948327 0.0400 0.051 0.50: 0.7692

46 0.0772017 0.0022 0.103 0.44 0.0597698 0.0540 0.050 0.47 0.7742

Table 2. Continued

No. Po Eo A M - m Pi Ei Ai M - m Pi/Po

47 0.07234961 0.0018 0.166 0.38 0.05599595 0.0144 0.024 0.57: 0.7740

48 0.0943323 0.0884 0.125 0.43 0.0730645 0.0595 0.068 0.44 0.7745

49 0.10182476 0.0668 0.097 0.38 0.07874227 0.0115 0.106 0.37 0.7733

50 0.1397989 0.0086 0.103 0.43 0.1074014 0.1004 0.071 0.44 0.7683

51 0.05679453 0.0170 0.119 0.40 0.04391921 0.0159 0.010 0.57: 0.7733

52 0.1700172 0.0244 0.146 0.34 0.1306380 0.1240 0.068 0.40 0.7684

53 0.0785583 0.0486 0.079 0.40 0.0607636 0.0042 0.023 0.35 0.7735

54 0.1763610 0.1380 0.063 0.46 0.1355230 0.0380 0.103 0.41 0.7684

55 0.07736643 0.0440 0.107 0.39 0.05999301 0.0164 0.038 0.48 0.7754

56 0.1151140 0.0122 0.054 0.41 0.0887887 0.0484 0.106 0.41 0.7713

57 0.10438148 0.0900 0.161 0.41 0.08063586 0.0038 0.030 0.40 0.7725

58 0.0958132 0.0123 0.121 0.38 0.0740858 0.0150 0.022 0.45 0.7732

59 0.1372574 0.0330 0.076 0.46 0.1058089 0.0545 0.074 0.43 0.7709

60 0.11232503 0.0460 0.136 0.36 0.08678022 0.0415 0.012 0.41 0.7726

61 0.09527799 0.0705 0.151 0.39 0.07358515 0.0315 0.009 0.45 0.7723

62 0.08486650 0.0133 0.057 0.43 0.06562550 0.0061 0.065 0.43 0.7733

63 0.06991782 0.0473 0.109 0.44 0.05419923 0.0150 0.022 0.50 0.7752

64 0.09210623 0.0508 0.121 0.40 0.07123833 0.0390 0.069 0.43 0.7734

65 0.11930159 0.1056 0.041 0.43 0.09233415 0.0752 0.086 0.41 0.7740

66 0.07236349 0.0598 0.066 0.44 0.05596885 0.0274 0.035 0.46: 0.7734

67 0.07228580 0.0541 0.120 0.39 0.05596237 0.0253 0.030 0.41 0.7742

68 0.06434355 0.0203 0.143 0.38 0.04979364 0.0298 0.040 0.47 0.7739

69 0.07320242 0.0264 0.144 0.39 0.05669563 0.0202 0.099 0.39 0.7745

70 0.09560174 0.0165 0.071 0.48 0.07394077 0.0040 0.068 0.42 0.7734

71 0.0772149 0.0317 0.126 0.41 0.0597305 0.0512 0.099 0.46 0.7736

72 0.0660610 0.0368 0.140 0.38 0.05113813 0.0305 0.039 0.46 0.7741

73 0.0803655 0.0124 0.231 0.34 0.0623454 0.0102 0.046 0.48 0.7758

74 0.1658014 0.0452 0.152 0.37 0.1270300 0.0644 0.027 0.44 0.7662

75 0.0708421 0.0708 0.124 0.35 0.05491516 0.0170 0.029 0.50 0.7752

76 0.06731149 0.0121 0.062 0.45 0.05214632 0.0169 0.025 0.48 0.7747

77 0.07758749 0.0199 0.122 0.40 0.06010818 0.0448 0.023 0.50 0.7747

78 0.05623590 0.0564 0.088 0.48 0.04359416 0.0335 0.093 0.43 0.7752

79 0.1122287 0.0190 0.100 0.40 0.08667595 0.0724 0.096 0.40 0.7723

80 0.0942969 0.0712 0.079 0.44 0.0727752 0.0632 0.021 0.43 0.7718

81 0.06365898 0.0580 0.107 0.45 0.04921392 0.0225 0.015 0.48 0.7731

82 0.1762366 0.1480 0.055 0.44 0.1393228 0.0115 0.029 0.43 0.7905

83 0.07991623 0.0670 0.066 0.45 0.06184430 0.0353 0.042 0.45 0.7739

84 0.08854017 0.0305 0.071 0.44 0.06860936 0.0345 0.061 0.38 0.7749

85 0.06807665 0.0480 0.092 0.42 0.05275110 0.0477 0.024 0.49 0.7749

86 0.06546138 0.0582 0.092 0.45 0.05063580 0.0100 0.025 0.45 0.7735

87 0.0858664 0.0660 0.090 0.45 0.0663899 0.0050 0.046 0.47 0.7732

88 0.15953655 0.1210 0.164 0.34 0.1219950 0.1250 0.041 0.47 0.7647

89 0.0662835 0.0605 0.096 0.50 0.0516362 0.0412 0.073 0.46 0.7790

90 0.0942627 0.0353 0.135 0.45 0.0729176 0.0163 0.062 0.46 0.7736

91 0.07978169 0.0733 0.115 0.45 0.06170619 0.0260 0.080 0.46 0.7734

92 0.09658696 0.0087 0.146 0.35 0.07474365 0.0752 0.064 0.44 0.7738

Table 2 (continued)

No. Pc Ec Ac M - m Pi Ei Ai M - m Pi/Pc

93 0.04885498 0.0208 0.177 0.35 0.03780310 0.0295 0.035 0.45 0.7738

94 0.06248116 0.0212 0.179 0.38 0.04838490 0.0073 0.060 0.44 0.7744

95 0.06410876 0.0045 0.094 0.40 0.04966603 0.0165 0.014 0.46 0.7747

96 0.07472371 0.0049 0.078 0.43 0.05780735 0.0430 0.034 0.50 0.7736

97 0.06811385 0.0272 0.069 0.47 0.05363685 0.0521 0.051 0.50 0.7875

98 0.14908390 0.0600 0.144 0.36 0.11490105 0.0840 0.014 0.53: 0.7707

99 0.1217307 0.1166 0.088 0.41 0.0937556 0.0488 0.080 0.45 0.7702

100 0.09814658 0.0150 0.100 0.41 0.07586985 0.0633 0.060 0.43 0.7730

101 0.06839633 0.0530 0.069 0.47 0.05352909 0.0350 0.041 0.48 0.7826

102 0.04895362 0.0051 0.111 0.43 0.03778475 0.0255 0.025 0.48 0.7718

103 0.09346032 0.0456 0.173 0.35 0.07230915 0.0167 0.052 0.44 0.7737

104 0.11699335 0.0055 0.096 0.40 0.09026021 0.0655 0.064 0.39 0.7715

105 0.0600389 0.0417 0.119 0.40 0.0464245 0.0053 0.018 0.45 0.7732

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106 0.09402636 0.0765 0.109 0.41 0.07266560 0.0436 0.097 0.39 0.7728

107 0.1284868 0.0660 0.024 0.49 0.0991545 0.0355 0.107 0.42 0.7717

108 0.0937395 0.0678 0.165 0.39 0.0725399 0.0090 0.059 0.43 0.7738

109 0.15146130 0.1400 0.134 0.37 0.11552007 0.0750 0.023 0.47 0.7627

110 0.08632393 0.0464 0.132 0.40 0.06671833 0.0150 0.026 0.50 0.7729

111 0.06287259 0.0574 0.036 0.47 0.04861082 0.0396 0.034 0.54: 0.7732

112 0.05206046 0.0240 0.209 0.47 0.04065318 0.0285 0.033 0.46 0.7809

113 0.06673347 0.0610 0.128 0.38 0.05186660 0.0421 0.014 0.44 0.7772

114 0.06897060 0.0496 0.105 0.47 0.05342869 0.0343 0.048 0.40 0.7747

115 0.07365285 0.0083 0.142 0.40 0.05703312 0.0520 0.022 0.49: 0.7744

116 0.07519871 0.0155 0.056 0.47 0.05812928 0.0240 0.091 0.41 0.7730

117 0.09865323 0.0480 0.041 0.48 0.07616441 0.0760 0.056 0.44 0.7720

118 0.07344838 0.0065 0.147 0.38 0.05674253 0.0090 0.037 0.47 0.7725

Table 3. Information on interaction frequencies

No. Pi+c Ai+c

1 - —

2 — —

3 0.04127931 0.017

4 0.02895127 0.017

5 0.03191389 0.023

6 0.02618844 0.026

7 — —

8 0.03790783 0.013

9 0.02501790 0.009

10 0.03274847 0.023

11 0.05727166 0.030

12 0.02668695 0.016

13 0.04754189 0.046

14 0.05071697 0.031

15 0.02452678 0.021

16 0.04033822 0.033

17 0.03241561 0.024

18 — —

Pi-c

A

i-c

Pi

i+2C

A

i+2C

Other

0.3223555 0.2277823 0.2535770 0.2146271

0.1963212

0.444659 0.215203 0.370099 0.3934226

0.3180588

0.166945

0.012 0.020 0.031 0.028

0.015

0.023 0.024 0.037 0.029

0.027

0.026

0.02015478

0.010

0.02807918

0.016

a

Table 3 (continued)

No. Pi+0 Ai+o

19 0.02512525 0.036

20 0.02824247 0.015

21 0.05173777 0.032

22 0.07081453 0.011

23 0.06311377 0.035

24 0.04147482 0.006

25 0.04343122 0.021

26 0.03737331 0.004

27 - -

28 - -

29 0.04472307 0.014

30 - -

31 0.04085031 0.038

32 0.04397993 0.020

33 - -

34 0.02838355 0.009

35 0.03103987 0.019

36 0.06973116 0.014

37 - -

38 - -

39 - -

40 0.0965268 0.033

41 - -

42 0.05752391 0.023

43 - -

44 0.02733252 0.008

45 - -

46 - -

47 0.03156553 0.019

48 0.04117343 0.030

49 0.04440433 0.041

50 0.06073909 0.020

51 - -

52 0.07387422 0.048

53 - -

54 - -

55 0.03379060 0.017

56 0.05012589 0.021

57 0.04549274 0.016

58 0.04178020 0.015

59 - -

60 0.04895718 0.008

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61 - -

62 0.03700804 0.022

63 0.03053147 0.008

64 0.04016972 0.025

Pi o

Ai_

io

Pi

1+20

A

1+20

Other

0.2040868

0.4036159

0.4867028 0.3242151

0.2933632

0.3498720 0.3200157 0.3231361 0.3426311

0.2219164 0.2437773

0.806864

0.2120846

0.347363 0.4634449

0.564029

0.2671553 0.3882436 0.3544606

0.3816054

0.3144267

0.031

0.027

0.026 0.005

0.003

0.013 0.007 0.028 0.017

0.011 0.020

0.020

0.008

0.037 0.020

0.025

0.016 0.032 0.022

0.006

0.024

0.03063056

0.014

0.03092047

0.015

b

0.02797070

0.011

Table 3 (continued)

No. Pi+0 Ai+o

65 0.05205001 0.011

66 - -

67 0.03154260 0.017

68 0.02807055 0.016

69 0.03195015 0.041

70 0.04169382 0.020

71 0.03367828 0.037

72 - -

73 0.2780419 0.036

74 0.07192454 0.018

75 0.03093503 0.023

76 0.02938308 0.006

77 - -

78 0.02455729 0.027

79 0.04890570 0.031

80 - -

81 - -

82 - -

83 - -

84 - -

85 0.02972092 0.008

86 - -

87 - -

88 0.06913216 0.025

89 - -

90 - -

91 0.03479446 0.023

92 0.04213677 0.031

93 0.02131213 0.015

94 0.02726850 0.032

95 - -

96 - -

97 0.03000743 0.012

98 0.06488945 0.010

99 - -

100 0.04279145 0.018

101 0.03002817 0.017

102 - -

103 0.04076761 0.039

104 0.05095159 0.022

105 0.02618066 0.010

106 0.04098874 0.028

107 - -

108 0.04089434 0.040

109 0.06553481 0.009

110 - -

Pi o

Ai

io

Pi

i+2o

A

i+2o

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Other

0.2478237 0.2202047 0.2514304

0.2637856

0.543220

0.1939254 0.3806920

0.5184505 0.2336731

0.2723680 0.3305004 0.1671060 0.2144556

0.5011128

0.3342642 0.2462620

0.3195103 0.3950135 0.204735 0.319869

0.3207267

0.016 0.013 0.042

0.048

0.009

0.018 0.031

0.014 0.031

0.017 0.042 0.014 0.035

0.007

0.013 0.017

0.027 0.010 0.010 0.018

0.033

0.02196017

0.009

0.04823154

0.014

A.V. Khruslov: A Study of Double-Mode HADS Variable Stars. I Table 3 (continued)

No. Pi+o Ai+o Pi-o Ai-o Pi+2o Ai+2o Other

111 - - - - - - -

112 0.02282749 0.019 0.1855331 0.022 - - -

113 0.02918404 0.012 - - - - -

114 - - 0.23709254 0.020 - - -

115 - - 0.2527484 0.012 - - -

116 0.03278564 0.016 0.2560867 0.012 - - -

117 - - - - - - -

118 0.03201187 0.019 0.2494732 0.011 - - -

Other modes: a fi + fo + 1, P = 0d 0243471,

A = 0.007.

b 2fi - fo, P2i-o = 0d06419139, A2i-o = 0.014; 2fi + fo, P2i+o = 0d02839295, A2i+o = 0.013; 2fo - fi, P2o-i = 0d 1440517, A2o-i = 0.013.

Acknowledgments: The author wishes to thank Dr. V.P. Goranskij for providing his software.

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