CC BY
0Peremennye Zvezdy ( Variable ¡Stars) 44, No. 4, 2024
Received 11 June; accepted 18 June.
DOI: 10.24412/2221-0474-2024-44-42-49
Unexplored Eclipsing Stars with Elliptical Orbits
Igor Volkov
Sternberg Astronomical Institute, Moscow University, Universitetsky Ave., 13, 119992 Moscow, Russia
This study presents parameters of several poorly studied eclipsing variable stars with elliptical orbits. The data were obtained from solution of our own long-term photometric observations.
The main goal of this work is to study the internal structure of stars. One of the ways of solving the problem is to measure the rotation speed of the apsidal line from observations of eclipsing stars with elliptical orbits. The rotation periods of the apsidal line can reach tens of thousands of years, and thus long series of observations of each star are required. In particular, our work has been going on for 35 years. Here we present a summary of our study.
The beginning of this study was first announced by Volkov and Volkova (2009), where the method of object selection was also described. The basis was the list of mainly northern stars obtained by Otero et al. (2006) from observations of ROTSE, ASAS, and Hipparcos. A number of stars were also selected that had previously avoided attention of observers due to difficulties of their observations: periods that are multiples of a day; eclipsing stars that are components of visual binary stars, etc.
We carried out observations with the 1.25-m and 60-cm reflectors at the Crimean Observatory of Sternberg institute; Zeiss-600 and Zeiss-1000 telescopes in the Simeiz INASAN observatory; 70-cm reflector of Moscow observatory of Sternberg Institute; 50-cm and 60-cm reflectors of Stara Lesna observatory, Slovakia. We mainly used CCD cameras, such as VersArray-512UV, VersArray-1300, ST-10XME, FLI PL09000; some others were also used, but not often. Observations were fulfilled in the Johnson-Cousins UBV RcR IcI system. For bright stars, a UBV photometer designed by I.M. Volkov, with an EMI9789 photomultiplier, was used (Volkov and Volkova, 2007).
The methods of our observations are described in detail in earlier publications: Bara-banov et al. (2021), Burlak et al. (2018), Volkov et al. (2021). Methods for processing observations and determining the relative and physical parameters of the systems are given in Volkov et al. (2010), Volkov et al. (2011), Bagaev et al. (2018).
Stellar temperatures were determined using Flower (1996) and Popper (1980) color index calibrations. Stellar magnitudes in the UBVRI system were determined by normalizing to standards from Kornilov et al. (1991), Moffett & Barnes (1979).
Table 1 presents the main observational parameters of the stars under study. Interstellar reddening was determined from our UBV photometry. If there is an asterisk, the interstellar reddening was determined from the survey by Green et al. (2015).
The B—V, U—B, V-R, R-I color indices corrected for interstellar reddening allowed us to determine spectral types of the components of eclipsing stars. In Table 1, we present only the B—V index as the most important one. The data in this Table are accurate to one half of the last significant digit.
Modern ephemeris of eclipsing stars given in Table 1 allow observers to pre-calculate minima with a high accuracy.
Table 1: Basic observational parameters of the stars
Star V B - V E(B - V) Spectrum Epoch HJD 2400000.0+ Period $ II
V871 Aql 12.51 1.06 1.19 B6V+B6V 52500.0229 2.952641 0.4451
V889 Aql 8.575 0.210 0.202 B9.5V+A0V 59060.3949 11.120760 0.3538
V645 Aur 9.72 0.01 0.11 B8V+B8V 52977.7382 10.8925082 0.7893
OO Cam 10.48 0.21 0.30* B8V+A0V: 55873.6014 8.1190455 0.4892
V347 Cam 10.96 0.26 0.09 A6IV+A6V 55314.4168 9.4545582 0.6944
V361 Cam 10.81 -0.06 0.10 B3IV+B9.5V 58561.2482 8.6385638 0.4727
V409 Cam 10.71 0.47 0.13 F0V+A9IV 57800.4846 6.676482 0.5231
V422 Cam 11.10 0.62 0.11 G0V+G1V 57803.3008 17.8705606 0.4904
V498 Cam 11.64 0.57 0.04 F7V+F7V 57795.3229 12.1102647 0.5653
KX Cnc 7.20 0.585 0.00 F9V+G0V 54162.7372 31.2198585 0.6432
DR CMi 11.06 0.13 0.0 A5IV 56644.5759 23.770030 0.6685
V1066 Cas 10.81 0.28 0.29 A3IV+A0V 58896.2402 8.4649440 0.5564
V1110 Cas 10.33 0.69 0.24 F5+F5: 58958.34515 24.849451 0.7063
V1141 Cas 11.93 0.19 0.49 B2V+B3V 59129.2382 6.9094135 0.4550
V1162 Cas 10.72 0.43 0.2? A0+A2: 59159.5948 29.0674505 0.2299
V750 Cep 11.26 0.68 0.76 B9V+A5V 58886.3278 18.8821656 0.438
V850 Cep 9.98 0.38 0.23* A0 51475.7273 12.914975 0.590
V880 Cep 10.27 0.28 0.32 A0V+A1V 58655.4035 27.330125 0.539
V897 Cep 11.44 0.71 0.3? KIII: 56235.5138 4.4871945 0.5118
V898 Cep 12.14 0.78 0.88 B9V+B9V? 55481.3576 2.8747704 0.6684
V921 Cep 11.69 0.87 0.61 F0IV+A8IV 58347.5032 13.7146644 0.4312
V922 Cep 11.41 0.42 0.5 B7V+B7V 55878.7002 3.57497303 0.5839
V944 Cep 10.92 0.95 1.03 B8V+B9V 58773.3625 6.56005423 0.5070
V1326 Cyg 11.44 0.22 0.23 B8V+B8V 55073.5052 16.681735 0.5302
V2544 Cyg 12.76 1.49 1.73 B2V+B2V 57927.3549 2.09381 0.5342
NS Dra 11.34 0.95 0.00 G5IV+K1III 58942.4806 50.54440 0.6321
V432 Dra 12.23 0.60 0.16 F5V+F5V 53278.3192 11.6281562 0.6985
UW Hya 13.19 0.53 0.0 F8V+F8V 47952.2502 2.11087916 0.5
IL Lac 12.47 0.26 0.35 B8V+B9V 55482.3025 7.395662 0.4354
V340 Lac 11.91 0.32 0.38 B9.5V+B9.5V 58350.5181 19.943091 0.7623
RU Mon 10.50 0.078 0.19 B8V+B9V 58921.1627 3.584690 0.3348
V501 Mon 12.31 0.501 0.22 A9V+F2V 52502.9358 7.0212043 0.4476
V521 Mon 10.055 0.135 0.249 B8V+B8V 59518.5547 2.970692 0.592
V2778 Ori 10.12 0.31 0.40 B6V+B9V 51629.65705 14.38759 0.4365
V751 Per 11.15 0.19 0.28 B8+B9 51508.6200 5.96134777 0.4487
V966 Per 13.08 0.06 0.24 B4V 54158.3045 4.3088431 0.3319
CR Sct 10.96 0.21 0.37 B5V+B5V 59365.5286 4.19235295 0.5112
V370 Sge 12.46 0.57 0.247 F0V+F2V 52734.9374 8.32628726 0.3790
EQ Vul 11.03 0.65 0.79 B6+B5III 60112.3244 9.297071 0.3214
V491 Vul 9.95 0.74 1.09 B0.5V 54648.4446 7.6697718 0.3348
Table 2: Relative parameters of the studied stars obtained from light curve solutions
Star e u i° ri r2 u obs °/year u theor °/year
V871 Aql 0.156(4) 236.90(2) 89.80(1) 0.172(1) 0.180(1) 1.37(9) 2.07
V889 Aql 0.368(4) 127.01(1) 89.21(1) 0.056(3) 0.052(3) 0.014(1) 0.016(2)
V645 Aur 0.5733(8) 320.04(1) 89.71(1) 0.0612(1) 0.0582(2) 0.020(5) 0.047
OO Cam 0.103(3) 260.62(1) 87.52(1) 0.0606(35) 0.0716(31) 0.008(2) -
V347 Cam 0.3110(1) 4.28(1) 87.59(1) 0.0728(1) 0.0441(5) - -
V361 Cam 0.128(3) 251.23(1) 89.49(1) 0.1339(7) 0.0544(3) 0.185 0.052
V409 Cam 0.043(2) 32.39(7) 84.92(1) 0.084(9) 0.105(6) 0.16(6) -
V422 Cam 0.035(3) 243.86(4) 89.57(1) 0.0324(1) 0.0244(1) - -
V498 Cam 0.259(9) 67.47(2) 87.54(1) 0.063(5) 0.050(7) 0.020(3) -
KX Cnc 0.4666(5) 63.80(1) 89.83(1) 0.0193(5) 0.0190(5) 0.0056(5)
DR CMi 0.562(3) 65.85(1) 88.32(1) 0.0492(6) 0.0548(5) 0.011(7) -
V1066 Cas 0.155(3) 55.34(1) 86.35(1) 0.1604(7) 0.0707(4) 0.193(4) -
V1110 Cas 0.512(20) 54.10(4) 87.68(1) 0.040(14) 0.036(17) 0.0088 0.0036:
V1141 Cas 0.365(2) 259.58(1) 89.14(1) 0.1135(3) 0.0919(2) 0.15(3) 0.235
V1162 Cas 0.522(2) 142.94(1) 89.71(1) 0.0268(6) 0.0263(6) 0.00043: 0.0028
V750 Cep 0.278(2) 109.86(1) 89.99(4) 0.0501(2) 0.0306(1) - 0.0050
V850 Cep 0.465(2) 74.20(1) 88.44(1) 0.0693(7) 0.0586(10) 0.010(3) -
V880 Cep 0.320(6) 79.55(1) 88.34(1) 0.0393(6) 0.0272(9) - -
V897 Cep 0.034(8) 57.8(2) 82.15(1) 0.12(4) 0.14(4) - -
V898 Cep 0.2670(1) 359.02(1) 85.15(1) 0.140(9) 0.149(9) 4.6(10) -
V921 Cep 0.469(2) 258.14(1) 89.68(1) 0.0868(2) 0.0699(2) 0.030(2) -
V922 Cep 0.1325(1) 3.56(1) 89.64(1) 0.1000(7) 0.0984(8) - -
V944 Cep 0.179(2) 86.33(1) 84.62(1) 0.1931(4) 0.1049(3) 0.44(3) 0.70
V1326 Cyg 0.396(9) 276.3(1) 89.12(1) 0.0403(2) 0.0502(1) 0.014(7)
V2544 Cyg 0.0827(9) 338.53(3) 85.97(1) 0.236(2) 0.190(3) 8.5(1) 8.9
NS Dra 0.349(9) 305.58(2) 88.09(1) 0.0245(3) 0.0674(8) 0.009(4) 0.0086
V432 Dra 0.377(1) 325.12(1) 89.19(1) 0.0389(4) 0.0388(4) 0.0265(10)
UW Hya 0.0 - 87.01(1) 0.196(3) 0.197(2) - -
IL Lac 0.1089(8) 158.83(2) 89.81(1) 0.0734(2) 0.0668(2) 0.047(20) 0.032
V340 Lac 0.4261(1) 4.35(1) 89.62(1) 0.0333(3) 0.0352(2) -
RU Mon 0.398(2) 128.87(1) 89.10(1) 0.129(2) 0.129(2) 1.00(2) 0.86(3)
V501 Mon 0.137(2) 233.22(1) 88.27(1) 0.0854(4) 0.0678(6) 0.021(6) 0.024
V521 Mon 0.192(5) 45.15(3) 86.82(1) 0.2075(12) 0.1255(9) 1.85(7) 1.60
V2778 Ori 0.164(2) 127.28(1) 89.24(1) 0.0689(2) 0.0487(2) 0.18(3) -
V751 Per 0.0809(1) 176.77(2) 88.72(1) 0.0942(2) 0.0761(4) 0.73: 0.05
V966 Per 0.2961(6) 206.52(1) 89.16(1) 0.1475(2) 0.1223(2) 0.68(2) 0.575
CR Sct 0.042(1) 65.7(1) 88.40(1) 0.1492(9) 0.1311(12) 0.57(1) 0.47(10)
V370 Sge 0.2189(4) 150.32(1) 89.02(1) 0.0945(1) 0.0756(1) 0.020(2) 0.025
EQ Vul 0.2906(6) 192.08(1) 88.88(1) 0.1543(6) 0.1282(6) 0.96(20) -
V491 Vul 0.3372(9) 220.63(1) 89.99(1) 0.1115(2) 0.1018(2) 0.340(5) 0.31
The algorithm of light curve solution used to obtain parameters in Table 2 is described in Khaliullin and Khaliullina (1984). In Volkov (2023), an algorithm of taking into account pulsations of components was added to the program. Parameters' errors are given in parentheses. The last two columns of Table 2 present the apsidal rotation velocities obtained from observations and their theoretical values. Theoretical values are given only for those stars for which we consider the observed values to be reliable. It can be seen that, for some systems, there is a significant discrepancy between the theoretical and
observed values. A possible explanation for this fact is lacking synchrc
between the
is lacking synchronism rotational and orbital moments. At this time, we do not have spectroscopic data on the axial rotation of the stars. Theoretical calculations are made under the assumption of synchronism at the periastron.
We obtained the absolute masses and radii of the components using the indirect method proposed by D.Ya. Martynov and described in Khaliullin (1985), Volkov et al. (2017). The results are presented in Table 3.
log M
1.5
1.0 0.5 -0.0
-0.5
-1.0
J_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_L
J_I_L
3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7
log T
Figure 1. Dependence of mass on temperature according to the data from Table 3. Blue circles are the primary components, the red ones are secondary components. Green curve is the zero age main sequence, ZAMS.
We plotted the obtained data from Table 3 in the diagrams presented in Figs. 1, 2. They are similar to such diagrams constructed for other objects by other authors and to theoretical ones. We can conclude that the indirect method works satisfactorily, the obtained sizes and masses are close to real ones, and our data are suitable for use in studying the structure and evolution of stars.
Table 3: Absolute parameters of the stars obtained from light curve solutions
Star TiK T2K Mi M2 Ri R2 log Li log L2 log 9i log 92
Mq Me Rq Rq Lq Lq cm/s2 cm/s2
V871 Aql 15500 15500 4.80 4.90 3.18 3.32 2.72 2.76 4.114 4.085
V889 Aql 10500 10120 2.49 2.42 1.97 1.84 1.58 1.46 4.245 4.275
V645 Aur 12000 11400 3.17 2.92 2.31 2.19 2.00 1.86 4.211 4.221
OO Cam 12000 9530 2.74 2.39 1.77 2.10 1.74 1.51 4.377 4.173
V347 Cam 7886 7950 1.97 1.55 2.08 1.26 1.18 0.75 4.095 4.426
V361 Cam 14852 11099 5.66 2.69 4.81 1.95 3.00 1.75 3.826 4.286
V409 Cam 7216 7399 1.74 2.00 1.94 2.43 0.96 1.20 4.104 3.967
V422 Cam 6453 5983 1.23 0.99 1.21 0.92 0.36 -0.017 4.359 4.510
V498 Cam 6198 6117 1.51 1.32 1.97 1.56 0.71 0.49 4.025 4.172
KX Cnc 6048 5994 1.138 1.131 1.057 1.043 0.127 0.099 4.446 4.455
DR CMi 8200 8200 2.44 2.57 2.93 3.26 1.55 1.64 3.892 3.822
V1066 Cas 9600 10000 3.80 2.64 5.21 2.29 2.32 1.68 3.584 4.137
V1110 Cas 6820 6725 1.74 1.63 2.16 1.95 0.96 0.84 4.009 4.070
V1141 Cas 21300 19000 7.59 6.39 4.21 3.39 3.51 3.22 4.069 4.184
V1162 Cas 9530 9140 2.17 2.06 1.72 1.69 1.34 1.25 4.301 4.295
V750 Cep 11240 8580 3.11 1.86 2.55 1.56 1.97 1.07 4.117 4.321
V850 Cep 8625 8454 2.45 2.21 2.68 2.27 1.55 1.37 3.971 4.071
V880 Cep 10200 9261 2.83 2.14 2.56 1.77 1.80 1.32 4.074 4.271
V897 Cep 5751 5819 1.41 1.50 2.01 2.22 0.60 0.71 3.981 3.921
V898 Cep 11376 11678 2.90 3.07 2.16 2.29 1.84 1.94 4.232 4.203
V921 Cep 7300 7650 2.36 2.22 3.47 2.80 1.49 1.38 3.730 3.890
V922 Cep 13197 13437 3.08 3.11 1.80 1.77 1.95 1.96 4.413 4.432
V944 Cep 12370 10200 5.16 3.13 5.76 3.13 2.84 1.98 3.629 3.943
V1326 Cyg 11238 11376 2.75 3.11 2.00 2.49 1.76 1.97 4.277 4.139
V2544 Cyg 21800 20500 7.5 6.3 3.90 3.13 3.49 3.19 4.130 4.247
NS Dra 5620 4767 1.42 2.00 2.12 5.83 0.61 1.20 3.935 3.206
V432 Dra 6587 6518 1.21 1.20 1.12 1.12 0.33 0.31 4.418 4.414
UW Hya 6158 6117 1.49 1.48 1.95 1.96 0.69 0.68 4.029 4.025
IL Lac 12008 11099 3.01 2.66 2.09 1.90 1.91 1.69 4.276 4.303
V340 Lac 10195 10011 2.32 2.34 1.72 1.82 1.46 1.47 4.333 4.288
RU Mon 12080 11736 3.21 3.07 2.35 2.35 2.02 1.95 4.202 4.183
V501 Mon 7319 6867 1.655 1.465 1.92 1.53 0.98 0.67 4.088 4.236
V521 Mon 14384 13867 4.77 3.58 3.65 2.21 2.71 2.21 3.992 4.303
V2778 Ori 12000 10000 3.71 2.60 3.17 2.24 2.27 1.65 4.006 4.152
V751 Per 11750 10500 3.10 2.49 2.31 1.87 1.96 1.58 4.201 4.292
V966 Per 15240 15240 4.86 3.43 3.32 2.74 2.74 2.58 4.082 4.096
CR Sct 16218 16218 5.30 4.97 3.54 3.12 2.89 2.78 4.063 4.147
V370 Sge 6964 7113 1.91 1.75 2.51 2.01 1.13 0.97 3.918 4.073
EQ Vul 14093 15488 6.34 6.35 6.69 5.56 3.20 3.20 3.588 3.750
V491 Vul 35900 34300 14.7 13.4 5.55 5.06 4.66 4.50 4.118 4.157
4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6
log T
Figure 2. Dependence of luminosity on temperature (Hertzsprung-Russell diagram) according to Table 3. Blue circles are the primary components, red circles are the secondary ones. Green curve is the zero age main sequence, ZAMS.
The obtained rates of apsidal rotation, both theoretical and observed, cannot yet be considered final. In some systems, the eccentricity turned out to be insignificant and therefore determined with a large error. This significantly degrades the accuracy of the calculated value. In other systems, the longitude of periastron is close to 0° or 180°, which makes determining the observed value extremely difficult. V751 Per is a prime example of this case. Its periastron longitude is u = 177°, and small errors in determining the periods led to a clearly erroneous overestimation of the rate of apsidal rotation, see Table 2.
However, for some stars both values were determined with good accuracy. For V889 Aql, V2544 Cyg, V501 Mon, V521 Mon, V966 Per, CR Sct, V370 Sge, V491 Vul, the observations do not contradict theory.
For V645 Aur and V944 Cep, apsidal rotation is slowed down and the reason may be lacking synchronism between rotational and orbital moments, just as we discovered earlier in the systems EQ Boo (Volkov et al., 2011) and V490 Sct (Volkov and Kravtsova, 2022). In V1103 Cas (Volkov and Kravtsova, 2022), the lack of synchronism accelerates the apsidal motion.
We pay special attention to the fact that the rate of apsidal rotation for CR Sct given in Wolf et al. (2004), uobs = 0.082(8)°/year, is 7 times lower than ours and is definitely wrong. The error is probably due to the use of photographic observations, which are not accurate enough. In addition, the orbital eccentricity turned out to be two times lower than Wolf et al. suggest, which leads to an underestimate of the apsidal rotation rate by
them.
The original observations in the V band on which this work is based are presented in the form of an electronic appendix to the html version of this paper, which contains headings with the name of the star and two columns: the heliocentric Julian date and the brightness of the star normalized to a constant level between minima. To get the real V magnitude of the star, one should add this value to the constant level between minima which is given in the second column of Table 1.
Original observations of some stars whose studies have already been published are added to this Table: BW Aqr (Volkov and Chochol, 2014), V1176 Cas (Bagaev et al., 2018), V798 Cep (Volkov et al., 2017), V541 Cyg (Volkov and Khaliullin, 1999), V2647 Cyg (Kravtsova et al., 2019), DI Her (Volkov, 2005), V577 Oph (Volkov and Volkova, 2010).
Currently, we continue observations of the objects, and the data presented in the Tables 1, 2, 3 may be refined over time.
Acknowledgements
This study has made use of the SIMBAD database of the Strasbourg Astronomical Data Center (France).
I express my sincere gratitude to A.S. Volkova for her help in processing the data and for valuable discussion.
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