A POSSIBLE DETECTION OF A SECOND LIGHT-TIME ORBIT FOR THE MASSIVE, EARLY-TYPE ECLIPSING BINARY STAR AH CEPHEI

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1 The Astronomical Journal, 129: , 2005 February # The American Astronomical Society. All rights reserved. Printed in U.S.A. A POSSIBLE DETECTION OF A SECOND LIGHT-TIME ORBIT FOR THE MASSIVE, EARLY-TYPE ECLIPSING BINARY STAR AH CEPHEI Chun-Hwey Kim, 1, 2 Il-Seong Nha, 3 and Jerzy M. Kreiner 4 Receivved 2004 April 12; accepted 2004 October 13 ABSTRACT All published and newly observed times of minimum light of the massive, early-type eclipsing binary star AH Cep were analyzed. After subtracting the light-time effect due to the well-known third body from the residuals of the observed times of minimum light, it was found that the second-order O C residuals varied in a cyclical way. It was assumed that the secondary oscillations were produced by a light-time effect due to a fourth body so all the times of minimum light were reanalyzed with a differential least-squares scheme in order to obtain the light-time orbits due to both the third and fourth bodies. The periods, eccentricities, and semiamplitudes of the light-time orbits for the third and fourth bodies were derived as P 3 ¼ 67:6 and P 4 ¼ 9:6yr,e 3 ¼ 0:52 and e 4 ¼ 0:64, and K 3 ¼ 0:0608 and K 4 ¼ 0:0040 days, respectively. The radial velocities of AH Cep published so far do not conflict with the hypothesis of the multiplicity of the system, but their accuracies are not high enough to support the interpretation. Other properties of the distant bodies are discussed for assorted possible inclinations of their orbits. Key words: binaries: eclipsing stars: individual (AH Cephei) 1. INTRODUCTION Since its discovery as a spectroscopic binary by Plaskett (Pearce 1927), AH Cep (HD , BD ) has been studied by many investigators. It is a relatively bright (m V ¼ 6:78 7:07 mag), massive (m p ¼ 15:3 M, m s ¼ 14:3 M ), and early-type (spectral type = B0.5 Vn + B0.5 Vn) eclipsing binary star. Many photoelectric (Huffer & Eggen 1947; Nekrasova 1960; Guarnieri et al. 1975; Mayer 1980; Hartigan & Binzel 1982; Bell et al. 1986; Oprescu et al. 1989; Harvig 1990) and spectroscopic (Pearce 1927; Harper et al. 1935; Bell et al. 1986; Holmgren et al. 1990, Burkholder et al. 1997) observations have been published. The photoelectric light curves were reanalyzed by Cester et al. (1978) and Drechsel et al. (1989, hereafter DLM89). Studies of the period changes of the system have been carried by Guarnieri et al. (1975), Mayer (1980, 1987), Mayer & Tremko (1983), Bell et al. (1986), Mayer & Wolf (1986), DLM89, Harvig (1990), and Mayer et al. (1991; 1998, hereafter M98). With regard to possible multiplicity of the system, Mayer & Wolf (1986) first suggested a light-time effect due to a third body in an orbit with the relatively large eccentricity e ¼ 0:534 and a period of P ¼ 62:3 yr. They also suggested fourth body with a period of 668 days and a semiamplitude of days. (See also Mayer 1987.) Subsequently, the light-time orbit of the suggested third body was improved by DLM89, who detected a third light contribution amounting to about 5% of the systemic light from reanalysis of the light curves of Huffer & Eggen (1947) and Bell et al. (1986). DLM89 also suggested a fourth body with a period of about 1 yr to explain changes of the orbital inclination and depth of minimum light of the eclipsing 1 Department of Astronomy and Space Science, College of Natural Science and Institute for Basic Science Research, Chungbuk National University, Cheongju , Korea; kimch@chungbuk.ac.kr. 2 Visiting Astronomer, Department of Astronomy and Astrophysics, Villanova University, Villanova, PA 19085; chun.kim@villanova.edu. 3 Nha Il-Seong Museum of Astronomy, Yechon-gun, Kyongbuk , Korea; slisnha@chollian.net. 4 Mount Suhora Observatory, Pedagogical University, ul. Podchorazych 2, Krakow, Poland; sfkreine@cyf-kr.edu.pl. 990 pair. Subsequently, Mayer et al. (1991) and M98 improved the third body light-time orbit. This paper reports the possible detection of a secondary modulation of the O C residuals by a short period of about 10 yr superposed on the primary light-time residuals with the long period of about 68 yr. There follows a discussion of the implications if this is a secondary light-time effect due to a fourth body. A reanalysis of all published radial velocity measurements is employed in this discussion. 2. NEW OBSERVATIONS AND PERIOD STUDY 2.1. New Observvations Our observations for eclipse timings of AH Cep were made at three different sites from 1982 to Data were first taken photoelectrically in the BV passbands with the 61 cm reflector at the Ilsan observing station of Yonsei University Observatory on the night of 1982 October 23. The photometer attached to the telescope was equipped with an uncooled 1P21 photomultiplier. The bandpasses correspond to the conventional UBV system. Each measurement was derived from a 1 minute trace on a strip recorder chart of the DC amplification of the multiplied photocurrent. The second set of observations was made in the BV passbands with the 61 cm reflector at Sobaek-san station of Korea Astronomy Observatory on the two nights of 1999 November 23 and 2000 January 26. A PM512 CCD imaging system of Photometric Instruments cooled by liquid nitrogen was used. Each measurement was integrated for 10 s. The reduction method of our CCD observations is well described elsewhere (Park 1993). The third data set was made only in the V passband using the 40 cm reflector equipped with a ST-8 CCD imaging system at the campus station of Chungbuk National University Observatory on the night of 1999 December 9. Each measurement was an integration of 10 s. DIPHAO in the IRAF S/W package 5 was used for the prereduction of these 5 IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the NSF.

2 TABLE 1 New Observed Times of Minimum Light for AH Cep K-W Method Quadratic Method Filter No. Weighted Mean Weighted Mean Mean V , , , B , V , , , , , B , , V , (7) 51, (7) 51, (10) 51, (10) 51, (7) V , , , (16) 51, (15) 51, B , , (14) TABLE 2 Observed Times of Minimum Light for AH Cep Epoch Error (O C) 1 (O C) 2 (O C) 3 (6) (O C) 4 (7) Method a (8) Type (9) Reference (10) 15, P II 1 24, VI I 2 24, SP I 1* 25, PE I 3 25, VI I 1 25, VI II 4* 26, VI I 1 26, VI I 1 26, VI I 1 34, PE I 5* 34, PE II 6 34, PE I 6 40, PE II 7 41, PE II 8* 42, PE I 8* 43, PE I 9 43, b PE I 8** 43, PE I 10* 43, PE II 10* 44, PE I 11 44, PE I 10* 44, PE II 10* 44, PE II 12* 45, PE I 10* 45, PE II 10* 45, PE II 11 45, PE II 11 45, PE II 13 45, PE II 11** 45, PE I 11** 45, PE II 14 46, PEB I 15** 46, PE II 16 46, PE I 16 46, PE II 16 46, PE II 10* 46, PE II 17 46, PE I 17 46, PE II 17 46, PEB I 18** 46, PE II 17 47, PE II 17 47, PE II 17 47, PE I 17 47, PE II 17** 991

3 992 KIM, NHA, & KREINER Vol. 129 TABLE 2 Continued Epoch Error (O C) 1 (O C) 2 (O C) 3 (6) (O C) 4 (7) Method a (8) Type (9) Reference (10) 47, PE I 17** 47, PE II 17 47, PE II 17 47, PE I 19 48, PEB I 20 48, PE I 21 48, PE I 21** 48, CCD II 22* 48, CCD II 22* 48, CCD II 22* 48, CCD I 22* 48, CCD I 22* 48, CCD I 22* 48, PE I 23 48, PE I 23** 49, PE II 24 50, PE II 25 51, CCD II 13 51, CCD II 13 51, CCD II 13 a P: Sky patrol plate; VI: visual; SP: spectroscopic; PE: multiplier photocell; PEB: multiplier photocell with blue filter only; CCD: electronic camera. b Redetermined in this paper. References. Zverev 1933; Moore 1936; Huffer & Eggen 1947; Zverev 1931; Nekrasova 1960; (6) Guarnieri et al. 1975; (7) Battistini et al. 1973; (8) Mayer 1980; (9) Hartigan & Binzel 1982; (10) Harvig 1990; (11) Mayer & Tremko 1983; (12) Diethelm 1981; (13) this paper; (14) Bell et al. 1986; (15) Diethelm 1985; (16) Mayer & Wolf 1986; (17) Mayer et al. 1991; (18) Diethelm 1987; (19) Isles 1992; (20) Diethelm 1990; (21) Hübscher et al. 1991; (22) this paper (Hipparcos); (23) Hübscher et al. 1992; (24) Agerer & Hübscher 1996; (25) Agerer & Hübscher measurements. (Individual observations made at the three different sites are available on request from the first author.) 6 Extinction coefficients for all observations were computed from the comparison star (HD , SAO 20214) measurements, and the differential magnitudes (variable comparison) in each color were left on the instrumental system. From our observations four times of minimum light, listed in Table 1, were obtained by the method of Kwee & van Woerden (1956) and also with quadratic curve fitting. The timings derived from the two methods agree within their internal errors. In addition, the Hipparcos photometric data of AH Cep were investigated in order to see if new timings could be determined. Unfortunately, there are no consecutive observations from which an eclipse light curve could be well defined. Nonetheless, six new times of minimum light from the Hipparcos photometric data were derived 7 and are listed in Table 2. These minima were obtained not in the time domain (i.e., magnitude vs. time) but in the phase domain (i.e., magnitude vs. phase) by superposing individual observed points onto an eclipse light curve formed from all Hipparcos data. Note that some of these minima have large internal errors, as listed in the third column of Table Period Study A total of 65 observed times of minimum light (one sky patrol, six visual, one spectroscopic, 48 photoelectric, and nine CCD) has been collected into a modern database (Kreiner et al. 2001). These are among those listed in Table 2. An O C 6 Also available at 7 One of us (J. M. K.) and his colleagues have completed determining times of minimum light from the photometric data for the eclipsing binaries observed by Hipparcos. diagram of AH Cep has been constructed in the upper part of Figure 1 with the linear ephemeris of M98: Min I ¼ 2; 434; 989:459 þ 1: E: The residuals from equation appear as (O C ) 1 in Table 2. The continuous curve in Figure 1 was drawn with the orbital elements of the light-time orbit due to a third body given by M98, which are given in the second column of Table 3. Clearly the times of minimum light after about 1990 have gradually deviated from the theoretical curve, implying that the lighttime orbit needed to be improved. Before that subsequent analysis, examination for timing uncertainties of photoelectric times of minimum light was made if the individual observations were available to us. One time of minimum light (JD 2,444,634.35) is certainly not useful because it has only two decimal places. A second timing at JD 2,441, was estimated without an ascending branch and another (JD 2,442, ) from only near the deepest part of the minimum with neither a descending nor an ascending branch. To these minima Mayer (1980) gave zero weight, whereas oddly Mayer et al. (1991) assigned a weight of 2 to them in their period study. JD 2,434, may be uncertain, because in their period study Guarnieri et al. (1975) gave zero weight to it. Because of these uncertainties we excluded all these minima from our subsequent analysis. Another timing (JD 2,443, ) was checked with the individual observations published by Mayer (1980). There were two gaps in the descending and ascending branches. Reanalysis of these individual measurements with the method of Kwee & van Woerden (1956) yielded a timing of JD 2; 443; 815:311(0:002), which is practically the same as that redetermined by Bell et al. (1986). Our result was used in our subsequent analysis. In ð1þ

4 No. 2, 2005 SECOND LIGHT-TIME ORBIT FOR AH CEP 993 Fig. 1. History of the timings of minimum light for AH Cep. Top: The (O C ) 1 residuals and continuous curve were constructed with the linear and light-time ephemeris for the third body given by M98, respectively. The residuals are coded by observational method. The times of minimum light after about 1990 have gradually deviated from the theoretical curve. Middle: The (O C ) 2 residuals and continuous curve with our improved linear and lighttime ephemeris given in col. of Table 3. Bottom: Residuals from our solution in Table 3. addition, two minima (JD 2,424, and JD 2,425,836.27) were discarded because the first of these was spectroscopically determined and the second one had been, in fact, discarded by the author (Zverev 1931). The Hipparcos timings were excluded from our analysis for a secondary modulation of the O C residuals (which is discussed below) because each of them was not determined from consecutive observations on the night to which each belongs. Also, seven timings determined by Harvig (1990) were checked carefully. Just as in the case of Hipparcos timings, six of these all except the last one (JD 2,446, ) were not determined from consecutive observations on the night to which the timing belongs, and even the last one was determined from insufficient data on the descending eclipse branch. Therefore, all of Harvig s timings are not sufficiently accurate for our purposes. His timings were ignored in our subsequent analysis for the secondary oscillation, by the criterion applied to the Hipparcos timings. Each of these discarded timings is marked by an asterisk in column (10) of Table 2. The weights for the visual and photoelectric minima were assigned to be 10 and 200, respectively, according to the inverse Fig. 2. Residuals between JD 2,443,000 and 2,450,000 from the light-time ephemeris given in col. of Table 3. A small oscillation with a period of about 9 yr and a semiamplitude of about days may be possible if the residuals indicated in numbers are removed. See the text for a detailed explanation. square of the scatter of their O C residuals. Weights of 1 for one sky patrol timing and of 20 for one photographic timing were given; these weights are ad hoc. In order to improve M98 s light-time orbit, we fitted the observed minima to the following light-time ephemeris: C ¼ T o þ PE þ 3 ; where 3 is the light-time term whose parametric form is taken from Irwin (1952). A more generalized treatment of equation would involve a differential corrections procedure (e.g., the Levenberg-Marquardt method; Press et al. 1992) to determine eight unknown parameters (Irwin 1959). Initial parameters for the light-time orbit were obtained from M98 s values. The parameters for our final solution are listed in column of Table 3 where 2 denotes value of the fit. The other elements are self-explanatory. The residuals from the linear terms of equation appear as (O C ) 2 in Table 2. The continuous curve in the middle of Figure 1 represents our theoretical light-time curve, and the residuals from it appear in the bottom panel of the figure. Although these (O C ) 2 residuals are well represented by our light-time curve, there exists scatter of about days in the photoelectric and CCD residuals. Such a value cannot be disregarded because of the accuracy ð2þ TABLE 3 Orbital Elements of Light-Time Orbits for the AH Cep System This Paper Without Outliers With Outliers Elements M98 Third Body Only Third Body Fourth Body Third Body (6) Fourth Body (7) T 0 ()... 2,434, ,434, (27) 2,434, (10) 2,434, (25) P (days) (41) (15) (37) K (days) e (18) 0.64 (16) (47) 0.91 (33)! (deg) (4.0) 73.3 (1.4) 161 (16) 79.2 (3.6) 168 (37) T ( )... 2,444,800 2,444,347(143) 2,443,968(52) 2,442,670(285) 2,444,192(136) 2,442,912(1172) P 0 ( yr) (1.9) 67.6 (6) (1.7) 9.2 (1.8)

5 994 KIM, NHA, & KREINER Vol. 129 Fig. 4. The O C diagrams of AH Cep. Top: Photoelectric and CCD residuals, with the linear ephemeris given in cols. and of Table 3. The continuous curve represents the theoretical light-time contributions due to the third and fourth bodies. Middle: Residuals after subtracting the third body lighttime term, with the theoretical light-time curve due to the suggested fourth body. Bottom: Residuals from the theoretical light-time contributions due to both the third and fourth bodies. Fig. 3. Power spectra of the photoelectric residuals between JD 2,443,000 and 2,450,000 from the improved light-time ephemeris for the third body. Panels (a) and(b) represent power spectra with and without the outliers, respectively. of modern photoelectric and CCD timings. In Figure 2 the residuals crowded between JD 2,443,000 and 2,450,000 are redrawn to see their behavior in detail. We noticed by careful examinations of the residuals that there may be a possible oscillation with a semiamplitude of about days and a period of about 9 yr if the residuals of nine timings (hereafter referred to as outliers ) are neglected. The outliers are indicated as arrows with numbers in Figure 2 and are marked in column (10) of Table 2 by double asterisks. In the figure we see that most residuals are contained in the residual scatter band of about except two timings numbered as 6 and 7, and the residuals of three timings (numbered as 5, 6, and 7) deviate by at least days from those of adjacent minima. We show below that these are more than 3 from our final solution. Period searches (Scargle 1982) were made for the two data sets with and without the nine outliers among the residuals from the light-time orbit obtained above. The resulting power spectra for the two data sets are shown in Figure 3. In Figure 3a, we see that there are no peaks at the frequencies corresponding to 668 days ( frequency) and 1 yr ( frequency), suggested by Mayer & Wolf (1986) and DLM89, respectively. Instead, the highest peak is at the frequency of 0.021, corresponding to a period of days. On the other hand, as shown in Figure 3b, the highest peak of the power spectrum obtained without outliers occurs clearly at a frequency of , corresponding to 9.13 yr. Therefore, we analyzed the times of minimum light for two long periods without and with the nine outliers. First, by excluding the outliers, we deduced the period and semiamplitude of the second oscillation to be about 9.13 yr and days, respectively, from the result of the period search and Figure 2. Then, after assuming that the second variation is due to a second light-time effect caused by a fourth body in the system, all the timings except the outliers were fitted to the following ephemeris: C ¼ T o þ PE þ 3 þ 4 ; where 3 and 4 are the light times due to the third and fourth bodies, respectively. The Levenberg-Marquardt method was used again to find the light-time orbits of the third and newly assumed fourth bodies simultaneously. In this case the parameters to be adjusted are 12 in number. The calculations converged quickly to yield the solution listed in columns and of Table 3. The (O C ) 3 residuals from the linear light elements in Table 3 are listed in column (6) of Table 2, and the photoelectric and CCD ones among them are drawn at the top of Figure 4 with the theoretical light-time curves due to the third and fourth bodies. In the middle part of Figure 4 the contributions to light times by only the fourth body are plotted. At the bottom of the figure there are plotted the (O C ) 4 residuals from the full ephemeris of equation, which appear in column (7) of Table 2. As shown in Figure 4, the residuals from the observed times of minimum light follow the theoretical light-time curves due to the assumed third and fourth bodies very well. As listed in Table 3, the semiamplitude, orbital period, and eccentricity for the light-time orbit of the mass center of the eclipsing pair due to the fourth body are days, 9.6 yr, and 0.64, respectively. The light-time semiamplitude due to the fourth body is about 15 times smaller than that due to the third one. The photoelectric and CCD residuals from only ð3þ

6 No. 2, 2005 SECOND LIGHT-TIME ORBIT FOR AH CEP 995 Fig. 5. Residuals from the middle panel of Fig. 4 phased with the suggested fourth-body ephemeris given in cols. and of Table 3. The continuous curve represents the projected light-time orbit of the barycenter of AH Cep caused by the fourth body. the fourth body are phased relative to the orbital period of the fourth-body in Figure 5, where the continuous line represents the theoretical light-time curve. Second, the same procedure was applied to all the minima, including the outliers. The final solution is listed in columns (6) and (7) of Table 3. As expected, 2 increased (about 8 times) when compared with that obtained without the outliers. In Table 3 we see that standard errors for the orbital elements obtained with outliers are systematically larger than those without the outliers. The photoelectric and CCD residuals from only the fourth body are phased at the top of Figure 6, where the outliers are marked as plus signs for the primary timing and crosses for the secondary, respectively, and the continuous line represents the theoretical light-time curve. For comparison, at the bottom of Figure 6 are plotted the photoelectric and CCD residuals including outliers due to only the fourth body phased according to the solution obtained without outliers. The deviations of the outliers from the theoretical fourth light-time curve aremorethan3 ( ¼ 0:00087 days). The (O C ) 4 residuals for two of seven timings of Harvig (1990) are within 1 and the rest are greater than 3, whereas half of the six Hipparcos timings are within 2 and half beyond 3. Third, the same procedures performed above were carried out for the period of days that had been found through the period search. No reasonable light-time orbits, however, were found. This possible period must be spurious. 3. REANALYSIS OF RADIAL VELOCITIES FOR THE AH CEP SYSTEM 3.1. Indivvidual Velocities of the Center of Mass of the EclipsinggPair We have now shown that there exists a second small modulation superposed on the light-time residuals due to a third body and that, if the variation is assumed to be due to a fourth body, it is well fitted by the light-time curve shown in Figure 5. In order to investigate whether or not our finding is supported by spectroscopic observations, we collected all the individual radial velocity measurements for AH Cep published so far. Table 4 lists all individual radial velocities. The spectroscopic solutions published so far are summarized in Table 5. Fig. 6. All photoelectric and CCD residuals phased with the suggested fourth-body ephemeris. The residuals and the curves in the top and bottom panels were drawn by using the solutions of Table 3 with and without the outliers, respectively. The outliers are marked as plus signs for the primary timing and crosses for the secondary. First, we calculated individual velocities for the center of mass of the eclipsing pair via the formula v cm ¼ M 1v 1 þ M 2 v 2 M 1 þ M 2 ; ð4þ where M 1 and M 2 (v 1 and v 2 ) denote the masses (the observed radial velocities) for the primary and secondary components of the AH Cep system, respectively. In the calculation, individual masses of the components used were M 1 ¼ 15:9 M and M 2 ¼ 13:7 M for the primary and secondary, respectively. These masses were calculated from the mean values (13.0 for the primary and 11.2 for the secondary) of Msin 3 i (M ) listed in Table 5 and the orbital inclination i of 69N3 found by Bell et al. (1986). The center-of-mass velocities are shown in column of Table 4, from which their mean value ( v cm ) and standard deviation () were calculated as 21.4 km s 1 and 12.6 km s 1, respectively. In Table 4 two data points (JD 2,446, and 2,446, ) published by Bell et al. (1986) deviate by more than 3 and were excluded in subsequent analysis. In their analysis of radial velocities Bell et al. (1986) also excluded one of these (JD 2,446, ) because it gave an abnormally high residual of 90 km s 1 for the velocity of the secondary component. The other measurement at JD 2,446, also shows the large radial velocity residual of approximately 51 km s 1 for the velocity of the primary (see Table 1 in Bell et al. 1986). After exclusion of the two indicated data points, v cm and were recomputed as 20.2 km s 1 and 10.1 km s 1, respectively. We note that v cm agrees with the systemic velocities of AH Cep obtained by previous investigators. (See Table 5.) The individual center-of-mass velocities are plotted at the top in Figure 7. In Figure 7 the continuous curve was drawnwiththeformula v cm ¼ v þ v 3 þ v 4 ; where v 3 and v 4 are the theoretical radial velocities due to the third and fourth bodies discussed in x 2 and are calculated with the solutions listed in columns and of Table 3, respectively. The quantity v denotes the systemic velocity of the eclipsing pair, and the v cm of 20.2 km s 1 determined above was used as v. In the figure we see that the theoretical curve is at least weakly correlated with the observed systemic velocity ð5þ

7 TABLE 4 Individual Radial Velocities of AH Cep v 1 v 2 v cm Ref. v 1 v 2 v cm Ref. 24, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , a 48, , , , a 48, , , , , , , , , , , , , , , , , , , , a Discarded. References. Pearce 1927; : Bell et al. 1986; : Holmgren et al. 1990; : Burkholder et al

8 SECOND LIGHT-TIME ORBIT FOR AH CEP 997 TABLE 5 Spectroscopic Solutions for the AH Cep System This Paper Parameters HA a BE a HO a BU a No Correction (6) Correction (7) Observing Interval Start ( )... 2,424, ,441, ,446, ,448, Observing Interval End ( )... 2,424, ,446, ,446, ,448, Number of Measurements T 0 ( )... 2,424, ,446, ,446, ,448, ,434, ,434, P (days) e ! (deg) K 1 (km s 1 ) K 2 (km s 1 ) (km s 1 ) (km s 1 ) (km s 1 ) M 1 sin 3 i (M ) M 2 sin 3 i (M ) q a 1 sin i (R ) a 2 sin i (R ) a HA: Harper et al. (1935); BE: Bell et al. (1986); HO: Holmgren et al. (1990); BU: Burkholder et al. (1997). of the eclipsing pair. The large scatter is due to the noise of the individual velocities. In the bottom of Figure 7 are plotted the differences between the observed and the theoretical systemic velocities due to only the third and fourth bodies. The mean value and standard deviation of the differences are 16.6 km s 1 and 9.3 km s 1,respectively.Theformernumber,whichmay be considered as a systemic velocity of the AH Cep system, is increased by about 4 km s 1 compared with that ( 20.2 km s 1 ) obtained without correction, whereas the latter, comparable to the standard errors of modern spectroscopic observations, is reduced by about 1 km s 1. One can argue that the analysis above is too weak to prove that the AH Cep system is a quadruple system. Although this is true, the analysis is not inconsistent with the proposition. Fig. 7. (a) Individual center-of-mass velocities of AH Cep. The continuous curve represents the theoretical radial velocity of the barycenter of AH Cep due to the third and fourth bodies and is drawn with the parameters given in cols. and of Table 3. (b) Residuals from the curve in (a). See the text for a detailed explanation. In (a) and(b) the triangles, open circles, squares, and downward-pointing triangles denote the individual data published by Pearce (1927), Bell et al. (1986), Holmgren et al. (1990), and Burkholder et al. (1997), respectively Spectroscopic Solutions of AH Cep As listed in Table 5, there have existed significant disagreements in the values of K 1 and K 2 among the spectroscopic solutions of AH Cep obtained from different data sets by different investigators, whereas the systemic velocities have remained nearly the same at about 21 km s 1. The latter fact can be easily understood from the previous discussions for individual center-of-mass velocities. The largest values of K 1 and K 2 are from Bell et al. (1986). They determined K 1 and K 2 from 15 radial velocities, among which five measurements were between JD 2,441,529 and 2,441,586 and the rest between JD 2,446,306 and 2,446,386. These intervals are separated by about 4789 days, or 13.1 yr. The former data set was near the top of the theoretical radial velocity curve of the center of mass of the eclipsing pair because of the third and fourth bodies, whereas the latter data set was near its bottom (see the top diagram in Fig. 7). It would be natural for the investigators to obtain the large values for K 1 and K 2 from those data since they did not consider effects due to multiplicity of the AH Cep system. It is useful to derive spectroscopic solutions taking account of the multiplicity of the AH Cep system, since previous investigators did not do so. For this purpose, spectroscopic solutions without and with consideration of the multiplicity were obtained using all the radial velocities listed in Table 4. For both cases a circular orbit of the eclipsing pair was assumed because there is no evidence for an eccentric orbit in our O C study. For the case of multiplicity, both light-time effects and radial velocity variations due to the third and fourth bodies were applied as corrections. Our solutions for the two cases are listed in columns (6) and (7) of Table 5. From these solutions the theoretical radial velocity curves were superposed on the individual data in Figures 8 and 9. As shown in Table 5, our solutions agree best with that of Holmgren et al. (1990). Comparison of our solutions for the two cases shows that K 1 and K 2 are in good agreement but that the systemic velocity shows a difference of about 4 km s 1. The standard error, when account is taken of multiplicity, is reduced slightly compared to the other case. In addition, the mean orbital period obtained

9 998 KIM, NHA, & KREINER Vol. 129 Fig. 8. Radial velocities as a function of orbital phase for the eclipsing pair of AH Cep. Open and filled symbols and plus signs denote the massive primary, the secondary, and the barycenter, respectively. Explanations for individual symbols are the same as for Fig. 7. The continuous curves were drawn with the spectroscopic solution in col. (6) of Table 5, which was obtained without any correction for the multiplicity of the AH Cep system. when multiplicity is considered is closer to that obtained from our O C study. From our solution obtained with correction for multiplicity, we obtained the absolute dimensions listed in Table 6. Because of the errors in the original plate measures, there is not much of a change in these values. 4. PROPERTIES OF THE THIRD AND FOURTH BODIES SUGGESTED IN THE AH CEP SYSTEM We calculated the mass functions for the fourth and third bodies with well-known formulae. For the fourth body, f (m) 4 ¼ (a 124 sini 124 ) 3 P 2 4 and for the third body, f (m) 3 ¼ (a 123 sini 123 ) 3 P 2 3 ¼ ¼ (M 4 sini 124 ) 3 (M 1 þ M 2 þ M 4 ) 2 ; ð6þ (M 3 sini 123 ) 3 (M 1 þ M 2 þ M 3 þ M 4 ) 2 : ð7þ Here a 124 sini 124 is the projected semimajor axis of the orbit for the eclipsing-pair mass center relative to the mass center of the eclipsing pair plus the fourth body, and a 123 sini 123 is that for the mass center of the eclipsing pair plus the fourth body relative to the mass center of the quadruple system. The mass functions were calculated with the parameters given in Table 3 and are listed in column of Table 7. Using these mass functions and the masses of the eclipsing pair given in Table 6, equations (6) and (7) were solved with a Newton-Rapson method to yield individual masses of the fourth and third bodies for various inclinations. For these calculations it was assumed that the orbits of the fourth and third bodies are coplanar. The results for three different orbital inclinations (i 123 ¼ i 124 ¼ 90,69N3, 60 ) are listed in columns of Table 7. The inclination of 69N3is the value for the eclipsing pair obtained by Bell et al. (1986). As shown in Table 7, the third body is about 2 times more massive Fig. 9. Radial velocities as a function of orbital phase for the eclipsing pair of AH Cep. Explanations for individual symbols are the same as for Fig. 7. The continuous curves were drawn with the spectroscopic solution in col. (7) of Table 5, which was obtained after correction for the multiplicity of the AH Cep system. than that of the fourth body for all the three different orbital inclinations. If the third and fourth bodies are main-sequence stars, their luminosities can be derived with the empirical mass-luminosity relation for main-sequence stars (Schmidt-Kaler 1982): for m=m > 0:2, log L=L ¼ 3:8logm=M þ 0:08: The results and resulting luminosity ratio [(L 3 þ L 4 )=(L 1 þ L 2 þ L 3 þ L 4 )] are listed from columns (6) (11) in Table 7. In Table 7 the sum of the minimum masses for the extra bodies in the AH Cep system is 11.1 M,whichis2M larger than that obtained for the mass of the third body by DLM89. As shown in Table 7, our light ratio of for an inclination of 90 is exactly the same as that for the third-light value obtained with the DC solution of DLM89, whereas the light ratio for an inclination of 69N3 is the same as that for the third-light value obtained with their SIMPLEX solution. The main-sequence spectral types corresponding to the supposed masses of the third and fourth bodies for an inclination of 69N3 areb2 B3 and B7 B8, respectively. The combined spectral type would be about B3, a few subclasses later than that of B0.5 for the binary components. From all the orbital parameters the spatial motion of the centerofmassoftheeclipsingpairrelativetothemasscenterof the four stars was calculated. It was assumed that all bodies are in the orbital plane (i ¼ 69N3) of the eclipsing pair. That TABLE 6 Absolute Dimensions of the Close Binary of the AH Cep System Star ð8þ Mass Radius Distance (M ) a (R ) b log (L/L ) c ( pc) d Primary Secondary a Determined with the photometric inclination of 69N3 given by Bell et al. (1986). b Determined using the mean radii of Drechsel et al. (1989). c Determined using temperatures given by Holmgren et al. (1990). d Given by Holmgren et al. (1990).

10 No. 2, 2005 SECOND LIGHT-TIME ORBIT FOR AH CEP 999 TABLE 7 Masses and Luminosities Deduced for the Third and Fourth Bodies in the AH Cep System Body f ( m) Mass (M ) log (L/L ) a L 34 /L total (M ) i =90 i =69N3 i =60 i =90 i =69N3 i =60 i =90 i =69N3 i =60 Fourth body Third body a L 34 ¼ L 3 þ L 4 ; L total ¼ L 1 þ L 2 þ L 3 þ L 4. schematized orbit projected on the celestial sphere is shown in Figure 10, where the origin marked by a cross is the center of mass of the four stars and the X Y plane is perpendicular to the Z-axis, directed to Earth. In the figure we see that the orbit of the mass center of the eclipsing pair is necessarily perturbed because of the presence of the fourth body. The absolute scales are about 22 and 8 AU in the X- andy-axes, respectively, corresponding to angular sizes of 0B028 and 0B011, if we adopt the distance of AH Cep as 725 pc. Therefore, since these maximum angular sizes are comparable to the observational limit of modern ground-based speckle interferometry, the spatial motion of the center of mass of the eclipsing pair is unlikely to be detected by that method. The individual light contributions from the third and fourth bodies for an inclination of 69N3 are, respectively, about 3.0 and 5.8 mag fainter than the combined maximum light of the eclipsing pair. The corresponding light ratios of the third and fourth bodies to the eclipsing pair are about 0.06 and 0.005, respectively. Hence, the third star s light might be isolated by careful observations with direct speckle imaging interferometry, as, e.g., in the case of ER Ori (Goecking et al. 1994). The third-body orbit relative to the mass center of the eclipsing pair in the X-Y plane is schematically drawn in Figure 11, where the origin marked by a cross is the center of mass of the eclipsing pair and the distance (50 AU) and the angular separation (0B07) from the origin to the third body are indicated for the epoch of SUMMARY AND DISCUSSION In this paper we analyzed all useful times of minimum light of the AH Cep system that have been observed from early 1901 to early It was found that the O C residuals from an improved eclipsing period have suffered an additional lighttime effect due to a fourth body, superposed on the well-known light-time effect due to a third star. The analysis of the radial velocities of AH Cep does not conflict with the hypothesis of the multiplicity of the system. The other short periodicity suggested by Mayer & Wolf (1986) and DLM89 was not confirmed. AH Cep may be considered a quadruple system with astrophysical parameters as follows: Eclipsing pair: M 1 ¼ 15:8 M, M 2 ¼ 13:7 M, P 12 ¼ 1:7747 days, Fourth body: e 4 ¼ 0:64, P 4 ¼ 9:6 yr,p 4 =P 12 ¼ 1976, Third body: e 3 ¼ 0:52, P 3 ¼ 67:6 yr, P 3 =P 12 ¼ 13; 913, P 3 =P 4 ¼ 7:0. Fig. 10. Schematized projected motion of the center of mass of the eclipsing pair relative to the mass center of the four stars of the AH Cep system. The X-Y plane (the celestial sphere) is perpendicular to the Z-axis directed toward Earth. The center of mass of the four stars is marked by a cross. Absolute scales were calculated after adopting the distance of AH Cep as 725 pc. The projected orbit of the mass center of the eclipsing pair is obviously perturbed by the fourth body. The trajectory is marked by a filled circle every 10 yr from 1900 to Fig. 11. Schematized projected motion of the third body relative to the mass center of the eclipsing pair of the AH Cep system. The explanations for the axes and scales are the same for Fig. 10. The linear and angular separations from the origin to the third body are indicated for the epoch of

11 1000 KIM, NHA, & KREINER This corresponds to hierarchy 2 (e.g., ab-c-d) according to the classification of Evans (1968). During the interval from the first time of minimum light to the last, the third body revolved about 1.5 cycles around the center of mass of the quadruple group, while the fourth body passed 10.6 cycles around the mass center of the eclipsing pair and itself. At this time there seem to be no apparent perturbations on their Keplerian orbits during the interval, but this possibility remains a likely one in the future. The ratios of the orbital periods and the approximate masses indicate that the quadruple system should be dynamically stable (Harrington 1977; Eggleton & Kiselev 1995). Future high-precision observations (e.g., observations of timings of minimum light, as well as astrometric and/or spectroscopic observations) will be needed to reveal more detailed properties of the AH Cep system. The authors wish to thank R. H. Koch for careful readings and corrections and for some helpful comments of the original version of the manuscript. We also express our gratitude to S. Sola for some critical suggestions. We would like to thank the anonymous referee for the useful comments and suggestions that certainly improved our original paper. Our appreciation also goes to V. Harvig and T. Pribulla for sending a copy of the paper of Harvig (1990), which was not known to the authors at our paper s first submission. One of us (C. H. K.) wishes to express his thanks for the hospitality of the staff of the department while a visiting professor at the Department of Astronomy and Astrophysics, Villanova University. This work was supported partly by grant R from the Basic Research Program of the Korean Science and Engineering Foundation and partly by grant M from the Korea Institute of Science and Technology and Planning. Agerer, F., & Hübscher, J. 1996, Inf. Bull. Variable Stars, , Inf. Bull. Variable Stars, 4562 Battistini, P., Bonifazi, A., & Guarnieri, A. 1973, Inf. Bull. Variable Stars, 817 Bell, S. A., Hilditch, R. W., & Adamson, A. J. 1986, MNRAS, 223, 513 Burkholder, V., Massey, P., & Morrell, N. 1997, ApJ, 490, 328 Cester, B., Fedel, B., Giuricin, G., Mardirossian, F., & Mezzetti, M. 1978, A&A, 62, 291 Diethelm, R. 1981, BBSAG Bull., , BBSAG Bull., , BBSAG Bull., , BBSAG Bull., 96 Drechsel, H., Lorenz, R., & Mayer, P. 1989, A&A, 221, 49 (DLM89) Eggleton, P. P., & Kiselev, L. G. 1995, ApJ, 455, 640 Evans, D. S. 1968, QJRAS, 9, 388 Goecking, K. D., Duerbeck, H. W., Plewa, T., Kaluzny, J., Schertl, D., Weigelt, G., & Flin, P. 1994, A&A, 289, 827 Guarnieri, A., Bonifazi, A., & Battistini, P. 1975, A&AS, 20, 199 Harper, W. E., Pearce, J. A., Petrie, R. M., & McKellar, A. 1935, JRASC, 29, 411 Harrington, R. S. 1977, Rev. Mex. AA, 3, 139 Hartigan, P., & Binzel, R. P. 1982, J. AAVSO, 11, 21 Harvig, V. 1990, Publ. Tartuskoj Astrofiz. Obs., 53, 115 Holmgren, D. E., Hill, G., & Fisher, W. 1990, A&A, 236, 409 Hübscher, J., Agerer, F., Wunder, Z., & Wunder, E. 1991, BAV Mitt., , BAV Mitt., 60 Huffer, C. M., & Eggen, O. J. 1947, ApJ, 106, 313 REFERENCES Irwin, J. B. 1952, ApJ, 116, , AJ, 64, 149 Isles, J. 1992, British Astron. Assoc. Variable Star Sect. Circ., No. 73, 16 Kreiner, J. M., Kim, C.-H., & Nha, I.-S. 2001, An Atlas of (O C) Diagrams of Eclipsing Binary Stars ( Krakow: Wydawn. Nauk. Akad. Pedagogicznej) Kwee, K. K., & van Woerden, H. 1956, Bull. Astron. Inst. Netherlands, 12, 327 Mayer, P. 1980, Bull. Astron. Inst. Czechoslovakia, 31, , Bull. Astron. Inst. Czechoslovakia, 38, 58 Mayer, P., Niarchos, P. G., Lorenz, R., Wolf, M., & Christie, G. 1998, A&AS, 130, 311 (M98) Mayer, P., & Tremko, J. 1983, Inf. Bull. Variable Stars, 2407 Mayer, P., & Wolf, M. 1986, Inf. Bull. Variable Stars, 2886 Mayer, P., Wolf, M., Tremko, J., & Niarchos, P. G. 1991, Bull. Astron. Inst. Czechoslovakia, 42, 225 Moore, J. H. 1936, Lick Obs. Bull., 483, 16 Nekrasova, S. V. 1960, Perem. Zvezdy, 13, 157 Oprescu, G., Suran, M. D., & Popescu, N. 1989, Inf. Bull. Variable Stars, 3312 Park, N. K. 1993, Publ. Korean Astron. Soc., 8, 185 Pearce, J. A. 1927, Pub. Dom. Astrophys. Obs. Victoria, 4, 67 Press, W., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. 1992, Numerical Recipes in FORTRAN (2nd ed.; Cambridge: Cambridge Univ. Press) Scargle, J. D. 1982, ApJ, 263, 835 Schmidt-Kaler, T. 1982, in Landolt-Bornstein, NS, Vol. 2, ed. K. Schaifers & H. H. Voigt (Berlin: Springer), 28 Zverev, M. 1931, Beob. Zirk. Astr. Nachr., 13, , Perem. Zvezdy, 4, 177