JOURNAL OF THE KOREAN ASTRONOMICAL SOCIETY Vol. 53 No. 2 JOURNAL OF THE KOREAN ASTRONOMICAL SOCIETY April 30, 2020 Vol. 53 No. 2, April 30, 2020 JOURNAL OF THE THE FORMATION OF THE DOUBLE GAUSSIAN LINE PROFILES OF THE SYMBIOTIC AG PEGASI ······················································································································································· Siek Hyung and Seong-Jae Lee 35 KOREAN POORLY STUDIED ECLIPSING BINARIES IN THE FIELD OF DO DRACONIS: V454 DRA AND V455 DRA ··························································································· Yonggi Kim, Ivan L. Andronov, Kateryna D. Andrych, Joh-Na Yoon, Kiyoung Han, and Lidia L. Chinarova 43 ASTRONOMICAL DECAY OF TURBULENCE IN FLUIDS WITH POLYTROPIC EQUATIONS OF STATE ················································································································································ Jeonghoon Lim and Jungyeon Cho 49 SOCIETY Vol. 53 No. 2, April 30, 2020

pISSN 1225-4614 eISSN 2288-890X THE KOREAN ASTRONOMICAL SOCIETY Officers of the Society (2020-2021)

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Published since 1968. Rev 2020/04 Journal of the Korean Astronomical Society https://doi.org/10.5303/JKAS.2020.53.2.35 53: 35 ∼ 42, 2020 April pISSN: 1225-4614 · eISSN: 2288-890X Published under Creative Commons license CC BY-SA 4.0 http://jkas.kas.org

THE FORMATION OF THE DOUBLE GAUSSIAN LINE PROFILES OF THE SYMBIOTIC STAR AG PEGASI Siek Hyung and Seong-Jae Lee Department of Earth Science Education (Astronomy), Chungbuk National University, Chungbuk 28644, Korea [email protected], [email protected] Received December 12, 2019; accepted February 8, 2020

Abstract: We analyze high dispersion emission lines of the symbiotic AG Pegasi, observed in 1998, 2001, and 2002. The Hα and Hβ lines show three components, two narrow and one underlying broad line components, but most other lines, such as H i, He i, and He ii lines, show two blue- and red-shifted components only. A recent study by Lee & Hyung (2018) suggested that the double Gaussian lines emitted from a bipolar conical shell are likely to form Raman scattering lines observed in 1998. In this study, we show that the bipolar cone with an opening angle of 74◦, which expands at a velocity of 70 km s−1 along the polar axis of the , can accommodate the observed double line profiles in 1998, 2001, and 2002. We conclude that the emission zone of the bipolar conical shell, which formed along the bipolar axis of the white dwarf due to the collimation by the accretion disk, is responsible for the double Gaussian profiles. Key words: : binaries: symbiotic — stars: individual: AG Peg – ISM: dynamics and kinematics — ISM: lines and bands

1. INTRODUCTION a GS transit cannot form the observed Gaussian line Symbiotic stars are identified as binary systems by their profiles. Using spectroscopic data obtained at phases peculiar spectra, which show contributions by emission φ ∼ 0.0 or 0.5 with the Lick observatory, LH18 investi- of both relatively low and very high temperature. How- gated whether the GS blocks the emission zone around ever, it is difficult to confirm the presence of two stars the hot WD, causing the observed Gaussian line profiles. in imaging studies. AG Pegasi was first reported as a They provided some kinematic inference that the GS Be star by Fleming (1907), after which Merrill (1916, passages were not the source of the double Gaussian line profiles. Instead, LH18 demonstrated that a bipolar 1942) observed He and Ca ii absorption lines, numerous conical shell formed double Gaussian lines and that a ionized or neutral emission lines, and TiO2 absorption bands. Considered a Cyg-Be type star at the time of its broad line wing was formed by Raman scattering in the discovery, AG Peg is now known as a symbiotic system neutral region of the shell. composed of a Wolf-Rayet type white dwarf (WD) and Raman scattering processes in symbiotic stars were a type M3 giant star (GS) (Merrill 1916; Hutchings et al. proposed first by Schmid (1989), and Raman wings were 1975). Spectroscopic monitoring of AG Peg has shown first discussed by Nussbaumer et al. (1989) and further variations of its brightness and spectral intensities on studied by Lee (2002). Lee & Hyung (2000) also showed time scales from months to decades. The orbital period that Raman scattering can occur in a dense planetary is about 800 days. The temperature of the WD is about nebula such as IC 4997. Symbiotic stars have been 100 000 K, while its cold component is a GS (Allen 1980; considered as candidates for type Ia supernovae. The Viotti 1988; Nussbaumer 1992). Since the beginning of kinematic structure, including the physical conditions astronomical observations, two major outbursts have of the ionized hydrogen region and the occurred: a first one in 1850, a second, shorter one in system, is of great interest. The mass inflow from the 2015. GS into the hot component might form an accretion disk. In a symbiotic binary system, variations of emission The fast stellar wind from the hot star is likely to develop line profiles might be caused by shielding effects partly bipolar conical shells that are responsible for both the due to the relative orientation of observer and binary sys- double and broad emission lines. As the orbital phase, tem. Some earlier studies noted the presence of double and thus the relative orientation of binary components Gaussian profiles or an absorption line at the center that and observer, changes, the shape of the line profiles might be associated with the GS passage at a specific changes accordingly. LH18 suggested that the bipolar phase (Contini 1997). However, the double Gaussian conical shell is the most likely source of double Gaussian profiles seemed to exist at all phases and the GS cannot line profiles, but they did not compute theoretical line always be in front of the emission zone surrounding the profiles for multiple phases. In this study, we fit the Hα WD. Lee & Hyung (2018), hereafter LH18, showed that or Hβ double Gaussian line profiles observed in arbitrary phases to a bipolar conical model. Corresponding author: S.-J. Lee Section 2 describes the observed spectra for three 35 36 Hyung & Lee

Table 1 matic subcomponents responsible for the full line profile, AG Peg observation log which involves a trial and error process that requires choosing the number of line components the observer can Observation date (UT) Julian Date Phase (φ) resolve. We used StarLink/Dipso provided by the Inter- 1998-09-17 2451073.70 10.24 active Data Language (IDL) software and the European 2001-08-30 2452151.60 11.56 Southern Observatory (ESO) for this deconvolution. 2002-08-11 2452498.92 11.98 Figure 1 shows the Hα lines observed in three epochs. To avoid saturation in the spectral line pro- files, we used the 3 min and 5 min exposures. LH18 epochs acquired by the Hamilton Echelle Spectrograph concluded that the Hα and Hβ hydrogen lines observed at Lick Observatory. Section 3 presents the predicted in 1998 are composed of three parts, a double line plus line profiles and the best parameters of a bipolar coni- a broad wing, while the previous studies assumed four cal shell suitable for three epochs. We investigate the to five separate components with different kinematics, geometric structure of the system, responsible for the with suggested radial velocities being 60, 120, 400, and double Gaussian line profiles, observed in 1998, 2001, 1000 km s−1 (Kenyon et al. 1993; Eriksson et al. 2004). and 2002. Theoretical line profiles corresponding to dif- At first glance, only two line components are obvious. ferent phases are provided to model the observed double However, our detailed analysis (using IDL) separates the line profiles. We conclude in Section 4. observed line profiles into three components. A detailed discussion of the three components is given in LH18 2. LICK OBSERVATORY SPECTRAL DATA based on the 1998 data. The narrow double lines consist Spectroscopic data were obtained by Lawrence H. Aller of blue- and red-shifted Gaussian components. and Siek Hyung in 1998, 2001, and 2002 at the Lick The Hα line observed in 1998 shows a red compo- observatory in the USA using the high-resolution Hamil- nent that is substantially stronger than the blue one, ton Echelle Spectrograph (HES) attached to the 3-meter while in 2001 the blue component is stronger than the Shane telescope. The spectral resolution of the HES red one. All observed line profiles in Figure 1 show is R ∼ 50 000. The slit width of the HES is 640 µm the two narrow red-shifted and blue-shifted line com- which corresponds to 1.200 on the sky, the wavelength ponents and one broad component. The full width at resolution is 0.1–0.2 A/pixel.˚ The observed spectral half maximum (FWHM) of the narrow lines is 60–120 range is 3470–9775 A,˚ and we selected only the H and km s−1 each, while the width of the broad line is 400– He lines from the various emission lines. Long exposure 500 km s−1. We adopted the kinematic center from the times of 1800 sec and 3600 sec were chosen for weak result of LH18 who analyzed the strong lines observed lines. Additional short-exposure spectra of 300 sec and in 1998. One expects to see He ii 6560 as well. Due to 180 sec were obtained to avoid saturation of the strong the relatively strong Hα emission, however, the He ii Hα line. The IRAF standard star HR 7596 served as 6560 line component(s) to the left of the Hα line cannot flux calibrator. be resolved. Table 1 shows the observation dates of AG Peg, The Hβ line spectra shown in Figure 2 display two Greenwich universal time (UT), Julian day, and phase. lines to the left of the Hβ components, corresponding The phase of AG Peg is defined as zero when the WD is to N iii 4858 and He ii 4559. Although the FWHMs of in front of the GS. The 2002 data (phase φ = 11.98, 0.98 the Hβ line components are slightly different from those hereafter), close to φ = 0.0, were secured when the WD of the Hα spectral line profiles, both consist of three was in front of the GS. The 2001 data (phase φ = 11.56, components. 0.56 hereafter) is close to φ = 0.5 when the GS was in We corrected the observed Hα and Hβ fluxes for front of the WD, i.e. when the GS could block the view interstellar extinction using a value of the extinction onto the emission zone near the WD. In this study, we coefficient C = log I(Hβ)/F (Hβ) = 0.04 (or color ex- also used data from 1998 obtained at a phase φ = 10.24 cess E(B − V ) = 0.027) adopted from Kim & Hyung (0.24 hereafter) when WD and GS were close to their (2008) (hereafter KH08). The extinction-corrected ratio maximum separation on the sky. A detailed description I(Hβ)/F (Hβ) for the (blue plus red) flux is 2.87 for of the 1998 spectral data is given in LH18. 1998, and those for 2001 and 2002 have similar values, The data were analyzed using the Image Reduction 2.97 and 2.86, respectively. In 2001, the red components and Analysis Facilities (IRAF) software package of the of both the Hα and Hβ lines are weak compared to the National Optical Astronomy Observatory (NOAO). The blue ones: the GS with its extended atmosphere may spectra, with wavelengths in units of A,˚ were corrected have blocked parts of the emission zone. for atmospheric extinction. We converted observed wave- Inconsistencies in flux intensity or line width exist lengths to emitter frame wavelengths using the radial between the Hα and Hβ fluxes observed in 1998 and 2002. velocity of AG Peg relative to the Sun of −7.09 km s−1 LH18 already noted their disagreement. Meanwhile, which we obtained from a large number of permitted H i the Hα and Hβ fluxes observed in 2001 show a better lines, namely the Balmer and the Paschen lines. Wave- agreement (see column 7 of Table 2). The discordance lengths were converted to speeds (in units of km s−1) to might partially be due to complex kinematics near the make the kinematic characteristics visible directly. The inner Lagrangian point and the GS with its extended final spectrum was further decomposed to find the kine- atmosphere. When the GS blocks the view toward the Formation of Double Gaussian Line Profiles of AG Pegasi 37

Figure 1. Hα spectral line profiles observed in 1998 (φ=0.24), 2001 (φ=0.56), and 2002 (φ=0.98), from left to right, respectively. Exposure times are 300 sec, 180 sec, and 300 sec, respectively. Radial velocities are given in units of km s−1, fluxes in units of 10−11erg s−1cm−2 (per km s−1). The kinematic center (location of 0 km s−1) has been adapted from LH18.

Figure 2. Like Figure 1 but for the Hβ spectral line profiles; see also LH18 for the 1998 Hβ line profile. The center of the broad line observed in 1998 is located at about 45 km s−1, red-shifted from those in 2001 and 2002.

WD at φ ∼ 0.5, it also hides the inner Lagrangian point As noted for other Balmer profiles by Lee et al. L1 and the extended atmosphere of the GS close to the (2017), the Raman scattering mechanism does not play WD. As a result, these regions are unlikely to contribute a role in the He ii or other H i lines, probably because the to the Hα and Hβ emission. corresponding Lyman lines are weak for gas densities of It is not straightforward to determine the kinematic order 1010 cm−3. The present study assumes an ionized center because three components are involved. However, bipolar cone extending from the WD (as in LH18) and the center of the broad line, which we assume to originate confirms that such a cone geometry accommodates the from Raman scattering, observed in 1998 seems to be observed double Gaussian line profiles. To do this, we about 45.0 km s−1 red-shifted relative to 2001 and 2002 first need to know whether the GS (or its extended (see Figure 2). If the emission zones are gravitationally atmosphere) blocks the view toward the receding cone bound to the WD, the kinematic center would be red- when the former is in front of the WD. We will provide shifted at φ ∼ 0.25 relative to those at φ ∼ 0.0 or 0.5 arguments in favor of this scenario from our examination (See Figure 4). of the double Gaussian line profiles. Table 2 lists the measurements of the observed Hα and Hβ lines for three observing runs. Note also that 3. DOUBLE GAUSSIAN LINE PROFILES FROM THE the ratio of blue-ward flux to red-ward flux is 33:67 (at BIPOLAR CONE φ = 0.24) and 35:65 (at φ = 0.98), respectively, whereas The Raman scattering is probably due to the small their strengths are reversed when the GS is in front, distance between the GS atmosphere and the WD. Fig- where the ratio of blue-ward and red-ward flux is 59:41 ure 3 schematically shows the size and relative position at φ = 0.56. Such a reversal of relative intensities is likely of both companions, assuming that AG Peg is a close to be caused by the positions of the two components binary system with both companions on circular orbits. relative to the observer. LH18 confirmed that the Hα The photons emitted in the ionized region enter into a and Hβ lines observed in 1998 both formed in the ionized neutral hydrogen region surrounding the GS where the gas (H ii) region, with the H atom number density being Raman scattering occurs (Heo & Lee 2015). ∼ 109.8 cm−3. LH18 showed that the broad line with As illustrated in Figure 3a, at φ ∼ 0.25 the observer a FWHM of 400–500 km s−1 is the result of Raman (located downward) can observe spectra from both the scattering. approaching GS and the receding WD. The spectral 38 Hyung & Lee

Table 2 Hα and Hβ observed in three epochs

Date φ Line Blue Red Raman wing blue:red (%) mean (%) 1998-09-17 10.24 Hα 1.09(−11) 3.20(−11) 8.15(−11) 25:75 Hβ 5.95(−12) 9.07(−12) 3.82(−12) 40:60 33:67 2001-08-30 11.56 Hα 2.29(−11) 1.33(−11) 6.16(−11) 59:41 Hβ 7.18(−12) 5.00(−12) 7.53(−13) 59:41 59:41 2002-08-11 11.98 Hα 5.17(−11) 7.99(−11) 1.08(−10) 39:61 Hβ 1.43(−11) 3.18(−11) 5.57(−12) 31:69 35:65

Fluxes are in units of erg−1s−1cm−2. Flux values are given in the form ‘mantissa(power of ten)’; e.g., 1.03(−11) denotes a flux of 1.03 × 10−11 erg−1s−1cm−2. See LH18 for measurement errors. data for φ = 0.56 observed in 2001 correspond to the 100 km s−1. case where the observer is located leftward (in case of Figure 6 shows theoretical double Gaussian line counter clock-wise revolution). At phases φ ∼ 0.0 or profiles calculated for a bipolar conical shell model with φ ∼ 0.5, the stars are aligned with the line of sight. It is the physical parameters given in Table 3. The physi- not possible to observe the emission line profiles when cal parameters are from Case A of Table 7 by LH18. the GS blocks the view onto the WD and the ionizing The bipolar conical shells comprise a H ii emission zone, gas. However, the observer can see parts of the bipolar which is responsible for the optical emission lines, sur- conical emission zone whenever the GS cannot block the rounded by the outer neutral H i zone. The bipolar view onto the ionizing gas completely. We conclude that conical shell is largely hollow with a thin H ii zone a semi-detached binary geometry describes the AG Per whose inner and outer radii are 3.16 × 1013 cm and system better than a close-binary geometry. Figure 4 3.18 × 1013 cm (∼2.1 au), respectively (see Model I by shows the relative positions of the GS and the WD in LH18). Since the bipolar conical shells expand radially a semi-detached binary system. Gas flows from the GS outwards from the WD, one can observe the double peak to the WD through the Lagrange point L1, filling some line profiles. Most of the material in the top conical of the WD-side Roche lobe (see Figure 3b). The Roche shell recedes from the observer (cf. Figure 5), causing lobe around the GS is about 20% larger than the GS the red-shifted line component. The material in the itself. A strong stellar wind with a terminal velocity of bottom conical shell approaches the observer, causing ∼1000 km s−1 is believed to be produced by the slow the blue-shifted component. The opening angle of the nova eruption of the WD beginning in 1850 (Eriksson cone, OA, determines the line width. The inclination of et al. 2004). Hence, one may assume the presence of a the polar axis, i, determines the separation between the mass-loaded strong stellar wind from the WD, which line components. would push the outer shell gas outward and form a The expansion speed of the bipolar cones pushed cone-shaped bipolar outflow. by the stellar wind of the hot WD was assumed to be Figure 5 shows a schematic diagram of the bipolar 70 km s−1. The opening angle is OA = 74◦, and the conical outflow, originating from the compact accretion inclination of the polar axis relative to the observer on disk around the WD. ‘HII’ indicates the H ii zone ionized Earth is i = 55◦; these parameter values give a FWHM by the hot WD star, while ‘HI’ indicates the neutral of 53 km s−1, consistent with the observed 44–55 km s−1. zone that surrounds the H ii zone. Kenyon et al. (1993) Although our model assumes that both conical shells are and Eriksson et al. (2004) presumed the presence of a identical, their sizes or opening angles could be different. collision region between the WD and the GS where the We smoothed the synthetic profile with a Gaussian kernel −1 −1 fast (velocity of ∼700 km s ) wind from the WD col- of width σG = 15 km s , narrower than the expected lides with the wind from the GS (which has a velocity of intrinsic line broadening which is ∼45 km s−1. The ∼60 km s−1), forming composite line profiles with ∼200- conical shell is related to the Roche lobe of the WD or km s−1 doublets. However, LH18 showed that the broad common envelope (CE), which expands along the polar line is actually a Raman scattering line, arising from the axis of the rotating WD. The radius of the photoionized high column density of the nearby H i zone. Both H i zone is larger than the radius of the GS or the orbital and H ii shells may exist within the conical shell, whose path, thus the emission zone responsible for the observed boundary is defined by the outer radius of the inner ion- emission lines is not obscured by them. ized H ii zone where the UV photons from the WD have The synthetic double Gaussian profiles in Figure 6b largely been absorbed. The photoionization (P-I) model consist of weaker (23%) blue-shifted and stronger (77%) by KH08 indicates that the temperature of the WD is red-shifted components. This asymmetry can be real- about 100 000 K. Such a high WD temperature would ized physically by different fractions of volume occupied produce mass-loaded fast stellar winds. The expanding by matter in the two shells, with the lower shell having outflows from the hot WD interact with parts of the a smaller fraction than the upper shell. Alternatively, CE and push them outwards, forming the expanding the lower shell might have a smaller opening angle than ionized bipolar cones with velocities ranging from 50 to the upper shell. The bipolar cones expanding in oppo- Formation of Double Gaussian Line Profiles of AG Pegasi 39

Figure 3. A close binary sys- tem viewed face-on. (a) The extended atmosphere of the GS (solid-line circle) arguably corre- sponds to the Raman scattering zone. Dashed circles indicate or- bital paths, the + sign marks the center of mass. The observer (marked by the arrow) is located downward, the orbital phase is φ ∼ 0.25. (b) The gas flowing from the GS to the WD through the Lagrange point L1 forms an ionized accretion disk around the hot WD. The dashed line indi- cates the Roche lobe.

Figure 4. Relative positions of the WD (plus accretion disk) and the GS in a semi-detached binary system at different orbital phases, viewed face-on. The observer is located downward. (a) φ ∼ 0.25: Both stars can be observed at the same time. (b) φ ∼ 0.5: The WD and the accretion disk are partially obscured, with the degree of obscuration depending on the angle between the line of sight and the orbital plane of the two stars. (c) φ ∼ 0: The WD and the accretion disk are observable. site directions form two peaks, whose separation varies and does not provide a self-consistent geometric model. depending on the inclination angle. We found that an The weaker flux intensities in the blue-shifted lines and inclination i = 55◦ gives two Gaussian peaks separated the stronger flux intensities in the red-shifted lines ap- by ±37 km s−1, consistent with our observations. At pear to be intrinsic, originating from the geometry itself. this stage, we do not yet consider a possible occultation Hence, the reversal of the flux ratio at φ ∼ 0.5 might be by the GS. The relatively large GS could eclipse some due to an eclipse of the emission zone by the extended portion of the CE and the conical shell at a specific line atmosphere of the GS. of sight. Figure 7 shows the relative positions of the bipolar As seen in Figures 1 and 2, the blue-shifted compo- conical shells and the GS, as well as the accretion disk nents of the H i line profiles observed in 1998 were weaker around the WD, as function of orbital phase (see Fig- than the red-shifted ones. The ratio of blue-shifted to ures 3 and 4 for the relative positions of the center of red-shifted flux intensity, derived from the mean of the mass and both stars). The H i and H ii zones are not Hα and Hβ fluxes, was 33%:67% in 1998 (φ ∼ 0.25), specified in this figure, but we assume that the inner which changed to 59%:41% in 2001 (at φ ∼ 0.5). The radius of the H ii zone is comparable to the size of the red component became weaker in 2001 (near φ ∼ 0.5), extended atmosphere of the GS. In Figure 7a, the GS is suggesting that the extended atmosphere of the GS, or slightly above the accretion disk plane at phase φ = 0.24, the GS with the CE, obscures the receding portion of while Figures 7b and 7c show their relative positions the conical shell and the WD itself. However, the red- at phases 0.56 and 0.98, respectively. At φ = 0.24 and shifted line component became stronger again in 2002 φ = 0.98, the GS does not obscure the conical shell. At (φ ∼ 0.0). Some earlier studies, e.g., Nussbaumer et al. φ = 0.56, the GS eclipses parts of the receding (top) (1995), Kenyon et al. (1993), and Eriksson et al. (2004), conical shell, provided that its atmosphere or the CE assumed various emission zones located between the are sufficiently extended. Figure 7 assumes that the GS and the WD to accommodate such a semi-periodic H ii zone is gravitationally bound to the WD. As shown variation. However, such an interpretation is ad hoc in Figure 2, the kinematic center of the line profiles 40 Hyung & Lee

Table 3 Model parameters for 1998 data

Parameter Value Unit −1 Vexp 70 km s Opening angle, OA 74 ◦ Inclination angle, i 55 ◦ FWHM 53 km s−1 Observed FWHM 44–55 km s−1 −1 σG 15 km s

Table 4 Model parameters for 1998, 2001, and 2002 data

Parameter Value Unit −1 Vexp 70 km s Opening angle, OA 74 ◦ Inclination angle, i 55 ◦ FWHM 53 km s−1 Observed FWHM 44–60 (55) km s−1 −1 Figure 5. Schematics of the bipolar conical shells and accre- σG 20 km s tion disk around the WD. The bipolar cones are formed by the mass-loaded outflow from the WD against the common envelope. OA: opening angle of the cone; i: inclination angle those in Table 3 except of the value of σG. of the pole relative to the observer located to the right. ‘HII’ Figure 9 shows theoretical profiles for (a) the case indicates the ionized emission zone in the conical shell, ‘HI’ that both conical shells are of equal , (b) a indicates the neutral zone. different expansion velocity of the shell, and (c) a wide opening angle. As our value of the inclination angle observed in 1998 is red-shifted by about +45.0 km s−1 correctly reproduces the observed line separation, we compared to the other two . We did not include did not vary it. The line profile in Figure 9a corresponds the orbital motion into our model. to a phase φ ∼ 0.35 − 0.40, while the other two profiles Figure 8 shows the theoretical line profiles for three (Figures 9b,c) are calculated at φ = 0.56. Note that a phases. Figure 8a shows the line profiles for both φ ∼ large opening angle not only widens the line profiles but 0.25 (1998) and φ ∼ 0.0 (2002). As discussed above, also produces a profile that appears to be composed of the blue-shifted line is weaker than the red-shifted one, three components. with a flux ratio of about 35%:65%, indicating that The hot accretion disk may cause an X-ray line the contribution by the two conical shells is different. with a blue and a red component, which is not dis- Note the position of the GS and the H ii conical shell at cussed in our study. Earlier studies, e.g., Nussbaumer φ = 0.98 and φ = 0.24 in Figure 7, where the GS does et al. (1995), Kenyon (1986), Kenyon et al. (1993), and not eclipse the H ii region. Figure 8b shows a relatively Eriksson et al. (2004), identified additional emission: strong (60%) blue component and a relatively weak (1) X-ray emission harder than a few 100 eV, possibly (40%) red component (for 2001). Hence, we conclude emitted by a hot plasma with a temperature of a few that the bipolar cone is asymmetric, with a relatively million K shock-heated by colliding winds; and (2) a luminous upper cone and a weaker lower cone. The blue-shifted absorption zone from the wind regions and radius of the GS appears to be smaller than the inner the surrounding nebula, affecting the UV lines and pro- radius of the H ii zone shell (Ri ∼ 2 au). However, the ducing profiles. The former could be related extended atmosphere (with radius Re) or the Roche lobe to the accretion disk or the shock interaction between of the GS located between the H ii zone and the observer two stars, while the latter could in part be due to the could hide a significant fraction of the H ii zone in the hot WD and its wind and the cool GS with its extended upper conical shell due to the relatively large inclination atmosphere (see also Figures 4 and 7). ◦ ◦ angle, i = 55 , because Ri < Re × (1 + tan(90 − i)). As shown in Figures 1 and 2, the two Gaussian line 4. CONCLUSIONS components are located at −37 km s−1 and +37 km s−1, Our analysis showed that an expanding bipolar conical and accordingly, the predicted peaks have the same sepa- shell can explain the observed Hα and Hβ double Gaus- ration. We applied a more realistic Gaussian smoothing sian spectral lines, assuming a semi-detached binary −1 kernel with a width of σG = 20 km s to the synthetic system as the most probable structure of AG Peg. The line profiles in Figure 8. Table 4 summarizes the param- observed H i spectral line profiles are consistent with eters for the bipolar conical shell model in Figure 7 from expanding bipolar conical shells having a polar axis with which we derive the theoretical line profiles in Figure 8. an inclination angle of i = 55◦ and an opening angle Note that the parameters in Table 4 are the same as OA = 74◦. The expansion velocity of the shell is prob- Formation of Double Gaussian Line Profiles of AG Pegasi 41

Figure 6. Model line profiles for the spectra obtained in 1998 (at φ ∼ 0.25). Fluxes are given in arbitrary units. (a) The red component only (same as Figure 8 in LH18). (b) A synthetic double Gaussian line profile composed of a weaker (23%) blue and a stronger (77%) red component. AG Peg is assumed to be a semi-detached binary system. Note that the center of the broad line in 1998 is red-shifted by about 45 km s−1 relative to the 2001 and 2002 centers, which are located at about 0 km s−1 (see Figure 2). Hence, the center of the line at φ ∼ 0.25 corresponds to λ = 6563.8 A˚ for the Hα line and 4862.1 A˚ for the Hβ line.

Figure 7. Relative location of bipolar conical shells, the accretion disk around the WD, and the GS (large circle) as function of orbital phase. The observer is located toward the right. (a) At φ = 0.24. (b) At φ = 0.56. (c) At φ = 0.98. At φ = 0.24 and φ = 0.98, the conical shells and the accretion disk are visible to the observer. At φ = 0.56, the extended atmosphere of the GS obscures parts of the receding conical shell. The proposed bipolar cones may be a part of the WD with the common envelope. See also Figure 4 for the relative positions of two stars and the center of mass. ably about 70 km s−1. The emission zone is probably large-scale structure of the expanding gas can be in- the result of the interaction of a fast stellar wind from ferred from radio images which indeed show a bipolar the WD and the gas flowing toward the WD from the shape. Within each cone, the H ii zone forms a thin shell GS. The accretion disk around the WD confines the fast which is surrounded by a relatively thick high-density stellar wind is confined to a bipolar conical outflow zone. H i shell. KH08 showed that the gas number density 9 10 −3 The geometrically thick accretion disk is the highly ex- of the H ii zone is nH = 10 − 10 cm , while LH18 cited zone responsible for the X-ray emission observed in showed that the outer H i zone has a column density of 19 −2 other studies. As shown by LH18, the Lyman lines can NH = 3 − 5 × 10 cm . Beyond the outer boundary likewise form double Gaussian profiles, which will trans- of the H i zone, there should be a relatively low-density form into the broad Balmer Hα and Hβ lines through bipolar H i zone on scales of 2000 to 10, in agreement with Raman scattering in the outer H i shell. 1.5 GHz and 5 GHz radio studies (Kenny et al. 1991).

Our proposed bipolar conical shells explain the ob- ACKNOWLEDGMENTS served double Gaussian profiles (FWHM ∼ 60 km s−1), This research was supported by a grant from the which are also responsible for the broad lines (∼400– National Research Foundation of Korea (NRF2015 500 km s−1) produced by Raman scattering. The upper R1D1A3A01019370; NRF2017 R1D1A3B03029309). We conical shell appears to differ in size from the lower one. are grateful to the late Professor Lawrence H. Aller However, when the GS is located between the observer (UCLA) who joined the HES observation at the Lick and the WD, the upper conical shell is obscured. The Observatory for this study. We express our gratitude 42 Hyung & Lee

Figure 8. Theoretical line pro- files. Flux scales are arbi- trary. (a) For φ = 0.24 or φ = 0.98. (b) For φ = 0.56. A velocity of 0 km s−1 corre- sponds to 6562.82 A˚ for the Hα line and 4861.33 A˚ for the Hβ line. We did not con- sider orbital motion. These spectra should be compared to Figures 2 and 6.

Figure 9. Theoretical line profiles for different values of model parameters. Fluxes are in arbitrary units. (a) For a phase φ = 0.35 − 0.40 when the GS partially obscures the H ii zone. (b) For a phase φ = 0.56 and a higher expansion velocity. (c) For φ = 0.56 and a cone opening angle OA = 120◦. to Dr. K.-H. Lee who helped to create Figures 1 and 2. Kim, H., & Hyung, S. 2008, Chemical Abundances of the We also thank the anonymous referees for reviewing this AG Pegasai, JKAS, 41, 23 (KH08) paper, and Eugenia H. for proof-reading. Lee, H.-W. 2002, Raman-Scattering Wings of Hα in Symbi- otic Stars, ApJ, 541, L25 REFERENCES Lee, H.-W., & Hyung, S. 2000, Broad Hα Wing Formation Allen, D. A. 1980, On the Late-type Components of Slow in the Planetary Nebula IC 4997, ApJ, 530, L49 Novae and Symbiotic Stars, MNRAS, 192, 521 Lee, K. H., Lee, S.-J., & Hyung, S. 2017, An Analysis of the Contini, M. 1997, The Evolving Structure of AG Pegasi, H Emission Line Profiles of the Symbiotic Star AG Peg, Emerging from the Interpretation of the Emission Spectra JKESS, 2017, 38, 1 at Different Phases, ApJ, 483, 887 Lee, S.-J., & Hyung, S. 2018, The Hα and Hβ Raman- Eriksson, M., Johansson, S., & Wahlgren, G. M. 2004, Mod- scattering Line Profiles of the Symbiotic Star AG Peg, eling the Wind Structure of AG Peg by Fitting of C iv MNRAS, 475, 5558 (LH18) and N v Resonance Doublets, A&A, 422, 987 Fleming, W. P. 1907, A Photographic Study of Variable Merrill, P. W. 1916, A Spectrum of the P Cygni Type, Pub. Stars, Ann. Astron. Obs. Harvard College, 47, 1 Astron. Obs. University of Michigan, 2, 71 Heo, J.-U., & Lee, H.-W. 2015, Accretion Flow and Disparate Merrill, P. W. 1942, Special Problems of Be Spectra, ApJ, Profiles of Raman Scattered O vi λλ 1032, 1038 in the 95, 268 Symbiotic Star V1016 Cygni, JKAS, 48, 105 Nussbaumer, H. 1992, The Outburst of Symbiotic Novae, Hutchings, J. B., Cowley, A. P., & Redman, R. O. 1975, Proc. IAU, 151, 429 Mass Transfer in the Symbiotic Binary AG Pegasi, ApJ, Nussbaumer, H., Schmutz, W., & Vogel, M. 1989, Raman 201, 404 Scattering as a Diagnostic Possibility in Astrophysics, Kenny, H. T., Taylor, A. R., & Seaquist, E. R. 1991, AG A&A, 211, L27 Pegasi: A Multi-shell Radio Source, ApJ, 366, 549 Nussbaumer, H., Schmutz, W., & Vogel, M. 1995, Mass Kenyon, S. J. 1986, The Symbiotic Stars, PhD Dissertation, Accretion onto Compact Objects in 2D, A&A, 301, 922 Harvard University (Boston: Cambridge University Press), 295 Schmid, H. M. 1989, Identification of the Emission Bands at Kenyon, S. J., Mikolajewska, J., Mikolajewski, M., et al. 1993, λλ 6830, 7088, A&A, 211, L31 Evolution of the Symbiotic Binary System AG Pegasi: The Viotti, R. 1988, The Symbiotic Novae, The Symbiotic Phe- Slowest Classical Nova Ever Recorded, AJ, 106, 573 nomenon Proceedings of the, IAU Coll., 103, 269 Journal of the Korean Astronomical Society https://doi.org/10.5303/JKAS.2020.53.2.43 53: 43 ∼ 48, 2020 April pISSN: 1225-4614 · eISSN: 2288-890X Published under Creative Commons license CC BY-SA 4.0 http://jkas.kas.org

POORLY STUDIED ECLIPSING BINARIESINTHE FIELDOF DODRACONIS: V454 DRAAND V455 DRA Yonggi Kim1, Ivan L. Andronov2, Kateryna D. Andrych2, Joh-Na Yoon1, Kiyoung Han1, and Lidia L. Chinarova2,3 1Chungbuk National University Observatory, Chungbuk National University, 361-763, Cheongju, Korea [email protected], [email protected] 2Department “Mathematics, Physics and Astronomy”, Odessa National Maritime University, 65029, Odessa, Ukraine tt [email protected], [email protected] 3Astronomical Observatory, Odessa I. I. Mechnikov National University, 65014, Odessa, Ukraine; [email protected] Received February 17, 2020; accepted February 27, 2020

Abstract: We report an analysis of two poorly studied eclipsing binary stars, GSC 04396-00605 and GSC 04395-00485 (recently named V455 Dra and V454 Dra, respectively). Photometric data of the two stars were obtained using the 1-m Korean telescope of the LOAO operated by KASI while monitoring the cataclysmic variable DO Dra in the frame of the Inter-Longitude Astronomy (ILA) project. We derived periods of 0.434914 and 0.376833 days as well as initial epochs JD 2456480.04281 and JD 2456479.0523, respectively, more accurate than previously published values by factors 9 and 6. The phenomenological characteristics of the mean light curves were determined using the New Algol Variable (NAV) algorithm. The individual times of maxima/minima (ToM) were determined using the newly developed software MAVKA, which outputs accurate parameters using “asymptotic parabola” approximations. The light curves were approximated using phenomenological and physical models. In the NAV algorithm, the phenomenological parameters are well determined. We derived physical parameters using the Wilson- Devinney model. In this model, the best-fit parameters are highly correlated, thus some of them were fixed to reasonable values. For both systems, we find evidence for the presence of a cool spot and estimate its parameters. Both systems can be classified as overcontact binaries of EW type. Key words: binaries: eclipsing — methods: data analysis — stars: individual: V454 Dra, V455 Dra

1. INTRODUCTION has motivated extensive observations of this star and All stars are variable at certain active stages during their the surrounding field. Because of the fast variability of evolution. Of special interest are objects exhibiting spe- DO Dra, observations were limited to one filter (Rc). cific types of variability like cataclysmic variables and Besides this very interesting object, there are other other interacting binaries, pulsating, and eruptive vari- variables in the field. Two of them, V454 Dra and V455 ables. Photometric monitoring of such objects requires Dra, were discovered in the frame of the ILA project by dozens or even hundred of nights and may not be done Virnina (2010) while searching for variability of stars in at large telescopes; instead, this is a proper task for various fields. Virnina (2011) presented finding charts meter and sub-meter class telescopes. There are several and preliminary parameters. Kim et al. (2019) found single- and multi-telescope projects monitoring selected the same variables from their time-series data obtained stars, including the Inter-Longitude Astronomy (ILA) by the Chungbuk National University observatory. The campaign (Andronov et al. 2003; see Andronov et al. systems were classified as W UMa-type (EW) objects. 2017a for a recent review of key results). The light elements were found to be One of the most interesting stars in our sample min I = 2455192.4773(5) + 0.434911(9) · E, (1) is DO Draconis which appears to represent an object category of its own, being simultaneously classified as min I = 2455191.7896(6) + 0.376832(6) · E (2) a magnetic dwarf nova and an outbursting intermedi- ate polar. It has shown a variety of new phenomena for V454 Dra and V455 Dra, respectively. (The number like quasi-periodic oscillations (QPOs), dependence of in brackets gives the accuracy in units of the last decimal the slope of the outburst decay on the brightness at digit). maximum, and a weak wave of the basic low bright- Recently, the two stars were officially named V454 ness between the outbursts, many of which seem to be Dra and V455 Dra in the General Catalog of Variable missing because of a short duration (Andronov et al. Stars (GCVS) (Kazarovets et al. 2015; Samus et al. 2017). 2008). A statistical study of the QPOs was presented Corresponding designations in other catalogs are 2MASS by Han et al. (2017). The peculiar nature of DO Dra J11403001+7111021 = GSC 4395.00485 = USNO-B1.0 1611-00091333 = USNO-B1.0 1611-00091333 Gaia DR2 Corresponding author: Y. Kim 1062691908235241216 = V0454 Dra = V454 Dra, and 43 44 Kim et al.

Figure 1. The phase light curve of V454 Dra together with Figure 3. The phase light curve of V454 Dra with the best-fit the best-fit NAV model (continuous line following the data). Wilson-Devinney model. The additional line above the eclipse corresponds to the “out of eclipse” continuum approximated by a trigonometric polynomial of order 2.

Figure 4. The phase light curve of V455 Dra with the best-fit Wilson-Devinney model. Figure 2. The phase light curve of V455 Dra together with the best-fit NAV model (continuous line following the data). The additional line above the eclipse corresponds to the focal ratio of f/2.92. In 2012, a new 4096 × 4096 pixel 0 0 “out of eclipse” continuum approximated by a trigonometric CCD providing a wide FOV of 72 × 72 was installed polynomial of order 2. in the camera; detailed information about the reduction process concerning data before and after the change of the CCD are given in Kim et al. (2019). 2MASS J11483649+7107507 = NOMAD-1 1611-0093251 All photometric observations were made in the = USNO-B1.0 1611-0091801 = V0455 Dra = V455 Dra. R filter band, using the star “Ref 2” from Virnina (It should be noted that, obviously, V454 Dra = V0454 (2011) as calibrator, assuming the magnitudes Rc = Dra, but an extra zero is a common designation in the 13.04m and V = 13.390m with a corresponding color of GCVS, which is used for sorting numerous stars in one V − R = 0.350m.1 The angular distances of V454 Dra .) The parallaxes for the two stars are and V455 Dra from DO Dra are 2028.4000 and 2463.2500, 0.7722±0.0201 mas and 0.8807±0.0186 mas (Gaia DR2 respectively (Samus et al. 2017). Simultaneous obser- 2018a,b), respectively. vations of both stars are possible using wide-field CCD 2. OBSERVATIONS images, applying a focal reducer. In total, we analyzed n = 1746 observations ob- Photometric data for DO Dra were obtained by the tained during 112 hours distributed over 35 nights from Mt. Lemmon Optical Astronomy Observatory (LOAO) JD 2456272 to JD 2456710, thus spanning 438d. This operated by the Korea Astronomy and Space Science time range was set by the availability of the focal reducer Institute (KASI) and the Chungbuk National University and is shorter shorter than the range spanned by the Observatory (CBNUO) for about 10 years from 2005 observations of the main target (i.e., DO Dra). till 2014. The robotic LOAO telescope, located at Mt. Lemmon in Arizona, has an aperture of 1.0 m and an 3. PHENOMENOLOGICAL MODELING effective focal ratio of f/7.5; it is mounted on a fork Phenomenological Modeling of the light curves of eclips- equatorial mount. The telescope is equipped with a ing systems using “special shapes” (also called “pat- 2048 × 2048 pixel CCD camera with a pixel scale of terns”), instead of trigonometric polynomials, was ini- 0.64 arcsec/pixel and a field of view (FOV) of 220.2 × 0 22 .2. The telescope at CBNUO is a Ritchey-Chr´etien 1ftp://ftp.aavso.org/public/calib/dodra.dat; data courtesy telescope with an aperture of 0.6 m and an effective AAVSO / Henden et al. (2007). Poorly Studied Eclipsing Binaries V454 Dra and V455 Dra 45

The phenomenological parameters obtained using the NAV algorithm are listed in Table 1. Their detailed description may be found in Andronov et al. (2015) and Tkachenko et al. (2016). The corresponding phase light curves and approximations are shown in Figures 1 and 2. A detailed description of the time series analysis is presented in Andronov (2020).

4. TIMESOF EXTREMA For further studies of the period changes, it is important to publish the times of individual extrema (for example, Figure 5. Model of V454 Dra with its cool spot (red area), Kreiner et al. 2003). Typically, only the moments of as seen at an orbital phase φ = 0.74. Red crosses mark minima are published for eclipsing binaries. However, for the centers of the stars and the position of the center of mass of the system. The red circle indicates the orbit of the both stars, the maxima are prominent, so we determined components around the common center of mass. To build the the maxima as well. model, 120 points per meridian and 240 points per parallel In the previous section, we modeled the complete were used for each star. phase curve based on all observations. In individual nights, the observation time is shorter than the period. Thus we take into account only the intervals close to an extremum. We determined the times of either minima or maxima using the “running parabola” approximations, initially proposed by Marsakova & Andronov (1996) and realized in the software package MAVKA introduced by Andrych et al. (2015) and Andrych et al. (2020). This method is among the most accurate ones for determi- nation of extrema from relatively short observing runs which only partially cover the descending and ascending parts of a light curve around an extremum. The “special Figure 6. Same as Figure 5 for V455 Dra, as seen at φ = 0.24. shapes” allow to avoid apparent waves, which are similar to the Gibbs phenomenon. For longer observing times, which completely cover tially proposed by Andronov (2012) for Algol-type (EA) ascending and descending segments of light curves, one systems with sharply defined beginnings and ends of may use many modifications of known approximations. eclipses. This approach resulted in the development Andronov et al. (2017b) compared almost 50 functions of the New Algol Variable (NAV) algorithm. The algo- and ranked them according to estimated accuracy. Some rithm can also be used for EB (β Lyr) and EW (W UMa) of them are improvements of phenomenological approx- type systems, including the prototypes of these classes imations proposed by Mikul´aˇsek (2015). Andrych & – β Per (Algol), (β Lyr), and EW UMa (Tkachenko et Andronov (2019) realized 21 approximations of 11 types al. 2016). For the eclipsing binary 2MASS J18024395 for the shorter intervals. + 4003309 = VSX J180243.9+400331 (also known as In Table 2, the moments of the individual minima V1517 Her) in the field of the intermediate polar V1323 and maxima are listed. Moments which correspond to Her, Andronov et al. (2015) estimated the physical pa- phases near 0 and 0.5 are primary (min I) and secondary rameters of the components using two-color photometry (min II) minima, respectively; the other extrema are and a statistical mass–radius– relation. maxima. These moments can be used for further analysis For the stars V454 Dra and V455 Dra, such a com- after adding further observations from other seasons; he plete analysis is not possible because our photometry typical timescale of period variations is on the order of is monochromatic. We computed periodograms using years for variations due to third bodies and much longer trigonometric polynomial (TP) approximations of vari- for variations due to the mass transfer (see, e.g., the ous orders, s, which correspond to the main wave and monograph by Kreiner et al. 2003). (s − 1) harmonics. The periods were corrected using differential equations (see Andronov 1994 for details) ac- 5. PHYSICAL MODELING cording to the algorithm described by Andronov (1994) The main purpose of studies of eclipsing binaries is to and Andronov and Marsakova (2006). The statistically build physical models. In contrast to spectroscopic, optimal degrees of the trigonometric polynomials are visual or astrometric binaries, it is possible to deter- s = 6 and s = 8 for V454 Dra and V455 Dra, respec- mine masses, , sizes, temperatures, surface tively. The corresponding number of parameters are brightness distributions and some parameters of the m = 2s + 2, i.e. m = 14 and 18, respectively; these num- component orbits for eclipsing variables. For modeling bers of parameters are higher than the ones required for and determination for these parameters we have to com- the NAV algorithm despite the curves being relatively bine photometric observations, curves, smooth. and spectroscopic observations. 46 Kim et al.

Table 1 Parameter values determined by phenomenological modeling

Parameter V454 Dra V455 Dra m m C1 13.5955 ±0.0011 14.2227±0.0022 m m C2 0.0061 ±0.0011 0.0299 ±0.0021 m m C3 -0.0047 ±0.0006 0.0032 ±0.0012 m m C4 0.1064 ±0.0017 0.1259 ±0.0032 m m C5 0.0022 ±0.0015 0.0054 ±0.0021 m m C6 0.1276 ±0.0042 0.3101 ±0.0088 m m C7 0.1326 ±0.0040 0.2464 ±0.0075 C8 0.1164 ± 0.0025 0.1159 ± 0.0025 C9 1.369 ± 0.079 1.357 ± 0.068 C10 1.539 ± 0.081 1.330 ± 0.069 d d T0 2456480.04281 ± 0.00030 2456479.05227 ± 0.00023 P 0.43491412 ± 0.00000091d 0.37683317 ± 0.00000097d d1 0.1109 ± 0.0034 0.2484 ± 0.0061 d2 0.1149 ± 0.0033 0.2031± 0.0055 d1 + d2 0.2258 ± 0.0058 0.4515± 0.0099 d1/d2 0.9646 ± 0.0291 1.2235± 0.0326 Max I 13.4845 ±0.0012m 14.0999±0.0020m Min I 13.8355 ±0.0032m 14.6886±0.0065m Min II 13.8284 ±0.0027m 14.5651±0.0055m

We use the Wilson-Devinney (WD) algorithm pro- the same potential on the surface, although they may posed by Wilson & Devinney (1971) and improved by have different temperatures (the temperature of one of Wilson (1979) and Wilson (1994). It determines physical them must be fixed while the other is simulated). Such parameters of a binary system from the phase curve and a system is already tight, the components are tidally the radial velocity curves. For this task, the user has deformed and have circular orbits. The co-latitude of to pre-set the intervals of the physical parameter values the center of a spot can vary from 0◦ (north pole of a and has to fix some of them for reasons related to the star) to 180◦ (south pole). The spot center longitude typical relationships for stars of corresponding type of varies from 0◦ to 360◦, counted counter-clockwise from variability. For example, the temperature of the first the line connecting the two stars. component at the pole, T1, was fixed to a reasonable Using the version of the original WD code created at value, as typically assumed in those models, whereas the Jagiellonian Astronomical Observatory in Krakow by the temperature of the second component, T2, is a free Prof. S. Zo laand others, we did a physical simulation parameter. The best fit estimate of T2 depends on T1 of the eclipsing binary systems V455 Dra and V454 monotonically but not exactly linearly. The surface po- Dra. For these objects, we have only the phase curve in tential Ω is expressed in units of the square of the orbital one filter, so some parameters and their intervals need velocity. As both stars overfill their Roche lobes and to be fixed to realistic values. Physical parameters of are at an overcontact stage, the potential is the same components in close binary systems were discussed by, for the common envelope, i.e., for both stars. (More e.g., Kreiner et al. (2003). The model light curves for details may be found in the monograph by Kallrath & both systems are shown in Figures 3 and 4 along the Milone 2009.) The algorithm is based on a Roche model data. The resulting physical parameters are shown in geometry for the construction of a binary system and Table 3. It should be noted that our solution is not uses Monte Carlo simulations to specify physical param- unique. Several parameters show a pairwise degeneracy eters and construct a model light curve. The WD-code with other parameters, notably the temperatures of both allows to simulate different types of eclipsing systems components, the inclination with the potential, and the from contact systems to exoplanet transits. radius of the spot with the temperature parameter. To determine the physical parameters of our target With Binary Maker 3 (Bradstreet 2005), we built binary systems, we used a modified version of the original a visual model of the system using the best-fit physical WD-code created at the Astronomical Observatory of parameters. In Figures 5 and 6, the respective system the Jagiellonian University in Krakow (see Zo la et al. models are shown. To build a model for a given binary, 1997, 2010 for a full description). This version can 120 points per meridian and 240 points per parallel are take into account up to two spots on the surface of taken for each star. the companion stars in a given binary. The differential code method in the original code has been replaced with 6. CONCLUSIONS the Price algorithm (controlled Monte Carlo method). Photometric time series for two poorly known binary The version we use is specialized on contact systems stars, V454 Dra and V455 Dra, were analyzed using where both objects fill their Roche lobes and both have phenomenological approximations of the complete phase Poorly Studied Eclipsing Binaries V454 Dra and V455 Dra 47

Table 2 Table 3 Moments of individual maxima and minima Parameter values derived from physical modeling

V454 Dra V455 Dra Parameter V455 Dra V454 Dra BJD−2400000 ± BJD−2400000 ± Orbit inclination 74.67 ± 0.06◦ 64.0 ± 0.03◦ Temperature of first star 5800 K 5500 K 56272.36895 0.0006 56272.35864 0.0009 Temperature of second star 5351 ± 5 K 5478 ± 3 K 56346.30781 0.0006 56346.31053 0.0019 Surface potential 3.62 ± 0.015 3.685 ± 0.0007 56355.22405 0.0008 56355.26561 0.0015 Mass ratio 0.95 ± 0.01 0.99 ± 0.005 56357.07647 0.0016 56357.06032 0.0012 Co-latitude of spot center 90◦ 133 ± 1.3◦ 56357.18191 0.0008 56357.14830 0.0005 Longitude of spot center 356.9 ± 0.2◦ 71 ± 1.3◦ 56357.28783 0.0013 56357.23665 0.0010 Spot radius 22.5 ± 0.2◦ 18.9 ± 0.74◦ 56360.22388 0.0016 56357.33556 0.0040 Spot temperaturea 0.8 0.6 56362.18030 0.0003 56360.16278 0.0009 56362.29182 0.0012 56362.23293 0.0005 aIn units of stellar temperature. 56363.26942 0.0009 56363.17623 0.0008 56366.20593 0.0007 56363.26870 0.0020 the accuracy of periods and initial epochs increases by 56373.05427 0.0013 56366.18734 0.0019 factors of 9 and 6 for the two stars, respectively. 56394.14774 0.0006 56373.06110 0.0017 We also applied physical modeling to our data using 56395.12243 0.0015 56394.07646 0.0007 56398.16881 0.0015 56394.16601 0.0005 the Wilson-Devinney algorithm. The set of physical 56401.10329 0.0012 56395.11733 0.0007 parameters which were determined should be regarded as 56409.04061 0.0020 56401.04896 0.0010 an initial approximation, which may be improved using 56411.97798 0.0020 56409.06225 0.0010 possible further spectral observations (to determine mass 56412.08446 0.0013 56411.97503 0.0019 ratios and temperatures) or well-calibrated multi-color 56413.06679 0.0006 56412.07457 0.0006 photometry (to determine temperatures). For both 56429.04618 0.0020 56413.00769 0.0013 systems, a cool spot is evident whose parameters we 56432.98510 0.0026 56413.10723 0.0011 estimated approximately. As we only have single-band 56434.04771 0.0049 56429.03185 0.0026 photometry available, several parameters and parameter 56650.31266 0.0015 56434.02193 0.0009 intervals needed to be fixed; thus, the formal errors for 56654.33204 0.0008 56650.32477 0.0043 these parameters drastically underestimate the actual 56658.35803 0.0013 56651.36037 0.0014 56659.33758 0.0007 56654.37722 0.0030 uncertainties. 56667.27487 0.0006 56658.23820 0.0015 Both stars were initially classified as being of EW 56686.30513 0.0017 56658.32863 0.0010 type by Virnina (2011). Current physical modeling 56710.22508 0.0019 56659.27614 0.0012 confirms this classification, as both systems are over- 56710.32959 0.0011 56659.36724 0.0011 contact binary systems. For subsequent improvements 56664.35500 0.0035 of physical models, multi-color observations, as well as 56667.28187 0.0007 an accurate calibration of the comparison stars which is 56686.31136 0.0005 currently absent for this field, will be needed. 56710.23717 0.0024 56710.33230 0.0032 ACKNOWLEDGMENTS The authors thank Prof. S. Zo la and Dr. B. D¸ebsky for allowing to use their software for physical modeling light curve (using the trigonometric polynomial and NAV and fruitful discussions. This work was supported by algorithms), as well as individual time series around the Overseas Dispatch Program of Chungbuk National maxima and minima (using the “asymptotic parabola” University in 2017. The data acquisition and analysis algorithm implemented in the recently developed soft- was partially supported by the Basic Science Research ware MAVKA), and a database containing almost 70 Program through the National Research Foundation moments was compiled. This database can be used in of Korea (NRF) funded by the Ministry of Education, combination with further monitoring to search for pos- Science and Technology (2011-0014954). This study is a sible period variations. We provide a table of individual part of the Inter-Longitude Astronomy (Andronov et al. extrema which may be used for future O-C analysis and 2003, 2017a) and Astroinformatics (Vavilova et al. 2017) searches for possible period variations. campaigns. The NAV algorithm achieves a better approxima- REFERENCES tion of the light curve minima using the special shape approach instead of trigonometric polynomials. The Andronov, I. L. 1994, (Multi-)Frequency Variations of Stars. Some Methods and Results, Odessa Astron. Publ., 7, 49 eclipses split the phase curve into four parts, meaning Andronov, I. L. 2012, Phenomenological Modeling of the that smooth trigonometric polynomial are a poor ap- Light Curves of Algol-type Eclipsing Binary Stars, Astro- proximation. This special shape approach, along with phys., 55, 536 a larger data set, has lead to a significant improvement Andronov, I. L. 2020, Advanced Time Series Analysis of of the accuracy of the model parameters compared to Generally Irregularly Spaced Signals: Beyond the Over- the discovery paper by Virnina (2011). For example, simplified Methods, in: P. Skoda et al. (eds.), Knowledge 48 Kim et al.

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DECAY OF TURBULENCEIN FLUIDSWITH POLYTROPIC EQUATIONS OF STATE Jeonghoon Lim1 and Jungyeon Cho1,2 1Department of Astronomy and Space Science, Chungnam National University, 99, Daehak-ro, Yuseong-gu, Daejeon 34134, Republic of Korea; [email protected], [email protected] 2Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea Received October 10, 2019; accepted March 18, 2020

Abstract: We present numerical simulations of decaying hydrodynamic turbulence initially driven by solenoidal (divergence-free) and compressive (curl-free) drivings. Most previous numerical studies for decaying turbulence assume an isothermal equation of state (EOS). Here we use a polytropic EOS, P ∝ ργ , with polytropic exponent γ ranging from 0.7 to 5/3. We mainly aim at determining the effects of γ and driving schemes on the decay law of turbulence energy, E ∝ t−α. We additionally study probability density function (PDF) of gas density and skewness of the distribution in polytropic turbulence driven by compressive driving. Our findings are as follows. First of all, we find that even if γ does not strongly change the decay law, the driving schemes weakly change the relation; in our all simulations, turbulence decays with α ≈ 1, but compressive driving yields smaller α than solenoidal driving at the same sonic Mach number. Second, we calculate compressive and solenoidal velocity components separately and compare their decay rates in turbulence initially driven by compressive driving. We find that the former decays much faster so that it ends up having a smaller fraction than the latter. Third, the density PDF of compressively driven turbulence with γ > 1 deviates from log-normal distribution: it has a power-law tail at low density as in the case of solenoidally driven turbulence. However, as it decays, the density PDF becomes approximately log-normal. We discuss why decay rates of compressive and solenoidal velocity components are different in compressively driven turbulence and astrophysical implication of our findings. Key words: ISM: general — hydrodynamics — turbulence

1. INTRODUCTION decades, the previous numerical results rely heavily on the isothermal condition. However, as long as various Supersonic turbulence in the interstellar medium (ISM) density and temperature phases in the ISM (Ferri`ere is a well-known phenomenon and plays an essential role 2001) are concerned, the use of a polytropic equation of in star formation processes (Larson 1981; Padoan & state (EOS) Nordlund 2002; Mac Low & Klessen 2004). Given that P = Kργ , (1) driving mechanisms of astrophysical turbulence are usu- ally intermittent in both space and time, it is natural where P is the pressure, ρ is the density, and both K for turbulence to decay. Earlier studies showed that and γ are constants, is a valid approach (see Vazquez- non-driven turbulence decays quickly in approximately Semadeni et al. 1996 and references therein). The poly- one large-eddy turnover time (for hydrodynamic turbu- tropic EOS has been used for many astrophysical prob- lence, see e.g., Lesieur 2008; for magnetohydrodynamic lems, such as complex chemical processes (Spaans & Silk turbulence, see Mac Low et al. 1998; Stone et al. 1998), 2000; Glover & Mac Low 2007a) or turbulence (Scalo which is consistent with the fact that an energy cascade et al. 1998; Li et al. 2003; Glover & Mac Low 2007b; occurs within one large-scale eddy turnover time even in Federrath & Banerjee 2015). the case of strongly magnetized turbulence (Goldreich Besides a variety of density and temperature phases, & Sridhar 1995). a wide range of driving agents of turbulence also char- It has been analytically suggested that turbulence acterizes interstellar turbulence (see Federrath et al. energy decays with a power-law of the form E ∝ t−α 2017 for a review). Based on its compressibility, we (see e.g., Chapter 7 of Lesieur 2008). Results from may consider two extreme types of driving: solenoidal previous numerical studies of turbulence have converged (divergence-free) and compressive (curl-free). Until re- to a value of α of approximately unity, and it does not cently, solenoidal driving had been mainly used for tur- strongly depend on the degree of magnetization and bulence studies. However, Federrath et al. (2010) showed compressibility (Mac Low et al. 1998; Stone et al. 1998; that compressive driving and solenoidal driving can have Biskamp & M¨uller1999; Ostriker et al. 2001; Cho et al. different statistics. For example, they showed that “the 2002). former yields stronger compression at the same RMS Even if the consensus that turbulence quickly de- Mach number than the latter, resulting in a three times cays has been numerically established for the last two larger standard deviation of volumetric and column density probability distributions.” To the best of our Corresponding author: J. Cho knowledge, scaling relations of decaying polytropic tur- 49 50 Lim & Cho

Table 1 Simulation runs and parameter values

a b c d e f g Run Driving γ Resolution kf Ms α1 α2 (t1, t2) SMS1-γ0.7 Solenoidal 0.7 5123 8.0 ∼1 1.185 0.279 (5,10) SMS1-γ1.0 Solenoidal 1.0 5123 8.0 ∼1 1.202 0.291 (5,10) SMS1-γ5/3 Solenoidal 5/3 5123 8.0 ∼1 1.313 0.289 (5,10) SMS3-γ0.7 Solenoidal 0.7 5123 8.0 ∼3 1.118 0.425 (5,10) SMS3-γ1.0 Solenoidal 1.0 5123 8.0 ∼3 1.140 0.469 (5,10) SMS3-γ5/3 Solenoidal 5/3 5123 8.0 ∼3 1.166 0.567 (5,10) SMS5-γ0.7 Solenoidal 0.7 5123 8.0 ∼5 0.971 0.644 (5,10) SMS5-γ1.0 Solenoidal 1.0 5123 8.0 ∼5 0.949 0.425 (5,10) SMS5-γ1.5 Solenoidal 1.5 5123 8.0 ∼5 1.073 0.650 (5,10) CMS1-γ0.7 Compressive 0.7 5123 8.0 ∼1 0.954 0.520 (5,10) CMS1-γ1.0 Compressive 1.0 5123 8.0 ∼1 0.825 0.599 (5,10) CMS1-γ5/3 Compressive 5/3 5123 8.0 ∼1 1.058 0.500 (5,10) CMS3-γ0.7 Compressive 0.7 5123 8.0 ∼3 0.777 0.522 (5,10) CMS3-γ1.0 Compressive 1.0 5123 8.0 ∼3 0.818 0.481 (5,10) CMS3-γ1.5 Compressive 1.5 5123 6.0 ∼3 0.839 0.461 (5,10) CMS5-γ0.7 Compressive 0.7 5123 8.0 ∼5 0.688 0.606 (5,10) CMS5-γ1.0 Compressive 1.0 5123 8.0 ∼5 0.767 0.513 (5,10)

We did not include a stiff EOS simulation (i.e., γ > 1) for compressively driven turbulence with Ms & 3 and for solenoidally driven turbulence with Ms & 5. This is because the simulations crashed before turbulence saturates due to the high Ms and large γ, which can lead to extremely low density values (see the PDFs in Figures 8 and 9 for γ > 1). Therefore, we could not run CMS3-γ5/3, CMS5-γ5/3, or SMS5-γ5/3. When γ = 5/3 simulations are unstable, we use γ = 1.5 instead for SMS5 and CMS3. Using different γ for the stiff EOS would not be appropriate for a comparison. Nevertheless, we think that the difference between γ = 1.5 and 5/3 would not significantly affect the decay law. Even γ = 1.5 is unstable for CMS5. Therefore, we do not have data for CMS5 with a stiff EOS. a Driving schemes – either solenoidal or compressive driving. b Polytropic exponent. c The driving wavenumber at which the energy injection rate peaks. d The sonic Mach number which is defined in Equation (5). e Average of power law exponent for v2 ∝ t−α1 . f −α2 Average of power law exponent for σρ/ρ0 ∝ t . g Time interval in units of large-eddy turnover time (ted) for averaging α1 and α2. bulence initially driven by compressive driving have not First, compressive driving yields more compressions been studied yet. at the same Mach number. Therefore, while decaying, The main goal of this paper is to examine whether compressed regions could generate additional kinetic the decay exponent α depends on the value of the poly- energy via expansion, which could affect the rate of tropic exponent γ. Here we concentrate on decay of poly- turbulence decay. tropic turbulence driven by either solenoidal or compres- Second, regarding the effects of γ on decaying tur- sive driving in both transonic and supersonic regimes. bulence, only limited regions of the parameter space Hence, we expect to demonstrate how polytropic EOS have been studied. Mac Low et al. (1998) found that and types of driving affect decaying turbulence. In addi- supersonic turbulence with γ = 1.4 decays with α ∼ 1.2. tion, we also investigate the probability density function For isothermal cases (i.e., γ = 1), they found that α is (PDF) of gas density and skewness of the PDF in decay- nearly unity. This suggests that the scaling exponent ing polytropic turbulence initially driven by compressive α in E ∝ t−α only weakly depends on γ as assumed driving. by Davidovits & Fisch (2017). However, Mac Low et The paper is organized as follows. We explain our al. (1998) used a random initial velocity perturbation motivation and numerical method in Section 2, and which follows a power-law, and a constant initial density. present the results from our numerical simulations in Therefore, it is necessary to test the decay law using Section 3. We discuss our finding and its astrophysical initial velocity and density data cubes from actual tur- implication and give a summary in Section 4. bulence simulations with both soft EOS (i.e., polytropic γ < 1) and stiff EOS (i.e., polytropic γ > 1). 2. MOTIVATION AND NUMERICAL METHOD Third, the effects of γ and the type of driving on density PDF of turbulence have been addressed in sev- 2.1. Motivation eral previous studies. For example, Federrath et al. As we described earlier, decay of solenoidally driven (2010) showed that solenoidal driving and compressive isothermal turbulence follows E ∝ t−α with α ≈ 1. The driving can produce different statistics of isothermal tur- type of driving or γ may affect this scaling relation. bulence. In addition, Federrath & Banerjee (2015) found Decay of Turbulence in Fluids with Polytropic Equations of State 51

Figure 1. Time evolution of hv2i in decaying turbulence initially driven by solenoidal driving with γ = 0.7 (blue), 1.0 (red), 1.5 (cyan), or 5/3 (green), where h· · ·i indicates spatial averaging. Left: Ms ∼ 1. Center: Ms ∼ 3. Right: Ms ∼ 5. Turbulence 2 2 starts decaying at tcode/ted = 0. We normalize hv i by v0 which is the value at tcode/ted = 0. The black dotted lines in all three panels are reference lines for different power-law exponents.

Figure 2. Same as Figure 1, but for the decay of the standard deviation of the density fluctuation σρ/ρ0 . We normalize σρ/ρ0 by (σρ/ρ0 )0 which is the value at tcode/ted = 0. that the density PDF of solenoidally driven turbulence driving wavenumber at which the energy injection rate with polytropic γ = 5/3 has a power-law tail at low peaks. Therefore, in our simulations, we generate tur- density, which is not observed in isothermal turbulence. bulence approximately at a scale kf times smaller than However, earlier studies have not addressed turbulence the computational box. We drive turbulence in Fourier with polytropic EOS and compressive driving. In this space using 100 forcing components, which are nearly paper, we use both compressive driving and polytropic isotropically distributed in the range kf /1.3 . k . 1.3kf . EOS to investigate density PDF and skewness of driven We consider solenoidal (∇ · f = 0) and compressive and decaying turbulence. (∇ × f = 0) drivings. In the former driving, we extract forcing components normal to k in Fourier space and 2.2. Numerical Method carry out an inverse Fourier transform to produce them 2.2.1. Numerical Code in real space. In the latter driving, the procedure is the We use an Essentially Non-Oscillatory (ENO) scheme same, but we extract forcing components parallel to k (see Cho & Lazarian 2002) to solve the ideal hydrody- in Fourier space. namic equations in a periodic box of size 2π (≡ L0): In both drivings, the driving vectors continuously change with a correlation time comparable to the large- ∂ρ eddy turnover time. We also adopt a polytropic EOS + ∇ · (ρv) = 0, (2) ∂t  ρ γ c2 ρ   ρ γ P = P = s0 0 (4) ∂v 0 ρ γ ρ + v · ∇v + ρ−1∇P = f, (3) 0 0 ∂t where P is the normalized pressure, and P0, cs0, and where f is a driving force, P is the pressure (see Sec- ρ0 are the initial pressure, sound speed, and density, tion 2.2.2), ρ is the density, and v is the velocity. The respectively. The sonic Mach number Ms is defined by density and velocity are set to 1 and zero at t = 0 to vrms assume a static medium with a constant density at the Ms ≡ , (5) c beginning. s0 where vrms is the rms velocity. We vary γ and the sonic 2.2.2. Simulations Mach number Ms to consider both soft and stiff EOS We use 5123 grid points in our periodic computational in transonic and supersonic regimes. box. The peak of energy injection occurs at k ∼ kf , Table 1 lists our simulation models. We use the no- where k is the wavenumber and kf (≈ 6 or 8) is the tation XMSY-γZ, where X = S or C refers to solenoidal 52 Lim & Cho

Figure 3. Same as Figure 1, but for compressively driven turbulence. Note that unlike Figure 1, hv2i slightly increases along the decay as indicated by a black arrow in each panel; this is most apparent in the case of Ms ∼ 1 (left panel).

Figure 4. Same as Figure 2, but for compressively driven turbulence.

(S) or compressive (C) driving; Y = 1, 3, or 5 refers to almost the same at the same Ms regardless of the value the sonic Mach number Ms; Z = 0.7, 1.0, 1.5, or 5/3 of γ (see Table 1 for measurements of α). The decay refers to the value of γ. We keep driving turbulence until is steeper in the case of Ms ∼ 1 (α ∼ 1.2) than in the system reaches saturation, after which the driving supersonic cases (α ∼ 1.0). Second, similar to the case 2 is turned off to let turbulence freely decay. In decaying of hv i, the decay of σρ/ρ0 is hardly affected by γ. In simulations, time is normalized like t = tcode/ted. Here, addition, except for the case Ms ∼ 1 (see Table 1), the tcode is the time in code units, ted = (L0/kf )/v0 is the power-law exponent for σρ/ρ0 is about half of that for 2 large-eddy turnover time, and v0 is the velocity at the hv i at the same Ms. For the cases with γ = 1 (i.e., moment turbulence starts decaying. isothermal cases), this result is consistent with the fact that the standard deviation of density fluctuations scales 3. RESULTS approximately linearly with the sonic Mach number (e.g., 3.1. Decay of Hydrodynamic Turbulence with Padoan et al. 1997; Passot & V´azquez-Semadeni1998), 2 1/2 −α/2 Polytropic EOS which implies σρ/ρ0 ∝ Ms ∝ hv i ∝ t . Our results imply that a similar argument holds true for 3.1.1. Decay of Turbulence Driven by Solenoidal Driving γ 6= 1. Here, we consider decaying polytropic turbulence ini- tially driven by solenoidal driving. Figures 1 and 2 show 3.1.2. Decay of Turbulence Driven by Compressive Driving the decay of hv2i1 and the standard deviation of den- We now deal with the decay of polytropic turbulence sity fluctuations σ , respectively, where h· · ·i denoted ρ/ρ0 initially driven by compressive driving. Figures 3 and spatial averaging. From left to right, the sonic Mach 4 show the decay of hv2i and σ , respectively. As number M is ∼ 1, ∼ 3, and ∼ 5, respectively. Blue, ρ/ρ0 s in the case of solenoidal turbulence,2 we use different red, cyan, and green curves in each panel correspond to values of M (from left to right panels) and γ (curves γ of 0.7, 1.0, 1.5, and 5/3, respectively. s with different colors). First of all, we clearly see that the decay of hv2i 2 −α Similar to the result from solenoidal turbulence, follows a power-law of the form hv i ∝ t , and α is 2 both hv i and σρ/ρ0 in compressively driven turbulence 1We are interested in the decay of h(δv)2i, where δv is the magni- exhibit power-law decay, and γ hardly affects the decay tude of the velocity fluctuations. If hvi is non-zero, h(δv)2i is dif- rate (see Table 1 for measurements of α). According to ferent from hv2i. That is, hv2i = h(δv)2i + hv i2 + hv i2 + hv i2, x y z Figure 3, the power-law exponent α is ∼1.0 for Ms ∼ 1, where hvii (i = x, y, z) is the mean velocity component over the whole computational box. In Runs SMS1-γ1.0, SMS3-γ1.0, and ∼0.8 for Ms > 1, which means that the decay of and SMS5-γ1.0, hvxi, hvyi, and hvzi are less than ∼0.01 at the compressively driven turbulence is slower than that of moment turbulence starts decaying. Therefore, we can ignore 2 the effect of non-zero hvii. We assume that this holds true also By “solenoidal turbulence” we mean turbulence initially driven for the other simulations presented in this paper. by solenoidal driving. Decay of Turbulence in Fluids with Polytropic Equations of State 53

Figure 5. Decay of the ratio of compressive (upper panels) and solenoidal (bottom panels) velocity components in compressively driven turbulence. Left panels: Ms ∼ 1. Center panels: Ms ∼ 3. Right panels: Ms ∼ 5. Blue, red, cyan, and green curves denote a γ of 0.7, 1.0, 1.5, and 5/3, respectively.

2 2 solenoidal turbulence at the same Ms. As can be seen pressive ratio hvcompi/hvtoti (upper panels) and the 2 2 from Figure 4, the power-law exponent α for the decay solenoidal ratio hvsoli/hvtoti (bottom panels) in com- of σ is about half of that for hv2i, which is consistent 2 2 2 ρ/ρ0 pressively driven turbulence. Here, vtot = vsol + vcomp, with the result from the previous section. However, and vsol and vcomp are the solenoidal and compressive the simulations with Ms ∼ 5 show similar values of α velocity components, respectively. Blue, red, cyan, and 2 between the decay of hv i and that of σρ/ρ0 . green curves indicate values of γ of 0.7, 1.0, 1.5, and Unlike in the case of solenoidal turbulence, γ slightly 5/3, respectively. affects the decay of compressively driven turbulence. First, Figure 5 shows that the compressive ratio 2 First, hv i in compressively driven turbulence shows decreases as turbulence decays. When Ms ∼ 1 or ∼ 3, bump-like features (indicated by the black arrow in the smaller γ, the larger the compressive ratio. How- 2 each panel of Figure 3). This slight increase of hv i is ever, when Ms ∼ 5, we do not see a dependence of most pronounced in the case Ms ∼ 1. Second, dip-like the compressive ratio on γ. This may be because as features (indicated by the black arrow in each panel of Ms increases, gas pressure becomes relatively less im- Figure 4) are visible in the evolution of σρ/ρ0 at about portant than turbulent pressure and so does the role the same time when the bump-like features in hv2i occur. of the polytropic EOS. Second, and more importantly,

Third, we can see from Figure 4 that the decay of σρ/ρ0 the solenoidal ratio increases as turbulence decays and for γ = 0.7 occurs earlier than for γ > 0.7 regardless of eventually becomes higher than the compressive ratio Ms. irrespective of γ and Ms, which means that the compres- sive velocity component decays faster. Figure 6 shows 3.2. Decay of Solenoidal and Compressive Velocity this for the case of isothermal turbulence initially driven Components in Turbulence Driven by Compressive by compressive driving, for which we plot the decay of and Solenoidal Drivings solenoidal and compressive velocity components. As Here, we deal with the differences in the decay of we can see from the reference lines (dotted lines in dif- solenoidal and compressive modes. We decompose 3D 2 ferent colors in each panel), hvcompi (magenta curves) velocity fields of both solenoidally and compressively 2 decays more quickly than hvsoli (cyan curves), result- driven turbulences into solenoidal and compressive com- ing in larger solenoidal velocity components at the late 3 ponents. Figure 5 illustrates the decay of the com- stages of decay irrespective of Ms. 3To be specific, we carry out Fourier transforms of total velocity While the compressive ratio in compressively driven fields first. Then, we take only the velocity components perpen- turbulence becomes less than the solenoidal ratio as the dicular to k, which correspond to solenoidal velocity component turbulence decays, Figure 7 shows that the compressive in Fourier space, and carry out inverse Fourier transforms to pro- ratio in solenoidal turbulence does not. In the figure, duce solenoidal velocity fields in real space. Finally, we subtract the solenoidal velocity components from the total velocities in blue, red, cyan, and green curves indicate γ of 0.7, 1.0, real space to yield the compressive velocity components. 1.5, and 5/3, respectively. As we can see from Figure 7, 54 Lim & Cho

Figure 6. Decay of solenoidal and compressive velocity components in compressively driven turbulence. Only the case of isothermal turbulence (i.e., γ = 1.0) is presented here. Left: Ms ∼ 1. Center: Ms ∼ 3. Right: Ms ∼ 5. Black, cyan, and magenta curves indicate total, solenoidal, and compressive velocity components, respectively. The dotted lines in different colors are reference lines for different power-law forms. Note that the compressive velocity component decays much faster than the solenoidal one.

Figure 7. Same as Figure 5 but for solenoidally driven turbulence and showing the compressive ratio only. the compressive ratio in solenoidal turbulence is always We first consider driven turbulence (black solid line less than 0.5 regardless of both Ms and polyrtropic γ, in Figures 8 and 9). Compressively driven turbulence implying that the solenoidal velocity component in such yields density PDFs which are not exactly log-normal turbulence is always dominant over the compressive one, even in the case γ = 1. The density PDF for γ = 0.7 is even at the late stages of decay. slightly right-skewed, and that for γ = 5/3 is strongly left-skewed. The latter has a pronounced power-law 3.3. Density PDF and Skewness tail at low density as shown in Figure 8c. Figure 9c In the following, we investigate the density PDF and its shows that the PDF of CMS3-γ1.5, which is for Ms ∼ 3, skewness for the case of compressively driven turbulence deviates more strongly from the log-normal form at low with polytropic EOS in driven and decay regimes. We density. However, it is not clear whether the low density define the skewness of the density PDF as tail follows a power-law. N  3 We can interpret the behavior of density PDF with 1 X si − hsi Skew(s) = (6) γ stated above as follows. When γ < 1 (γ > 1), compres- N σ i=1 s sion decreases (increases) internal temperature and thus local sound speed, which allows (hinders) high density where N is the total number of data points, s ≡ ln(ρ/ρ0) regions to develop easily (Federrath & Banerjee 2015). is the natural logarithm of the density fluctuations, and Since compression can commonly occur in our simula- h· · ·i denotes the spatial average. Skewness measures tions of transonic/supersonic turbulence with compres- the asymmetry of a probability distribution. When the sive driving, the density PDFs with γ < 1 (γ > 1) be- distribution is left-skewed (right-skewed), the skewness come right-skewed (left-skewed). In addition, although takes a negative (positive) value. density PDFs with γ = 0.7 are slightly right-skewed, Figures 8 and 9 show the density PDFs of polytropic we expect that they become more right-skewed as γ turbulence with Ms ∼ 1 and ∼ 3, respectively. Each decreases further. In fact, Scalo et al. (1998) showed a colored solid curve corresponds to the density PDF at a clear power-law tail toward high density regions in their different time along the decay. We carry out log-normal one-dimensional simulation with γ = 0.3 and Ms ∼ 3. fitting for the PDFs using the relation Note that our results are for compressively driven 1  (s − hsi)2  turbulence with polytropic EOS. Earlier studies are avail- √ ps(s) = exp − 2 . (7) able for solenoidal turbulence with polytropic EOS and 2πσs 2σs Ms ∼ 10 (Federrath & Banerjee 2015), and for com- In the figures, fitting lines are indicated by dotted lines. pressively driven turbulence with γ = 1 and Ms ∼ 5 Decay of Turbulence in Fluids with Polytropic Equations of State 55

Figure 8. PDF of the logarithmic density s = ln(ρ/ρ0) of decaying polytropic turbulence initially driven by compressive driving with Ms ∼ 1. Left: polytropic γ = 0.7. Center: γ = 1.0. Right: γ = 5/3. Note that the p(s) axis is logarithmic. Black, purple, and orange curves represent different times along the decay. Dotted lines are log-normal fitting line.

Figure 9. Same as Figure 8, but for Ms ∼ 3. Left: γ = 0.7. Center: γ = 1.0. Right: γ = 1.5.

(Federrath et al. 2010). Our current result is consis- bottom panels of Figure 10 show the skewness for Ms ∼ 1 tent with that of the latter reference in that the PDF and ∼ 3, respectively. At t = 0, the PDF for γ > 1 has a is slightly left-skewed when γ = 1. Our result is also negative skewness (see green curve in the top panel and consistent with that of the former reference in that the cyan curve in the bottom panel). Likewise, the skewness PDF is strongly left-skewed with a power-law tail when for other values of γ indicates that the density PDFs γ = 5/3. However, the PDF of solenoidal turbulence is presented in the figure deviate from a log-normal form more or less symmetric when γ = 0.7 or 1.0 (Federrath at t = 0. & Banerjee 2015), while that of compressively driven Next, we consider the decay regime. As can be seen turbulence is clearly right-skewed for transonic turbu- from the purple and orange solid curves in Figures 8 lence and slightly right-skewed in supersonic turbulence and 9, as turbulence decays, the density PDFs become when γ = 0.7 (see Figures 8a and 9a). narrow and approach log-normal forms in all cases. We Figure 10 shows the time evolution of the skewness can confirm this trend in Figure 10. The skewness for of the density PDF for compressively driven turbulence. Ms ∼ 1 and ∼ 3 approaches, and fluctuates around, The horizontal axis denotes the elapsed decay time nor- zero as turbulence decays, which is consistent with the malized by the large-eddy turnover time. The top and temporal change of the PDFs shown in Figures 8 and 9.

4. DISCUSSIONAND SUMMARY Table 2 The purpose of this study is to investigate the effects of Skewness of the PDFs presented in Figures 8 and 9 the equation of state (i.e., the value of the polytropic in- Run ta tb tc dex γ) and driving schemes (i.e., solenoidal and compres- 1 2 3 sive drivings) on decaying turbulence and its statistics. CMS1-γ0.7 0.260 0.084 0.294 In this paper, we demonstrated that the scaling relation CMS1-γ1.0 −0.335 0.290 0.260 for the decay law (hv2i ∝ t−α) does not show a strong CMS1-γ5/3 −1.576 0.168 0.260 dependence on γ and the driving schemes. Throughout CMS3-γ0.7 0.204 0.197 0.093 2 CMS3-γ1.0 −0.301 −0.149 0.070 our simulations, hv i decays with 0.8 . α . 1.2. The CMS3-γ1.5 −0.784 −0.382 −0.013 range is nearly same as the one found by Mac Low et al. (1998) (0.85 < α < 1.2). a Time marked by black solid lines in Figures 8 and 9. For γ > 1 cases, α ranges from 1.0 to 1.2 in b Time marked by purple solid lines in Figures 8 and 9. solenoidal turbulence and from 0.8 to 1.0 in compres- c Time marked by orange solid lines in Figures 8 and 9. 56 Lim & Cho

feature for the case γ = 0.7. Lastly, σρ/ρ0 decays more quickly for γ = 0.7 than for γ > 0.7 in both transonic and supersonic turbulence driven by compressive driv- ing. This is beuase when γ is less than one, expansion increases the internal temperature, which dissipates den- sity structures quickly. Therefore, the density decays faster for smaller γ. More interestingly, the compressive velocity compo- nent decays faster than the solenoidal one in the case of turbulence initially driven by compressive driving as shown in Section 3.2. We can interpret this as follows. When turbulence initially driven by compressive driv- ing decays, the energy of the compressive component is dissipated through both turbulent cascades and dissipa- tion at shocks, and a fraction of the energy is converted into solenoidal energy. For the solenoidal component, only turbulent cascades can dissipate the energy. As it has fewer channels for energy dissipation available than the compressive component, the solenoidal velocity component in turbulence initially driven by compressive driving decays more slowly. However, a detailed analysis addressing, e.g., what fraction of compressive kinetic energy changes into solenoidal kinetic energy, is beyond the scope of this paper; further studies are required to understand this issue quantitatively. Our study uses polytropic EOS to quantify decaying astrophysical turbulence. A fluid with polytropic EOS (P = Kργ ) is different from an ideal gas with adiabatic index γ > 1 as follows. First, we do not need to solve the energy conservation equation for the former, while we have to for the latter. Note that the EOS for the latter becomes P = (γ − 1)E, where E is the internal energy Figure 10. Time evolution of the skewness of the density density. Second, they provide different descriptions PDFs shown in Figure 8 (top panel) and Figure 9 (bottom of shocks. The polytropic EOS is only applicable to panel). Blue, red, cyan, and green curves correspond to γ of isentropic flows; the entropy is constant everywhere, 0.7, 1.0, 1.5, and 5/3, respectively. even in shocked regions. The entropy is not constant for an ideal gas. Although entropy is conserved in regions with adiabatic expansion/compression, it increases when sively driven turbulence, with the largest value of α a parcel of gas passes through a shock front in an ideal being obtained for the case Ms ∼ 1 in both driving gas. Therefore, a fluid with polytropic EOS and an ideal schemes. This result confirms the assumption of Davi- gas with adiabatic index γ have different descriptions dovits & Fisch (2017) that for γ = 5/3, α falls into the of shocks and their shock jump conditions are different range 1.0–1.5 with a weak dependence on the initial (see Bisnovatyi-Kogan & Moiseenko 2016 for shock jump Mach number. conditions for adiabatic and isentropic flows). Even though no relationship between polytropic in- Our results have astrophysical implications. The dex γ and scaling of the decay law (hv2i ∝ t−α) is found polytropic exponent γ is useful to describe a variety in our study, there are several noticeable characteristics of components in the ISM. A polytropic EOS with in case of compressively driven turbulence. First, a slight γ ' 0.8 can represent the density range of 10cm−3 ≤ 2 4 −3 increase of hv i and an associated decrease of σρ/ρ0 are n ≤ 10 cm , where n is the hydrogen number den- found in Figures 3 and 4, respectively. As discussed sity (Glover & Mac Low 2007b); this corresponds to earlier, those effects can be interpreted as being due to the density range for giant molecular clouds (Ferri`ere additional energy released from compressed regions via 2001). Furthermore, the EOS with γ ∼ 1.4 describes expansion. Second, in the case Ms ∼ 1, the effect is the conditions in the centers of protostellar cores, with a most pronounced. This may be due to the pressure being corresponding density range 1012cm−3 ≤ n ≤ 1017cm−3 strong compared to that of supersonic cases, resulting in (Masunaga & Inutsuka 2000). Accordingly, our results stronger expansion. Third, for the same Ms, bump and suggest that once driving of turbulence ceases to act, dip like features are more prominent for γ = 0.7. This turbulence quickly decays with a dynamical timescale is because, for compressive driving, turbulent gas with that is characteristic of a certain system irrespective of γ = 0.7 is more easily compressed. Thus, expansion is its spatial scale. Moreover, even if turbulence is initially easier when turbulence decays, leading to the stronger driven by compressive driving like supernova explosions, Decay of Turbulence in Fluids with Polytropic Equations of State 57 solenoidal motions will dominate as the turbulence de- Federrath, C. & Banerjee, S. 2015, The Density Structure cays due to the much faster decay of compressive mo- and Star Formation Rate of Non-isothermal Polytropic tions. Turbulence, MNRAS, 448, 3297 In summary, we have studied the influence of poly- Federrath, C., Rathborne, J. M., Longmore, S. N., et al. tropic EOS and driving schemes on decaying turbulence 2017, The Link between Solenoidal Turbulence and Slow and its statistics and found the following results. Star Formation in G0.253+0.016, Proc. IAU Symp. 322, 123 1. There is no significant correlation between the scal- ing of the decay law (E ∝ t−α) and polytropic Ferri`ere,K. M. 2001, The Interstellar Environment of our exponent γ in the case of solenodially driven turbu- Galaxy, Rev. Mod. Phys., 73, 1031 lence. Glover, S. C. O. & Mac Low, M.-M. 2007, Simulating the Formation of Molecular Clouds. I. Slow Formation by Grav- 2. Driving schemes have a non-negligible effect on the itational Collapse from Static Initial Conditions, ApJS, decay rate of turbulence: the power-law index α for 169, 239 turbulence initially driven by compressive driving is smaller than that for turbulence initially driven Glover, S. C. O. & Mac Low, M.-M. 2007, Simulating the Formation of Molecular Clouds. II. Rapid Formation from by solenoidal driving. Turbulent Initial Conditions, ApJ, 659, 1317 3. There is no significant effect of γ on the decay rate Goldreich, P. & Sridhar, S. 1995, Toward a Theory of Inter- of velocity in compressively driven turbulence. stellar Turbulence. II. Strong Alfvenic Turbulence, ApJ, 4. The polytropic exponent γ has a small effect on 438, 763 the density fluctuations in compressively driven Larson, R. B. 1981, Turbulence and Star Formation in Molec- turbulence: the smaller γ is, the faster the standard ular Clouds, MNRAS, 194, 809 deviation of density fluctuations of the turbulence Lesieur, M. 2008, Turbulence in Fluids (Dordrecht: Springer) decays. Li, Y., Klessen, R. S., & Mac Low, M.-M. 2003, The For- 5. When considering the decay of solenoidal and mation of Stellar Clusters in Turbulent Molecular Clouds: compressive velocity components in compressively Effects of the Equation of State, ApJ, 592, 975 driven turbulence separately, the energy of the com- Mac Low, M.-M., Klessen, R. S., Burkert, A., & Smith, pressive velocity component decays much faster. M. D. 1998, Kinetic Energy Decay Rates of Supersonic 6. Regarding the statistics of compressively driven and Super-Alfv´enicTurbulence in Star-Forming Clouds, turbulence, the density PDF deviates from a log- Phys. Rev. Lett., 80, 2754 normal distribution, especially for γ > 1. In addi- Mac Low, M.-M. & Klessen, R. S. 2004, Control of Star tion, we have found that the skewness of the density Formation by Supersonic Turbulence, Rev. Mod. Phys., PDF of the turbulence becomes zero as it decays. 76, 125 Masunaga, H. & Inutsuka, S. 2000, A Radiation Hydrody- ACKNOWLEDGMENTS namical Model for Protostellar Collapse. II. The Second This paper has been expanded from a chapter of Collapse and the Birth of a Protostar, ApJ, 531, 350 Jeonghoon Lim’s MSc thesis. This work is supported Ostriker, E. C., Stone, J. M., & Gammie, C. F. 2001, Den- by the National R&D Program through the National sity, Velocity, and Magnetic Field Structure in Turbulent Research Foundation of Korea Grants funded by the Ko- Molecular Cloud Models, ApJ, 546, 980 rean Government (NRF-2016R1A5A1013277 and NRF- Padoan, P., Nordlund, A., & Jones, B. J. T. 1997, The 2016R1D1A1B02015014). 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Printed on April 30, 2020 JOURNAL OF THE KOREAN ASTRONOMICAL SOCIETY Vol. 53 No. 2 JOURNAL OF THE KOREAN ASTRONOMICAL SOCIETY April 30, 2020 Vol. 53 No. 2, April 30, 2020 JOURNAL OF THE THE FORMATION OF THE DOUBLE GAUSSIAN LINE PROFILES OF THE SYMBIOTIC STAR AG PEGASI ······················································································································································· Siek Hyung and Seong-Jae Lee 35 KOREAN POORLY STUDIED ECLIPSING BINARIES IN THE FIELD OF DO DRACONIS: V454 DRA AND V455 DRA ··························································································· Yonggi Kim, Ivan L. Andronov, Kateryna D. Andrych, Joh-Na Yoon, Kiyoung Han, and Lidia L. Chinarova 43 ASTRONOMICAL DECAY OF TURBULENCE IN FLUIDS WITH POLYTROPIC EQUATIONS OF STATE ················································································································································ Jeonghoon Lim and Jungyeon Cho 49 SOCIETY Vol. 53 No. 2, April 30, 2020

pISSN 1225-4614 eISSN 2288-890X THE KOREAN ASTRONOMICAL SOCIETY