Mode Identification in the Pulsating Subdwarf B Star KIC 2697388 A.S
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ACTA ASTRONOMICA Vol. 62 (2012) pp. 179–200 Mode Identification in the Pulsating Subdwarf B Star KIC2697388 A.S. Baran Mt. Suhora Observatory of the Pedagogical University, ul. Podchor ˛azych˙ 2, 30-084 Cracow, Poland email: [email protected] Missouri State University, Department of Physics, Astronomy, and Materials Science, 901 S. National Av., Springfield, MO 65897, USA Received June 28, 2012 ABSTRACT We present our results on mode identification in the pulsating subdwarf B star KIC2697388 observed with the Kepler spacecraft. We detected 148 frequencies of which five were attributed to p-modes while 122, to g-modes. The remaining 21 frequencies are also likely g-modes. We used multiplets and asymptotic period spacing to constrain degrees of 89 peaks to either l = 1 or 2. Using splittings in multiplets we derived the rotation period of this star to be nearly 45days. The average period spacing between l = 1 and 2 overtones are nearly 240 s and 139 s, respectively. Our results show that combining tentative identification based on the presence of multiplets and asymptotic period spacing is indeed useful in mode identification. Mode degree consistently inferred from independent methods make the results reliable and will help to construct accurate models of subdwarf B stars. Key words: subdwarfs – Stars: oscillations – Asteroseismology 1. Introduction Subdwarf B stars (sdB) are located on the blue extension of the horizontal branch. They are compact objects of half a solar mass, with typical radii of 0.2 R and typical effective temperatures of 30000K. We infer that an sdB progenitor on⊙ the main sequence has a mass similar to our Sun. After the hydrogen in its core is exhausted it moves from the main sequence heading toward the red giant branch. Initial masses of sdB progenitors indicate that they should pass through a helium flash and according to our best knowledge of sdB evolution progenitors have to lose almost all of their hydrogen envelopes prior to, or during, that flash. Only if this happened, can an object settle down on the blue edge of the horizontal branch, fusing helium into carbon in its core. After the helium is depleted, the stars head to the white dwarf cooling stage. 180 A. A. The first sdB stars were discovered by Humason and Zwicky (1947) who search- ed for blue objects around the North Galactic Pole. Half century later Kilkenny et al. (1997) discovered that sdB stars can pulsate. This opened an asteroseismic way to study sdB stars and attracted others to search for pulsating subdwarf B stars (sdBV). The first observations of sdBV stars were conducted from the ground. In total more than 70 sdBV stars have been found including both p- and g-mode dominated pulsators. However, most sdBV stars discovered from the ground are p-mode dom- inated. We consider it to be a selection effect since p-modes oscillate on shorter time scales and typically have bigger amplitudes than g-modes. Space telescopes allowed us to monitor pulsating stars on longer time scales and detect g-modes with low amplitudes. CoRoT and Kepler are recent examples. Non-radial pulsations can be described by means of spherical harmonics iden- tified by three natural numbers: n-radial order, l-degree and m-azimuthal order. It is not trivial to derive these three numbers for distant stars but they have to be known when making stellar models to compare with observations. Some efforts to identify modes were undertaken with ground-base data but with limited success. It includes multicolor photometry (Baran et al. 2008), spectroscopy (Telting and Østensen 2004) or multiplets (Baran et al. 2009). Typically, only a few modes have been identified. In the best case, three were identified in Balloon 090100001 (Baran et al. 2009), with two of them independently confirmed using another method of mode identification (Baran et al. 2008). Since only a few modes in sdB stars could be identified, an application of asteroseismology has been limited to straightfor- ward method of comparing model frequencies with the observed ones, assuming trial values of natural numbers. This method can easily produce a degenerate solu- tion (Charpinet et al. 2011a). Space-age asteroseismology provides us with an opportunity to make better mode identifications, sometimes using two independent methods allowing for cross- correlation. In such cases we may always go through assignment of quantized num- bers reducing the number of free parameters significantly. Resulting models should be more accurate. In this paper we present a mode identification using two independent methods which combined together make our conclusions even stronger. We used rotational multiplets and asymptotic period spacings between g-mode overtones. 2. Two Methods of Mode Identification 2.1. Rotational Multiplets Preliminary work on KIC2697388 based on one month of data was published by Reed et al. (2010). In the current paper we present results obtained from nine months of data. This results in nine times better frequency resolution and three times lower average noise level. A longer coverage works in favor of detecting Vol. 62 181 more peaks and resolving those split by rotation. Since our goal is to make mode identifications we hoped to detect enough peaks to easily recognize sequences of consecutive overtones and detect multiplets to cross-correlate mode degrees. Non-radial modes of degree l have 2l + 1 components differing in m number. However, in the absence of rotation, they all have the same frequency becoming indistinguishable in an amplitude spectrum. Stellar rotation can lift this degeneracy, shifting frequencies of m = 0 components according to the following relation: 6 1 Cn l ν = ν + ∆ν = ν + m − , n,l,m n,l,0 n,l,m n,l,0 P where: ∆νn,l,m is a frequency splitting, P is a rotation period of a star and Cn,l expresses the Ledoux constant. This expression is valid for slow rotators and os- cillations with periods shorter than the stellar rotation. According to this equation, components differing in m will be shifted in frequency proportionally to a rotation frequency. This will produce an equidistant (in frequency) pattern of peaks with a number of components which depends on the l value, i.e.,three peaks for l = 1, five peaks for l = 2 and so on. Detecting evenly spaced peaks in frequency may help to identify l values without using advanced methods for mode identifications. Unfortunately, the structure of multiplets depends on the inclination of the pul- sation axis to the observer’s line of sight. A good example is provided in Fig.8.5 in the Supplementary material of Charpinet et al. (2011b). An unconstrained angle can result in two different explanations of a detected multiplet. It may be that we, for example, see three peaks creating a triplet at moderate angle or a quintuplet with central and two outermost peaks. In such a case independent information is needed to correctly identify l values. Even though we expect that stars rotate, it was a challenge to use multiplets for mode identification in ground based data. Balloon090100001 (Baran et al. 2009) is an exception with two multiplets detected. The rotation period estimated from frequency splitting is close to 7days. If a period of days or weeks is representative for sdB stars it can explain why multiplets are rarely seen in amplitude spectra. It requires long (weeks or months) monitoring to resolve close peaks in multiplets. We expected that nine months monitoring of KIC2697388 should help to resolve and detect multiplets, which then could help with mode identification. Identifying multiplets allows us to estimate a rotation period. From the above equation we derive: 1 Cn l P = − , . ∆νn,l,m If we know the splitting, m and the Ledoux constant we can calculate the stellar rotation period. A splitting ∆νn,l,m is our measurement for given m. The Ledoux constant is the only quantity we have to derive from theoretical models. From a theoretical work by Charpinet et al. (2002) we know that this constant has values close to zero for p-modes. The exceptions are the fundamental and first overtone 182 A. A. l = 2, 3 modes which may have values up to 0.3. For g-modes the Ledoux constant depends on the l parameter and may be calculated from the expression: Cn,l . 2 1 (l +l)− . It is close to 0.5 and 0.15 for l = 1 and 2 modes, respectively. The ratio between l = 2 and 1 splittings will be close to 1.7. 2.2. Asymptotic Period Spacing Another way to identify mode degree is a period spacing of consecutive over- tones of the same degree. In the asymptotic limit n l , consecutive overtones of oscillation modes should be evenly spaced in period.≫ In sdB stars high order g-modes are usually detected, which makes period spacing useful for identifying g-modes. For given n and l values the period of oscillation modes is: P0 Pl,n = n + ε pl(l + 1) where P0 is the period of the fundamental radial mode and ε represents a small value (Unno et al. 1979). The period spacings between two consecutive overtones are P0 ∆Pl = Pl,n+1 Pl,n = . − pl(l + 1) We can use the above equation to estimate the ratio between spacings of overtones of different degree. The ratio between l = 1 and 2 is equal to 1/√3 and between l = 2 and 3 equals to 1/√2. Since the surface cancellation effect is decreasing with smaller l , we expect that modes of low degree (l 3) will be detected easier ≤ with l = 1 modes to be detected the easiest. The above asymptotic relation is valid only for a homogeneous star.