Asteroseismic Analysis of NASA s Kepler Space Telescope K2 Mission Data of Compact Stars Matt Yeager Department of Physics, Astronomy, and Materials Science Missouri State University Advisor: Dr. Mike Reed Abstract NASA s K2 mission has provided, and continues to provide, a wealth of data on numerous subdwarf B (sdb) pulsators for asteroseismology. However, the data is not as clean as the original Kepler mission and requires significant processing before useful analysis can be performed. After a clean light curve is extracted from the data, we use various tools, including Fourier transforms, the Kolmogorov Smirnov test, and Echelle diagrams for seismic studies. Asteroseismology is the study of stars pulsations to probe their inner structure which we cannot directly observe. Comparing the observed pulsations of a star with models can lead to new discoveries about sdb stars, including additional physics to be incorporated into models. The goal of this report is to provide an overview of what asteroseismology is and the methods and tools used to apply it to sdb stars. Subdwarf B Stars The average subdwarf B star has a core mass of around 0.5 M, a radius of around 0.2 solar radii, and an effective temperature near 30,000K. They have a very thin hydrogen envelope, generally less than 0.01 M 1. It was discovered in 1996 that sdb stars can pulsate 2 and since then two different pulsation types have been determined, short period and long period. Short period pulsations have periods of less than about 15 minutes and are pressure (p-) mode pulsations. Long period pulsations are characterized by periods of 45 minutes to 8 hours and are gravity (g-) mode pulsations. Due to their longer periods, ground-based observations of g-mode pulsators over multiple pulsation periods was difficult, but with the advent of the Kepler mission, asteroseismology of g-mode pulsators became much easier. The Kepler mission The Kepler space telescope was launched in 2009 for a four year mission to observe exoplanetary transits around their stars 3. In 2013 a second of the four reaction wheels used to keep the spacecraft aligned on its target failed bringing an end to the primary mission of the spacecraft. A secondary mission was developed, termed K2, which observes in the ecliptic plane for approximately 80-day increments using radiation pressure from the Sun to balance the spacecraft along with its remaining two reaction wheels. K2 observes in short cadence and long cadence modes. Short cadence sums nine 6.02 second exposures (each with 0.52 seconds of overhead) into a total of 58.85 seconds 4. Long cadence has 270 6.02 second exposures combined for a total of 29.43 minutes. Since the spacecraft is observing along the ecliptic, it must change fields around every 80 days to prevent the Sun s light from entering the telescope, allowing for 4 observing campaigns a year 5. Every month, Kepler must transmit data back to Earth due to storage constraints on board the spacecraft, leading to short breaks in data collection. Asteroseismology Asteroseismology is the method of using observations of stellar pulsations to determine a star s physical characteristics and to match stellar models to observations. The deviation of observations from the models can help in the discovery of new physics and to refine stellar models.
The categorization of non-radial pulsations uses the quantized numbers n, l, and m which represent the number of radial(n), surface(l), and azimuthal surface nodes(m). Radial pulsations have l and m equal to zero. In the asymptotic limit of n>>l, g-mode pulsations are equally spaced in period for sequential values of n (for an idealized homogeneous star) according to: Π Π,Π = Π Π Π(Π + 1) Π + є Where Π Π and є are constants in units of seconds (Unno et al. 1979). The spacing between two consecutive overtones is: Π Π Π Π = Π(Π + 1) The geometry of the pulsations on the surface can lead to a reduction of their observed amplitudes and the pulsation modes with the least geometric cancellation are the l=1 and l= 2 modes. The asymptotic relationship between these two modes is: Π Π,Π=2 = Π Π,Π=1 3 + Π Where C is a small constant and is zero if є 1 = є 2. Azimuthal modes are degenerate unless a force, usually Coriolis caused by rotation, is applied. Stars which complete multiple revolutions over the course of observations will have azimuthal degeneracy lifted and form frequency multiplets. The multiplets have 2l+1 components with a spacing of: Π Π,Π,Π = Π Π,Π,0 + ΠΠ(1 Π Π,Π ) Where Ω is the rotational frequency and C n,l is the Ledoux constant (Ledoux 1951, Aerts et al. 2010) which for g-modes is given by Π Π,Π 1 Π(Π + 1) Data Processing and Analysis We acquire K2 data of prospective sdb pulsators from MAST (Mikulski Archive for Space Telescopes), and then process them to produce a light curve using aperture photometry. Two main steps in that process are de-trending (or flattening ) the light curve, and de-correlating the periodic thruster firing signal from the data. The thrusters on Kepler fire only at intervals of six hours, imparting a signal on the data which can be in the middle of the g-mode period range. We then make a Fourier transform (FT) of the light curve to better see the pulsation mode frequencies of the star. Figure 1 shows the FT of the sdb star V1405 Ori, which is a hybrid pulsator, showing both pressure (rightmiddle peaks) and gravity (far left peaks) pulsation modes.
Figure 1. Fourier transform of V1405 Ori s lightcurve A noise cutoff is determined, usually around 4-4.5σ, which allows for the determination of real pulsation frequencies from noise. Asymptotic period spacings in g-modes were discovered in Kepler data by Reed et al. (2011) 6. The primary tool used to determine if there are period spacings is the Kolmogorov Smirnov(KS) test. In figure 2, the KS test for J19155-2333 is seen to have a large trough around 250 seconds, indicating the period spacing.
Figure 2. KS test for J19155-2333 The KS test, however, does not show what periods fall on the period spacing sequence. To accomplish that task, Echelle diagrams are used, which fold the periods over the suspected asymptotic spacing. An echelle diagram is shown in figure 3 for J19348-1855 l = 1 and l = 2 period sequences in the g-mode region of a star. Figure 3. Echelle diagram for J19348-1855. The left panel is folded appropriate for l=2 and the right panel for l=1. Black circles indicate l=1, blue triangles l=2, and red stars are periods which do not fit either sequence. Periods which fit both sequences are indicated with outlines.
Finally, since K2 data are obtained over a period of multiple months, we can use the sliding Fourier transform (SFT) to help visualize the data and determine how constant the pulsation amplitudes are through time. An SFT takes a slice of the data and creates a Fourier transform and slides it along the to cover full timespan of the data. In figure 4 we see an SFT for the star J19155-2333. The SFT runs from about 260 to 285 mhz and spans the observations, nearly 70 days. There is an obvious high amplitude pulsation frequency that runs the whole length of the observation period, along with two fainter, lower amplitude frequencies. The furthest right line, however, is not constant, the amplitude cutting out in the middle before rising back up. Figure 4. Sliding Fourier transform for J19155-2333 used to evaluate amplitude stability with time. Conclusion Kepler and K2 data have brought significant changes to asteroseismic applications of sdb stars. While most of the tools were previously known, their application to ground-based data were not successful. They required the precision and uninterrupted duration of space-based data. Their application to Kepler and K2 data have resulted in hundreds of pulsations with mode identification to constrain models, as well as new discoveries, such as subsynchronous rotation in short-period binaries, and radially differential rotation. With continued examination of K2 data, significant advances in understanding of late-evolution compact stars is possible. Acknowledgements Funding for this research was provided by the Missouri Space Grant Consortium, funded by NASA. Data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-hst data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts. Biography Matt Yeager is a junior at Missouri State University studying in the Physics, Astronomy, and Materials Science Department. His major is Physics with an emphasis in Astronomy/Astrophysics. After graduation he hopes to pursue a PhD in physics.
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