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Inference on Radii and Masses of Red Giants that host exoplanets from Solar-like Oscillations Yifan Chen Abstract Asteroseismology is the study of natural oscillations in stars. By observing a star's oscillation frequencies, it is possible to directly infer its stellar properties. Here we measure the solar-like oscillations in 6 planet- hosting red giants using data from the Transiting Exoplanet Survey Satellite (TESS) mission. Fourier transformation could be applied to the light curves obtained and passed into the pysyd pipeline to measure the frequency of maximum power (νmax) and made into an ´echelle diagram to measure the large separation of oscillations (∆ν) By using the scaling relations of νmax and ∆ν, the radius and mass of the stars can be obtained and compared with existing published values. Overall, the results demonstrated radius and mass obtained using asteroseismology to have a lower uncertainty compared to other methods. 1 Introduction 1.1 Asteroseismology A star is a ball of gas consisting of mainly hydrogen, some helium, and a bit of other elements (which as- tronomers call metals). Whie seismology is the study of earthquakes and seismic waves that move through the Earth, the study of Asteroseismology looks at \starquakes", or oscillations of stars. Spherical waves form in the interior of a star, forming oscillation modes and patterns of compression (similar to how air would form standing waves in a wind instrument). These oscillations would cause energy to rise up to the surface, resulting in the intensity of light emitted to change (Bedding 2009). The first discovery of oscillations in stars date back to 1596, yet not many oscillations (apart for the Sun's, which has its own branch of astronomy called helioseismology) can be studied closely for hundreds of years due to technological limitations. With the help of satellite telescopes deployed throughout the years (COROT, Kepler), we were able to detect these oscillations in stars beyond our solar systems, and turn our attention to the much larger sample space that is the Milky Way. 1.2 Incentive for Measuring Mass and Radius of Red Giants \Know your star, know your planet". One of the largest motivations for finding out mass and radius of a star is when the star hosts planets. Being able to accurately measure the mass and radius of the star allows us to know the planets' properties better. For stars whose planets transit the star (that is, its orbit is oriented such that the planet \eclipses" between us and the star), we can use the sudden drop in light intensity when the planet is in front of the stars, along with the star's radius, to measure the radius of the planet. This is, however, a rare orientation and most planets do not transit the star. The alternative method which do not require such orientation, the Radial Velocity Dopplershift method, requires knowledge of the mass of the star. The star is dragged by the mass of the star, causing light emitted from the star to experience Dopplershift. This shift can then be used to determine the planet's mass (with a sinusoidal factor corresponding to the inclination). However, neither radius nor mass are straightfoward to measure when stars are light years away and only consist of faint dots of light even to the most sensitive instruments. 2 Methodology 2.1 Traditional stellar property measurement The traditional method to measure mass and radius involves photometry and spectroscopy. However, both of these methods are influenced significantly by the effective temperature (or surface temperature, Teff) as well as the metallicity (such as amount of iron relative to hydrogen [Fe=H]), which have a relatively large uncertainty. From the width of spectral lines, the Teff as well as metallicity could be determined. Using parallax we can 1 obtain the distance to the star, and with apparent luminosity this gives us the absolute luminosity. Using Stefan-Boltzmann law, we can then calculate radius of the star. By using luminosity and Teff, the star could be placed on the Hertzprung-Russel diagram, and the mass can be obtained by fitting evolutionary models. Note that this process has a particularly large uncertainty when measuring the mass of red-giants, as main sequence stars' evolution paths converge as they develop into red-giants. 2.2 Asteroseismology Asteroseismology provides an alternative path to measuring properties of a star through two important quan- tities { the maximum oscillations power (νmax) and the large frequency separation (∆ν). By obtaining these two quantities from the light coming from stars (and with considerable effort), we can compute the radius and mass of a star to a much lower uncertainty compared to the traditional methods (Lundkvist et al. 2018). To understand what the two quantities mean, it is helpful to discuss the data we have, namely the light coming from the stars. The data used came from the TESS mission, a NASA mission with the aim to search for planets outside of the solar system, along with the use of the Python package lightkurve (Lightkurve Collaboration et al. 2018). Upon processing and cleansing, we obtain a light curve { a graph of flux (intensity) against time (days). By converting to fractional flux (dividing the flux by its mean), we obtain a graph such as in fig. 1a. From there we perform discrete fourier transformation (Lomb-Scargle periodogram) on the light curve, producing a power spectrum { a plot of frequency versus power, which decomposes the frequency components of the star's oscillations such as in fig. 1b. (a) Example of a light curve. This particular light curve (b) Power spectrum (logarithmic x axis) of the exam- contains 5 sectors of data ple light curve Figure 1: Light curve and Power Spectrum of HD120084, a red giant in the constellation of Ursa Minor. This red giant is discovered to have a planet with at least 4.5 times the mass of Jupiter in its orbit (Sato et al. 2013). The νmax value is the frequency (usually measured in µHz) at which the maximum power of oscillation is detected. The ∆ν value is the equal spacings between peaks of the same mode of oscillation. In fig.2, we can roughly estimate the νmax value to be around the l = 0 peak in the middle (but not exactly on that peak) at νmax ≈ 71 µHz, and the ∆ν ≈ 11 µHz. 2 Figure 2: Power spectrum (linear scale axis) zoomed into the region around νmax. The value of ∆ν is defined to be the interval between the l = 0 modes (or between the l = 1 or l = 2 modes themselves) The above gives an example of how to roughly estimate νmax and ∆ν from the graph by eye (which indeed has high uncertainty). There are more well-defined and accurate methods of measuring these quantities. 2.3 Measuring νmax using pySYD Measuring νmax accurately is not trivial, since it does not simply take the position at which the (often times) highest l = 0 mode peak occurs. I have used pySYD, the automated pipeline to extract oscillation parameters as described in Huber et al.(2009). Observe the power spectrum in fig. 1b, it can be noticed that the peaks around νmax (which we are interested in) lies on a large slope that increases to the left. This is due to the granulation background, where hot plasma forms convection cells on the surface and produce a periodic change in light intensity, resulting in frequency signals which increase in intensity as frequency decreases (Mathur et al. 2011). The pipeline performs background-correction by fitting and subtracting power law components (Harvey models) that remove the granulation background, as well as removing the white noise component. The pipeline smoothens the power spectrum, calculates the absolute autocorrelation function (ACF), and fits the collapsed ACF with a Gaussian distribution. The peak of this Gaussian curve is the measurement of νmax. For all stars processed, the following options were used to produce results: pysyd -verbose -mc 200 where the number of bootstrap sampling was set to 200 to estimate the uncertainty. 2.4 Measuring ∆ν using ´echelle diagrams Although the pySYD pipeline described in section 2.3 also produces results for ∆ν (and is a good estimate), the method of using ´echelle diagrams allow more direct and validatable measurements of ∆ν to be produced. An ´echelle diagram is a different form of the power spectrum. The power spectrum is divided up into equal partitions along the x axis (frequency), and then stacked on top of one another. The intensity of the frequency is usually transformed into a color scale, where darker corresponds to larger intensity. Since the peaks of the modes are equally spaced by ∆ν value, if the power spectrum was divided in partitions of length ∆ν, the dark parts of the graph which correspond to peaks in the power spectrum would line up in the ´echelle diagram. 3 Figure 3: Echelle diagram of HD222076. The pair of l = 0 and l = 2 modes are lined up nicely on the left, with a rough l = 1 modes to the right. The x axis shows the partition width to be 15:5 µHz, the likely ∆ν value of this star. So to obtain the ∆ν value we would need to line the dark regions of the Echelle diagram in (approximately) a straight line. The l = 1 are usually disregarded, since mode bumping can occur, producing multiple l = 1 lines on the ´echelle, or mode suppression could occur (which do happen in red giants). Looking at the l = 0 and l = 2 lines and lining them up by adjusting the partition width give quite an accurate measurement of ∆ν values, and looking at how much the width could be increased or decreased without affecting the alignment of the modes give an estimate of uncertainty in the ∆ν measurement.

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