Inference on Radii and of Red Giants that host from Solar-like Oscillations

Yifan Chen

Abstract Asteroseismology is the study of natural oscillations in . By observing a ’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 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 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 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 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 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 (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 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 in the of . 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.

The Python library echelle developed by Hey & Ball(2020) was used to produce ´echelle diagrams. The interactive Echelle diagram allowed convenient adjustment of partition width and lining up of the l = 0 and l = 2 modes.

2.5 Scaling Relations for the νmax and the ∆ν The two most important relations we are interested in described in Lundkvist et al.(2018) are r M ∆ν ∝ (1) R3 M ν ∝ √ (2) max 2 R Teff which are results that arise through analysis of the Which allows us to derive the following expressions for mass and radius M  ν 3 ∆ν −4 T 1.5 =∼ max eff (3) M νmax, ∆ν Teff, R  ν  ∆ν −2 T 0.5 =∼ max eff (4) R νmax, ∆ν Teff, Note that this is an approximate relation, with intrinsic scatter of about 1.4% for eq. (4) and 0.4% for eq. (3) as determined by Li et al.(2021).

2.6 Source and specifications of data The data used came from the TESS mission, and in particular those preprocessed by the Science Processing Operations Center (SPOC) with exposure time of 120 seconds, where the exposure time is the time interval at

4 which data is sampled at. This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al. 2018), to retrieve the data programatically. Since we are taking the Fourier Transform, it is useful to allow for a sufficiently large Nyquist frequency in order to guarantee that the νmax is below it. The TESS mission observes parts of the sky in sectors of about 27 days, so data of the same star from multiple sectors (if available) are combined by first individually taking the fractional flux (dividing mean throughout), and then concatenating to form a larger dataset.

3 Results and Discussion

This section highlights some of the stars listed in Wittenmyer et al.(2020), which were confirmed to be planet- hosting red giants. By following the method described above, we are able to calculate the stellar properties of various stars. The Teff values used in the calculations of mass and radius were obained from the source papers in order to have a more valid comparison between the published values in the source papers.

Star νmax(µHz) ∆ν(µHz) Teff(K) R M Source paper HD11343 71.8 ± 0.2 7.05 ± 0.03 4670 ± 100 7.677 ± 0.217 1.23 ± 0.029 Jones, M. I. et al.(2016) HD33844 165.4 ± 1.0 12.7 ± 0.1 4861 ± 100 5.598 ± 0.156 1.54 ± 0.036 Wittenmyer et al.(2016) HD47205 180.9 ± 1.8 13.65 ± 0.1 4792 ± 100 5.219 ± 0.164 1.453 ± 0.036 Luque, R. et al.(2019) HD121056 132.0 ± 2.9 11.1 ± 0.1 4805 ± 100 5.742 ± 0.471 1.285 ± 0.074 Wittenmyer et al.(2015) HD135760 197.2 ± 3.6 14.6 ± 0.05 4850 ± 100 5.01 ± 0.206 1.469 ± 0.043 Jones, M. I. et al.(2016) HD222076 204.4 ± 0.9 15.5 ± 0.1 4806 ± 100 4.59 ± 0.126 1.272 ± 0.029 Wittenmyer et al.(2017)

Table 1: The maximum oscillation power (νmax) and large frequency power (∆ν) of planet-hosting red-giants, the Teff as indicated in the source paper, and the calculated (R ) and (M )

5 Figure 4: Radius and Mass of HD11343, HD33844, HD47205, HD121056, HD135760, and HD222076, determined by asteroseismology (blue) compared with published values (black).

From fig.4 we can deduce that in general asteroseismology measurements give results for mass and radius that are consistent with the published work. In particular, compared to the published values (which do not involve asteroseismology), the asteroseismological method gives a much lower uncertainty for both radius and mass. Note that the initial source paper for HD47205 (Wittenmyer et al. 2011) reported a radius of 2.3 ± 0.1, which deviated substantially from the asteroseismology value as well as other published values and is considered an error (thus omitted from fig.4. The exceptions are for radius of HD47205 and HD121056, which have a slightly larger uncertainty than the published values (+0.12, −0.11 in Luque, R. et al.(2019) for HD47205 and ±0.29 in Wittenmyer et al.(2015) for HD121056).

For HD47205, the percentage uncertainty for radius is around 3%, which is acceptable and very similar to the uncertainty indicated by Luque, R. et al.(2019). A potential major reason the asteroseismology uncertainties were relatively high for HD121056 and HD135760 may be due to the fact that both had only 1 sector of data available. It’s possible that there is simply not enough data to produce a power spectrum that represents its

6 actual oscillations, and hence the mass and radius calculated were of high uncertainty. Since the light curve data was needed to have an exposure time of 120 s, limitations in quantity of data is a problem since not all stars have been observed by TESS with such short exposure time for multiple sectors. This may be a potential issue when trying to apply the asterosesimology method to a large amount of stars, as only a limited amount of stars have data of such quality.

4 Conclusion and Suggestions for Future Work

Using asteroseismology to analyze the light curves of planet-hosting red giants and comparing the calculated mass and radius to published results, we were able to obtain results that largely agreed with existing literature, while obtaining lower uncertainties compared to non-asteroseismology methods of measurement. This gives us confidence in the results obtained using asteroseismology, and suggests that the asteroseismology method has a higher accuracy in determining stellar radius and mass due to eliminations of large influencing factors.

This work demonstrated the ability to use asteroseismology to measure stellar properties on a relatively small sample of planet-hosting red giants. Future works may include the mass application of such method on a larger sample of stars, to reveal potential biases in the methodology that could not be demonstrated for a small samples size. We may also expect that stellar properties determined by asteroseismology to be used to give more accurate measures of exoplanets and help shape the study of exoplanets. Applying this method on stars other than red giants may also reveal whether the method works better or worse for particular types of stars compared to other methods.

5 Acknowledgement and Reflection

This project was conducted under the supervision of Professor Tim Bedding of The University of Sydney. He has provided great help in terms of both theoretical knowledge as well as computational procedures.

The project was aimed at using the pySYD pipeline on stars and understand its procedures in order to test its functionality. Upon the code being functional (despite it still being actively developed), 6 stars that were confirmed to host planets from Wittenmyer et al.(2020) were chosen to be processed by the pipeline and produce results – with great success. Echelle´ diagrams and the formulae to obtain mass and radius from asteroseismology quantities were also used to produce results for the stellar properties, which were then compared to published values in the source papers.

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