This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Andrew Ridden-Harper, Department of Astronomy and Carl Sagan Institute, Cornell University & Las Cumbres Observatory;
(2) Stevanus K. Nugroho, Astrobiology Center & Japan & National Astronomical Observatory of Japan;
(3) Laura Flagg, Department of Astronomy and Carl Sagan Institute, Cornell University;
(4) Ray Jayawardhana, Department of Astronomy, Cornell University;
(5) Jake D. Turner, Department of Astronomy and Carl Sagan Institute, Cornell University & NHFP Sagan Fellow;
(6) Ernst de Mooij, Astrophysics Research Centre, School of Mathematics and Physics & Queen’s University Belfast;
(7) Ryan MacDonald, Department of Astronomy and Carl Sagan Institute;
(8) Emily Deibert, David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto & Gemini Observatory, NSF’s NOIRLab;
(9) Motohide Tamura, Dunlap Institute for Astronomy & Astrophysics, University of Toronto
(10) Takayuki Kotani, Department of Astronomy, Graduate School of Science, The University of Tokyo, Astrobiology Center & National Astronomical Observatory of Japan;
(11) Teruyuki Hirano, Astrobiology Center, National Astronomical Observatory of Japan & Department of Astronomical Science, The Graduate University for Advanced Studies;
(12) Masayuki Kuzuhara, Las Cumbres Observatory & Astrobiology Center;
(13) Masashi Omiya, Las Cumbres Observatory & Astrobiology Center;
(14) Nobuhiko Kusakabe, Las Cumbres Observatory & Astrobiology Center.
Table of Links
- Abstract & Introduction
- Observation
- Data Reduction and Processing
- Model Spectra
- Cross-Correlation Method
- Results
- Clouds and Hazes
- Implications for JWST
- Conclusion and Acknowledgement
- Appendix & References
5. CROSS-CORRELATION METHOD
To search for GJ 486b’s atmosphere, we crosscorrelated the models described in Section 4 with the processed data from Section 3. We cross-correlated each model with one processed frame at a time, to produce a cross-correlation function (CCF). We phase-folded these CCFs by shifting them into the planet rest frame. GJ 486b’s radial velocity as a function of time, relative to its host star, is given by
To optimally combine our three data sets, we mapped our cross-correlation functions to log-likelihoods (following, e.g., Brogi & Line 2019; Gibson et al. 2020; Herman et al. 2022). This approach allows the log-likelihoods from each data set to be trivially summed.
The log-likelihood for each data set, ln LD, is calculated according to
where the D subscript refers to the data set (e.g., IRD, IGRINS, and SPIRou), N is the total number of intransit data points in the data set, and χ 2 is related to the cross-correlation function, CCF, via
where fi is the mean-subtracted spectrum, σi is the outer product of the standard deviation of each wavelength and exposure bin (normalized by the standard deviation of the spectra in each order), mi is the meansubtracted Doppler-shifted model, α is a scaling factor to allow uncertainty in the scale of the model, and the CCF is defined as
We determined which frames were in transit by generating model transit light curves with the BAsic Transit Model cAlculatioN in Python (BATMAN[8] ) package (Kreidberg 2015), using the system parameters reported by Trifonov et al. (2021), and weighted all in-transit frames equally. We converted the summed log-likelihood to a normalized likelihood, Lnorm, according to Lnorm = exp(lnL − max(lnL)).
[8] https://github.com/lkreidberg/batman