Appendix

NON-TRANSITING HOT JUPITER HD187123B

D: Star-Only Correlation

3.8 Appendix

One- and Two-Dimensional Cross Correlations

When there is only one dominant spectral component in the data, the data can be described by the model

𝑓(𝑛) =𝑎𝑔(𝑛−𝑠) +𝑑_𝑛, (3.4) where𝑎is a scaling factor,𝑔(𝑛)is a template spectrum in the same reference frame as the data, 𝑠 is a wavelength shift, and 𝑑_𝑛 is the noise at bin 𝑛. In this case, a one-dimensional cross correlation function𝐶(𝑠) is sufficient to match the model to the data and can be computed as

𝐶(𝑠) = Í

𝑛 𝑓(𝑛)𝑔(𝑛−𝑠) 𝑁

q 𝜎2

𝑓

𝜎2 𝑔

, (3.5)

where 𝑓(𝑛) and 𝑔(𝑛) are the target and template spectra, respectively, and the variances of the target (𝜎_𝑓) and the template (𝜎_𝑔) are given by

𝜎²

𝑖 = 1

𝑁 Õ

𝑛

𝑖²(𝑛). (3.6)

When there is more than one spectral component in the data, however, as is the case in the multi-epoch technique, the model described by Equation 3.4 can no longer accurately describe the data. Rather, a model considering two components is necessary,

𝑓(𝑛) =𝑎[𝑔

1(𝑛−𝑠

1) +𝛼𝑔

2(𝑛−𝑠

2)] +𝑑_𝑛. (3.7)

As above,𝑎is a scaling factor and𝑑_𝑛is the noise at bin𝑛. The two spectral templates are given by𝑔

1and𝑔

2with wavelength shifts of𝑠

1and𝑠

2, respectively. The scaling factor𝛼accounts for the intensity ratio between the two template models. For this work, we set 𝛼 equal to 0.0014, which is the spectroscopic contrast given by our stellar and planetary models and assuming a planetary radius of 1 𝑅_𝐽. We have found, however, that the shape of the resulting log likelihood surfaces, from both data and simulations, is independent of𝛼in the range of 1.4×10⁻³ to 10⁻⁹. This is consistent with what was seen by Lockwood et al. 2014 and Piskorz et al. 2016.

Zucker & Mazeh 1994 showed that a 2D CC𝑅(𝑠

1, 𝑠

2, 𝛼)could be calculated as 𝑅(𝑠

1, 𝑠

2, 𝛼) = Í

𝑛 𝑓(𝑛) [𝑔

1(𝑛−𝑠

1) +𝛼𝑔

2(𝑛−𝑠

2)]

𝑁 𝜎_𝑓𝜎_𝑔(𝑠

1, 𝑠

2) , (3.8)

where𝜎_𝑓 is the same as described above, but𝜎_𝑔(𝑠

1, 𝑠

2)can now be calculated as 𝜎_𝑔 =

q 𝜎²

𝑔1+2𝛼𝜎_𝑔

1𝜎_𝑔

2𝐶

12(𝑠

2−𝑠

1) +𝛼2𝜎²

𝑔2. (3.9)

𝐶12is the correlation between the two templates.

In all of the CC-to-log(L) approaches described below, we combine 2D CCs rather than 1D CCs. This involves replacing (𝐶(𝑠)) with (𝑅(𝑠

1, 𝑠

2, 𝛼)) and using 𝜎_𝑔 calculated by Equation 3.9 rather than by Equation 3.6.

Once we have calculated the 2D log(L) surface for each epoch, we reduce to the one-dimensional log likelihood functions (e.g., as seen in Figure 3.3) by taking a cut along the maximum stellar velocity, which we check matches the expected stellar velocity from the combined systemic and barycentric velocities.

Zucker (2003) log(L) Approach

First, all correlations from a single night (segments from all orders after the saturated tellurics are removed) are combined using the approach from Zucker (2003). This considers the observed spectrum 𝑓(𝑛)and a model𝑔(𝑛)with a scaling factor (𝑎), a shift (𝑠), and random white Gaussian noise (𝜎). Expressions for𝑎,𝜎, and𝑠can be found that maximize the log(L) between the observed spectrum and the model(s).

By substituting these expressions in to log(L) equation, Zucker 2003 showed that cross correlations can be related to log likelihoods (log(L)) as

log(𝐿) =−𝑁

2 log(1−𝑅²). (3.10)

The individual cross correlations are converted to log likelihoods and summed for each epoch. The fact that the cross correlation𝑅is squared in this operation means that a negative correlation would provide the same log likelihood as a positive cor- relation. In other words, a model would give the same log likelihood when fit to the data at a given velocity whether it were multiplied by 1 or -1. This could be concerning because, while planetary absorption and emission lines are not merely related by a sign-flip, correlation between an absorption line in the data with an emission line in the model, or vice versa, would produce an anticorrelation, which, if the baseline correlation were at zero, would be given the same likelihood as a corresponding positive correlation by Equation 3.10. The pressure/temperature pro- file of a planet’s atmosphere, whether inverted or non-inverted, determines whether lines will show up in absorption vs. emission, and so not being able to distinguish between the two cases would severely limit our ability to understand atmospheres.

In our multi-epoch data sets, however, stellar lines are the dominant component and the real planetary signal must correspond with the correct stellar velocity. In

other words, we can only detect the planetary signal once the model and data stellar lines are matched up. Therefore, the variation in the planetary correlation is around the mean stellar correlation peak, which is well above zero. Because the planetary correlation values will never reach down to, or below, zero, anticorrelation between the planetary lines in the data and model will be distinguishable from correlation because it will result in smaller (i.e., below the stellar correlation baseline), but still positive, correlation values. We demonstrate the technique’s ability to distinguish between inverted and non-inverted planetary atmospheres in Section 6.4.

As a further step in processing the correlations, we correct any negative correlation values to zero. These negative correlation values correspond to incorrect stellar velocities, and so correcting them will not affect the planetary curves. This negative correlation correction is done by calculating log(𝐿) as

𝑦_𝑖(𝑠) = (

𝑁_𝑖log(1−𝑅_𝑖(𝑠)²) 𝑅_𝑖(𝑠) ≥0 𝑁_𝑖log(1+𝑅_𝑖(𝑠)²) 𝑅_𝑖(𝑠) < 0

)

log(𝐿(𝑠)) = ( −¹

2

Í

𝑖𝑦_𝑖(𝑠) Í

𝑖𝑦_𝑖(𝑠) < 0

0 Í

𝑖𝑦_𝑖(𝑠) ≥0 )

.

(3.11)

Applying this correction after summing the𝑦_𝑖functions, rather than for each negative 𝑅_𝑖, accounts only for heavily weighted negative correlations. That is, we do not set negative values in the individual 𝑅_𝑖 functions equal to zero before combining them because we wish to retain the information from negative 𝑅_𝑖 functions that arise from noise or uncertainty in the spectra. By waiting until the 𝑦_𝑖 functions are combined to make this cut, we avoid automatically losing both small negative values in the𝑅_𝑖 functions or negative values in an𝑅_𝑖 function that have very small relative weighting (𝑁_𝑖). This correction creates the horizontal portions at zero of the stellar log likelihood curve in Panel A of Figure 3.3. This method of correcting negative correlations has been used in previous multi-epoch analyses (e.g., Piskorz et al., 2016, 2017, 2018), and we describe it here for transparency.

We want to stress that negative correlations should not be corrected when using a one-dimensional cross correlation or when the two spectral components in a two- dimensional cross correlation are of similar strength. Doing so would artificially alter the distribution of likelihood values which would invalidate the uncertainties given by the resulting likelihood surface.

Then, the log(L) from different nights of data are converted from 𝑣_{𝑠𝑒 𝑐} to𝐾_𝑝 space according to Equation 3.2. Finally, the log likelihoods are summed to find the most likely𝐾_𝑝.

Zucker (2003) ML Approach

The Zucker ML method follows the Zucker log(L) method up to Equation 3.10.

However, rather than combining the likelihoods at this point, Zucker 2003 shows that individual correlations can be combined into an “effective” correlation value, ML, as follows:

𝑁_{𝑡 𝑜𝑡}log[1−ML²(𝑠)] =Õ

𝑖

𝑁_𝑖log[1−𝑅²

𝑖(𝑠)], (3.12) where the right side is the sum of the log(L)’s of individual segments and the left side is the log(L) of the full data set (from a single night where the planetary velocity is constant). The 𝑅_𝑖’s and 𝑁_𝑖’s are the 2D cross correlations and number of pixels of each of the segments, respectively, and 𝑁_{𝑡 𝑜𝑡} is the total number of pixels. By analogy, ML is the effective correlation of the full data set. Because ML is an effective correlation, we rename it𝑅(𝑠)and evaluate it as,

𝑅(𝑠) = s

1−exp 1

𝑁_{𝑡 𝑜𝑡} Õ

𝑖

𝑁_𝑖log[1−𝑅²

𝑖(𝑠)]

. (3.13)

This gives us an effective correlation for each epoch. We correct for negative correlation values here in an analogous fashion to that described for the Zucker log(L) approach. The effective cross correlations can then be converted to log(L) following Lockwood et al. 2014:

log(𝐿) =const+𝑅(𝑠). (3.14) Finally, the log(L)’s from different nights are converted from𝑣_{𝑠𝑒 𝑐}to𝐾_𝑝space, as in the other approaches, and summed.

This was the CC-to-log(L) approach used in the previous NIRSPEC multi-epoch detection papers (Lockwood et al., 2014; Piskorz et al., 2016, 2017, 2018).

Brogi & Line (2019) Approach

Brogi & Line (2019) recently presented a new approach to converting cross cor- relations to log(L). Instead of substituting the expression for 𝑎 that maximizes the log(L) between an observed spectrum and a model, they set𝑎 equal to 1. Setting𝑎 to 1 allows for discrimination between correlation and anticorrelation, or between emission and absorption lines, in a 1D cross correlation routine.

By setting𝑎 =1, Brogi & Line (2019) derive the expression log(𝐿) =−𝑁

2

log(𝜎_𝑓𝜎_𝑔) +log 𝜎_𝑓

𝜎_𝑔 +

𝜎_𝑔 𝜎_𝑓

−2𝑅(𝑠) . (3.15)

We stress that since our approach uses two-dimensional cross correlations, in ap- plying this conversion to our data, 𝑅(𝑠) and 𝜎_𝑔 are the two-dimensional variants described in Equations 3.8 and 3.9, rather than the one-dimensional 𝐶(𝑠) and 𝜎_𝑔 described in Equations 3.5 and 3.6.

We note too, as above, that in our 2D case, where there are both stellar and planetary signals in the data, a negative 𝑎 would invert the stellar absorption lines as well as the planetary lines. Our multi-epoch data have high enough S/N on the stellar lines that flipping the stellar model would produce a strong anticorrelation, which would be corrected to zero as described above. Therefore, while allowing𝑎to vary in 1D CC routines on low S/N planetary data could certainly present challenges, 2D routines on data with high S/N stellar features would not run into the same obstacles as negative 𝑎 values would be ruled out by the stellar spectrum. Thus, even without setting𝑎 to 1, the Zucker methods would not confuse planetary (and stellar) emission and absorption lines.

As in the Zucker 2003 approach, the log(L) functions from a single night are summed, then the summed log(L) for each night is converted from𝑣_{𝑠𝑒 𝑐}to𝐾_𝑝space and summed.

References

Alonso-Floriano, F. J., Sánchez-López, A., Snellen, I. A. G., et al. 2019,A&A, 621, A74, doi:10.1051/0004-6361/201834339

Benneke, B. 2015, ArXiv e-prints. https://arxiv.org/abs/1504.07655 Birkby, J. L., de Kok, R. J., Brogi, M., Schwarz, H., & Snellen, I. A. G. 2017,AJ,

153, 138, doi:10.3847/1538-3881/aa5c87

Boogert, A. C. A., Blake, G. A., & Tielens, A. G. G. M. 2002, ApJ, 577, 271, doi:10.1086/342176

Borysow, A. 2002,A&A, 390, 779, doi:10.1051/0004-6361:20020555

Brogi, M., de Kok, R. J., Albrecht, S., et al. 2016, ApJ, 817, 106, doi: 10.3847/

0004-637X/817/2/106

Brogi, M., de Kok, R. J., Birkby, J. L., Schwarz, H., & Snellen, I. A. G. 2014,A&A, 565, A124, doi:10.1051/0004-6361/201423537

Brogi, M., Giacobbe, P., Guilluy, G., et al. 2018, A&A, 615, A16, doi:10.1051/

0004-6361/201732189

Brogi, M., & Line, M. R. 2019,AJ, 157, 114, doi:10.3847/1538-3881/aaffd3

Brogi, M., Snellen, I. A. G., de Kok, R. J., et al. 2012, Nature, 486, 502, doi:10.

1038/nature11161

—. 2013,ApJ, 767, 27, doi:10.1088/0004-637X/767/1/27

Burrows, A., & Volobuyev, M. 2003,ApJ, 583, 985, doi:10.1086/345412 Butler, R. P., Marcy, G. W., Vogt, S. S., & Apps, K. 1998, PASP, 110, 1389,

doi:10.1086/316287

Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, 2246

Esteves, L. J., de Mooij, E. J. W., Jayawardhana, R., Watson, C., & de Kok, R. 2017, AJ, 153, 268, doi:10.3847/1538-3881/aa7133

Feng, Y. K., Wright, J. T., Nelson, B., et al. 2015,ApJ, 800, 22, doi:10.1088/0004- 637X/800/1/22

Fulton, B. J., Petigura, E. A., Blunt, S., & Sinukoff, E. 2018, PASP, 130, 044504, doi:10.1088/1538-3873/aaaaa8

Furtenbacher, T., Császár, A. G., & Tennyson, J. 2007, Journal of Molecular Spec- troscopy, 245, 115, doi:10.1016/j.jms.2007.07.005

Guilluy, G., Sozzetti, A., Brogi, M., et al. 2019,A&A, 625, A107, doi:10.1051/

0004-6361/201834615

Hawker, G. A., Madhusudhan, N., Cabot, S. H. C., & Gandhi, S. 2018,ApJL, 863, L11, doi:10.3847/2041-8213/aac49d

Husser, T. O., Wende-von Berg, S., Dreizler, S., et al. 2013,A&A, 553, A6, doi:10.

1051/0004-6361/201219058

Kausch, W., Noll, S., Smette, A., et al. 2014, in Astronomical Society of the Pacific Conference Series, Vol. 485, Astronomical Data Analysis Software and Systems XXIII, ed. N. Manset & P. Forshay, 403

Konopacky, Q. M., Barman, T. S., Macintosh, B. A., & Marois, C. 2013, Science, 339, 1398, doi:10.1126/science.1232003

Lockwood, A. C., Johnson, J. A., Bender, C. F., et al. 2014, ApJL, 783, L29, doi:10.1088/2041-8205/783/2/L29

Madhusudhan, N., Knutson, H., Fortney, J. J., & Barman, T. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 739 Martin, E. C., Fitzgerald, M. P., McLean, I. S., et al. 2018, in Society of

Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10702, Ground-based and Airborne Instrumentation for Astronomy VII, 107020A

Martins, J. H. C., Santos, N. C., Figueira, P., et al. 2015, VizieR Online Data Catalog, 357

McLean, I. S., Becklin, E. E., Bendiksen, O., et al. 1998, inProc. SPIE, Vol. 3354, Infrared Astronomical Instrumentation, ed. A. M. Fowler, 566–578

Nugroho, S. K., Kawahara, H., Masuda, K., et al. 2017, AJ, 154, 221, doi: 10.

3847/1538-3881/aa9433

Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, ApJL, 743, L16, doi: 10.

1088/2041-8205/743/1/L16

Piskorz, D., Benneke, B., Crockett, N. R., et al. 2016,ApJ, 832, 131, doi:10.3847/

0004-637X/832/2/131

—. 2017,AJ, 154, 78, doi:10.3847/1538-3881/aa7dd8

Piskorz, D., Buzard, C., Line, M. R., et al. 2018,AJ, 156, 133, doi:10.3847/1538- 3881/aad781

Piskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Jeffery, C. S. 1995, A&AS, 112, 525

Rothman, L. S., Gordon, I. E., Barbe, A., et al. 2009,J. Quant. Spec. Radiat. Transf., 110, 533, doi:10.1016/j.jqsrt.2009.02.013

Schwarz, H., Ginski, C., de Kok, R. J., et al. 2016,A&A, 593, A74, doi:10.1051/

0004-6361/201628908

Snellen, I. A. G., Brandl, B. R., de Kok, R. J., et al. 2014, Nature, 509, 63, doi:10.1038/nature13253

Snellen, I. A. G., de Kok, R. J., de Mooij, E. J. W., & Albrecht, S. 2010, Nature, 465, 1049, doi:10.1038/nature09111

Soubiran, C., Jasniewicz, G., Chemin, L., et al. 2013, A&A, 552, A64, doi: 10.

1051/0004-6361/201220927

Takeda, G., Ford, E. B., Sills, A., et al. 2007,ApJS, 168, 297, doi:10.1086/509763 Tennyson, J., & Yurchenko, S. N. 2012,MNRAS, 425, 21, doi:10.1111/j.1365-

2966.2012.21440.x

Valenti, J. A., & Fischer, D. A. 2005,ApJS, 159, 141, doi:10.1086/430500 Zhang, J., Kempton, E. M. R., & Rauscher, E. 2017,ApJ, 851, 84, doi:10.3847/

1538-4357/aa9891

Zucker, S. 2003,MNRAS, 342, 1291, doi:10.1046/j.1365-8711.2003.06633.x Zucker, S., & Mazeh, T. 1994,ApJ, 420, 806, doi:10.1086/173605

C h a p t e r 4

PRIMARY VELOCITY AND ORBITAL PHASE EFFECTS ON