Instrumental Considerations Wavelength RegionWavelength Region

CHOICES ALONG THE MULTI-EPOCH ANALYTIC PATHWAY

6.3 Instrumental Considerations Wavelength RegionWavelength Region

6.3 Instrumental Considerations

Moving beyond NIRSPEC, Finnerty et al. 2021 showed that doubling the spectral grasp of an observation nearly doubles the planetary detection strength. This implies that instruments like IGRINS (simultaneous 𝐻 𝐾 coverage) or SPIRou (simultane- ous 𝐽 𝐻 𝐾 coverage) could offer significant improvements to multi-epoch detection strengths from NIRSPEC (fraction of a band at once), even on smaller, less sta- ble telescopes. As described in Section 6.3, our current NIRSPEC observations are dominated by structured noise rather than shot noise. A smaller telescope would add to the shot noise, but broader coverage would likely decrease the structured noise.

The structured noise that we have focused on comes from correlation between the stellar signal in the data and the planetary template. With broader coverage, more of both spectra are seen, and because the two spectra are different, the correlation between the two, and the resulting structured noise, will decrease. Thus, where broad simultaneous wavelength coverage is possible, it should be taken, and where not, simulations should be used to direct the choice of wavelength range.

Spectral Resolution

Instrumental spectral resolution is a crucial factor in any spectroscopic study of a planetary atmosphere. At high resolution, spectral lines do not blend, so we can measure the true depth, width, and plurality of lines (e.g., see Figure 2 in Birkby, 2018), all of which are dependent on three-dimensional atmospheric struc- ture, including rotation and winds. Further, each IR-active molecule has a distinct rovibrational spectrum across the near-infrared wavelengths. By fully resolving the near-IR spectrum, we can elucidate the distinct signatures of each molecule, and be confident in the detection/non-detection of each molecular species.

The multi-epoch approach has been applied to data in the 𝑅 ≈ 25,000−40,000 range and the single-night, or CRIRES-style, approach has been applied to data from 𝑅 ≈ 50,000 (with GIANO, e.g., Guilluy et al., 2019) to 100,000 (with CRIRES, e.g., Brogi et al., 2012). Recall that the main difference between these approaches is that the multi-epoch version utilizes observations during which the planetary signal does not shift across detector pixels, while the CRIRES-style version wants the planetary signal to move. Whether the planetary signal crosses detector pixels during an observation depends on the instrumental resolution and the change in planetary line-of-sight motion over the observation (as described in Section 6.2).

The multi-epoch technique requires the change in planetary line-of-sight velocity be smaller than the velocity resolution of the instrument, and the CRIRES-style technique requires it to be larger. In fact, the CRIRES-style technique often requires

the planetary signal to cross multiple pixels for principal component analysis based techniques to cleanly separate the planetary signal from the telluric and stellar spectral components. As an example, the signal from Tau Boo b crossed around 15 SPIRou pixels in a 5-hour observation (Pelletier et al., 2021). In mathematical terms, the multi-epoch approach aims for

Δ𝑣_{𝑠𝑒 𝑐} < 1 pixel, (6.4)

while the CRIRES-style approach requires

Δ𝑣_{𝑠𝑒 𝑐} & 15 pixels. (6.5)

We can estimate the size of a detector resolution element as𝑐/𝑅, where𝑐is the speed of light. The number of pixels per resolution element, which we will call𝑏, varies by instrument, but is typically around 2−3 for these near-infrared high-resolution spectrometers. Then, plugging in Equation 6.2 for the change in planetary line-of- sight velocity, we can relate the number of pixels we want the planetary signal to cross (𝑛_𝑝𝑖𝑥) to the necessary resolution (𝑅), given parameters of the system (𝐾_𝑝, 𝑃_ℎ𝑟) and observation (ℎ_{𝑜 𝑏 𝑠}, 𝑀^{𝑐𝑒𝑛𝑡}

𝑜 𝑏 𝑠 ).

𝑅= 𝑛_𝑝𝑖𝑥𝑐 Δ𝑣_{𝑠𝑒 𝑐}𝑏

=

𝑛_𝑝𝑖𝑥𝑐

2𝐾_𝑝sin(𝜋 ℎ_{𝑜 𝑏 𝑠}/𝑃_ℎ𝑟)cos(2𝜋 𝑀^{𝑐𝑒𝑛𝑡}

𝑜 𝑏 𝑠 )𝑏

. (6.6)

The multi-epoch approach works best when𝑅

instr < 𝑅(𝑛_𝑝𝑖𝑥 =1), and the CRIRES approach works best when𝑅

instr & 𝑅(𝑛_𝑝𝑖𝑥 =15).

For a 5-hour observation of a typical hot Jupiter, such as the example we describe above, with an orbital period of 3 days and a 𝐾_𝑝 of 100 km/s, either technique could be applied. For the planet to cross 15 pixels (𝑏 =3 pixels/resolution element) when it is near quadrature, the instrument would need a resolution of nearly 3×10⁵. With the current maximum resolution of ground based near-infrared instruments of

∼1×10⁵, the multi-epoch approach would be much more capable of detecting this planet near quadrature. On the other hand, near conjunction, an instrument would need a resolution below about 2.3×10³ to prevent the planet from crossing any detector resolution elements. The CRIRES-style approach would be better suited for near conjunction epochs.

This does imply, though, that not all orbital positions for hot Jupiter-like planets could be appropriately studied via the multi-epoch technique. Epochs near quadrature would need to be broken apart and considered separate epochs. If not, significant

error could arise from the uncertainty in orbital position (see Section 6.4). In the case of the HD 88133 b multi-epoch data set published by Piskorz et al. 2016, if HD 88133 b actually had a 𝐾_𝑝 of 40 km/s (see Buzard et al. 2021b), over the course of the six NIRSPEC1.0 observations, the planet would cross roughly 0.3 to 1.2 resolution elements. If it had the maximum 𝐾_𝑝 of 153 km/s (calculated with parameters from Luhn et al. 2019), which would correspond to an edge-on orbit, the planet would cross 0.8 to 4.5 resolution elements in the six observations.

In this case, the planetary signal would cross pixels in all six epochs. Analyzing these epochs with the multi-epoch assumption that the planetary signal would not shift may have significantly reduced the planetary signal in two ways: by averaging it across multiple detector resolution elements and by allowing the time-variable signal to be picked up and removed by the PCA-based telluric correction routine.

In short, while𝐾_𝑝 is often not known ahead of time, researchers should take care to ensure that the planetary signal truly does not shift before analyzing the data as such.

As we move to planet populations on longer orbits, though, the CRIRES-style approach will struggle. For instance, for an Earth twin, with an orbital period of 365.25 days and a𝐾_𝑝of 30 km/s, in an full night observation (8 hours), even near conjunction, an instrument would need a resolution of 5.8×10⁵to allow the planet signal to cross even one pixel; 8.7×10⁶for the planet to cross 15 pixels (comparable to what has been used in CRIRES-style detections to date, e.g., Birkby et al. 2017;

Pelletier et al. 2021), more than 80× larger than the instrumental resolution of current near-infrared spectrometers. For these planet populations, the multi-epoch approach is the only currently available high-resolution technique.

Figure 6.1 illustrates the number of pixels on a 𝑅 =100,000 spectrometer (𝑏 = 3 pixels/resolution element) that a signal from each of the known planets (as of October 20, 2021, exoplanets.org) would cross during an 8-hour observation centered at inferior conjunction. These estimates assume that the planets transit and that they are on circular orbits. It is important to remember, though, that most planets do not transit. Since𝐾_𝑝depends on orbital inclination, a non-transiting planet would have a smaller value of𝐾_𝑝 than would a comparable mass transiting planet, resulting in the planetary signal that crosses fewer pixels than predicted by Figure 6.1. Since the value of 𝐾_𝑝 is rarely known for non-transiting planets, and in fact is typically the parameter being measured, it would be challenging to determine before observing whether a planet would cross enough pixels for the CRIRES-style technique to

10 ² 10 ¹ 10⁰ 10¹ 10²

Semi-Major Axis [AU]

10 ⁴ 10 ³ 10 ² 10 ¹ 10⁰ 10¹

Pla ne t M as s [ M

Jup

]

0 5 10 15 20 25

Number of Pixels Crossed

Figure 6.1: Known exoplanets from Exoplanets Data Explorer (www.exoplanets.org; Han et al. 2014) as of October 20, 2021 plotted in terms of their planet mass and semi-major axis. The colors represent the number of pixels on a 𝑅 = 100,000 (3 pixels/resolution element) instrument each planet would cross during an 8-hour observation centered at conjunction. These estimates assume that the planets transit and that they are on circular orbits. Non-transiting systems would cross fewer pixels than estimated here, and very eccentric orbits may cross more pixels.

be effective. Very eccentric orbits, on the other hand, may cross more pixels than predicted here, but high-eccentricity orbits are much less common that non- transiting orbits. From this figure, we can see that with a 𝑅 =100,000 instrument, the CRIRES-style approach is really only applicable to planets within ∼ 0.1 AU.

The multi-epoch approach, on the other hand, with an effective telluric correction procedure and a sufficiently large number of epochs, could work for any of these systems.

While the𝑅 ∼25,000−100,000 near-infrared spectrometers currently being used for exoplanet characterization are both applicable to hot Jupiters at different parts of the planets’ orbits, as we aim to study planets on longer orbital periods and with smaller Keplerian orbital velocities, only the multi-epoch approach will suffice.

Length of Observation, e.g., Signal-to-White Noise per Observation

The length of an observation will affect the total change in planetary line-of-sight velocity seen, as described in Sections 6.2 and 6.3, the level of total shot noise, and the effectiveness of principal component analysis telluric correction procedures.

Planetary signals should increase proportionally to𝑡_{𝑒𝑥 𝑝} and shot noise increases as

√

𝑡_{𝑒𝑥 𝑝}, so the signal-to-shot noise should increase as √

𝑡_{𝑒𝑥 𝑝} as well. Through the simulations presented throughout this thesis, however, we have seen that in a small (. 10) epoch limit, structured noise arises in the cross correlation space that far outweighs the shot noise. Buzard et al. 2020 analyzed seven 𝐿 band NIRSPEC epochs on HD 187123 b, with a total signal-to-shot noise of 5874 taken over 8.9 hours. We found, both with simulations and with the data, that the signal-to-shot noise could have been reduced to 3968 overall without a significant change in the shape of the final log likelihood result. Below this level, the log likelihood curve began to change shape, implying that around a signal-to-shot noise around 4000 was the limit where shot noise contributions became comparable to structured noise. Reducing to the total signal-to-shot noise could have saved a factor of 2.2 in exposure time. Since we have also seen that many, lower S/N epochs are better at reducing structured noise than fewer, higher S/N epochs, it may be advantageous to take shorter observations rather than waste time reducing the total shot noise level when it is already far below the level of structured noise.

Piskorz et al. 2016 introduced a principal component analysis (PCA) approach for correcting telluric contamination from multi-epoch data. We will expand more on this approach in Section 6.4, but in short, it assumed the telluric atmosphere changes over the course of the observation and used PCA to remove these time- varying components from the dataset. With a longer baseline spent on a target, the telluric atmosphere would be able to vary more and so be easier for PCA to identify and remove. The time relevant to PCA is the total time spent on each target, or the full time from the beginning to the end of the observation. The exposure time relevant to the shot noise level described above is just a portion of this total time, excluding time spent between exposures or reading out frames, for instance. If a different telluric correction approach could be used that did not require the same baseline, shorter observations would be beneficial in both reducing the change in planetary line-of-sight velocity and permitting the observation of more targets per night while still remaining in the regime whether structured noise outweighs shot noise.

6.4 Analytic Considerations