Measuring the Obliquities of the TRAPPIST-1 Planets with MAROON-X

There are several stellar and planetary parameters that drive the exact shape and amplitude of the RM curve. We describe them in this section and justify the priors used in the final Markov Chain Monte Carlo (MCMC) fit.

In the RM curve, there is a well-known degeneracy between the stellar rotational velocity and the obliquity, our parameter of interest. The amplitude of the RM effect, at small values of R_p /R_⋆ and ignoring limb darkening, goes as (Winn 2010):

$$\begin{eqnarray}&&{K}_{\mathrm{RM}}\propto v\,\sin (i){\left(\displaystyle \frac{{R}_{p}}{{R}_{\star }}\right)}^{2}\sqrt{1-{b}^{2}},\end{eqnarray} \tag{ 1 }$$

where $v\,\sin \,i$ is the projected rotational velocity of the star and b is the impact parameter of the planet. The signal amplitude is related in a more complex fashion to λらむだ; typically, objects with polar or near-polar orbits will cause no RV deviations. It is thus helpful to constrain TRAPPIST-1's $v\,\sin \,i$, though we do note that this degeneracy is broken if the planetary inclination is not 90°, as in that case the obliquity also has an effect on the shape of the RM curve.

While TRAPPIST-1 has been extensively studied, its rotational velocity is uncertain. Reiners et al. (2018), directly measuring line broadening with CARMENES, found that TRAPPIST-1 had a $v\,\sin \,i$ < 2 km s⁻¹, consistent with it being a slow-rotating star. A photometric rotation period measurement of 3.3 days from Luger et al. (2017) and Dmitrienko & Savanov (2018) pointed to a $v\,\sin \,i$ = 1.8 km s⁻¹, in agreement with the CARMENES measurement. However, Morris et al. (2018a) concluded that this 3.3 day signal may be an activity timescale and not a rotation period. We define a conservative upper velocity limit of $v\,\sin \,i\lt 10$ km s⁻¹ and explore how that impacts our results in later sections. Ideally, as none of the TRAPPIST-1 planets have inclinations of exactly 90°, we should be able to break the degeneracy between the obliquity and rotational velocity, allowing us to use a less informative velocity prior.

Another important parameter to consider is stellar limb darkening. As the RM effect is the result of a planet covering up different portions of the star's surface, the variation in stellar brightness across the surface will have some impact on the final results. We make use of the quadratic limb-darkening parameters for TRAPPIST-1 from Claret et al. (2012), using the parameters for a T_eff = 2600 K, logg = 5.0 star in the Sloan Digital Sky Survey (SDSS) z bandpass, which is the listed bandpass that is the most similar to the highest-S/N MAROON-X orders.

We did not make an effort to estimate or model the surface convective blueshift of TRAPPIST-1. The surface convective blueshift of a star has been shown to be relevant to RM curve calculations by Cegla et al. (2016). However, they found that the difference between a model that does and does not account for convective blueshift is ≤0.5 m s⁻¹ for stars with $v\,\sin \,i$ = 2 km s⁻¹. Given that our typical RV error estimates are on the order of 5 m s⁻¹, such an effect will not be detectable in our results, so we avoided modeling the convective blueshift, as this would have introduced additional model complexity.

Given the large-amplitude transit time variations in our data sets and our lack of simultaneous photometry, we estimated the actual transit times in our data sets by making use of the times forecasted by Agol et al. (2021), along with their forecast errors, which are all less than five min. We corrected the times recorded by MAROON-X to the times at the solar system barycenter, using light travel times calculated by astropy, to allow for direct comparison to the forecast times, which are recorded in BJD_TDB. The observed RM curves (especially for planets where the signal is especially clear, such as b and f) appear to roughly coincide with the Agol et al. (2021) transit time predictions, supporting our choice to make use of their forecasts in our analysis.

TRAPPIST-1's multiple transiting planets (with impact parameters varying between 0 and 0.4; Agol et al. 2021) allow us to measure the differential rotation velocity αあるふぁ across the star's surface. In general, a higher absolute value of αあるふぁ (which varies between −1 and +1) indicates a more latitude-dependent rotation rate, while a value of αあるふぁ close to zero indicates that the stellar rotation is at the same rate across the entire surface of the star. As the TRAPPIST-1 planets have different impact parameters, it may be possible to measure αあるふぁ because different planets scan different latitudes of the host star. Vida et al. (2017) found an additional P = 2.9 day signal in the K2 TRAPPIST-1 light curve that could potentially be related to some form of surface differential rotation, but noted that that would result in an unphysically high surface shear, meaning that that signal is likely related to some other phenomenon. Thus, the potential presence of surface differential rotation on TRAPPIST-1 is mostly unconstrained. Typically, stars with rapid rotation tend to have low αあるふぁ values and cooler stars tend to have higher αあるふぁ (e.g., Reinhold et al. 2014; Küker et al. 2019), though the surface differential rotation of late M dwarfs is relatively poorly studied.

The planetary periods, radii, and inclinations, as well as the stellar radius, are all constrained in Agol et al. (2021). While all of these parameters have an impact on the observed obliquity, they have been constrained to the point that allowing them to vary will be unlikely to impact our fits meaningfully (besides making them take much longer). As a point of reference, the typical radius error of a TRAPPIST-1 planet (and the star itself) is on the order of 1%–2%, which would translate to a roughly 2%–4% change in the RM amplitude. Given that our typical RV error bars are on the order of 30%–50% of the RM curve amplitude, fitting the radius would not meaningfully impact our results. The period error (which would affect the transit duration) is also on the order of tenths of a minute, which is substantially less than the other timescales considered in this analysis, such as the errors on the forecast transit times and the exposure lengths. Finally, perturbing the individual planets' inclinations by their error bars only results in changes in RV on the order of <1 m s⁻¹, which is much less than the typical RV errors. These deviations are even smaller for the outer planets, with more well-constrained inclinations, and for systems with low obliquities. Thus, we fix these parameters at the medians of the distributions quoted by Agol et al. (2021) in our fits. The planetary eccentricities are all known to be low and thus we fix them at e = 0 in our fits.

The RM curve for an object with an obliquity of λらむだ is identical to that with an obliquity of −λらむだ for an edge-on planet. However, the TRAPPIST-1 planets are slightly inclined, breaking this degeneracy. Thus, we allow λらむだ to vary in between −180° and 180° in our fits.

Additionally, we found linear trends in our RV baselines that cannot be explained by the known planetary system around TRAPPIST-1. We observe slopes on the order of 10 m s⁻¹ over three hours, which are not in agreement with the relatively small RV amplitudes of the known planets. We did not notice any long-term or large-amplitude trends in RV over our entire data set over two years of observations, so these trends appear to be incoherent over long timescales. They are likely due to short-term stellar activity, and only obscure what we are actually trying to study. Thus, we simultaneously fit a linear slope term alongside the RM curve fit for each night of transit observations.

In our final analysis, the parameters allowed to vary were the stellar rotation velocity $v\,\sin \,i$, the planetary obliquities λらむだ, the stellar surface differential rotation αあるふぁ, the transit times T, and the nuisance linear trend. We performed the fits assuming the planets all have a shared obliquity and, alternatively, with all of the obliquities calculated separately. The former model is a reasonable assumption, as it would be a tremendous coincidence for all of the planets to transit the star at the same inclination but have noncoplanar orbits. The latter model is useful for showing which planets are the most effective at constraining the overall obliquity. Unfortunately, it is difficult to estimate the obliquities of some of the low-signal planets individually (such as d and e) with this method. The values used for each parameter, along with the priors used (if relevant), are shown in Table 2.

Table 2. The Parameters Used in the MCMC Fit of the RM Effect of the TRAPPIST-1 Planets

Parameter	Value	Prior	Source
Eccentricity	0	fixed
αあるふぁ		${ \mathcal U }$(-1, 1)
Obliquity λらむだ (degrees)		${ \mathcal U }$(-180, 180)
Stellar radius (R⊙)	0.1234	fixed
vsini (m s−1)		${ \mathcal U }$(0, 10000)
u1,LD	0.6542	fixed	b
u2,LD	0.2834	fixed	b
Fit offset (m s−1)		${ \mathcal U }$(-1000, 1000) for each transit
Fit slope (m s−1/day)		${ \mathcal U }$(-1000, 1000) for each transit
Transit Time (BJD—2,450,000)
Tb		${ \mathcal N }$(9520.849564, 0.000331)	a
Tc		${ \mathcal N }$(9104.003866, 0.000285), ${ \mathcal N }$(9786.951957, 0.000656)	a
Td		${ \mathcal N }$(9783.965213, 0.003294), ${ \mathcal N }$(9788.015726, 0.003325)	a
Te		${ \mathcal N }$(9526.829202, 0.001351), ${ \mathcal N }$(9782.991182, 0.001571)	a
Tf		${ \mathcal N }$(9788.943700, 0.001198)	a
Tg		${ \mathcal N }$(9184.835364, 0.000714)	a
Planet Radius (R⊕)
Rb	1.116	fixed	a
Rc	1.097	fixed	a
Rd	0.788	fixed	a
Re	0.920	fixed	a
Rf	1.045	fixed	a
Rg	1.129	fixed	a
Planet Inclination (°)
ib	89.728	fixed	a
ic	89.778	fixed	a
id	89.896	fixed	a
ie	89.793	fixed	a
if	89.740	fixed	a
ig	89.742	fixed	a
Planet Period (d)
Pb	1.510826	fixed	a
Pc	2.421937	fixed	a
Pd	4.049219	fixed	a
Pe	6.101013	fixed	a
Pf	9.207540	fixed	a
Pg	12.352446	fixed	a
Semi-Major Axis (10−2 au)
ab	1.154	fixed	a
ac	1.580	fixed	a
ad	2.227	fixed	a
ae	2.925	fixed	a
af	3.849	fixed	a
ag	4.683	fixed	a

Note. Reference (a) refers to Agol et al. (2021) and (b) refers to Claret et al. (2012) .

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We modeled the RM effect using the python code starry (Luger et al. 2021), which simulates a rotating limb-darkened stellar surface and estimates the observed RV shifts as a function of transiting planet position. Adopting the priors from Section 3.1, we then fit for $v\,\sin \,i$, λらむだ, αあるふぁ, and T using the MCMC sampler emcee (Foreman-Mackey et al. 2013). As our exposures were long compared to the timescale of the transit, we modeled the expected RV of each exposure by calculating the RV at four times within each exposure and then averaging out the values. This methodology is valid because the RVs evolve smoothly at all times except at the very beginning and the very end of the transit.

Initially, we considered two separate models: one in which we assumed all of the planets had the same obliquity (see Table 3), and one in which we fit the individual planet obliquities separately (see the table in the Appendix). The model in which all of the obliquities are fixed to the same value are preferred by the Bayesian information criterion (BIC; Schwarz 1978), with ΔでるたBIC = −16.2. This indicates that we lack statistical evidence to conclude that TRAPPIST-1's planets do not share a common obliquity. This is an expected result, as it would be unlikely for these planets to all have different orbital orientations but transit the host star at the same inclination. For many of the lower-amplitude and short-period planets (such as c, d, and e), the observed posterior distributions of λらむだ were multimodal, reflecting the fact that the RM curve of a planet with a positive obliquity looks extremely similar to that of a planet with a negative obliquity of equal magnitude when the impact parameter is low. The planets with the longest orbital periods, TRAPPIST-1f and TRAPPIST-1g, had the most constraining individual transits, with roughly ±20° constraints on their obliquities from a single transit. Future efforts at constraining the TRAPPIST-1 system should likely focus on these planets. We caution against quoting the obliquities derived using single transits, as the limited time resolution can result in systematic deviations from models that feature multiple transits. As an example, TRAPPIST-1f when fit alone appears to have a strongly negative (λらむだ ≈ 45°) obliquity instead of an obliquity closer to zero, and this can be attributed to the phase coverage of the transit "missing" the positive RV deviation expected from a zero-obliquity planet.

Table 3. The MCMC Best-fit Parameters of the RM Model for an αあるふぁ = 0 Star and a System of Planets That All Share the Same Obliquity

Parameter	Planet	Value
$v\,\sin \,i$ (km s−1)	⋯	2.10 ± 0.29
λらむだ (°)	⋯	$-{2}_{-19}^{+17}$
T (BJD—2,450,000)
	b	9520.8496 ± 0.0003
	c	9104.0038 ± 0.0003, 9786.9523 ± 0.0007
	d	9783.9642 ± 0.0026, 9788.0135 ± 0.0021
	e	9526.8290 ± 0.0013, 9782.9909 ± 0.0015
	f	9788.9437 ± 0.0011
	g	9184.8350 ± 0.0007

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In our shared-obliquity model (plotted in Figure 1), we find that the obliquity of the system is equal to $\lambda =-2{^\circ }_{-{19}^{\circ }}^{+{17}^{\circ }}$ from the RM effect alone, which is consistent with the planets having no spin–orbit misalignment. This is an improvement over the RM fits performed on the three planet data set studied by Hirano et al. (2020a), which had a precision of ±28°, but still does not allow us to conclude whether or not the TRAPPIST-1 system has a tidally damped obliquity. We also measure $v\,\sin \,i$ = 2.1 ± 0.3 km s⁻¹, which would correspond to a rotation period (assuming $\sin \,i\approx 1$) of about 2.5–3.3 days. This is consistent within 2σしぐま with the velocity of 1.5 ± 0.4 km s⁻¹ from Hirano et al. (2020a) and also agrees with the rotation period of 3.3 days from Luger et al. (2017) and Dmitrienko & Savanov (2018). The Agol et al. (2021) transit time forecasts describe our results well, though we do note that the g transit appears to occur slightly earlier than the forecast predicts.

Figure 3 highlights the observed RM signal by shifting, stacking, and stretching the data set according to the best-fit model of each individual planet's transit duration and signal amplitude. It is obvious that the smallest planets (d and e) had substantial RV errors relative to the expected RM signal and thus were not very helpful in constraining our results.

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Figure 4 shows a corner plot of this model. There is a slight degeneracy between $v\,\sin \,i$ and λらむだ, but this only becomes a problem at large values of λらむだ, which are inconsistent with the data.

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We also investigated the impact of αあるふぁ on our fits (shown in the Appendix). Given the degeneracies between $v\,\sin \,i$ and λらむだ, we only fit for αあるふぁ in our model where all of the planets have a shared λらむだ. We found that the posterior distribution of αあるふぁ ($\alpha =-{0.1}_{-0.6}^{+0.7}$) is essentially just the priors (with only a slight preference for low values of αあるふぁ), and allowing αあるふぁ to vary causes a slight increase in the BIC over a model in which it is simply fixed at zero, ΔでるたBIC = 2.7. In addition, the fit values for $v\,\sin \,i$ (2.1 ± 0.3 km s⁻¹) and λらむだ (−2° ± 18°) are very similar to those fit in the case where αあるふぁ is fixed at 0. Future efforts to measure αあるふぁ will likely require far more precise data or a system featuring planets with a broader range of impact parameters.

As we have access to high-resolution spectra from MAROON-X, we can estimate the star's rotational broadening directly from the spectra by examining the broadening of the lines. MAROON-X has a resolution of roughly 85,000 (Seifahrt et al. 2018), meaning that we cannot expect to measure the rotational velocities below about 2–3 km s⁻¹ accurately with this method. If the measured $v\,\sin \,i$ is at or below this limit, it would provide additional support for the low rotational velocity measured in Section 3.2.

To estimate the rotational velocity directly, we used the CCF comparison method described in Gray (2005) and employed in Reiners et al. (2012). Instead of comparing our TRAPPIST-1 spectral line broadening directly to model spectra (which may introduce systematic biases due to differences between the model and stellar spectra), this method compares it to the spectrum of a slow-rotating star collected using the same instrument. This method is described more detail in the following paragraphs.

First, we selected a spectrum of a star that has been observed by MAROON-X that is similar in spectral type to TRAPPIST-1 but is known to be a slow rotator (and thus possess little rotational broadening). We used several different stars as templates, as there are no other stars observed by MAROON-X with the same spectral type as TRAPPIST-1 (M7.5; Gizis et al. 2000) and we wanted to investigate the impact of varying spectral types on our final results. The template stars are listed in Table 4, but all of them have rotation periods suggesting a $v\,\sin \,i\leqslant 0.1$ km s⁻¹.

Table 4. The Various Stars Used as Templates for the Purposes of Estimating the $v\,\sin \,i$ of TRAPPIST-1

Star	Type	Prot (days)	Period Reference	TRAPPIST-1 $v\,\sin \,i$ (km s−1)
Teegarden's Star	M7	99.6 ± 1.4	Terrien et al. (2022)	1.9 ± 0.7
LP 791-18	M6	>100	Crossfield et al. (2019)	2.7 ± 0.7
Ross 128	M4	>100	Bonfils et al. (2018)	2.4 ± 0.5
Barnard's Star	M4	145 ± 15	Terrien et al. (2022)	2.4 ± 0.8

Note. The last column shows the derived $v\,\sin \,i$ of TRAPPIST-1, using the given star as a template.

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We simulated rotational broadening of each template spectrum for a grid of different values of $v\,\sin \,i$ (0–10 km s⁻¹, with a grid spacing of 0.5 km s⁻¹) using the rotational convolution kernel from Gray (2008). To simulate the rotational broadening, we assumed a linear limb-darkening coefficient of 0.8446 (Claret et al. 2012), corresponding to the linear limb-darkening coefficient of a 2600 K, logg = 5.0 star in the SDSS z filter. This grid of artificially broadened spectra effectively allows us to derive a relationship between the width of the template's stellar lines and the rotational velocity.

After generating this grid of broadened spectra, we masked the tellurics out of the template's spectrum (using the same line list described in Section 2) and calculated the CCFs between the broadened template and the original unbroadened template spectrum for each given rotational velocity. We then fit the center of each CCF with a Gaussian and determined the CCF FWHM. As the FWHM of each CCF is a proxy for the average line width, we expect it to be directly correlated with the template $v\,\sin \,i$. We thus fitted the relationship between the FWHM and the $v\,\sin \,i$ with a simple quadratic interpolator to create a function that can estimate the $v\,\sin \,i$ of an unbroadened stellar spectrum given its CCF FWHM. This analysis is done on an order-by-order basis, such that each template MAROON-X order has an associated function, as shown in Figure 5. We dropped any orders with nonmonotonic functions between $v\,\sin \,i$ and the FWHM, as that is an indication that the CCFs in these orders are dominated by systematics that confuse the Gaussian fitting process. This frequently occurred in orders with low signals or many masked telluric regions.

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As shown in Figure 5, each template star has a slightly different relationship between FWHM and $v\,\sin \,i$, even when compared to other stars of the same spectral type. As an example, LP 791-18 has a much narrower CCF than Teegarden's Star, despite them both being late M dwarfs that rotate slowly. Teegarden's Star did appear to be fairly active over the course of our observations, which could account for its broader CCF, though we selected a spectrum that was not taken during a flare for this analysis. These differences could also be merely due to the differences in spectral type or telluric absorption between these spectra. Figure 5 also shows that, below about $v\,\sin \,i$ = 2 km s⁻¹, the relationship between FWHM and $v\,\sin \,i$ approaches a vertical line, meaning that, as expected, this method would have difficulty discriminating between $v\,\sin \,i$ values significantly below 2 km s⁻¹.

We then calculated the CCF of the observed TRAPPIST-1 spectrum with each unbroadened template, and estimated the $v\,\sin \,i$ for each order using the relationships for each template shown in Figure 5, assuming the CCF broadening comes primarily from the rotation of TRAPPIST-1. In some orders, the FWHM of the TRAPPIST-1-to-template CCF was smaller than the FWHM of the autocorrelation function of the unbroadened template, which is an unphysical result that yields a negative $v\,\sin \,i$ calculation. This issue is likely caused by systematic differences between the spectra due to differing host star properties, as well as the low S/N of the TRAPPIST-1 spectrum in the bluest orders. We thus excluded these orders from our analysis. We also excluded orders fit to rotational velocity values higher than 10 km s⁻¹, which likely suffer from similar problems. We took the weighted mean and standard deviations of the $v\,\sin \,i$ values of the remaining orders to estimate the final v sin i and its associated errors (rightmost column of Table 4).

For our final result, we quote the $v\,\sin \,i$ estimated using the Teegarden's Star template, which is both the result with the lowest final $v\,\sin \,i$ value and the one with the latest-type spectral template. Thus, with the CCF method, we find that the rotational velocity of TRAPPIST-1 is around 1.9 ± 0.7 km s⁻¹ (corresponding to a rotation period of 2.3–5.0 days given ${\sin }\,i\approx 1$), which agrees with the 2.1 ± 0.3 km s⁻¹ measurement performed in Section 3.2.

There are several factors that introduce additional errors into our calculations. Differences between the template spectra and TRAPPIST-1 in terms of activity levels, metallicity, and spectral type likely broaden the calculated CCFs, resulting in an overestimated rotation velocity. This effect is apparent in Table 4, which shows that the template closest in type to TRAPPIST-1 has the lowest measured rotational velocity, though the relationship between spectral type and measured velocity is not monotonic. In addition, our calculations also assumed that the template stars had no rotational broadening. As these stars do rotate, our estimate of TRAPPIST-1's rotation is an underestimate, though the magnitude of this underestimation is smaller than the quoted error bars (0.5–0.8 km s⁻¹) due to the slow rotation (<0.1 km s⁻¹) of these stars. Additionally, our quoted limb-darkening coefficient was likely an overestimate, as most of the template stars are hotter and bluer than TRAPPIST-1 and thus have slightly lower linear limb-darkening coefficients. However, we found that decreasing the limb-darkening coefficient by as much as 0.5 tended to only have a minor effect on the resulting calculated rotational velocities, decreasing them by around 0.1 km s⁻¹, which is significantly less than the quoted $v\,\sin \,i$ error and thus does not warrant a more precise prescription.

Another method we can use to measure the obliquity is the Doppler tomography (also known as the Doppler shadow) technique (e.g., Cameron et al. 2010), in which the obliquity is inferred from line shape perturbations caused by the planet passing over the rotating star's surface. As the planet eclipses the rotating stellar disk, the planet distorts the stellar line shape, which often manifests as a "bump" which moves across the stellar line profile. If the planet is measured with sufficient time coverage, we can watch how this bump moves in time and velocity space and infer the system's obliquity. As the Doppler shadow technique and the RM effect are both observable with high-precision spectra, both can be performed independently using the same data set to constrain a given star's obliquity.

We attempted to perform a Doppler tomography analysis with the MAROON-X data. The overall line profile is calculated by estimating the CCF of the given spectrum with a template or mask. serval uses a least-squares fitting technique instead of the cross-correlation method, which necessitated that we use different software for this analysis stage. We adapted the publicly available raccoon code (Lafarga et al. 2020; which was originally produced with the intention to perform CCF analyses with CARMENES data) to be able to process the MAROON-X spectra. We also adapted raccoon to generate a mask template with ≈1000 lines out of the coadded MAROON-X TRAPPIST-1 spectrum using all available data, both in- and out-of-transit, to achieve the maximal possible spectral S/N for the purpose of identifying appropriate lines. We found that using the publicly available template for Teegarden's Star (which is an M7 dwarf) did little to change our results, even though the template contained many more (≈5000) lines. Using our template, we calculated the line profile for each TRAPPIST-1 exposure after deblazing the spectra. We ignored the contributions to the CCF from orders blueward of around 730 nm and from orders with no identified template lines. We performed this calculation for each of our MAROON-X spectra to find the average line profile of each observation.

The CCF profile was binned in increments of 3.5 km s⁻¹, to match the expected velocity resolution of MAROON-X. Not binning the CCF profile in velocity space would result in excess correlations between the line profile values. However, this does mean that the expected Doppler shadow of TRAPPIST-1 (which is expected to travel from −2 to +2 km s⁻¹) will only be encompassed by around two independent points in each CCF profile.

We normalized each CCF to unity by fitting a linear term to each line profile's baseline (>10 km s⁻¹ from the line center) and then dividing each profile by that fit line. For each night, we calculated an average out-of-transit profile by averaging together all of the line profiles from observations not taken during the transit (according to the predictions of Agol et al. 2021). We then estimated the residuals by subtracting the night's out-of-transit line profile from each individual CCF profile. For a well-aligned low-obliquity system, the residuals are expected to show a small bump, traveling from the left side of the line profile to the right side of the line profile, as the planet transits in front of its host.

To describe the Doppler shadow signal, we turn to Cameron et al. (2010), who modeled it as a Gaussian perturbation added to a Gaussian line profile with linear limb darkening. In this model, we adopted the same parameters for the planetary properties as described in Table 2, but instead adopted a linear limb-darkening term of 0.8446 (Claret et al. 2012) and fit for $v\,\sin \,i$, CCF FWHM, and (shared) planetary obliquity using the methodology described in Cameron et al. (2010), with fits performed using emcee. The FWHM describes the width of the Gaussian residual bump in the CCF profile as the planet transits, $v\,\sin \,i$ describes the extent of the bump's motion in velocity space, and λらむだ describes the actual direction in which the bump moves. Due to the low time resolution of our data, much like in Section 3.2, we calculated the model CCF at four evenly spaced times encompassed by each observation and averaged them together. To speed up our fits, we calculated a set of line profiles (referred to as h(x) in Cameron et al. 2010) based on a grid of planet positions and stellar rotational velocities before running the MCMC fit and performed linear grid interpolation to estimate the line profile at each step. This dramatically sped up the fit by avoiding the repeated numerical integration that is necessary in their methods, in return for a slight loss in accuracy (typically on the order of one part in 10⁵ or less).

We assumed a fixed relative system velocity of 0 km s⁻¹, as the CCF is generated by referencing the spectra against a template generated from the same spectral data. The spectra are shifted according to the expected barycentric velocity each night, but we did not include any planet-induced RV shifts because they are substantially smaller than the 3.5 km s⁻¹ resolution. We used the same obliquity prior given in Table 2, but adopted an upper limit on $v\,\sin \,i$ as that of the line profile FWHM, as the FWHM is by definition equal to or larger than $v\,\sin \,i$ in the Cameron et al. (2010) models.

After processing the data, we compared our simulations to the models described in Cameron et al. (2010). Overall, we found that the expected signal amplitude of the TRAPPIST-1 planets is comparable to (or less than) the observed noise in the nightly CCFs outside of the line profile (where we do not expect to see any planetary signals). The data (see Figure 6) show a trend of increased noise in the in-transit line profiles and a general tendency for the in-transit line profiles to be shallower than the out-of-transit line profiles. This could be explained with planetary signals but could also be the result of slight normalization errors, which are difficult to correct given the small number of data points and the significant noise. We found that allowing the $v\,\sin \,i$ to vary freely resulted in fits that strongly preferred extremely low values of $v\,\sin \,i$ (≈1 km s⁻¹), in disagreement with our results from the canonical RM effect modeling in Section 3.2 and consistent with a velocity of zero given our instrumental resolution. Fixing $v\,\sin \,i$ at 2.1 km s⁻¹ typically resulted in fit obliquity measurements of around 60°–90°. This apparent degeneracy between $v\,\sin \,i$ and λらむだ is a sign that we have failed to detect the signature of TRAPPIST-1, as a low $v\,\sin \,i$ and a λらむだ close to 90° both manifest as a bump traveling up the center of the CCF profile, which can easily be reproduced by slight errors in CCF normalization that are difficult to correct for with our current data set.

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This analysis shows that we are unable to detect the Doppler shadow of TRAPPIST-1 with MAROON-X. This is primarily due to a combination of the low S/N of the TRAPPIST-1 spectra, the insufficient resolution of the instrument, and the small expected signal amplitude. It is obvious from Figure 6 that the typical CCF noise can be far stronger than the anticipated signal, and our instrument's resolution is too low for us to resolve the planet-induced Doppler shadow clearly. Additionally, the center of the CCF profile is sensitive to errors in normalization (primarily introduced by our low resolution), making it difficult to distinguish a Doppler tomographic signal from noise unless it is fairly strong.

Hirano et al. (2020a) claimed a detection of this signal based on Subaru IRD data, measuring an obliquity of $19{^\circ }_{-{15}^{\circ }}^{+{13}^{\circ }}$. We note that the IRD is sensitive to redder wavelengths than MAROON-X, which theoretically would make late M dwarf profile characterization more straightforward due to the decreased amount of line blending (Önehag et al. 2012). This would make the generation of an accurate CCF template easier. In addition, while their derived RVs are lower in precision than those obtained via serval, the MAROON-X RVs derived via CCF (using raccoon) have much larger errors, likely due to the relatively small number of lines included in the template. They also have shorter exposure times (300 s versus 600 s) during the transits, which translate to better temporal resolution of the signal. However, the slow rotational velocity of the star (≈2 km s⁻¹) and the velocity resolution of IRD (≈4 km s⁻¹) would likely result in similar problems with regards to fitting the signal as to what we faced with the MAROON-X data set. They also found planet signals which were similar in magnitude, if not weaker, than the typical noise in their CCF profiles. These issues are reflected somewhat by the high false alarm probability of >1% quoted in their paper. Overall, we recommend using obliquity and velocity measurements derived from the RM effect instead of those derived from Doppler tomography for this system until a less ambiguous detection is made.