CONSTRAINING HIGH-SPEED WINDS IN EXOPLANET ATMOSPHERES THROUGH OBSERVATIONS OF ANOMALOUS DOPPLER SHIFTS DURING TRANSIT

We present four models of hot Jupiter circulation here, two that are drag-free and two that include a simple treatment of magnetic drag in the atmosphere. The drag-free models differ in that one uses the same hyperdissipation strength from Rauscher & Menou (2010), while the other uses a hyperdissipation timescale an order of magnitude shorter, in order to reduce the numerical noise that is apparent at low pressures in the original model.⁴ The two magnetic drag models differ in the implementation of drag at low pressures, as discussed below. The 3D models analyzed in this paper are all basic extensions of models presented previously in Rauscher & Menou (2010) and Perna et al. (2010), and details can be found in those papers. Briefly, they were all calculated using the same dynamical core and Newtonian relaxation scheme for radiative forcing as presented in Menou & Rauscher (2009). Each model was run for 2000 planet days at a resolution of T31L45 (corresponding to a horizontal resolution of ∼4° and 45 vertical levels in log pressure). Two modifications have been made to the setup of these models in order to better facilitate the calculation of transmission spectra: a different specific gas constant was used and the upper boundary has been extended up to 10 μbar.

The models from Rauscher & Menou (2010) and Perna et al. (2010) used a specific gas constant of R = 4593 J K⁻¹ kg⁻¹, which corresponds to a mean molecular mass of 1.81 g mol⁻¹ and a value for R/c_p of 0.321 (c_p is the specific heat capacity at constant pressure). For better consistency with the transmission spectroscopy modeling, which is highly sensitive to atmospheric scale height and therefore mean molecular weight (e.g., Miller-Ricci et al. 2009), we chose to use R = 3523 J K⁻¹ kg⁻¹ and R/c_p = 0.286 in the circulation models presented here. This now corresponds to a mean molecular weight of 2.36 g mol⁻¹, a value more appropriate for solar composition, while the value for R/c_p now matches that for a purely diatomic gas.

The other significant change over the previously published models is that we have extended the upper boundary of the atmosphere to 10 μbar, necessary because transmission spectra probe pressures well above the 1 mbar top boundary of the original models. This extension to lower pressures meant that we had to extrapolate our radiative forcing and drag prescriptions.

The Newtonian relaxation scheme used for the radiative forcing requires the choice of equilibrium temperatures and radiative times (for details see Rauscher & Menou 2010). In order to extend these up to 10 μbar we chose to continue the 1000 K day–night temperature difference, which is constant down to 100 mbar. The night-side temperature was then set in the same manner as in our previous work, so that at each level the integrated T⁴ matched the profile from Figure 1 of Iro et al. (2005). The radiative times were taken from Figure 4 of Iro et al. (2005), which includes these lower pressures.

For a simplified treatment of magnetic drag in the atmospheres of hot Jupiters, the models presented in Perna et al. (2010) employed a Rayleigh drag with a time constant that is horizontally uniform, but varies with pressure: dv/dt = −v/τ_drag(P). Here we use the strongest drag strengths from Perna et al. (2010), which ranged from 2 × 10⁴ s at 1 mbar to 8 × 10⁶ s at 100 bar. Since this scheme is already simplifying the underlying physics of magnetic drag, we choose two simple forms for the extrapolation of drag times to lower pressure: either we maintain a constant τ_drag above 1 mbar, or we maintain a constant τ_drag/τ_rad ratio above 1 mbar. Figure 1 shows the radiative and drag times used throughout our models.

Zoom In Zoom Out Reset image size

To summarize the four atmosphere models, they are

• 1.  
  Drag-free: canonical hot Jupiter model from Rauscher & Menou (2010), but extended to an upper boundary at 10 μbar and using an updated gas constant.
• 2.  
  Drag-free + hyperdissipation: same as the drag-free model but with a hyperdissipation timescale that is an order of magnitude shorter to reduce numerical noise in the model.
• 3.  
  Magnetic drag (a): magnetic drag model using the strongest drag strengths from Perna et al. (2010) below 1 mbar and maintaining a constant τ_drag above 1 mbar.
• 4.  
  Magnetic drag (b): same as magnetic drag model (a) but with a constant τ_drag/τ_rad ratio above 1 mbar. Since this means that the drag timescale continues to decrease with pressure, this model experiences stronger drag than model (a).

The atmospheric structure for all four models is qualitatively very similar to the results presented in Rauscher & Menou (2010) and Perna et al. (2010). High in the atmosphere the basic flow pattern is SSAS, meaning that winds blow from day to night across all latitudes of the terminator, with transonic wind speeds (although weaker for the magnetic drag models). At moderate pressures (hundreds of mbar) the advective timescales become comparable to the radiative timescales and the flow is able to alter the temperature structure from a hot-day/cold-night pattern to one where the hottest region of the atmosphere is advected eastward of the substellar point. Finally, at deep pressures the winds are much slower (and subsonic), but are easily able to minimize day–night temperature differences. Some details differ between the models analyzed here and those in Rauscher & Menou (2010) and Perna et al. (2010). While similar, the advected temperature structures are not identical. In addition, similar features between the models tend to occur at slightly lower pressure in the original models than the ones here, likely a result of the change in the specific gas constant and its effect on the pressure scale height.

The four models in this paper have generally similar temperature structures throughout the atmosphere, as shown by the temperature–pressure profiles in Figure 2. Although the behavior at high pressure varies between the drag and drag-free models, all of the models have the same day–night temperature difference at low pressures (due to the same radiative forcing setup and very short radiative times). The models with magnetic drag or extra hyperdissipation do not show the effect of small-scale numerical noise seen in the original drag-free model. The temperature structure around the west terminator (θ = 270°) is nearly the same as the east terminator (θ = 90°) for the magnetic drag models, but has a profile that has been further altered by advection in the drag-free models.

Zoom In Zoom Out Reset image size

The main difference between the four models is the flow structure, especially at the low pressures probed by transmission spectroscopy, where the Doppler effect could be observed. The differences between the high-altitude flow in our four models can be seen in more detail in Figure 3, which shows flow patterns across the planet at the 60 μbar level, representative of the high-altitude regime. In all cases the winds are directed away from the substellar point toward the anti-stellar point across the terminator. This results in a net blueshifted wind directed toward the observer during transit for all models, although the magnitude of the wind speed and the details of the wind pattern vary from model to model.

Zoom In Zoom Out Reset image size

First we compare the no-drag models. Enhanced hyperdissipation in the second drag-free model reduces the numerical buildup of noise on small scales, which has the effect of making the flow more coherent, but reduces the maximum wind speeds. Nevertheless, the mean eastward wind speed across the terminator is nearly the same for both models and remains nearly identical between the two models across a wide range of pressures (∼5 km s⁻¹ at 60 μbar). The models with magnetic drag have much slower winds at this pressure. The flow in model (b) is strongest in a narrow region on either side of the terminator, which is also the area probed by transmission spectroscopy, but it has an average eastward wind speed at the terminator of only ∼2.3 km s⁻¹, compared to ∼3.8 km s⁻¹ in model (a). The magnetic drag models have similar average terminator winds throughout much of the atmosphere (where they have identical drag times), but at pressures less than 1 mbar the winds in the weak-drag model increase with altitude, while the winds in the strong-drag model decrease.

The transmission spectrum is obtained by dividing the spectrum obtained during transit by the stellar spectrum, thus revealing the excess absorption from the species that make up the planetary atmosphere. We calculate the attenuation in intensity of a beam of stellar light passing through the planet's atmosphere according to

$$\begin{equation} I(\lambda) = I_{0} e^{-\tau }, \end{equation} \tag{ 1 }$$

where I₀ is the incident intensity from the star. We ignore any effects of additional scattering into or out of the beam or of refraction, which have been found to be of minimal effect to hot Jupiter transmission spectroscopy (Hubbard et al. 2001). The slant optical depth τ is calculated as a function of wavelength λ, height H, and latitude ϕ according to

$$\begin{equation} \tau = \int \kappa ds, \end{equation} \tag{ 2 }$$

where ds is the differential path length through the planet's atmosphere along the observer's line of sight, and κ is the opacity calculated for solar composition gas evaluated at the local temperature and pressure, which are each in turn dependent on the local height H, latitude ϕ, and longitude θ according to the 3D dynamics model. The opacity is furthermore Doppler shifted by the local line-of-sight velocity as outlined in the following paragraphs. The total in-transit flux passing through the planet's atmosphere is then calculated by integrating the intensity from Equation (1) over the solid angle subtended by the atmosphere.

We include opacities from gas phase CH₄, CO, CO₂, H₂O, and NH₃ along with collision-induced H₂ opacities. Abundances of all species are calculated assuming the atmosphere resides in a state of thermochemical equilibrium. Further details on our opacity tables and equilibrium chemistry calculations for solar composition gas can be found in Miller-Ricci et al. (2009) and Miller-Ricci & Fortney (2010). It is important to note for this work that our opacity tables for the molecular absorption come in the form of individual line lists, which allow us to calculate opacities at arbitrary spectral resolution. We calculate line broadening using Voigt profiles, with Doppler and H₂ pressure-broadening components. Our standard broadening prescription does not include microturbulence, yet additional small-scale turbulence can further broaden the spectral lines affecting the height at which the line cores form. To determine the magnitude of this effect, we have performed limited tests with additional line broadening from microturbulence of up to 4 km s⁻¹ and have found that this does not significantly alter the results presented in the following sections.

As photons pass through the planetary atmosphere during transit, they encounter gas moving at the local velocity v, whose line-of-sight component will induce a Doppler shift on the absorption signature. The Doppler shift through the planet's atmosphere is given by

$$\begin{equation} \frac{\Delta \lambda }{\lambda } = \frac{v_{{\rm LOS}}}{c}, \end{equation} \tag{ 3 }$$

where v_LOS is the line-of-sight component of the velocity. There are three components that contribute to the line-of-sight velocity that a photon "sees" as it passes through the planet's atmosphere. These are (1) the wind speed, (2) the rotational velocity, and (3) the orbital speed, all evaluated locally in the planet's atmosphere. The net line-of-sight component to the velocity is then given by

$$\begin{eqnarray} v_{{\rm LOS}} &=& {-}(u \sin \theta \cos \phi + v \cos \theta \sin \phi + (R_{p} + z) \nonumber\\ && \times \Omega \sin \theta \cos \phi + v_{{\rm orb}} \sin \varphi). \end{eqnarray} \tag{ 4 }$$

The first two terms in Equation (4) give the contribution to the line-of-sight velocity from the winds in the planet's atmosphere. The east–west and north–south components of the wind are given by u and v, respectively. The third term in the equation gives the contribution from the planet's rotation, where R_p is the planet's radius at 1 bar (=1.32 R_Jup for all models), z is the height in the atmosphere above the 1 bar level, and Ω is the planet's rotational speed in rad s⁻¹. The final term in Equation (4) gives the contribution from the orbital motion of the planet. The orbital speed is denoted by v_orb, and φ is the phase angle of the orbit, defined as φ = 0 at the center of transit. For the results presented in this paper we assume a circular orbit resulting in a constant v_orb. To calculate the rotation and orbital speeds we assume a tidally locked planet (P_orb = P_rot) with an orbital period of 3.53 days. Equation (4) does not include the additional contribution from the gravitational redshift, due to the planet lying relatively deep within the potential well of its host star. For the hot Jupiter presented in this paper, the gravitational redshift will impart an additional systematic 60 m s⁻¹ redshift to the transmission spectrum that is not included in our subsequent modeling.

Figure 4 shows the Doppler-shifted wind speeds along the planet's terminator that result from each of the four atmospheric dynamic models described in Section 2.1. The opacity κ from Equation (2) is finally evaluated at wavelength

$$\begin{equation} \lambda = \lambda _0\left(1-\frac{v_{{\rm LOS}}}{c}\right), \end{equation} \tag{ 5 }$$

where λ₀ is the unshifted wavelength, to produce the properly Doppler-shifted absorption.

Zoom In Zoom Out Reset image size

High-resolution spectroscopy at a spectral resolution of ∼10⁵ along with sufficiently high signal-to-noise is necessary to measure Doppler shifts in transmission spectra at the km s⁻¹ level. Here we calculate all of our transmission spectra at a spectral resolution of 10⁶, which is higher than the resolution of most currently available high-resolution spectrographs by at least an order of magnitude. (For example, the CRIRES spectrograph used by Snellen et al. 2010 has a working resolution of ∼10⁵). We compute spectra at such high resolution to clearly show the effects of Doppler shifts on our transmission spectra and also to show the power of very high spectral resolution for future instrumentation. All of our spectra can easily be degraded to lower spectral resolution by convolution with a Gaussian of the appropriate width.

We calculate all of our spectra from 2291 to 2350 nm as a representative wavelength range over which high-resolution transmission spectra can be obtained from the ground. This wavelength range covers the 2.3 μm first overtone (Δν = 2) band of CO. This is also the wavelength coverage of the observations by Snellen et al. (2010), which facilitates comparisons between our model spectra and their results.

It is important to note that the stellar spectrum experiences velocity shifts of its own during transit. These shifts result from both the induced motion from the planet's orbit (stellar RV) along with the Rossiter–McLaughlin effect by which the planet blocks out a portion of the blue- or redshifted limb of the star during transit. Both effects produce a zero net Doppler shift at the center of transit, provided that the planet's orbit is circular and the stellar spin axis is aligned with the normal axis of the planet's orbit. However, when either of these effects becomes non-zero, it is necessary to divide out the appropriately Doppler-shifted stellar spectrum from the in-transit spectrum to obtain a transmission spectrum that only has the effects of the planetary Doppler shift imprinted on it. For the spectra that we present in the following section, we assume that the stellar Doppler shift is known and has been appropriately accounted for.

The entire 2291–2350 nm section of the hot Jupiter transmission spectrum for the drag-free atmosphere model is shown in Figure 5 with no velocity shifts. In the unshifted spectrum, the first overtone band of CO is clearly visible at the model spectral resolution of 10⁶. Many additional spectral features from water are also present in this region of the spectrum along with weaker CH₄ features that originate from cooler regions in the atmosphere. The comb of strong and well-separated CO lines around 2.3 μm is particularly useful for measuring Doppler shifts as long as the spectral resolution exceeds ∼10⁴. At the much lower spectral resolution of most transmission spectrum measurements to date, the 2.3 μm CO features are unresolved in a single broad absorption band, and RV measurements at a precision of ∼1 km s⁻¹ are not possible.

Zoom In Zoom Out Reset image size

Figure 6 shows a representative 2.3 nm segment of the fully Doppler-shifted spectra from 2308.0 to 2310.3 nm. (The rest of the spectrum shows similar RV shifts from winds, but it is not feasible to show the entire spectral range of our model in a single plot and to still see the Doppler shifts by eye.) A net blueshift in all four models is clearly apparent in the top panel of Figure 6. Additionally, a significant amount of line broadening occurs in the transmission spectra from the joint effects of winds and rotation, which effectively weakens the peak strength of each of the individual spectral lines and also causes many of the adjacent lines to blend together. The effect of broadening due to rotation alone is shown in the bottom panel of Figure 6, clearly indicating the extent to which the Doppler broadening is due to rotation as opposed to winds. Rotation is the dominant source of broadening (quite similar to the effect of rotational broadening on stellar spectra). However, the winds themselves also cause some further broadening of the transmission spectra, since the wind patterns are not entirely coherent and instead some considerable variation in the line-of-sight wind speeds occurs around the terminator as shown in Figure 4. Some of the models clearly show a higher level of incoherence in the wind pattern than others (e.g., the standard drag-free model as opposed to the model with enhanced hyperdissipation), which has a small effect on the width of the broadened spectral lines from model to model.

Zoom In Zoom Out Reset image size

By cross-correlating the Doppler-shifted spectra against the unshifted transmission spectra, we can obtain the average velocity shift for the entire 2291–2350 nm region of the calculated spectrum. The cross-correlation functions for each of the four atmosphere models taken at the center of transit (phase φ = 0) are shown in Figure 7. Each model gives a different overall velocity shift with the magnetic drag models producing the smallest shifts and the drag-free models producing the largest shifts. The net blueshifts for the drag-free, drag-free with added hyperdissipation, and magnetic drag models (a) and (b) are 2.22, 2.40, 1.48, and 1.15 km s⁻¹, respectively. The blueshifts in all cases can be attributed to the SSAS flow pattern at altitudes above 1 mbar. The line cores sample pressures in the 0.01–1 mbar range, and the hot Jupiter atmosphere is optically thick to transmission at pressures greater than ∼10 mbar, resulting in a spectrum that purely samples the SSAS flow and is insensitive to the deeper zonal flow pattern. The shifts are as expected with the highest drag models producing the lowest net velocity shifts. However, given that the difference in the magnitude of the Doppler shifts between models only varies by 1.25 km s⁻¹, observations with RV precision of better than 1 km s⁻¹ will be necessary to differentiate between atmospheres with and without magnetic drag. At the 1 km s⁻¹ RV precision of the Snellen et al. (2010) measurements, all four of our model atmospheres are consistent with the measured winds in HD 209458b's atmosphere of 2 ± 1 km s⁻¹.

Zoom In Zoom Out Reset image size

It is interesting to note that even though the drag-free model has higher maximum wind speeds than the drag free + hyperdissipation model, the largest Doppler shifts take place in the latter case. This is because the extra hyperdissipation reduces the small-scale numerical noise observed in the original model and results in a more coherent flow. Although the average wind speed across the terminator is the same in both models, the winds are better aligned in the drag free + hyperdissipation model and contribute more strongly to the blueshifted signal. Although the effect is small, on the order of ∼200 m s⁻¹, this demonstrates the observational uncertainty associated with the choice of hyperdissipation strength in our 3D models.

When the effects of winds are ignored and only the RV effect of rotation is considered, a zero net Doppler shift is expected because the blueshifting of the eastern limb of the planet should counteract the redshifting of the western limb, resulting in Doppler broadening but no net shift. However, cross-correlating the rotationally broadened spectra against the unshifted spectra (not shown) results in a very small net blueshift for all of our models. This results from the fact that the eastern (blueshifted) limb of the planet tends to be hotter and therefore more puffed up than the western limb. As a consequence, the blueshifted limb plays the dominant role in the rotational broadening signature. This effect is on the order of several to several tens of m s⁻¹, depending on the model atmosphere, with the no-drag models producing the larger signature. While interesting, this effect requires measuring the rotational asymmetry in the transmission line profiles at an RV prevision that is two orders of magnitude higher than what has previously been obtained by Snellen et al. (2010). Even if this were possible, it would be nearly impossible to disentangle the rotational Doppler shifts from the Doppler shifts that result from the planet's winds.

Currently, measurements of Doppler-shifted winds via transmission spectroscopy require that the transmission spectra be integrated over the full transit just to attain the signal-to-noise necessary for making a 1 km s⁻¹ measurement. However, future observations that can measure Doppler shifts as a function of orbital phase throughout transit will be a very powerful tool for resolving the spatial structure of the winds across a planet's terminator. These types of measurements will likely necessitate next generation ground-based 30 m class telescopes to attain high signal-to-noise over shorter exposure times. Of particular interest is the measurement of Doppler shifts during transit ingress and egress when only one limb of the planet is in front of the star, which will allow for a straightforward mapping of the eastward and westward flows on either limb of the planet.

We calculate Doppler shifts as a function of orbital phase throughout transit for each of the four model atmospheres in Figure 8. We separately show the cumulative effects of rotation, winds, and orbital motion. Generally, the orbital motion is the dominant effect on the Doppler-shifted signal. When that motion is subtracted off, as in Snellen et al. (2010), the effects of high-altitude winds become apparent.

Zoom In Zoom Out Reset image size

Some details of the calculation resulting in Figure 8 are as follows. Throughout the analysis we use a single snapshot from each model, safely assuming that the amount of temporal variation is negligible and its effect is small compared to our other simplifying assumptions. We account for rotation of the planet away from phase φ = 0 by shifting the longitudes in our model by an angle equivalent to the orbital phase. This step is important because the planet rotates by an angle of 16° over the duration of the transit relative to an observer on Earth, whereas the frame of reference of the model atmosphere calculations always has a 0° line of longitude at the planet's substellar point. The planet's rotation relative to the observer has only a small effect on the calculated velocities, but the effect of the slow eastward motion of both the hottest and coldest points on the planet does induce small velocity shifts throughout transit. For simplicity, we assume that the planet transits exactly across the middle of the star at an orbital inclination of 90°. We do not include any effects of limb darkening. We also ignore the geometric effects of curvature on the limb of the star, in effect treating the edge of the star as a straight line, tangential to the orbital direction of the planet. These last three effects should minimally impact the results shown in Figure 8, but the qualitative results are still instructive.

During full transit, between the second and third contact points when the planet is completely in front of the star, the wind and rotation RVs are fairly constant, despite a very small effect from the rotation of the planet relative to the observer, which can cause shifts of up to 200 m s⁻¹. In contrast, the orbital RV signature varies considerably during transit as it sweeps out the φ = 0 region of a sine curve with peak amplitude 146 km s⁻¹, equivalent to the orbital speed of the planet on its circular orbit.

However, during transit ingress and egress, between the first and second, and third and fourth contact points, respectively, the induced Doppler shifts from winds and rotation both vary considerably as first one limb of the planet and then the other appears/disappears in front of the stellar disk. The effect of the planet's rotation is to blueshift the eastward limb and redshift the westward limb of the planet, thus creating an anomalous redshift during ingress and blueshift during egress. Since the rotation is symmetric, the rotational RV signature is also approximately (anti-)symmetric about the center of transit. The magnitude of the Doppler shift due to rotation alone increases from ∼0 km s⁻¹ during the phase of full transit to ∼2 km s⁻¹ at the very beginning of ingress and at the end of egress, which is equivalent to the rotational speed of the planet, as seen in the bottom panel of Figure 8.

Winds display similar behavior to rotation during ingress and egress where the eastward limb tends to be more strongly blueshifted than the westward limb. This effect varies from model to model and also becomes stronger with depth, as shown in Figure 4. In general, most of the terminator is predicted to be blueshifted at mbar pressures. However, the models with magnetic drag tend to have more uniform velocities, whereas the drag-free models show a stronger tendency towards blueshifted winds at the eastern terminator and redshifted winds at the western terminator. In terms of the ingress and egress velocities, the canonical drag-free model gives the largest differences between the Doppler shifts from winds at the start and finish of transit. At the very beginning of ingress the drag-free model gives an additional 1.6 km s⁻¹ blueshift on top of the 2 km s⁻¹ redshift from the planet's rotation (for a net redshift of 0.4 km s⁻¹). At the very end of egress this same model gives an additional 3.3 km s⁻¹ blueshift on top of the 2 km s⁻¹ rotational blueshift. This results in an overall 1.7 km s⁻¹ measured difference between the line-of-sight wind speeds for opposing limbs of the planet. In contrast, the winds in the models with magnetic drag each only produce a 0.3 km s⁻¹ difference in the Doppler shifts between the leading and trailing limbs. If this effect can be measured, it allows for another constraint to be placed on the amount of drag present in a hot Jupiter atmosphere. It is worth pointing out that correctly measuring the Doppler shifts that arise exclusively from winds requires that both the orbital and rotational effects be subtracted off, which necessitates an assumption of tidal locking along with knowledge of the orbital speed.

Measuring velocity shifts during ingress and egress presents two key challenges. The first, which has already been mentioned, is that high signal-to-noise spectra must be obtained using fairly short exposures since transit ingress/egress may each only last for 10–30 minutes in total. A second major challenge to obtaining high signal-to-noise during ingress/egress is that the excess absorption from the planet's atmosphere that creates the transmission spectrum decreases in magnitude as the planet moves off the stellar disk. The amount of excess absorption scales linearly with the cross-sectional area of atmosphere that is in front of the star at a given time, which quickly reduces to zero at the beginning and end of transit. With almost no photons passing directly through the planet's atmosphere near first and fourth contacts, it would be impossible to measure any meaningful Doppler shifts, even if the winds at the edge of the planet's limb were particularly strong. Doppler shift measurements during ingress/egress should still be possible when a larger fraction of the planet is in front of its host star, but the signal-to-noise requirements will almost certainly require next-generation telescopes and instrumentation.

Another interesting constraint can be placed on the spatial structure of the winds by measuring Doppler shifts from individual spectral lines. Stronger spectral lines in transmission result from the planet becoming optically thick at larger radii, so these lines originate higher in the atmosphere. Therefore, by measuring velocity shifts in individual spectral lines as a function of their peak strength, a measurement can be made of the vertical structure of the exoplanet winds. If stronger lines produce larger Doppler shifts, then the winds higher in the atmosphere are stronger, whereas weaker Doppler shifts at altitude imply that wind speeds fall off as a function of height.

Our drag-free atmosphere models tend to produce higher line-of-sight wind speeds at altitude, whereas the magnetic drag models have very little wind shear or even opposite wind shear in the case of magnetic drag model (b) (see Figure 4). To test the extent to which this could be an observable effect, we determine Doppler shifts for 134 individual spectral lines relative to their rest wavelengths. The results for each of the four atmosphere models are shown in Figure 9. As predicted, the drag-free models show increasing velocity (blue) shifts as a function of increasing line strength, whereas the magnetic drag models both show a slight trend toward decreasing velocity shifts at altitude (more so for model (b)). The line shifts were all calculated by measuring the shift in the wavelength of peak intensity from the unshifted model to the fully Doppler-shifted model. For this reason, velocity shifts are only measured in integer multiples of the velocity resolution of our R = 10⁶ spectra, resulting in the vertical alignment seen among groups of data points in Figure 9.

Zoom In Zoom Out Reset image size

Linear fits to the line-shift data for each model are also shown in Figure 9 to guide the eye, as there is considerable scatter in the measurements. The 134 lines sampled include spectral lines from CO, H₂O, and CH₄, which likely increases the scatter in the line strength versus velocity relationship. This is particularly true for CO and CH₄, because lines from these two molecules tend to originate from different regions of the atmosphere—CO from regions of hotter gas, likely weighted toward the eastern terminator, and CH₄ from regions of colder gas on the western terminator. Measuring the line strengths for only the 76 first-overtone lines of CO (not shown) results in a tighter linear relationship for line strength versus velocity. However, this particular set of lines samples a far smaller range of altitude in the atmosphere, since all of the CO spectral lines are quite strong and therefore originate from regions of high altitude and low pressure.

Measuring Doppler shifts of individual spectra lines in transmission is another effect that will almost certainly require next generation instrumentation to realize, due to the very high signal-to-noise requirements. The only measurements of Doppler shifts in transmission to date (Snellen et al. 2010) required the full spectrum from 2291 to 2349 nm to detect a Doppler shift at km s⁻¹ precision. Individual spectral lines were not present in their spectra at close to sufficient signal-to-noise to measure RV shifts, even when integrating over a full transit.