CHOICES ALONG THE MULTI-EPOCH ANALYTIC PATHWAY
6.2 Physical Considerations Planetary Orbital PositionPlanetary Orbital Position
C h a p t e r 6
Quadrature Epochs
There are several factors to consider in selecting orbital positions. In the multi-epoch technique, a two-dimensional cross correlation uses line-of-sight velocities to tease apart the stellar and planetary signals. From this perspec- tive, quadrature (π = 0.25,0.75) epochs seem the most appealing as they provide the largest velocity separation between the stellar and planetary sig- nals. Additionally, at quadrature, the planetβs line-of-sight acceleration is the slowest. This means that at quadrature, we can afford to take slightly longer observationsβto either build up a higher signal-to-noise or to obtain a longer baseline to support telluric correctionsβwithout worrying that the planetary signal will shift across detector pixels, leading to a weaker signal overall and a broadenedπΎπdetection.
We can estimate the change in planetary line-of-sight velocity at an epoch with the equation,
Ξπ£π π π =2πΎπsin
π βπ π π πβπ
cos(2π ππ πππ‘
π π π ), (6.2)
whereβπ π π is the length of the observation in hours,πβπis the orbital period is hours, andππ πππ‘
π π π is the orbital phase at the center of the observation. We note that this equation measures the difference between the start and end positions of the planet. If the center of the observation is at quadrature, Ξπ£π π π = 0 because the planet starts and ends at a common velocity. If an observation passes through either π =0.25 or 0.75, then, this equation should be broken up into the portions before and after quadrature.
This equation makes a few assumptions. First, that the planet is on a circular orbit. Second, that the change in the barycentric velocity over the observation is negligible, which is a very reasonable assumption since the maximum change in barycentric velocity over 12 hours, longer than any ground-based observation, is around 0.3 km/s.
As an example, for a planet with a 3-day orbital period and a Keplerian orbital velocity,πΎπ, of 100 km/s, in an 5-hour observation centered around conjunc- tion, the planetary line-of-sight orbital velocity would change by 43.3 km/s.
The smallest change in velocity would happen during an epoch centered at quadrature, when the planetary signal would shift in one direction during half of the observation and back during the other half. For the same system, the smallest change in planetary velocity, from 2.5 hours before quadrature to
quadrature itself, before the planet signal starts shifting back, would be only 2.4 km/s. The signal-to-white noise per observation and telluric correction procedures benefit from longer observations, while the instrumental resolu- tion limits the acceptable change in planetary velocity over the observation, therefore limiting the observation length. We will discuss each of these factors in future sections. These calculation do show, though, that quadrature epochs offer not only the largest separation between planetary and stellar signals, but also an opportunity to take the longer observations without having the planetary signal diminish by shifting over detector pixels.
Day-Side Epochs
Recent high-resolution spectroscopic investigations of hot Jupiters have con- sidered how the signatures of tidally locked planets may change versus orbital position. In other words, if their day- and night-sides show different chemical and physical properties, their day- and night-side spectra may look very dif- ferent. Brogi et al. 2012, and other CRIRES-style detections, have targeted the day-side, seen at secondary eclipse, under the assumption that the brighter day-side should be easier to detect that the fainter night-side. From this per- spective, day-side epochs (between π = 0.25 and 0.75) may allow stronger detections, though simulations could be run to determine how the increase from a larger fraction of visible day-side moving towards superior conjunc- tion (π = 0.5) and the decrease from smaller relative velocity between the planetary and stellar features trade off.
Further, high-resolution spectroscopy can also be used to gain information about planetary atmospheric motion. In high-resolution CRIRES data from the hot Jupiter, HD 209458 b, Snellen et al. 2010 found hints of weak day- to-night side winds. Beltz et al. 2021 asked whether the atmosphereβs three- dimensional structure could be further constrained by fitting CRIRES data of the HD 209458 b with three-dimensional atmospheric circulation models that considered temperature structure and atmospheric motion, such as winds and planetary rotation, rather than one-dimensional models. They found an increase in the detection significance of at least 1.8πwith three-dimensional models, with the primary improvement coming from the inclusion of a 3D temperature structure which varies spectral feature depths relative to what one would expect from a 1D temperature structure, and secondary improve-
ments from chemistry and Doppler effects. The multi-epoch analysis and simulation framework has not, to date, considered differences in the planetary spectra as a function of planetary orbital phase. Inclusion of such planetary spectral variations as a function of orbital phase in the simulation framework could more strongly indicate certain orbital positions as the most effective for planetary detection, and thus support the planning of future observations.
Three-dimensional atmospheric models could also be considered in cross correlating the data; we will describe how this could be done in Section 6.4.
Barycentric Velocity
The barycentric velocity is the portion of the Earthβs orbital motion in the direction of the target system. It depends on the targetβs right ascension and declination, as well as the time of observation. We incorporate the barycentric velocity into the primary velocity, as
π£ππ π =π£π π¦ π βπ£π ππ π¦, (6.3)
whereπ£π π¦ π is the systemic velocity, the relative velocity between the systemβs center- of-mass and the Solar Systemβs center-of-mass. We do not consider the stellar reflex motion because it is below the velocity resolution of most near-infrared high resolution instruments (NIRSPEC2.0βΌ 3 km/s), typically on the order of 0.1β0.01 km/s.
In Buzard et al. 2021a, we found that primary velocities near 0 km/s lead to the strongest planetary detections, on average more than twice the significance of de- tections made with randomly selected primary velocities. This trend grew even stronger with cooler host stars. Observing nights should be chosen then, when possible, when the barycentric velocity (nearly) cancels out the systematic velocity.
During such epochs, the host stellar spectrum is aligned with the telluric frame.
These results came from simulations of a hot Jupiter-like planet with a Keplerian orbital velocity,πΎπ, of 75 km/s. As we look to planets on longer orbits, with smaller values of πΎπ, or to cooler planet temperatures, near-zero primary velocity epochs may no longer be as advantageous. Smaller values ofπΎπ would bring the planetary spectrum closer to the stellar (and telluric) frame. This could make telluric correction more difficult, especially for cooler planets which are more spectroscopically similar to our telluric atmosphere. The suggestion of near-zero primary velocity epochs should, therefore, be taken with care as we move on to new planet populations.
6.3 Instrumental Considerations