EXPRES. I. HD 3651 as an Ideal RV Benchmark

Following the early detections of gas giant planets around Sun-like stars, radial velocity (RV) surveys saw a steady stream of discoveries, punctuated by regular improvements in instrumental precision. With the introduction of the environmentally stabilized High Accuracy Radial velocity Planet Searcher (HARPS) spectrograph in 2003 (Mayor et al. 2003), single-measurement RV precision reached ∼1 ${\rm{m}}\ {{\rm{s}}}^{-1}$. Multidecade campaigns continued to push to lower mass planets and longer period orbits, but the state-of-the-art RV precision (Fischer et al. 2016) has remained at this level for more than a decade.

The long-term nature of RV surveys has enabled the collection of thousands of high-fidelity measurements for a range of spectral types. Many of these stars are RV standards without detected planets. The RV rms scatter of standard stars or the residuals after fitting a simple Keplerian model has been used to evaluate RV measurement precision. The quietest stars have shown a scatter of just under ∼2 ${\rm{m}}\ {{\rm{s}}}^{-1}$ (Huang et al. 2018; Soubiran et al. 2018; Brems et al. 2019). However, it is unclear how this RV scatter is apportioned between astrophysical, instrumental, and analysis error sources.

The Kepler and K2 transit surveys have demonstrated that nearly every star hosts at least one planet (Burke et al. 2015; Hsu et al. 2018). The most commonly detected transiting planet has a radius between 1 and 4 ${R}_{\oplus }$ (Burke et al. 2015) and many of the transiting planet architectures contain tightly packed systems of small planets (Winn & Fabrycky 2015) that would produce short-period, low-amplitude reflex velocities in the host stars. Analogs of the Kepler rocky planets and compact multiplanet systems are largely missing from RV surveys. This implies that at least some of the RV scatter in standard stars is likely caused by undetected low-mass planets. Both improved RV precision and higher observing cadence are required to tease out these signals. Since a reliable sample of stars without planets does not exist, a new type of standard star is needed to evaluate improvements in RV precision.

The Extreme Precision Spectrometer (EXPRES) is one of the first in a new generation of Extreme Precision Radial Velocity (EPRV) instruments delivering high-fidelity data with the goal of disentangling photospheric velocities from Keplerian velocities. EXPRES is located at the Lowell Discovery Telescope (LDT; Levine et al. 2012; DeGroff et al. 2014). The instrumental stability is at least 10 cm s⁻¹ (Blackman et al. 2020) with single-measurement uncertainties of about 30 cm s⁻¹ in spectra with a signal-to-noise ratio (${\rm{S}}/{\rm{N}}$) ∼ 250 per pixel at 550 nm (Petersburg et al. 2020). The primary science mission for EXPRES is the 100  Earths  Survey to identify low-mass planets in habitable zone orbits around Sun-like stars. The combination of high-precision measurements and high observing cadence will enable the detection of planets that were commonly found with the Kepler mission, but that have been missed in previous RV surveys.

In this paper, we highlight our RV data for HD 3651 b as a way to evaluate the long-term, on-sky precision of EPRV instruments. The high eccentricity of this planetary orbit dynamically clears out most simulated test particles within and slightly outside its orbit, as expected from stability theory (Gladman 1993). Similar orbital parameters are known for only a handful of detected exoplanets. HD 3651 is especially well suited as a standard star for demonstrating RV precision because this bright star is accessible by all current EPRV instruments.

EXPRES was fully commissioned in 2019 February and has been used to collect science observations for the 100  Earths  Survey since that time. The high-resolution optical spectrograph is a fiber-fed echelle design (Jurgenson et al. 2016) with double scrambling and active agitation (Petersburg et al. 2018). It covers 3800–7800 Å and has a median resolving power of R = 137,500. EXPRES is located in a vibration isolated vacuum enclosure, in a temperature controlled room. The front-end module contains an atmospheric dispersion compensator and a fast tip-tilt system keeps the star focused on the 09 octagonal input fiber. The overall seeing-dependent throughput is ∼8%–15% (Blackman et al. 2020). Wavelength calibration is carried out with a Menlo Systems laser frequency comb (LFC; Probst et al. 2016), and we have demonstrated an instrumental precision better than 10 cm s⁻¹ (Blackman et al. 2020). A chromatic exposure meter picks off 2% of the light to monitor photon arrival times (Blackman et al. 2017, 2019). The current single-measurement precision is 30 cm s⁻¹ at ${\rm{S}}/{\rm{N}}$ of 250 per pixel (Petersburg et al. 2020), meeting the spectrograph design goals. Further work is underway to mitigate the impact of photospheric noise on the RVs. All instrument adjustments and observations are handled through a python-based messaging server and associated database with a web front-end, enabling high-cadence observations with minimal overhead.

2.1. The Science Goals

2.2. Stellar Targets for the 100 Earths Survey

2.3. Exposure Times

2.4. Cadence

2.5. Data Reduction and Analysis

2.6. Validating On-sky Precision

To evaluate our measurement precision, we wanted to rule out contamination from low-amplitude, short-period planets. Instead of focusing on RV quiet stars, we selected a star with a known planet in a very eccentric and moderately long-period orbit. Our simulations show that short-period planets should not be able to survive (Section 5), leaving the exoplanetary system free of RV scatter from undetected exoplanets.

HD 3651 is an old nearby K dwarf with stellar parameters summarized in Table 2. The star hosts a Saturn-mass planet ($M\sin i$ ∼ 70M_⊕) on a ∼62 day orbit with an eccentricity of 0.61. Since its discovery (Fischer et al. 2003), additional observations have been obtained using the Keck High Resolution Echelle Spectrometer (HIRES; Butler et al. 2017). Using 161 archival observations taken over 17 yr, we fit a single-planet model to the data with an rms scatter to the residuals of 3.4 m s₋₁ (Table 4). This is a few times larger than the single-measurement precision of HIRES and close to the low end of the distribution of RV rms scatter for Keck HIRES (Fischer et al. 2016). The low activity of the star (${R}_{\mathrm{HK}}^{{\prime} }$ = −5.01) makes this an ideal target with low intrinsic stellar jitter (Isaacson & Fischer 2010). The rms in the HIRES RV residuals suggests that instrumental or analytical uncertainties may dominate the error budget. For comparison, Cosentino et al. (2014) found that a single-planet fit to two years of data using the HARPS-N spectrograph gave an rms scatter of 1.82 ${\rm{m}}\ {{\rm{s}}}^{-1}$ for HD 3651.

Table 2.  HD 3651 Parameters

Parameter	Value
Identifier	54 Psc
	HR 166
	HIP 3093
V mag	5.88
B–V	0.85
dist (pc)	11.137 (0.007)
L	0.52
${M}_{{\rm{V}}}$	6.11
Sp Type	K0V
Age (Gyr)	8.2 (3.0)
${T}_{\mathrm{eff}}$ (K)	5210 (30)
$\mathrm{log}g$	4.45 (0.15)
$[\mathrm{Fe}/{\rm{H}}]$	0.05 (0.05)
$v\sin i$ (km s−1)	1.7 (0.5)
Mass (${M}_{\odot }$)	0.8 (0.05)
Radius (${R}_{\odot }$)	0.88 (0.02)
RV (km s−1)	−33.00 (0.16)
$\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$	−5.01
Prot (days)	44.5

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Table 3.  EXPRES RVs of HD 3651

BMJD	Vel (${\rm{m}}\ {{\rm{s}}}^{-1}$)	Err (${\rm{m}}\ {{\rm{s}}}^{-1}$)
18714.482110	−10.090	0.329
18714.490311	−9.811	0.334
18715.476876	−11.452	0.377
18715.485466	−11.262	0.396
18716.417348	−12.623	0.311

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 4.  Keplerian Model for HD 3651 b

Parameter	EXPRES	Keck HIRES
(1)	(2)	(3)
P (days)	61.88 ± 0.55	62.26 ± 0.075
Tp (days)	58726.2 ± 1.2	58726.68 ± 0.5
e	0.606 ± 0.09	0.612 ± 0.12
ω	243.8 ± 23.4	231.9 ± 41
$K({\rm{m}}\ {{\rm{s}}}^{-1})$	16.93 ± 0.22	17.15 ± 0.9
$M\sin i\,({M}_{\oplus })$	69.04 ± 4.1	66.88 ± 5.9
arel (au)	0.284 ± 0.002	0.285 ± 0.001
$\mathrm{RMS}\ ({\rm{m}}\ {{\rm{s}}}^{-1})$	0.58	3.4

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3.1. Observations

We obtained 61 EXPRES RV measurements between 2019 August and 2020 February, which are presented in Table 3. The velocities were derived from optimally extracted spectra with forward modeling, as described in Petersburg et al. (2020). A histogram showing the distribution of tabulated single-measurement errors is shown in Figure 1. The relatively high cadence of the 100  Earths  Survey allowed for excellent phase coverage over two orbital periods for HD 3651 b. On a night when we did not have telescope time, Lowell astronomer Maxime Devogele kindly yielded about 20 minutes of his time so that we could obtain a set of four spectra that allowed us to catch the rapid velocity change during periastron passage in 2019 November.

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3.2. Keplerian Fitting

Keplerian modeling of the velocities for HD 3651 was carried out using a Levenburg–Marquardt algorithm to fit the linearized Keplerian equations (Wright & Howard 2009) that are built into the IDL widget Keplerian Fitting Made Easy developed by Giguere et al. (2012). The best-fit model yields an orbital period of 61.88 ± 0.55 days, consistent with the better constrained orbital period of 62.26 ± 0.075 days modeled with the 17 yr time baseline of Keck HIRES data (Butler et al. 2017). Fixing the orbital period to 62.26 days gives an equally good fit in the EXPRES data (Figure 2) with a residual velocity rms of 58 cm s⁻¹. There is no apparent periodicity in the residuals and a Lomb–Scargle periodogram shows no significant peaks.

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To obtain uncertainties in the orbital parameters, we ran 1000 bootstrap Monte Carlo (MC) trials. For each MC trial, we fit a Keplerian orbit to the data, subtracted the best-fit model, scrambled the residuals (seeding with a random number generator), and added the scrambled residuals back to the best-fit model Keplerian velocities. We also carried out 1000 bootstrap MC trials with the Keck HIRES data (Butler et al. 2017). The bootstrap MC errors on the orbital period and the time of periastron passage were smaller with the Keck HIRES data set because of the longer time baseline; however, the errors on all other orbital parameters were somewhat larger with the Keck HIRES data because of the larger error bars on those RV measurements. The independently fit model parameters for HD 3651 b using EXPRES and HIRES data are summarized in Table 4.

The phased orbital fit for the EXPRES velocities is shown in Figure 3 where we also include the fit to the archival Keck data phased and plotted in the same way for comparison. The EXPRES data show much less scatter in the RVs with scatter in the residuals that is about six times smaller than the Keck HIRES data.

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To analyze the variability and look for serendipitous transits of the planet, we present 1192 photometric observations of HD 3651 acquired over an interval of 25 yr from the 1993–1994 to the 2017–2018 observing seasons with the T4 0.75 m Automatic Photoelectric Telescope (APT) at Fairborn Observatory in southern Arizona. Our observations include the 10 observing seasons presented in the HD 3651b discovery paper of Fischer et al. (2003). The T4 APT is equipped with a single channel photometer that uses an EMI 9124QB bi-alkali photomultiplier tube to measure the difference in brightness between the program star and three nearby comparison stars in the Strömgren b and y passbands. To improve the photometric precision, we combine the differential b and y magnitudes into a single $(b+y)/2$ passband. The precision of a single observation with T4, as measured from pairs of constant comparison stars, is around 0.0015 mag on good nights. The T4 APT is described in Henry (1999), where further details of the telescope, precision photometer, and observing and data reduction procedures can be found.

Table 5 gives a summary of the photometric results of HD 3651. We computed the differential magnitudes in the sense HD 3651 minus HD 3690, the best of our three comparison stars. All magnitudes in the table refer to the average $(b+y)/2$ passband. The standard deviations of a single observation from the seasonal means, given in column 4, range from 0.00105 to 0.00197 mag, so the night-to-night scatter in the observations is similar to the typical measurement uncertainty. Period analysis found no significant variability within any observing season. However, the seasonal means given in column 5 exhibit a range of ∼0.003 mag. HD 3651 is a quiet star with a very low value of $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }=-5.01$ and so probably exhibits no spot variability measurable at our precision. The K0 Iab comparison star is the likely source of the long-term variability.

4.1. APT Photometric Analysis

For further analysis, the 25 observing seasons of APT photometry are normalized such that all 25 seasons have the same mean magnitude as the first. This removes the long-term variability in the comparison star and in HD 3651, if any. The 1192 normalized observations are plotted in the top panel of Figure 4, where we note that the standard deviation of all observations from the normalized mean is 0.00153 mag, consistent with our measurement precision. A period search of the complete normalized data set shows no significant variability between 1 and 100 days, as expected from HD 3651's low value of $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ and the low scatter in the observations. In particular, we find no signal in the vicinity of the estimated 44.5 day rotation period from Fischer et al. (2003).

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The observations are phased with the epoch of transit center and the orbital period of the planet and are plotted in the middle panel of the figure. A least-squares sine fit of the normalized observations to the planetary orbital period gives a peak-to-peak amplitude of 0.000025 ± 0.000125 mag. This extremely low limit to any variability on the planetary orbital period is strong confirmation that the observed Doppler shifts are due to the planetary reflex motion of HD 3651 b.

Finally, in the bottom panel of Figure 4, we show the photometric observations near the transit epoch predicted from the RVs. Given the orbital period and orientation of the orbit, the transit probability is only about 1%. The transit duration is computed from the orbital elements and properties of the star, while the transit depth is estimated to be around 0.01 mag. The horizontal error bar below the transit window is the uncertainty in the time of transit. It is clear there remains no evidence for transits of HD 3651 b.

From the photometric observations, we can see that HD 3651 is a chromospherically inactive star. The high-eccentricity planet also makes it unlikely that there are interior planets that might contribute to additional low-amplitude, high-frequency noise in RV fitting. We test the stability of various orbits using N-body simulations of the HD 3651 b system. We use the MERCURIUS hybrid symplectic integrator included in the REBOUND package (Rein et al. 2019). We circularly distributed zero-mass test particles throughout the HD 3651 system at 21 logarithmically spaced values of semimajor axis. Orbits ranged from $(1-e)/2$ to $2(1+e)$ times the semimajor axis of HD 3651 b. The simulation was integrated with 1 day timesteps for 10 Myr, with post-Newtonian precessional effects (Nobili & Roxburgh 1986; Tamayo et al. 2020) taken into account. Particles that collided with the planet or host star were removed from the simulation. The outcomes are depicted in Figure 5. HD 3651 b clears out most particles within and slightly outside of its orbit, as expected from stability theory (e.g., Gladman 1993). Exceptions tend to be in mean-motion resonance with the planet, but their orbits are significantly disturbed. Although some particles survive within ∼0.09 au, their orbits are disturbed by the planet as seen by fractional changes in semimajor axis. Remaining particles may not last on longer timescales; indeed, a few particles within this boundary were cleared out within 10 Myr. If a close-in Earth-mass planet could survive, it would leave a semi-amplitude signature of no more than ∼34 cm s⁻¹. This value is above our measurement precision, however, our sampling might not be sufficient for such a planet to be detectable in the periodogram. More massive or closer-in planets would leave more significant signals.

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5.1. Detectability Limits

Given the potential for at least marginally stable orbits at very short periods, we ran simulations to place mass limits on planets that might remain undetected with the data at hand. From the dynamics, planets within 0.09 au of the host star, or with less than 11 day orbits, may survive. We therefore simulate possible planets out to 15 day periods and up to 15 Earth masses.

For each planet, we simulate Keplerian RVs with identical temporal sampling as the observed data sets, preserving any window functions in the observations. Representative white and red noise is added, scaled to the rms of each data set's planet fit and the calculated activity level of HD 3651 respectively. For each simulated planet, 1500 independent realizations of noise are generated. We then compare the periodogram power of just the noise against the periodogram power of the noise with the injected RV signal at the injected period.

The p-value for each simulated planet gives the probability that the noise has equally or more significant signal as the injected signal plus noise, meaning the injected signal was buried in the noise. A p-value of less than 0.01 is deemed a successful detection. The results are shown in Figure 6, where contours are drawn in p-value space. Blue areas indicate that if such a planet existed, it would have already been detected. Only planets smaller than two or three Earth masses may still be hidden in the data.

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Although we are most concerned about short-period signals, our dynamical simulations also showed possible stable orbits beyond the 2:1 mean-motion resonance of HD 3651 b. We performed the same detectability simulations as above for periods between 120 and 900 days with planet masses of 2M_⊕ ≤ M ≤ 150M_⊕ (see Figure 5 right). We find that planets with masses above 6 ${M}_{\oplus }$ are excluded out to 160 days, M > 8 ${M}_{\oplus }$ is excluded out to 210 days, and M > 10 is excluded out to 300 days. There is a window near 150 days where a lack of the higher cadence EXPRES data would permit planets up to 10 ${M}_{\oplus }$ to remain undetected. All planets above 25-${M}_{\oplus }$ are excluded out to 900 day orbits. The higher precision of EXPRES allows the EXPRES data to exclude more short-period planets with lower masses than even the 20 yr time baseline of the HIRES data. However, as expected, the long time baseline of Keck data surpasses the 4 month time baseline of EXPRES data for orbital periods longer than a year.

The EXPRES 100  Earths  Survey has completed its first year of science operations. The instrumental precision of EXPRES has been measured to be 10 cm s⁻¹ (Blackman et al. 2020) and the single-measurement precision on stars is about 30 cm s⁻¹ for spectra with ${\rm{S}}/{\rm{N}}$ of 250 per pixel near 550 nm (Petersburg et al. 2020). In addition to instrumental errors, we expect that other contributors to the RV error budget will include photospheric velocities, undetected low-mass planets in short-period orbits, telluric contamination, and errors from our analysis methods.

To eliminate some of the possible terms in the EPRV error budget we have used a new benchmark: the chromospherically quiet HD 3651 star, which hosts an eccentric, Saturn-mass planet in a ∼62 day orbit. We carried out N-body simulations to demonstrate that planets interior to the Saturn-mass planet, HD 3651 b, would be dynamically unstable; this eliminates undetected short-period, low-mass planets as one possible source of RV scatter in our data. The rms scatter after fitting a Keplerian model is 58 cm s⁻¹ over ∼6 months. This suggests that added (roughly in quadrature) to our single-measurement errors of 30 cm s⁻¹, the remaining error terms (stellar activity, imperfectly modeled telluric contamination, and long-term instrumental drifts) contribute no more than about 50 cm s⁻¹. The residuals to this single-planet fit give a good measure of the true long-term RV precision for chromospherically quiet stars observed with EXPRES. If similarly quiet stars exhibit more than 50 cm s⁻¹ RV scatter, then undetected, short-period planets are good candidates for those RV variations.

Importantly, the result that the intrinsic long-term precision for chromospherically quiet stars is ∼50 cm s⁻¹ helps to answer the question of whether RV precision in previous-era spectrographs was limited by the instrument, the analysis methods, or stability of the stellar photosphere. Several astronomers have long argued that photospheric velocities were the tall pole in the RV error budget. However, this work shows that in the case of quiet stars, the photospheric velocities were not dominating the error budget and this is validation for our decade-long effort to design EXPRES as a next-generation EPRV spectrograph. While it is certainly true that photospheric velocities from active stars will be a strong contributor to the RV error budget, it is also likely that the high-fidelity data from next-generation EPRV spectrographs offer the best chance for ultimately disentangling those photospheric velocities. The EPRV spectrographs will help the community to take the next big step along the path toward detecting smaller amplitude signals. This is a new parameter space for RV surveys. The statistical results from the Kepler mission suggest that this new parameter space will be a rich source of previously undetected exoplanets.

EXPRES is not the only new EPRV instrument, and users of other EPRV spectrographs will want to evaluate their on-sky precision to track down instrumental issues that may affect their planet detection capability. HD 3651 b is an ideal benchmark for demonstrating long-term precision and for showing improvements relative to EXPRES. At a moderate northern decl., the star is observable by all current EPRV spectrographs and its brightness makes it an ideal standard for comparing RV precision and instrumental stability.

These results made use of the Lowell Discovery Telescope at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the LDT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University, and Yale University. We thank Lowell astronomer, Maxime Devogele, for generously yielding telescope time so that we could obtain the radial velocity point for HD 3651 b near periastron passage. We gratefully acknowledge ongoing support for telescope time from Yale University, the Heising-Simons Foundation, and an anonymous donor in the Yale community. We especially thank the NSF for funding that allowed for precise wavelength calibration and software pipelines through NSF ATI-1509436 and NSF AST-1616086 and for the construction of EXPRES through MRI-1429365. G.W.H. acknowledges long-term support from NASA, NSF, Tennessee State University, and the State of Tennessee through its Centers of Excellence program. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under grant No. DGE1122492 (R.R.P., L.L.Z., and A.B.D.).