M2K. II. A TRIPLE-PLANET SYSTEM ORBITING HIP 57274^*

HIP 57274 (GJ 439) has an apparent magnitude of V = 8.96, color B − V = 1.111, and parallax of 38.58 ± 1 mas according to the Hipparcos catalog (ESA 1997; van Leeuwen 2007). This yields a distance of 25.9 pc and absolute visual magnitude of M_V = 6.89. We carried out spectral synthesis modeling of the iodine-free template spectrum using Spectroscopy Made Easy (SME; Valenti & Piskunov 1996; Valenti & Fischer 2005) to determine stellar parameters. Following the method described in Valenti et al. (2009), the initial parameters derived with SME were used as input for interpolation of the Yonsei-Yale (Y2) isochrones (Demarque et al. 2004), which returns a new value for log g. We then ran an iterative loop, fixing log g to the isochrone value and running a new SME model fit. The other (free) stellar parameters change in response to the fixed surface gravity, so subsequent isochrone interpolations produce a slightly different value for log g. We continue the iteration until the output log g from the isochrones does not change by more than 0.001 dex from the previous iteration. This provided the following spectroscopic parameters: T_eff = 4640 ± 100 K, vsin i = 0.5 ± 0.5 km s⁻¹, [Fe/H] = +0.09 ± 0.05, log g = 4.71 ± 0.1, and Y2 isochrone models for a stellar mass of 0.73 ± 0.05 M_☉, a radius of 0.68 ± 0.03 R_☉, and an age of 7.87 ± 5 Gyr. The brightness is consistent with any main-sequence age. The spectral classification is listed as K8V in the Hipparcos catalog, although the B − V color, spectroscopic temperature, and derived mass are more consistent with a somewhat earlier spectral type. One of us (S.L.) has obtained a medium-resolution spectrum with the Mark III spectrograph at the MDM 1.3 m telescope; the absence of a clear TiO absorption band head redward of 7000 Å rules out spectral classifications of K5V or later. The hint of a weak TiO band absorption is consistent with the K4V classification of Gray et al. (2003), and we adopt this spectral classification here. The stellar parameters for HIP 57274 are summarized in Table 1.

Isaacson & Fischer (2010) determined the chromospheric activity of 2630 stars observed at Keck by measuring the emission in the cores of the Ca ii H and K lines relative to adjacent continuum regions. These S_HK values were calibrated to the long-standing S_HK values from the Mt. Wilson program (Duncan et al. 1991). Using their technique, we measure a mean S_HK = 0.38 for HIP 57274. The individual measurements of S_HK are listed in the last column of Table 2, along with the radial velocity measurements.

Table 3. Orbital Parameters for HIP 57274

Parameter	b	c	d
P (days)	8.1352 (0.004)	32.03 (0.02)	431.7 (8.5)
Tp − 2.44 × 106 (JD)	14801.015 (1.3)	15785.208 (9.5)	15108.116 (14)
ecc	0.187 (0.10)	0.05 (0.02)	0.27 (0.05)
ω (deg)	81 (59)	356.2 (120.0)	187.2 (5)
K1 (m s−1)	4.64 (0.47)	32.4 (0.6)	18.2 (0.5)
Msin i (M⊕)	11.6 (1.3)	130 (3)	167.4 (8)
arel (AU)	0.07	0.178	1.01
Tc (HJD − 2.44e6)	15801.779 (0.271)	15793.035 (0.176)
Tduration (hr)	3.08 (0.35)	5.88 (0.1)
tprob	6.5%	2.7%
Trend (${\rm m\ s^{-1}\ {\rm day}^{-1}}$)	−0.026 (0.002)
Avg S/N	225
rms (m s−1)	3.15
Nobs	99
χ2ν	1.06
Jitter (m s−1)	2.8

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Cooler stars typically have larger S_HK values than solar-type stars because of weaker continuum in the near-UV. Therefore, S_HK values should not be directly compared for stars of different spectral types. Noyes et al. (1984) correct for the photospheric contributions to produce a normalized activity metric, log R'_HK. Chromospheric activity is tied to dynamo-driven magnetic fields and decreases as the star ages and spins down. Noyes et al. (1984) made use of this activity–rotation correlation and calibrated log R'_HK to rotational periods for stars in open clusters of different ages. Using the Noyes relation, we derive log R'_HK = −4.89, indicating low activity for HIP 57274 and P_rot ∼ 45 days, implying a relatively old age in the broad range allowed by the observed brightness. This is also in agreement with the periodogram of the S_HK activity indicator (Figure 1), which has power at 45.9 days. We caution that both log R'_HK and P_rot were calibrated by Noyes et al. (1984) using stars with 0.4 < B − V < 1.0 and rotational periods shorter than 30 days; HIP 57274 falls outside both of these properly calibrated ranges and therefore our derived log R'_HK and rotational period should be considered to be rough estimates. The rotation period will need to be verified with photometric observations during the next observing season for this star.

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Because the activity calibration for stars redward of B − V = 1.0 is an extrapolation, Isaacson & Fischer (2010) established a differential activity measurement, ΔS_HK, and evaluated the impact of chromospheric activity on radial velocities in four separate ranges of B − V. Following their method, we plot the S_HK index for the 170 stars observed on the M2K program with B − V color from 0.8 to 1.6 (Figure 2). The dashed line in this plot is taken from Isaacson & Fischer (2010) and indicates the baseline S_HK values for low-activity stars. ΔS_HK is the difference between this baseline activity level and the mean S_HK for a given star. Active stars float high above the baseline values while chromospherically quiet stars are closer to the dashed line in Figure 2.

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In Figure 3, we plot velocity rms as a function of excess chromospheric activity, ΔS_HK, for all of the observed M2K stars. We fit a linear function to the lower 20th percentile velocity scatter and interpret this (red solid line overplotted in Figure 3) as the quadrature sum of internal errors and jitter (where jitter is both instrumental and astrophysical). The rms scatter for inactive stars with ΔS_HK ∼ 0.0 is 2.38 m s⁻¹, and given the typical internal errors of 1.5 m s⁻¹, this implies a minimum jitter of 1.45 m s⁻¹. We measure ΔS_HK = 0.03 for HIP 57274, suggesting a low stellar jitter of ∼1.5 m s⁻¹.

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We have obtained 99 observations of HIP 57274 with a mean signal-to-noise ratio of 200 and an average exposure time of 360 s. The mean formal measurement errors are 1.23 m s⁻¹ (Table 2). Figure 4 shows the time series data, overplotted with our Keplerian model for a triple-planet system. We used the partially linearized Levenberg–Marquardt algorithm (Wright & Howard 2009) built into the Keplerian Fitting Made Easy (KFME) program (Giguere et al. 2011) to model the data. The best-fit Keplerian model for three planets includes a trend of −0.026 m s⁻¹ day⁻¹. Parameter uncertainties were calculated with a bootstrap Monte Carlo analysis (Marcy et al. 2005) in KFME (Giguere et al. 2011), which iteratively fits the data with a best-fit model, then adds the scrambled residuals back to the theoretical velocities before refitting.

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The planet with the shortest period completes one orbit in 8.135 ± 0.004 days and induces a velocity amplitude of 4.64 ± 0.46 m s⁻¹. With a stellar mass of 0.73 M_☉, we derive a planet mass Msin i = 11.6 M_⊕ and a semimajor axis of 0.07 AU. The orbital eccentricity is 0.187 ± 0.10 and the argument of periastron passage ω ∼ 82°. Because the velocity amplitude is small compared to the uncertainties and stellar jitter, the eccentricity for this planet is poorly constrained, however the signal itself is unambiguous. Figure 5 shows the periodogram of the residual velocities of HIP 57274 after removing the other two planets and linear trend described below. We carried out a Monte Carlo test to determine the false alarm probability (FAP) of the periodogram power. In this test, 10,000 trials were carried out where the best-fit (triple Keplerian and trend) model was subtracted and the residual velocities were scrambled before being added back to the theoretical velocities and refitting. In these 10,000 trials, a peak at least as high as observed was never found in the residuals to the fit of the two more massive planets, yielding a FAP less than 10⁻⁴. Figure 6 shows the phase-folded velocities of HIP 57274b overplotted with the theoretical Keplerian model after removing the signals from the two additional planets and the linear trend.

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We calculated the prospective time of transit, transit duration, and transit probability using KFME (Giguere et al. 2011). The transit ephemeris is 2455801.779 ± 0.27 HJD or 2011 August 28 06:41:40.7 UT. The next transit observable from Mauna Kea occurs at 3 AM Hubble Space Telescope (HST) on 2012 January 13, except that the uncertainty in the transit time is more than 6 hr. The duration of the prospective transit would be 3.08 ± 0.356 hr and the geometric probability that this planet will transit is 6.5%.

The middle planet in this system has an orbital period of 32.0  ±  0.02 days. The best-fit model for this planet has a nearly circular orbit, with an eccentricity of 0.05 ± 0.02. The velocity amplitude is 32.4 ± 0.6 m s⁻¹, implying a planet with Msin i = 130 M_⊕ or 0.41 M_Jup and an orbital radius of 0.18 AU. The phase-folded data and model for HIP 57274c is shown in Figure 7 after subtracting theoretical velocities for the linear trend and the inner and outer planets. The prospective ephemeris time is 2454793.035 ± 0.176 HJD, although the longer orbital period for the middle planet results in a lower 2.7% transit probability.

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The third planet, HIP 57274d, has an orbital period of 432  ±  8 days, a velocity amplitude of 18.2 ± 0.5 m s⁻¹, and an orbital eccentricity of 0.27 ± 0.05. The planet mass is Msin i = 0.53 M_Jup, and the semimajor axis of the orbit is a familiar 1.01 AU. Figure 8 shows the time series velocities and Keplerian model for this planet after removing the inner two planets and the linear trend. The velocity rms for the triple-planet fit is 3.15 m s⁻¹. However, if we adopt the predicted jitter of 1.5 m s⁻¹, we find that χ²_ν = 2.7, indicating that the model does not fully describe our observations. The Keplerian models for planets b, c, and d are summarized in Table 3.

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To check for additional planets in the system, we subtracted the theoretical velocities for the linear trend and three Keplerian models. Figure 9 shows a periodogram of the residual velocities, with significant peaks at 1.019 days and 52.996 days that could be aliases of each other, due to the diurnal cadence: 1.0 + 1./52.996 = 1.019 days. This predicts the presence of a second peak at 1.0 − 1./52.996 − 0.981, and we see a second peak in the periodogram at 0.981 days. Interestingly, the ∼53 day peak has been increasing in strength. Unlike the periodograms for the three planets that we modeled (HIP 57472b, c, and d), the 53 day peak is flanked by two additional peaks. We suspect that this signal may be caused by spots rotating on the surface of the star which reduce flux on the approaching blueshifted edge of the star and then on the receding redshifted edge of the star. Physically, this would produce a line profile asymmetry that could be spuriously modeled as reflex motion due to a planet. We checked to see if the residual velocities to the full triple-planet fit were correlated with the activity measurements but found only an insignificant trend (Figure 10, dashed line). We tried detrending the velocities with the linear fit shown in Figure 10, however this only slightly reduced the periodogram power in the residual velocities. If we blindly fit this periodic signal with a Keplerian model, we derive a period of 53.2 days with an amplitude of 2.6 m s⁻¹, and the residuals drop to 2.63 m s⁻¹ with a χ²_ν = 1.28. Interpreting this signal as an additional noise source from coherent spots, we add the 2.63 m s⁻¹ signal in quadrature with the expected jitter of 1.5 m s⁻¹ to obtain a revised jitter estimate of 2.8 m s⁻¹. This changes χ²_ν to 1.06 for the model of a triple-planet system plus a linear trend. If the 53 day signal originates from star spots, then it should have a detectable photometric signal. We have started a photometric campaign to check for spot modulation and to search for the transit signal of the inner 8.135 day planet. In addition to the photometric observations, we will continue to obtain Doppler measurements to better understand the origin of the 53 day signal.

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To assess the stability of the HIP 57274 system, we ran an ensemble of dynamical simulations of the triple-planet system with the hybrid symplectic integrator code Mercury 6 (Chambers 1999). Orbital parameters for each body were calculated from the values in Table 1 assuming Gaussian-distributed errors (with a truncation at 0.0 for eccentricity). We ran 20 simulations with sin i = 1 (minimum mass case) and 15 simulations with sin i = 0.3 (95% of random orientations will have a sin i greater than this value) and a mass of 0.73 M_☉ for the central star. For radii and thus collision probabilities, we assumed mean densities of 6.0, 1.0, and 1.0 g cm⁻³ for the b, c, and d planets, which would be typical for the densities of rocky and gas giant planets. The time step was set to 0.365 days (4.5% of the orbital period of HIP 57274b) and each simulation was run for 10 Myr. In none of the 35 independent simulations did collisions or ejections occur. We conclude that the system is stable regardless of its inclination.

The orbital periods of planets B and C are within 1.6% of a 1:4 mean-motion commensurability and thus we examined the possibility that they might be in resonance. The resonance conditions are that the libration angle

$$\begin{equation} \phi = pL_B - qL_C - m\omega _B - n\Omega _B - r\omega _C - s\Omega _C, \end{equation} \tag{ 1 }$$

where L is the mean anomaly, ω is the longitude of periastron, Ω is the longitude of the ascending node, and p, q, m, n, r, s are small integers, osculates around a fixed value rather than circulates over 0 − 2π. In this case, p = 1 and q = 4. We performed 10 simulations as previously described, varying the initial orbital elements according to the solution uncertainties and recording the osculating elements every 10³ years. In every simulation ϕ orbits with a period of ∼1.5 years. We also examined apsidal resonance between B and C, but found that ω_B − ω_C also circulates, with a period of ∼650 years. Thus there is no evidence that B and C are in an orbital resonance, but the question should be revisited when additional radial velocity data permit more precise orbit determinations.

To further evaluate a planetary explanation for the signal at 53 days, we performed 20 additional dynamical simulations spanning 10 Myr with a four-planet solution, including an "e" planet with an orbital period of 53.3 days and a minimum mass of 12.3 Earths. Other details of the simulations were the same as before. In ten simulations, sin i = 1, and in the other ten, sin i = 0.31. In eight of the first set, e collided with either c or d (four instances each) within 10 Myr. In all ten runs of the second set, e collided with either c (six instances) or d (six instances) in under 10⁵ years. These results strongly disfavor the existence of a fourth planet on a 53 day orbit, at least with a mass sufficient to explain the Doppler signal at that period.