NEW OBSERVATIONAL CONSTRAINTS ON THE υ ANDROMEDAE SYSTEM WITH DATA FROM THE HUBBLE SPACE TELESCOPE AND HOBBY-EBERLY TELESCOPE^*

Spectroscopic observations were obtained with the High Resolution Spectrograph (HRS; Tull 1998) at the HET at McDonald Observatory using the absorption iodine cell method (Butler et al. 1996). A detailed description of our reduction of HET HRS data is given in Bean et al. (2007), which uses the REDUCE package (Piskunov & Valenti 2002). Our observations include a total of 237 high-resolution spectra which were obtained between August of 2004 and July of 2008. Usually, three observations are made in less than one hour per night and those observations are combined with a robust estimation technique using GaussFit (Jefferys et al. 1988), thus creating 80 epochs of observation.

Our RV data was combined with velocities from four other sources to produce a total data set that spans 14 years and includes a new reduction from Lick Observatory (Wright et al. 2009b) provided by Giguere and Fischer (D. Fischer et al. 2010, in preparation), the European Southern Observatory Elodie (Naef et al. 2004), the McDonald Observatory HJS (Wittenmyer et al. 2007), and the Smithsonian Astrophysical Observatory Whipple 60 inch telescope's AFOE Spectrograph (Butler et al. 1999) re-reduced by Korzennik (available from skorzennik@cfa.harvard.edu by e-mail request). The total RV data set contained 974 observations of υ And. Figure 1 shows the times of all RV and astrometric observations. Table 3 contains reduced HET data for the observed epochs.

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FGS-1r a two-axis, white-light interferometer aboard HST was used to make the astrometric observations in position (POS) "fringe-tracking" mode. A detailed description of this instrument is found in Nelan (2007).⁷ Our data sets were reduced and calibrated as detailed in Benedict et al. (2007). This data set used a new improved optical field angle distortion (OFAD) calibration (McArthur et al. 2002), as yet unpublished. The astrometric data used in this research is available from the HST Program Schedule and Information Web site,⁸ in proposal numbers 9407, 9971, and 10103. The two-step pipeline used to reduce the raw data to the values used in this modeling is available with the latest calibration parameters from the Space Telescope Science Institute in IRAF STSDAS and in a standalone version available from one of the coauthors, the HST FGS Instrument Scientist at STScI Ed Nelan (nelan@stsci.edu).

Data are downloaded from the HST online archival retrieval system and processed through the two-stage pipeline system. The initial calibration pipeline extracts the astrometry measurements (typically 1–2 minutes of fringe position information acquired at a 40 Hz rate, which yields several thousand discrete measurements) and the median (after outlier removal), and estimates the errors. The second calibration stage applies the OFAD which is variant over time, and corrects the velocity aberration, processes the time tags, and uses the JPL Earth orbit predictor Standish (1990) to calculate the parallax factors. The methodology of the OFAD is discussed in several calibration papers (McArthur et al. 1997, 2002, 2006). Ongoing stability tests (LTSTABs) maintain this calibration. Systematics introduced by our instrument (such as intra-orbit drift and color and filter effects) and their corrections are discussed below. After 18 years of calibration, we have not uncovered additional systematics in our data which are above our detection limits. Regression analysis between Hipparcos and HST parallax measurements has shown not only good agreement between Hipparcos and HST parallaxes (with the exception of the Pleiades), but an overestimation of error of HST astrometric measurements (Benedict & McArthur 2004; Benedict et al. 2007).

Fifty-four orbits of the HST were used to make 13 epochs of astrometric observations between December of 2001 and August of 2006. Every orbit contains several observations of υ And and surrounding reference stars. The distribution of the reference stars in the υ And field is shown in the Digital Sky Survey image in Figure 2. The FGS measures the position of each star position sequentially. Each epoch contains multiple visits, alternating between the target (υ And) and the 10 reference stars, providing x(t) and y(t) positions in the HST reference frame in seconds of arc. Observations during an orbit were corrected for slow FGS intra-orbit positional drift by using an adaptable polynomial fitting routine (Benedict et al. 2007). Because υ And is a bright star (V = 4.09), a neutral density filter (F5ND) was used for observing it. For the reference stars, the F583W filter was used. The dates of observation, the epoch group of the observation, the number of measurements of υ And for each date, and the HST orientation angles are listed on Table 4. For the day of the year, letters following the day indicate more than one observation set for a single day. The HST astrometric data for υ And and its reference stars is available only online as a machine-readable table (Table 5). For the most current calibration, the data should be retrieved from the HST online archival retrieval system and processed through the two-stage pipeline system.

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Because the parallax determined for υ And is measured with respect to reference frame stars which have their own parallaxes, we must either apply a statistically derived correction from relative to absolute parallax (van Altena et al. 1995, Yale Parallax Catalog, hereafter YPC95), or estimate the absolute parallaxes of the reference frame stars. In principle, the colors, spectral type, and luminosity class of a star can be used to estimate the absolute magnitude, M_V, and V-band absorption, A_V. The absolute parallax is then simply,

$$\begin{equation} \pi _{\rm abs} = 10^{-(V-M_V+5-A_V)/5}. \end{equation} \tag{ 1 }$$

3.3.1. Reference Star Photometry

Our bandpasses for reference star photometry include V (from FGS-1r) and JHK from 2MASS.⁹ The JHK values have been transformed to the Bessell & Brett (1988) system using the transformations provided in Carpenter (2001). Table 6 lists VJHK photometry for the target and reference stars indicated in Figures 2. Figure 3 contains a (J − K) versus (V − K) color–color diagram with reference stars and υ And labeled. Schlegel et al.(1998) find an upper limit A_V∼ 0.3 toward υ And. In the following, we adopt 〈A_V〉 = 0.0 for all but ref-105, ref-108, and ref-112. We increase the error on those reference star distance moduli by 1 mag to account for absorption uncertainty.

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Table 6. V and Near-IR Photometry of υ And A and B and Astrometric Reference Stars

ID	V	K	(J − H)	(J − K)	(V − K)
υ And A	4.24 ± 0.01	2.90 ± 0.27	0.27 ± 0.28	0.34 ± 0.35	1.34 ± 0.27
7	14.22 ± 0.01	12.52 ± 0.02	0.33 ± 0.03	0.41 ± 0.03	1.70 ± 0.03
102	13.11 ± 0.01	11.77 ± 0.02	0.33 ± 0.03	0.35 ± 0.03	1.35 ± 0.02
103	13.77 ± 0.01	12.53 ± 0.03	0.28 ± 0.03	0.29 ± 0.03	1.24 ± 0.03
104	14.51 ± 0.01	12.62 ± 0.02	0.38 ± 0.03	0.52 ± 0.03	1.89 ± 0.02
105	14.82 ± 0.01	12.73 ± 0.02	0.47 ± 0.03	0.56 ± 0.03	2.09 ± 0.03
106	11.41 ± 0.01	9.92 ± 0.02	0.35 ± 0.03	0.36 ± 0.02	1.49 ± 0.02
108	14.62 ± 0.01	12.05 ± 0.02	0.67 ± 0.02	0.76 ± 0.02	2.57 ± 0.02
111	15.82 ± 0.01	14.23 ± 0.06	0.34 ± 0.04	0.34 ± 0.06	1.59 ± 0.06
112	11.78 ± 0.01	9.31 ± 0.02	0.60 ± 0.02	0.70 ± 0.02	2.48 ± 0.02
υ And B	13.86 ± 0.01	8.55 ± 0.02	0.62 ± 0.04	0.92 ± 0.02	4.77 ± 0.02

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The derived absolute magnitudes are critically dependent on the assumed stellar luminosity, a parameter impossible to obtain for all but the latest type stars using only Figure 3. To confirm the luminosity classes, we obtain NOMAD proper motions (Zacharias et al. 2005) for a one-degree-square field centered on υ And, and then iteratively employ the technique of reduced proper motion (Yong & Lambert 2003; Gould & Morgan 2003) to discriminate between giants and dwarfs. The end result of this process is contained in Figure 4.

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3.3.2. Estimated Reference Frame Absolute Parallaxes

3.3.3. Estimated Reference Frame Proper Motions

The υ And reference frame contained a very generous field of 10 (nine usable) reference stars plus υ And B. With the positions (x', y') measured by FGS-1r, we build an overlapping plate model that accounts for scale, rotation, and offset relative to an arbitrarily adopted constraint or "master" plate. The astrometric model also accounts for the time-dependent movements of each star, given by the absolute parallax π_abs and the proper motion components, μ_α and μ_δ, and the corrections for the cross-filter and lateral color positional shifts. Therefore, the model is given by the standard coordinates ξ and η in these equations of condition:

$$\begin{eqnarray} x^\prime & = & x + lc_x(B-V) - \Delta XFx, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} y^\prime & = & y + lc_x(B-V) - \Delta XFy, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \xi & = & Ax^\prime + By^\prime + C - \mu _\alpha \Delta t\nonumber\\ &&- P_\alpha \pi - (\mbox{ORBIT}_yc + \mbox{ORBIT}_yd), \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \eta & = & {-}Bx^\prime + Ay^\prime + F - \mu _\delta \Delta t\nonumber\\ &&- P_\delta \pi - (\mbox{ORBIT}_xc + \mbox{ORBIT}_xd), \end{eqnarray} \tag{ 5 }$$

where x and y are the measured coordinates from HST; lc_x and lc_y are the lateral color corrections, and B − V are the B − V colors of each star; ΔXFx and ΔXFy are the cross-filter corrections in x and y, applied only to the observations of υ And. A and B are scale and rotation plate constants, C and F are offsets, μ_α and μ_δ are proper motions, Δt is the epoch difference from the mean epoch, P_α and P_δ are parallax factors, and π is the parallax. We obtain the parallax factors from a JPL Earth orbit predictor Standish (1990), upgraded to version DE405. In order to find a global solution, we used a program written in the GAUSSFIT language (Jefferys et al. 1988). ORBIT is a function using Thiele–Innes constants (Heintz 1978) of the traditional astrometric and RV orbital elements. Table 9 shows the resulting astrometric catalog from the combined orbital modeling.

There are additional equations of condition relating an initial value (an observation with associated error) and final parameter value. For the reference stars, there are equations in the model for proper motion and spectrophotometric parallax. For the target star (υ And), we add equations for cross filter. Both target and reference stars have equations for lateral color parameters. The roll of the plates also has a condition equation. Through these additional equations of condition, the χ² minimization process is allowed to adjust parameter values by amounts constrained by the input errors. In this quasi-Bayesian approach, prior knowledge is input as an observation with associated error, not as a hardwired quantity known to infinite precision. For υ And A and B, no priors were used for parallax or proper motion. These values and the orbital parameters were determined independently without bias.

3.4.1. Astrometric Reference Frame Residual Assessment

Before the target star, υ And, is modeled simultaneously with the reference frame, the reference frame is independently modeled many times to assess the which plate model is appropriate, prior knowledge of spectrophotometric parallaxes, catalog proper motions, and stability as a reference star. Poor fits of the reference frame can lead to reassessment of the spectrophotometric parallaxes, derivation of independent proper motion estimates, and removal of "bad" reference stars. In this case, reference Star 111 was not used in the modeling because this 15.82 mag star either had false locks in the interferometer (which can occur with faint stars), or it is a double star.

The OFAD calibration (McArthur et al. 2002) reduces the as-built HST telescope and FGS-1r distortions with amplitude of more than 1 arcsec to below 2 mas over much of the FGS-1r field of regard. From histograms of the υ And field astrometric residuals shown in Figure 5, we conclude that we have obtained satisfactory correction. The resulting reference frame "catalog" in ξ and η standard coordinates was determined with average position errors 〈σ_ξ〉 = 0.17 and 〈σ_η〉 = 0.16 mas.

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To determine whether there might be unmodeled, but possibly correctable, systematic instrumental effects at the 1 mas level, we plotted reference frame x and y residuals against a number of spacecraft, instrumental, and astronomical parameters. These included x, y position within our total field of view, radial distance from the field-of-view center, reference star V magnitude and B − V color, and epoch of observation. We saw no obvious trends.

We model the radial ($\hat{z}$) component of the stellar orbital movement around the barycenter of the system. This is given by the projection of a Keplerian orbital velocity to observer's line of sight plus a constant velocity offset.

$$\begin{eqnarray} \cos vw &=& \displaystyle\frac{\cos (E) - e}{1 - (e \times \mbox{cos}(E))} \times \cos (\omega)\nonumber\\ &&- \displaystyle\frac{\sqrt{1-e^2}\times \sin E}{1 - (e \times \cos (E)) }\times \sin (\omega), \end{eqnarray} \tag{ 6 }$$

$$\begin{equation} {\rm orb} = \gamma + K1 \times (\mbox{ecc} \times \cos (\omega) + \mbox{cos}vw), \end{equation} \tag{ 7 }$$

where e is eccentricity, ω is the longitude of periastron, K1 is the semi-amplitude of the RV signal, E is the eccentric anomaly from Kepler's equation, and γ is the constant velocity offset. We model three planetary orbits, one slope (indicating an additional long-period companion) and five γ's, one for each data set. Additionally, a systematic slope found only in the AFOE data was modeled.

We use an unperturbed Keplerian orbit with linear combination of Keplerian orbits, which is an acceptable first-order approximation of the orbital elements. Our long-term stability tests perform a full computation of perturbed orbits, which is considered in detail in Section 5. Additionally, we constrain a relationship between the astrometry and the RV through this equation (Pourbaix & Jorissen 2000):

$$\begin{equation} \displaystyle {{\alpha \sin i \over \pi _{\rm abs}} = {P K (1 - e^2)^{1/2}\over 2\pi \times 4.7405}}, \end{equation} \tag{ 8 }$$

where quantities only derivable from the astrometry (parallax, π_abs, primary perturbation orbit size, α, and inclination, i) are on the left, and quantities derivable from both (the period, P, and eccentricity, e), or radial velocities only (the RV amplitude of the primary, K), are on the right.

We investigated the use of alternative orbital modeling software (Systemic¹⁰) that considered perturbations, but found that they included assumptions, such as an inclination of 90°and coplanarity (S. Meschiari 2009, private communication), that would have presented a less realistic interpretation of our data. We did create an orbital model which included N-body perturbations, which is discussed in Section 4.2.1.

Serendipitously, one of our reference stars (ref-113) was υ And B (= 2MASS J01365042+412332). This gave us the opportunity to assess its status as the stellar companion to υ And. υ And B was first announced in early 2002 (Lowrance et al. 2002) as a distant stellar companion at an apparent projected separation of ∼750 AU. Its association was determined by the calculation of a similar proper motion from the POSS I and POSS II digitized images. Later in 2002, an attempt was made to detect a binary companion to stars with planets including υ And using high-resolution and wide-field imaging with Keck and Lick (Patience et al. 2002). The conclusion of this study was that because there should have been a detection of υ And B, but there was not, that it was a background object. In 2004, it was included in the study of stellar multiples of Eggenberger et al. (2004). In 2006, Raghavan et al. (2006) proposed a projected separation of 702 AU between components A and B, and noted that they were not gravitationally bound.

We find a parallax of 73.71 ± 0.10 mas for υ And and 73.45 ± 0.44 mas for υ And B (shown along with proper motions in Table 10, including the HIPPARCOS values for comparison). This is a separation in parallax between υ And and υ And B of 0.26 mas which is about 0.048 parsecs or ∼9900 AU. The previous estimates of separations are within our errors which support separations less than ∼30,000 AU. The parallax and proper motions uncertainties of υ And B are higher than normal because a significant portion of the observations could not be used because diffraction spikes from υ And contaminated the data. HST proper motions are relative to the reference frame that we observed.

The updated orbital parameters of υ And presented in Butler et al. (2006) were based upon 268 observations from Lick Observatory. For our combined orbital solution, we had 974 RV observations, including the new data from the HET. We ran periodograms on the residuals from these fits: (1) only γ's fitted, (2) γ's and planet b fitted, (3) γ's and planets b and c fitted, and (4) γ's and planets b, c, and d fitted. We looked for evidence of additional planetary periodic signals in these periodograms (see Figure 6). In the periodogram of the residuals to a Keplerian model that contains the planet b, c, and d, we see a weak signal at around one month and a presumed systematic signal at around 180 days that originated in the Lick data, as mentioned in Butler et al. (1999). We found no indications of additional planetary objects from these periodograms. We also examined the residuals in phase space and time plots and did note a large slope (∼35 ms^-1) spanning the ∼ 6.3 year AFOE data set. After applying a linear correction to this data, the residuals of the AFOE velocities decreased by 30%.

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If we find no additional periodic signals to model from the periodograms or inspection of the residuals, we add a slope to the model to test for a longer period planet. Table 11 shows the χ², degrees of freedom (dof), and rms of the various υ And independent RV planet modeling. We find a significant χ² improvement (from 745 down to 606) for the additional 1 degree of freedom, which we interpret as evidence of this long-period object. The amplitude of the trend is approximately 2.5 ms^-1 over the 14 year data set.

From previous work (Benedict et al. 2002; Bean et al. 2007; Martioli et al. 2010) and exploratory analysis, we have confidence in conservatively detecting astrometric signals down to around 0.25 mas. Of the three well-determined planets around υ And, only planet d would have an astrometric signal that would be detectable by the HST FGS at all inclinations (see Figure 7 noting that α is the half-amplitude of the astrometric signal). Planet b could be detected if its inclination was less than 12, planet c if its inclination was less than ∼44° and planet d could be detected at any inclination.

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Simultaneous astrometric and RV modeling was carried out using GaussFit. (For a review of the use of GaussFit for problems of linear regression with errors in both variables and the errors it produces see Murtagh 1990.) Astrometric modeling of the target and reference frame consisted of scale, lateral color, cross filter, proper motion, parallax, and the astrometric orbital elements of either planet d alone or planets c and d or planets b, c, and d, and was carried out as detailed in Section 3.4, with RV modeling of a slope and planets b, c, and d. Our modeling program GaussFit allows us to set a tolerance that prevents a solution being found in a shallow minima. That setting combined with the simultaneous modeling of free parameters (no parameters are held as constants) minimizes the chance of false detection.

Our attempt at the simultaneous astrometric modeling of planets b, c, and d failed. With all parameters free, the program iterated endlessly rather than settling on a false detection, indicating that there was no detectable astrometric signal for planet b. However, when we only modeled the astrometric signals of planets c and d with the same method, we detected the astrometric signal of planets c and d. The modeling of the orbital elements determined simultaneously with GaussFit, using robust estimation, including the astrometric signals of planets c and d had improved χ² and lowered residuals over the same model which did not include the astrometric orbit of planets c and d. In Table 12, we show the χ² improvement after the cumulative addition of the astrometric modeling components. These are stepwise astrometric models with the RV elements as constants for illustrative purposes, not the simultaneous model of our solution. The final addition of the orbital elements includes the six astrometric elements of planets c and d, with the other orbital elements input as constants from our simultaneous solution. The dof in this modeling include not only the astrometric data and the parameters solved for but also the constraint equations (to control the scale of the master plate and for Pourbaix's relation) and the a priori Bayesian input data related to the proper motions and parallax of the reference frame and the lateral color and cross-filter instrument calibrations.

The χ² of 360 for 1624 dof indicates that the astrometric data has errors that are overestimated as we have verified in comparisons with Hipparcos data (see Section 3.2). The HST astrometric error estimation is based upon complex statistical examination of the raw data, and we have chosen to continue to use these usually overestimated errors because we are not able to assign the overestimation to a particular component of the complex error estimation process. The orbital elements from this solution are shown in Table 13 along with the derived elements. Because of the overestimation of the astrometric data input errors, the actual errors of the astrometric parameters are most likely smaller than listed in the table.

We find the inclination of planet c to be 787 ± 10 and the inclination planet d to be 2376 ± 13. These values are shown in Figure 7 with error bars. The measured ascending nodes are Ω_c= 236853 ± 75 and Ω_d= 407 ± 33. These inclinations and ascending nodes permit a determination of the mutual inclination (Φ) of components c and d of the υ And system. Using Equation (9) (Kopal 1959):

$$\begin{equation} \cos (I_{\Phi }) = \cos (i_c) \cos (i_d) + \sin (i_c) \sin (i_d) \cos (\Omega _c - \Omega _d), \end{equation} \tag{ 9 }$$

we find the mutual inclination of υ And c and d to be 299 ± 10. The mutual inclination of the orbits of components c and d are shown schematically in Figure 8. (The error on Φ was determined by recalculating Equation (9) with inclination and ascending node values at ±1 − σ extrema and comparing the resulting Φ values.) These are measurements, not quantities derived from dynamical stability analyses. The astrometric motions of planets υ And c and d against time are shown in Figure 9, while they are shown on the sky in Figure 10. The astrometric data shown in these plots (the dark filled circles) are normal points made from the υ And residuals to an astrometric fit of the target and reference frame stars of scale, lateral color, cross filter, parallax, and proper motion of multiple observations (light open circles) at each epoch.

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For calculations of the masses, we use an M_☉ of 1.31^+0.02_−0.01 for υ And, which is a Bayesian-derived determination by Takeda et al. (2007), who note that though there is a probability (which is low) of the star being older and lower mass, they conclude that isochrone analyses favors the younger main-sequence model instead of the older turnoff or post-main-sequence models. We then find a minimum mass for planet b of 0.69 ± 0.016 M_JUP and actual masses for planet c of 13.98^+2.3_−5.3 M_JUP and planet d of 10.25^+0.7_−3.3 M_JUP. The actual masses were derived from the HST astrometrically measured α's of planet c (0.619 ± 0.078) and planet d (1.385 ± 0.072). The probable actual mass range of planet b suggested by dynamical analysis is discussed in Section 5.

Figure 11 shows the RV of companions b, c, and d (with the other component velocities removed) plotted against orbital phase. The γ adjusted velocities with the combined orbital fit of the five RV data sources are shown in the top panel of Figure 12 and the velocity residuals are shown in the lower panel. Table 14 shows the number of observations and rms of the five RV data sources with an average rms of 10.66. The histogram in Figure 13 shows the Gaussian distribution of the RV residuals of the combined orbital model which include residuals from five different sources spanning 14 years. Figure 14 shows the distribution of the HET residuals alone.

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4.2.1. Orbital Solution using N-body Integrations

In addition to the above simultaneous Keplerian model orbital solution, we also performed an orbital solution using N-body integration. We used the Mercury code (Chambers 1999) with the RADAU integrator (Everhart 1985) for the integration of the equations of motion in our model. All bodies were considered not as point masses, but planets with actual mass (non-zero radii). This is a preliminary modeling process which does not include all relativistic effects, but disk loses the difference in solutions when you include the planet–planet interaction and indirect forces. The orbital elements determined with the method are listed in Table 15. Using this method, we find the mutual inclination of υ And c and d to be 309, which is within the errors to the 299 found with the simple Keplerian model. In Figure 15, we show the Keplerian and perturbed orbital solutions plotted together. While the small microarcsecond difference affects our determination of mutual inclination within the errors, it is clear that with data from the Space Interferometry Mission (SIM), orbital modeling will have to enter a new level of precision, and current methods of determining gravitational and relativistic effects will have to be enhanced.

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Table 15. υ Andromedae A: Orbital Parameters and Masses using N-body Integration

Parametera	υ And b	υ And c	υ And d
RV
K (m s−1)	70.47	52.74	66.08
Astrometry
α (mas)		0.51	1.4
i (deg)		9.3	23.14
Ω (deg)		224.03	5.28
Astrometry and RV
P (days)	4.61796	237.7	1302.61
Tb (days)	50033.64	49956.109	50008.67
e	0.012	0.25	0.32
ω (deg)	44.14	250.7	250.36
Derivedc
a (AU)	0.059	0.822	2.55
MA (deg)d		270.479	266.0659
Msin i(MJ)e	0.66	1.92	4.25
M(MJ)		11.59	10.29
Planets c and d
Mutual i (Φ) (deg)			30.94

Notes. ^aEpoch for these osculating elements = 2452274.0. ^bT = T − 2400000.0. ^cAn υ And mass of 1.31 ± 0.02 M_☉ (Takeda et al. 2007) was used in these calculations. ^dMean anomaly. ^eThe quantity referred to in radial velocity studies as mass, but actually is minimum mass.

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HIPPARCOS (HIP) made 27 visits to υ And, all one-dimensional measurements that were projected star positions on its instantaneous reference circle. The HIP intermediate astrometric data (IAD) has been used in several studies to estimate the mass or upper mass limits of some of the υ And planets. These studies typically held the published RV orbits (of the time) and the HIP parallax as constants, and then used various techniques to derive astrometric orbital elements (e.g., inclination) that were not measurable in the HIP data (as they are in the HST data).

In the first HIP study of υ And d by (Mazeh et al. 1999, MZ99), a valley in χ² was found for α at 1.4 mas across all possible inclinations. The α was found to be 1.4 mas ± 0.6 mas, and the deduced (astrometrically undetectable) inclination was 156°⁺⁸₋₂₅ for a one-sigma mass estimate of 10.1^−4.6_+4.7 M_JUP.¹² The inclination (because it could not be measured) was derived from an equation using K and the inclination predicted from the minimum χ² of that equation. In 2001, a similar process was used by Han et al. (2001) for mass estimations limits, except that in this process the asin i derived from the RV was fixed from the beginning. They also ignored the error in the spectroscopic elements. They found an α of 1.38 mas ± 0.64 mas and an inclination of 1555 (no error) and a mass of 108 M_JUP. It is unclear how such a large mass was derived for υ And d, unless it was a typo.

Later in 2001, (Pourbaix 2001, P01) re-examined these earlier results, and found that some small inclinations found were merely artifacts of the fitting procedure that was used. P01 found that fitting (i, Ω) to the HIP IAD when the A_asin i is much smaller than the astrometric precision always yields low values of sin i, regardless of the true inclination. He found for υ And d using A_asin i as a constraint an inclination of 287 ± 168 and for υ And c an inclination of 172° ± 38; his F-test rejected both of these values at 20% and 23%, respectively. His test was the probability of obtaining an F-value greater or equal to the value found in the absence of a signal in the IAD (a false positive).

The parallax and proper motion results for υ And (see Table 10) illustrate that for this object, HST has achieved significantly more precise results (a parallax error of 0.1 mas for HST versus 0.72 mas for HIP) than HIP. We have enough data in this study to have confidence in signals that are as small as 0.25 mas. For our analysis, we do use a constraint in the relationship between the astrometry and the RV (see Equation (8)). Our modeling is significantly different than the modeling of the HIP IAD, however. We hold no orbital or astrometric parameters as constants. Our solutions do not converge unless there is a measurable signal, as discussed in Section 4.2. It is meaningful to note that the value of α for υ And d found by MZ99 (1.4 mas ± 0.6 mas) is within the errors equal to our HST determined value (1.385 mas ± 0.072 mas) and the mass estimate from MZ99, 10.1^+4.7_−4.6 M_JUP, is very close to ours, 10.25^+0.7_−3.3 M_JUP. Though MZ99, found an inclination of 156°⁺⁸₋₂₅ for υ And d, P01 found 287 ± 168, a value he did not have confidence in, but which compares favorably with the HST value of 23758 ± 1316. We note that for the MZ99 inclination of 156°, 180°−156°= 24°, the HST value. Our determination of astrometric elements and mass for υ And d confirm the much earlier determination by MZ99 and P01.

In retrospect, it is worth noting that almost all of the stability analyses carried out on the υ And system did not consider or mention the mass of υ And d determined in 1999, but instead used the significantly smaller minimum masses derived from the RV.

υ And b is a hot Jupiter (a minimum mass of ∼ 0.7 M_JUP) in a near-circular orbit. Nagasawa et al. (2008) found that an inner planet such as υ And b can re-enter a Kozai cycle, and that this repeated mechanism can improve the chances for tidal circularization to occur. υ And b is thought to have a hot day side and a cool night side, a "pM" class planet, in contrast to the "pL" class planets (represented by HD189733 b; pM–pL reference Fortney et al. 2008). Its infrared brightness was measured by Spitzer Space Telescope at five epochs and analyzed by Harrington et al. 2006, in which they proposed a consistent picture of the atmospheric energetics as long as the inclination is >30°.

While it was not possible to measure the inclination or astrometric signal α from υ And b (see Figure 7; which is most certainly in the low-microarcsecond regime) with HST, our dynamical stability analysis suggests possible masses (see Figure 16). υ And b may have a mass of M_P= 1.4 M_JUP (the minimum mass from RV is ∼0.7 M_JUP) with an inclination around 25°. Without knowledge of Ω, we cannot estimate the mutual inclination.

The υ And system presented itself, through its RV orbital elements, as an intriguing and theory-provoking system in 1999. It was a system with three Jupiter-mass-sized objects, the outer two having higher eccentricities, the inner surprisingly circular, which appeared to be on the edge of instability from dynamical analyses (Barnes & Quinn 2001). Studies focused on the existence of planet b in its circular orbit (Nagasawa & Lin 2005; Adams & Laughlin 2006), and the mechanism that excited the eccentricities of planets c and d (Lissauer & Rivera 2001; Chiang et al. 2001, 2002).

While early studies (Chiang et al. 2002) declared the gravitational interactions between υ And c and d are to "excellent approximation" secular, more recent studies (Migaszewski & Gozdziewski 2009) have found that "there is no simple and general recipe to predict the behavior of the secular system," that even in systems without extreme parameters the phase space of the modeling changes, and more branches of stable solutions are revealed. The finding by Adams & Laughlin (2006) that the addition of general relativity to the dynamic modeling damps the evolutionary growth of eccentricity in planet b is fundamental. This showed that incomplete stability modeling could erroneously makes an observed system seem improbable, if not impossible.

While most studies suggested that an outside player—another planet, a star, or the circumstellar disk—was directly responsible for the evolution of the eccentricities of planets c and d, there was a difference in interpretation based upon assumptions researchers made about the system. Assuming an Occum's razor position that the system is coplanar, Chiang et al. (2002) considered that interactions of the outer planet d with the circumstellar disk of gas provided for the amplification of eccentricities. Most others (Rivera & Lissauer 2000; Ford et al. 2005; Barnes & Greenberg 2006a) regard a planet as the likely accelerant. While a planet-scattering model (Ford et al. 2005) was found to be the simple explanation of the eccentricity amplification, a more explicit "rogue planet model" (Barnes & Greenberg 2007) was found to be necessary for the "near-separatix" state of the system. But both of these scenarios were based upon assumptions about the masses of planets c and d, based upon the RV minimum mass estimates (Msin i_c = 1.8898^+0.12_−0.12M_JUP and Msin i_d = 4.1754^+0.26_−0.27M_JUP), and therefore assumed that the mass of planet d was larger than planet c. In MMR crossing between planets on divergent orbits (Chiang et al. 2002) and in close encounters between planets (Ford et al. 2001), the greater eccentricity is bestowed upon the least massive planet, and because of this assumption of a more massive planet d, an external source was needed. But, our astrometric measurements have shown that it is planet c (M_c = 13.98^+2.3_−5.3 M_JUP) that is likely the more massive planet (M_d = 10.25^+0.73_−3.27 M_JUP) ¹³ thus altering the foundation for the theories used to explain the system that was "seen" with RV orbit alone (higher eccentricity of higher Msin i companion) but not expected (higher eccentricity on lower mass companion). The astrometric orbital elements turned the seen from unexpected to expected.

We find inclinations of υ And c to be 79 ± 1° and υ And d to be 238 ± 1°. For the first time, measurements of astrometric orbital elements of an extrasolar planetary system have been made to find the mutual inclination between planets. This is not a value derived from dynamical analysis as that of Bean & Seifahrt (2009); the signal of GL876 b is in the microsecond of arc range, far too small to have been detected or measured by HST astrometry, but it could be detected in the future by SIM.

The mutual inclination of υ And c and d is 299 ± 1°. Gas disk processes probably cannot pump mutual inclinations much above the opening angle of ≲5° (see, e.g., Ida & Lin 2004), but previous investigations of planet–planet scattering (Marzari & Weidenschilling 2002; Chatterjee et al. 2008) have easily produced such mutual inclinations. However, we cannot rule out a Kozai-type interaction (see, e.g., Kozai 1962; Takeda et al. 2008; Libert & Tsiganis 2009) with υ And B. Its orbit is poorly constrained, but if it is near the apocenter of an extremely eccentric and significantly inclined orbit, then it may be able to induce the observed mutual inclination. As the Kozai origin requires specific, low probability events, but the scattering naturally and easily explains the tight packing (Raymond et al. 2009) and large mutual inclination, we suggest that planet–planet scattering more likely produced the observed configuration.

We now have two planetary systems in which the actual mass and spatial relationship between two planets have been measured, our own solar system and the υ And system. These systems contain very different planetary objects at different distances and display contrasting architecture. The υ And planetary system seems to lie even closer to the stability boundary than previously thought (see, e.g., Laughlin & Adams 1999; Barnes & Quinn 2001; Barnes & Raymond 2004; Goździewski et al. 2001). Such a proximity to instability may seem startling, but could fit well with the planet–planet scattering models (Raymond et al. 2009).

Questions remain. Does the model of planet–planet scattering (Ford et al. 2005) provide a mechanism for excitation of eccentricity and mutual inclination that we measured in the υ And system, or does a modified theory of late-stage scattering of protoplanets (Barnes & Greenberg 2007) explain both the υ And system and our own solar system better? Could the divergence of their current state have been seeded during late formation or is it a product of the evolution of these systems? During the formation of the giant planets in υ And, could the interactions which formed the cores of these planets drive them into a higher mutual inclination state? Does the residual gas disk after planet formation also contribute to what we see? What mechanism keeps this system that appears to lie on the edge of stability stable?

The astrometric determination of the mass of a low-mass companion can decisively characterize it as a planet and reveal its hierarchical position in a planetary system. A good illustration of this fact can be seen from the results of our group for three objects that were previously listed as extrasolar planet candidates: Gliese 876 b, HD 136118 b, and HD 33636 b. Surprisingly, each object has been found to belong to a different class: a giant planet, a brown dwarf, and an M dwarf star, respectively, Benedict et al. (2002), Martioli et al. (2010), and Bean et al. (2007). These results demonstrate the importance of the application of complementary techniques in observing extrasolar planetary systems.

The measurements of the actual masses and the inclination of υ And c and d provide a new foundation for future theoretical and dynamical studies of our first uncovered extrasolar planetary system.