# The architecture of the GJ876 planetary system. Masses and orbital coplanarity for planets b and c

**ABSTRACT** We present a combined analysis of previously published high-precision radial velocities and astrometry for the GJ876 planetary system using a self-consistent model that accounts for the planet-planet interactions. Assuming the three planets so far identified in the system are coplanar, we find that including the astrometry in the analysis does not result in a best-fit inclination significantly different than that found by Rivera and collaborators from analyzing the radial velocities alone. In this unique case, the planet-planet interactions are of such significance that the radial velocity data set is more sensitive to the inclination of the system through the dependence of the interactions on the true masses of the two gas giant planets in the system (planets b and c). The astrometry does allow determination of the absolute orbital inclination (i.e. distinguishing between i and 180-i) and longitude of the ascending node for planet b, which allows us to quantify the mutual inclination angle between its orbit and planet c's orbit when combined with the dynamical considerations. We find that the planets have a mutual inclination of 5.0 +3.9 -2.3 degrees. This result constitutes the first determination of the degree of coplanarity in an exoplanetary system around a normal star. That we find the two planets' orbits are nearly coplanar, like the orbits of the Solar System planets, indicates that the planets likely formed in a circumstellar disk, and that their subsequent dynamical evolution into a 2:1 mean motion resonance only led to excitation of a small mutual inclination. This investigation demonstrates how the degree of coplanarity for other exoplanetary systems could also be established using data obtained from existing facilities. Comment: 9 pages, accepted for publication in A&A

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**ABSTRACT:**We use full available array of radial velocity data, including recently published HARPS and Keck observatory sets, to characterize the orbital configuration of the planetary system orbiting GJ876. First, we propose and describe in detail a fast method to fit perturbed orbital configuration, based on the integration of the sensitivity equations inferred by the equations of the original $N$-body problem. Further, we find that it is unsatisfactory to treat the available radial velocity data for GJ876 in the traditional white noise model, because the actual noise appears autocorrelated (and demonstrates non-white frequency spectrum). The time scale of this correlation is about a few days, and the contribution of the correlated noise is about 2 m/s (i.e., similar to the level of internal errors in the Keck data). We propose a variation of the maximum-likelihood algorithm to estimate the orbital configuration of the system, taking into account the red noise effects. We show, in particular, that the non-zero orbital eccentricity of the innermost planet \emph{d}, obtained in previous studies, is likely a result of misinterpreted red noise in the data. In addition to offsets in some orbital parameters, the red noise also makes the fit uncertainties systematically underestimated (while they are treated in the traditional white noise model). Also, we show that the orbital eccentricity of the outermost planet is actually ill-determined, although bounded by $\sim 0.2$. Finally, we investigate possible orbital non-coplanarity of the system, and limit the mutual inclination between the planets \emph{b} and \emph{c} orbits by $5^\circ-15^\circ$, depending on the angular position of the mutual orbital nodes.Celestial Mechanics and Dynamical Astronomy 05/2011; 111. · 2.32 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**We describe statistical methods for measuring the exoplanet multiplicity function - the fraction of host stars containing a given number of planets - from transit and radial-velocity surveys. The analysis is based on the approximation of separability - that the distribution of planetary parameters in an n-planet system is the product of identical 1-planet distributions. We review the evidence that separability is a valid approximation for exoplanets. We show how to relate the observable multiplicity function in surveys with similar host-star populations but different sensitivities. We also show how to correct for geometrical selection effects to derive the multiplicity function from transit surveys if the distribution of relative inclinations is known. Applying these tools to the Kepler transit survey and radial-velocity surveys, we find that (i) the Kepler data alone do not constrain the mean inclination of multi-planet systems; even spherical distributions are allowed by the data but only if a small fraction of host stars contain large planet populations (> 30); (ii) comparing the Kepler and radial-velocity surveys shows that the mean inclination of multi-planet systems lies in the range 0-5 degrees; (iii) the multiplicity function of the Kepler planets is not well-determined by the present data.The Astronomical Journal 06/2011; · 4.97 Impact Factor - SourceAvailable from: Enrico Gerlach[Show abstract] [Hide abstract]

**ABSTRACT:**Prior to the detection of its outermost Uranus-mass object, it had been suggested that GJ 876 could host an Earth-sized planet in a 15-day orbit. Observation, however, did not support this idea, but instead revealed evidence for the existence of a larger body in a $\sim$125-day orbit, near a three-body resonance with the two giant planets of this system. In this paper, we present a detailed analysis of the dynamics of the four-planet system of GJ 876, and examine the possibility of the existence of other planetary objects interior to its outermost body. We have developed a numerical scheme that enables us to search the orbital parameter-space very effectively and, in a short time, identify regions where an object may be stable. We present details of this integration method and discuss its application to the GJ 876 four-planet system. The results of our initial analysis suggested possible stable orbits at regions exterior to the orbit of the outermost planet and also indicated that an island of stability may exist in and around the 15-day orbit. However, examining the long-term stability of an object in that region by direct integration revealed that the 15-day orbit becomes unstable and that the system of GJ 876 is most likely dynamically full. We present the results of our study and discuss their implications for the formation and final orbital architecture of this system.Celestial Mechanics and Dynamical Astronomy 02/2012; 113(1). · 2.32 Impact Factor

Page 1

arXiv:0901.3144v1 [astro-ph.EP] 20 Jan 2009

Astronomy & Astrophysics manuscript no. ms

January 20, 2009

c ? ESO 2009

The architecture of the GJ876 planetary system

Masses and orbital coplanarity for planets b and c⋆

J. L. Bean1and A. Seifahrt1

Institut f¨ ur Astrophysik, Georg-August-Universit¨ at G¨ ottingen, Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany

e-mail: bean@astro.physik.uni-goettingen.de

Accepted January 12, 2009

ABSTRACT

We present a combined analysis of previously published high-precision radial velocities and astrometry for the GJ876 planetary

system using a self-consistent model that accounts for the planet-planet interactions. Assuming the three planets so far identified in

the system are coplanar, we find that including the astrometry in the analysis does not result in a best-fit inclination significantly

different than that found by Rivera and collaborators from analyzing the radial velocities alone. In this unique case, the planet-planet

interactions are of such significance that the radial velocity data set is more sensitive to the inclination of the system through the

dependence of the interactions on the true masses of the two gas giant planets in the system (planets b and c). The astrometry does

allow determination of the absolute orbital inclination (i.e. distinguishing between i and 180 − i) and longitude of the ascending

node for planet b, which allows us to quantify the mutual inclination angle between its orbit and planet c’s orbit when combined

with the dynamical considerations. We find that the planets have a mutual inclination Φbc= 5.0◦ +3.9◦

determination of the degree of coplanarity in an exoplanetary system around a normal star. That we find the two planets’ orbits are

nearly coplanar, like the orbits of the Solar System planets, indicates that the planets likely formed in a circumstellar disk, and that

their subsequent dynamical evolution into a 2:1 mean motion resonance only led to excitation of a small mutual inclination. This

investigation demonstrates how the degree of coplanarity for other exoplanetary systems could also be established using data obtained

from existing facilities.

−2.3◦. This result constitutes the first

Key words. stars: individual: GJ876 – planetary systems – astrometry – methods: data analysis

1. Introduction

A planetary system’s “architecture” consists of the census,

masses, and orbits of the objects in the system. The formation

and evolutionary history of a planetary system are encoded in

these characteristics. The architecture of the Solar System is

broadly consistent with the theory of planet formation in a cir-

cumstellar disk, and its deviations from the expected pattern are

interpreted as evidence of evolutionary processes. It is impor-

tant to determine the architecture of exoplanetary systems so as

to have broader constraints on the formation and evolution of

planetary systems than can be obtained from study of the Solar

System alone.

The GJ876 planetarysystem is oneexoplanetarysystem that

has beenthe focus of muchattention to ascertain its architecture.

GJ876 itself is an M4 dwarf star with an essentially solar metal-

licity (Bonfils et al. 2005; Bean et al. 2006) that is at least older

than 1Gyr as evidenced by its low activity and slow rotation

(Marcy et al. 1998). It is at a distance of 4.7pc (Perryman et al.

1997) and appears to be a singleton as no stellar companions

havebeenreported.Thestardoesnotappeartoalsoharborasub-

stantial debris disk (Trilling et al. 2000; Shankland et al. 2008).

⋆Based on observations made with the NASA/ESA Hubble Space

Telescope, obtained at the Space Telescope Science Institute, which

is operated by the Association of Universities for Research in

Astronomy, Inc., under NASA contract NAS 5-26555 (programs GO-

8102, 8775, and 9233); and on observations obtained at the W. M. Keck

Observatory, which is operated jointly by the University of California

and the California Institute of Technology.

A gas giant planet was found to orbit GJ876 by two groups

independentlyin 1998 using Doppler spectroscopy(Marcy et al.

1998; Delfosse et al. 1998). This was the first convincing detec-

tion of a planet around an M dwarf, and GJ876 still remains one

of the few known planet hosting stars of this type. Subsequent

observations by Marcy et al. (2001) revealed that the system

contains a second, lower-mass gas giant that is in a 2:1 mean

motion resonance with the first detected planet.

Soon after the discovery of the second planet in the GJ876

system, Laughlin & Chambers (2001) and Rivera & Lissauer

(2001) pointed out that the two identified planets are experienc-

ing mutual interactions on an unprecedentedly short timescale.

They found that the resulting perturbations are of such signif-

icance that a model based on Keplerian orbits was not accu-

rate enough to match the radial velocity data, and instead di-

rect integration of the equations of motion (i.e. Newtonian or-

bits) for the three body configuration was needed. Those au-

thors also found that, although they introduced significant ad-

ditional complexity in the modeling of the observational data,

the occurrence of short-timescale interactions could allow in-

ference of the true masses of the planets from radial velocity

data alone. This is in contrast to the usual situation where only

planets’ minimum masses are determinable from radial veloc-

ity data owing the orbital inclination degeneracy. The key to the

additional insight is that the non-Keplerian perturbations are de-

pendent on the true masses of the interacting bodies. Therefore,

if these perturbations could be characterized well enough with

radial velocity data then the planet masses could be con-

strained. Laughlin & Chambers (2001) and Rivera & Lissauer

Page 2

2J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system

(2001) both concluded that tight limits on the planet masses

could not be set given the data available at the time, but that fu-

tureanalysisofthecontinuingDopplermeasurementsforGJ876

would possibly give better results.

Benedict et al. (2002) carriedout astrometricobservationsof

the GJ876system using the Fine Guidance Sensor (FGS) instru-

ment on the Hubble Space Telescope (HST) beginning around

the time the first planet was announced and continuing for 2.5

years. Analysis of these data revealed a residual perturbation

with semimajor axis of 0.3±0.1mas in phase with the orbit

of planet b expected from modeling radial velocity data. This

was the first definitive astrometric detection of an exoplanet,

and it is still one out of only a few such successful detections.

Modelingtheastrometrytogetherwiththeradialvelocitiesavail-

able yielded an estimate of the orbital inclination of planet b

(ib = 84◦± 6◦), and thus the planet’s true mass after assum-

ing a mass for GJ876 (mb= 1.9 ± 0.3 MJup). In their analysis,

Benedict et al. (2002) used the standard Keplerian rather than

Newtonian orbital calculations for modeling the radial velocity

and astrometry data.

Continuing the trend of exoplanet firsts and rarities in the

GJ876 system, Rivera et al. (2005) proposed the existence of a

third, very low-mass planet (mdsin id= 5.9 ± 0.5M⊕) based on

a relatively extensive set of new high-precision radial velocities.

At the time, this was possibly (allowing for the inclination ambi-

guity) the lowest mass planet yet found around a main sequence

star, and it is still one of only a few known planets with a mass

potentially in the “Super-Earth” regime (i.e. 1<∼m<∼10 M⊕).

A comprehensive photometric search for transits of this planet

by the discovery team did not result in a detection, although

grazing transits could not be ruled out.

Rivera et al. (2005) modeled their radial velocity data with

self-consistent Newtonian four-body orbits assuming the three

identified planets were in coplanar orbits. This modeling of the

new data set yielded seemingly tight constraints on the inclina-

tion of the system as foreseen by Laughlin & Chambers (2001)

and Rivera & Lissauer (2001). Rivera et al. (2005) found that

the coplanar system inclination appeared to be ∼ 50◦, and the

masses for planets b, c, and d were 2.3 MJup, 0.8 MJup, and

7.5 M⊕respectively.Althoughno formaluncertaintyin the incli-

nation was given, inspection of their reported χ2map (see their

Fig.3) suggests a standard error of ∼ 2◦.

Rivera et al. (2005) also argued that the orbits of the two gas

giants must have a small or even zero mutual inclination (i.e.

they are coplanar) because the radial velocity fit quality of their

model deteriorated when the inclinations of each of the planets

were pushed away from 50◦. However, quantification of the mu-

tual inclination of the planets’ orbits was not possible due to the

incomplete characterization of their full three dimensional or-

bits.Specificallymissingwasneededinformationontheplanets’

absolute orbital inclinations (i or 180 − i) and orbital longitudes

of the ascending nodes. Rivera et al. (2005) did not consider the

HST astrometryortheresults fromBenedict et al. (2002)intheir

analysis.

The findings of Benedict et al. (2002) and Rivera et al.

(2005) with regards to the planets’ orbital inclinations and

masses are inconsistent, and this has led to confusionaboutwhat

is the “best” model of the GJ876 system. Numerous theoretical

studies have been carried out since the discovery of the second

planet to determine what physical processes gave rise to the sys-

tem’s unique architecture (e.g. Lee & Peale 2002; Ji et al. 2002;

Kley et al. 2004, 2005; Zhou et al. 2005; Crida et al. 2008) be-

cause such specific consideration has the potential for con-

straining general theories of planet formation and evolution.

Determining what mechanisms could have led to the current

system arrangement depends critically on knowing the masses

of the planets involved, and what the current arrangement itself

even is. Therefore, continued refinement of the GJ876 system

model would be valuable.

In this paper we present new constraints on the architecture

of the GJ876 planetary system based on reanalysis of the previ-

ously published high precision radial velocities and astrometry

for the system. We aimed to resolve the discrepancy between

the results of Benedict et al. (2002) and Rivera et al. (2005), and

study the degree of coplanarity in the system by the first com-

bined analysis of the data sets using a self-consistent Newtonian

orbit model. Our study represents a synergy in the conceptual

advances made by Benedict et al. (2002) in their analysis of

a combined radial velocity and astrometry data set for an ex-

oplanetary system, and that of Laughlin & Chambers (2001),

Rivera & Lissauer (2001), and Rivera et al. (2005) in their use

of Newtonian orbits to account for the signature of multi-body

interactions in radial velocity data and infer more information

than it is normally possible to obtain from such data. The paper

is organizedas follows. In §2 we describethe data utilized in our

analysis. We present our analysis in §3. We conclude in §4 with

a discussion of the implications of the results obtained and the

potential for continued work in this area.

2. The data

2.1. Radial velocities

We utilized the time series radial velocities for GJ876 presented

by Rivera et al. (2005) in our analysis. These velocities were

measuredfromhigh-resolutionspectroscopicobservationsmade

with the HIRES instrument equipped with an iodine absorp-

tion cell and fed by the KeckI telescope at the W. M. Keck

Observatory. This data set contains 155 measurements that have

a median uncertainty of 4.1ms−1and were obtained over 7.6yr.

No adjustments were made to the uncertainties to account for

the potential affect of stellar “jitter” (a loose term referring

to changes in stellar spectra arising from variation of inhomo-

geneities on the surface of stars that can be misconstrued as a

radial velocity change) because Rivera et al. (2005) achieved a

best-fit reduced χ2for the radial velocities very close to 1.0,

which indicates that there is little or no additional noise in the

data. More details about these data can be found in Rivera et al.

(2005) and references therein.

Other relevant radial velocity data for GJ876 were pre-

sented by Delfosse et al. (1998, data from the ELODIE and

CORALIE spectrographs) and Marcy et al. (2001, data from

Lick Observatory). We elected not to include these data sets in

our analysis because their consideration would have been more

confusing than illuminating. As all the radial velocities are rel-

ative in nature, each data set included therefore requires the ad-

dition of a free offset parameter in the orbit fitting. Furthermore,

there is the issue of how to treat the estimated uncertainties

when using inhomogeneous data sets. Each group has their own

method of error estimation based on some combination of the

photon statistics in the obtained spectra, stability of the in-

strument used, and accuracy of the Doppler shift measurement

methodemployed.Also,theuncertaintiesforthevelocitiesinthe

three neglected data sets are all > 10ms−1and none approach

the time baseline of the Keck velocities . Therefore, our choice

to use onlythis later data set simplifies the analysis andbypasses

a number of potential issues without ignoring very useful data.

Page 3

J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system3

We note that Rivera et al. (2005) also chose to focus exclusively

on the Keck velocities for GJ876 in their analysis.

2.2. Astrometry

We also included in our study the astrometry for GJ876 that was

obtained with the FGS3 instrument on the HST and that was

presented by Benedict et al. (2002). The data set is made up of

observations of GJ876 and five reference stars obtained over 27

HST orbits and distributed in 9 epochs. The measurements span

2.5yr and are coincident with the Keck radial velocities. The

majority of the observations were obtained near the periastron,

apastron, and subsequent periastron times of planet b duringone

of its orbits (a time span of ∼60d). Further observations were

scheduled to allow breaking the degeneracybetween the orbital,

proper, and parallactic motions.

We used the exact same data as Benedict et al. (2002) with

only one modification. We multiplied the nominal uncertain-

ties estimated by their data reduction pipeline for the X and Y

axis position measurementsby0.34and 0.50respectively.These

re-weightings were motivated when we obtained a reduced χ2

significantly below 1.0 for the data during preliminary model-

ing. The weightings were iteratively adjusted to yield a reduced

χ2= 1.0 for the best-fit coplanar model (see below). The sep-

arate re-weightings for the X and Y axis data are appropriate

because the FGS instrument has two separate arms for mea-

suring the apparent positions in the two axes. Thus, the data

from the axes are essentially independent. The median uncer-

tainty in the position measurements for both axes is 0.95mas

after re-weighting.More details about these data, and FGS mea-

surements in general, can be found in Benedict et al. (2002) and

references therein.

3. Analysis

3.1. Modeling

Our analysis consisted of modeling the Keck radial velocities

and HST astrometry described in §2 simultaneously. A self-

consistent model for a four-body system (three planets and the

host star) was used to account for the orbital motion of GJ876

in both data sets. This model was generated using the Mercury

code (Chambers 1999) to integrate the equations of motion. All

the bodies were assumed to be point masses and the only force

consideredwas Newtoniangravity.We havepreviouslyusedthis

same general method to simultaneously model radial velocities

and eclipse times for an exoplanetary system with consideration

of possible planet-planet interactions (Bean & Seifahrt 2008).

We assumed the mass of GJ876 (mA) is 0.32 M⊙, as sug-

gested by Rivera et al. (2005) based on consultation with empir-

ical Mass – Luminosityrelationships for low mass stars. The un-

certainty in the estimate is probably ∼10%. We did not attempt

to account for this uncertainty because it would be prohibitively

time consuming to repeat the analyses numerous times with dif-

ferent assumed values. With this assumed mass, the model for

the orbital motion of GJ876 depended on the input masses and

osculating orbital elements for the three planets. The orbital el-

ements are the six usual ones: semi-major axis (a), eccentric-

ity (e), argument of periastron (ω), mean anomaly (M), incli-

nation (i), and longitude of the ascending node (Ω). The ref-

erence epoch for the osculating elements was taken to be HJD

2452490.0 as Rivera et al. (2005) did so that our results may be

directly compared to theirs. Including the masses, there were 7

parameters for each planet and 21 parameters total in the orbit

model.

Following Rivera et al. (2005), we always fixed the eccen-

tricity and argument of periastron for planet d to zero. This

planet is expected to be in a nearly circular orbit owing to tidal

torques from the host star. Furthermore, it induces a modulation

with semi-amplitude of only 6.5ms−1on the radial velocities.

Therefore, the signature of non-zero eccentricity is negligible

given the quality of the data and may be ignored. The treatment

of the remaining orbit model parameters varied in the different

analyses described below.

Comparison of the orbit model with the radial velocities re-

quired one additional step. A single correction factor was added

to all themodelradialvelocitiesto shiftthem tothe relativescale

of the observed velocities. This offset was always a free param-

eter in the analyses described below. The equation of condition

for the radial velocity model was thus

∆γ= RV − (γ + ORBITR),

(1)

where RV is the measured relative radial velocities, ORBITRis

the model radial velocity component of GJ876’s orbital motion,

γ is the offset, and ∆γis the residual.

Our approach for generating the astrometric model closely

followedthe methodsusedbyBenedict et al. (2002), whichhave

also beenused to analyze FGS astrometryforother exoplanetary

systems (McArthur et al. 2004; Benedict et al. 2006; Bean et al.

2007) and binary star systems (e.g. Benedict et al. 2001). The

model included the orbital motion of GJ876, parallactic and

proper motion for GJ876 and the five reference stars, and plate

adjustments for the 27 epochs of data.

We selected the observations made during epoch 22 to serve

as the astrometric constraint “plate.” FGS observations of differ-

ent stars during an epoch are carried out sequentially rather than

simultaneously so there isn’t an actual plate in the traditional

sense. However, the sequential observationsduring a single HST

orbit may be combined to form an effective plate due to stability

of thetelescope andinstrumentresponseduringthat time period.

We refer to this combination of sequential observations as sim-

ply a plate below for brevity. The choice of the constraint plate

does not have a significant impact on the results and our specific

choice was made for consistency with Benedict et al. (2002).

Thepositiondeviationsofthe stars duetoparallactic,proper,

and orbital (GJ876 only) motion were calculated in the usual

right ascension – declination reference frame first. These devi-

ations were then rotated about the roll angle of the FGS during

the constraint plate observations to place them in the X – Y ref-

erence frame of the instrumentat that epoch.The equationsused

for these calculations were

Dα= Pαπ + µα∆t + ORBITα,

(2)

Dδ= Pδπ + µδ∆t + ORBITδ,

(3)

Dξ= Dαcos θ + Dδsin θ,

(4)

Dη= −Dαsin θ + Dδcos θ,

(5)

where D are the motion displacements, P are the parallax fac-

tors, π is the parallax, µ are the proper motions, ∆t is the time

difference from the reference epoch, ORBIT are the orbital mo-

tions, and θ is the roll angle of the constraint plate. The α and

δ subscripts refer to the right ascension and declination compo-

nents respectively. The ξ and η subscripts refer to the X and Y

componentsin the referenceframeofthe constraintplate respec-

tively.

Page 4

4J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system

Table 1. Parameters from the coplanar analysis.

Orbital Parameters

ParameterPlanet bPlanet cPlanet d

m

2.57+0.06

−0.08MJup

0.80+0.02

−0.02MJup

8.17+0.95

−0.93M⊕

a (AU)0.20688+0.00005

−0.00004

0.13062+0.00004

−0.00004

0.0208069+0.0000001

−0.0000004

e

0.0376+0.0022

−0.0019

0.2657+0.0022

−0.0017

0.0 (Fixed)

ω (◦)184.0+2.8

−3.3

197.3+0.4

−0.6

0.0 (Fixed)

M (◦) 167.3+3.8

−3.2

311.6+1.0

−0.8

311.8+4.4

−5.9

ParameterValue

i (◦)48.9+1.8

−1.6

Ω (◦)251+16

−16

Fit information

ParameterValue

χ2

860.0

DOF843

radial velocity rms (ms−1)4.2

astrometry rms (mas)0.9

Note: The orbital parameters are osculating and valid at HJD2452490.0.

The roll angle of the constraint plate was estimated to be

26.01◦± 0.05◦based on comparison to ground based astrome-

try catalogs. The actual roll angle we used for calculating the

model was always a free parameter and the estimated value was

treated as an observation with error to provide a constraint (i.e.

a comparisonof the estimated value to the actual value used was

included in the overall χ2calculation).

We always solved for the parallaxes and proper motions of

GJ876 and the five reference stars in each of the analyses de-

scribed below. We used as observations with error the same es-

timated parallaxes and previously measured proper motions for

the reference stars also utilized by Benedict et al. (2002). Unlike

Benedict et al. (2002), we also used the Hipparcos parallax and

proper motion for GJ876 (π = 212.69 ± 2.10mas, µα= 960.31

± 3.77masyr−1, µδ= -675.61 ± 1.58mas yr−1, Perryman et al.

1997) as observations with error. This is justified because the

HipparcossolutionforGJ876is unlikelyto be affected byits or-

bital motion due to the small size (∼0.3mas) and short timescale

(∼60d) of the perturbations relative to the precision (∼5mas)

and time span (2.2yr) of the Hipparcos observations of GJ876.

We used the same six parameter model as Benedict et al.

(2002) to account for changes in the plate scale, rotation, and

offset during the different observational epochs. The equations

of condition for the astrometry model were

∆ξ= Ax + By + C + Rx(x2+ y2) − (ξ + Dξ),

(6)

∆η= −Bx + Ay + F + Ry(x2+ y2) − (η + Dη),

(7)

where ∆ are the residuals, x and y are the measured positions;

A, B, C, F, Rx, and Ryare the plate parameters; and ξ and η

nominal positions of the stars at the reference epoch. For the

observationsmadeduringtheadoptedconstraintepoch,theplate

parameter A was fixed to 1, and B, C, F, Rx, and Rywere fixed

to 0. These were free parametersfor each of the other 26 epochs.

The nominal positions for each star at the reference epoch were

also free parameters.

In total, and excluding the orbital motion component for

GJ876, there were always five free parameters for each of the

six stars, six free parameters for each of 26 epochs, plus the one

roll angle for a total of 186 astrometric only parameters. There

were 436 FGS observations in each X and Y, as well as three

constraints for each of the five stars and one constraint for the

roll angle.

We used the usual χ2parameteras the goodness-of-fitmetric

in our analyses. The χ2of the model comparisons to all the data

was calculated from Eq. (1), (6), and (7) along with the compar-

ison of the certain astrometric parameters mentioned above to

their input values.

3.2. Coplanar study

We carried out an analysis of the data assuming that the plan-

ets were in coplanar orbits (i.e. we set i = ib = ic = id and

Ω = Ωb= Ωc= Ωd). We used a combination of grid search and

local minimization algorithms to find the parameters that min-

imized the χ2between the model and the observed data. The

parameter uncertainties were estimated by stepping out from

the best-fit values of each parameter in turn while marginal-

izing over the remaining parameters until the χ2increased by

1.0 from the minimum value. The identified orbital parameters

and their uncertainties along with some fit quality statistics are

given in Table 1. The identified astrometric parameters (paral-

laxes, proper motions, and positions) are essentially the same as

for the non-coplanar analysis (§3.3), so we give only the results

from the later investigation because we consider them more ro-

bust (see Table 2).

Page 5

J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system5

For the orbital orientations, we find i = 48.9◦+1.8◦

251◦+16◦

very similar to that suggested by Rivera et al. (2005) based on

analysis of the radial velocities alone. It is also completely con-

sistent with the astrometric perturbation size that Benedict et al.

(2002) measured, but is not consistent with their estimated incli-

nation for planet b (ib= 84◦± 6◦). The difference between our

result and theirs arises from the radial velocity data. We have

more and better quality data than was available then. In addi-

tion, we have the benefit of hindsight that the dynamical model-

ing is crucial for the radial velocity data, and that this modeling

gives more constraints on the planets’ orbital orientations than

can usually be obtained.

We find that the planet-planet perturbations are of such sig-

nificance that the radial velocity data set is actually more sensi-

tive than the astrometry to the inclination of the system in this

unique case. This is due to the dependence of the interactions,

which are visible in the radial velocity data, on the true masses

of the two gas giant planets (planets b and c). The situation is

illustrated in Fig. 1, where we show the best-fit χ2for the radial

velocity and astrometry data components along with the total

for fixed inclinations. The perturbation due to planet b is clearly

detected in the astrometry (with false alarm probability of 1.5

x 10−5), but its inclusion does not alter the fit quality response

so much that the best-fit inclination is significantly different be-

cause of the steep response of the radial velocity fit quality.

The astrometry fit quality component does trend lower for

smaller inclinations, but this should not be interpreted to mean

that the astrometry would favor smaller inclinations were it in-

dependentof the radial velocities. This is because the astrometry

data are not sufficient to uniquely determine all of the necessary

planet orbital parameters. Therefore, they cannot be divorced

from the constraints given by radial velocities and an “astro-

metric only” inclination cannot be determined. For example, the

astrometry data are better fitted for i = 30◦, but only with the

majority of the orbital parameters determined from the fit to the

radial velocities. In the later case, the fit is very bad (∆χ2> 100

from the best-fit) and, thus, the orbital parameters determined

for this inclination and used to generate the astrometric orbit are

not physical. As the fit to the astrometry is still acceptable for

i ∼ 50◦(we have only a 0.05mas larger rms than the best-fit of

Benedict et al. 2002), we concludethat the astrometryand radial

velocities are not in disagreement and that the model giving the

best-fit to the combined data set is the optimal one.

Although the radial velocities are sensitive to the inclination

of the system, there remains a degeneracybetween i and 180◦−i

pairsthatis unresolvablewith thatdataalone.Thisis becausethe

radial velocities are sensitive to the inclination through their de-

pendenceonthe true masses ofthe planets andthe masses would

be the same for i and 180◦− i. The inclusion of the astrometry

resolves this degeneracy, and we find that the inclination of the

system is near 50◦rather than 130◦.

−1.6◦and Ω =

−16◦. The inclination we determine for a coplanar model is

3.3. Constraining the degree of coplanarity

3.3.1. Fitting the data

As discussed in §1, Rivera et al. (2005) found that the radial ve-

locities are sensitive to differences in the orbital inclinations of

planets b and c, and they reached a tentative conclusion that the

orbits were likely coplanar. However, the mutual inclination an-

gle (Φ) of two orbits depends not only on the inclinations, but

also on the longitudes of the ascending nodes. In this case, the

Fig.1. Relative change in the best-fit total χ2(pluses), radial ve-

locity component χ2(triangles), and astrometry component χ2

(circles) as a function of the coplanar model inclination. The in-

set shows a region around the minimum for the total χ2.

mutual inclination of planets b’s and c’s orbits is given by the

equation

cos Φbc= cos ibcos ic+ sin ibsin iccos(Ωb− Ωc).

(8)

Because of their lack of the constraint on the absolute orbital

inclinations and orbital longitudes of the ascending nodes that

is offered by the astrometry, Rivera et al. (2005) were unable to

quantify the mutual inclination of the two planets’ orbits.

Inourcase,theinclusionoftheastrometryhelpscharacterize

the full three dimensional orbit of planet b. Therefore,we have a

benchmark to measure misalignment relative to. This motivated

us to use the combined data set and dynamical model to deter-

minetheorbitalorientationsofplanetsbandcuniquely,andthus

set limits on their orbital mutual inclination.

To do this, we fit the data using an expanded version of the

coplanar model. Instead of solving for a single inclination and

longitude of the ascending node, we allowed planets b and c to

have their own independent values (ib, Ωb, ic, and Ωc). The data

are not very sensitive to the orientation of planet d’s orbit, so we

set its inclination and longitude of the ascending node to that of

planet b (i.e. id= iband Ωd= Ωb), which is the most massive of

the three planets.

For our initial analysis, we used the same grid search and

localminimizationtechniqueas forthecoplanarstudy.However,

wefoundtheχ2surfacetobeverychaotic.Thishinderedreliable

identification of the minimum χ2and error estimation because

there were many valleys in the surface containing local minima

that were statistically indistinguishable from each other.

The difficulty with the previous method led us to ultimately

use a Markov Chain Monte Carlo (MCMC) analysis to iden-

tify the most likely parameter values and their confidence inter-

vals for the non-coplanar model. The details and advantages of

MCMCaredescribedextensivelyelsewhere(fordiscussioninan

astronomical context see e.g. Tegmark et al. 2004; Ford 2006).

Our implementation used 10 Markov chains of 107points each.

For each chain step, a jump in one of the parameters was consid-

ered. If the jump resulted in a lower χ2than the previous point

the jump was accepted. Otherwise, the jump was accepted with

Page 6

6 J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system

Table 2. Parameters from the non-coplanar analysis.

Orbital Parameters

Parameter Planet bPlanet cPlanet d

m

2.64+0.11

−0.09MJup

0.78+0.05

−0.03MJup

8.41+0.78

−0.75M⊕

a (AU)0.20700+0.00010

−0.00009

0.13062+0.00005

−0.00005

0.0208069+0.0000004

−0.0000004

e

0.0363+0.0028

−0.0026

0.2683+0.0058

−0.0052

0.0 (Fixed)

ω (◦)188.2+4.9

−4.0

200.4+1.8

−1.9

0.0 (Fixed)

M (◦)163.1+4.0

−4.9

309.1+1.9

−1.7

312.2+4.9

−5.0

i (◦)47.2+2.2

−2.4

51.1+3.6

−3.9

47.2 (Tied)

Ω (◦) 252.3+8.4

−7.7

249.4+7.1

−8.4

252.3 (Tied)

Astrometric Parameters

Star

ξηπabs

(mas)

µα

µδ

(arcsec)(arcsec) (masyr−1)(masyr−1)

GJ87651.9001+0.0004

−0.0004

730.3929+0.0003

−0.0003

215.5+0.4

−0.5

955.7+1.7

−1.7

-673.4+1.1

−1.1

Ref-2-27.8463+0.0003

−0.0003

767.6817+0.0005

−0.0005

1.0+0.3

−0.3

3.1+0.4

−0.4

2.3+0.6

−0.6

4.7+0.6

−0.7

1.9+0.3

−0.3

5.5+1.6

−1.7

14.2+2.0

-16.0+1.2

−1.1

Ref-3-226.2855+0.0005

−0.0005

759.8752+0.0007

−0.0007

−2.0

2.4+1.6

−1.7

-43.0+1.2

Ref-4-297.4706+0.0005

−0.0005

639.3754+0.0005

−0.0005

-39.8+2.2

−2.2

−1.2

Ref-5450.9640+0.0006

−0.0006

635.2999+0.0004

−0.0004

7.3+3.0

−3.3

-13.1+2.4

-1.2+2.4

−2.2

Ref-6351.6679+0.0005

−0.0005

594.2625+0.0005

−0.0005

−2.7

-3.8+2.3

−2.1

Fit information

ParameterValue

χ2

855.8

DOF841

radial velocity rms (ms−1)4.2

astrometry rms (mas) 0.9

Note: Planet d’s inclination and longitude of the ascending node were tied to those of planet b. The orbital parameters are osculating and valid

at HJD2452490.0. The coordinates (ξ and η) are relative positions in the reference frame of the constraint plate. To convert to the RA – DEC

reference frame, the coordinates should be rotated about the inverse of the determined roll angle (26.07◦ +0.04◦

best-fit model.

−0.04◦). The χ2and rms shown are for the

a probability of exp(−∆χ2/2). The characteristic jump sizes for

each parameter were tuned to give a 20 – 40% acceptance rate.

Each chain was initialized with a different combination of

parameter values well dispersed from the region of parameter

space thought to contain the lowest χ2. Initial tests indicated a

typical correlation length of ∼2000 points, or roughly 10 times

the number of parameters. Therefore, we elected to record the

parameters only every 2000 steps to save memory. Each chain

took30CPU days computationtimeon anaveragedesktopcom-

puter.

All the chains converged to the same region of parameter

space, or “burned in”, within ∼18000 steps. To provide a sta-

tistical check that the probability distributions had been thor-

oughly sampled, we computed the Gelman & Rubin (1992) R

statistic for the parameter values among the chains. The statistic

was within 10% of unity for all the parameters, which indicates

the chains were likely long enough for robust inference.

After trimming the burn-in points, we combined the data

from the 10 chains to give parameter distributions with 49910

points. We adopted the medians of the MCMC distributions as

the best estimates of the parameter values. The 1σ uncertainties

were taken to be the range of values that encompassed 68.3%

of the parameter distributions on each side of the corresponding

median. The results are given in Table 2. It should be noted that

the errors from the MCMC analysis are correlated because they

are calculated fromthe distributions arising from a simultaneous

determinationof all the parameters.This explains whythe errors

on our astrometric parameters are larger by factors of 2 – 3 than

the uncorrelated errors given by Benedict et al. (2002) despite

our achieving a similar astrometric fit quality (rms = 0.9mas).

Further support for the validity of the MCMC analysis is

the fact that the orbital model formed by the adopted param-

eter values (medians of the MCMC parameter distributions) is

essentially the same as the one we initially identified as the best

non-coplanar model using the grid search with local minimiza-

tion technique (∆χ2= 0.2). The main advantages of the MCMC

method are that the parameter confidence limits account for the

irregularχ2surface, and allow calculation of composite parame-

ter uncertainties when including correlations among the param-

eters (see below).

The MCMC parameter distributions for the masses, inclina-

tions, and longitudes of the ascending nodes for planets b and c

Page 7

J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system7

Fig.2. Probability distributions for the masses, inclinations, and longitudes of the ascending nodes for planets b and c from the

MCMC analysis. The medians and two-sided 68.3% confidence limits are given by the dashed and dotted lines respectively.

Fig.3. Probability distribution for the mutual inclination of the

orbits of planets b and c computed from the probability distribu-

tionsforib,Ωb,ic,andΩcusingEq.8.Themedianandtwo-sided

68.3%confidencelimits are givenby the dashedand dotted lines

respectively.

are shown in Fig. 2. As these plots demonstrate, the combined

analysis of the radial velocity and astrometry data with the dy-

namical model yielded tight constraints on the masses and or-

bital orientations of the two planets. Using these probability dis-

tributions, we may calculate the probability distribution for the

mutual inclination angle between their orbits directly. The result

is shown in Fig. 3, and we find Φbc= 5.0◦ +3.9◦

−2.3◦.

3.3.2. Secular behavior

The orbital parameters we identified are only valid for the refer-

ence epoch (HJD2452490.0) because the system configuration

is varying with time. Therefore, the mutual inclination angle we

determined is only a snapshot of the system and could be mis-

leading about its normal characteristics. For example, we might

have caughtthe system whenplanets b’s and c’s orbits were near

their minimum or maximum mutual inclination.

To study the time-dependency of the configuration implied

by our model for the GJ876 system, we integrated the planets’

orbital motion forward for 1Myr. A short segment of the results

for the inclinations, longitudes of the ascending nodes, and mu-

tual inclination for the orbits of planets b and c are shown in

Fig. 4. We find that the orbital projection angles for the plan-

ets are varying regularly, but with a variety of different frequen-

cies andamplitudes.Themutualinclinationbetweenthe planets’

orbits varies with a main period of 4.8yr and amplitude 0.15◦.

There are also two other coherent lower-amplitude periodicities

around 60d, which is similar to the outer planet’s orbital period.

These low-amplitude variations are slightly out of phase with

one another and their interference leads to beat patterns over

10yr timescales.

Aside from mutual inclination changes, the two planets’ or-

bital orientation angles relative to the plane of the sky librate

with a period of 101.9yr. This is a projection effect arising from

the libration of the planets’ orbital nodes. Their mutual incli-

nation is not affected by this variation and so the planets’ or-

bital nodes must be librating together. We conclude from this

exploratory investigation that our measured mutual inclination

for the orbits of planets b and c is likely representative of the

long-term status of the system as the measurement uncertainties

are an order of magnitude larger than the potential variations.

A more thorough examination of the dynamical qualities of our

model would be interesting, but is beyond the scope of the cur-

rent paper.

Page 8

8 J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system

Fig.4. Long-term evolution of the projected orbital elements and mutual inclination for planets b (solid line) and c (dashed line).

The variation is regular and continues in a similar fashion for at least 1Myr. The bars on the left indicate the uncertainties in the

osculating orbital elements used to initialize the simulation.

4. Discussion

Our analysis has revealed the full three-dimensional orbits, and

thus the degree of coplanarity, of two planets in an exoplanetary

systemaroundanormalstarforthefirsttime1.Broadlyspeaking,

we find the orbits of GJ876b and c to be coplanar. However,

our results also imply a small, but potentially significant (∼95%

confidence), non-zero orbital mutual inclination that could be

important.

Dynamical friction from planetesimals during the late stages

of planet formation is expected to result in orbits for gas gi-

ants that are coplanar (Kokubo & Ida 1995; Pollack et al. 1996;

Goldreich et al. 2004). Therefore, our general result provides

further evidence for planet formation in a circumstellar disk,

and suggests that the evolution of planetary systems might

not lead to excitation of extreme inclinations for planetary or-

bits relative to the original plane of the disk. Additional ev-

idence from observations of exoplanetary systems for planet

formation in a disk and little inclination from the original

plane includes the coplanarity of an exoplanet’s orbit with

a debris disk (Benedict et al. 2006) and the stellar spin –

planet orbit alignment of a number of transiting planet sys-

tems (Winn et al. 2005, 2006; Wolf et al. 2007; Winn et al.

2007; Narita et al. 2007; Bouchy et al. 2008; Winn et al. 2008;

Loeillet et al. 2008; Johnson et al. 2008; Cochran et al. 2008),

although see H´ ebrard et al. (2008) for one possible exception to

this trend.

Our more subtle finding of a possible small degree of non-

coplanarityin the GJ876system is a complementtothe observa-

tions that planetary system evolution often leads to eccentricity

excitation and displacement of planets from their birthplace. As

has been previously noted, the eccentricity of planet c is signifi-

cantly non-zero (0.27), and it is unlikely that both planets b and

c could have formed in situ (Laughlin et al. 2005). Therefore, it

1A previous coplanarity measurement was obtained for two planets

orbiting a pulsar (Konacki & Wolszczan 2003).

seems probable that the system has undergone some significant

evolutionary changes.

Lee & Peale (2002) have suggested that convergent migra-

tion of planets b and c due to disk torques led to resonance cap-

ture and eccentricity excitation. This seems to be the most likely

explanation for the system’s configuration, but there is still an

open question of how the planets’ eccentricities were kept from

being excited to even higher values while the planets were mi-

grating (Kley et al. 2004, 2005; Laughlin et al. 2005). Either the

planets actually didn’t migrate very far, or there was effective

eccentricity damping during the migration.

Along this same line, Thommes & Lissauer (2003) have

shown that resonance capture of two planets can also result in

an inclination-type mean motion resonance that quickly leads to

excitation of mutual inclinations of 30◦. Mutual inclinations of

60◦or more can be achieved if the system experiences this si-

multaneously with an eccentricity-type mean motion resonance.

However,entry into the inclination-typemean motion resonance

requires the eccentricity of the inner planet to be>≈0.6, which is

a condition that was likely not met in the GJ876 system. Our

finding of only a small mutual inclination for planets b and c is

therefore a further constraint on the system’s evolutionary his-

tory. The nearly coplanar configuration of the planets’ orbits

is fully consistent with the scenario that they did not experi-

ence the inclination-type mean motion resonance because of the

only moderate eccentricity excitation during migration. Thus,

the question of why the planets’ eccentricities were not excited

to higher values becomes more important. It would be interest-

ing to investigate whether hydrodynamic simulations of differ-

ential migration and resonance capture due to disk interactions

(e.g. Kley et al. 2004) could reproduce the small degree of non-

coplanarity we have found when extended to three dimensions.

As the GJ876 system is the only planetary system other

than the Solar System for which we have tight constraints

on the degree of coplanarity it is interesting to compare the

two. Surprisingly, we find that they share some similarities de-

Page 9

J. L. Bean and A. Seifahrt: The architecture of the GJ876 planetary system9

spite their obvious differences. GJ876b and c have a mass ra-

tio (mb/mc = 3.38) very similar to the Jupiter – Saturn pair

(mJup/mSat = 3.34). This seems like a coincidence because it

is unclear how such a property of neighboring gas giants could

be maintained in different formation environments. Gas giants

are thought to form via runaway gas accretion on to a solid core

so such a property would require exact timing uniquely for each

case.

More interestingly, Tsiganis et al. (2005) hypothesized that

Jupiter and Saturn experienced an encounter with the 2:1 mean

motion resonance due to migration. They suggested that this

encounter led to eccentricity and mutual inclination excitation

for the planets in the outer part of the Solar System, and is

the reason for their currently non-circular and non-coplanar or-

bits. In contrast to the GJ876 system though, the Tsiganis et al.

(2005) model for the Solar System has Jupiter and Saturn pass-

ing through the resonance owing to their diverging migration.

As a result of being caught in the resonance, GJ876c’s eccen-

tricity was pumped up to at least three times the value that any

of the Solar System giant planets’ orbits reach. Additionally,

our results indicate that the GJ876 b-c orbital mutual inclina-

tion is potentially a few times larger as well. Furthermore, the

Jupiter – Saturn 2:1 resonance encounter interactions also in-

volved Uranus and Neptune, and the planets’ final orbits de-

pended on the details of the complex four body scattering. The

GJ876 system is known to harboran additional low-mass planet

in a short period orbit. It is unclear how this object was involved

in the dynamical evolution of the system, to say nothingof other

still-to-be-discovered planets that could potentially exist in the

system.

Ultimately, our determination of the orbital mutual inclina-

tion forGJ876bandc is just onemore pieceof the planetforma-

tion and evolutionpuzzle.It would be useful to obtainsuch mea-

surements for many other exoplanetarysystems to see if most or

all systems tend to be fairly coplanar, but with a small amount

of mutual inclination. Systems with planets in low-order reso-

nances are particularly interesting targets due to the constraints

on disk interactions knowledge of their architecture provides.

In this regard, our analysis illustrates how such measurements

could be achieved using data obtained from existing facilities.

The GJ876 system is unique for the size and timescale of the

planet-planet interactions, and so radial velocity data for other

systems will not be as sensitive to their architecture. However, a

numberof other moderatelyinteracting multi-planetsystems are

better astrometric targets than GJ876 because one of the planets

in them induces a host star perturbationlarger than 1mas, which

is the typical measurement uncertainty of the HST FGS. If ro-

bust astrometric characterization of one planet in a moderately

interacting system can be obtained, then dynamical considera-

tions could be used to constrain the degree of coplanarity for the

other planets.

Acknowledgements. We thank Ansgar Reiners and an anonymous referee for

comments that helped us improve this paper. J.B. and A.S. acknowledge funding

for this work from the DFG through grants GRK 1351 and RE 1664/4-1.

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