Properties of the single Jovian planet population and the pursuit of Solar system analogues
MMNRAS , 1–13 (2017) Preprint 06 November, 2018 Compiled using MNRAS L A TEX style file v3.0
Properties of the single Jovian planet population and thepursuit of Solar system analogues
Matthew T. Agnew, Sarah T. Maddison, and Jonathan Horner Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia University of Southern Queensland, Toowoomba, Queensland 4350, Australia
Accepted 2018 April 2. Received 2018 March 15; in original form 2018 January 17
ABSTRACT
While the number of exoplanets discovered continues to increase at a rapid rate, we arestill to discover any system that truly resembles the Solar system. Existing and nearfuture surveys will likely continue this trend of rapid discovery. To see if these systemsare Solar system analogues, we will need to efficiently allocate resources to carry outintensive follow-up observations. We seek to uncover the properties and trends acrosssystems that indicate how much of the habitable zone is stable in each system toprovide focus for planet hunters. We study the dynamics of all known single Jovianplanetary systems, to assess the dynamical stability of the habitable zone around theirhost stars. We perform a suite of simulations of all systems where the Jovian planet willinteract gravitationally with the habitable zone, and broadly classify these systems.Besides the system’s mass ratio ( M pl / M star ), and the Jovian planet’s semi-major axis( a pl ) and eccentricity ( e pl ), we find that there are no underlying system propertieswhich are observable that indicate the potential for planets to survive within thesystem’s habitable zone. We use M pl / M star , a pl and e pl to generate a parameter spaceover which the unstable systems cluster, thus allowing us to predict which systems toexclude from future observational or numerical searches for habitable exoplanets. Wealso provide a candidate list of 20 systems that have completely stable habitable zonesand Jovian planets orbiting beyond the habitable zone as potential first order Solarsystem analogues. Key words: methods: numerical – planets and satellites: dynamical evolution andstability – planets and satellites: general – planetary systems – astrobiology
A key goal of exoplanetary science is to find Earth analogueplanets - planets that might have the right conditions for lifeto both exist and be detectable. Given that the only locationwe know of that hosts life is the Earth, that search is stronglybiased towards looking for planetary systems that stronglyresemble our own - Solar system analogues. While we haveseen an explosion of exoplanet discoveries in the last decadethat is sure to continue (e.g. Sullivan et al. 2015; Dressinget al. 2017), the discovery of Solar system analogues stillproves to be a decidedly challenging goal. While Jupiter-sized planets have been detected for over 20 years, theseare often very close to their host stars (e.g. Mayor & Queloz1995; Charbonneau et al. 2000). It has only been much morerecently that we have begun to detect Jupiter-sized planetson decade long orbital periods; the so-called Jupiter ana-logues (Boisse et al. 2012; Wittenmyer et al. 2014, 2016;Kipping et al. 2016; Rowan et al. 2016). Similarly, discover- ies of lower mass planets have become more common (Wrightet al. 2016; Vogt et al. 2015; Gillon et al. 2017), thanks in alarge part to the Kepler survey (Borucki et al. 2010; Mortonet al. 2016). The current count of confirmed exoplanets nowexceeds 3500 . As a result, we can begin to consider the ex-oplanet population as a whole in order to better understandany overarching properties of the sample, and to also pro-vide a means to exclude existing systems from further followup in our search for Solar system analogues.Due to the observational biases inherent to the radialvelocity (RV) method (Wittenmyer et al. 2011; Dumusqueet al. 2012), a great deal of work has gone into attemptingto theoretically predict where additional exoplanets couldremain stable in existing systems, via both predictions ofregions of stability and/or instability (Jones et al. 2001;Jones & Sleep 2002; Jones et al. 2005; Jones & Sleep 2010; As of 18 January 2018 (NASA Exoplanet Archive, exoplan-etarchive.ipac.caltech.edu). © a r X i v : . [ a s t r o - ph . E P ] A p r M. T. Agnew et al.
Giuppone et al. 2013) and dynamical simulations (Rivera &Haghighipour 2007; Thilliez et al. 2014; Kane 2015; Thilliez& Maddison 2016). The large size of Jovian planets meansthey are often easier to detect and can dominate RV sig-nals. For this reason, it has been suggested that the seemingabundance of single Jovian planet systems is the result ofan observational bias rather than a true reflection of theexoplanet population (Marcy et al. 2005; Cumming et al.2008).In the Solar system, Jupiter is thought to have playedan integral role in determining the Solar system architecturethat we see today (e.g. Gomes et al. 2005; Horner et al. 2009;Walsh et al. 2011; Izidoro et al. 2013; Raymond & Morbidelli2014; Brasser et al. 2016; Deienno et al. 2016). A number ofauthors have investigated the role Jupiter may have playedin nurturing the right environment on Earth for life to haveprospered (e.g. Bond et al. 2010; Carter-Bond et al. 2014;Carter-Bond et al. 2012; Martin & Livio 2013; Quintana& Lissauer 2014; O’Brien et al. 2014) although this is stillan active area of research for which debate continues (e.g.Horner & Jones 2008, 2009, 2010, 2012; Horner et al. 2014a;Grazier 2016). However, without a clear answer, the searchfor Solar system analogues and the search for habitable exo-planets remains tightly coupled. Finding a true Solar systemanalogue is inherently challenging due to the small transitand radial velocity signals of the inner rocky planets, andthe large decade to century long orbits of the outer giantplanets. Because of this, we begin with a simplified defini-tion of a Solar system analogue, that being: a Sun-like starwith a rocky planet in the habitable zone (HZ), and a Jo-vian planet orbiting beyond the outer boundary of the HZ.Thus, searching for single Jovian systems that are capableof hosting hidden Earth-like planets in the HZ becomes anatural starting point in the search for habitable exoplanetsand Solar system analogues.Agnew et al. (2017) took a sample of single Jovianplanet systems and used N-body simulations to produce acandidate list of systems that could host a M ⊕ planet ona stable orbit within the system’s habitable zone, and thatcould be detected with current or near-future instruments.Here we expand upon that earlier work by examining allcurrently known single Jovian planet systems to (1) identifyany overarching trends (that may be the result of formationor evolution scenarios) within the single Jovian planet popu-lation, (2) exclude planetary architectures within which thesystem’s HZ would be unstable, and (3) provide a candidatelist to guide future observing efforts in the search for Solarsystem analogues.In section 2, we describe the method used to calculatethe boundaries of the HZ, how we select the single Jovianplanet systems which we wish to simulate, and detail the nu-merical simulations used to dynamically analyse these sys-tems. We then discuss our results in section 3, and presenta candidate list of Solar system analogues for use by futureplanet hunters. We summarise our findings in section 4. Table 1.
The constants used to calculate the HZ for our sim-ulations, assuming the Earth-like planet we are searching for is M ⊕ , as presented in Kopparapu et al. (2014)Runaway Greenhouse Maximum Greenhouse a . × − . × − b . × − . × − c − . × − − . × − d − . × − − . × − S eff (cid:12) .
107 0 . We first consider the existing single Jovian population fromthe NASA Exoplanet Archive , removing all systems withincomplete stellar or planetary properties. We then calcu-late the HZ boundaries for each system using the methodoutlined in Kopparapu et al. (2014). This allows us to es-timate those systems for which the HZ will likely be stable(due to the distance of the Jovian planet from the HZ). Forthose systems for which the Jovian is located close enoughto the HZ to potentially perturb the region, we then peformdynamical simulations to ascertain the degree to which thisoccurs.For our analysis of the sample of the single Jovian planetpopulation, we accept the stellar and planetary properties asthey are presented in the relevant databases, acknowledgingthat there may be uncertainties associated with these pa-rameters. We calculate the HZ boundaries using the method outlinedby Kopparapu et al. (2014), which is only valid for stars with K (cid:54) T eff (cid:54) K. They present an equation for theastrocentric distance of different regimes for the inner andouter boundary of the HZ as d HZ = (cid:115) L / L (cid:12) S eff au , (1)where L is the luminosity of the star, and S eff is calculatedas S eff = S eff (cid:12) + aT (cid:63) + bT (cid:63) + cT (cid:63) + dT (cid:63) , (2)where T (cid:63) = T eff − K, and a , b , c , d and S eff (cid:12) are con-stants depending on the planetary mass considered, M pl ,and the HZ boundary regime being used. Here, we assumea M ⊕ planet, and use a conservative HZ boundary regimeutilising the Runaway Greenhouse boundary for the inneredge, and the Maximum Greenhouse boundary for the outeredge (Kopparapu et al. 2014). This corresponds with theconstants shown in Table 1. Kane (2014) found that theseboundaries are significantly influenced by uncertainties ofthe stellar parameters, but we use the best fit values as pre-sented in the NASA Exoplanet Archive. Our sample of single Jovian planets was obtained from theNASA Exoplanet Archive, exoplanetarchive.ipac.caltech.edu, on27 March 2017 which gave an initial sample of 771 planets.MNRAS000
107 0 . We first consider the existing single Jovian population fromthe NASA Exoplanet Archive , removing all systems withincomplete stellar or planetary properties. We then calcu-late the HZ boundaries for each system using the methodoutlined in Kopparapu et al. (2014). This allows us to es-timate those systems for which the HZ will likely be stable(due to the distance of the Jovian planet from the HZ). Forthose systems for which the Jovian is located close enoughto the HZ to potentially perturb the region, we then peformdynamical simulations to ascertain the degree to which thisoccurs.For our analysis of the sample of the single Jovian planetpopulation, we accept the stellar and planetary properties asthey are presented in the relevant databases, acknowledgingthat there may be uncertainties associated with these pa-rameters. We calculate the HZ boundaries using the method outlinedby Kopparapu et al. (2014), which is only valid for stars with K (cid:54) T eff (cid:54) K. They present an equation for theastrocentric distance of different regimes for the inner andouter boundary of the HZ as d HZ = (cid:115) L / L (cid:12) S eff au , (1)where L is the luminosity of the star, and S eff is calculatedas S eff = S eff (cid:12) + aT (cid:63) + bT (cid:63) + cT (cid:63) + dT (cid:63) , (2)where T (cid:63) = T eff − K, and a , b , c , d and S eff (cid:12) are con-stants depending on the planetary mass considered, M pl ,and the HZ boundary regime being used. Here, we assumea M ⊕ planet, and use a conservative HZ boundary regimeutilising the Runaway Greenhouse boundary for the inneredge, and the Maximum Greenhouse boundary for the outeredge (Kopparapu et al. 2014). This corresponds with theconstants shown in Table 1. Kane (2014) found that theseboundaries are significantly influenced by uncertainties ofthe stellar parameters, but we use the best fit values as pre-sented in the NASA Exoplanet Archive. Our sample of single Jovian planets was obtained from theNASA Exoplanet Archive, exoplanetarchive.ipac.caltech.edu, on27 March 2017 which gave an initial sample of 771 planets.MNRAS000 , 1–13 (2017) roperties of the single Jovian planet population The single Jovian planet population as of 27 March 2017is made up of 771 systems . We remove from this samplethose systems that are missing planetary or stellar proper-ties, which excludes 175 systems. From the 596 that remain,54 systems feature stellar temperatures that fall outside therange K (cid:54) T eff (cid:54) K required by the Kopparapuet al. (2014) HZ calculation, and so these too are removedfrom our sample. This yields the final sample of 542 systems.For all systems for which the Jovian planet is greaterthan 10 Hill radii from the midpoint of the HZ, we expectlittle to no gravitational stirring within the HZ (Jones et al.2005; Jones & Sleep 2010; Giuppone et al. 2013). In suchsystems, computational resources are wasted on simulatingcompletely stable HZs. Of the 542 systems in our sample,a total of 360 systems fell into this category, of which 355had an interior Jovian, and 5 an exterior Jovian. We ex-cluse these systems from our suite of simulations, and sim-ply tag them as having wholly stable HZs. In our sample, allof the systems remaining that are expected to gravitation-ally stir the HZ have a Jovian planet with an orbital period . T HZ,mid (cid:54) T Jovian (cid:54) T HZ,mid , and so we use this cri-teria as a slightly more conservative cut than a pl > R Hill .A histogram illustrating the 542 systems with the orbitalperiod cuts overlaid can be seen in Figure 1.Using the Jovian orbital period criterion outlined aboveleaves a total of 182 single Jovian systems that could, poten-tially, exhibit a degree of instability within the HZ. In thiswork, we simulate this sample to investigate the impact ofthe Jovian planets on the stability across the HZ.
In order to assess whether a system with a known Jovianplanet could host an Earth-like world in its habitable zone,we carry out a suite of detailed N -body simulations. Wedistribute a large number of massless test particles (TPs)through the HZ of the systems in which we are interested,and integrate the evolution of their orbits forwards in timefor a period of 10 million years. This is a computationally in-tensive endeavour, and the simulations we present below re-quired a total of six months of continuous integration acrossthe several hundred computing cores available to us. In orderto facilitate a timely analysis, in this work we solely examinethe scenario of co-planar systems - in which the orbits of ourputative exo-Earths are always set to move in the same planeas the Jovian planet. This focus on co-planar orbits is com-mon in exoplanetary science, being the standard assumptionin the modelling of the orbits of newly discovered multipleexoplanet systems (e.g. Robertson et al. 2012; Horner et al.2013, 2014b), where no information is currently held on themutual inclination between the orbits of the known planets.In some studies (e.g. Horner et al. 2011; Wittenmyer et al.2012; Wittenmyer et al. 2013b), we have shown that mutualinclination between exoplanet orbits typically acts to rendera system less stable. As such the assumption of co-planarityis a mechanism by which we maximise the potential for agiven system to exhibit a dynamically stable habitable zone. Those systems with only one planet that has R pl > Earth radiior M pl > Earth masses in lieu of available radius data.
Table 2.
The range of orbital parameters within which the testparticles were randomly distributed throughout the HZ.Min Max a (au) HZ min HZ max e i ( ° ) 0.0 0.0 Ω ( ° ) 0.0 0.0 ω ( ° ) 0.0 360.0 M ( ° ) 0.0 360.0 Both our own Solar system and those multiple exoplanetsystems have demonstrably very low mutual inclinations(Lissauer et al. 2011a,b; Fang & Margot 2012; Figueira et al.2012; Fabrycky et al. 2014). However, we acknowledge thatthose systems are not perfectly flat. Once a small amountof mutual inclination is added to a previously co-planar sys-tem, it opens up the possibility for the excitation of boththe inclination and eccentricity of objects that would other-wise have been moving on mutually stable orbits. As such,in the future, we plan to expand this work to investigate theimpact of small, but non-zero, mutual inclinations on thestability (or otherwise) of those systems for which this workpredicts a dynamically stable outcome. In this work, our fo-cus remains on the study of the most optimistic scenario,that being perfect co-planarity, the aforementioned caveatmust be kept in mind.To test the dynamic stability of each system, we use the swift N -body software package (Levison & Duncan 1994)to run a series of simulations with massless TPs. We ran-domly distribute 5000 TPs throughout the HZ of each sys-tem within the ranges shown in Table 2. The upper boundof the TP eccentricity of 0.3 was selected as a reasonable up-per value for an orbit to remained confined to the HZ (Joneset al. 2005). While a planet may be considered habitable ateccentricities as high as . < e < . depending on the re-sponse time of the atmosphere-ocean system (Williams &Pollard 2002; Jones et al. 2005), we are interested in planetsand systems that more closely resemble the Earth and theSolar system.The simulation of each system was run for a total in-tegration time of T sim = years, or until all TPs wereremoved from the system. TPs ejected beyond an astrocen-tric distance of au are removed from the simulations.The time step of each simulation was calculated to be / of the smallest initial orbital period of the TPs, or the Jovianplanet if it was interior to the HZ. For each of the 182 systems simulated, the lifetime of eachTP and the resulting number of TPs that survived wasrecorded, as well as their initial semi-major axis and eccen-tricity. The resulting stability of individual systems, as wellas the entire population, can then be analysed.
Our 182 systems are broadly be divided into 6 dynamicalclassifications based on the apsides of the Jovian planet rel-ative to the boundaries of the HZ. These classes are:
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M. T. Agnew et al.
Normalised Orbital Period, ( T / T HZ , mid )05101520253035 S y s t e m C o un t Figure 1.
All 542 currently known single Jovian planet systems that have the required stellar and planetary parameters, and that satisfythe K (cid:54) T eff (cid:54) K criterion for the HZ calculation. The x -axis is the orbital period of each Jovian planet, normalised by the theperiod in the centre of the HZ. The red, hashed area represents the cut-off for systems that do not satisfy the . T HZ (cid:54) T Jovian (cid:54) T HZ criterion. Table 3.
The location of the Jovian planet’s periastron, r peri , andapastron, r ap , relative to the HZ for each dynamical classification.Class r < a HZ , in a HZ , in < r < a HZ , out a HZ , out < r I-a r peri , r ap - -I-b r peri r ap -II-a - - r peri , r ap II-b - r peri r ap III - r peri , r ap -IV r peri - r ap I-a
Interior,
I-b
Interior & touching,
II-a
Exterior,
II-b
Exterior & touching,
III
Embedded, and IV Traversing.The locations of the Jovian planet’s periastron, r peri , andapastron, r ap , relative the HZ are listed in Table 3. Theseare demonstrated in the schematic in Figure 2. The numberof single Jovian systems in each dynamical class is listed inTable 4.In Figure 3 we show the entire population of single Jo-vian planet systems simulated. Figure 3 very clearly demon-strates the importance of e pl on HZ stability where thosesystems with high eccentricities – represented by large errorbars – have far less stable HZs. Class I and II systems havelarge regions of stability within the HZ, and this decreaseswith the Jovian planet’s increasing eccentricity as the systembegins to approach becoming a class IV (traversing) system.We would intuitively expect class III and IV systems to be Table 4.
The number of single Jovian systems within each dy-namical class.Class Sub-count TotalI-a Interior 64 88I-b Interior & Touching 24II-a Exterior 34 60II-b Exterior & Touching 26III Embedded 11IV Traversing 23Total 182 unstable. However, we do see TPs surviving for the durationof the simulation in some cases, and thus we will investigatethese classes in greater detail.In Figures 4 and 5 we again show the entire popula-tion of single Jovian planet systems simulated, but this timeconsidering how the orbital eccentricities of the stable TPsevolve. Figure 4 demonstrates the level of eccentricity ex-citation of each TP, shown by the change in eccentricity( ∆ e ) of the simulated TPs, where a change of 1 is used toindicate complete removal of the TP from the system. Asthere are 5000 TPs per simulation, we bin the TPs in 182equally spaced bins, and take the mean eccentricity changeof all TPs within each bin ( ∆ e = / n (cid:205) ∆ e n ) so as to notlose information from excessive stacking of points. We plotthe systems in the same order as they appear in Figure 3to allow comparison. Unsurprisingly, TPs with higher ini-tial eccentricities are removed in systems where the Jovianplanet is located near the HZ, as the apsides of their orbitsmean they begin to experience close encounters with the Jo- MNRAS000
The number of single Jovian systems within each dy-namical class.Class Sub-count TotalI-a Interior 64 88I-b Interior & Touching 24II-a Exterior 34 60II-b Exterior & Touching 26III Embedded 11IV Traversing 23Total 182 unstable. However, we do see TPs surviving for the durationof the simulation in some cases, and thus we will investigatethese classes in greater detail.In Figures 4 and 5 we again show the entire popula-tion of single Jovian planet systems simulated, but this timeconsidering how the orbital eccentricities of the stable TPsevolve. Figure 4 demonstrates the level of eccentricity ex-citation of each TP, shown by the change in eccentricity( ∆ e ) of the simulated TPs, where a change of 1 is used toindicate complete removal of the TP from the system. Asthere are 5000 TPs per simulation, we bin the TPs in 182equally spaced bins, and take the mean eccentricity changeof all TPs within each bin ( ∆ e = / n (cid:205) ∆ e n ) so as to notlose information from excessive stacking of points. We plotthe systems in the same order as they appear in Figure 3to allow comparison. Unsurprisingly, TPs with higher ini-tial eccentricities are removed in systems where the Jovianplanet is located near the HZ, as the apsides of their orbitsmean they begin to experience close encounters with the Jo- MNRAS000 , 1–13 (2017) roperties of the single Jovian planet population (a) I-a Interior (b) II-a Exterior (c) III Embedded(d) I-b Interior & touching (e) II-b Exterior & touching (f) IV Traversing Figure 2.
The six dynamical classes of single Jovian planet systems: (a) I-a Interior, (b) II-a Exterior, (c) III Embedded, (d) I-b Interiorand touching, (e) II-b Exterior and touching, and (f) IV Traversing. The green annulus represents the HZ of the system, while the blueellipse represents the orbit of the Jovian planet. The black cross represents the star. vian planet while lower eccentricity TPs may not. In the 1Dhistogram in lower panel of Figure 4, we combine the initialeccentricity data of the survivors across all 182 simulatedsystems. This more clearly highlights that, over the entirepopulation, lower initial eccentricity TPs are more likely tosurvive.Figure 5 shows the final eccentricities of the survivingTPs against their initial eccentricity for all systems. We seethat the majority of the final eccentricities of surviving TPsare low ( e < . ), suggesting little eccentricity excitation.However, there are clear examples of higher eccentricity TPssurviving, shown by the green and yellow points scatteredacross the plot. We combine the final eccentricity data of allthe survivors across the 182 systems in the 1D histogram inthe lower panel. This clearly shows that TPs with final ec-centricities less than 0.3 ( . ) dominate the surviving TPpopulation. Combined with the level of excitation shown inthe top panel, this demonstrates that surviving TPs experi-ence low levels of excitation, and that when TPs are excitedthey tend to be removed entirely.In the search for an Earth-like planet, we focus specifi-cally on TPs that have a final eccentricity of less than 0.3,as this is a value that generally leads to a HZ confined orbit(Jones et al. 2005). However, studies suggest that a planetmay still receive sufficient luminosity to be considered habit- able with eccentricities as high as 0.7, depending on a rangeof planet properties (Williams & Pollard 2002; Jones et al.2005). Taking the cut off of e < . indicates that . of the surviving TPs would be considered potentially habit-able, based solely on this criterion, while the more optimisticcut of e < . takes that total up to . . A Jovian planet whose orbit is embedded within the HZwould seem to suggest a completely unstable HZ. It is nat-ural to suspect that, in such cases, the planet would not beable to coexist with a M ⊕ planet within the HZ. How-ever, our simulations reveal several systems which demon-strate stability via mean-motion resonances (MMRs) witha HZ-embedded Jovian planet, including those in the formof planets trapped at the stable L4 and L5 Lagrange points,commonly referred to as Trojans. An example of such a classIII system is HD 19994 shown in Figure 6. Figure 6a showsthe Cartesian view of the system at the conclusion of thesimulation, clearly highlighting the stable Trojan compan-ions that survive on the same orbit as the Jovian planet. Wealso see TPs that survive in the 2:3 and low eccentricity 3:5MMRs. Figure 6b shows the position of every particle testedon a semi-major axis versus eccentricity ( a − e ) parameter MNRAS , 1–13 (2017)
M. T. Agnew et al. ( a / a HZ , mid ) S y s t e m Habitable ZoneSurviving TPsClass I - InteriorClass II - ExteriorClass III - EmbeddedClass IV - Traversing
Figure 3.
All 182 systems simulated plotted on a normalised semi-major axis ( a / a HZ , mid ) x -axis. The coloured points represent theJovian planet and error bars the apsides of its orbit. The green region represents the HZ, and the black points are those TPs of the initial5000 that are still surviving at the end of the simulation. space, where the colour corresponds to the survival time ofthe particle on a logarithmic scale. This more clearly demon-strates the influence of stabilising MMRs (overlaid in green).The 1:1 and 2:3 MMRs offer particularly strong protectionfor TPs, but the influence of both the 3:4 and 3:5 MMRscan also be seen to result in a number of stable outcomes.It should be kept in mind that these simulations usedmassless TPs, and so the mutual gravitational interactionsbetween a possible M ⊕ planet and the Jovian planet have not been taken into account. However, based on the findingsof Agnew et al. (2017), it is often the case that M ⊕ planetsare also able to survive in such simulations. Furthermore,the L4 and L5 Lagrangian points are stable for cases wherethe mass ratio of the Jovian planet to its host star is µ < / which holds true for all the star–Jovian planet systemsconsidered in this work (Murray & Dermott 1999).An inherently challenging issue in the detection of plan-ets that share an orbit with a Jupiter mass planet is that MNRAS000
All 182 systems simulated plotted on a normalised semi-major axis ( a / a HZ , mid ) x -axis. The coloured points represent theJovian planet and error bars the apsides of its orbit. The green region represents the HZ, and the black points are those TPs of the initial5000 that are still surviving at the end of the simulation. space, where the colour corresponds to the survival time ofthe particle on a logarithmic scale. This more clearly demon-strates the influence of stabilising MMRs (overlaid in green).The 1:1 and 2:3 MMRs offer particularly strong protectionfor TPs, but the influence of both the 3:4 and 3:5 MMRscan also be seen to result in a number of stable outcomes.It should be kept in mind that these simulations usedmassless TPs, and so the mutual gravitational interactionsbetween a possible M ⊕ planet and the Jovian planet have not been taken into account. However, based on the findingsof Agnew et al. (2017), it is often the case that M ⊕ planetsare also able to survive in such simulations. Furthermore,the L4 and L5 Lagrangian points are stable for cases wherethe mass ratio of the Jovian planet to its host star is µ < / which holds true for all the star–Jovian planet systemsconsidered in this work (Murray & Dermott 1999).An inherently challenging issue in the detection of plan-ets that share an orbit with a Jupiter mass planet is that MNRAS000 , 1–13 (2017) roperties of the single Jovian planet population . . . . . . . . . . . e , | e final- e initial | .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . Initial Eccentricity, e S y s t e m .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . Initial Eccentricity, e . . . . . . P e rc e n t a g eo f t o t a l s u r v i v o r s , [ % ] Figure 4.
The change in TP eccentricities for all 182 systemssimulated. We plot the initial TP eccentricity against the sys-tems ordered along the y -axis as in Figure 3. The colour repre-sents the mean change in eccentricity of all TPs within each bin( ∆ e = / n (cid:205) ∆ e n ). The lower panel shows a 1D histogram of thepercentage of all surviving TPs that particular initial eccentricityvalues make up. . . . . . . . . . . . Final Eccentricity, e .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . Initial Eccentricity, e S y s t e m . . . . . . Final Eccentricity, e P e rc e n t a g eo f t o t a l s u r v i v o r s , [ % ] Figure 5.
Similar to Figure 4, but colour now represents finaleccentricities of the TP in all 182 systems. The lower panel showsthe 1D histogram of the percentage of all surviving TPs that havea particular final eccentricity. The various cuts for habitabilityfor an e < . and e < . are overlaid in green and orangerespectively, while the percentage of surviving TPs that fall withinthese cuts are listed above.MNRAS , 1–13 (2017) M. T. Agnew et al. − − − x [au] − − − y [ a u ] HD 19994 (a) Cartesian Plot . . . . . . . . Semi-major axis, a [au] . . . . . . . E cc e n t r i c i t y , e : : : : : : L i f e t i m e , l o g h t y r i (b) a − e Map
Figure 6.
The HD 19994 system demonstrating the MMRs andL4 and L5 Lagrange points providing a means to stabilise an oth-erwise inherently unstable HZ. The Cartesian plot is a snapshotat the end of the simulation, with the existing Jovian planet’sorbit shown in blue. they represent a degenerate scenario for the radio velocitysignal, and so would be indiscernible from the signal of asingle Jupiter mass planet. This degeneracy would be bro-ken if one (or both) planets transit or via differences in theplanets long-term librations.
Due to the higher eccentricities, a planetary architecturewith a more unstable HZ is likely that of a HZ-traversingJovian planet. As was the case with the example shown insection 3.1.1, MMRs can again provide stability in such ascenario. Figure 7 demonstrates one such system, HD 43197.We can see in Figure 7a that there exist stable TPs at theconclusion of the simulation on orbits that straddle the HZ.Even though some of these particles may move on orbits thatexit the HZ, it may still be possible for such high eccentric-ity planets ( e < . ) to remain habitable depending on theresponse rate of the atmosphere-ocean system (Williams & − . − . − . − . . . . . . x [au] − . − . − . − . . . . . . . y [ a u ] HD 43197 (a) Cartesian . . . . . . . Semi-major axis, a [au] . . . . . . . E cc e n t r i c i t y , e : : : : : : L i f e t i m e , l o g h t y r i (b) a − e Map
Figure 7.
The HD 43197 system demonstrating the 1:2 MMRproviding a means to stabilise an otherwise inherently unstableHZ. The Cartesian plot is a snapshot at the end of the simulation,with the existing Jovian planet’s orbit shown in blue.
Pollard 2002; Jones et al. 2005). Figure 7b demonstrates thatthe source of stabilisation in this scenario is the 1:2 MMRwith the existing Jovian planet.While it is interesting to demonstrate that a HZ-traversing Jovian planet can coexist with bodies in the HZ ofthe system, it should be noted that a high eccentricity Jovianplanet would most likely be the result of gravitational inter-actions with other massive bodies during the planetary sys-tem’s evolution, and so it seems highly unlikely that a rockyplanet could remain in the HZ after such dynamical interac-tions (Carrera et al. 2016; Matsumura et al. 2016). In addi-tion, as a result of eccentricity harmonics and aliasing, a frac-tion of published eccentric single planets are actually multi-ple systems (Anglada-Escud´e et al. 2010; Anglada-Escud´e &Dawson 2010; Wittenmyer et al. 2013a) and so such systemsmust be further examined to confirm they are high eccentric-ity single systems. The result nevertheless demonstrates thatseemingly destructive systems are certainly capable of har-bouring other bodies on stable orbits within the HZ throughthe influence of stabilising resonances.
MNRAS000
MNRAS000 , 1–13 (2017) roperties of the single Jovian planet population We also search for correlations between the stability of theHZ and the observable system parameters. Since it is gravi-tational interactions that will determine the stability of theHZ, one might expect M pl , e pl , and a pl to have an effect.Other observables include T eff and stellar metallicity, forwhich we would not expect any correlation, though T eff iscorrelated with M (cid:63) , which likely affects the mass ratio.We examine all the systems simulated and plot theirsemi-major axis, eccentricity, mass ratio, metallicity and ef-fective temperature against one another in Figure A1, withthe colour of each point indicating the number of survivingTPs. Other than some very slight clustering with respectto mass ratio, e pl , and a pl , no clear trends are revealed byour analysis. The clustering, however, does emphasise theexpected dependence that mass ratio, semi-major axis andeccentricity have on the stability of the HZ. As such, we usethe semi-analytic criterion from Giuppone et al. (2013) inorder to introduce a parameter that incorporates these pa-rameters. The equation they use for the reach of the chaoticregion around a planet is δ = C µ / a pl , (3)where C was calculated to be a constant equal to . (Dun-can et al. 1989; Giuppone et al. 2013), µ = M pl / M (cid:63) is themass ratio between the planet and its parent star, and a pl is the semi-major axis of the planet. While the equation wasoriginally formulated by Wisdom (1980) for circular orbits,Giuppone et al. (2013) mention that it offers an approxima-tion for eccentric orbits (as an eccentric orbit will precessand sweep out the entire annulus bound by the apsides ofthe orbit). As such, we can calculate the chaotic region as a pl ( − e ) − δ (cid:54) Chaotic Region (cid:54) a pl ( + e ) + δ, (4)where e is the planet’s eccentricity, and δ is defined as inequation 3. Hence, the width of the chaotic region can becalculated by a chaos = (cid:0) a apastron + δ (cid:1) − (cid:0) a periastron − δ (cid:1) = ( a ( + e ) + δ ) − ( a ( − e ) − δ ) and substituting δ from equation 3 yields a chaos = (cid:16) a ( + e ) + C µ / a (cid:17) − (cid:16) a ( − e ) − C µ / a (cid:17) , a chaos a = (cid:16) e + C µ / (cid:17) . (5)This value, a chaos / a , we refer to as the chaos value for thefollowing plots and discussion.In Figure 8 we plot all the systems by normalised semi-major axis ( a / a HZ , mid ) against the chaos value. It is immedi-ately apparent that there is a very obvious “desert” where noTPs survive in the systems considered. As our simulationsuse massless TPs, this plot cannot be used to predict thestability of massive bodies, such as a M ⊕ planet. However,this plot can be used to exclude systems from further ob-servational searches for planets in the HZ, since if TPs aredynamically unstable, then, in general, one would expectplanets to also be dynamically unstable. However, there willbe exceptions to this rule. Whilst adding an additional mas-sive body (or, indeed, changing the mass of the giant planetin the system) will not affect the location of mean-motion − Normalised Semi-major axis, ( a/a HZ , mid ) . . . . . . . C h a o s . . . . . . . . . . . N u m b e r o f S u r v i v o r s , % (a) Individual Systems(b) Classification Regions Figure 8.
The stability of all 182 single Jovian planet systemssimulated in this study. The existing Jovian planet is plotted on anormalised semi-major axis ( a / a HZ , mid ) against chaos value (equa-tion 5) parameter space. Fig 8a shows each individual system,where the colour of each point represents the percentage of sur-viving TPs at the end of the simulation. Contours have beeninterpolated and underlaid based on the 182 data points. Fig 8bdivides the parameter space into different dynamical regions. resonances, the secular dynamics of the system will be im-pacted by such changes (Barnes & Raymond 2004; Raymond& Barnes 2005; Horner & Jones 2008). In rare cases, thismight lead to an otherwise unstable orbit being stabilised.Newly discovered systems can be plotted on this map topredict whether it is worthwhile undertaking follow up ob-servations, or exhaustive numerical simulations, to furtherinvestigate whether a M ⊕ planet could be hidden withinthe HZ on a stable orbit. As the number of planetary sys-tems discovered continues to grow, having a quick methodby which systems with unstable HZs can be removed fromfurther studies will be beneficial. From our investigation of the entire single Jovian popula-tion, we are able to provide a candidate list of potential
Solar system analogues for future planet hunters. These are
MNRAS , 1–13 (2017) M. T. Agnew et al. shown with respect to the single Jovian planet populationin Figure 8b, and schematically in Figure 9. There are sev-eral ways in which a Solar system analogue can be defined. (cid:15)
Eridani (Sch¨utz et al. 2004; Backman et al. 2009; Greaveset al. 2014; Lestrade & Thilliez 2015; MacGregor et al. 2015;Su et al. 2017) and HR 8799 (Marois et al. 2008; Rhee et al.2007; Su et al. 2009; Marois et al. 2010; Matthews et al.2014; Contro et al. 2016) are two such examples, both hav-ing multiple asteroid belts and hosting (or are proposed tohost) several giant planets, just as we find in the Solar sys-tem. However, none have thus far been found to host rockyplanets in their HZs. Another example is the recent discov-ery of the eighth planet in the Kepler-90 system (Shallue& Vanderburg 2017), just as we find 8 planets in our ownsystem. However, these planets are all on far smaller, tighterorbits than planets in our Solar system. For this work, weuse the term
Solar system analogue to encompass those sys-tems that have a rocky Earth-like planet in the HZ, and aJovian planet beyond the HZ with orbital periods similar tothe Earth and Jupiter respectively . Given that we use mass-less TPs in our simulations, we cannot constrain the stablesemi-major axes of the HZ, as gravitational interactions be-tween any putative exo-Earth and the Jovian planet are nottaken into account. Thus we seek only those systems witha Jovian planet beyond the HZ that also have an entirely,or predominantly, unperturbed HZ. Figure 9 shows the 20systems found in this study that fit this definition, with ourSolar system (Earth and Jupiter only) for comparison.Following Agnew et al. (2017), we can compute the mag-nitude of the Doppler wobble that a M ⊕ planet wouldinduce on its host star using the equation K = (cid:18) π GT ⊕ (cid:19) M ⊕ sin I ( M (cid:63) + M ⊕ ) (cid:113) − e ⊕ , (6)where G is the gravitational constant, M (cid:63) is the mass of thehost star, I is the inclination of the planet’s orbit to ourline of sight, and T ⊕ , e ⊕ and M ⊕ are the period, mass andeccentricity of the M ⊕ planet respectively.We next ask: what is the minimum radial velocity reso-lution required to detect an exo-Earth in the HZ if it exists? K is larger for smaller orbital periods, so we calculate K for the outer boundary of the HZ, which can be consideredthe “conservative view”, i.e. the weakest Doppler wobble arocky body would induce on its host star. Thus we calcu-late the semi-amplitude of the Doppler shift produced atthe outer edge of the HZ to provide the minimum radial ve-locity resolution required to detect the exo-Earth if it exists .These values are presented in Table 5. We also provide theresolution required for larger M ⊕ and M ⊕ planets, butacknowledge that the boundaries of the HZ will vary slightlyfor a larger planet, as noted by Kopparapu et al. (2014).We also consider the dynamical evolution of the system,and what impact that may have had on the HZ. We usethe definition of resilient habitability introduced by Carreraet al. (2016), which defines the ability of a planet to avoidbeing removed from a system (by collision or ejection) andremain within the HZ. Carrera et al. (2016) considered thedynamical interactions an existing Jovian planet would havewith objects in the HZ during its evolution to the orbitalparameters seen today. If a planet has resilient habitability,the HZ was not completely disturbed during the dynami- Semi-major axis, (au)Solar SystemHD 10442HD 30669HD 95872HD 164922HD 290327HD 30177HD 70642HD 32963HD 114613HD 27631HD 6718HD 24040HD 86226HD 216435HD 220689psi Dra BHD 13931HD 25171HD 72659HD 222155 S y s t e m Figure 9.
The 20 single Jovian planet systems that have beenloosely classified as potential
Solar system analogues . This cor-responds to a system that has a Jovian planet orbiting beyondthe HZ, and a completely stable HZ within which a M ⊕ planetcould be hidden. The error bars represent the apsides of the Jo-vian planet’s orbit. The size of the point is proportional to m / . cal evolution of the system. Carrera et al. (2016) simulatea large suite of systems, and by scaling the results of thesesimulations to various different semi-major axes, create an a − e map that can be used to infer the probability that aplanet has resilient habitability, given the a and e values ofan existing Jovian planet. We plot the Jovian planet fromeach of our Solar system analogues on the resilient habit-ability plots presented by Carrera et al. (2016), and providethe probability bin each system falls into in Table 5. Thisprovides another parameter by which to prioritise systemsfor observational follow-up to hunt for potentially habitableexo-Earths. Of the candidates we put forward, we are par-ticularly interested in those that have a greater than probability of having resilient habitability.We find that four systems have a resilient habitabil-ity probability of greater than : HD 222155, HD 24040,HD 95872 and HD 13931, which has an almost prob-ability. We suggest these should be the priority candidatesfor follow-up observation with ESPRESSO as they have notonly dynamically stable HZs, but also have a greater than probability that the dynamical evolution of the Jovian MNRAS000
Solar system analogues . This cor-responds to a system that has a Jovian planet orbiting beyondthe HZ, and a completely stable HZ within which a M ⊕ planetcould be hidden. The error bars represent the apsides of the Jo-vian planet’s orbit. The size of the point is proportional to m / . cal evolution of the system. Carrera et al. (2016) simulatea large suite of systems, and by scaling the results of thesesimulations to various different semi-major axes, create an a − e map that can be used to infer the probability that aplanet has resilient habitability, given the a and e values ofan existing Jovian planet. We plot the Jovian planet fromeach of our Solar system analogues on the resilient habit-ability plots presented by Carrera et al. (2016), and providethe probability bin each system falls into in Table 5. Thisprovides another parameter by which to prioritise systemsfor observational follow-up to hunt for potentially habitableexo-Earths. Of the candidates we put forward, we are par-ticularly interested in those that have a greater than probability of having resilient habitability.We find that four systems have a resilient habitabil-ity probability of greater than : HD 222155, HD 24040,HD 95872 and HD 13931, which has an almost prob-ability. We suggest these should be the priority candidatesfor follow-up observation with ESPRESSO as they have notonly dynamically stable HZs, but also have a greater than probability that the dynamical evolution of the Jovian MNRAS000 , 1–13 (2017) roperties of the single Jovian planet population Table 5.
The minimum required radial velocity sensitivities re-quired to detect a M ⊕ , M ⊕ or M ⊕ planet in the HZ. We alsoinclude the probability that each system has resilient habitabilitydescribed by Carrera et al. (2016). K min (m s − ) Probability ofSystem M ⊕ M ⊕ M ⊕ Resilient HabitabilityHD 222155 0.0504 0.1008 0.2016 − HD 72659 0.0565 0.1129 0.2259 − HD 25171 0.0581 0.1162 0.2323 − HD 13931 0.0613 0.1226 0.2452 − psi Dra B 0.0600 0.1200 0.2400 − HD 220689 0.0643 0.1286 0.2573 − HD 216435 0.0589 0.1179 0.2357 − HD 86226 0.0654 0.1308 0.2616 − HD 24040 0.0630 0.1260 0.2520 − HD 6718 0.0701 0.1402 0.2805 − HD 27631 0.0717 0.1433 0.2866 − HD 114613 0.0596 0.1193 0.2385 − HD 32963 0.0717 0.1435 0.2870 − HD 70642 0.0681 0.1361 0.2723 − HD 30177 0.0705 0.1410 0.2820 − HD 290327 0.0751 0.1501 0.3002 − HD 164922 0.0752 0.1503 0.3006 − HD 95872 0.0755 0.1511 0.3022 − HD 30669 0.0795 0.1590 0.3179 − HD 10442 0.0615 0.1229 0.2458 − planet did not completely destabilise the HZ. The candidatelist presented in Table 5 represents those systems we testednumerically to possess dynamically stable HZs. There arealso those systems that we tagged as having a stable HZ (asdiscussed in section 2.2), 5 of which have a Jovian planetexterior to the HZ that may also be considered Solar sys-tem analogues , however, we present only those systems forwhich numerical simulations were carried out. We also do notpresent those systems labelled as “Stable Interior Jupiters”in Figure 8b. These systems are shown to have very sta-ble HZs. However, as they are interior to the HZ it raisesanother question regarding Jovian planet formation and mi-gration scenarios, and what effect these may have on amountof material remaining for terrestrial planet formation (Ar-mitage 2003; Mandell & Sigurdsson 2003; Fogg & Nelson2005; Mandell et al. 2007).As discussed in Section 2.3, our investigation and sub-sequent results are for perfectly co-planar systems. Mutuallyinclined planets may exchange angular momentum, resultingin the excitation of the orbital inclination and/or eccentric-ity of an otherwise potentially habitable world, which wouldclearly affect its HZ stability. As our Solar system analoguesare defined such that the Jovian planet does not gravita-tionally perturb the HZ, it may also be the case that theJovian planet is at a sufficient distance from the HZ that itwould not strongly disturb the HZ at shallow mutual incli-nations either. Regardless, we recommend further analysisof these systems of interest to more robustly prioritise themfor observational follow-up.
We simulated the dynamical stability of the entire knownsingle Jovian population for which stellar and planetary properties are available, that satisfy the criterion K (cid:54) T eff (cid:54) K, and for which the Jovians are located within . T HZ (cid:54) T Jovian (cid:54) T HZ . We then investigated both thedynamical properties of individual systems as well as of theentire population with the aim of providing a guide for whereto focus resources in the search for Earth-like planets andSolar system analogues.We divide the 182 single Jovian systems into dynamicalclasses based on the apsides of the Jovian planet relative tothe boundaries of the HZ: systems for which the Jovian isinterior to the HZ, exterior to the HZ, embedded in the HZ,and traverses the HZ. Perhaps somewhat surprisingly, wefind that there are regions of stability in the HZ even whenthe Jovian is embedded in or traverses the HZ. For such dy-namical classes, we find that stabilising MMRs are capableof providing regions of stability, including the L4 and L5Lagrange points in the case of HZ-embedded Jovians. Whilethese stable regions have been demonstrated with masslesstest particles in this study, Agnew et al. (2017) has shownthat this is usually a strong indication that the region is alsostable for a M ⊕ . In the case of the L4 and L5 Lagrangianpoints, the regions are stable as long as the mass ratio ofthe Jovian planet is µ < / . It should be noted that theseresults are based on the systems as we see them today and ithas been shown that, particularly with the high eccentricity,HZ-traversing Jovian planets, the dynamical evolution mayhave already destroyed or ejected any bodies from the HZ(Carrera et al. 2016; Matsumura et al. 2016).Examining the entire population, as expected we findthat the main indicators of HZ stability are semi-major axis,mass ratio and eccentricity of the Jovian planet. Other ob-servable quantities such as T eff and stellar metallicity showno overarching trends within the single Jovian planet pop-ulation. However, by using the semi-analytic criterion ofGiuppone et al. (2013), we find a “desert” in the chaos–normalised semi-major axis parameter space which we canuse to exclude single Jovian planet systems that are unableto coexist with other bodies in their HZ. This map of the chaos value is useful for rapidly excluding newly discoveredsystems from further observational or numerical follow-upin the search for habitable exoplanets and Solar system ana-logues.Systems with a completely stable HZ suggest that theJovian planet’s gravitational influence is not strong enoughto interact with bodies within the HZ. In this work, we cansimply define Solar system analogues as systems with an ex-terior Jovian planet and an entirely, or predominantly, un-perturbed HZ. We find that there are 20 systems which wecan therefore define as Solar system analogues, with a Jo-vian planet exterior of the HZ, and for which the HZ is leftcompletely, or nearly completely, unperturbed by its grav-itational influence. These systems are ideal candidates tosearch for an Earth-like planet in the HZ of the system, andare shown in Figure 9 and Table 5. We also present the prob-ability that each of these systems has resilient habitabilityas outlined by Carrera et al. (2016). Specifically, we find thatHD 222155, HD 24040, HD 95872 and HD 13931 all have agreater than probability (and in fact HD 13931 has analmost probability) that a terrestrial planet could sur-vive in the HZ during the dynamical evolution of the existingJovian. As a result, we suggest that these systems should bea priority for observational follow-up with ESPRESSO.
MNRAS , 1–13 (2017) M. T. Agnew et al.
While we were unable to find overarching propertiesamongst the single Jovian planet population that can beused to indicate HZ stability, we were able to generate amap over a parameter space which shows a clear clusteringof systems with unstable HZs. This can be used to excludenewly discovered systems from an exhaustive suite of simu-lations or observational follow up in the search for habitableexoplanets. Furthermore, we have been able to provide acandidate list of potential Solar system analogues for useby planet hunters, which should assist in focusing resourcesin the search for habitable Earth-like planets amongst theever-growing exoplanet population.
ACKNOWLEDGEMENTS
We wish to thank the anonymous referee for helpful com-ments and suggestions that have improved the paper.MTA was supported by an Australian Postgraduate Award(APA). This work was performed on the gSTAR nationalfacility at Swinburne University of Technology. gSTAR isfunded by Swinburne and the Australian Governmentˆa ˘A´ZsEducation Investment Fund. This research has made use ofthe Exoplanet Orbit Database, the Exoplanet Data Explorerat exoplanets.org and the NASA Exoplanet Archive, whichis operated by the California Institute of Technology, undercontract with the National Aeronautics and Space Admin-istration under the Exoplanet Exploration Program.
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APPENDIX A: SINGLE JOVIAN PLANETSYSTEM TRENDS
This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS , 1–13 (2017) M. T. Agnew et al.
Semi-major Axis . . . . . . E cc e n t r i c i t y Semi-major Axis − − − − M a ss R a t i o Eccentricity M a ss R a t i o Semi-major Axis − . − . − . − . − . . . . . . M e t a lli c i t y Eccentricity M e t a lli c i t y Mass Ratio M e t a lli c i t y Semi-major Axis E ff e c t i v e T e m p e r a t u r e . . . . . . Eccentricity E ff e c t i v e T e m p e r a t u r e − − − − Mass Ratio E ff e c t i v e T e m p e r a t u r e − . − . − . − . − . . . . . . Metallicity E ff e c t i v e T e m p e r a t u r e . . . . . . . . . . . Number of Survivors, % Figure A1.
Plots of all simulated single Jovian planet systems comparing the semi-major axes, eccentricities, mass ratios, metallicitiesand effective temperatures against one another. The colours indicate the number of survivors. As can be seen, other than some minorclustering with respect to semi-major axis, eccentricity and mass ratio, there is very little trending. MNRAS000