Eccentricity generation in hierarchical triple systems: the planetary regime
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Eccentricity generation in hierarchical triple systems: theplanetary regime
Nikolaos Georgakarakos
100 Delphon Str., Thessaloniki 546 43, Greeceemail: [email protected]
ABSTRACT
In previous papers, we developed a technique for estimating the inner eccentricityin hierarchical triple systems, with the inner orbit being initially circular. We consid-ered systems with well separated components and different initial setups (e.g. coplanarand non-coplanar orbits). However, the systems we examined had comparable masses.In the present paper, the validity of some of the formulae derived previously is testedby numerically integrating the full equations of motion for systems with smaller massratios (from 10 − to 10 , i.e. systems with Jupiter-sized bodies). There is also dis-cussion about HD217107 and its planetary companions. Key words:
Celestial mechanics, planetary systems, binaries:general.
A hierarchical triple system consists of a binary system and a third body on a wider orbit. The motion of such a systemcan be pictured as the motion of two binaries on slowly perturbed Keplerian orbits: the binary itself (inner binary) and thebinary which consists of the third body and the centre of mass of the binary (outer binary). Hierarchical triple systems arewidely present in the galactic field and in star clusters and studying the dynamical evolution of such systems is a key tounderstanding a number of issues in astronomy and astrophysics, such as tidal friction and dissipation, mass transfer andmass loss due to a stellar wind, which may result in changes in stellar structure and evolution (for systems with close innerbinaries, where the separation between the components is comparable to the radii of the bodies). But even in systems withwell-separated inner binary components, the perturbation of the third body can have a devastating effect on the triple systemas a whole (e.g. disruption of the system).For most hierarchical triple stars, the period ratio X is of the order of 100 and these systems are probably very stabledynamically. However, there are systems with much smaller period ratios, like the system HD 109648 with X = 22 (Jha et al.2000), the λ Tau system, with X = 8 . X = 7 . At this point, we would like to remind the reader of the formulae that were derived in HTS1 and HTS2. The formula forthe circular outer binary case is (with the addition of two more short period terms, i.e. the next order terms in P and P Legendre polynomials; those terms which were included in HTS2 for greater accuracy and were denoted as P and P , canbe obtained by setting e = 0 in equations (9), (11), (13) and (15) of HTS2): e = m M X (cid:18)
434 + 225128 m ∗ X + 3659 1 X + 83612048 m ∗ X + 1223 1 X (cid:19) + 158 m M m ∗ X CA − B + 2 (cid:16) CA − B (cid:17) (1)with c (cid:13) Nikolaos Georgakarakos A = β, B = 1 + 758 γ, C = 54 αe T , e T = 34 m m ( m + m ) M X . The formula for the eccentric outer binary case is (slightly different from the one in HTS2, as some coefficients have beencorrected, see Georgakarakos 2005): e = m M X (1 − e ) (cid:20)
438 + 1298 e + 12964 e + 1(1 − e ) ( 438 + 64516 e + 193564 e + 215128 e ) + 1 X (1 − e ) [ 36518 ++ 44327144 e + 119435192 e + 2561051152 e + 683359216 e + 1(1 − e ) ( 36518 + 768316 e + 2823116 e + 295715192 e + 24158 e ++ 129012048 e )] + 1 X (1 − e ) [ 613 + 3052 e + 9158 e + 30548 e + 1(1 − e ) ( 613 + 8543 e + 21354 e + 213512 e ++ 2135384 e )] + m ∗ X (1 − e )[ 225256 + 33751024 e + 76252048 e + 292258192 e + 4842516384 e + 8252048 e + 1(1 − e ) ( 225256 ++ 29251024 e + 775256 e + 22258192 e + 25512 e )] + m ∗ X (1 − e ) [ 83614096 + 1254158192 e + 37624532768 e + 4180565536 e ++ 1(1 − e ) ( 83614096 + 585272048 e + 87790516384 e + 29263516384 e + 292635524288 e )] (cid:21) + 2( CB − A ) . (2)with A = β (1 − e ) , B = 1(1 − e ) + 258 γ e (1 − e ) and C = 54 α e (1 − e ) . In both cases: α = m − m m + m a a , β = m m M m ( m + m ) ( a a ) , γ = m M ( m + m ) ( a a ) ,e is the outer eccentricity and a and a are the inner and outer semi major axes respectively. Also, M is the total massof the system, m ∗ = m − m ( m + m ) M and X is the period ratio of the two orbits. In order to test the validity of the formulae derived in the previous papers, we integrated the full equations of motionnumerically, using a symplectic integrator with time transformation (Mikkola 1997).The code calculates the relative position and velocity vectors of the two binaries at every time step. Then, by usingstandard two body formulae, we computed the orbital elements of the two binaries. The various parameters used by the code,were given the following values: writing index
Iwr = 1, method coefficients a a icor = 1.The average number of steps per inner binary period NS , was given the value of 60 for testing the short period terms and thelong term circular case. For the long period eccentric cases, we set NS = 5 in order to accelerate the execution of the code,without any precision cost, as the motion was mainly dominated by secular evolution.For our simulations, we also defined the two mass ratios K m m + m and K m m + m , with 0 . ≤ K ≤ . . ≤ K ≤ X f , defined as the ratio ofthe period that the outer binary would have on a circular orbit with a semi major axis equal to its periastron distance over theperiod of the inner binary. In all cases X f ≥
10. We also used units such that G = 1 and m + m = 1 and we always startedthe integrations with a = 1. In that system of units, the initial conditions for the numerical integrations were as follows: r = 1 , r = 0 , r = 0 R = a cos φ, R = a sin φ, R = 0˙ r = 0 , ˙ r = 1 , ˙ r = 0˙ R = − r Ma sin φ, ˙ R = r Ma cos φ, ˙ R = 0 , for the circular case ( φ is the initial relative phase of the two binaries) and c (cid:13) , 000–000 ccentricity generation in HTS Figure 1. K K e = 0 and for which A − B = 0. The right graph is the same as the left one, but for K ≤ . r = 1 , r = 0 , r = 0 R = R cos ( f + ̟ ) , R = R sin ( f + ̟ ) , R = 0˙ r = 0 , ˙ r = 1 , ˙ r = 0˙ R = − r Ma (1 − e ) sin ( f + ̟ ) , ˙ R = r Ma (1 − e ) cos ( f + ̟ ) , ˙ R = 0 . for the non circular case, where f and ̟ are the initial true anomaly and longitude of pericentre respectively of theouter orbit. First, we present the results from testing equations (1) and (2) for long term behaviour. The formulae were compared withresults obtained from integrating the full equations of motion numerically.For the circular case, each system was numerically integrated for φ = 0 ◦ − ◦ with a step of 45 ◦ . After each run, e was averaged over time using the trapezium rule and after the integrations for all φ were done, we averaged over φ by usingthe rectangle rule. The integrations were also done for smaller steps in φ (e.g. 1 ◦ ), but was not any difference in the outcome.For the non circular case, each system was numerically integrated for ̟ = 0 ◦ − ◦ and f = 0 ◦ − ◦ with a step of60 ◦ . For a given value of ̟ and f we integrated our system. After each run, e was averaged over time using the trapeziumrule and then we integrated the system for a different f . After the integrations for all f were done, we averaged over f byusing the rectangle rule. Then, the same procedure was applied for the next value of ̟ and when the integrations for all ̟ were done, we averaged over ̟ by using the rectangle rule. The integrations were also done for smaller steps in ̟ and f ,but the difference in the outcome was insignificant.These results are presented in Tables 1, 2 and 3 ( e = 0 , . , .
75 respectively), which give the percentage error between theaveraged numerical e and equations (1) and (2). The error is accompanied by the period of the oscillation of the eccentricity,which is the same as the integration time span (when there is no period given, we integrated for an outer orbital period, i.e2 πX , as there was not any noticeable secular evolution). There are four values per ( K − K
2) pair, corresponding, from topto bottom, to X = 10 , , ,
50 respectively (of course, for the eccentric outer binary case, X is replaced by X f ).Generally, it appears that the theory is in agreement with the numerical integrations. There are some cases (larger K e , which meansthat it is about double than the one in e ), but it dropped as the period ratio increased. There are also some cases wherethe formulae do not seem to work well; in fact, for some systems (e.g. e = 0 . K . K . A − B gets small, i.e. our approximate secularsolution is near resonance. This is demonstrated in figures 1,2 and 3, which are plots of K K A − B = 0.Finally, for large eccentricities, there is a significant error for K . X f = 10, which is due to terms omitted from theapproximate secular solution. Those terms become insignificant as X f increases. c (cid:13)000
50 respectively (of course, for the eccentric outer binary case, X is replaced by X f ).Generally, it appears that the theory is in agreement with the numerical integrations. There are some cases (larger K e , which meansthat it is about double than the one in e ), but it dropped as the period ratio increased. There are also some cases wherethe formulae do not seem to work well; in fact, for some systems (e.g. e = 0 . K . K . A − B gets small, i.e. our approximate secularsolution is near resonance. This is demonstrated in figures 1,2 and 3, which are plots of K K A − B = 0.Finally, for large eccentricities, there is a significant error for K . X f = 10, which is due to terms omitted from theapproximate secular solution. Those terms become insignificant as X f increases. c (cid:13)000 , 000–000 Nikolaos Georgakarakos
Table 1.
Percentage error between the averaged numerical e and equation (1). For all systems, e = 0. K \ K .
001 0 .
005 0 .
01 0 .
05 0 . . . ) -3 (4 . ) 6.9 (210 ) 9.4 (3 . ) 9.7 (1 . ) 12.2-1.5 (510 ) -15.7 (3 . ) 2.3 (1 . ) 3 (1 . ) 3.1 (10 ) 3.70.4 (1 . ) -16.8 (1 . ) 2.4 (310 ) 1.7 (510 ) 1.7 (2 . ) 1.9-1.2 (310 ) -29.4 (510 ) -0.4 (1210 ) 0.9 (1 . ) 0.9 (8 . ) 0.80.01 10.2 8.3 (10 ) -4 (1 . ) -1.7 (5 . ) 8.3(2 . ) 12.22.1 3.9 (3 . ) -0.9 (510 ) -15.9 (410 ) 2.8 (1 . ) 3.80.3 2.7 (810 ) -0.5 (1 . ) -17 (1 . ) 1.6 (3 . ) 1.9-0.8 0.5 (2 . ) 0.7 (2 . ) -37 (6 . ) -2.2 (1 . ) 0.80.1 13.8 13.3 12.6 11.4 (910 ) 7 (1 . ) 12.85.4 4.7 5.1 (3 . ) 4.6 (410 ) 2.8 (510 ) 43.3 2.5 3 (910 ) 3.1 (910 ) 2.2 (1 . ) 2.11.8 1 1.8 (2 . ) 1.9 (2 . ) 1.2 (310 ) 11 26.7 26.6 26.4 25 23.5 17.312.8 12.7 12.6 11.5 10.2 5.98.8 8.7 8.6 7.6 6.8 (1 . ) 3.25.6 5.5 5.4 4.5 4.3 (4 . ) 1.610 36 35.9 35.7 34.5 32.9 26.214.7 14.7 14.6 13.8 12.8 8.49.8 9.7 9.6 8.9 8.2 4.66.2 6.2 6.1 5.6 5.4 2.2100 31.4 31.4 31.4 30.9 30.4 28.310.9 10.8 10.8 10.5 10.2 8.96.5 6.4 6.4 6.2 5.9 4.83.7 3.7 3.6 3.5 3.2 2.31000 29.3 29.4 29.2 29.1 29 28.59.4 9.4 9.4 9.3 9.2 8.95.2 5.2 5.2 5.2 5.1 4.82.7 2.7 2.7 2.6 2.6 2.4 In order to be consistent with papers HTS1 and HTS2, and for completeness reasons, we also present results from comparingthe numerical and theoretical inner eccentricity, on shorter timescales. The results, presented in Tables 4,5 and 6, show thepercentage error between the averaged numerical and theoretical inner eccentricity (equations (30) and (31) of HTS1 with themodification mentioned in section 2 and equations (28) and (29) of HTS2). Again, there are four entries per ( K − K
2) pair,corresponding, from top to bottom, to X = 10 , , ,
50 respectively. The integrations were done with φ = 90 ◦ for a circularouter binary and with f = 90 ◦ and ̟ = 0 ◦ for a system with an eccentric outer binary. However, this does not affect thequalitative understanding of the problem at all; similar results are obtained for different initial conditions and an example ofthat is given in Table 7. In the past decade, 155 planets have been discovered orbiting stars other than our Sun, with properties that are somehowdifferent compared to our solar system (e.g. eccentric orbits are rather common among exoplanets). Among them, multipleplanetary systems have been detected around 17 stars, by use of the Doppler technique. For a summary of those developmentsup to date, see Marcy et al. 2005. Here, we discuss some possible applications of our formula on exoplanets, although thatwas not our initial intention when we started this work, firstly presented in HTS1.From the multiple exoplanetary systems that have been discovered so far, we picked up HD217107, which has a dynamicalsetup similar to the one we study in our paper (well separated components, no mean motion commensurabilities). So far,two planets have been detected orbiting HD217107 (Fischer et al. 1999, 2001; Vogt et al. 2005). The orbital elements of thesystem are given in Table 8. c (cid:13) , 000–000 ccentricity generation in HTS Table 2.
Percentage error between the averaged numerical e and equation (2). For all systems, e = 0 . K \ K .
001 0 .
005 0 .
01 0 .
05 0 . . . ) -35.8 (2 . ) -8.1 (9 . ) -0.9 (1 . ) 0.2 (8 . ) 12.9 (1 . )-2.8 (1 . ) -40.6 (1 . ) -4.9 (5 . ) -0.6 (8 . ) -0.1 (4 . ) 3.8 (810 )-2 (3 . ) -55.5 (6 . ) -3.7 (1 . ) -0.3 (2 . ) -0.2 (1 . ) 2 (210 )-0.9 (9 . ) -171.6 (3 . ) -2.8 (5 . ) -0.2 (7 . ) -0.2 (3 . ) 1.1 (6 . )0.01 0.4 (310 ) -2.6 (3 . ) -7.8 (4 . ) -36.3 (2 . ) -5.1 (1 . ) 12.9(1 . )-0.3 (1 . ) -0.3 (1 . ) -1.7 (1 . ) -42.4 (2 . ) -3.5 (6 . ) 3.8 (8 . )-0.1 (2 . ) -0.6 (3 . ) -0.9 (3 . ) -65.5 (7 . ) -2.8 (1 . ) 2.2 (2 . )-0.2 (7 . ) -0.4 (8 . ) -0.7 (9 . ) -333.2 (3 . ) -2.7 (6 . ) 1.6 (7 . )0.1 -0.6 (310 ) -1 (3 . ) -0.4 (3 . ) -2.6 (3 . ) -5.5 (4 . ) 14.6 (1 . )1.6 (1 . ) 1.6 (1 . ) 1.1 (1 . ) 0.4 (1 . ) -0.4 (1 . ) 6.8 (2 . )1.4 (2 . ) 1 (2 . ) 1 (2 . ) 1.1 (3 . ) 0.3 (3 . ) 3.6 (4 . )0.9 (8 . ) 0.6 (8 . ) 0.5 (8 . ) 0.6 (9 . ) 0.5 (1 . ) 1.7 (9 . )1 1.4 (4 . ) 4.2 (4 . ) 5.4 (410 ) 3.5 (4 . ) 4 (4 . ) 17.7 (2 . )8 (210 ) 7.9 (210 ) 7.1 (2 . ) 9 (210 ) 7.9 (2 . ) 5.6 (1 . )6.3 (4 . ) 6.4 (4 . ) 6.5 (4 . ) 7.5 (4 . ) 6.8 (4 . ) 3.2 (2 . )4.2 (1 . ) 4.3 (1 . ) 4.4 (1 . ) 4 (1 . ) 4.4 (1 . ) 2 (7 . )10 16.8 (3 . ) 16.5 (410 ) 16.6 (410 ) 17 (410 ) 17 (410 ) 24.3 (4 . )12.4 (10 ) 14.6 (9 . ) 16.3 (9 . ) 14.7 (9 . ) 13.1 (9 . ) 8 (510 )10.9 (2 . ) 12.5 (2 . ) 10.4 (2 . ) 10.9 (2 . ) 12.8 (2 . ) 4.7 (1 . )8.1 (7 . ) 0.2 (7 . ) 8.1 (7 . ) 8.2 (7 . ) 8.4 (7 . ) 2.7 (3 . )100 21.5 (1 . ) 20.7 (1 . ) 20.8 (1 . ) 21.3 (1 . ) 21.9 (1 . ) 27.113 (910 ) 12.5 (910 ) 12.5 (910 ) 14.1 (8 . ) 9.3 (9 . ) 8.4 (510 )8.5 (2 . ) 12 (2 . ) 12 (2 . ) 12 (2 . ) 12 (2 . ) 5 (1 . )9.3 (6 . ) 7.8 (6 . ) 7.8 (6 . ) 7.9 (6 . ) 7.8 (6 . ) 2.8 (3 . )1000 24.5 (1 . ) 24 (1 . ) 24 (1 . ) 24.2 (1 . ) 24.5 (1 . ) 27.111 (910 ) 13.7 (8 . ) 13.6 (8 . ) 12.7 (8 . ) 10.7 (8 . ) 8.3 (4 . )9 (2 . ) 13.9 (2 . ) 12.2 (2 . ) 13.6 (2 . ) 10.1 (2 . ) 4.8 (1 . )9.6 (6 . ) 8.2 (6 . ) 8.2 (6 . ) 8.2 (6 . ) 9.4 (6 . ) 2.7 (3 . )Assuming that the two binaries move in coplanar orbits and substituting the HD217107 orbital elements into formula (2),we get that q e ≈ .
025 (Keck) and q e ≈ .
028 (Lick) (with sin i = 1, i being the inclination of the orbital plane with theplane of the sky). Fig. 4 is a plot of inner eccentricity against time for a system with the orbital parameters of HD217107 andit is obtained by integrating the full equations of motion numerically using our symplectic code ( e = 0, f = 0 ◦ , ̟ = 0 ◦ ).As one can see, the maximum eccentricity is around 0.04, a value that is much smaller than the observed one of around 0.13(our theoretical model produces the same graphs). We would probably be able to have an orbit with a larger eccentricity if westarted the system with a small but non zero eccentricity, e.g. 0.08 would give us a maximum eccentricity around 0.13. However,this may not be a good assumption to make, as the planet is very close to the star ( a = 0 . AU ) and tidal circularisationwould be expected to take place. Therefore, we would still need a not so small perturbation to the inner orbit, capable ofmaintaining an eccentricity of at least 0.13 (we say at least, because we do not really know which phase of the eccentricityoscillation we currently see). We also performed integrations varying sin i , but with very little effect on the eccentricity. Whatseemed to affect things, was to consider a system with a larger outer eccentricity (for example, for e out = 0 .
8, the maximum e in was around 0.12).Another possibility of course, is that the two orbits are not coplanar. In the case of low mutual inclination I ( I < . ◦ or I > . ◦ ), our past experience from circular binaries (Georgakarakos 2004) and some quick numerical integrations ofthe secular equations of motion with a 4th-order Runge-Kutta method with variable stepsize (Press et al. 1996; for the secularequations see Marchal 1990), led us to the conclusion that there was little difference between the eccentricity of a coplanarand a non coplanar orbit, with the rest of the orbital elements being the same (an exception here might be the occurence ofa secular resonance between the two pericentre frequencies, which can increase the amplitude of the eccentricity oscillation).On the other hand, in the case of high mutual inclination (39 . ◦ < I < . ◦ ), it is known that the eccentricity can reachsignificant values, it can even become one for I = 90 ◦ ( e max = p − cos I ; more about the high and low inclination regimesin Kozai 1962). c (cid:13) , 000–000 Nikolaos Georgakarakos
Table 3.
Percentage error between the averaged numerical e and equation (2). For all systems, e = 0 . K \ K .
001 0 .
005 0 .
01 0 .
05 0 . . . ) -54.8 (1 . ) -10.8 (5 . ) -2.6 (9 . ) -0.8 (4 . ) 81.8 (8 . )1.3 (8 . ) -84.9 (1 . ) -5.9 (3 . ) -0.5 (510 ) 0.1 (2 . ) 5.6 (4 . )0.3 (1 . ) -176.2 (3 . ) -4.6 (9 . ) -0.2 (1 . ) -0.2 (6 . ) 2.6 (1 . )0.3 (4 . ) -8233.2 (1 . ) -4.2 (3 . ) -0.3 (4 . ) -0.1 (2 . ) 1.1 (3 . )0.01 3.6 (1 . ) 4 (1 . ) 6.6 (2 . ) -55.7 (1 . ) -7.4 (6 . ) 70 (910 )0.5 (6 . ) 0.8 (7 . ) 1.4 (8 . ) -100.5 (1 . ) -4.3 (410 ) 5.5 (510 )0.7 (1 . ) 0.3 (1 . ) 0.7 (1 . ) -262.9 (4 . ) -3.8 (1 . ) 2.7 (1 . )0.1 (410 ) 0.3 (4 . ) 0.1 (4 . ) -4855 (2 . ) -3.8 (4 . ) 1.2 (4 . )0.1 1.8 (1 . ) 2.1 (1 . ) 2.1 (1 . ) 1.8 (1 . ) 1.7 (2 . ) 37.8 (3 . )1.3 (6 . ) 0.9 (6 . ) 1.1 (6 . ) 1.3 (7 . ) 0.9 (8 . ) 5.7 (10 )0.8 (1 . ) 1.1 (1 . ) 0.7 (1 . ) 0.9 (1 . ) 0.9 (1 . ) 2.8 (210 )0.4 (4 . ) 0.6 (4 . ) 0.5 (4 . ) 0.4 (4 . ) 0.5 (5 . ) 1.3 (4 . )1 -3.4 (2 . ) -3.3 (2 . ) -3.3 (2 . ) -5.5 (2 . ) -5.5 (2 . ) 36.7 (1 . )2.6 (1 . ) 3.6 (1 . ) 4.6 (1 . ) 2.6 (1 . ) 2.6 (1 . ) 7.1 (5 . )3.1 (2 . ) 3.6 (2 . ) 3.7 (2 . ) 3.8 (2 . ) 3.2 (2 . ) 3.7 (1 . )2.5 (7 . ) 2.6 (7 . ) 2.3 (7 . ) 2.9 (7 . ) 2.5 (7 . ) 2.1 (3 . )10 2.7 (910 ) -6.3 (10 ) -7 (10 ) -2.6 (9 . ) 3.2 (910 ) 35 (4 . )8.3 (5 . ) 6.6 (5 . ) 6.6 (5 . ) 6.6 (5 . ) 6.6 (5 . ) 9.4 (2 . )7 (1 . ) 5.6 (1 . ) 5.6 (1 . ) 6.4 (1 . ) 6.5 (1 . ) 5.2 (610 )4.8 (3 . ) 4.8 (3 . ) 4.8 (3 . ) 4.2 (3 . ) 4.1 (3 . ) 2.7 (210 )100 11 (8 . ) 10.8 (8 . ) 10.9 (8 . ) 12.2 (8 . ) 16.4 (8 . ) 31.8 (410 )6.3 (4 . ) 4.4 (4 . ) 4.4 (4 . ) 6.3 (4 . ) 2.7 (4 . ) 10 (2 . )6.4 (1 . ) 7.2 (1 . ) 7.2 (1 . ) 6.4 (1 . ) 5.7 (1 . ) 5.8 (5 . )5.3 (3 . ) 5.3 (3 . ) 5.3 (3 . ) 5.3 (3 . ) 5.3 (3 . ) 3 (1 . )1000 33.5 (810 ) 33.6 (810 ) 33.8 (810 ) 35.3 (810 ) 37 (810 ) 38.8 (410 )4.1 (4 . ) 4.1 (4 . ) 4.1 (4 . ) 4.3 (4 . ) 4.6 (4 . ) 10 (2 . )6.2 (1 . ) 7 (1 . ) 7 (1 . ) 7 (1 . ) 7 (1 . ) 5.7 (5 . )4.9 (3 . ) 4.9 (3 . ) 4.9 (3 . ) 4.9 (3 . ) 4.9 (3 . ) 3 (1 . )Thus, as a conclusion, we could say that with the current information about the HD217107 system, the planets thathave been detected there so far is not very likely to move on the same plane. Unfortunately, the two formulae presented inthis paper (eqns. (1) and (2)) are only applicable for hierarchical triple systems on coplanar orbits (or orbits with a mutualinclination of a few degrees) and hence, they can not be used to place further constraints to the system.We would like to point out here that equations (1) and (2) are purely classical and do not take into account any relativisticeffects. However, in situations where a planet is very close to the host star, relativistic apsidal precession of the inner orbitcould affect the evolution of eccentricity (e.g. see Holman et al. 1997, Ford et al. 2000, Mardling & Lin 2002). In two previous papers, we derived formulae for estimating the inner eccentricity in hierarchical triple systems with coplanarorbits and with the inner eccentricity being initially zero. However, those calculations were done for systems with comparablemasses. In the present paper, we tested the formulae for systems with mass ratios from 10 − to 10 . The theoretical modelsappeared to work well for most of the cases. There were a few cases for which the theory did not work well, but no theory isperfect. Finally, we applied the theory on the exosolar planetary system HD217107, in an attempt to obtain more informationabout its orbital characteristics, although that kind of application was not part of the initial motivation for the constructionof the models. We concluded that the two planets orbiting the host star are probably on non coplanar orbits. The derivationof a three dimensional formula for eccentric binaries is one of our future aims, as it would definitely prove more helpful forvarious dynamical problems, especially with the constant discovery of more and more exosolar planets every day. c (cid:13) , 000–000 ccentricity generation in HTS Figure 2. K K e = 0 . A − B = 0. The right graph is the same as the left one, but for K ≤ . Figure 3. K K e = 0 .
75 and for which A − B = 0. The right graph is the same as the left one, but for K ≤ . Figure 4.
Inner eccentricity against time for HD217107. The graphs come from the numerical integration of the full equations of motion(the left graph is based on the Keck Observatory data, while the left one uses data from Lick Observatory). The initial conditions are e = 0, f = 0 ◦ and ̟ = 0 ◦ . The time t is in yrs (the period of oscillation for the left graph is 1 . yrs and 6 . yrs for the rightone).c (cid:13) , 000–000 Nikolaos Georgakarakos
Table 4.
Percentage error between the averaged numerical and averaged theoretical e . The theoretical model is based on equations(30) and (31) of HTS1. For all systems, e = 0 and φ = 90 ◦ . K \ K .
001 0 .
005 0 .
01 0 .
05 0 . . ACKNOWLEDGMENTS
The author wants to thank Douglas Heggie for the useful discussion concerning this paper and Seppo Mikkola, who kindlyprovided the code for integrating hierarchical triple systems.
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Percentage error between the averaged numerical and averaged theoretical e . The theoretical model is based on equations(28) and (29) of HTS2. For all systems, e = 0 . f = 90 ◦ and ̟ = 0 ◦ . K \ K .
001 0 .
005 0 .
01 0 .
05 0 . . c (cid:13)000
05 0 . . c (cid:13)000 , 000–000 Nikolaos Georgakarakos
Table 6.
Percentage error between the averaged numerical and averaged theoretical e . The theoretical model is based on equations(28) and (29) of HTS2. For all systems, e = 0 . f = 90 ◦ and ̟ = 0 ◦ . K \ K .
001 0 .
005 0 .
01 0 .
05 0 . . Table 7.
Percentage error between the averaged numerical and averaged theoretical e , for a system with K . K e = 0. X \ φ ◦ ◦ ◦ ◦
10 7.2 17.6 5.3 17.720 3.4 7.2 4.5 7.230 2.3 4.6 4.4 4.650 1.5 2.7 3.4 2.7 c (cid:13) , 000–000 ccentricity generation in HTS Table 8.
Orbital elements for the planetary system HD217107, taken by Vogt et al. 2005. The periods are in days, the planetary massesin Jupiter masses and the the semimajor axes in AU. Keep in mind that the planet masses given in the table are not the actual masses,but M planet sin i , where the angle i is the inclination of the orbital plane with the plane of the sky. Object Mass Period Semimajor axis EccentricityHD217107 1 . M ⊙ — — —HD217107b 1 . M J . M J . M J c (cid:13)000