Dynamics of planets
in
binary systems
N. A.
A. Solovaya
Solovaya and
and E.
E. M.
M. Pittich
Pittich
N.
Astronomical Institute, Slovak Academy of Sciences
Dynamics of planets in binary systems. We investigated dynamical behavior of extrasolar planets in binary systems with high inclinations and large eccentricity orbits in the
frame of the general three-body problem. Also we investigated the dynamical stability of
such orbits, which understand as the conservation of the configuration of the system over
an astronomically long time interval – the eccentricity of the planetary orbit remains much
more less than 1, the mutual inclination of the orbits changes in small limits, and there are
no close approaches among bodies, which can lead to the destruction of the system.
The planet in a binary system revolves around one of the components. The ratio of the
semi-major axes of a planet and of the distant (secondary) star is less than or equals 0.1.
The motion is considered in the Jacobi’s coordinate system and the invariant plane is taken
as the reference plane. We used the canonical Delaunay elements Lj , Gj , and gj (j = 1, for
the planet’s orbit, j = 2 for the secondary star’s orbit). They can be expressed through the
Keplerian elements as
√
Lj = β j a j ,
r
Gj = Lj 1 − e2j ,
gj = ω j ,
(1)
(m0 + m1 ) m2
,
β2 = k √
m0 + m 1 + m 2
(2)
where
m0 m1
,
β1 = k √
m0 + m 1
and m0 , m2 – the masses of the stars, m1 – the mass of the planet, k – the Gaussian constant,
aj – the semi-major axis, ej – the eccentricity, and ωj – the argument of the perigee.
We used the Hamiltonian of the system without the short-periodic terms. The shortperiodic perturbations in the motion of the both components are very small (Solovaya,
1972). Their values are entirely insensible to the contemporary precision of the definition
of the elements or are on the boundary of that precision. Expanded in terms of the Legendre
polynomials and truncated after the second-order terms the Hamiltonian has the form
i
γ1
L41 h
γ2
2
2
2
2
F =
+
−
γ
1
−
3
q
5
−
3
η
−
15
1
−
q
1
−
η
cos
2
g
3 3 3
1 ,
2L21 2L22
L 2 G2
(3)
where the coefficients γ1 , γ2 , and γ3 depend on mass as follows
γ1 =
µ1 =
m0 m1
,
m0 + m 1
β14
,
µ1
µ2 =
γ2 =
β24
,
µ2
(m0 + m1 ) m2
,
m0 + m 1 + m 2
γ 3 = k 2 µ1 µ2
q=
β26
,
β14
c2 − G21 − G22
,
2G1G2
q
η = 1 − e21 .
(4)
c is the constant of the angular momentum, g1 is the argument of the perigee of the planet
orbit in the invariable plane, and q is cosine of the mutual inclination of the orbits.
The canonical system of the equations of motion, corresponding to the Hamiltonian (3),
divides into following mutually combined equations with regards to the eccentricity and the
argument of perigee of the planet:
dG1
15
L4 = − γ3 3 1 3 1 − q 2 1 − η 2 sin 2 g1 ,
dt
8
L 2 G2
(5)
dg1
3
L31 1
1
= γ3 3 3
−η 2 + 5 q 2 +
η q 5 − 3 η2
dt
8 L 2 G2 η
G2
(
"
2
+5 η − q
2
1
2
−
η q 1 − η cos 2 g1 .
G2
#
)
(6)
These equations have the equilibrium solutions. The right part of the Eq. (5) converts to
zero in one of the following cases:
q = ±1 ,
η = 1,
sin 2 g1 = 0 .
The first case, for which q = ±1, belongs to the planar case. The second case is circular
orbits when η = 1. For our purpose consider the qthird case, when sin 2 g1 = 0, e1 > 0.
The eccentricity may have the meaning e1 = 1 − ξ, where ξ is variable value and the
dependence between ξ and t is defined by the following equation (Orlov and Solovaya, 1988):
1 2Z ξ 1
1
m2
B3
r
√ dξ =
G
+
γ
n1 (t − t0 ) ,
3
12 2 ξ1 ∆
A1 16
2
(1 − e2)
(7)
where
v
v
u m0 + m 1
k u
t
,
n1 =
a1
a1
u m0 + m 1 + m 2
k u
t
n2 =
,
a2
a2
m=
n2
,
n1
γ=
m2
.
m0 + m 1 + m 2
∆ is the polynomial of the fifth order. It can be separated to two polynomials of the second
and the third order, which have the form:
2
2
f2 (ξ) = ξ − 2 c +
2
3 G2
2
ξ+ c −
2 2
G2
+
2
2
(10 + A3) G2 ,
3
(8)
5 2
5 2
2
2 2
2
f3 (ξ) = ξ − 2 c +
+
ξ +
c + G2 + c − G2 −
4
2
#
1 2
5 2
2 2
− G2 (10 + A3) ξ −
c − G2 ,
6
4
3
2
!
2
G2
"
(9)
where
A3 = 2 − 6 η02 q02 − 6 1 − η02 2 − 5 1 − q02 sin2 g10 ,
h
i
G2 =
G2
,
L1
c=
c
.
L1
(10)
For qualitative investigation of motion it is necessary to know the roots of the equations
f2(ξ) = 0 and f3(ξ) = 0. The subscript or superscript 0 denotes initial values of all
parameters.
The solution of this system of equations has a meaning in the region where f2(ξ) f3 (ξ) > 0 .
All roots are real and positive, but only two of them, ξ1 and ξ2 , are always less than unit.
The variable ξ, which is a function of the eccentricity, ξ = 1 − e21 ,. must change within the
interval ξ1 ≤ ξ ≤ ξ2 .
In the case when g1 = 0, the expression in the curly braces of Eq. (6) is equal to
2η
1
2+
ηq .
G2
!
2
(11)
This expression converts to zero if η = 0. The case when the initial value of e1 = 1
(parabolic orbit) we do not consider.
In the case when g1 = π/2, the expression in the curly braces of Eq. (6) is equal to
1
2 5q − 3η +
η q 5 − 4 η2 .
G2
"
2
#
2
(12)
This expression converts to zero if
q=
"
2
η 4η −5 ±
r
2
60 G2
10 G2
+ (5 −
4 η 2 )2
#
,
(13)
which we designed as q01 for the sign minus before the root term, and q02 for the sign plus
before the root term.
In the case g1 = π/2 we can rewrite the equations the second and the third orders as
η02
η02
f2 (ξ) = ξ −
f3 (ξ) = ξ −
"
ξ−
η02
5
ξ−
4
− 4 G2 η0 q0 +
!
ξ−
η02
−
2
8 G2
2
3 G2 ξ
+
2
4 G2 1
−
5 q02
1−
+ G2 η0 q0 (5 − 4 ξ) +
η02
+
2
5 G2 q02
#
2
16 G2
.
, (14)
(15)
If the expression (12) is negative then q01 < q0 < q02 and ξ = η02 is the least root
of the
q
equation of the second order (Eq. 14), which is less than 1. In this case e1 min = 1 − η02 .
The value of e1 max can increase. In this case in the perigee the large tidal forces can
influence on the motion of the planet.
If the expression (12) is positive, then q0 < q01 or q0 > q02 and ξ is the least
root of the
q
equation of the third order (Eq. 15), which is less than 1. In this case e1max = 1 − η02 . The
maximum value of the eccentricity of the planet’s orbit cannot exceed the initial value of
the eccentricity. The motion of the planet is stable.
Elements
Planet
Secondary
star
a
e
i
ω
Ω
P
2.1 AU
0.2
64.61◦
95.0 ◦
75.6 ◦
2.47 yrs
22 AU
0.439
51.3◦
162.1◦
10 AU
5 AU
70 yrs
5 AU
Motion of the planet in binary system γ Cephei around the primary star within
50 000 years, projected to the plane perpendicular to the line of sight. Right top – one of
possible stable planetary orbits (Ω2 = 75◦), right down – one of unstable planetary orbirs
(Ω2 = 240◦). Masses: m0 = 1.59 m, m2 = 0.58 m, and m1 = 1.76 mJ . m1 sin i1 = 1.59 mJ .
1.0
q
q02
0.8
0.6
0.4
Zone of
stability
0.2
q0 < q01
or
q0 > q02
0.0
q0 = cos I
−0.2
−0.4
−0.6
q01
−0.8
−1.0
0
60
120
180
240
300
Ω2 ◦ 360
Theoretical boundaries of stable zone. The stable motion of the planet of γ Cephei
is limited with values of the longitude of ascending node of the secondary star Ω2 from the
interval 30◦ < Ω2 < 122◦.
1.0
90
I◦
e1
Theoretical minimum, maximum,
Theoretical minimum, and maximum
values of e1
and mean values of I
60
0.8
Zone of
stability
Zone of
stability
30
0.6
0
0
60
120
Ω2 ◦ 180
0.4
0.2
0.0
0
60
120
Ω2 ◦ 180
Conditions for dynamical stability of the planetary orbit. The stable motion of the
planet depends on the angular momentum of the system. It is function of the longitudes of
ascending node of the planet (Ω1 ) and the secondary star (Ω2 ). The stability of the motion
of the planet preserves when the mutual inclination I between the planet and the secondary
star and the eccentricity e1 of the planetary orbit change a little. Its is one of the most
important conditions for the origin, evolution, and conservation of the life. For the planet
of γ Cephei this condition carries out around the interval 60◦ < Ω2 < 90◦.
e
e
.75
.75
.50
.50
.25
.25
q [AU]
q [AU]
2.00
2.00
1.50
1.50
1.00
1.00
.50
.50
I [ o]
150
I [ o]
150
120
120
90
90
γ Cep (Ω2 = 75◦ )
60
60
30
0
γ Cep (Ω2 = 240◦ )
30
0
1000
2000
3000
4000
t [yr]
0
0
1000
2000
3000
4000
t [yr]
Orbital evolution of the planet. The interval of 5000 years covering several revolutions
of the distant star. Left for Ω2 = 75◦, right for Ω2 = 240◦. Dots represent results obtained
by the numerical integration, the boundary of the blue zones by the analytical theory. Very
good agreement between the results obtained by the theory and the numerical integration
was received.
20 AU
5 AU
The planet in binary system 61 Cygni. Motion of the planet and the secondary star
around the primary star within 50 000 years, projected to the plane perpendicular to the line
of sight. Masses: m0 = 1.11 m, m2 = 1.00 m, and m1 = 10.5 mJ . The orbital elements:
a1 = 3 AU, e1 = 0.53, i1 = 134◦, ω1 = 295◦, Ω1 = 94◦, a2 = 80 AU, e2 = 0.48, i2 = 54◦,
ω2 = 146◦, Ω2 = 176◦. The elements and masses of the planet and the binary components
are from Sixth Catalog of Orbits of Visual Binary Stars (Hartkopf and Masson, 2004).
e
.75
.50
.25
q [AU]
1.50
1.00
.50
I [o ]
150
120
90
60
61 Cyg
30
0
0
10 000
20 000
30 000
40 000
t (yr)
Orbital evolution of the planet 61 Cyg. The interval of 50 000 years covering several
revolutions of the distant star. Dots represent results obtained by the numerical integration,
the boundary of the blue zones by the analytical theory.
Conclusion
• Theory allows us to calculate minimum and maximum values of I and e1 and predicts
the behavior and the stability of the motion of extra-solar planets over an astronomically
long time interval.
• Theory has been demonstrated and verified by numerical integration of the γ Cephei
and 61 Cygni systems.
• The planet of γ Cephei has a stable orbit for 30◦ < Ω2 < 122◦, then 0.176 < e1 < 0.423.
• The planet of γ Cephei has a stable orbit for 30◦ < Ω2 < 122◦, then 0.176 < e1 < 0.423.
• The planet of 61 Cygni has a perturbed orbit. The mutual inclination
eccentricity e1 change in limits: 109◦ < I < 148◦ and 0.518 < e1 < 0.943
period of 22 412 years. For e1 maximum the perigee distance of the planet
AU. In this system the tidal phenomena can occur. The high inclination
orbit permits to preserve its stability.
I and the
within the
rπ = 0.170
of planet’s
Acknowledgements. This work was supported by the Slovak Academy of Sciences Grant
VEGA No. 2/4002/04. Authors are gratefull to the Organizing Committee of the JENAM
2004 for the grant.
References
Extrasolar Planet Guide: 2004, http://wwww.extrasolar.net/star.asp?StarID=196.
Fuhrmann, K.: 2004, Stellar data. http://youngstars.mpe.mpg.de/ PlanetenAGB/PUT/
gamma Cep facts.html.
Hartkopf, W.I. and Mason, B.D.: 2004, Sixth Catalog of Orbits of Visual Binary Stars.
http://ad.usno.navy.mil/wds/orb6.html.
Orlov, A.A. and Solovaya, N.A.: 1988, The stellar problem of three bodies and applications,
In The Few Body Problem, edited by M.J. Valtonen, Kluwer Acad. Publish., Dordrecht,
pp. 243–247.
Solovaya, N.A.: 1972, Application of von Zeipel’s method to the stellar three-body problem.
Trudy GAISH, XLIII, P. 38–51.
Schneider, J.: 2004, Extra-solar Planets Catalogue. http://www.obs.pm.fr/encycl/
catalog.html.