3 QSO Solutions and Stability
3.3. Influence of the Second Primary
3.3.2. Approximate Solutions in the Osculating Elements
which can be reduced to a quadratic polynomial by the variable changeκ=λ2 P(κ) =aκ2+bκ+c=κ2+
4− 3
√1−e2 − µ
¯
α3(A1+A2)
κ+µ2
¯
α6A1A2+ 3
√1−e2 µ
¯
α3A2 (3.49) For the system (3.47) to be stable we need the four roots of the characteristic polynomial in (3.48) to have non-positive real part which can only be achieved if all of the four roots are pure imaginaries. From the variable change, this translates in having two real negative roots in (3.49). These roots are given by
κ1,2= −b±√ b2−4c
2 (3.50)
witha= 1. The roots of (3.48) are real negative for
κ1+κ2<0 κ1κ2>0 b2−4c >0
⇒
b >0 c >0 b2−4c >0
⇒
4− 3
√1−e2− µ
¯
α3(A1+A2)>0 µ2
¯
α6A1A2+ 3
√1−e2 µ
¯ α3A2>0
4− 3
√1−e2− µ
¯
α3(A1+A2) 2
−4 µ2
¯
α6A1A2+ 3
√1−e2 µ
¯ α3A2
>0 (3.51) Notice that the second condition is always satisfied sinceA1, A2>0. The conditions are then reduced to two4
α > µ(A1+A2) 4−√ 3
1−e2
!1/3
4− 3
√
1−e2 − µ
¯
α3(A1+A2) 2
−4 µ2
¯
α6A1A2+ 3
√ 1−e2
µ
¯ α3A2
>0
(3.52)
For given values ofeandµa minimum value of the amplitude for which it is possible to find sufficiently stable orbits is defined by conditions (3.52). The maximum limit is defined by the validity of the approxi- mationx, y, z <<1which can be studied numerically. Remember that this region of stability is valid for quasi-synchronous orbits. It is possible to find sufficiently stable orbits below this limit with a different periodicity (Wiesel, 1993). These, however, are not the main focus of our work.
The solutions for Hill’s unperturbed case are
x=g1(c, f) =α(1 +ecosf) cos (f+φ) +δx x˙ =h1(c, f) =−α(sin (f+φ) +esin (2f+φ)) y=g2(c, f) =−α(2 +ecosf) sin (f+φ) +δy y˙ =h2(c, f) =−α(2 cos (f+φ) +ecos (2f +φ)) z=g3(c, f) =γcos (f +ψ) z˙ =h3(c, f) =−γsin (f+ψ)
(3.53) or, in short form
r=g(c, f) r˙=h(c, f) (3.54) with the functionhibeing the derivatives in respect to the independent variablef
h= ∂g
∂f
c=const.
(3.55) andcithe osculating elements
c=
α φ δx δy
γ ψ
(3.56)
withδx= 0for the unperturbed case.
The method of variation of the arbitrary constants used to derive the differential equations that describe the variation ofccan be found in (Danby, 1962). The equations of motion (3.4) can be written as
˙
xi =fi(x, f) +pi(x, f) (3.57)
with|f|>>|p|. Let the unperturbed system
˙
xi=fi(x, f) (3.58)
have general solution
xi=Xi(c1, c2, . . . , cn, f) =Xi(c, f) (3.59) with derivative
˙
xi= ∂Xi
∂f +
n
X
j=1
∂Xi
∂cj c˙j (3.60)
Substituting in equation (3.57) we get
∂Xi
∂f +
n
X
j=1
∂Xi
∂cj
˙
cj =fi(xi, f) +pi(xi, f) (3.61)
but we know that Xi is the general solution of the unperturbed system defined in (3.58), i.e., X˙i =
fi(xi, f). This way, we have
n
X
j=1
∂Xi
∂cj
˙
cj =pi(xi, f) (3.62)
The variations of the osculating elements in equation (3.62) can be obtained in matrix form (Schaub and Junkins, 2003)
˙
c= [L]−1 ∂R
∂c T
(3.63) where[L]is the anti-symmetric matrix defined by the Lagrangian brackets
Lij = ∂g
∂ci
T∂h
∂cj
− ∂h
∂ci
T∂g
∂cj
(3.64)
withLij = −Lji, Lii = 0, andgand hdefined by (3.53). R is the perturbation function written as a function of the osculating elements
R= 1
1 +ecosf
µ
(g12+g22+g32)1/2 (3.65)
This way, the differential equations onciare obtained
˙
α=µα[3 + 2ecosf−cos (2(f+φ))]esinf+ 2 [2 sin (f+φ) +esin (2f+φ)]δαx+ 2 [cos (f+φ) +ecos (2f+φ)]δαy 2r23(1 +ecosf)(2 +e2+ 3ecosf)
φ˙=µ4 +e[6 cosf+ sinfsin (2(f+φ)) +e+ecos (2f)] + 2 [2 cos (f+φ) +ecos (2f+φ)]δαx+ 2 [sin (f+φ)−esin (2f+φ)]δαy 2r32(1 +ecosf)(2 +e2+ 3ecosf)
δ˙x=µα(1 +ecosf)(2 sin (f φ) +esinφ)−esinfδαx−(1 +ecosf)δαy r32(2 +e2+ 3ecosf)
δ˙y=µα(2 +ecosf)(cos (f+φ) +ecosφ) + (2 +ecosf)δαx−esinfδαy r32(1 +ecosf)(2 +e2+ 3ecosf)
˙
γ=µγcos (f+ψ) sin (f+ψ) r32(1 +ecosf)
ψ˙=µ cos2(f+ψ) r23(1 +ecosf)
(3.66)
These equations are too complex to conclude on the stability of the system. Nevertheless, if we assume e << 1, and the amplitude of the motion in the z coordinate and the displacements of the
’ellipse’ origin, δx andδy, to be much smaller than the amplitude of the motion in the x-y plane, i.e., δx/α, δy/α, γ/α <<1, we can expand equations (3.66) in a Taylor series up to the first order about zero
on these quantities. By doing so we obtain the approximated differential equations
˙
α = µ
3 sinf −cos (2(f+φ)) sinf
4d3 α+sin (f +φ)
d3 δx+cos (f+φ) 2d3 δy
φ˙ = µ 1
d3 +sinfsin (2(f+φ))−4 cosf
4d3 e−3cosfcos2(f+φ) + 2 cosfsin2(f+φ)
d5 eα2
+cos (f+φ)(3 sin2(f+φ)−2)
d5 αδx−sin (f+φ)(3 sin2(f+φ)−11)
d5 αδy
δ˙x = µ
sin (f +φ)
d3 α−cos (f+φ) sinf
2d3 eα−3 sin (f+φ)(cosfcos2(f+φ) + 2 cosfsin2(f+φ))
d5 eα3
−3 cos (f+φ) sin (f+φ)
d5 α2δx+9 sin2(f+φ)−1 2d5 α2δy
δ˙y = µ
2 cos(f +φ)
d3 αcosφ−3 cos (2f+φ)
2d3 eα−6 cos (f+φ)(cosfcos2(f+φ) + 2 cosfsin2(f+φ))
d5 eα3
+4(2−3 cos2(f+φ))
d5 α2δx+12 cos (f +φ) sin (f +φ) d5 α2δy
˙
γ = µcos (f+ψ) sin (f+ψ)
d3 γ
ψ˙ = µ
cos2(f+ψ)
d3 −cos2(f+ψ)(4 cosfcos2(f+φ) + 10 cosfsin2(f+φ))
d5 eα2
−3cos (f +φ) cos2(f+ψ)
d5 αδx+ 6sin (f+φ) cos2(f +ψ)
d5 αδy
(3.67) withdas defined by (3.39).
We can not solve equations (3.67) analytically and these are too complex for any interpretation. How- ever, we can average their periodic coefficients which provides enough insight to assess the stability of the system. This way, we get the averaged differential equations on the osculating elements
˙¯
α= 0 φ˙¯= µ
π¯α3E δ˙¯x= µ
6πα¯3(K−E)¯δy− µ
9πα¯2e(5K−8E) sin ¯φ δ˙¯y=− 2µ
3π¯α3(4E−K)¯δx− µ
9π¯α2e(70E−19K) cos ¯φ
˙¯
γ=− µ
6πα¯3γ(5E¯ −2K) sin (2 ¯β) ψ˙¯= µ
6π¯α3(3E+ (5E−2K) cos (2 ¯β)) β˙¯= ˙¯ψ−φ˙¯= µ
6πα¯3(−3E+ (5E−2K) cos (2 ¯β))
(3.68) whereK and E are complete elliptic integrals of modulek =√
3/2. The differential equation on φ¯is independent from the other osculating elements and its solution is easily derived
φ¯= µ
π¯α3E f+φ0 (3.69)
After substitution ofφ, the system composed by¯ δ¯xandδ¯ybecomes separable from the other variables and can be represented as a second-order system
d df
" δ¯x δ¯y
#
=
"
0 a
−b 0
# " δ¯x δ¯y
# +
"
−csin ¯φ
−dcos ¯φ
#
, a, b, c, d >0 (3.70)
The solution of a non-homogeneous system of differential equations with constants coefficients, x˙ = Ax+u(f), can be written as (Russell, 2007)
x(f) =M(f)M(f0)−1x0+
Z
ff0
M(f)M(s)−1u(s)ds (3.71)
where, for a second-order system
M(f) =
v1eλ1f v2eλ2f
(3.72) withλ1,2andv1,2being the eigenvalues and eigenvectors, respectively, of matrixA, andx0the system’s initial conditions. The solution of the system is of the form
δx = δx0+
φc˙¯ +ad ab−φ˙¯2
cos ¯φ0
! cos (√
abf) + ra
b δy0−bc+ ˙¯φd ab−φ˙¯2
sin ¯φ0
! sin (√
abf)
−φc˙¯ +ad ab−φ˙¯2
cos ¯φ δy = δy0−bc+ ˙¯φd ab−φ˙¯2
sin ¯φ0
! cos (√
abf)− rb
a δx0+
φc˙¯ +ad ab−φ˙¯2
cos ¯φ0
! sin (√
abf) +bc+ ˙¯φd
ab−φ˙¯2 sin ¯φ
(3.73)
where c and dare the only factors depending on the orbital eccentricity of the second primary. The system (generalized in (3.70)) has characteristic polynomial
P(λ) =λ2+a b, a, b >0 (3.74)
and, as the eigenvaluesλ1,2are a pure imaginary conjugate pair, the system is stable. Solutions (3.73) are composed by two distinct periodic motions. The first has an amplitude composed by both terms that depend and not depend on the orbital eccentricity. This motion has frequencyω1=√
a b(from the argument of the trigonometric functions) and period
P1= 2π
√a b = 6π2
(5EK−4E2−K2)1/2
¯ α3
µ ≈37.1483α¯3
µ (3.75)
the other distinguishable motion has an amplitude that depends on the orbital eccentricity of the second primary and has the same frequency and period asφ¯—ω2= ˙¯φand
P2= 2π φ˙¯
= 2π2 E
¯ α3
µ ≈16.2992α¯3
µ (3.76)
The system (3.68) is also independent on the variableβ¯and its solution is β¯=−arctan
rm−n
m+ntanp
m2−n2(f+Cβ¯)
(3.77) withm= 3µE/(6πα¯3)andn = 3µ(5E−2K)/(6π¯α3). The frequency and period of this oscillation are obtained in the same fashion,ωβ¯=√
m2−n2and
Pβ¯= 2π
√m2−n2 = 6π2
(5EK−4E2−K2)1/2
¯ α3
µ =P1≈37.1483α¯3
µ (3.78)
After substitution ofβ¯the solution onγ¯is obtained
¯
γ=Cγ¯ m−ncos (2 ¯β)−1/2
(3.79) which oscillates two times faster thanβ¯and, thus,Pγ¯=Pβ¯/2.
The relation between the orbital mean motions of the QSO,nQSO, and of the second primary,n, in their orbits around the first primary can also be obtained. The QSO has orbital mean motionnQSO = 1 + ˙¯φ andnis 1.
nQSO
n = 1 + ˙¯φ= 1 + µE
π¯α3 ≈1 + 0.385491 µ
¯
α3 (3.80)
The second primary, located on the origin of the reference frame, acts as a restoring force on the the third-body. This force varies with the inverse of the distance between these two bodies. If the third- body gets too far from the origin, there is a chance that this restoring force will not be strong enough to maintain the third-body in orbit around the second primary.
The inclination of the QSO influences the distance of the third body to the second primary as the motions in the z direction and on thex-y plane are independent. Consequently, the inclination of the QSO also influences the restoring capability of the second primary. A critical value for the ratioγ/α, for which this restoring capability vanishes, can be derived.
The motion of the third-body in thex-yplane is nearly elliptic and the distance to the second primary is maximum when it crosses the y-axis in y = ±2α±δy. The motion in the z-direction has also to be considered. The maximum distance between the second and third bodies is achieved when the third-body achieves the maximum height (in absolute value)z=±γin the same point that achieves the maximum distance to the second primary in thex-yplane,y=±2α±δy. This is the worst-case scenario for the analysis of the restoring capability of the second primary with the inclination of the QSO and it is defined by an angleβ=±π/2(angle between the intersection line of the QSO plane with the primaries’
orbital plane and the positivexsemi-axis).
If we give up on the assumption that the quantityq=γ/αis small, the system composed byδ¯xand¯δy
in (3.70) maintains the same form but now with
a= 4Kq−(q4+ 3q2+ 4)Eq
π(q2+ 3)(q2+ 4)3/2 µ
¯ α3
b= 4(q2+ 4)Eq−4Kq π(q2+ 3)(q2+ 4)1/2
µ
¯ α3
(3.81)
where the elliptic integrals Kq and Eq have module k = p
(q2+ 3)/(q2+ 4). The period of the main motion in this case is
P = 2π
√ab = 2π2(q2+ 3)(q2+ 4)
[(4Kq−(q4+ 3q2+ 4)Eq)(4(q2+ 4)Eq−4Kq)]1/2
¯ α3
µ (3.82)
We now want to compute the critical ratioqc for which separation occurs. Analytically, the period is infinite for ejected orbits, hence, the quantityqccan be computed numerically by finding the root of the denominator in (3.82)
P|q=q
c =∞ →
(4Kq−(qc4+ 3qc2+ 4)Eq)(4(qc2+ 4)Eq−4Kq)1/2
= 0→qc= 0.961073 (3.83)
This conclusion is based on the averaged equations of motion where the peaks of periodic effects are neglected. Thus, it is expected that separation occurs before the value found forqc. The computed value is merely a statement that QSOs withq > qcwill suffer orbit separation. The numerical exploration of the problem will tell us how accurate is this prediction.