# Variation of the Equator due to a Highly Inclined and Eccentric Disturber

Authors: Clair do Nascimento; Tadashi Yokoyama
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The New Horizons project is currently in progress, but until recently years ago, a
mission to Neptune-Triton System predicted in the mid term 2008–2013 was one of the
priorities of NASA's Solar System Exploration theme. Any way, it is important to
increase our knowledge on the dynamics of the inner satellites of Neptune, since
according to some authors, this was a key question in the mission to Neptune-Triton
system. In a previous work, we presented the expansion of the disturbing function for
the dynamics of this system. Here we derive the averaged classical equations of the
precession of the equator for this problem. The highly inclined and retrograde orbit of
Triton makes this problem very unusual. Therefore, the classical truncations in the
inclinations are not acceptable, so that the precession equations must be obtained in
closed form for the inclination. With a significant mass and due to its distance from
Neptune, which is continuously decreasing, Triton should exert in the future, important
precession on Neptune's equator. The effects of this precession on the inner satellites
are shown, including some resonant cases predicted in the future. Although Triton's
orbit is almost circular, no expansion in the eccentricity is needed.


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## Body

## 1. Introduction

Some years ago, the exploration of the Neptune system in the next ten years was posed as an important priority of NASA's Solar System Exploration theme  [1]. Although the New Horizons project is now the top priority for the far reaches of the solar system, most probably, to design a mission to explore the Neptune-Triton system will be just a mater of time. The authors  [1] raised some important questions about the dynamics (stability) of the arcs and inner satellites of Neptune. In order to give support to the mission, we must increase our knowledge on the dynamics of this system. In this work we derive the classical equations of the precession for the equator of Neptune. Basically, this equator is disturbed by the presence of Triton since it has a remarkable mass: mT/MN≈2.09×10-4. (mT, MN are the masses of Triton and Neptune, resp.). Moreover, Triton's inclination with respect to Neptune's equator is quite unusual (≈157.350). In a short time span, this precession should not be so important on the dynamics of the inner satellites. However, for a longer time (more than 600 years), this effect must be investigated. This work is organized in the following way: in Section 2 the averaged precessional equations for Neptune's equator considering arbitrary inclination obtained. Some authors [2] hypothesize that in the past, the eccentricity of Triton (eT), might have been very high. Therefore, the precessional equations are also obtained for arbitrary eT. In Section 3, we discuss the definition of the invariable plane and the existence of a corresponding equilibrium solution when we assume circular orbit for Triton. Considering the current dynamics of Neptune-Triton system, several authors have improved the ephemerides and data of the small satellites (e.g., Owen et al. [3]). Here, some numerical simulations showing the effects of the precession , particularly for the future 2:1 and 3:1 resonances involving Triton and Proteus, are presented. Section 4 is devoted to the conclusions.


## 2. TheĨ and Ω̃ equations

LetXYZ be an arbitrary reference system (inertial) and a moving system xyz fixed on the equator of Neptune (see Figure 1). We define Ĩ= inclination of the equator and Ω̃= node of the equator [4]. Due to the presence of a massive satellite (Triton), the equator of Neptune will be disturbed. Let T be the associated torque due to Triton and let WT be the corresponding oblateness potential of Neptune:Figure 1
XYZ: inertial system, xyz: moving system fixed on the equator.(2.1)WT=-GMNmTJ2RN2rT3(-12+3sin2β2),
where G is the gravitation constant, MN and mT are the masses of Neptune and Triton, respectively. The oblateness constant is J2, β is the Triton's latitude and rT is the radial distance Neptune-Triton in the xyz system. The equatorial radius of Neptune is indicated by RN. Then:
(2.2)T=(L,M,N)=-rT×∇TWT,
where ∇T is the usual gradient operator with respect to Triton coordinates and L, M, and N are the components of the torque T, that is:(2.3)L=(∂WT∂yT)zT-(∂WT∂zT)yT,M=-(∂WT∂xT)zT+(∂WT∂zT)xT,N=(∂WT∂xT)yT-(∂WT∂yT)xT,
where xT,yT and zT are the cartesian coordinates of Triton with respect to the moving system xyz. On account of the rotational symmetry, we can show that N=0. To this end, since β is the latitude we have: sin(β)=zT/rT. Then a direct calculation shows that WT depends on Triton coordinates only through rT and zT/rT, therefore:(2.4)WT=WT(rT,zTrT),rT=(xT2+yT2+zT2)1/2,∂WT∂xT=∂WT∂rTxTrT,∂WT∂yT=∂WT∂rTyTrT.
Substituting these last two relations in N given in (2.3), we get N=0. Now, let w=(pN,qN,wN) be the rotation vector of Neptune. According to Euler's dynamical equations:(2.5)ṗN+C-AAwNqN=LA,q̇N-C-AAwNpN=MA,ẇN=NC=0,
where A (= B) and C are the moments of inertia of the planet. From the geometric equations, w can be expressed in terms of the inertial XYZ system:(2.6)pN=sin(Ĩ)sin(χ)Ω̃̇+cos(χ)Ĩ̇,qN=sin(Ĩ)cos(χ)Ω̃̇-sin(χ)Ĩ̇,wN=cos(Ĩ)Ω̃̇+χ̇.
Taking the time derivative of pN,(2.7)ṗN=cos(Ĩ)Ĩ̇(sin(χ)Ω̃̇)+sin(Ĩ)(χ̇cos(χ)Ω̃̇+sin(χ)Ω̃̈)-sin(χ)χ̇Ĩ̇+cos(χ)Ĩ̈.
Assuming that the polar component of the rotation of the planet is dominant, the following quantities: Ĩ̇Ω̃̇, Ĩ̈ e Ω̃̈ e Ω̃̇2 can be neglected when compared to χ̇  [5, 6]. Then,(2.8)ṗN=sin(Ĩ)χ̇cos(χ)Ω̃̇-sin(χ)χ̇Ĩ̇.
Substituting ṗN e qN in the first Euler's equation,(2.9)LA=(Ω̃̇χ̇cos(χ)sin(Ĩ)-Ĩ̇χ̇sin(χ))+CAwN(Ω̃̇sin(Ĩ)cos(χ)-sin(χ)Ĩ̇)-wN(Ω̃̇sin(Ĩ)cos(χ)-sin(χ)Ĩ̇).
Substituting wN=cos(Ĩ)Ω̃̇+χ̇ and neglecting terms factored by Ω̃̇2 and Ĩ̇Ω̃̇, we have(2.10)CAwN(sin(Ĩ)cos(χ)Ω̃̇-sin(χ)Ĩ̇)=LA.
Similarly, putting q̇N and pN into the second Euler dynamical equation, we obtain:(2.11)-CAwN(sin(Ĩ)sin(χ)Ω̃̇+cos(χ)Ĩ̇)=MA.
Now we can eliminate L and M in the above equations. Defining W0=-GmTMNJ2RN2/2, we have(2.12)WT=W0(3zT2rT5-1rT3),∂WT∂yT=W0[∂∂rT(3zT2rT5-1rT3)]yTrT,∂WT∂zT=W0[∂∂rT(3zT2rT5-1rT3)]zTrT+Wz,
and so,(2.13)L=∂WT∂yTzT-∂WT∂zTyT=-WzyT,
where Wz=6W0zT/rT5. Similarly, M=WzxT. From (2.10) and (2.11),(2.14)Ω̃̇sin(Ĩ)=-WzCwN(yTcos(χ)+xTsin(χ)),Ĩ̇=WzCwN(-xTcos(χ)+yTsin(χ)).
If a sequence of three rotations, Ω̃, Ĩ, is χ are applied on the system XYZ, it will coincide with xyz. Therefore we can write xT, yT, and zT in terms of the well-known elliptic orbital elements of Triton.(2.15)xT=rTcos(fT+wT)cosΩT-rTsin(fT+wT)sinΩTcosIT,yT=rTcos(fT+wT)sinΩT+rTsin(fT+wT)cosΩTcosIT,zT=rTsin(fT+wT)sinIT,
where rT, fT, wT, ΩT, wT, and IT refer to classical orbital elements of Triton referred to (X,Y,Z) inertial system.Finally, averaging the resulting equations with respect to the mean anomaly of Triton, the precessional equations, without any restriction on the inclinationIT or on the eccentricity eT, are
(2.16)Ĩ̇=F04aT3(1-eT2)3/2[sin(Ĩ)sin(IT)2sin(2Ω̃-2ΩT)+cos(Ĩ)sin(2IT)sin(Ω̃-ΩT)],Ω̃̇=F04aT3(1-eT2)3/2[(1-3cos(IT)2)cos(Ĩ)+2sin(IT)cos(IT)cos(2Ĩ)cos(Ω̃-ΩT)sin(Ĩ)+sin(IT)2cos(Ĩ)cos(2Ω̃-2ΩT)2sin(IT)cos(IT)cos(2Ĩ)cos(Ω̃-ΩT)sin(Ĩ)],
where
(2.17)F0=3GmTMNRN2J2CwN.
Notice that, as expected in this approximation, these two equations are decoupled from χ equation.


## 3. The Invariable Plane

In this section, we define the reference plane. Lett1 and n1 be the orbital angular momentum of Triton and rotation angular momentum of Netpune, respectively. Then R=t1+n1 is a constant vector. Therefore, the plane perpendicular to the direction of R is invariable (see Figure 2), and it will be adopted as our reference plane.t1 is perpendicular to the orbital plane of Triton, n1 is perpendicular to Neptune's equator, and R is the resulting vector which defines the invariable plane.
(a)(b)The dynamics of the system involvingĨ, Ω̃, ΩT, and IT shows an interesting solution in this invariable plane. To see that, let us assume Triton in a circular Keplerian orbit. In this case, we have t1=mTnTaT2, where nT is the mean motion, and mT is the Triton's mass. On the other hand, n1=CwN, where C is the polar moment of inertia of Neptune and wN is its angular velocity of rotation (spin rate). Now, from Figure 2, we have(3.1)sin(π-IT)=smTnTaT2,sinĨ=sCwN
that is, the inclination Ĩ of the invariable plane is given by:(3.2)sinĨ=mTnTaT2sinITCwN.
In order to have the complete set of equations that govern the motion of the equator, we have to consider (2.16) and also the corresponding equations for IT and ΩT. They are(3.3)dITdt=-1nTaT2sin(IT)∂RJ2∂ΩT,dΩTdt=1nTaT2sin(IT)∂RJ2∂IT,
where RJ2 is the disturbing function due to the oblateness of Neptune which is written with respect to the following invariable plane:(3.4)RJ2=S0[12(1-32sin2Ĩ)(1-32sin2IT)+38sin(2Ĩ)sin(2IT)cos(ΩT-Ω̃)+38sin2Ĩsin2ITcos(2ΩT-2Ω̃)],
where(3.5)S0=GMNRN2J2[aT3(1-eT2)3/2].
For Ω̃-ΩT=π, we easily see that(3.6)İT=Ĩ=0,Ω̃̇=-S2sin(Ĩ)sin(2Ĩ+2IT),Ω̇T=-34GMNRN2J2nTaT5sinITsin(2Ĩ+2IT),
where
(3.7)S2=3GmTMNRN2J24CwNaT3.
Therefore,(3.8)Ω̃̇-Ω̇T=3GMNRN2J24aT3sin(2Ĩ+2IT)[1nTaT2sinIT-mTCwNsin(Ĩ)],
that is, the term inside the brackets is automatically zero provided the reference plane is the invariable plane (see (3.2)). In this case; Ω̃̇-Ω̇T=0, IT and Ĩ are constants; Ω̃ and ΩT are synchronous. This means that the precession rate of the Neptune's equator is the same of the Triton's node on the invariable plane (≈688 years). It is also easy to show that Ω̃-ΩT=π defines a stable periodic solution of the system (simple calculations show that such solution does not exist for Ω̃-ΩT=0). This means that it is very convenient to choose the invariable plane as the reference for our purposes.


## 4. Some Effects on the Inner Satellites

Initially we tested our (2.16) for the case of Earth and Mars. As expected, the results agree quite well with very known values (≈26000 and ≈175000 years for Earth and Mars, resp.). For the Neptune's equator, considering aT=14.063RN, our numerical experiments show ≈688 years for the period of precession, referred to the invariable plane Neptune-Triton. This period is rather sensitive for the adopted value of the semimajor axis of Triton. In this section we test the effect of the precession of Neptune's equator on an inner satellite of this planet. In Figure 3, we always integrate two orbits. The lighter points in the left panels, correspond to the orbits when we include the precessional effect, while the thick points come from the orbit when we neglect (2.16). The inclinations are referred to the invariable plane. We consider a test particle (Proteus) disturbed by Triton.(a), (c): the lighter points were obtained taking into account theĨ and Ω̃̇ equations in the integration, while in the thick points these contributions were neglected.  (b), (d): time variation of the eccentricity (b) and inclination (d) for two orbits. Initially their paths are similar, but when their semimajor axes enter the 2:1 resonance region, a jump appears in the eccentricity (b), first in the orbit which contains the effect of the precession. The same occurs for the inclination (d), that is, the precession accelerates the onset of the jump of the inclination.
(a)(b)(c)(d)In the integration of the equations of the motion, we used our averaged disturbing functionR as described Yokoyama et al.  [7] which is very efficient for this case. The initial conditions for Proteus are a=4.751, e=0.0005, w=00, Ω=900, I=10Figure 3(a) and I=100Figure 3(c), and λ=900, while for Triton we consider a=14.325, e=0.000016, w=00, Ω=2700, I=1570, and λ=2700. The effect of the precession is very clear: Figure 3(a), shows the variation of Proteus' inclination which remains confined in a very narrow interval when precession equations are considered. In Figure 3(c), in opposition to the above panel, we take Proteus at high inclination (100). Then, although the periods are different, the inclinations of both orbits undergo a large variation. Moreover, there is quite clear a periodic oscillation of about 670 years, which is related to the precession of the node (ΩT). Panels on the right, refer to the variation of the eccentricity Figure 3(b) and inclination Figure 3(d) of Proteus in the case of future 2:1 resonance. As described by Yokoyama [8] and Yokoyama et al. [7], due to the tides, Triton is falling down to the planet, so that, its orbital period will be in a 3:1 or 2:1 orbital resonance with Proteus in the future. For these two panels, instead of the averaged equations, we integrated the exact equations and we applied a slow dissipation on the semimajor axis of Triton in order to simulate the tidal effect. As pointed in  Yokoyama et al. [7] the eccentricity suffers a strong variation and Proteus can be ejected. The effect of the precession is to anticipate the ejection. However, this is not a general rule, since some tests performed in the 3:1 resonance showed different results, that is, the precession can delay the ejection. More tests must be carried on in order to clarify these details.


## 5. Conclusion

In this work, we derived the classical averaged equations for the precession of the Neptune's equator, extending for the case of arbitrary inclination and eccentricity of the disturber. Due to the high retrograde inclination of Triton, these equations must be obtained in closed form for this inclination. If the invariable plane is chosen as the reference plane, the averaged equations of the precession in the case ofeT=0, satisfy an equilibrium solution where Ω̃ and ΩT remain synchronous while Ĩ and IT are constants. The numerical experiments show that, when the precessional equations are considered, the inclination of an inner satellite of Neptune (Proteus) remains confined in a very narrow range, that is, the variation of the inclination with respect to the invariable plane is very small. However for high initial inclination (≥100) of the inner satellite, this confinement is not preserved anymore, that is, even considering the precessional effect, the amplitude of the variation of the inclination is significant. Additional simulations show that the precessional effects are also important in the future 2:1 and 3:1 resonances, since the jump in inclination or eccentricity can be anticipated or delayed.



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