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

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.

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: 𝑚_𝑇/𝑀_𝑁≈2.09×10⁻⁴. (𝑚_𝑇, 𝑀_𝑁 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 (𝑒_𝑇), might have been very high. Therefore, the precessional equations are also obtained for arbitrary 𝑒_𝑇. 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. The 𝐼̃ and Ω equations

Let 𝑋𝑌𝑍 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 Ω= 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 𝑊_𝑇 be the corresponding oblateness potential of Neptune:

Figure 1 

𝑋𝑌𝑍: inertial system, 𝑥𝑦𝑧: moving system fixed on the equator.

𝑊_𝑇=−𝐺𝑀_𝑁𝑚_𝑇𝐽₂𝑅²_𝑁𝑟³_𝑇1(−2+3sin²𝛽2), (2.1) where 𝐺 is the gravitation constant, 𝑀_𝑁 and 𝑚_𝑇 are the masses of Neptune and Triton, respectively. The oblateness constant is 𝐽₂, 𝛽 is the Triton's latitude and 𝑟_𝑇 is the radial distance Neptune-Triton in the 𝑥𝑦𝑧 system. The equatorial radius of Neptune is indicated by 𝑅_𝑁. Then: 𝐓=(𝐿,𝑀,𝑁)=−𝐫_𝐓×∇_𝑇𝑊_𝑇, (2.2) 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:

𝐿=(𝜕𝑊_𝑇𝜕𝑦_𝑇)𝑧_𝑇−(𝜕𝑊_𝑇𝜕𝑧_𝑇)𝑦_𝑇,𝑀=−(𝜕𝑊_𝑇𝜕𝑥_𝑇)𝑧_𝑇+(𝜕𝑊_𝑇𝜕𝑧_𝑇)𝑥_𝑇,𝑁=(𝜕𝑊_𝑇𝜕𝑥_𝑇)𝑦_𝑇−(𝜕𝑊_𝑇𝜕𝑦_𝑇)𝑥_𝑇, (2.3) where 𝑥_𝑇,𝑦_𝑇 and 𝑧_𝑇 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)=𝑧_𝑇/𝑟_𝑇. Then a direct calculation shows that 𝑊_𝑇 depends on Triton coordinates only through 𝑟_𝑇 and 𝑧_𝑇/𝑟_𝑇, therefore:

𝑊_𝑇=𝑊_𝑇(𝑟_𝑇,𝑧_𝑇𝑟_𝑇),𝑟_𝑇=(𝑥²_𝑇+𝑦²_𝑇+𝑧²_𝑇)^1/2,𝜕𝑊_𝑇𝜕𝑥_𝑇=𝜕𝑊_𝑇𝜕𝑟_𝑇𝑥_𝑇𝑟_𝑇,𝜕𝑊_𝑇𝜕𝑦_𝑇=𝜕𝑊_𝑇𝜕𝑟_𝑇𝑦_𝑇𝑟_𝑇. (2.4) Substituting these last two relations in 𝑁 given in (2.3), we get 𝑁=0. Now, let 𝐰=(𝑝_𝑁,𝑞_𝑁,𝑤_𝑁) be the rotation vector of Neptune. According to Euler's dynamical equations:

𝑝̇_𝑁+𝐶−𝐴𝐴𝑤_𝑁𝑞_𝑁=𝐿𝐴,𝑞̇_𝑁−𝐶−𝐴𝐴𝑤_𝑁𝑝_𝑁=𝑀𝐴,𝑤̇_𝑁=𝑁𝐶=0, (2.5) 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:

𝑝_𝑁̃̇̇̃𝑞=sin(𝐼)sin(𝜒)Ω+cos(𝜒)𝐼,_𝑁̃̇̇̃𝑤=sin(𝐼)cos(𝜒)Ω−sin(𝜒)𝐼,_𝑁=cos(𝐼)̃Ω̇+𝜒̇. (2.6) Taking the time derivative of 𝑝_𝑁,

𝑝̇_𝑁̃̇̃̇̃̇̇̈𝜒̇̇̃̈̃=cos(𝐼)𝐼(sin(𝜒)Ω)+sin(𝐼)(𝜒cos(𝜒)Ω+sin(𝜒)Ω)−sin(𝜒)𝐼+cos(𝜒)𝐼. (2.7) Assuming that the polar component of the rotation of the planet is dominant, the following quantities: 𝐼̇Ω̃̇, 𝐼̈̃ e Ω̈ e Ω̇² can be neglected when compared to 𝜒̇ [5, 6]. Then,

𝑝̇_𝑁=sin(𝐼)̃𝜒̇cos(𝜒)Ω̇−sin(𝜒)𝜒̇𝐼̇.̃ (2.8) Substituting 𝑝̇_𝑁 e 𝑞_𝑁 in the first Euler's equation,

𝐿𝐴Ω̇̇̃𝐼̇̃̇+𝐶=(𝜒cos(𝜒)sin(𝐼)−𝜒sin(𝜒))𝐴𝑤_𝑁(Ω̇sin(𝐼)̃cos(𝜒)−sin(𝜒)𝐼̇)̃−𝑤_𝑁(Ω̇sin(𝐼)̃cos(𝜒)−sin(𝜒)𝐼̇)̃. (2.9) Substituting 𝑤_𝑁=cos(𝐼)̃Ω̇+𝜒̇ and neglecting terms factored by Ω̇² and 𝐼̇Ω̃̇, we have

𝐶𝐴𝑤_𝑁̃̇̇̃𝐿(sin(𝐼)cos(𝜒)Ω−sin(𝜒)𝐼)=𝐴. (2.10) Similarly, putting 𝑞̇_𝑁 and 𝑝_𝑁 into the second Euler dynamical equation, we obtain:

−𝐶𝐴𝑤_𝑁̃̇̇̃𝑀(sin(𝐼)sin(𝜒)Ω+cos(𝜒)𝐼)=𝐴. (2.11) Now we can eliminate 𝐿 and 𝑀 in the above equations. Defining 𝑊₀=−𝐺𝑚_𝑇𝑀_𝑁𝐽₂𝑅²_𝑁/2, we have

𝑊_𝑇=𝑊₀(3𝑧²_𝑇𝑟⁵_𝑇−1𝑟³_𝑇),𝜕𝑊_𝑇𝜕𝑦_𝑇=𝑊₀[𝜕𝜕𝑟_𝑇(3𝑧²_𝑇𝑟⁵_𝑇−1𝑟³_𝑇𝑦)]^𝑇𝑟_𝑇,𝜕𝑊_𝑇𝜕𝑧_𝑇=𝑊₀[𝜕𝜕𝑟_𝑇(3𝑧²_𝑇𝑟⁵_𝑇−1𝑟³_𝑇𝑧)]^𝑇𝑟_𝑇+𝑊_𝑧, (2.12) and so,

𝐿=𝜕𝑊_𝑇𝜕𝑦_𝑇𝑧_𝑇−𝜕𝑊_𝑇𝜕𝑧_𝑇𝑦_𝑇=−𝑊_𝑧𝑦_𝑇, (2.13) where 𝑊_𝑧=6𝑊₀𝑧_𝑇/𝑟⁵_𝑇. Similarly, 𝑀=𝑊_𝑧𝑥_𝑇. From (2.10) and (2.11),

Ω̇sin(𝐼)̃=−𝑊_𝑧𝐶𝑤_𝑁(𝑦_𝑇cos(𝜒)+𝑥_𝑇̇̃𝑊sin(𝜒)),𝐼=^𝑧𝐶𝑤_𝑁(−𝑥_𝑇cos(𝜒)+𝑦_𝑇sin(𝜒)). (2.14) If a sequence of three rotations, Ω, 𝐼̃, is 𝜒 are applied on the system 𝑋𝑌𝑍, it will coincide with 𝑥𝑦𝑧. Therefore we can write 𝑥_𝑇, 𝑦_𝑇, and 𝑧_𝑇 in terms of the well-known elliptic orbital elements of Triton.

𝑥_𝑇=𝑟_𝑇cos(𝑓_𝑇+𝑤_𝑇)cosΩ_𝑇−𝑟_𝑇sin(𝑓_𝑇+𝑤_𝑇)sinΩ_𝑇cos𝐼_𝑇,𝑦_𝑇=𝑟_𝑇cos(𝑓_𝑇+𝑤_𝑇)sinΩ_𝑇+𝑟_𝑇sin(𝑓_𝑇+𝑤_𝑇)cosΩ_𝑇cos𝐼_𝑇,𝑧_𝑇=𝑟_𝑇sin(𝑓_𝑇+𝑤_𝑇)sin𝐼_𝑇, (2.15) where 𝑟_𝑇, 𝑓_𝑇, 𝑤_𝑇, Ω_𝑇, 𝑤_𝑇, and 𝐼_𝑇 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 inclination 𝐼_𝑇 or on the eccentricity 𝑒_𝑇, are 𝐼̇̃=𝐹₀4𝑎³_𝑇(1−𝑒²_𝑇)^3/2[sin(𝐼)̃sin(𝐼_𝑇)²sin(2Ω−2Ω_𝑇)+cos(𝐼)̃sin(2𝐼_𝑇)sin(Ω−Ω_𝑇̇𝐹)],Ω=⁰4𝑎³_𝑇(1−𝑒²_𝑇)^3/2[(1−3cos(𝐼_𝑇)²)cos(𝐼)̃+2sin(𝐼_𝑇)cos(𝐼_𝑇)cos(2𝐼)̃cos(Ω−Ω_𝑇)sin(𝐼)̃+sin(𝐼_𝑇)²cos(𝐼)̃cos(2Ω−2Ω_𝑇)], (2.16) where 𝐹₀=3𝐺𝑚_𝑇𝑀_𝑁𝑅²_𝑁𝐽₂𝐶𝑤_𝑁. (2.17) 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. Let 𝐭_𝟏 and 𝐧_𝟏 be the orbital angular momentum of Triton and rotation angular momentum of Netpune, respectively. Then 𝐑=𝐭_𝟏+𝐧_𝟏 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.

(a)

(b)

Figure 2 

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.

The dynamics of the system involving 𝐼̃, Ω, Ω_𝑇, and 𝐼_𝑇 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=𝑚_𝑇𝑛_𝑇𝑎²_𝑇, where 𝑛_𝑇 is the mean motion, and 𝑚_𝑇 is the Triton's mass. On the other hand, 𝑛₁=𝐶𝑤_𝑁, where 𝐶 is the polar moment of inertia of Neptune and 𝑤_𝑁 is its angular velocity of rotation (spin rate). Now, from Figure 2, we have

sin(𝜋−𝐼_𝑇𝑠)=𝑚_𝑇𝑛_𝑇𝑎²_𝑇,sin𝐼̃=𝑠𝐶𝑤_𝑁 (3.1) that is, the inclination 𝐼̃ of the invariable plane is given by:

sin𝐼̃=𝑚_𝑇𝑛_𝑇𝑎²_𝑇sin𝐼_𝑇𝐶𝑤_𝑁. (3.2) 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 𝐼_𝑇 and Ω_𝑇. They are

𝑑𝐼_𝑇1𝑑𝑡=−𝑛_𝑇𝑎²_𝑇sin(𝐼_𝑇)𝜕𝑅_𝐽2𝜕Ω_𝑇,𝑑Ω_𝑇=1𝑑𝑡𝑛_𝑇𝑎²_𝑇sin(𝐼_𝑇)𝜕𝑅_𝐽2𝜕𝐼_𝑇, (3.3) where 𝑅_𝐽2 is the disturbing function due to the oblateness of Neptune which is written with respect to the following invariable plane:

𝑅_𝐽2=𝑆₀[123(1−2sin²̃3𝐼)(1−2sin²𝐼_𝑇)+38sin(2𝐼)̃sin(2𝐼_𝑇)cos(Ω_𝑇−Ω)+38sin²𝐼s̃in²𝐼_𝑇cos(2Ω_𝑇−2Ω)], (3.4) where

𝑆₀=𝐺𝑀_𝑁𝑅²_𝑁𝐽₂[𝑎³_𝑇(1−𝑒²_𝑇)^3/2]. (3.5) For Ω−Ω_𝑇=𝜋, we easily see that

𝐼̇_𝑇=̃̇𝑆𝐼=0,Ω=−²̃̃sin(𝐼)sin(2𝐼+2𝐼_𝑇Ω̇),_𝑇3=−4𝐺𝑀_𝑁𝑅²_𝑁𝐽₂𝑛_𝑇𝑎⁵_𝑇sin𝐼_𝑇sin(2𝐼̃+2𝐼_𝑇), (3.6) where 𝑆₂=3𝐺𝑚_𝑇𝑀_𝑁𝑅²_𝑁𝐽₂4𝐶𝑤_𝑁𝑎³_𝑇. (3.7) Therefore,

Ω̇−Ω̇_𝑇=3𝐺𝑀_𝑁𝑅²_𝑁𝐽₂4𝑎³_𝑇sin(2𝐼̃+2𝐼_𝑇)[1𝑛_𝑇𝑎²_𝑇sin𝐼_𝑇−𝑚_𝑇𝐶𝑤_𝑁sin(𝐼)̃], (3.8) that is, the term inside the brackets is automatically zero provided the reference plane is the invariable plane (see (3.2)). In this case; Ω̇−Ω̇_𝑇=0, 𝐼_𝑇 and 𝐼̃ are constants; Ω and Ω_𝑇 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 Ω−Ω_𝑇=𝜋 defines a stable periodic solution of the system (simple calculations show that such solution does not exist for Ω−Ω_𝑇=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 𝑎_𝑇=14.063𝑅_𝑁, 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)

(b)

(c)

(d)

Figure 3 

(a), (c): the lighter points were obtained taking into account the 𝐼̃ and Ω̇ 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.

In the integration of the equations of the motion, we used our averaged disturbing function 𝑅 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 (Ω_𝑇). 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 of 𝑒_𝑇=0, satisfy an equilibrium solution where Ω and Ω_𝑇 remain synchronous while 𝐼̃ and 𝐼_𝑇 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.

Acknowledgments

The authors thank FAPESP, CNPQ, and FUNDUNESP.