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The self-gravity model of the longitudinal span of the neptune arc fraternite.

1 Introduction

From the very first observations of the Neptune Adams ring arcs [6,12], plus the subsequent observations [2,11], the Adams arcs seemed to change in arc locations and in brightness. More recently, these dynamic natures of the arcs, Fraternite, Egalite (2,1), Liberte, and Courage, have been confirmed beyond any doubt in another ground observation [1]. Measuring from the center of the main arc Fraternite, they extend a total of about 40[degrees] ahead of Fraternite. Occasionally, some arcs flare up and others fade away. Furthermore, the arc configuration appears to be changing in time as well. The leading arc Courage appears to have leaped over to another CER site recently [1]. Although the twin arc Egalite (2,1) is small, it is a very bright arc. According to de Pater et al [1], its relative intensity to Fraternite varied from 17 percent higher in 2002 to seven percent lower in 2003 totaling a 24 percent relative change over a short period of time. The angular span of the twin arc Egalite appeared to be 30 percent larger in 2005 and 1999 publications than in 1989 Voyager 2 results. This widening of Egalite was accompanied by a corresponding narrowing of Fraternite, which indicated a likely exchange of material between the two. As for Liberte, 1999 data showed it was about 3[degrees] ahead of its position in Voyager 2 pictures. For the 2005 results, the 2002 data appeared to show Liberte as a twin arc separated by about 4.5[degrees] with the leading twin at the original Voyager 1989 location, while in 2003 it returned again as one single arc at the Voyager location. With respect to the normally low intensity arc Courage, it flared in intensity to become as bright as Liberte in 1998 indicating a possible exchange of material between the two arcs. Most interestingly, it was observed in the 2005 data that Courage has moved 8o ahead from 31.2[degrees] to 39.7[degrees] [1].

According to the prevailing theories, based on the restricted three-body framework (Neptune-Galatea-arcs) with a conservative disturbing potential, these arcs are radially and lon gitudinally confined by the corotation resonance potential of the inner moon Galatea. In order to account for these arcs, the 84/86 corotation resonance due to the inclination of Galatea (CIR) had been invoked to give a potential site of 4.18[degress] [4]. Later on, because of its eccentricity (CER), the 42/43 resonance was considered giving a resonant site of 8.37[degrees] on the Adams ring arcs [3,5,10]. The arc particles librate about the potential maximum imposed by the corotational resonance satellite Galatea. Dissipated energy of the particle is replenished by the Lindblad resonance. Nevertheless, well established as it is, there are several difficulties. Firstly, with Fraternite centered at the potential maximum spanning approximately 5[degrees] on each side, it crosses two unstable potential points which ought to reduce the angular spread. Secondly, the minor arcs leading ahead of Fraternite are mislocated with the CIR or CER potential maxima. Furthermore, should the arcs were confined by the corotation potential, there ought to be arcs in other locations along the Adams ring distributed randomly instead of clustered near Fraternite.

2 Time-dependent arcs

Recently, there is a model that considers Fraternite as being captured by the CER potential of Galatea. With Fraternite having a finite mass, the minor arcs are clustered at locations along the Adams ring where the time averaged force vanishes under the corotation-Lindblad resonances [13,14]. The finite mass of Fraternite has been suggested by Namouni [9] and Porco [10] to pull on the pericenter precession of Galatea to account for the mismatch between the CER pattern speed and the mean motion of the arcs. The arc locations are determined by the Lindblad resonance reaction of the arc itself. Because the force vanishes only on a time averaged base, as comparing to the stationary CER potential in the rotating frame, the arc material could migrate on a long time scale from one site to another leading to flaring of some arcs and fading of oth ers. This could also generate twin arcs (Egalite, Liberte) and displace Courage from 31.2[degrees] to 39.7[degrees] (resonant jump) [1], as required by observations. Although there are only arcs in the leading positions ahead, arcs in the trailing positions behind could be allowed in this model. According to this Lindblad reaction model, only Fraternite f is confined by the externally imposed CER potential of Galatea x which reads

[[PHI].sub.c] = [Gm.sub.x]/ax 1/2 (2n+[a.sub.x] [partial derivative]/[partial derivative][a.sub.x]) 1/[a.sub.x] [b.sup.(n).sub.1/2 ([alpha]) [e.sub.x] cos [[phi].sub.fx], (1)

where [[??].sub.x] = ([r.sub.x], [[theta].sub.x]) and f = (r, 0) are the position vectors of Galatea x and Fraternite mass distribution, [a.sub.x] and a are the respective semi-major axes, [[theta].sub.x] and [e.sub.x] are the arguments of perihelion and eccentricity of x, [[theta].sub.fx] = ([n.sub.[theta]] - (n - 1)[[theta].sub.x] - [[theta].sub.x]) is the corotation resonance variable, [b.sup.(n).sub.(1/2)] [alpha] is the Laplace coefficient, [alpha] = [a.sub.x]/a < 1, and n = 43. With [a.sub.x] = 61952.60 km, a = 62932.85 km, and [alpha] = 0.98444 [2,11], the CER potential is

[[PHI].sub.c] = [Gm.sub.x]/[a.sub.x] 34 [e.sub.x] cos [[phi].sub.fx]. (2)

To complement this model, we consider the self-gravity of Fraternite, which has a distributed mass, on the CER potential to account for its longitudinal 10[degrees] arc span. We first consider a qualitative spherical self-gravity physical model to grasp the 10[degrees] arc span. We begin with the Gauss law of the gravitational field


[??] = +[nabla][PHI]. (4)

Under a qualitative physical model of arc span, we take a spherical uniform mass distribution of radius [r.sub.0]. Solving for the potential [PHI]([r.sub.*]) inside the sphere with [rho]([??]) = p0 and outside the sphere with [rho](r) = 0 respectively, where r. is measured from the center of Fraternite, and matching the potential and the gravitational field across the boundary, we get

[[PHI].sub.f] = - 1/2 [Gm.sub.f]/[r.sub.0] [([r.sub.*]/[r.sub.0]).sup.2]+ 3/2 [Gm.sub.f]/[r.sub.0], 0 <[r.sub.*] <[r.sub.0], (5)

[[PHI].sub.f] = [Gm.sub.f] /[r.sub.*], [r.sub.0] [r.sub.*] < [infinity]. (6)

This potential shows a normal 1/[r.sub.*] decaying form for [r.sub.0] < [r.sub.*], but a [r.sup.2.sub.*] form for [r.sub.*] < [r.sub.0]. Writing in terms of [a.sub.x] and [m.sub.x], we have for 0 < [r.sub.*] < [r.sub.0], [delta][theta] < [delta][[theta].sub.0],


and for [r.sub.0] < [r.sub.*] < [infinity], [delta][[theta].sub.0] < [delta][theta],

[[PHI].sub.f] = + [Gm.sub.f]/[a.sub.x] [a.sub.x]/[r.sub.*] = + [Gm.sub.x]/[a.sub.x] [m.sub.f] [m.sub.x] 1/[delta][theta], (8)

where [r.sub.*] is now taken on the longitudinal direction along the arc, so that we can write [r.sub.*] = a[delta][theta] and [r.sub.0] = a600 with [delta][theta] as the angular span in radian. Taking [m.sub.f] /[m.sub.x] = [10.sup.-3], [e.sub.x] = [10.sup.-4], and [delta][[theta].sub.0] = 5[degrees] = 0.087 rad, which are within the estimates of the arc parameters [9], we have plotted in Fig. 1 the sinusoidal CER potential in thick line with a minimum around [delta][theta] = 4[degrees] and the self-potential in thin line in units of [Gm.sub.x]/[a.sub.x]. The superposition of the two in thick line is also shown in the same figure. The superimposed potential has a maximum at the center and a minimum around [delta][theta] = 5[degrees]. Although self-gravity is resulted from all the particles of the arc, each individual particle will see the self-potential as an external potential. The particles will girate in stable orbit about the central maximum of the superpositioned CER potential and self-potential.

3 Self-gravity

We now present an elongated ellipsoid model of self-gravity. For an ellipsoidal mass distribution with uniform density [[rho].sub.0] over a volume

[(x/a1).sup.2] + [(y/[a.sub.2]).sup.2] + [(z/[a.sub.3]).sup.2] = 1, (9)

where [a.sub.1] > [a.sub.2] > [a.sub.3], the potential in space for the gravitational field [??]([??]) have been addressed in honorable treatises such as Kellogg [7] and Landau and Lifshitz [8]. Here, we follow the celebrated original work of Kellogg [7] especially in Section 6 of Chapter 7. The potential in space of this homogeneous ellipsoid is given by

[[PHI].sub.f] (x,y, z) = G[[rho].sub.o] [pi] [a.sub.1][a.sub.2][a.sub.3] x



[sigma]([lambda]) = ([a.sup.2.sub.1]+ [lambda]) ([a.sup.2.sub.2] + [lambda]) ([a.sup.2.sub.3] + [lambda]), (11)

and where [lambda] parameterizes a family of ellipsoids. Consider a prolate ellipsoid with [a.sub.1] > [a.sub.2] = [a.sub.3]. This ellipsoid has a circular cross section on the y-z plane and an axis of symmetry in x. The y-z plane of x = 0 is the equatorial plane. In this prolate case, the self-potential inside and outside the ellipsoid is given respectively by [7, Exercise 6, p.196]



f is the distance between the two foci, r is the perpendicular distance to the axis of symmetry, s is the sum of distances from the two foci to the point of interest [??]. The inside potential can be obtained from the outside potential by using s = 2[a.sub.1]. To evaluate the potential on the axis of symmetry, we take r = 0. Denoting [m.sub.f] = [[rho].sub.0] (4[pi]/3)[a.sub.1][a.sub.2][a.sub.3] and considering [a.sub.1] [much greater than] [a.sub.2], we get



for the self-potential inside and outside the ellipsoid respectively. Taking again [m.sub.f] /[m.sub.x] = [10.sup.-3], with [a.sub.x] = 61952.60 km for Galatea, and semi-major axes [a.sub.1] = 5500 km and [a.sub.2] = 55 km, the CER potenial, the self-potential, and the superposition of the two with a minimum around [delta][theta] = 5[derees] are shown in Fig. 2.

With Fraternite 1 x [10.sup.-3] of the mass of Galatea, the selfpotential actually exceeds the CER potential in magnitude, as shown in Fig. 2. Each test mass would be librating around the potential maximum, dominated by the self-gravity of the collective mass distribution. Should Fraternite be elongated further while maintaining the total mass, it would increase the semi-major axis [a.sub.1] of the ellipsoid. This would reduce the amplitude of the self-potential of (14) through the ([a.sub.x]/4[a.sub.1]) factor in the constant term, and weaken the self-potential. The elongation would feed the minor arcs. With this self-gravity model, not just the minor arcs are dynamically changing [1], the main arc Fraternite could be under a dynamical process as well.

4 Conclusions

In order to explain the 10[degrees] arc span of Fraternite, we draw attention to the fact that Fraternite, as an arc, has a significant mass. This mass is a distributed mass, instead of a point-like mass, such that its self-gravity should be taken into considerations to account for its angular span. We have used two models to evaluate the self-potential in the longitudinal direction. First is the tutorial spherical model, as a proof of principle study, with a uniform mass distribution over a sphere of radius [r.sub.0]. Second is the elongated ellipsoidal model for a more realistic evaluation. Using the accepted range of Fraternite parameters, the ellipsoid model shows that the selfpotential of the arc could be the cause of its angular span. For a longer arc, the ellipsoid gets longer and the ratio [a.sub.1] /[a.sub.2] becomes larger. Eventually, for a complete ring, the ellipsoid is infinitely long and the self-potential in the longitudinal direction becomes constant. The effects of self-gravity are felt only in the transverse direction for a planetary ring.

Submitted on March 4, 2013 / Accepted on April 9, 2013


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Instituto de Fisica - Universidade Federal Fluminense, Campus da Praia Vermelha,

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Author:Tsui, K.H.; Souza, J.A.; Navia, C.E.
Publication:Progress in Physics
Article Type:Author abstract
Date:Jul 1, 2013
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