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The configuration of Fraternite–Egalite2–Egalite1 in the Neptune ring arcs system
ARTICLE IN PRESS
Planetary and Space Science 55 (2007) 237–242 www.elsevier.com/locate/pss
The configuration of Fraternite–Egalite2–Egalite1 in the Neptune ring arcs system K.H. Tsui Instituto de Fisica, Universidade Federal Fluminense, Campus da Praia Vermelha, Av. General Milton Tavares de Souza s/n, Gragoata, 24.210-340, Niteroi, Rio de Janeiro, Brazil Received 22 February 2005; received in revised form 20 June 2006; accepted 24 June 2006 Available online 7 August 2006
Abstract By considering the finite mass of Fraternite, although small, it is shown that there are two time-averaged stationary points in its neighborhood due to the reaction of the test body to the fields of Neptune, Galatea, and Fraternite. These two locations measuring 11:7 and 13:8 from the center of Fraternite could correspond to the locations of Egalite 2 and Egalite 1. This model accounts for the 10 span of Fraternite and estimates its mass at mf ¼ 6:4 1016 kg. The eccentricities of Egalite 2 and Egalite 1 are believed to be about e ¼ 5 104 . r 2006 Elsevier Ltd. All rights reserved. Keywords: Planets; Rings
1. Introduction Ever since the discovery of the Neptune arcs (Hubbard et al., 1986), the constant monitoring of their evolution has revealed much of their dynamic properties (Smith et al., 1989; Sicardy et al., 1999; Dumas et al., 1999). Nevertheless, a complete model to account for them is still not available. Currently, there are two approaches that attempt to explain their structures. The first is the two-satellite or multi-satellite approach. This consists of Galatea for radial confinement of the arcs and some hypothetical Lagrange moons for their azimuthal confinement (Lissauer, 1985; Sicardy and Lissauer, 1992). These Lagrange moons could be quite small, say 10–20 km in diameter (Showalter, 1999) with mass from 5 1014 to 5 1015 kg. However, with the detection limit of 6 km objects for Voyager data and continued ground-based observations, such Lagrange moons have not yet been detected. The second approach is the one-satellite model where Galatea and the arcs are in the 42/43 corotation-Lindblad resonance (LR) due to inclinations (CIR) or eccentricities (CER) of Galatea and arcs(Goldreich et al., 1986; Porco, 1991; Foryta and Sicardy, 1996). However, recent measurements have E-mail address: [email protected]. 0032-0633/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2006.06.012
indicated that the arcs are off the corotation resonance location by 0.3 km, and the corotation velocity has a slight mismatch from the arc velocity (Horanyi and Porco, 1993; Sicardy et al., 1999; Dumas et al., 1999; Namouni and Porco, 2002). In order to close the mean motion mismatch, it is proposed to take into consideration the finite mass of the arcs that pulls on the epicycle frequency of Galatea. A combination of eccentricity resonance and inclination resonance, etc. has been examined in Table 1 of their paper (Namouni and Porco, 2002). Another problem independent of the radial offset is the angular spread of each arc and the spacing among them. The two minor arcs Egalite 2 and Egalite 1 trail behind the main arc Fraternite by about 10 and 13 , respectively, measuring from center to center. Fraternite has a span of about 10 , Egalite 2 spans over 3 while Egalite 1 spans about 1 only (Smith et al., 1989; Sicardy et al., 1999; Dumas et al., 1999). Although Fraternite’s spread appears to match approximately a 42/43 corotation-Lindblad site of eccentricity type which has a 8:4 site, the minor arcs and their spacing from the main arc fail to fit into this scheme (Goldreich et al., 1986; Porco, 1991; Foryta and Sicardy, 1996). Neither can this configuration be explained by the multi-satellite approach (Lissauer, 1985; Sicardy and Lissauer, 1992)in a reasonable manner.
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Here, we attempt to address this second question of arc spacings. Our model follows the general approach of Goldreich et al. (1986) and Porco (1991) which consists of the central body S (Neptune), the primary body X (Galatea), a minor body F (Fraternite) which has a finite mass although it is much smaller than that of X, and a test body s (Egalite 2 and Egalite 1). Furthermore, F is in orbit–orbit resonance with X, and s is coorbital with F. The bodies X ; F ; s are assumed to have eccentricities to provide corotation resonance and LR, although they could be of residue level. Nevertheless, we have to remark that this four-body model attempts to explain the mean motion mismatch of the radial offset problem. Inspired by their approach, we will use this four-body model to examine the arc spacing problem, which has been studied so far only under the scope of corotation-LRs. With the intuition of applying this four-body model elsewhere, for example to Saturn system, the analytic model is developed under general designations of S2X 2F 2s. Identification to the Neptune system parameters is only made in the final stage. Before we present our approach, let us first recall that the corotation and the LRs are studied through the disturbing potential which is a function of X and F. The corotation potential of X can be derived by expanding the orbit parameters of X in powers of its eccentricity, while keeping the lowest order of F (circular F), generating the corotation sites (Goldreich et al., 1986; Foryta and Sicardy, 1996). Should inclination be considered, similar corotation potential can also be derived with twice as many sites. By the same token, the Lindblad potential of F is obtained by expanding the orbit parameters of F in powers of its eccentricity, while keeping the lowest order of X (circular X), generating the reactions of F to the corotation resonance of X (Goldreich et al., 1986; Foryta and Sicardy, 1996). Since s is coorbital with F, the test body s is also in corotation-LR with X. Unfortunately, observations of the minor arcs do not confirm this 42/43 corotation-Lindblad scenario based on eccentricity nor its inclination variants. Under the framework of Goldreich et al. (1986) and Porco (1991), these observations of minor arcs (s) could mean that eccentricity of Galatea (X) is sufficiently small that there might be some other as yet unknown factors that would be more important than the corotation potential of X. We believe these factors could come from Fraternite (F) which has a small mass. Since we are looking for new dynamic features in this resonant four-body system where inclination may not be essential, we take a simple coplanar model for more transparency and easier handling. With this coplanar model, the LR of F reacts to the corotation eccentricity resonance of X. Inclination resonance is absent, and is not needed for the development of our model. Our goal is not to fine tune the current model including orbit inclination effects, but to uncover new dynamics. Instead of working with the disturbing potential, we consider the equations of motion of s directly and expand them in powers of eccentricity for its reactions to X and especially to F. By taking a long time-average, we will
show that the equations of motion have two time-averaged stationary points in the vicinity of F, that could correspond to the positions of the minor arcs. 2. Lindblad resonance We designate M; mx ; mf as the masses of the central body S, the primary body X, and the minor body F, respectively. Also ~ rx ¼ ðrx ; yx Þ, ~ rf ¼ ðrf ; yf Þ, and ~ rs ¼ ðrs ; ys Þ are the position vectors of X, F, and s measured from S with respect ~¼~ to a fixed reference axis in space. Furthermore, R rs ~ rx 0 ~ and R ¼ ~ rs ~ rf are the position vectors of s measured from X and F, respectively as in Fig. 1. We consider all the bodies moving on the same plane by neglecting the orbit inclinations. With respect to a coordinate system centered at the central body S, the equations of motion of s are d2 rs GM 2 ¼ þ r o s s r2s dt2 Gmx Gmx ½r r cosðDy Þ þ r cosðDy Þ s x sx x sx r3x R3 Gmf 0 3 ½rs rf cosðDysf Þ, ð1aÞ R 1 d2 Gmx Gmx rs o s ¼ rx sinðDysx Þ rs dt r3x R3 Gmf 0 3 rf sin Dysf . ð1bÞ R Here, Dysx;sf ¼ ðys yx;f Þ, whereas os ¼ dys =dt is the angular velocity of s about the central body S with respect to a reference axis. We have used y to denote the longitude and o to denote the angular velocity aided by a subscribe to distinguish each body. We note that should we expand the orbit parameters of X in these equations of s, we would get the effects of corotation potential. Since observations do not reveal structures of minor arcs compatible to such corotation potential, we believe that the eccentricity of X is sufficiently small that other new contributions override the corotation potential. A possible source of new contributions is the minor body F which has a finite mass and is in resonance with X. Under the believe that the eccentricity of X is sufficiently small, we will simplify our analysis by neglecting it all together, and seek to identify the new contributions. Denoting L2 ¼ GMa with a as the semi-major axis of s, 2 Lx ¼ GMrx and L2f ¼ GMrf , we write the coefficients in terms of angular momentum to read ( d2 rs GM mx GM 2 2 ¼ þ r o s s r2s L dt2 M a mx GM 2 3 ½rs rx cosðDysx Þ þ M Lx R mf GM 2 1 cosðDysx Þ rx M Lf rf 0 3 ½rs rf cosðDysf Þ, ð2aÞ R
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) mx GM 2 a mx GM 2 1 L r2x M R3 M Lx mf GM 2 rx sinðDysx Þ M Lf rf 0 3 rf sinðDysf Þ. ð2bÞ R In these equations, there are even functions f ðysx Þ like f 1 ¼ 1=R3 and f 2 ¼ cosðDysx Þ=R3 due to the orbital resonance between s and X. These terms can be expanded in cosine series which reads X f ¼ ðb0 þ f n Þ ¼ b0 þ 2 bn cosðnDysx Þ. 1 d2 r os ¼ rs dt s
(
In terms of f n , these equations become d2 rs GM mx GM 2 2 ¼ þ rs os 2 ars ðb0 þ f n Þ1 rs L dt2 M mx GM 2 þ arx ðb0 þ f n Þ2 L M mx GM 2 1 cosðDysx Þ rx M Lx mf GM 2 rf ½rs rf cosðDysf Þ, M Lf R0 3 1 d2 mx GM 2 r os ¼ arx ðb0 þ f n Þ1 sinðDysx Þ rs dt s L M mx GM 2 1 þ sinðDysx Þ rx M Lx mf GM 2 rf rf sinðDysf Þ. M Lf R0 3
1 d2 mx GM 2 rs os ¼ arx 2ebn2 sinðFsxL Þ rs dt L M mf GM 2 a2 sinðDysf Þ, L M R0 3
ð4aÞ
ð4bÞ
where FsxL ¼ ½ðn þ 1Þys nyx fs is the LR variable of s with s and F outside of X. The coefficients b01 , bn1 , b02 , and bn2 are defined through the Laplace coefficients as b01 ¼ ð1=2Þð1=a3 Þbð0Þ bn1 ¼ ð1=2Þð1=a3 ÞbðnÞ b02 ¼ ð1=2Þ 3=2 ðaÞ, 3=2 ðaÞ, h i ðnþ1Þ ðn1Þ 3 ð1=a3 Þbð1Þ ðaÞ, b ¼ ð1=4Þð1=a Þ b ðaÞ þ b ðaÞ , where n2 3=2 3=2 3=2 ð3aÞ
ð3bÞ
We expand the parameters of s on the right sides of these in powers of its eccentricity to calculate the reaction of s to X. Since s is coorbital with F, we also take rf ¼ a. As for the parameters of other bodies, there is no need to expand them here, and will be referred to their circular orbit values for simplicity. Taking a time average over a long interval compared to the orbital period of s, we have d2 rs GM 2 1 2 mx GM 2 2 2e ¼ þ a b01 L a L dt2 M 2 mx GM þ a2 e cosðys fs Þf n1 L M mx GM 2 þ arx b02 L M mf GM 2 a2 ½1 cosðDysf Þ, L M R0 3 1 d2 mx GM 2 r os ¼ arx f n1 cosðDysx Þdy rs dt s L M 2 2 mf GM a sinðDysf Þ. L M R0 3
These equations can be written as d2 rs GM 2 1 2 mx GM 2 2 2e ¼ a b01 L a L dt2 M mx GM 2 2 þ a ebn1 cosðFsxL Þ L M mx GM 2 þ arx b02 L M mf GM 2 a2 ½1 cosðDysf Þ, L M R0 3
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a ¼ rx =ao1. Rewriting FsxL ¼ ½FfxL þ ðn þ 1ÞDysf ðfs ff Þ in terms of the FX corotation-LR variable FfxL , the above equations become i d2 rs GM 2 1 mx 3 h rx 2 2e a b ¼ b 01 02 L a dt2 M a mx 3 a ebn1 cos½FfxL þ ðn þ 1ÞDysf ðfs ff Þ þ M mf a 3 ½1 cosðDy Þ , ð5aÞ sf M R0 3 1 d2 GM 2 1 mx 2 rs o s ¼ a rx 2ebn2 rs dt L a M sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ m f a3 sinðDysf Þ . þ M R0 3
ð5bÞ
By setting the right sides to zero, the angular positions of s where the time-averaged force acting on it vanishes are given by mx 3 a ebn1 cos½FfxL þ ðn þ 1ÞDysf ðfs ff Þ M i m a3 mx 3 h rx f a b01 b02 ½1 cosðDysf Þ ¼ 0, M a M R0 3
2e2 þ
ð6aÞ m f a3 1 3 2 0 mx R a rx 2bn2 sinðDysf Þ . sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ
e¼
ð6bÞ
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These are the general conditions for vanishing timeaveraged force for two coorbital objects s and F that are in corotation-LR with an interior X. 3. Fraternite–Egalite2–Egalite1 Let us now apply these conditions to the Neptune ring arcs. With the Neptune system parameters, the last term, the fourth term, on the left side of the first equation can be neglected unless R0 =a is exactly zero which amounts to a collision. To justify this assertion, let us compare the fourth term to the third term. Taking Dysf ¼ 10 as the center-to-center angular distance between Fraternite and Egalite, and anticipating 2a3 ½b01 ab02 ¼ 42:9 in the numerical calculations after Eqs. (7), these two terms have the ratio ð2=43Þðmf =mx Þ ðp=18Þ3 which is very much less than unity. Also, considering the mass ratio mx =M of the Neptune system much less than the expected eccentricity such that the linear eccentricity term can be neglected. We, therefore, keep only the quadratic term in eccentricity in the first equation which is balanced by the third term. Substituting the eccentricity of Eq. (6b) to Eq.(6a), the stationary locations are given by 0 3 sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ R a sinðDysf Þ 1=2 1 mf a 1 1 M 1 1 ¼ 2 . ð7aÞ 2 mx rx a3 bn2 mx a3 ½b01 ðrx =aÞb02 For ðR0 =aÞ51, we can write ðR0 =aÞ ¼ sinðDysf Þ, or with better precision, we choose to write ðR0 =aÞ ¼ 2 sinðDysf =2Þ. In this case, Eq. (7a) reads 3 sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ 1 2 sin Dysf 2 sinðDysf Þ 1 1 ¼ 4 tan Dysf sin Dysf 2 2 sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ 1=2 1 mf a 1 1 M 1 1 2 . ¼ 2 mx rx a3 bn2 mx a3 ½b01 ðrx =aÞb02
ð7bÞ
According to observations (Sicardy et al., 1999; Dumas et al., 1999), we take rx ¼ 61; 952:60 km and a ¼ 62; 932:85 km. This renders a ¼ rx =a ¼ 0:98444, and the ð1Þ 4 Laplace coefficients are bð0Þ 3=2 ¼ 0:26487 10 , b3=2 ¼ ð41Þ ð42Þ 4 4 0:26470 10 , b3=2 ¼ 0:20168 10 , b3=2 ¼ 0:19975 104 , 4 bð43Þ 3=2 ¼ 0:19782 10 . With these coefficients, we get 3 2a b01 ¼ 0:26487 104 , 2a3 b02 ¼ 0:26470 104 , 4a3 bn2 ¼ 0:39950 104 , and 2a3 ½b01 ab02 ¼ 42:9. The right side of the above equation can be calculated to give 4 tan 12 Dysf sin 12 Dysf sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ mf M 1=2 ¼ 1:5528 104 mx mx m f . ð8Þ ¼ 0:5490 108 M
The second equality is reached by taking M ¼ 1 1026 kg for the central body Neptune, and mx ¼ 2 1018 kg for the primary body Galatea, so that M=mx ¼ ð1=2Þ 108 . Considering the center of Fraternite to be at the maximum of the corotation site, we can take FfxL ¼ p=2. Furthermore, with the intuition that the minor arcs are being tugged along by the main arc, we take fs ¼ ff . In other words, the minor arcs are strongly coupled to the dynamics of the main arc. The positions where the timeaveraged force vanishes are given by 4 tan 12Dysf sin 12Dysf cos½ðn þ 1ÞDysf mf . ð9Þ ¼ 0:5490 108 M The left side of Eq. (9) as the y coordinate is plotted in Fig. 2 as a function of Dysf which shows two minima. To understand the first minimum, we note that the cosine function in Eq. (9) starts out with a central maximum at ðn þ 1ÞDysf ¼ 0 and reaches its first minimum at ðn þ 1ÞDysf ¼ p on each side forming a complete site of 8:4 with n ¼ 42. However, due to the other factors on the left side, the central maximum of cosine is replaced by a broad null. The numerical solution in Fig. 2 shows that the first minimum is slightly shifted outwards to 4:85 on each side spanning an angular width of 9:7 . We believe this corresponds to the observed extension of Fraternite. As for the second minimum, it is located at 12:8 from the center. The roots of Eq. (9) are given by the intercepts of the left side with the right side. There are either two intersects around this second minimum or none of them. Naturally, there are more intersects as Dysf is increased. Since mf is small, we consider only short-range gravitational interactions, and take only the first two intersects. We take the mass ratio mf =M ¼ 6:4 1010 or mf =mx ¼ 0:032 which gives mf ¼ 6:4 1016 kg for Fraternite, so that the right side corresponds to 3:5 102 . This gives two intercepts at 11:7 and 13:8 , which are approximately where the minor arcs are observed. There is a slight difference of 1 between our calculated and the observed positions. Also the two intercepts are separated by 2 in our model as compared to 3 in observations. These small discrepancies are probably caused by our representation of Fraternite’s mass as a point mass at its center. In reality, Fraternite’s mass is distributed over its own extension. Interestingly, this model gives an upper limit of Fraternite’s mass. If we take mf =M ¼ 1:0 109 or mf =mx ¼ 0:05,so that the right side corresponds to 5:5 102 , there will be no more intersects. The stationary points would disappear should Fraternite be more massive. Besides giving a mass estimate of Fraternite and the positions of Egalite 2 and Egalite 1, we can also estimate the eccentricity of the two minor arcs, as a result of LR of s in response to the corotation resonance of Galatea, by using Eq. (5) or Eq. (6). Calculations from both equations give e ¼ 5 104 approximately. This eccentricity is
ARTICLE IN PRESS K.H. Tsui / Planetary and Space Science 55 (2007) 237–242
generated by the LR of s. We now recall that we have taken the assumption that the eccentricity of Galatea is sufficiently small to neglect it all together in analyzing the equations of motion of s for new contributions from the finite mass minor body Fraternite. We have uncovered two time-averaged stationary points and have predicted an eccentricity e ¼ 5 104 for the minor arcs. This gives our → R’
s
F → R
X
241
model a constraint that ex 55 104 ,
(10)
which has to be checked for its veracity. We recall that Namouni and Porco (2002) have investigated the radial offset problem of the Neptune arcs as mentioned in the Introduction. In order to close the mean motion mismatch, they have proposed a finite mass to Fraternite to pull on the epicycle frequency of Galatea. The results in Fig. 2 of their paper indicate a range from mf =mx ¼ 0:128 for ex ¼ 104 to mf =mx ¼ 0:002 for ex ¼ 106 . The estimate derived from our model to account for arc spacings gives mf =mx ¼ 0:032 which is within their cited range. Furthermore, with mf =mx ¼ 0:032, we would have ex ¼ 105 for Galatea according to their Fig. 2. This eccentricity estimate happens to be consistent to Eq. (10). 4. Conclusions
rf rs rx
θs θf θx S Fig. 1. This shows the configuration of the coplanar S2X 2F 2s fourbody system where S is the central body, X is the primary body, F is a minor body, and s is a test body. The longitudes are denoted by y with appropriate subscripts.
To conclude, we have studied a resonant coplanar fourbody system where the center of mass is set by the central and primary bodies. A minor body with finite mass is in corotation-LR with the primary body, and a test body is coorbital with the minor body at close distances. Through the equations of motion of the test body, we have shown that there are stationary locations where the time-averaged force vanishes. These points are located behind the minor body as well as in front of it. They differ from the Lagrangian points of a restricted three-body system in that the averaged force is zero, and that they are self-generated dynamically by the test body’s reaction to the resonant field between the minor and primary bodies. We have applied
0.06
0.04
y
0.02
0.00
−0.02
−0.04
−0.06 0
2
4
6
8 Δθsf
10
12
14
16
Fig. 2. The left side of Eq. (9), denoted by the y label, is plotted as a function of Dysf in degree. The right side of Eq. (9) is a constant and is represented by a horizontal line. The intercepts give the roots of Eq. (9) that define the locations where the time-averaged force vanishes.
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these points to account for the arcs’ configuration in the Neptune–Galatea system. Using this model, it is able to explain the 10 extension of Fraternite. By requiring Fraternite’s mass be 6:4 1016 kg, two time-averaged stationary points are located at 11:7 and 13:8 from the center of Fraternite which seem to be compatible with the observed positions of Egalite 1 and Egalite 2. These locations would disappear should Fraternite be more massive. This model also estimates the eccentricity of Egalite 2 and Egalite 1 at 5 104 . Our model requires that the eccentricity of Galatea be much smaller than this value. Acknowledgments This work was supported by the Conselho Nacional de Desenvolvimentos Cientifico e Tecnologico (CNPq, The Brazilian National Council of Scientific and Technologic Developments) and the Fundacao de Amparo a Pesquisa do Estado do Rio de Janeiro (FAPERJ, The Research Fostering Foundation of the State of Rio de Janeiro).
References Dumas, C., et al., 1999. Stability of Neptune’s ring arcs in question. Nature 400, 733–735. Foryta, D.W., Sicardy, B., 1996. The dynamics of the Neptunian Adams ring’s arcs. Icarus 123, 129–167. Goldreich, P., Tremaine, S., Borderies, N., 1986. Towards a theory for Neptune’s arc rings. Astron. J. 92, 490–494. Horanyi, M., Porco, C.C., 1993. Where exactly are the arcs of Neptune? Icarus 106, 525–535. Hubbard, W.B., et al., 1986. Occultation detection of a neptunian ring like arc. Nature 319, 636–640. Lissauer, J.J., 1985. Shepherding model for Neptune’s arc ring. Nature 318, 544–545. Namouni, F., Porco, C.C., 2002. The confinement of Neptune’s ring arcs by the moon Galatea. Nature 417, 45–47. Porco, C.C., 1991. An explanation for Neptune’s ring arcs. Science 253, 995–1001. Showalter, M.R., 1999. Neptune’s misbehaving rings. Nature 400, 709–710. Sicardy, B., Lissauer, J.J., 1992. Dynamic model of the arcs in Neptune’s 63 K ring. Adv. Space Res. 12, 81–95. Sicardy, B., et al., 1999. Images of Neptune’s ring arcs obtained by a ground based telescope. Nature 400, 731–732. Smith, B.A., et al., 1989. Voyager 2 at Neptune: imaging science results. Science 246, 1422–1449.
Planetary and Space Science 55 (2007) 237–242 www.elsevier.com/locate/pss
The configuration of Fraternite–Egalite2–Egalite1 in the Neptune ring arcs system K.H. Tsui Instituto de Fisica, Universidade Federal Fluminense, Campus da Praia Vermelha, Av. General Milton Tavares de Souza s/n, Gragoata, 24.210-340, Niteroi, Rio de Janeiro, Brazil Received 22 February 2005; received in revised form 20 June 2006; accepted 24 June 2006 Available online 7 August 2006
Abstract By considering the finite mass of Fraternite, although small, it is shown that there are two time-averaged stationary points in its neighborhood due to the reaction of the test body to the fields of Neptune, Galatea, and Fraternite. These two locations measuring 11:7 and 13:8 from the center of Fraternite could correspond to the locations of Egalite 2 and Egalite 1. This model accounts for the 10 span of Fraternite and estimates its mass at mf ¼ 6:4 1016 kg. The eccentricities of Egalite 2 and Egalite 1 are believed to be about e ¼ 5 104 . r 2006 Elsevier Ltd. All rights reserved. Keywords: Planets; Rings
1. Introduction Ever since the discovery of the Neptune arcs (Hubbard et al., 1986), the constant monitoring of their evolution has revealed much of their dynamic properties (Smith et al., 1989; Sicardy et al., 1999; Dumas et al., 1999). Nevertheless, a complete model to account for them is still not available. Currently, there are two approaches that attempt to explain their structures. The first is the two-satellite or multi-satellite approach. This consists of Galatea for radial confinement of the arcs and some hypothetical Lagrange moons for their azimuthal confinement (Lissauer, 1985; Sicardy and Lissauer, 1992). These Lagrange moons could be quite small, say 10–20 km in diameter (Showalter, 1999) with mass from 5 1014 to 5 1015 kg. However, with the detection limit of 6 km objects for Voyager data and continued ground-based observations, such Lagrange moons have not yet been detected. The second approach is the one-satellite model where Galatea and the arcs are in the 42/43 corotation-Lindblad resonance (LR) due to inclinations (CIR) or eccentricities (CER) of Galatea and arcs(Goldreich et al., 1986; Porco, 1991; Foryta and Sicardy, 1996). However, recent measurements have E-mail address: [email protected]. 0032-0633/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2006.06.012
indicated that the arcs are off the corotation resonance location by 0.3 km, and the corotation velocity has a slight mismatch from the arc velocity (Horanyi and Porco, 1993; Sicardy et al., 1999; Dumas et al., 1999; Namouni and Porco, 2002). In order to close the mean motion mismatch, it is proposed to take into consideration the finite mass of the arcs that pulls on the epicycle frequency of Galatea. A combination of eccentricity resonance and inclination resonance, etc. has been examined in Table 1 of their paper (Namouni and Porco, 2002). Another problem independent of the radial offset is the angular spread of each arc and the spacing among them. The two minor arcs Egalite 2 and Egalite 1 trail behind the main arc Fraternite by about 10 and 13 , respectively, measuring from center to center. Fraternite has a span of about 10 , Egalite 2 spans over 3 while Egalite 1 spans about 1 only (Smith et al., 1989; Sicardy et al., 1999; Dumas et al., 1999). Although Fraternite’s spread appears to match approximately a 42/43 corotation-Lindblad site of eccentricity type which has a 8:4 site, the minor arcs and their spacing from the main arc fail to fit into this scheme (Goldreich et al., 1986; Porco, 1991; Foryta and Sicardy, 1996). Neither can this configuration be explained by the multi-satellite approach (Lissauer, 1985; Sicardy and Lissauer, 1992)in a reasonable manner.
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Here, we attempt to address this second question of arc spacings. Our model follows the general approach of Goldreich et al. (1986) and Porco (1991) which consists of the central body S (Neptune), the primary body X (Galatea), a minor body F (Fraternite) which has a finite mass although it is much smaller than that of X, and a test body s (Egalite 2 and Egalite 1). Furthermore, F is in orbit–orbit resonance with X, and s is coorbital with F. The bodies X ; F ; s are assumed to have eccentricities to provide corotation resonance and LR, although they could be of residue level. Nevertheless, we have to remark that this four-body model attempts to explain the mean motion mismatch of the radial offset problem. Inspired by their approach, we will use this four-body model to examine the arc spacing problem, which has been studied so far only under the scope of corotation-LRs. With the intuition of applying this four-body model elsewhere, for example to Saturn system, the analytic model is developed under general designations of S2X 2F 2s. Identification to the Neptune system parameters is only made in the final stage. Before we present our approach, let us first recall that the corotation and the LRs are studied through the disturbing potential which is a function of X and F. The corotation potential of X can be derived by expanding the orbit parameters of X in powers of its eccentricity, while keeping the lowest order of F (circular F), generating the corotation sites (Goldreich et al., 1986; Foryta and Sicardy, 1996). Should inclination be considered, similar corotation potential can also be derived with twice as many sites. By the same token, the Lindblad potential of F is obtained by expanding the orbit parameters of F in powers of its eccentricity, while keeping the lowest order of X (circular X), generating the reactions of F to the corotation resonance of X (Goldreich et al., 1986; Foryta and Sicardy, 1996). Since s is coorbital with F, the test body s is also in corotation-LR with X. Unfortunately, observations of the minor arcs do not confirm this 42/43 corotation-Lindblad scenario based on eccentricity nor its inclination variants. Under the framework of Goldreich et al. (1986) and Porco (1991), these observations of minor arcs (s) could mean that eccentricity of Galatea (X) is sufficiently small that there might be some other as yet unknown factors that would be more important than the corotation potential of X. We believe these factors could come from Fraternite (F) which has a small mass. Since we are looking for new dynamic features in this resonant four-body system where inclination may not be essential, we take a simple coplanar model for more transparency and easier handling. With this coplanar model, the LR of F reacts to the corotation eccentricity resonance of X. Inclination resonance is absent, and is not needed for the development of our model. Our goal is not to fine tune the current model including orbit inclination effects, but to uncover new dynamics. Instead of working with the disturbing potential, we consider the equations of motion of s directly and expand them in powers of eccentricity for its reactions to X and especially to F. By taking a long time-average, we will
show that the equations of motion have two time-averaged stationary points in the vicinity of F, that could correspond to the positions of the minor arcs. 2. Lindblad resonance We designate M; mx ; mf as the masses of the central body S, the primary body X, and the minor body F, respectively. Also ~ rx ¼ ðrx ; yx Þ, ~ rf ¼ ðrf ; yf Þ, and ~ rs ¼ ðrs ; ys Þ are the position vectors of X, F, and s measured from S with respect ~¼~ to a fixed reference axis in space. Furthermore, R rs ~ rx 0 ~ and R ¼ ~ rs ~ rf are the position vectors of s measured from X and F, respectively as in Fig. 1. We consider all the bodies moving on the same plane by neglecting the orbit inclinations. With respect to a coordinate system centered at the central body S, the equations of motion of s are d2 rs GM 2 ¼ þ r o s s r2s dt2 Gmx Gmx ½r r cosðDy Þ þ r cosðDy Þ s x sx x sx r3x R3 Gmf 0 3 ½rs rf cosðDysf Þ, ð1aÞ R 1 d2 Gmx Gmx rs o s ¼ rx sinðDysx Þ rs dt r3x R3 Gmf 0 3 rf sin Dysf . ð1bÞ R Here, Dysx;sf ¼ ðys yx;f Þ, whereas os ¼ dys =dt is the angular velocity of s about the central body S with respect to a reference axis. We have used y to denote the longitude and o to denote the angular velocity aided by a subscribe to distinguish each body. We note that should we expand the orbit parameters of X in these equations of s, we would get the effects of corotation potential. Since observations do not reveal structures of minor arcs compatible to such corotation potential, we believe that the eccentricity of X is sufficiently small that other new contributions override the corotation potential. A possible source of new contributions is the minor body F which has a finite mass and is in resonance with X. Under the believe that the eccentricity of X is sufficiently small, we will simplify our analysis by neglecting it all together, and seek to identify the new contributions. Denoting L2 ¼ GMa with a as the semi-major axis of s, 2 Lx ¼ GMrx and L2f ¼ GMrf , we write the coefficients in terms of angular momentum to read ( d2 rs GM mx GM 2 2 ¼ þ r o s s r2s L dt2 M a mx GM 2 3 ½rs rx cosðDysx Þ þ M Lx R mf GM 2 1 cosðDysx Þ rx M Lf rf 0 3 ½rs rf cosðDysf Þ, ð2aÞ R
ARTICLE IN PRESS K.H. Tsui / Planetary and Space Science 55 (2007) 237–242
) mx GM 2 a mx GM 2 1 L r2x M R3 M Lx mf GM 2 rx sinðDysx Þ M Lf rf 0 3 rf sinðDysf Þ. ð2bÞ R In these equations, there are even functions f ðysx Þ like f 1 ¼ 1=R3 and f 2 ¼ cosðDysx Þ=R3 due to the orbital resonance between s and X. These terms can be expanded in cosine series which reads X f ¼ ðb0 þ f n Þ ¼ b0 þ 2 bn cosðnDysx Þ. 1 d2 r os ¼ rs dt s
(
In terms of f n , these equations become d2 rs GM mx GM 2 2 ¼ þ rs os 2 ars ðb0 þ f n Þ1 rs L dt2 M mx GM 2 þ arx ðb0 þ f n Þ2 L M mx GM 2 1 cosðDysx Þ rx M Lx mf GM 2 rf ½rs rf cosðDysf Þ, M Lf R0 3 1 d2 mx GM 2 r os ¼ arx ðb0 þ f n Þ1 sinðDysx Þ rs dt s L M mx GM 2 1 þ sinðDysx Þ rx M Lx mf GM 2 rf rf sinðDysf Þ. M Lf R0 3
1 d2 mx GM 2 rs os ¼ arx 2ebn2 sinðFsxL Þ rs dt L M mf GM 2 a2 sinðDysf Þ, L M R0 3
ð4aÞ
ð4bÞ
where FsxL ¼ ½ðn þ 1Þys nyx fs is the LR variable of s with s and F outside of X. The coefficients b01 , bn1 , b02 , and bn2 are defined through the Laplace coefficients as b01 ¼ ð1=2Þð1=a3 Þbð0Þ bn1 ¼ ð1=2Þð1=a3 ÞbðnÞ b02 ¼ ð1=2Þ 3=2 ðaÞ, 3=2 ðaÞ, h i ðnþ1Þ ðn1Þ 3 ð1=a3 Þbð1Þ ðaÞ, b ¼ ð1=4Þð1=a Þ b ðaÞ þ b ðaÞ , where n2 3=2 3=2 3=2 ð3aÞ
ð3bÞ
We expand the parameters of s on the right sides of these in powers of its eccentricity to calculate the reaction of s to X. Since s is coorbital with F, we also take rf ¼ a. As for the parameters of other bodies, there is no need to expand them here, and will be referred to their circular orbit values for simplicity. Taking a time average over a long interval compared to the orbital period of s, we have d2 rs GM 2 1 2 mx GM 2 2 2e ¼ þ a b01 L a L dt2 M 2 mx GM þ a2 e cosðys fs Þf n1 L M mx GM 2 þ arx b02 L M mf GM 2 a2 ½1 cosðDysf Þ, L M R0 3 1 d2 mx GM 2 r os ¼ arx f n1 cosðDysx Þdy rs dt s L M 2 2 mf GM a sinðDysf Þ. L M R0 3
These equations can be written as d2 rs GM 2 1 2 mx GM 2 2 2e ¼ a b01 L a L dt2 M mx GM 2 2 þ a ebn1 cosðFsxL Þ L M mx GM 2 þ arx b02 L M mf GM 2 a2 ½1 cosðDysf Þ, L M R0 3
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a ¼ rx =ao1. Rewriting FsxL ¼ ½FfxL þ ðn þ 1ÞDysf ðfs ff Þ in terms of the FX corotation-LR variable FfxL , the above equations become i d2 rs GM 2 1 mx 3 h rx 2 2e a b ¼ b 01 02 L a dt2 M a mx 3 a ebn1 cos½FfxL þ ðn þ 1ÞDysf ðfs ff Þ þ M mf a 3 ½1 cosðDy Þ , ð5aÞ sf M R0 3 1 d2 GM 2 1 mx 2 rs o s ¼ a rx 2ebn2 rs dt L a M sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ m f a3 sinðDysf Þ . þ M R0 3
ð5bÞ
By setting the right sides to zero, the angular positions of s where the time-averaged force acting on it vanishes are given by mx 3 a ebn1 cos½FfxL þ ðn þ 1ÞDysf ðfs ff Þ M i m a3 mx 3 h rx f a b01 b02 ½1 cosðDysf Þ ¼ 0, M a M R0 3
2e2 þ
ð6aÞ m f a3 1 3 2 0 mx R a rx 2bn2 sinðDysf Þ . sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ
e¼
ð6bÞ
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These are the general conditions for vanishing timeaveraged force for two coorbital objects s and F that are in corotation-LR with an interior X. 3. Fraternite–Egalite2–Egalite1 Let us now apply these conditions to the Neptune ring arcs. With the Neptune system parameters, the last term, the fourth term, on the left side of the first equation can be neglected unless R0 =a is exactly zero which amounts to a collision. To justify this assertion, let us compare the fourth term to the third term. Taking Dysf ¼ 10 as the center-to-center angular distance between Fraternite and Egalite, and anticipating 2a3 ½b01 ab02 ¼ 42:9 in the numerical calculations after Eqs. (7), these two terms have the ratio ð2=43Þðmf =mx Þ ðp=18Þ3 which is very much less than unity. Also, considering the mass ratio mx =M of the Neptune system much less than the expected eccentricity such that the linear eccentricity term can be neglected. We, therefore, keep only the quadratic term in eccentricity in the first equation which is balanced by the third term. Substituting the eccentricity of Eq. (6b) to Eq.(6a), the stationary locations are given by 0 3 sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ R a sinðDysf Þ 1=2 1 mf a 1 1 M 1 1 ¼ 2 . ð7aÞ 2 mx rx a3 bn2 mx a3 ½b01 ðrx =aÞb02 For ðR0 =aÞ51, we can write ðR0 =aÞ ¼ sinðDysf Þ, or with better precision, we choose to write ðR0 =aÞ ¼ 2 sinðDysf =2Þ. In this case, Eq. (7a) reads 3 sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ 1 2 sin Dysf 2 sinðDysf Þ 1 1 ¼ 4 tan Dysf sin Dysf 2 2 sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ 1=2 1 mf a 1 1 M 1 1 2 . ¼ 2 mx rx a3 bn2 mx a3 ½b01 ðrx =aÞb02
ð7bÞ
According to observations (Sicardy et al., 1999; Dumas et al., 1999), we take rx ¼ 61; 952:60 km and a ¼ 62; 932:85 km. This renders a ¼ rx =a ¼ 0:98444, and the ð1Þ 4 Laplace coefficients are bð0Þ 3=2 ¼ 0:26487 10 , b3=2 ¼ ð41Þ ð42Þ 4 4 0:26470 10 , b3=2 ¼ 0:20168 10 , b3=2 ¼ 0:19975 104 , 4 bð43Þ 3=2 ¼ 0:19782 10 . With these coefficients, we get 3 2a b01 ¼ 0:26487 104 , 2a3 b02 ¼ 0:26470 104 , 4a3 bn2 ¼ 0:39950 104 , and 2a3 ½b01 ab02 ¼ 42:9. The right side of the above equation can be calculated to give 4 tan 12 Dysf sin 12 Dysf sin½FfxL þ ðn þ 1ÞDysf ðfs ff Þ mf M 1=2 ¼ 1:5528 104 mx mx m f . ð8Þ ¼ 0:5490 108 M
The second equality is reached by taking M ¼ 1 1026 kg for the central body Neptune, and mx ¼ 2 1018 kg for the primary body Galatea, so that M=mx ¼ ð1=2Þ 108 . Considering the center of Fraternite to be at the maximum of the corotation site, we can take FfxL ¼ p=2. Furthermore, with the intuition that the minor arcs are being tugged along by the main arc, we take fs ¼ ff . In other words, the minor arcs are strongly coupled to the dynamics of the main arc. The positions where the timeaveraged force vanishes are given by 4 tan 12Dysf sin 12Dysf cos½ðn þ 1ÞDysf mf . ð9Þ ¼ 0:5490 108 M The left side of Eq. (9) as the y coordinate is plotted in Fig. 2 as a function of Dysf which shows two minima. To understand the first minimum, we note that the cosine function in Eq. (9) starts out with a central maximum at ðn þ 1ÞDysf ¼ 0 and reaches its first minimum at ðn þ 1ÞDysf ¼ p on each side forming a complete site of 8:4 with n ¼ 42. However, due to the other factors on the left side, the central maximum of cosine is replaced by a broad null. The numerical solution in Fig. 2 shows that the first minimum is slightly shifted outwards to 4:85 on each side spanning an angular width of 9:7 . We believe this corresponds to the observed extension of Fraternite. As for the second minimum, it is located at 12:8 from the center. The roots of Eq. (9) are given by the intercepts of the left side with the right side. There are either two intersects around this second minimum or none of them. Naturally, there are more intersects as Dysf is increased. Since mf is small, we consider only short-range gravitational interactions, and take only the first two intersects. We take the mass ratio mf =M ¼ 6:4 1010 or mf =mx ¼ 0:032 which gives mf ¼ 6:4 1016 kg for Fraternite, so that the right side corresponds to 3:5 102 . This gives two intercepts at 11:7 and 13:8 , which are approximately where the minor arcs are observed. There is a slight difference of 1 between our calculated and the observed positions. Also the two intercepts are separated by 2 in our model as compared to 3 in observations. These small discrepancies are probably caused by our representation of Fraternite’s mass as a point mass at its center. In reality, Fraternite’s mass is distributed over its own extension. Interestingly, this model gives an upper limit of Fraternite’s mass. If we take mf =M ¼ 1:0 109 or mf =mx ¼ 0:05,so that the right side corresponds to 5:5 102 , there will be no more intersects. The stationary points would disappear should Fraternite be more massive. Besides giving a mass estimate of Fraternite and the positions of Egalite 2 and Egalite 1, we can also estimate the eccentricity of the two minor arcs, as a result of LR of s in response to the corotation resonance of Galatea, by using Eq. (5) or Eq. (6). Calculations from both equations give e ¼ 5 104 approximately. This eccentricity is
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generated by the LR of s. We now recall that we have taken the assumption that the eccentricity of Galatea is sufficiently small to neglect it all together in analyzing the equations of motion of s for new contributions from the finite mass minor body Fraternite. We have uncovered two time-averaged stationary points and have predicted an eccentricity e ¼ 5 104 for the minor arcs. This gives our → R’
s
F → R
X
241
model a constraint that ex 55 104 ,
(10)
which has to be checked for its veracity. We recall that Namouni and Porco (2002) have investigated the radial offset problem of the Neptune arcs as mentioned in the Introduction. In order to close the mean motion mismatch, they have proposed a finite mass to Fraternite to pull on the epicycle frequency of Galatea. The results in Fig. 2 of their paper indicate a range from mf =mx ¼ 0:128 for ex ¼ 104 to mf =mx ¼ 0:002 for ex ¼ 106 . The estimate derived from our model to account for arc spacings gives mf =mx ¼ 0:032 which is within their cited range. Furthermore, with mf =mx ¼ 0:032, we would have ex ¼ 105 for Galatea according to their Fig. 2. This eccentricity estimate happens to be consistent to Eq. (10). 4. Conclusions
rf rs rx
θs θf θx S Fig. 1. This shows the configuration of the coplanar S2X 2F 2s fourbody system where S is the central body, X is the primary body, F is a minor body, and s is a test body. The longitudes are denoted by y with appropriate subscripts.
To conclude, we have studied a resonant coplanar fourbody system where the center of mass is set by the central and primary bodies. A minor body with finite mass is in corotation-LR with the primary body, and a test body is coorbital with the minor body at close distances. Through the equations of motion of the test body, we have shown that there are stationary locations where the time-averaged force vanishes. These points are located behind the minor body as well as in front of it. They differ from the Lagrangian points of a restricted three-body system in that the averaged force is zero, and that they are self-generated dynamically by the test body’s reaction to the resonant field between the minor and primary bodies. We have applied
0.06
0.04
y
0.02
0.00
−0.02
−0.04
−0.06 0
2
4
6
8 Δθsf
10
12
14
16
Fig. 2. The left side of Eq. (9), denoted by the y label, is plotted as a function of Dysf in degree. The right side of Eq. (9) is a constant and is represented by a horizontal line. The intercepts give the roots of Eq. (9) that define the locations where the time-averaged force vanishes.
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these points to account for the arcs’ configuration in the Neptune–Galatea system. Using this model, it is able to explain the 10 extension of Fraternite. By requiring Fraternite’s mass be 6:4 1016 kg, two time-averaged stationary points are located at 11:7 and 13:8 from the center of Fraternite which seem to be compatible with the observed positions of Egalite 1 and Egalite 2. These locations would disappear should Fraternite be more massive. This model also estimates the eccentricity of Egalite 2 and Egalite 1 at 5 104 . Our model requires that the eccentricity of Galatea be much smaller than this value. Acknowledgments This work was supported by the Conselho Nacional de Desenvolvimentos Cientifico e Tecnologico (CNPq, The Brazilian National Council of Scientific and Technologic Developments) and the Fundacao de Amparo a Pesquisa do Estado do Rio de Janeiro (FAPERJ, The Research Fostering Foundation of the State of Rio de Janeiro).
References Dumas, C., et al., 1999. Stability of Neptune’s ring arcs in question. Nature 400, 733–735. Foryta, D.W., Sicardy, B., 1996. The dynamics of the Neptunian Adams ring’s arcs. Icarus 123, 129–167. Goldreich, P., Tremaine, S., Borderies, N., 1986. Towards a theory for Neptune’s arc rings. Astron. J. 92, 490–494. Horanyi, M., Porco, C.C., 1993. Where exactly are the arcs of Neptune? Icarus 106, 525–535. Hubbard, W.B., et al., 1986. Occultation detection of a neptunian ring like arc. Nature 319, 636–640. Lissauer, J.J., 1985. Shepherding model for Neptune’s arc ring. Nature 318, 544–545. Namouni, F., Porco, C.C., 2002. The confinement of Neptune’s ring arcs by the moon Galatea. Nature 417, 45–47. Porco, C.C., 1991. An explanation for Neptune’s ring arcs. Science 253, 995–1001. Showalter, M.R., 1999. Neptune’s misbehaving rings. Nature 400, 709–710. Sicardy, B., Lissauer, J.J., 1992. Dynamic model of the arcs in Neptune’s 63 K ring. Adv. Space Res. 12, 81–95. Sicardy, B., et al., 1999. Images of Neptune’s ring arcs obtained by a ground based telescope. Nature 400, 731–732. Smith, B.A., et al., 1989. Voyager 2 at Neptune: imaging science results. Science 246, 1422–1449.