We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting resulting in resonance overlap which causes the chaos. A single degree of freedom model captures the essential features of the chaotic dynamics. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 deg yr^−1 and 3.1 deg yr^−1, respectively

Goldreich, Peter

Rappaport, Nicole

English

arXiv

Caltech Authors - Main

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Origin of Chaos in the Prometheus-Pandora
System
Peter Goldreich
Institute for Advanced Study
Princeton NJ 08540
E-mail: pmg@sns.ias.edu
&
California Institute of Technology
Pasadena CA 91125
E-mail: pmg@tapir.caltech.edu
and
Nicole Rappaport
Jet Propulsion Laboratory
California Institute of Technology
Pasadena CA 91109
E-mail: Nicole.J.Rappaport@jpl.nasa.gov
Submitted to Icarus on June 18, 2003
Number of pages: 23
Number of tables: 3
Number of figures: 7
1
Proposed Running Head: Origin of Chaos in the Prometheus / Pandora System
Editorial correspondence to:
Dr. Nicole J. Rappaport
Jet Propulsion Laboratory
MS 301-150
Pasadena CA 91109
Phone: (818) 354-8211
Fax: (818) 393-6388
E-mail: Nicole.J.Rappaport@jpl.nasa.gov
2
ABSTRACT
We demonstrate that the chaotic orbits of Prometheus and Pandora are due to inter-
actions associated with the 121:118 mean motion resonance. Differential precession
splits this resonance into a quartet of components equally spaced in frequency. Libra-
tion widths of the individual components exceed the splitting resulting in resonance
overlap which causes the chaos. A single degree of freedom model captures the es-
sential features of the chaotic dynamics. Mean motions of Prometheus and Pandora
wander chaotically in zones of width 1.8 deg yr−1 and 3.1 deg yr−1, respectively.
Key Words: Satellites of Saturn, Orbits, Chaos
3
1 INTRODUCTION
Goldreich and Rappaport (2003) (hereafter abbreviated as GR) showed that the motions
of Prometheus and Pandora are chaotic. The chaos arises from their mutual gravitational
interactions, which explains why their longitude discrepancies have comparable magnitudes
and opposite signs (French et al. 2002). Numerical integrations that account for the full
mutual interactions and Saturn’s gravitational oblatenss yield a Lyapunov exponent of order
0.3 yr−1. Although the results reported by GR assumed satellite masses based on a nominal
density of 0.63 g cm−3, the Lyapunov exponent is insensitive to the assumed density above a
critical value of approximately 0.3 g cm−3.
GR’s integrations also reproduce qualitative features of the discrepancies between the
longitudes of the satellites derived from analysis of recent HST data and predictions based
on orbits fit to Voyager images (French et al. 2002). Sudden changes in the mean motions of
Prometheus and Pandora are a striking feature of the numerical integrations. These occur
at intervals of 6.2 yr when the satellites’ apses are anti-aligned. It is notable that the only
clearly documented changes in the mean motions occurred around the time of the most
recent apse anti-alignment (cf. GR).
The plan of this paper is as follows. In §2 a quartet of 121:118 mean motion resonances is
identified as the probable cause of the chaos.1 Then we describe two new programs in which
interactions between the satellites are limited to those due to this quartet. The simpler of
1Differential apsidal precession splits each mean motion resonance into a multiplet of closely spaced
components.
4
these reduces the resonant dynamics to a system with one degree of freedom. Results from
these programs are compared in §3 with those obtained from integrations that account for
the full gravitational interactions. §4 is devoted to a discussion of the width of the chaotic
zone.
2 ORIGIN OF CHAOS
2.1 Resonant Quartet
A systematic search for j : j−k mean motion resonances with k ≤ 4 turned up j = 121, k =
3.
Following Murray and Dermott (2001), we write the disturbing function for the action
of Pandora on Prometheus as
R =
Gm′
a′
RD , (1)
and that for the action of Prometheus on Pandora as
R′ =
Gm
a′
RD . (2)
Here m and a denote mass and semi-major axis, and G is the gravitational constant.2 To
lowest order in the eccentricities, the terms in the literal expansion of the disturbing function
associated with a k = 3 resonance take the form
RD = e
3 f82 cos [121λ
′ − 118λ− 3̟] +
2Primed and unprimed symbols refer to Prometheus and Pandora, respectively.
5
e2e′ f83 cos [121λ
′ − 118λ− 2̟ −̟′] +
ee′2 f84 cos [121λ
′ − 118λ−̟ − 2̟′] +
ee′3 f85 cos [121λ
′ − 118λ− 3̟′] , (3)
where e, λ, and ̟ stand for eccentricity, mean longitude, and argument of periapse. The
f8n are expressed in terms of Laplace coefficients evaluated at α = a/a
′.3
f82 =
1
48


(
−26j + 30j2 − 8j3
)
b
(j)
1/2(α) +
(
−9 + 27j − 12j2
)
α
db
(j)
1/2(α)
dα
+
(6− 6j)α2
d2b
(j)
1/2(α)
dα2
− α3
d3b
(j)
1/2(α)
dα3

 , (4)
f83 =
1
16


(
−9 + 31j − 30j2 + 8j3
)
b
(j−1)
1/2 (α) +
(
9− 25j + 12j2
)
α
db
(j−1)
1/2 (α)
dα
+
(−5 + 6j)α2
d2b
(j−1)
1/2 (α)
dα2
+ α3
d3b
(j−1)
1/2 (α)
dα3

 , (5)
f84 =
1
16


(
8− 32j + 30j2 − 8j3
)
b
(j−2)
1/2 (α) +
(
−8 + 23j − 12j2
)
α
db
(j−2)
1/2 (α)
dα
+
(4− 6j)α2
d2b
(j−2)
1/2 (α)
dα2
− α3
d3b
(j−2)
1/2 (α)
dα3

 , (6)
f85 =
1
48


(
−6 + 29j − 30j2 + 8j3
)
b
(j−3)
1/2 (α) +
(
6− 21j + 12j2
)
α
db
(j−2)
1/2 (α)
dα
+
(−3 + 6j)α2
d2b
(j−3)
1/2 (α)
dα2
+ α3
d3b
(j−3)
1/2 (α)
dα3

 . (7)
Tables 1 and 2 list values for the parameters used in this paper. Satellite masses are given
as fractions of Saturn’s mass based on an assumed density of 0.63 g cm−3. Initial values for
3In the following, j should be viewed as a shorthand for 121.
6
mean longitudes, apsidal angles, mean motions, and eccentricities are based on orbits fit
to Voyager images by Jacobson (2001). Precession rates are calculated from the Saturnian
gravitational field (Campbell and Anderson 1989).
Table 1. Masses, Initial Mean Longitudes, & Mean Motions.
Satellite m/M Mean Longitude (◦) Mean Motion (◦/s)
Prometheus 5.80× 10−10 188.53815 6.797331× 10−3
Pandora 3.43× 10−10 82.14727 6.629506× 10−3
Table 2. Eccentricities, Initial Apsidal Angles, & Precession Rates.
Satellite Eccentricity Apsidal Angle (◦) Precession Rate (◦/s)
Prometheus 2.29× 10−3 212.85385 3.1911× 10−5
Pandora 4.37× 10−3 68.22910 3.0082× 10−5
Rates of change of the arguments, corresponding periods, and coefficients for the four
terms in equation (3) are given in Table 3.
Table 3. Resonance Arguments, Rates of Change, Periods, and Coefficients.
7
Argument Rate of Change (◦/s) Period (year) Coefficient
121λ′ − 118λ− 3̟ −1.058× 10−5 1.078 e3f82 = −0.001
121λ′ − 118λ− 2̟ −̟′ −0.875× 10−5 1.303 e2e′f83 = 0.006
121λ′ − 118λ−̟ − 2̟′ −0.692× 10−5 1.648 ee′2f84 = −0.01
121λ′ − 118λ− 3̟′ −0.509× 10−5 2.239 e′3f85 = 0.007
2.2 Numerical Integrations
To demonstrate that the quartet of 121 : 118 resonances is the cause of chaos in the
Prometheus Pandora system, we develop two new programs to integrate the satellites’ equa-
tions of motion. Interactions between the satellites are restricted to the four resonant inter-
action terms in the Fourier expansion of the disturbing function RD. Each program evolves
propagates the satellites’ orbital elements rather than their cartesian coordinates and veloc-
ities as is done by the “old program” FSHEP used in GR.
We adopt epicyclic elliptic elements instead of the more standard osculating elliptic el-
ements since, unlike the latter, they do not require short period terms to describe elliptic
orbits around oblate planets (cf. Borderies-Rappaport and Longaretti 1994; henceforth, re-
ferred to as BRL.). BRL derive a modified version of Gauss’ equations for the elements ae,
ee, ̟e = ωe + Ωe, and λe = ̟e +Me.
4 From these, it is a straightforward exercise to derive
a restricted version of Lagrange’s equations that is valid in the planar case. We work with
a simplified set appropriate for low eccentricity orbits about a modestly oblate planet. The
4Hereafter we drop the subscript e.
8
equations read:
dλ
dt
= Ω , (8)
da
dt
=
2
κa
∂R
∂λ
, (9)
d̟
dt
= Ω− κ , (10)
de
dt
= −
1
κa2e
∂R
∂̟
, (11)
where
Ω2 =
GM
a3
[
1 +
3
2
(
Rp
a
)2
J2 −
15
8
(
Rp
a
)4
J4 +
35
16
(
Rp
a
)6
J6 ...
]
, (12)
κ2 =
GM
a3
[
1−
3
2
(
Rp
a
)2
J2 +
45
8
(
Rp
a
)4
J4 −
175
16
(
Rp
a
)6
J6 ...
]
. (13)
2.3 New Programs
In the planar approximation, the Prometheus-Pandora system has four degrees of freedom
and preserves two integrals, total energy and total angular momentum. Thus each phase
space trajectory lies on a six dimensional hypersurface embedded in the eight dimensional
phase space.
FSHEPRES integrates the four, first-order equations (8)-(11) for each satellite. Thus
it differs from FSHEP mainly because it limits the interactions between the satellites to
9
resonant terms.5 Other minor differences arise because FSHEPRES integrates a simplified
set of Lagrange’s equations. In particular, the conservation laws are only approximately
satisfied.
FSHEPSIM integrates only the first two equations (8) and (9) for each satellite. This
drastic simplification is reasonable because, as a consequence of the rapid differential pre-
cession caused by Saturn’s oblateness, interactions between the satellites produce negligible
effects on their apsidal angles and orbital eccentricities (GR). A further simplification arises
because conservation of energy implies6
m
a2
da
dt
=
m′
a′2
da′
dt
. (14)
Thus the resonant dynamics of the Prometheus-Pandora system reduces to a single degree
of freedom system. It proves convenient to define the variable
ψ = 121λ′ − 118λ , (15)
so that RD is expressed as
RD =
4∑
k=1
Ck cos(ψ + δk) , (16)
with each δ˙k = constant. The evolution of ψ is governed by
d2ψ
dt2
= 3
[
(121Ω′)2
m
M
+ α(118Ω)2
m′
M
]
4∑
k=1
Ck sin(ψ + δk) (17)
= 3× (121Ω′)2
m
M
[1 + α(m′/m)]
4∑
k=1
Ck sin(ψ + δk) , (18)
5We view as an unimportant difference the use of orbital elements by FSHEPRES and cartesian positions
and velocities by FSHEP.
6In the following equation we ignore the interaction energy which is only significant near conjunctions.
10
where in writing the second form of equation (18), we have applied the mean motion reso-
nance relation Ω′/Ω ≈ 118/121 and emphasized the contribution from the lighter body, m′.
Individual mean longitudes follow from the relations
λ(t) =
−α(m′/m)ψ(t) + 118[λ(0) + λ˙(0)t] + 121α(m′/m)[λ′(0) + λ˙′(0)t]
121[1 + α(m′/m)]
, (19)
λ′(t) =
ψ(t) + 121α(m′/m)[λ′(0) + λ˙′(0)t] + 118[λ(0) + λ˙(0)t]
121[1 + α(m′/m)]
. (20)
Although we are left with a system described by a single degree of freedom, the absence of
an energy integral still allows for chaos.
3 COMPARISON OF RESULTS
In this section we compare result obtained using FSHEPRES and FSHEPSIM with those
obtained with FSHEP. As in GR, all our simulations are initialized with orbital elements
for Prometheus and Pandora taken from Jacobson’s ephemerides at epoch 1981 August 23
04:02:12 UTC. Comparisons among similar calculations done with each of the three programs
are presented in Figures 1-6. As a consequence of chaos, qualitative similarities are the best
that can be expected. These are apparent in each set of figures. However, there is a hint that
FSHEPSIM yields a slightly smaller Lyapunov exponent than either FSHEP or FSHEPRES.
The similarity between the 20 year runs of longitude variations displayed Figures 1 and
2, while consistent with a Lyapunov exponent of approximately 0.3 yr−1 as shown in Figure
3, probably also reflects the fact that at the Voyager epoch the mean motions of Prometheus
11
and Pandora were close to their respective maximum and minimum. This accounts for the
negative values of the rates of each resonant argument quoted in Table 3.
Figure 4 shows that over 3000 years the net variation of 121λ′ − 118λ − 2̟′ − ̟ is
much smaller than that of the other phases. Together with the constraint imposed by the
conservation of energy on relative variations of n and n′, this implies that over this time
interval the average mean value of n was smaller than its initial value by about 0.67 deg yr−1
and that of n′ was larger by about 1.14 deg yr−1. The relatively small value of 121n′ −
118n−2 ˙̟ ′− ˙̟ has a plausible dynamical explanation. The term with this phase rate is the
one with the largest amplitude. Moreover, the amplitudes of terms with neighboring phase
rates are about half as large and have opposite signs to that of the dominant term, whereas
the amplitude of the term with the slowest phase rate is much smaller. Support for this
explanation is provided by observing that interchanging the values of e and e′ results in the
phase 121λ′−118λ−̟′−2̟ assuming the special status of being the one with the smallest
net variation.
Figures 5 and 6 display longitude variations over 3000 years relative to the longitude
based on the average mean motion over this interval. These are seen to be bounded by
±180 degrees.
4 Discussion
A closer examination of the one-degree of freedom model developed for FSHEP provides
additional insight regarding chaos in the Prometheus-Pandora system.
12
Overlapping resonances are known to produce chaos. Frequencies of individual mem-
bers of the resonant quartet are spaced by ˙̟ − ˙̟ ′ ≈ 1.0 rad yr−1. This is smaller than
the half widths of the individual resonance components.7 Half widths computed from
equation (18) and the data in Tables 1-3 are, in order of increasing resonance frequency,
1.5, 3.7, 5.1, 4.1 rad yr−1.
Figure 7 shows surfaces of section based on data from 3000 year integrations using
FSHEPSIM. A single point with coordinates ψ − ̟ − 2̟′, ψ˙ − ˙̟ − 2 ˙̟ ′ is plotted each
time the apses align (every 6.2 yr when ̟ − ̟′ = 0 modulo 2π).8 Nominal values for the
satellites’ masses were used for the upper panel. The scattering of points over an area in the
phase plane is a signature of chaos. The balance in the number of points above and below the
horizontal axis and the overall vertical width of their distribution are a consequence of the
dominance of the resonance component with phase 121λ′−118λ−̟−2̟′. Satellite masses
were reduced by a factor 10 below their nominal values to obtain the integrable example
whose surface of section is shown in the lower panel.
Variations of n and n′ are related to those of ψ˙ by
dn
dt
=
1
121[1 + α(m′/m)]
dψ
dt
, (21)
dn′
dt
=
−α(m′/m)
118[1 + α(m′/m)]
dψ
dt
. (22)
Thus the total width in ψ˙ corresponds to full width variations ∆n ≈ 1.8 deg yr−1 and
7The half width is the maximum angular velocity achieved during motion on the separatrix.
8We chose apse alignment to minimize the effects of the interaction energy.
13
∆n′ ≈ 3.1 deg yr−1.
5 Acknowledgments
This research was supported by NASA Planetary Geology and Geophysics grant 344-30-53-02
and by NSF grant AST-0098301.
14
6 References
Borderies-Rappaport N., and P.-Y. Longaretti 1994: Test Particle Motion around an
Oblate Planet, Icarus 107, 129-141.
Campbell, J.K., and J.D. Anderson 1989: Gravity Field of the Saturnian System from
Pioneer and Voyager Tracking Data, Astron. J. 97, 1485-1495.
French, R.G., C.A. McGhee, L. Dones, and J.J. Lissauer 2003: Saturn’s wayward
shepherds: the perigrinations of Prometheus and Pandora, Icarus 162, 144-171.
Goldreich, P. and N. Rappaport 2003: Chaotic Motions of Prometheus and Pandora,
Icarus 162, 391-399.
Jacobson 2001: Tables of Prometheus and Pandora Planetocentric Mean Elements at
Julian Ephemeris Date 2444839.6682 Referred to the Earth Mean Equator and Equinox of
J2000 System, Personal Communication.
Murray, C.D., and S.F. Dermott 2001: Solar System Dynamics, Cambridge Univer-
sity Press, 592 pp.
15
7 Figure Captions
FIGURE 1: Prometheus longitude in degrees from numerical integration as a function of
time over 20 years. A drift based on the initial mean motion has been subtracted from the
longitudes. Dashed lines indicate the times of periapsis antialignment. Results shown in the
top, middle, and bottom panels were obtained with the programs FSHEP, FSHEPRES, and
FSHEPSIM.
FIGURE 2: Pandora longitude in degrees from numerical integration as a function of time
over 20 years. A drift based on the initial mean motion has been subtracted from the
longitudes. Dashed lines indicate the times of periapsis antialignment. Results shown in the
top, middle, and bottom panels were obtained with the programs FSHEP, FSHEPRES, and
FSHEPSIM.
FIGURE 3: Lyapunov exponent in yr−1 for the Prometheus-Pandora system over a period
of 3 × 103 years. The results shown in the top, middle, and bottom panels were obtained
with the programs FSHEP, FSHEPRES, and FSHEPSIM.
FIGURE 4: Phases ψ+ δk, k = 1, . . . , 4, in radians along the solution. Results shown in the
top, middle, and bottom panels were obtained with the programs FSHEP, FSHEPRES, and
FSHEPSIM.
FIGURE 5: Prometheus longitude in degrees from numerical integration as a function of
time over 3000 years. A drift based on the mean motion averaged over 3000 years has been
subtracted from the longitude. Results shown in the top, middle, and bottom panels were
obtained with the programs FSHEP, FSHEPRES, and FSHEPSIM.
FIGURE 6: Pandora longitude in degrees from numerical integration as a function of time
over a 3000 years. A drift based on the mean motion averaged over 3000 years has been
subtracted from the longitude. Results shown in the top, middle, and bottom panels were
obtained with the programs FSHEP, FSHEPRES, and FSHEPSIM.
FIGURE 7: Surfaces of section obtained by plotting (ψ−̟−2̟′, ψ˙− ˙̟ −2 ˙̟ ′) at each time of
periapsis alignment over 3,000 years. Units are radians and radians per year. Computations
were made with FSHEPSIM, for the top panel with the nominal value of 0.63 g cm−3 for the
satellite density. For the bottom panel, the density was reduced by a factor of 10 in order
to obtain an integrable example to contrast with the chaotic one shown above.
16
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Origin of chaos in the Prometheus–Pandora system

Gravity Field of the Saturnian System from Pioneer and Voyager Tracking Data,

Solar System Dynamics,

Tables of Prometheus and Pandora Planetocentric Mean Elements at Julian Ephemeris Date 2444839.6682 Referred to the Earth Mean Equator and Equinox of J2000 System,

Test Particle Motion around an Oblate Planet,

https://authors.library.caltech.edu/2431/1/GOLicarus03b.pdf

Origin of chaos in the Prometheus–Pandora system

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