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**Saturn System**- 7.9 Mbytes
- Produced using
*Starry Night Deluxe* - A cosmic "zoom" from the Earth to Saturn starting from a view of the night sky from London on 17 March 1999. After zooming in on the planet and showing the motion of the inner moons, the movie shows a polar view of the same moons tracing their orbits before moving outwards to show the orbits of the more distant moons.

**Orbital Precession**- 1.9 Mbytes
- Produced using
*Mathematica®* **Chapter 2, Section 2.3 and Chapter 6, Section 6.11**- An illustration of the difference between orbital motion and precessional motion. The first movie shows the Keplerian motion of a satellite (green) in an elliptical orbit about a spherical planet (red); notice how the motion is faster at pericentre (closest distance to the planet) and slowest at apocentre (furtherest distance from the planet). The second movie shows how the elliptical orbit changes due to an oblate planet. The oblateness causes the orbit to rotate slowly or precess in space.

**Guiding Centre**- 0.8 Mbytes
- Produced using
*Mathematica®* **Chapter 2, Section 2.6**- An illustration of the principles behind the guiding centre
approximation in the two-body problem. A satellite (green disk) moves in
an elliptical orbit (green path) about a planet (magenta disk). However,
the satellite's path can be thought of as being composed of
- a uniform epicyclic motion around a small centred ellipse (yellow) with axes in the ratio 2 to 1, and
- a uniform circular motion (magenta circle) of the epicentre or guiding centre about the planet.

**Stability**- 0.1 Mbytes
- Produced using
*Mathematica®* **Chapter 3, Section 3.7.1**- This shows how the eigenvalues resulting from a linear stability
analysis around the L
_{1}Lagrangian equilibrium point change as the mass ratio (top left-hand corner) decreases from 0.1 to 0.01. The plot shows the eigenvalues in the Argand diagram where a complete number, say*a + ib*, is represented as a point*(a,b)*in the plane. Note that the quartic gives rise to four eigenvalues. For L_{1}these always occur in pairs of the form*±a*and*±ib*regardless of the mass ratio, although the actual values of*a*and*b*do change. Because the eigenvalues are never purely imaginary, the L_{1}point is always linearly unstable to small displacements.

**Stability**- 0.2 Mbytes
- Produced using
*Mathematica®* **Chapter 3, Section 3.7.2**- This shows how the eigenvalues resulting from a linear stability
analysis around the L
_{4}or L_{5}Lagrangian equilibrium point change as the mass ratio (top left-hand corner) decreases from 0.1 to 0.01. The plot shows the eigenvalues in the Argand diagram where a complete number, say*a + ib*, is represented as a point*(a,b)*in the plane. Note that the quartic gives rise to four eigenvalues. For L_{4}and L_{5}these always occur in pairs, either- of the form
*±a ± ib*, or - of the form
*±ib*and_{1}*±ib*,_{2}

*±a ± ib*but change to*±ib*and_{1}*±ib*when the mass ratio becomes sufficiently small. In the animation the switch occurs between 0.039 and 0.038. The actual critical value is (27 - sqrt(621))/54 or approximately 0.0385. When the eigenvalues are purely imaginary, i.e. when the mass ratio is less than the critical value of 0.0385, the L_{2}_{4}and L_{5}points are linearly stable to small displacements. - of the form

**Motion Around the L**_{4}Point- 1.9 Mbytes
- Produced using
*Mathematica®* **Chapter 3, Section 3.8, Figures 3.14 and 3.15**- The analytical solution for small displacement motion around the
L
_{4}point shows that two types of motion combine to give a complicated path. The long-period motion (magenta path) of the epicentre (small yellow dot) around the L_{4}point (yellow cross) and the short-period, epicyclic motion (2:1 cyan-coloured centred ellipse) representing the Keplerian motion of the particle (large yellow dot) around the central mass. The resulting path of the particle in this rotation frame appears complicated and yet it can be described by a simple analytical solution.

**Coorbital**- 1.5 Mbytes
- Produced using
*Mathematica®* **Chapter 3, Sections 3.9 and 3.12, Figures 3.17b and 3.26**- A schematic representation of the dynamics of a coorbital system consisting of a central mass, an orbiting mass, and a test particle in a horseshoe orbit coorbital with the orbiting pass. The same system is shown in both the inertial non-rotating frame (on the left) and the rotating frame (on the right).

**Cruithne**- 49.2 Mbytes
- Produced using
*Mathematica®* **Chapter 3, Section 3.11**- The path of the guiding centre of asteroid (3753) Cruithne's motion
from -24,590 to +10,602 years centred on the present. The animation shows
the complicated coorbital nature of Cruithne's orbit. The data are derived
from the results presented in Fig.2a,b of the paper by Namouni, Christou
& Murray in
*Physical Review Letters*,**83**, 2506-2509 (1999). The Sun and Earth are denoted by red and green circles - their sizes are exaggerated. The semi-major axis of the Earth's orbit is denoted by a yellow circle. The semi-major axis of the guiding centre is exaggerated by a radial factor of 40 in order to make the nature of the coorbital motion easier to see. The types of motion detectable and their approximate times are as follows:- external passing orbit (-24,000 to -22,000),
- displaced L4 tadpole orbit (-22,000 to -16,000),
- tadpole-retrograde satellite-tadpole orbit (-16,000 to -4,000),
- internal passing orbit (-4,000 to -1,000),
- horseshoe-retrograde satellite orbit (-1,000 to 6,000),
- retrograde satellite orbit (6,000 to 8,000) and
- passing orbit (8,000 to 11,000).

**Radial Tide**- 0.3 Mbytes
- Produced using
*Mathematica®* **Chapter 4, Section 4.10, Figure 4.15**- The changing orientation in the equatorial plane for the equipotential curves arising from the radial tide on a satellite in synchronous rotation on an elliptical orbit around a planet. The red circle denotes the equilibrium, zero tide configuration. The radial tide is due to the varying distance of the satellite from the planet.

**Librational Tide**- 0.3 Mbytes
- Produced using
*Mathematica®* **Chapter 4, Section 4.10, Figure 4.15**- The changing orientation in the equatorial plane for the equipotential curves arising from the librational tide on a satellite in synchronous rotation on an elliptical orbit around a planet. The red circle denotes the equilibrium, zero tide configuration. The librational tide is due to the fact that the satellite keeps one face pointed towards the empty focus of its orbit.

**Radial and Librational Tide**- 0.4 Mbytes
- Produced using
*Mathematica®* **Chapter 4, Section 4.10, Figure 4.15**- The combined effect of a radial and librational tide is shown in this animation. The red circle denotes the equilibrium, zero tide configuration.

**Secular Evolution of Eccentricity**- 0.4 Mbytes
- Produced using
*Mathematica®* **Chapter 7, Section 7.5, Figure 7.7a**- The evolution of 250 test particles under perturbations from Jupiter
and Saturn. The particles were started with the same semi-major axis
(11.8 AU), free eccentricity (0.049) and free inclination (2.12°), but
with randomised free longitudes of perihelion and longitudes of ascending
node. The orbits of the particles, Jupiter and Saturn were integrated for
70,000 years. The radial distance from the origin is the particle's
eccentricity and the angular coordinate is its longitude of perihelion.
The actual eccentricity of a particle can be thought of as the vector sum of
- its forced eccentricity (determined by its semi-major axis and the orbits of the perturbing planets) and
- its free or proper eccentricity.

**Secular Evolution of Inclination**- 0.3 Mbytes
- Produced using
*Mathematica®* **Chapter 7, Section 7.5, Figure 7.7b**- The evolution of 250 test particles under perturbations from Jupiter
and Saturn. The particles were started with the same semi-major axis
(11.8 AU), free eccentricity (0.049) and free inclination (2.12°), but
with randomised free longitudes of perihelion and longitudes of ascending
node. The orbits of the particles, Jupiter and Saturn were integrated for
70,000 years. The radial distance from the origin is the particle's
eccentricity and the angular coordinate is its longitude of perihelion.
The actual eccentricity of a particle can be thought of as the vector sum of
- its forced eccentricity (determined by its semi-major axis and the orbits of the perturbing planets) and
- its free or proper eccentricity.

**Inner Brouwer**- 3.7 Mbytes
- Produced using
*Mathematica®* **Chapter 7, Section 7.8, Figure 7.9**- The secular evolution of the orbits of Mercury (yellow), Venus (green), Earth (blue) and Mars (purple) over a period of 2.5 million years according to the secular theory of Brouwer and van Woerkom.

**Outer Brouwer**- 3.7 Mbytes
- Produced using
*Mathematica®* **Chapter 7, Section 7.8, Figure 7.10**- The secular evolution of the orbits of Jupiter (yellow), Saturn (green), Uranus (blue) and Neptune (purple) over a period of 2.5 million years according to the secular theory of Brouwer and van Woerkom.

**Asteroid Families**- 7.2 Mbytes
- Produced using
*Mathematica®* **Chapter 7, Section 7.10**- The location of some 11,500 asteroids in
*a-e-sin i*space (where*a*is the proper semi-major axis,*e*is proper eccentricity and*i*is proper inclination). The long (*x*-) axis is the semi-major axis, the*y*-axis is proper eccentricity and the vertical (*z*-) axis is sine of the proper inclination. The locations of members of ten prominent families (Themis, Eos, Koronis, Maria, Eunomia, Gefion, Nysa, Flora, Vesta and Dora) as defined by Zappala et al (*Icarus*,**116**, 291-314 (1995)) are identified by coloured dots whereas other asteroids are denoted by white dots. Because families are identified by clusterings in orbital elements they are easy to identify in such plots of proper elements. As the display rotates note the clear gaps (the Kirkwood gaps) in the distribution of asteroids at certain locations close to resonances with Jupiter. In reality these are not as wide as suggested here because asteroids near them have not been included.

**Resonant Capture (Case 1)**- 0.5 MBytes
- Produced using
*Mathematica®* **Chapter 8, Section 8.12.1, Figure 8.22**- Capture into resonance for the case where
*delta*(the parameter denoting the distance from exact resonance) is increasing from negative to positive values and the initial eccentricity is smaller than the critical value. In this case capture into resonance is certain. The value of*delta*is shown in the top left-hand corner. The yellow curve shows the trajectory and it always encloses a constant area even though its shape changes. This is because the enclosed area (the action) is a constant in this system. For*delta*>= 0, the red curve shows the separatrix of the resonance. Note the shift towards the right as*delta*increases. When libration begins and*delta*continues to increase, the banana-shaped path becomes narrower and moves further to the right, implying larger eccentricity.

**Orbital Evolution**- 1.0 Mbytes
- Produced using
*Mathematica®* **Chapter 8, Section 8.15**- The movie illustrates the gradual expansion of three satellite orbits as they slowly evolve due to tidal forces. When the inner and middle orbits encounter a 2:1 resonance, they get captured (colour of orbit changes from yellow to cyan) and evolve together.

**Interior Surface of Section**- 1.0 Mbytes
- Produced using
*Mathematica®* **Chapter 9, Section 9.4**- The changing nature of surfaces of section in the circular restricted
three-body problem for test particle orbits interior to the perturber.
The number in the top right-hand corner is the value of the Jacobi constant,
the horizontal axis is the value of
*x*and the vertical axis is the value of*xdot*when*y*= 0 with*ydot*> 0. The mass ratio is 0.001 throughout. (See textbook for a more detailed explanation.)

**Exterior Surface of Section**- 1.6 Mbytes
- Produced using
*Mathematica®* **Chapter 9, Section 9.4**- The changing nature of surfaces of section in the circular restricted
three-body problem for test particle orbits exterior to the perturber.
The number in the top left-hand corner is the value of the Jacobi constant,
the horizontal axis is the value of
*x*and the vertical axis is the value of*xdot*when*y*= 0 with*ydot*> 0. The mass ratio is 0.001 throughout. (See textbook for a more detailed explanation.)

**Asteroids**- 20.8 Mbytes
- Produced using
*Mathematica®* **Chapter 9, Section 9.8.1, Figure 9.27**- A year in the life of the asteroid belt. This animation shows the evolution of approximately 8000 asteroids over the course of a year starting in December 1997. Note that for the purposes of the movie each asteroid is assumed to move on an unperturbed Keplerian orbit.

Copyright © 1999-2003 by Carl D. Murray

Last modified on 8 September 2006

Maintained by Carl D. Murray

Designed by Lynne Marie Stockman