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Credit: NASA/JPL/Space Science Institute

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## Animations

• 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

• 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

• 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.

• At pericentre the two motions are in the same direction, resulting in maximum speed while at apocentre the motions are in opposite directions giving minimum speed. The straight lines illustrate what happens when a satellite is in synchronous rotation (i.e. its spin period is equal to its orbital period) about a planet. Concentrate on the straight line joining the planet to the epicentre as it sweeps out its uniform circular motion at a constant angular rate equal to the mean motion of the planet. Now look at the straight line joining the satellite to the empty focus (small magenta dot) of its elliptical path. Imagine this as a line coming out of the surface of the satellite indicating its rotation. Note that as the satellite moves around, the two straight lines are parallel, showing that they move with the same angular velocity. This shows that a satellite in synchronous rotation always keeps one face pointed towards the empty focus of its orbit. This helps to explain why we are able to see more than 50% of the lunar surface from our viewpoint on Earth.

Stability

• Chapter 3, Section 3.7.1

• This shows how the eigenvalues resulting from a linear stability analysis around the L1 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 L1 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 L1 point is always linearly unstable to small displacements.

Stability

• Chapter 3, Section 3.7.2

• This shows how the eigenvalues resulting from a linear stability analysis around the L4 or L5 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 L4 and L5 these always occur in pairs, either

• of the form ±a ± ib, or

• of the form ±ib1 and ±ib2,

• depending on the mass ratio. Note that the eigenvalues start as pairs of the form ±a ± ib but change to ±ib1 and ±ib2 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 L4 and L5 points are linearly stable to small displacements.

• Chapter 3, Section 3.8, Figures 3.14 and 3.15

• The analytical solution for small displacement motion around the L4 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 L4 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.

• 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).

• 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:

1. external passing orbit (-24,000 to -22,000),

2. displaced L4 tadpole orbit (-22,000 to -16,000),

4. internal passing orbit (-4,000 to -1,000),

5. horseshoe-retrograde satellite orbit (-1,000 to 6,000),

6. retrograde satellite orbit (6,000 to 8,000) and

7. passing orbit (8,000 to 11,000).

• 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​

• 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.

• 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​

• 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.

• In this coordinate system particles having the same free eccentricity form a circle of radius equal to the free eccentricity and with centre displaced from the origin by a distance equal to the forced eccentricity. As the system evolves note how the centre moves. This is because Jupiter and Saturn interact to change the value of the forced eccentricity. However, although the eccentricity of each particle (i.e. the distance from the origin) changes, they all remain on a circle. This shows that the free eccentricity remains constant to a good approximation. The centre of the circle (magenta dot) was actually calculated using secular perturbation theory. The numerical integration was carried out by Jer-Chyi Liou.

Secular Evolution of Inclination​

• 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 inclination of a particle can be thought of as the vector sum of (i) its forced inclination (determined by its semi-major axis and the orbits of the perturbing planets) and (ii) its free or proper inclination.

• its forced inclination (determined by its semi-major axis and the orbits of the perturbing planets) and

• its free or proper inclination.

• In this coordinate system particles having the same free inclination form a circle of radius equal to the free inclination and with centre displaced from the origin by a distance equal to the forced inclination. As the system evolves note how the centre moves. This is because Jupiter and Saturn interact to change the value of the forced inclination. However, although the inclination of each particle (i.e. the distance from the origin) changes, they all remain on a circle. This shows that the free inclination remains constant to a good approximation. The centre of the circle (magenta dot) was actually calculated using secular perturbation theory. The numerical integration was carried out by Jer-Chyi Liou.​

Inner Brouwer

• ​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​

• 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

• ​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)​

• 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​

• 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​

• 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​

• 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.)

• 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.

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