This document summarizes Colin D'Elia's senior project applying Suzuki's fourth-order symplectic integration scheme to numerically integrate a simplified solar system model over 165 million years. Key findings include:
1) The estimated Lyapunov time for test particle Plutos of greater than 6.3 million years agrees with previous research estimating the Lyapunov time between 6.5-20 million years.
2) In contrast to previous non-symplectic integrations that showed linear energy growth, this symplectic integration exhibited periodic energy fluctuations with the energy virtually unchanged after 200 million years.
3) Recovering the initial conditions after forward and backward integration for 165 million years found the positions
Optimal trajectory to Saturn in ion-thruster powered spacecraftKristopherKerames
In this document, I derive the equations of motion for an ion-thruster powered spacecraft and use numerical methods to calculate its optimal trajectory to Saturn. I did this work within 48 hours for the University Physics Competition in 2020.
Using the inclinations_of_kepler_systems_to_prioritize_new_titius_bode_based_...Sérgio Sacani
Artigo descreve como cientistas aplicaram a relação de Titius-Bode nos dados do Kepler para prever a existência de bilhões de exoplanetas parecidos com a Terra na Via Láctea.
Optimal trajectory to Saturn in ion-thruster powered spacecraftKristopherKerames
In this document, I derive the equations of motion for an ion-thruster powered spacecraft and use numerical methods to calculate its optimal trajectory to Saturn. I did this work within 48 hours for the University Physics Competition in 2020.
Using the inclinations_of_kepler_systems_to_prioritize_new_titius_bode_based_...Sérgio Sacani
Artigo descreve como cientistas aplicaram a relação de Titius-Bode nos dados do Kepler para prever a existência de bilhões de exoplanetas parecidos com a Terra na Via Láctea.
A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
A young astronomer’s by now ten years old
results are re-told and put in perspective. The implications are
far-reaching. Angular-momentum shows its clout not only in
quantum mechanics where this is well known, but is also a
major player in the space-time theory of the equivalence
principle and its ramifications. In general relativity, its
fundamental role was largely neglected for the better part of a
century. A children’s device – a friction-free rotating bicycle
wheel suspended from its hub that can be lowered and pulled
up reversibly – serves as an eye-opener. The consequences are
embarrassingly far-reaching in reviving Einstein’s original
dream
The Equation Based on the Rotational and Orbital Motion of the PlanetsIJERA Editor
Equations of dependence of rotational and orbital motions of planets are given, their rotation angles are calculated. Wave principles of direct and reverse rotation of planets are established. The established dependencies are demonstrated at different scale levels of structural interactions, in biosystems as well. The accuracy of calculations corresponds to the accuracy of experimental data
A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
A young astronomer’s by now ten years old
results are re-told and put in perspective. The implications are
far-reaching. Angular-momentum shows its clout not only in
quantum mechanics where this is well known, but is also a
major player in the space-time theory of the equivalence
principle and its ramifications. In general relativity, its
fundamental role was largely neglected for the better part of a
century. A children’s device – a friction-free rotating bicycle
wheel suspended from its hub that can be lowered and pulled
up reversibly – serves as an eye-opener. The consequences are
embarrassingly far-reaching in reviving Einstein’s original
dream
The Equation Based on the Rotational and Orbital Motion of the PlanetsIJERA Editor
Equations of dependence of rotational and orbital motions of planets are given, their rotation angles are calculated. Wave principles of direct and reverse rotation of planets are established. The established dependencies are demonstrated at different scale levels of structural interactions, in biosystems as well. The accuracy of calculations corresponds to the accuracy of experimental data
Direct Measure of Radiative And Dynamical Properties Of An Exoplanet AtmosphereSérgio Sacani
Two decades after the discovery of 51Pegb, the formation processes and atmospheres of short-period gas giants
remain poorly understood. Observations of eccentric systems provide key insights on those topics as they can
illuminate how a planet’s atmosphere responds to changes in incident flux. We report here the analysis of multi-day
multi-channel photometry of the eccentric (e ~ 0.93) hot Jupiter HD80606b obtained with the Spitzer Space
Telescope. The planet’s extreme eccentricity combined with the long coverage and exquisite precision of new
periastron-passage observations allow us to break the degeneracy between the radiative and dynamical timescales
of HD80606b’s atmosphere and constrain its global thermal response. Our analysis reveals that the atmospheric
layers probed heat rapidly (∼4 hr radiative timescale) from<500 to 1400 K as they absorb ~20% of the incoming
stellar flux during the periastron passage, while the planet’s rotation period is 93 35
85
-
+ hr, which exceeds the predicted
pseudo-synchronous period (40 hr).
Key words: methods: numerical – planet–star interactions – planets and satellites: atmospheres – planets and
satellites: dynamical evolution and stability – planets and satellites: individual (HD 80606 b) – techniques:
photometric
Eccentricity from transit_photometry_small_planets_in_kepler_multi_planet_sys...Sérgio Sacani
Artigo descreve estudo que mostra que a órbita dos exoplanetas terrestres são na sua maioria órbitas circulares, o que é bom para se procurar por vida e o que vem causando uma revolução no entendimento sobre os sistemas de exoplanteas.
Von Neumann worked on the cellular automata in in late 1940’s and 1950's as an abstraction of self replication. Von Neumann's ideas of propagation of information from parent cells to next cycles in a cellular automaton, could be an explanation of the geometry of space-time grid, limitation on the speed of light, Heisenberg’s Uncertainty principle, principles of Quantum theory, Relativity, elementary particles of physics, why universe is expanding,… and the list goes on. If this simple mechanism could explain so many things, why was it not a prominent field of research? The answer is very simple but at the same time quite unexpected: one of the applications of this post war study of Von Neumann on Cellular Automata was cryptography; therefore his results were classified and still kept as top secret by USA government. However it’s time to look at this subject from a different point of view: Can this mechanism be used to explain the physical universe? we are more interested in the secret of existence than encryption-decryption of text or data; where did all these galaxies, stars, cosmological objects come from, when did it start, what it was like at the beginning of time and space
Astronomy. 1511 Laboratory Manual In studying the physics and chemist.pdferodealainz
Astronomy. 1511 Laboratory Manual In studying the physics and chemistry of our solar system,
we learn more about its origins and how it will evolve in time. In light of the discoveries of extra
solar planets and the stars they orbit around, we can use information we learn about them to
understand more about our own planets to reach a deeper understanding. Calculating the Mass of
the Moon Mass is the dominant factor in the initial formation of a solar system, so we: carefully
study the masses of objects to understand how it has an affect during formation and after the
system is stable over a period of time. When you learned about Kepler's Three Laws of Planetary
Motion and Newton's Law of Gravity, you were provided the tools to understand the
relationships between the orbital properties of the planets and the Sun. Because these physical
laws are universal, they also apply to all orbiting objects. As long as the objects are allowed to
move in a natural way, Kepler's Third Law enables us to calculate the mass of the source of
gravity for the orbit. You are going to use a portion of the ephemeris (position and time) data
collected by the Explorer 35 spacecraft when it orbited the Moon to draw its orbit. From this
drawing you will use the properties of the ellipse (refer to your lab on Kepler's Laws and your
textbook for assistance) to determine its semi-major axis. With data from Table 1 and the semi-
major axis measurement, you will use the equation for Kepler's Third Law to calculate the mass
of the Moon. Activity: 1. Plot the data from Table 1 on the graph. Please note that only the
position data is used for the graph. 2. Use a ruler to find the major and minor axis of the ellipse.
This requires you to use your best judgment to determine where they are located. Recall that the
longest distance in the ellipse is the major axis and the shortest distance is the minor axis. Lay
the ruler across the graph and rotate it around until you find them, then draw each line. The
intersection of the two lines should be the geometric center of the ellipse. 3. The center of the
Moon is located on the graph at the coordinates (0,0) in lunar radii. Since the Explorer 35
spacecraft orbited around the Moon, according to Kepler's 1 ut Law, the Moon must be at one
focus of the ellipse. 4. Note that the scale of the grid on the graph is approximately: 1 Lunar
Radii =10 small grid boxes a 23.7mm.
Astronomy 1511 Laboratery Manua! Table 1 - Explorer 35 Ephemeris Data
5. Find and record the following properties of the ellipse in terms of Lunar Radii: (These
properties require either measurement and calculation or just calculation. You will need your
calculator to convert from millimeters to Lunar Radii.) a. The Semi-Major Axis (a): b. The Semi-
Minor Axis (b): c. Measure the distance from the intersection of the lines for the major and
minor axis to the center of the Moon at (0,0). This distance is labeled, c. The Distance from the
Center of the Ellipse to the Focus (c).
Study Unit
Ill Engineerin M
Part4
an1cs
By
Andrew Pytel, Ph.D.
Associate Professor, Engineering Mechanics
The Pennsylvania State University
When you complete this study unit, you'll be able to
• Calculate the mass moment of inertia
• Calculate the kinetic energy of a body
• Determine the linear impulse and momentum of a body
• Analyze the equations and conditions used to determine the forces involving rectilinear
translation
• Describe centripetal and centrifugal force
• Describe the forces that impact the rotation of a rigid body without translation
• Explain the motion of a wheel, and calculate the magnitude of the linear acceleration and
friction forces
• Analyze the work-energy method as it applies to the motion and action of a body
iii
PRELIMINARY EXPLANATIONS PERTAINING TO KINETICS .
FORCE-MASS-ACCELERATION METHOD .....
Translation of Rigid Body
Rotation of Rigid Body without Translation
General Plane Motion of Rigid Body
23
WORK-ENERGY METHOD . . . . . . . . . . . . . . . . . . . . . . . . . 53
Application of Method for Translation
Other Applications of Work-Energy Method
IMPULSE-MOMENTUM METHOD . . . . . . .
Rectilinear Translation of Single Body
Collision of Two Bodies
PRACTICE PROBLEMS ANSWERS
EXAMINATION . . . . . . . . . .
. ........... 77
93
95
Engineering Nlechanics, Part 4
PRELIMINARY EXPLANATIONS PERTAINING
TO KINETICS
Scope of This Text
1
1 • In the preceding texts on engineering mechanics, we have discussed
separately the relations of forces in a system and the conditions of mo-
tion of bodies. In this text, we shall consider the relation between the
motion of a body and the force or forces acting on the body to produce
the motion. The basis for the relationship between motion and force
is Newton's second law of motion. However, there are three different
methods of applying this law. These are commonly called the force-
mass acceleration method, the work-energy method, and the impulse-
momentum method. Each method is most useful for solving certain
types of problems.
Statement of Newton's Second Law of Motion
2 • In Engineering Mechanics, Part 1, Newton's second law of motion was
stated as follows:
If a resultant force acts upon a particle, the particle will be accelerated
in the direction of the force. Furthermore, the magnitude of the accel-
eration will be directly proportional to the magnitude of the resultant
force and inversely proportional to the mass of the particle.
Newton's second law can be expressed mathematically by the following
equation:
F
a=k-
m
in which a = magnitude of the acceleration of a particle
k = a numerical factor
F = magnitude of the force acting upon the particle
m = mass of the particle
(1)
The mass of a particle is a measure of the exact amount of matter in
the particle. Any body is composed of a number of particles, and the
mass of a body is the sum of the masses of all the particles.
posting this here so no one wastes their time on making another stupid ppt lol. this was my presentation on differential calculus and it's uses in real life as holiday homework. feel free to use it :)
1. Suzuki's Symplectic Integrator
applied to the Long-term Evolution
of the Solar System
Colin D. D’Elia
Advisor: Vincent Moncrief
Senior Project completed April 24, 1995
At Yale University
2. Suzuki's Symplectic Integrator applied to the
Long-term Evolution of the Solar System
Colin D. D'Elia
Suzuki (1991) has recently shown how to compose symplectic integration
schemes of arbitrarily high order. Here, his fourth order scheme has been
applied to a pared-down version of the solar system consisting of the Sun
and the outer five planets. This system was numerically integrated over a
165 million-year period. An estimate of the Lyapunov time for two test
particle Plutos was in line with those of other researchers. Non-
symplectic integrations of the solar system have shown a linear growth in
Energy while this one does not.
The question of the stability and predictability of the solar
system has intrigued scientists, including such notables as Laplace,
Newton, and Euler for more than 200 years. Today, computers are
allowing researchers to examine this question in a new light. In
1951, Eckert, Brouwer, and Clemence completed a 400-year
numerical integration of the outer five planets. Cohen and
Hubbard published a 120,000-year numerical integration of the
outer planets in 1965 which was soon advanced to 1 million years
by Cohen, Hubbard, and Oesterwinter (1973). In 1984 Kinoshita
and Nakai integrated the outer planets for 5 million years The
Digital Orrery, a special purpose computer designed to
numerically integrate systems with a small number of bodies
which move in nearly circular orbits, allowed Applegate, Douglas,
Gursel, Sussman, and Wisdom (1986, hereafter referred to as
ADGSW) to integrate the outer planets for 200 million years This
was exceeded by an 845 million year integration by Sussman and
Wisdom (1988) using that same computer Each of these
computations was performed with a non-symplectic integrator
and involved only the five outer planets.
3. The most notable long-term integration of the solar system
utilizing a symplectic scheme was performed by Wisdom and
Holman (1991). Theirs was a 1.1-billion-year integration of the
outer planets which utilized a second order symplectic scheme.
Sussman and Wisdom (1992) integrated the entire solar system for
100 million years using that same integrator.
Symplectic Integrators
Symplectic integrators have a number of advantages over
traditional numerical methods. These are enumerated below:
(1) Volume in phase-space is preserved satisfying
Liouville's Theorem.
(2) Both the total angular momentum and the total linear
momentum are invariant.
(3) Symplectic schemes are time reversible.
Each of these is subject to computer round-off error. Round-off
error enters into any numerical calculation since only a finite
number of digits may be kept from one iteration to the next. The
most well-known second-order symplectic scheme is the Trotter
formula;
𝑆2( 𝑡 + 𝜀) = 𝑆 (
𝐴𝜀
2
) ∘ 𝑆( 𝐵𝜖) ∘ 𝑆 (
𝐴𝜀
2
)
where the Hamiltonian is divided into two separably integrable
parts A and B. The most natural division of the n-body
gravitational Hamiltonian, which is the one used in this
integration, is to separate the kinetic and potential energy terms
so that
4. 𝐴 = ∑
𝜌𝑖
2
2𝑚 𝑖
𝑛−1
𝑖=0 and B = - ∑
𝐺𝑚 𝑖 𝑚 𝑗
𝑟𝑖𝑗
𝑖<𝑗
To evolve the dynamical variables forward a time-step ε, a "leap-
frog" method is used. In canonical variables the positions are
evolved according to
𝑞( 𝑡 + 𝜀) = 𝑞 ( 𝑡) + 𝜀̇ 𝑞̇(𝑡)
and the momenta by
𝑝( 𝑡 + 𝜀) = 𝑝 ( 𝑡) + 𝜀̇ 𝑝̇( 𝑡)
𝑞̇ and 𝑝̇ are obtained from the Hamiltonian. Hence;
𝑞̇ =
𝜕𝐻
𝜕𝑝
and 𝑝̇ =
𝜕𝐻
𝜕𝑞
In summary, this means that first the momentum of each body in
the system is evolved with a half time-step while keeping the
positions constant. Then the positions are evolved a whole-step
while the momenta are kept constant and then once again the
momenta are evolved another half time-step keeping the
positions constant. This completes one iteration and has been
shown to agree to second order with an expansion of the full
solution. The final positions and momenta of each iteration are
then used to begin the next iteration. Until recently, there had not
been a general method for obtaining symplectic integration
schemes greater than second order. Suzuki (1991) showed how to
write a symplectic scheme to any desired order by composing the
n order scheme from the (n-2) scheme in the following way;
𝑠 𝑛( 𝑡 + 𝜀) = 𝑠 𝑛−2[ 𝑘 𝑛−1 𝜀] ∘ 𝑠 𝑛−2[(1 − 2𝑘 𝑛−1) 𝜀] ∘ 𝑠 𝑛−2[ 𝐾 𝑛−1 𝜀]
5. Every even ordered scheme n agrees with the n-1 scheme. S3 = S4
for example. It is remarkable that whenever a new scheme is
composed, two orders of accuracy are gained instead of just one.
This lends itself very naturally to numerical computations. The
integration being reported here utilized Suzuki's fourth order
scheme. I began with the invariable frame center of mass
coordinates and momenta listed in ADGSW (1986) for
“JD2430000.5”. The masses were also copied from this source. The
masses of the inner planets were added to that of the Sun with
Pluto integrated as a test particle. This 165-million-year run was
performed on the Vax alpha machine at Yale University and took
about two weeks of computation time. A somewhat arbitrary time-
step of 40 days was chosen with the output recorded at
approximately 5 million year intervals.
The following three questions were investigated;
(1) Does the estimate of the Lyapunov time for Pluto agree with
that of other researchers?
(2) How well is the Energy of the system conserved and how
does this compare with that of other long-term solar system
integrations?
(3) How accurate can the initial conditions be recovered after
running the system forward a time t and then backwards
that same time t?
The Lyapunov Time
The Lyapunov time is defined as the time it takes for the paths of
two bodies to become separated in phase-space by a factor of e. A
positive Lyapunov time is considered to be the hallmark of Chaos.
Formally, this definition of the Lyapunov time can only be applied
when the paths are diverging exponentially and in the limit as
6. time -> ∞ Any numerical estimate then, can only give an
approximation of the Lyapunov time. Furthermore, it may be that
the estimate is sensitive to uncertainties in the initial positions and
momenta though this has not yet been investigated.
Phase-space distance is measured by the Euclidean six-
dimensional norm. Since the positions are quoted in Astronomical
Units (A.U.) and the velocities as A.U./day, the velocities are small
compared to the distances in these units. On that basis, I have
chosen to estimate the Lyapunov time by the time it takes for the
test-particle Plutos to become separated in Cartesian 3-Space by a
factor of e.
In this case the Lyapunov time for the trajectory of Pluto is what
was investigated. The Sun, Jupiter, Saturn, Uranus, Neptune, and
two test particle Plutos were integrated by point particle
Newtonian gravitation in rectangular coordinates. The initial and
final positions of the test particle Plutos (in A.U.) are included
below. The number of digits does not reflect the machine accuracy
and are included for completeness only.
time = 0 days
x y z
Pluto 1 -21.3858977531572 A.U. 32.0719104740000 A.U. 2.49245689556095 A.U.
Pluto 2 -21.3858977531572 A.U. 32.0719104739886 A.U. 2.49245689556095 A.U.
time = 6.0225x 108
days
x y z
Pluto 1 -18.9969402226207 A.U. 38.9892747647021 A.U. -7.79468473359073 A.U.
Pluto 2 -21.3858977531572 A.U. 32.0719104739886 A.U. -7.07607867883915 A.U.
7. The initial positions of the two test-particle positions differ by only
1.14x 10-10
A.U. in the y-coordinate. This is about 1.7 meters. The
final positions differed by about 2.66 A.U. or 3.99x109
meters.
Practically, this means that after 165 million years the position of
Pluto cannot be ascertained to within 3.99x109
meters unless the
initial position of Pluto is known to within 1.7 meters.
It is estimated here that the Lyapunov time is greater than 6.3
million years. This is in quite reasonable agreement with the
results of ADGSW (1986) who showed a Lyapunov time of greater
than 6.5 million years for a 110 million year run and Wisdom,
Sussman (1988) who estimated the Lyapunov time as greater than
20 million years in their 845-million-year integration. Wisdom and
Holman (1991) have also estimated Pluto's Lyapunov time as
about 20 million years in their 1.1-billion-year integration. Fig. 1
shows the Lyapunov time plotted vs. time. It should be noted that
the Lyapunov time has not levelled off here and is still growing. A
longer run will have to be performed to see whether this method
is in strict agreement with the results reported earlier in this
paper. That all numerical computations have given a Lyapunov
time of the same order of magnitude is significant and seems to
confirm the chaotic trajectory of Pluto though Wisdom and
Holman (1991) caution that until the dynamical mechanisms are
understood we cannot be certain.
8. Energy Conservation
Traditionally, one method of testing the reliability of Solar system
integrations has been to observe the change in energy with time.
In a "perfect" integration, energy should, of course, be strictly
conserved. Sussman and Wisdom (1988) reported the relative
energy error, defined at time t as the energy at time t minus the
initial energy divided by the initial energy, for a number of non-
symplectic integrations. These included Sussman and Wisdom
(1988), ADGSW (1986), Cohen, Hubbard, and Oesterwinter (1973),
Kinoshita and Nakai (1984), and Project LONGSTOP (A.E. Roy et.al).
Each of these reported a nearly linear relative energy error with
magnitudes ranging from 3.0x10-19
per year to 5x10-16
per year.
It has been expected that long-term symplectic integrations of the
solar system would result in a periodic energy (see Liu, Liao, Zhao,
and Wang (1994)) and this seems to be confirmed here (FIG.2).
9. It is quite impressive that after 200 million years the energy is
virtually where it started. It appears to be cyclic with a period of
about 120 million years. It is not known whether that particular
length of time has a special significance. The difference between
the maximum and minimum energies is only 3.75x10-14
S.M.*(A.U.)2
/(day). FIG. 3 shows the relative energy error vs. time. It
is clear that unlike the results of non-symplectic integrations, the
relative energy error is not linear.
FIG. 3. Relative Energy Error vs. Time
10. There were actually two separate runs with the only difference
being the initial positions of the test-particle Plutos. These were
run in separate batches to speed the integration process. As a
result, one of the runs was actually 200 million years long and it
was from this run that FIG. 2 and FIG.3 were composed.
Forward/Backward Run
A separate run was performed in order to see how well the initial
positions could be recovered after integrating forward and then
backward three million years. This is an excellent indicator of the
degree to which round-off error enters into the calculation since
the scheme is otherwise time-reversible.
At the end of the run the error in recovering the positions of the
planets ranged from 1.7x104
A.U. for Pluto to 8.4x10-3
A.U. for
Jupiter. Sussman and Wisdom (1988) performed a
forward/backward run of three million years and were able to
recover the initial position for each of the planets with an error of
about 10-5
for each planet. Most of this difference is probably due
to the defined machine precision. Sussman and Wisdom used a
special purpose computer with pseudo-quadrupole precision
variables. All the variables in this calculation were defined as
double precision.
Conclusions
This research, as well as that of others, shows symplectic schemes
to be performing well. To this author's knowledge, no long-term
evolution of the solar system has been performed utilizing a sixth
order or higher symplectic scheme.
11. Until higher order symplectic schemes are used to integrate the
solar system, direct comparison between symplectic schemes and
non-symplectic schemes will be difficult. All but one of the non-
symplectic integrations reported in this paper employed a 12th-
order integrator.
Additionally, Wisdom and Holman (1991) suggest that the division
of the Hamiltonian which was used in this paper is not the optimal
one. Instead they recommend H=HKEPLER + HINTERACTION where the
first term is the Keplerian motion of each planet with respect to
the sun and the second includes the perturbations which the
planets have upon each other. This division, they claim, will allow
larger time-steps to be used thus saving valuable computation
time. Wisdom and Holman were able to perform their 2nd-order
symplectic integration with a time-step of one year.
Since the dynamical variables at each time t are calculated from
the time (t-ε) positions and momenta, this integration method is
well suited for parallel processing where a different processor
could be assigned to each planet. I estimate that this could speed
the computation time by a factor of four to six times. Each two
orders of magnitude one wishes to gain using Suzuki's method
increases the computation time by a factor of three.
Given that the 165 million year 4th-order integration reported
here took roughly 14 days, it is conceivable that with the above
mentioned improvements one could repeat Wisdom and
Holman's 1.1 billion year, second order integration of the outer
planets at 8th-order on the Yale Vax in a little over two weeks’
time. This is the same time scale in which they performed their
2nd-order calculation
12. As judged by the 3 million year forward and backward runs the
integration method used here still falls short of the best solar
system integrations. Since this scheme is time reversible, the only
difference must be in the precision with which the variables are
defined. Keeping more digits from one iteration to the next,
though, costs computation time. Combining the methods of
reducing computation time mentioned above should effectively
allow greater machine precision to be defined. The energy seems
to be conserved to a greater degree of accuracy here than with
integrations using non-symplectic methods. It is encouraging that
the energy seems to be bounded with this integrator. With
traditional methods the relative energy error will continue to
increase or decrease roughly linearly with time. All integrations
indicate a chaotic trajectory for Pluto. The effect which this has on
the rest of the solar system must still be fully investigated. It is
extremely unlikely that integrating Pluto as a full particle would
have much of an effect upon the conclusions here though this
should still be investigated. We are just beginning to see long-term
integrations of the entire solar system which use the full equations
of motion. These are allowing researchers to investigate the
stability of the solar system as a whole. In short, there are still
many interesting questions which can be investigated and as top-
speed computers become increasingly accessible it can only be
expected that they will.
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