Numerical Simulations of the
Orbits for Planetary Systems:
from the Two-body Problem to Orbital Resonances in Three-body
NTHU
Wang
Supervisor
Tanigawa, Takayuki
Gu, Pin-Gao
Outline
• Orbital Elements
• Numerical Error & Numerical Method
(check it in two-body)
• Gas Disk Model
• Three-Body Motion
• Resonance Phenomenon
Introduction to job
• Orbital Elements
• Numerical Error & Numerical Method
(check it in two-body)
• Gas Disk Model
• Three-Body Motion
• Resonance Phenomenon
Orbital Elements
•
•
•
•
•
“Semi-major axis”  “ a “
Half the major axis of an orbit's ellipse
“Eccentricity”  “ e “
Defines the shape of the orbit
“Inclination”  “ i “
Angle between the orbital plane and a
reference plane.
“Argument of Pericentre”  “ω”
Angle between the “ascending node”
and “pericentre of orbit” in the orbital
plane
“Longitude of the ascending node”
 “Ω”
Angle between “line of ascending
nodes” and “the zero point of
longitude” in the reference plane.

• In my simulation, i=0 ,
because I simplified the
model.
i.e. reference plane = orbital plane
HELP : pericentre = pericenter = perihelion
Orbital Elements
Orbital elements we check
when we analyzed the evolution
of orbit.
1.
Semi-major axis
a
2.
Eccentricity
e
3. Longitude of pericentre

Its definition:
  
even the orbital plane is
different from reference
plane.
At these kind of cases,  is
a “dogleg” angle.
• In my simulation, the reference
plane and orbital plane are the
same plane. We can use
  
formula without worrying about
different plane.
Numerical Error & Method
(for two-body)
• Get numerical error
Error amplitude
The error amplitude will fluctuate between a region.
Secular error
The error amplitude may have an increasing or decreasing trend with time.
• Choose a Numerical Method
Modified Hermite Method
A method had been developed recently.
Its features: error with periodic oscillation, no secular error, algorithm is simple
Runge-Kutta Method
The basic numerical method we know. It is better relative with another
methods like Euler, Modified Euler …
Numerical Error & Method
(for two-body)
• Result of “Get numerical error”
Error amplitude of Modified Hermite Method
At dt = 2-5 , “Δa” fluctuates between”10-8“ [unit is reduce]
“Δe” fluctuates between”10-8“
“Δ “fluctuates between”10-7“ [unit is radian]
Error amplitude of Runge-Kutta Method
At dt = 2-5 , “Δa” fluctuates between”10-9“ [unit is reduce]
“Δe” fluctuates between”10-8“
“Δ “fluctuates between”10-7“ [unit is radian]
Secular error of Modified Hermite Method (a, e have no secular err)
At dt = 2-5 , “Δ“decreases ”10-5 “during 30,000 cycles [unit is radian]
Secular error of Runge-Kutta Method
At dt = 2-5 , Δ a decreases “2*10-3” during 30,000 cycles
, Δ e decreases “10-3” during 30,000 cycles
, Δ increases “10-1” during 30,000 cycles
Numerical Error & Method
(for two-body)
• Result of “Choose a Numerical Method”
Modified Hermite Method – better 
Its error amplitude is bigger, but nearing amplitude of Runge-Kutta Method.
Semi-major axis and eccentricity have no secular error.
Longitude of pericentre has much smaller secular error.
Runge-Kutta Method – worst 
The error amplitude of oscillation is smaller (but not much smaller).
Secular error makes the error amplitude grow with time. The secular error
of Runge-Kutta Method is too much larger than secular error of Modified
Hermite Method.
Gas Disk Model
• If the planet is large
enough, it will form a gap
in disk.
• Evolution of the planet
orbit will follow evolution
of gas disk after the gap
form.
[Lin and Papaloizou 1993 & Takeuchi et al. 1996]
• Consider the simplest
model ~> gas move
inward caused from
viscosity.
• So the gas pull on planet
to move inward….
• We check it in two-body
motion, the result shows
planet move inward.
Three-Body Motion
• The most important – conservation of
energy and angular momentum.
The numerical method is not good enough if these
quantities have too big fluctuation or secular error.
• Result
At dt = 10-3 ,
For angular momentum, the amplitude is about “10-8”
For energy, the amplitude is about “10-8”
[unit is reduce unit]
If there is no secular growing error, it means
conservation is O.K.!!
Three-Body Motion
Result : dt = 10-3
1.For angular momentum,
the amplitude is about
10-8 [unit is reduce unit]
2.For energy, the
amplitude is about 10-8
[unit is reduce unit]
angular
momentum
angular momentum of 3-body system ; 1000000 cycles
2.280490E-03
2.280485E-03
2.280480E-03
2.280475E-03
2.280470E-03
2.280465E-03
2.280460E-03
cycles
0
energy
-9.6800E-05
-9.6810E-05
250000
500000
750000
1000000
total energy of 3-body system ; 1000000 cycles
0
250000
500000
750000
1000000
-9.6820E-05
3.No secular growing error,
it means that
conservation is O.K.!!
-9.6830E-05
-9.6840E-05
-9.6850E-05
cycles
Gas Disk Model
• Gas pull on planet to
move inward….
Resonance Phenomenon
Tinner
Touter
• Resonance occurs when
is rational
fraction, like 1/2 . 1/3 ……
• Then we can use Kepler’s 3rd law to know
the “ratio of semi-major axis”.
• After resonance occurs, the ratio between
planets maintain the same value .
Resonance Phenomenon
• We consider a simplest model
<1> A 3-body motion in a gas disk, and we assume three
bodies are at the same plane.
<2> Outer planet is affected by gas disk.
Inner planet isn’t affected by gas disk.
• Then outer planet will move inward.
• The resonance will occur when outer
planet move to “suitable position” that ratio
of periods is rational fraction.
• Check a, e,
Resonance Phenomenon
Semi-major
6axis
• Check semi-major axis, we
can see that the resonance
occurs at about 134000 cycles.
Semi-major axis during 160000 cycles
inner
outer
5
4
3
2
1
cycles
0
0
• Here, the ratio fluctuates
between 1.57~1.60.
• The corresponsive ratio of
periods is 1:2.
• Check if ratio between planets
maintain constant or not.
50000
ratio
6
100000
150000
200000
The ratio between semi-major axis of two planets
5
4
3
2
1
cycles
0
0
50000
100000
150000
200000
Resonance Phenomenon
• Check the eccentricity
• It fluctuates more and
more with time.
• We can see that
“eccentricity of inner
planet” grows quickly
after resonance occurring.
Eccentricity
Eccentricity
inner
1
outer
0.8
0.6
0.4
0.2
cycles
0
0
50000
100000
150000
200000
Resonance Phenomenon
• Check longitude of
pericentre  .
•  of outer planet is
almost constant before
resonance occurring. It
means “the direction of
pericentre of outer
planet’s orbit.”
•  of inner planet changes
regular first,but fluctuates
when distance between
planets decreases.
The longitude of pericentre
The longitude of
pericentre
difference
inner
4
outer
3
2
1
cycles
0
0
-1
-2
-3
-4
50000
100000
150000
200000
Conclusion
1. Modified Hermite Method is good enough to calculate motion of
planetary system. The error amplitude is much smaller than
quantities we calculate, and the conservation of energy and angular
momentum are both O.K.
2. In three-body motion, the orbit of planet interacts to each other
strongly when resonance occurs.
Future Work
1. Compare theses results with perturbation theories.
2. Calculate the resonance problem by different initial conditions and
see the evolution of the orbit.
3. Calculate other quantities and check their evolution.
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Thanks for your endurance
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Thanks for all in ASIAA
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
Gas Disk Model
• Assuming the axissymmetric viscous
disk, the radial
velocity of the disk
v r , gas
 c 
 3 (2  p  q) 0 
 v K ,0 
2
1
 r  2 q
  v K , 0
 r0 
• The force acted on
the planet in
azimuthal direction
 c 
j
3
 dj 
f     / r  2 v r , gas    (2  p  q) 0 
2
2r
 dt 
 v K ,0 
2
r
 
 r0 
1 q
v K ,0  K ,0
f 
F
F  ma

a
m
m