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 ” A ngle 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” “ [unit is reduce] “Δe” fluctuates between” “ “Δ “fluctuates between” “ [unit is radian] Error amplitude of Runge-Kutta Method At dt = 2 -5, “Δa” fluctuates between” “ [unit is reduce] “Δe” fluctuates between” “ “Δ “fluctuates between” “ [unit is radian] Secular error of Modified Hermite Method (a, e have no secular err) At dt = 2 -5, “Δ “decreases ” “ during 30,000 cycles [unit is radian] Secular error of Runge-Kutta Method At dt = 2 -5, Δ a decreases “2* ” during 30,000 cycles, Δ e decreases “ ” during 30,000 cycles, Δ increases “ ” 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 = For angular momentum, the amplitude is about [unit is reduce unit] 2.For energy, the amplitude is about [unit is reduce unit] 3.No secular growing error, it means that conservation is O.K.!!

Gas Disk Model Gas pull on planet to move inward….

Resonance Phenomenon Resonance occurs when is rational fraction, like 1/2. 1/3 …… Then we can use Kepler’s 3 rd 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 A 3-body motion in a gas disk, and we assume three bodies are at the same plane. 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 Check semi-major axis, we can see that the resonance occurs at about cycles. 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.

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.

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.

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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Gas Disk Model Assuming the axis- symmetric viscous disk, the radial velocity of the disk The force acted on the planet in azimuthal direction