Computational Physics Orbital Motion
Equations of MotionComputational PhysicsOrbital Motion
OutlineFourth Order Runge-Kutta MethodEquation of motion in 3 dimensionsProjectile Motion ProblemOrbit Equations
Second Order Runge-KuttaDiferentialEquationEstimate valueof y at half-step(Euler Method)Use value athalf-step tofnd new estimateof derivative
Fourth Order Runge-KuttaEstimate of derivative in intervalValue at beginningof intervalTwo estimates ofvalue at mid-pointEstimate of value atend of interval
Motion in Three DimensionsIndependent Equations Single Vector Equationfor each dimensionin Python we write vectors:r [x, y, z]F [ Fx , Fy , Fz ]
3D Solution in PythonIndependent Equations - ionComponentsX[0] 0.Y[0] 0.Z[0] 0.VX[0] 0.VY[0] 0.VZ[0] 0.# FX, FY, FZ are components of forcefor i in range(n):VX[i i] VX[i] FX[i]/m*dtVY[i i] VY[i] FY[i]/m*dtVZ[i i] VZ[i] FZ[i]/m*dtX[i i] X[i] VX[i]*dtY[i i] Y[i] VY[i]*dtZ[i i] Z[i] VZ[i]*dt
3D Solution in PythonVector Equations - Euler MethodX0X0 [0.,0.,0.][0.,0.,0.]V0V0 [0.,0.,0.][0.,0.,0.]InitializationVelocity VectorPosition VectorXX np.zeros(np.zeros((nsteps,3)(nsteps,3)) )VV np.zeros(np.zeros((nsteps,3)(nsteps,3)) rsvectorsX[0,:]X[0,:] X0X0V[0,:]V[0,:] cheachstepstepi iforfori iininrange(nsteps):range(nsteps):V[i 1,:]V[i 1,:] V[i,:]V[i,:] F[i,:]/m*dtF[i,:]/m*dtX[i 1,:]X[i 1,:] X[i,:]X[i,:] V[i,:]*dtV[i,:]*dt
Projectile Motion ProblemMotion of particleunder gravity, andeventually otherrealistic forces.Initial Conditions:specify location ofbeginning oftrajectoryspecify initialvelocity
Equation of MotionGravity OnlyGravity is only force:Acceleration in -z direction
Constants of the MotionGravity OnlyConstants of motion are useful forevaluating whether your programworks!No Force in X and Y directions:momentum in X and Y conservedForce of gravity depends on positiononlytotal energy is conservedpotential energy m g zkinetic energy ½ m v 2total energy: E ½ m v 2 m g z
Orbit ProblemEquation of MotionSecond Order ODERadial Forcedependent onposition only:Angular Momentumconserved; Motion ina plane. ObjectrEnergy conserved.Sun
The r unit vector.Gravitational Force is radial, so needunit vector in r direction to derive force.A convenient way to look at this, forpython programs is:Magnitude of r is numpy.linalg.norm(r)
Initial Conditions3 Dimensional, Second Order D.E.6 Numbersinitial position: r [x, y, z] at time 0initial velocity: rdot [vx, vy, vz] at time 0Each set of initial conditions has uniqueorbit. Can characterize orbit with any sixnumbers that will describe it.Astronomers use "Orbital Elements" tospecify and describe orbits.
Orbital ElementsSize and Shape of OrbitSemimajor Axis - aEccentricity - eOrientation of Orbital Plane in SpaceInclination wrt EclipticLongitude of Ascending NodeArgument of PerihelionTime of Perihelion Passageeaa
Constants of the MotionSpecifc Energy (Energy per mass) ConservedPotentialKineticSpecifc Angular Momentum Conserved
Orbital Elements andConstants of the MotionSemimajor Axis - aSemimajor Axisdependsonly on energyEccentricity - eEccentricity dependson angular momentumand energyNote: Eccentricity Vector abovepoints towards periapsis
Computational Physics Orbital Motion. Outline Fourth Order Runge-Kutta Method Equation of motion in 3 dimensions Projectile Motion Problem Orbit Equations. Second Order Runge-Kutta Diferential Equation Estimate value of y at half-step
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