Computational Physics Project 3 Comet Dust Tails Outline ● Comet Dust Tails ● Forces acting on Dust Particles ● Two basic elements of calculation – Acceleration of dust grain near nucleus – Subsequent motion of dust grain in orbit ● Calculation Parameters and Questions to be considered Comet Tails ● Ion Tail – ionized gas driven away from sun by solar wind Ion Tail ● Dust Tail – Dust particles pushed by solar radiation pressure Dust – Curved shape with respect Tail to the nucleus is result of orbital motion of particles responding to the radiation pressure. Dust Tail Features ● Anti-tail – Large dust particles an move sunward before turning around under radiation pressure. – Most in plane of orbit ● Structures in Tail – Produced by changes in the amount of dust with time. ● We are going to try to explain these features. Dust Tail Forces ● Gravity of Sun – Nucleus in orbit around Sun, no other forces will be considered in calculating position of nucleus. – Dust grain in orbit around Sun, so solar gravity must be considered. ● Gravity of Comet – Dust grain feels gravity of comet. ● Drag of outflowing gas from the comet – Dust grain pushed by outflowing gas ● Solar Radiation Pressure – Dust grain pushed by radiation pressure of sunlight Gravity Force Geometry This shows where the dust Is with respect to the nucleus NUCLEUS r - r d n r DUST n GRAIN r d SUN Drag A density of air in cylinder: ρ Projectile distance travelled in ∆t = ∆L Mass of air molecules encountered in time ∆t: m = ρ A ∆L = d A v ∆t Energy imparted to air molecules through collision: ∆E ~ ½ m v2 ~ ½ ρ A v2 ∆L Energy comes from projectile: Work done = Drag Force X ∆L ∆E = Drag Force ∆L ~ ½ ρ A v2 ∆L Drag Force ~ ½ ρ A v2 Drag on Dust Grain C is drag coefficient drag v is gas velocity – gas is directed radially outward from nucleus gas A is cross section of grain ρ is gas density gas Q is number of water molecules per second MH2O is mass of water Where r is distance from comet nucleus molecule and R is radius of comet nucleus. Vgas is velocity of gas Radiation Pressure (Revised) P P Change = 2P S is solar constant = 1361 W/m^2 ʘ NOTE: Must Multiply by A, A the cross sectional area of the dust grain C = 2 for reflection or 1 for absorption rad A Basic Problem ● The dust acceleration step takes place near the nucleus in a relatively short time, so first part of calculation requires a small time step. ● The orbital behavior develops over longer times, so we'd like to take bigger time steps to make the program run faster. ● Suggested Approach: – Use a smaller time step for the initial period while the dust particle is still accelerating outward close to the nucleus. – Then change step size to a larger value to follow the particle into the tail. Nucleus Parameter Assumptions ● Mass of Nucleus = 9.982 E 12 kg ● Radius of Nucleus = 2000 m ● Production Rate: Q = 1 E27 molecules per second ● Vgas = 1000 m/sec ● Mass of H2O = 3 E-26 kg ● Orbit: Put the nucleus in a circular orbit around the Sun at 1 AU in the x-y plane. Dust Parameter Assumptions ● Particle size – lets call the radius “a” – We wish to consider a range of sizes from, say, 1 mm to 0.1 micrometers. ● Particle density – lets call this ρ and assume a d typical value of 2000 kg/m^3 ● Particle drag coefficient – adopt 2 ● Initial Conditions: – Particles start on surface of nucleus at zero velocity. – Particles can start from any location on the surface and will initially be sent off perpendicular to the surface. Particle Start Geometry z y R Θ Subsolar point is To Theta = 0 Phi = 0 Sun Φ (-x) So x = -R y=0 z=0 Steps in our investigation ● STEP 1: Make sure that you do Exercise 14 – the orbit – to be sure that your basic program works. It will be helpful if you have a function to compute the gravitational force. ● STEP 2: Write functions to compute the various other forces in the problem: – Gravity from Comet – Drag from gas – Radiation Pressure – CHECK THESE FUNCTIONS CAREFULLY Steps in our Investigation ● STEP 3: Combine the individual forces acting on the dust particle into a single function which can be called during the Runge-Kutta integration. – How can we test the program at this point?? – Be sure to discuss ways that you have verified that the program is working in your report. ● STEP 4: Investigate Behavior by carrying out runs of the program. See next slides for advice and key questions. Investigation of Acceleration ● We'd like to look at the initial behavior of dust grains close to the nucleus as they are accelerated by gas drag. ● I recommend trying some short runs of the program with different grain size to address the questions on the next slide, since there is no need to follow the particles into the tail for this. ● Experiment with starting the particle from different positions on the nucleus. You should see that answers are similar during the acceleration phase. Questions about acceleration phase ● Is there a maximum size of particle than can be lifted from the surface of the nucleus? (You might do best answering this question by just refering to the equations.) ● The final velocity of a particle depends on the grain size. – Why? (another one where looking at the equations will help to answer the question.) – What is the dependence of final speed on grain size? Investigation of “orbit phase” ● Next turn attention to what happens when the particle goes into the tail. ● It is fun to see what trajectories are followed if you start at different places on the nucleus, but it can get a bit confusing. ● Therefore, I think it is simplest to investigate behavior by having all particles start at the same place … I suggest the subsolar point … and then investigating what happens with different particle sizes. Questions about orbit phase ● In inertial space, what orbit do the dust particles follow? How does this depend on the grain size? Would you expect to find particles along the comet's orbit? ● Plot the trajectory of a particle in a frame of reference which is fixed on the comet nucleus with x axis along Sun-Nucleus direction. Y Xc Yc Nucleus Dust Θ X Questions about orbit phase ● In the comet frame: – Does the trajectory of particles look like a dust tail? – How does trajectory depend on grain size? Do you expect to see grains of different size in different parts of the tail? – Show the location of particles of different sizes after a fixed amount of time. How do they line up? Does this look like an explanation of the tail structures seen? – Can you make an “antitail”? What particle sizes are favored in the antitail? .