Welcome to CAMCOS Reports Day Fall 2010
and Dynamics in Electrically Charged Binary Asteroid Systems Doug Mathews, Lara Mitchell, Jennifer Murguia, Tri Nguyen, Raquel Ortiz, Dave Richardson, Usha Watson, Macken Wong Dr. Julie Bellerose, NASA Ames Center Dr. Jared Maruskin, Faculty Advisor December 10, 2010
CAMCOS Team 2010 Group Leaders Lara Mitchell Macken Wong Coding Team David Richardson Usha Watson Team Raquel Ortiz Tri Nguyen Calculation Team Doug Mathews Jennifer Murguia
Historical Context December 1972: rays of sunlight observed along the Moon s horizon during Apollo 17 Similar to shafts of light observed on Earth as light scatters off particles in the atmosphere. This phenomenon is known as Lunar Horizon Glow (LHG).
Historical Context LHG is believed to be caused by electrically charged dust particles levitating above the lunar surface. These observations began the modern study of dust levitation on airless bodies.
Motivation Lunar charging and dust levitation remains a topic of interest in space science: Lunar Atmosphere and Environment Explorer (LADEE) - 2013 mission to characterize the atmosphere and lunar dust environment But what is a new avenue of reseach within this topic? Near Earth Asteroid Rendezvous-Shoemaker (NEAR-Shoemaker) probe Dawn spacecraft Hayabusa spacecraft Orion Asteroid Mission - manned exploration of an asteroid
Our Questions What happens when electrical charging occurs in a binary asteroid system? Under what conditions do charged particles in this system levitate? What fraction of levitated particles escape, transfer, and return? What do particle trajectories look like? These are the types of questions we set out to answer in our project.
Project Overview Project Overview: Create a model for the motion of a charged particle in a charged binary asteroid system ly simulate trajectories of charged particles Investigate long-term behavior of the system Understand mass transfer between asteroids
Previous Previous research on related topics: Electrical charging on a single body (Moon) Electric field formation due to charging 3-Body - describes motion of a small object in the presence of two larger objects (gravitational interaction only) New features of our model: Electrical charging on asteroids Charging in a multiple body system 4-Body with gravitational and electric forces
Presentation Overview Surface charging mechanism 3-Body (R3BP) Intermission model Electric force from asteroids Gravitational force from the Sun model Overview of programs Graphical respresentation of potential energy Charged particle levitation When and where does this occur? Charged particle trajectories
Overview of Goal: incorporate electrostatic fields into binary asteroid systems, study levitation on asteroids and its effect on mass transfer. Definition of dust levitation Mechanism of dust levitation Electrostatic field production conditions of charged particles
What is an Asteroid? An asteroid is not a single solid body. It is composed of a main body plus loose rocks, particles, and rubble that rest on the surface Asteroid Itokawa - September 12, 2005
What Is? Asteroid has low gravity enviroment Electrical forces could be significant The interaction of the gravitational force and electrical forces on dust particle on asteroid surface may result in the particle loses contact with the asteroid surface and is levitated.
Mechanism of Photoelectric effect s on asteroid surface are bombarded by photons from the Sun Electrons in those particles gain energy and escape s lose electrons and become positively charged
Electrostatic Field Production Result: Creates charge differential between sunlit and dark hemispheres that gives an electrostatic field around the asteroid Creates charge differential between sunlit and dark part of a feature near the terminator region resulting a smaller (local) electrostatic field around the terminal region.
What Factors Affect Charging Asteroid s size: larger size means more charging time for that particles on asteroid s surface resulting a stronger field Composition: conductivity on the surface affects the ability of particle to hold charge. Higher resistivity also means stronger field Spin rate: slower spin rate means longer charging time for the particles, and thus gives a stronger field
Conditions of free particles Pascal Lee s Conditions: F e > F g General Condition: F n > 0 F n : net outward force acting on loose particle F n includes contributions from Electrical force acting on the particle Gravitational forces of other bodies in the system Centripetal force due to spinning asteroid Inertial forces due to rotation of frame, centripetal and Coriolis forces.
Conditions of free particles
The Three Body The Three Body - To understand the motion of three mutually gravitating bodies in space. The Planar Circular Three Body (PCR3BP) - We build in a number of physical assumptions to simplify the model. Two main bodies - primaries Primaries are in circular orbit around their center of mass. Third body is small - does not affect the orbit of primaries. The third body remains in the plane of the primaries orbit.
Examples There are many common astrodynamical examples where the PCR3BP accurately models the motion: Earth - moon - satellite Sun - Earth - moon Two asteroids and a satellite Two asteroids and a particle Two men and a baby PCR3BP gives us physcial insight into physical systems we otherwise would not have.
Rotating frame In the PCR3BP the non-intertial, rotating frame produces fictitious forces. Coriolis Forces - Force perpendicular to the particle s velocity. Centripetal Acceleration - Outward inertial force.
Libration Points There exist five fixed points, Libration or Lagrange points. If a particle with zero initial velocity is located at one of these points it will remain there for all time. L 1, L 2, L 3, are colinear with the two asteroids on the x-axis. The other two, L 4 and L 5 form equilateral triangles with the two primaries.
Location of Libration Points in Sun-Jupiter System Clusters of asteroids have been found at the L 4 and L 5 libration points of the Sun-Jupiter system. The asteroids located at the L 4 point are called the Greeks. The asteroids located at the L 5 point are called the Trojans.
Integral of Motion An integral of motion is a quantity which: Depends on the particle s position and velocity Remains constant in time when evaluated along the particle s trajectory A common example of this is total energy. When you toss a ball, E = KE + PE remains constant along the ball s path.
Total Energy of System J = KE + PE In the 3-Body, we have a function which acts like the potential energy, but contains terms related to centripetal acceleration. The sum J = KE + PE is conserved. J can be determined solely based on the particle s initial position and velocity.
Permissible and Initial condition = J o = KE o + PE o. Along trajectory = J = J o = KE + PE(x, y) KE = J o PE(x, y) Fundamental fact = KE 0 This divides the space into different regions: Admissible region: J o PE(x, y) 0 Zero Velocity Curve: J o PE(x, y) = 0 region: J o PE(x, y) < 0 Example: tossing a ball
Dots represent Earth and Moon in 3BP Shaded region is forbidden at given energy level
5 minute Intermission
Our Goal: model the levitation and motion of a charged particle in a charged binary asteroid system Building the : Develop a 4-Body the motion of the charged particle Investigating the Results: Contour maps of potential energy Asteroid levitation conditions Charged particle trajectories Animation of time-dependent forbidden regions
Building the contains 4 bodies: Sun, two asteroids, particle Binary asteroid system dynamics: Binary asteroid system orbits the sun (Ω) The two asteroids orbit their center of mass (ω) Each asteroid rotates about its own axis (Ω s )
Building the dynamics: Physical forces acting on charged particle: ma = F g1 + F g2 + F e1 + F e2 Fictitious forces from noninertial reference frames Resulting expression for particle acceleration: a = R + ω (ω ρ) + ( ρ) r + 2ω ( ρ) r 1. centripetal acceleration due to revolution about Sun 2. centripetal acceleration due to rotation of frame 3. particle s acceleration relative to frame (we solve for this to obtain equations of motion) 4. Coriolis acceleration due to rotation of frame
Electric Potential How to determine the electric force on the particle? Charging assumed to result in a step potential: +5 V on sunlit hemisphere -1000 V on dark hemisphere Use LaPlace s Equation to find the electric potential of the system
Electric Potential Binary System Electric Potential
Electric Field Electric field obtained from potential: E = V Electric force on charged particle: F = qe Binary System Electric Field
Potential Energy The potential energy governs levitation conditions and particle trajectories Our model for PE includes Gravitation from the sun and asteroid bodies Electrical forces from the charge on the asteroids Centripetal and Coriolis forces due to rotation of frame Centripetal force due to revolution about sun Electric potential includes superposition of two Legendre Series solutions of Laplace s equation. We ve computed 500,000 coefficients of the Legendre Series to estimate electric potentials
Potential Complexity ẍ 2ωẏ = U x, U = 1 2 (x 2 + y 2 ) Π 1 [ Π 2 [ V 1 (r 1, θ 1 ) + V 2 (r 2, θ 2 ) B 2n+1 = 4n + 3 2 V (r, θ) = B 0 r U ÿ + 2ωẋ = y, U z = z x sin ( (1 ω i )t ) + y cos ( (1 ω i )t ) ] ] [ 1 µ Π 3 + µ ] r 1 r 2 + n=0 B 2n+1 r 2n+2 P 2n+1(cos θ) ( 1) n (2n)! (α + β) 2 2n+1 (n!) 2 (n + 1) α + β as n πn Π 1 = d R 2 = d Ω 2 R ω 2, Π 2 = q m = q q 0 M m, Π 3 = GM ω 2 R 3
Motivation for Simulations How do we visualize this? We turned to computer simulations.
Motivation for Simulations Computation of potential energy, its gradient, and particle trajectories is computationally intensive. We ve worked hard to optimize our code, but computation time continues to be a constraint. Our graphing software helped us gain physical insight into system, validate the model We can use simultaneous plots of the various formulas to check the model and code for consistency.
Programs and Features Our program produces: contour plots of electrical potential Charge distribution on asteroid s surface computation of derivatives of potential energy - necessary for equations of motion integration yields individual particle trajectories time dependent forbidden regions animations of trajectories/potential contours/forbidden regions
on the surface of the asteroid floats off the surface. In general this happens because: The surface and the dust get charged simultaneously They become charged with the same type of charge(+) This causes the dust to be repelled from the surface Prevous work has modeled this for single bodies. Once the particle has levitated, it becomes subject to the equations of motion for our model (leading to trajectories) Generating these maps is non-trivial as we increase the complexity of the model.
in Our Since we have more than one asteroid we have several new factors influencing levitation. The contributing factors new to our model are: Centripetal force caused by the rotation of the asteroid. Centripetal force caused by the orbit of the system around the sun. Gravity from the second asteroid. Corolis term that is a result of our rotating frame. Electric force from the second asteroid.
Map will Levitate when F n > 0 (the outward normal force is greater than zero) map drawn as a Mercator map of the asteroid s surface, like a flat map of the Earth. occurs in shaded regions.
Map Over Time Our system changes as a function of time, therefore our levitation map changes too.
Outward Force We are not only interested in where this could happen, we also need the actual force on the particles. With the Force we can calculate initial velocities once the particle is levitated
Our computational methods The trajectories were calculated using the ODE45 differential equations solver in MATLAB. To differentiate our potential-like function, we used Richardson s extrapolation on a Simpson s method divided difference. Our model is a three-dimensional model, planar graphs and initial conditions were used in this talk for visibility. But three dimensions are necessary for the levitation maps.
3 Types of Orbits We can separate trajectories of levitated dust particles into three categories, based on their ultimate motion: Return trajectory: A particle trajectory that ultimately returns to the asteroid of origin. Transfer trajectory: A particle trajectory that ultimately impacts the other asteroid. Escape trajectory: A particle trajectory that ultimately escapes to infinity.
A Sample Transfer Orbit 1.5 y 1 0.5 0 0.5 1 1.5 2 2.5 3 Initial Conditions [1.2, 0.6, 0, 4.5, 5.75, 0] 3.5 4 3 2 1 0 1 2 3 x
A Sample Return Orbit y 3 2 1 0 1 2 3 4 5 Initial Conditions [ 2.75, 0.7003, 0, 10, 8.3, 0] 3 2 1 0 1 2 3 4 5 x
A Sample Escape Orbit y 4 3 2 1 0 1 2 3 Initial Conditions [ 1.2, 0.62, 0, 3.6, 5, 0] 6 4 2 0 2 4 x
regions In the R3BP, the potential is independent of time. J o PE(x, y) < 0 In our model the potential energy is time-dependent. J(t) is not conserved along the trajectory. Therefore the forbidden region varies with time. J(t) PE(x, y, t) < 0 If we could freeze the current value of J(t) and the potential energy function, the instantaneous forbidden region would be the region which the particle could not reach with that fixed value of energy.
Run trajectory simulations for a large number of particles, based on time-varying levitation maps, in order to obtain statistical data on the probability of escape, transfer and return. Analyze whether there are special characteristics to orbits initially at the instantaneous Lagrange points. Study the mutual electrostatic force of primary bodies (asteroids) on each other. Incorporate levitation and mass transfer into non-restricted binary asteroid systems with irregularly shaped asteroids.
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