Monte Carlo Model of Comet Dust Computational Physics Monte Carlo Model of Comet Dust Project 1
About Comets... Coma (Head) Dust Tail Ion Tail Nucleus 10^4-10^5 km 10^5-10^6 km 10^5-10^6 km 1 km All the comet activity starts with the nucleus and sublimation of ices. We'd like to understand that better, so we need to look at the process up close. The materials in the comet nucleus are some of the least altered materials left from formation of the solar system.
Project Overview Submillimeter-wave Observations of Comet Churyumov-Gerasimenko (67P) by the MIRO instrument on the Rosetta Spacecraft show enhanced emission above the limb of the comet nucleus. MIRO Team thinks this is due to large (several millimeters) dust particles lifted off of the nucleus as ice sublimates and streams away from the surface. Our project will be to construct a model of this process and see if we can match the observations.
MIRO Instrument 30 cm Observe at 0.5 mm 1.6 mm
MIRO Observation
What's going on here?? SubMM Channel Map
Basic Dust Observation DUST EMISSION
log TA (K) Analysis of Radial Profiles log r (km) Dust coming from sunlit side of the nucleus -1.6 Radial fall-off goes as r
Our Problem Simple model of uniform spherical expansion at constant velocity would expect that dust signal -1-1.6 would decrease as r. So, why r? Ideas: Dust grains accelerate as they are pushed away from the nucleus by the gas. Large dust grains seen by MIRO get smaller as they go out. Eventually they are too small to be seen at millimeter wavelengths. The flow of material from the nucleus is highly nonuniform. Perhaps a different pattern of dust flow can account for the difference?
How can we test this? Construct a model of the outflow and compare to observations Given the radial distribution of dust around the nucleus, we may integrate the density distribution along the line of sight. Our signal is proportional to this quantity. s r b x0
For outflow at constant speed, number density, n, ( number of particles per cubic meter): Integrate number density to get column density total number along line of sight per area. s r R Expect: b x0 N(b) goes as 1/b
Problems In our real comet, the function n(r) will be complicated: Speed is not constant Particles destroyed n(r) not symmetric but could depend on lots of horrible stuff. When confronted with lots of complexity, sometimes the best bet is to simulate what is going on and observe it to model the result.
Simulation Particles originate at surface of spherical nucleus (radius R). We can choose to start particles from anywhere on the nucleus. Particles go in some direction with a velocity profile that might include acceleration. We integrate the velocity profile to get location of particle as a function of time. Particles may be destroyed on a particular efolding time scale.
Geometry X Particle Location: r Z to Sun Y to MIRO
How do we specify location? Need to determine r, theta, and phi of a lot of particles. r depends on velocity of the particles and time that particle has been traveling before we observe it. We need to know the time since creation before observation. We need to know r(t) profile consistent with velocity model. theta, phi must be selected according to where the particle originated.
r location Time of Simulation t1 Time of Flight, t Observation Time Time of creation of particle 0 Given time of simulation, t1, we will say that a particle could originate at any time between 0 and t1 with equal probability. A UNIFORM distribution. t1 must be selected so that we properly sample all particles along our line of sight. Since we need to look out to infinity, it follows that t1 will be big enough that the distance travelled by the particle will be pretty large compared to the distance we'd like to calculate column density. Given t1, we select t, the time of flight for a particular particle. Then we need to know r(t) to know its distance from the nucleus.
Choice of t1 MIRO 100km r = v t1 We are interested in column densities out to 100 km from nucleus. t1 must be chosen so that maximum distance a particle can travel is very large compared to 100 km in order to compute good column densities.
r(t) models NOTE: r(t) assumes v(t) Constant Speed Accelerating Speed We will consider three models: MODEL Particle Size v0 (m/s) tau0 (s) A Small 2.45 1600 B Medium 1.15 3300 C Large 0.25 14000
Direction Models Select theta and phi according to probability that emission will be into a particular solid angle. For isotropic emission All phi's are equally likely: Uniform Distribution All values of cos(theta) are equally likely:
Procedure Given t1 and R (assume 2000 m): For each particle: Draw 3 random numbers: r1, r2, r3. Compute time: t = t1*r1 Compute cos(theta) = 1-2*r2 Compute phi = 2pi *r3 Use r(t) formula to find r. Location is then -----------> Do this LOTS of times
Observables By LOTS I really mean a lot certainly millions, maybe a billion. It is NOT advisable to keep all locations in memory. Better to compute the observables as particles are calculated. Observables How many particles in a bin at a particular radius? How many particles in a bin of b the impact parameter? How many particles in bins in the X-Z plane this makes an image of the emission.
Binning Results Accumulate particle into Bin in radius. r R R+dr R+2dr R+3dr R+4dr r Accumulate total number of particles in each radius bin. This gives the radial distribution of the particles in the simulation. Note that for the radial distribution you are counting numbers in concentric shells around the nucleus.
Other Bins For simplicity, we will assume MIRO at infinite distance. The plane of the sky seen from MIRO Is the X-Z plane In our coordinate system. X Column Density is found by counting in annular rings around nucleus. b Bin for Ring b + db Z X An image may be computed by accumulating points in pixels. A pixel is a small box in the x-z plane. z+dz x+dx Bin for pixel z+dz, x+dx Z
What to do with Numbers in Bins? Total number of particles divided by time of simulation is proportional to rate of emission. Note that the number of particles that wind up in a bin divided by the total number of particles in the simulation is the fraction of all particles in the bin. Number Density (n)versus radius is proportional to fraction in a radial shell divided by volume of shell. Column Density (N) is fraction in annular ring divided by area of ring. Image Intensity (I) in image is fraction of particles in a pixel divided by area of the pixel.
How Many Particles? That's ultimately up to you. Justify your choice. Things to consider: In the simulation, the number in a particular bin will change from one simulation to the next. There will be noise on our radial profiles. How does amount of noise depend on total numbers? What is acceptable level of noise in the result?
Example Uniform Distribution 10 bins Two experiments with 1000 particles. Note change and noise in simulation. Two additional experiments with many more particles. Note decrease in noise.
Development Steps Isotropic Particle Direction at Constant Velocity Provides case where answer is known (see previous slides). number density proportional to 1/r^2 column density proportional to 1/b Compute constant velocity cases for models: A,B,C How do quantities depend on value of velocity? Isotropic Particle Direction using r(t) model for acceleration. How do the number density and column density change compared to constant velocity case? Do any of the column density models match the MIRO observation that column density falls as 1/b^1.6??
Development Steps (continued) Add particle destruction to the model: Particles may breakup or evaporate with time. The probability of a particle existing at time t, P(t), will decrease exponentially with some lifetime L: To consider this effect, accumulate particles into bins with a weight P(t). Consider constant velocity and accelerated velocity models. How do results change when this effect is added to the calculation? How does radial falloff in column density change with lifetime? Can we do a better job of matching the observed radial falloff in column density?
More things to try (extra credit) Observation shows that dust is coming from the day side of the nucleus. Modify particle production so that particles come from day side only. Consider what happens to images of the emission when we get production from day hemisphere or from even smaller angles. How does the program need to change to consider the true observation from a finite distance of, say 150 km, from the nucleus? Does this affect our results computed for infinite distance?
Still more things to try... How would we handle time variability of the emission in the model? Comet rotates with 12.4 hour period. What happens if there is an active spot which turns on when it is in the sun? What happens if the rate of emission changes during the time of the simulation?
What am I supposed to do? Write Python program to do the simulation. Compute observables for basic cases: Constant V models (use V0 for models A,B,C) Accelerating V models (use models A,B,C) Particle destruction models. For each model characterize r behavior of n and b behavior of N as a power law and derive exponent for comparison to MIRO observation. Consider other suggested calculations if inclined to do so...
Write Report Background Describe Numerical Method and Rationale for selecting this method. Demonstrate checks you have made to be sure that the program is correct. Write description of results; answer any questions posed. Include copy of scripts.