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 beconsidered

Comet Tails

● Ion Tail – ionized gas driven away

from sun by solar wind

● Dust Tail – Dust particles pushed by

solar radiation pressure

– Curved shape with respectto the nucleus is result oforbital motion of particlesresponding to the radiationpressure.

IonTail

DustTail

Dust Tail Features● Anti-tail

– Large dust particles anmove sunward beforeturning around underradiation pressure.

– Most in plane of orbit

● Structures in Tail– Produced by changes in

the amount of dust withtime.

● We are going to try toexplain these features.

Dust Tail Forces● Gravity of Sun

– Nucleus in orbit around Sun, no other forces will beconsidered in calculating position of nucleus.

– Dust grain in orbit around Sun, so solar gravity mustbe 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

NUCLEUS

DUSTGRAIN

SUN

rn

rd

rd - r

n

This shows where the dustIs with respect to the nucleus

Drag

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

distance travelled in ∆t = ∆L

A density of airin cylinder: ρ

ProjectileProjectile

Drag on Dust Grain

Cdrag

is drag coefficient

vgas

is gas velocity – gas is directed radially outward from nucleus

A is cross section of grain

ρgas

is gas density

Q is number of water molecules per secondMH2O is mass of water moleculeVgas is velocity of gas

Where r is distance from comet nucleusand R is radius of comet nucleus.

Radiation Pressure(Revised)

P

P

Change = 2P

Crad

= 2 for reflection or 1 for absorption

Sʘ is solar constant = 1361 W/m^2

A NOTE: Must Multiply by A,the cross sectional areaof the dust grain

A Basic Problem● The dust acceleration step takes place near the

nucleus in a relatively short time, so first part ofcalculation requires a small time step.

● The orbital behavior develops over longertimes, so we'd like to take bigger time steps tomake the program run faster.

● Suggested Approach:– Use a smaller time step for the initial period while

the dust particle is still accelerating outward close tothe nucleus.

– Then change step size to a larger value to followthe 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, 1mm to 0.1 micrometers.

● Particle density – lets call this ρd and assume a

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 surfaceand will initially be sent off perpendicular to thesurface.

Particle Start Geometry

RΘ

Φ

ToSun(-x)

z

y

Subsolar point isTheta = 0 Phi = 0

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 programworks. It will be helpful if you have a function tocompute the gravitational force.

● STEP 2: Write functions to compute the variousother 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 actingon the dust particle into a single function whichcan be called during the Runge-Kuttaintegration.– How can we test the program at this point??

– Be sure to discuss ways that you have verified thatthe program is working in your report.

● STEP 4: Investigate Behavior by carrying outruns of the program. See next slides for adviceand key questions.

Investigation of Acceleration

● We'd like to look at the initial behavior of dustgrains close to the nucleus as they areaccelerated by gas drag.

● I recommend trying some short runs of theprogram with different grain size to address thequestions on the next slide, since there is noneed to follow the particles into the tail for this.

● Experiment with starting the particle fromdifferent positions on the nucleus. You shouldsee that answers are similar during theacceleration phase.

Questions about acceleration phase

● Is there a maximum size of particle than can belifted from the surface of the nucleus? (Youmight do best answering this question by justrefering to the equations.)

● The final velocity of a particle depends on thegrain size. – Why? (another one where looking at the equations

will help to answer the question.)

– What is the dependence of final speed on grainsize?

Investigation of “orbit phase”

● Next turn attention to what happens when theparticle goes into the tail.

● It is fun to see what trajectories are followed ifyou start at different places on the nucleus, butit can get a bit confusing.

● Therefore, I think it is simplest to investigatebehavior by having all particles start at thesame place … I suggest the subsolar point …and then investigating what happens withdifferent particle sizes.

Questions about orbit phase

● In inertial space, what orbit do the dust particlesfollow? How does this depend on the grainsize? Would you expect to find particles alongthe comet's orbit?

● Plot the trajectory of a particle in a frame ofreference which is fixed on the comet nucleuswith x axis along Sun-Nucleus direction.

X

YXc

Yc

ΘDust

Nucleus

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 youexpect to see grains of different size in differentparts of the tail?

– Show the location of particles of different sizes aftera fixed amount of time. How do they line up? Doesthis look like an explanation of the tail structuresseen?

– Can you make an “antitail”? What particle sizes arefavored in the antitail?