Comet Dynamics and Implications For Orbiting Spacecraft

So it's been a while since my last update. I've completed my masters program, gotten a job, moved to Tucson Az, and haven't had much time to MatLab. Also, my MatLab subscription ran out! OH NO! I use it at work but I can't use it for my own uses. So I have been learning c++ and will continue to update this blog with whatever I create using that.

So I thought I would add my thesis to this blog. Below is a link to download the PDF file. Let me know what you think! (Honestly, I thought I skimped out a little bit on the orbiting satellite analysis. If I had another month, it would have been epic.)

Comet Dynamics and Implications For Orbiting Spacecraft

Quick Update

Thesis is done and has been defended. Once it has been published I will include a link here. Now I just need to finish the semester and find a job...

A few things have been left out of the thesis analysis since I ran out of time, such at the stability of the comet's rotation and conduction between discretized faces. In time I will include them into the comet simulation and ultimately into SpaceSim, which I plan on restructuring to allow for more modularity and less glitchiness.

I haven't written anything about the satellites interaction with the outgassing jets. Below is a figure of the orbiting satellite, but the body of the text is in the thesis, I don't feel like writing it all out right now. There is also a video that goes along with it, but it isn't optimized since it wasn't needed in the thesis (damn paper not displaying moving images) and I just needed a snapshot (as seen below).

Orbit of a satellite with/without interaction with outgassing jets. 

Other then that, I helped some colleagues develop an app to diagnose anemia by looking at the color of eyes/nails. I did the coding and created a GUI for it, but I can't discuss more then that since there is some no-talking-about-it thing, but it's pretty cool and works well. 

Thesis and things

For the last while I have been hard at work on my thesis, which mainly encompasses what what I have written about. Right now I am sitting at a solid 29 pages (written, 40 total including all the random bull-stuff), with 3 of the 5 chapters are done, with chapter 1 being the intro, 2 is the comet model with sublimating jets and whatnot, and 3 is the dynamic expressions used. The 4th chapter will be the simulated results and the 5th is analyzing a satellite orbiting the comet.

So, I will only talk about the new stuff. 

Going off the last post, I added in a variable shape to go along with the change in inertia and mass. Essentially I just made the face vertices into a variable. It's a little buggy right though. The way it is written, the faces translate, but do not change area. Therefore after many orbits, it becomes very inaccurate. I have been playing around with ways to fix that, like changing the area as a ratio of volume to surface lost, however there are implications for the other faces which use that vertex. So that is going on the back burners for now.

 I also have written a way to describe the outgassing jets, which can be seen in the figures below. 

Left: Density/Pressure ratio throughout the outgassing jet 

Right: Geometry of the outgassing jet

To "get around" the fact that the faces are triangles, a circle of equal area is placed at the center, and is considered the pseudo-face. Beta functions (alpha=beta=2) are then used to describe the distribution through the jet. While the density at the face reaches 150%, it does not make a difference in the far field. It can be remedied with a beta function varying with distance from the face (starting at alpha=beta=1 which is a square wave), but I decided it was more trouble than it is worth. Using these definitions, the effect of outgassing jets on a orbiting satellite can be found with a different simulation I have also been working on, but that results from that are for a different time.

Comet Dynamics II

So I finished the comet model for now (for a second time). I have reduced how long the model takes to run again from 33 seconds to 14 seconds by taking out the cross function and replacing it with the direct equation.

Changes from the last post:

• Variable inertia matrix

The inertia matrix now changes with the ejected mass. This makes the simulation more realistic. However, the shape of the comet does not change (ie. the change in mass does not change the comet thickness). 

• Random off-gassing jet placement and variable composition

Instead of off-gassing jets being able to be produced on every face, now the chance that a jet will be on a specific face is randomized (I currently have it at a 5% chance). The composition of those faces are then randomized at a maximum of 5% of the face being volatilizes. This also effects the initial inertia matrix.

One implication of this is with constant face jets, the comet usually will spin up along a specific rotation axis. This can be seen in the following video. The great thing is that the axis it settles on does not have the be a major or minor axis due to the off-gassing thrust.

Black line: Rotation axis.

Yellow line: Sun vector .

Comet Color: Proportional to temperature (Red is hot).

Dark blue line: Total force on body.

Rings: Torque direction (green is +, red is -) and normalized magnitude. 

Light blue lines: Direction and normalized magnitudes of off-gassing jets.

Off-Gassing Dynamics

The off-gassing of an asteroid also has implications in satellite dynamics. However, different size and velocity particles are created and can effect a spacecraft in different ways, ie. it can totally destroy it, puncture a solar panel, provide a torque, etc.. All of which would make precise data acquisition difficult. The second part of my thesis project will be analyzing this.

• Off-Gassing

On top of the trust off-gassing provides to the asteroid, it also provides a pressure on spacecraft, as well as accelerates dust particulate off the surface of the comet. To be able to analyze this, I have made a simple model to find the velocity that specific size dust would travel at. The velocity of each particle is a function of the radius of the comet, the coefficient of drag, the solar flux density, a diffusion coefficient, radius of the comet, the gas ejecta density and velocity, and the particle density and radius. 

In most literature I have seen, out-gas jets are taken to have a half-angle and not diverge from that. However, since the pressure will not be constant through the coma, I created the diffusion coefficient. There is no empirical data behind this except that a high pressure jet would not stay at a constant half-angle due to pressure differentials between the jet and a low density area / vacuum, ie. it would defuse. Also, literature says the majority of mass will be at 20% of the velocity of the jets and usually be between 20 and 100 m/s. My off-gassing model adheres to that for dust larger than 1 mm. I believe this makes sense, but that's because I made it this way. 

The following plot is the results from this model:

Dust Radius: [0.0001;0.0005;0.001;0.0025;0.005;0.01;0.02;0.03;0.04;0.05] (m)

Gas Velocity: 400 (m/s)     Gas Density: 0.004 (kg/m^3)     Dust Density: 300 (kg/m^3)     

Diffusion Ratio: 1.5           Coefficient of drag: 0.5              Comet Radius: 500 (m)

Left: Velocity of 10 different particles

Middle: Distance from initial location of particles

Right: Median acceleration of the particles

Conclusions:

While I showed one run of the model here, it can be seen that the particles settle on a certain velocity relatively quick (withing 15 seconds for any "reasonable" parameter combination). I need to do further research or develop a formula to find the velocity from the above parameters without having to run a quick simulation. On top of that, at close quarters to the comet (within 1000 meters, the acceleration due to the gas dominates, whereas the acceleration due to gravity and solar radiation pressure gain prominence after a while. 

So whats next:

• Satellite Dynamics

Here is a quick model of the satellite I will be using (it's the same as past satellite pictures, but now it has an "engine" in the back and a "camera" in the front). Behold the JimSat in all it's glory:

JimSat

I will be using this to analyze a satellites ability to acquire data in highly dynamic offgassing/asteroid environment.

• Coma and Tail Model

I'm not too sure if I will be able to do this, but with the given trajectory and composition I want to predict the tail and brightness of the coma. Maybe even predict if the comet will give the Earth a meteor shower, I don't know.

Comet Dynamics

So I finished the comet model for now. It works well for me to move on and work on other parts of the model. I have verified it with literature and it gets similar results (which was a nice thing to see), but it still needs some work. It should be noted that the only "volatile" taken into account so far is ice/water vapor. 

Before I get into the bulk of this post, I found a few edits which make the model run magnitudes faster (~33 seconds opposed to 40 minutes). 

   - Due to the differential equation solver (ode113), if the axes lengths of the ellipse are equal or if the rotation axis is along a ellipse axis, the model will take forever (about 40 minutes, depending on simulation time) to run. This is because the differential equation solver wants to keep the angular velocity along the same axis, or having an angular acceleration of [0;0;0]. By making the rotation axis slightly offset from an ellipse axis, this is avoided since it neither wants to keep a constant rotation axis as well as constant angular velocity. I currently have the rotation axis at [pi/1000;pi/1000], or 0.18 degrees, and it works amazingly.

• Comet Model (Asteroid model with off-gassing)

The difference between the asteroid model and comet model is the thrust and energy use from off-gassing. The temperature of the face is used to predict the sublimation rate. Since the comet is in a vacuum, the velocity of the gas should be mach 1, which is found easily using values for water vapor for molar mass and heat capacity ratio. The force is also found and is shown to greatly effect the rotation of the comet.

So without further explanation, here are the same plots and videos from the last two posts, but with off-gassing. The first plot is a comet with axes a=5000, b=3000, and c=2000;, the second is scaled down by 100. 

Large Comet:

Small Comet: 

Conclusions: 

The time scale for both the first and second simulations was the same (just under 1 day's time). Notice the relatively constant temperature of the fast rotating comet, as shown in the video, this makes the force tend to be in the direction of travel, instead of along the anti-sun vector. This has some large implications for small comets because this means that they would (rather quickly) increase their semi-major axis until the off-gassing reduced to a negligible amount. Also, it can be concluded that small comets either will break apart due to their spin rate, become tumblers, or break apart from internal pressure build up. 

YORP Update

This will be quick post. I have found a few errors in the YORP model and corrected them. I was wondering why the force plot looked as it did, the reason is because it was only recording one face. As you can see below, this has been fixed.

Following are a few things to note and the new output plot as well as a video I made to validate some aspects of the model.

Things to note:

• The torques and forces are in the body frame. To make it useful it needs to be put into the orbital frame (check the video).
• The average temperature reaches steady state as the number of faces increases and the asteroid becomes circular about the rotation axis (if the rotational and orbital angular momentum axes also align).
• The warmest time on the asteroids is in the afternoon (around 2-4).

For the video below, the yellow line is the direction of solar rays, the red line is the direction of the force due to solar radiation pressure and thermal emittance. The color of the asteroid is relative to it's temperature with the colormap [1 0 0] (red) being the total hottest temperature and [0 0 0] being absolute zero. 

Asteroid Rotation Video viewed along the Z-axis

(Sorry about the resolution, uploading it requires compressing)

New output plot