Travis Fischer UWWhitewater fischertc13 at uww.edu 
REU programSummer 2006 
Large spiral galaxies, such as the Milky Way, grow in part due to the accretion of smaller systems such as the Magellanic galaxies. My task for the summer was to better understand how this accretion affects a galaxy such as our own, by determining:
Dallas Parr (CSIRO)
 Whether these clouds will be accreted by our galaxy or not?
 How long will it take to be accreted?
 Where do the clouds end up in the galaxy after being accreted?
To the left is an illustration of our galaxy, the Milky Way, and to neighboring galaxies, the Large and Small Magellanic Clouds. A large amount of their gas and dust is ripped away and accreted by the Milky Way, this ribbon of debris is called the Magellanic Stream.
Everything I have done this summer has sprouted from one program, Swarm5, which created data that simulated gaseous highvelocity clouds passing by galaxies.
Each cloud was given an initial mass and velocity, thus giving each cloud an initial momentum. The shape of the cloud falling into a galaxy is shown below as a cylinder, much like the actual shape of a rain falling through our atmosphere. As the cloud falls toward the galaxy, it may squish to be a thinner cylinder, but the mass to area ratio (column density) will remain constant.
Taking the derivative of the momentum, where the mass is held constant, in reality takes the derivative of the velocity. This is equal to the acceleration of the cloud. The acceleration of the cloud multiplied by the constant mass of the cloud gives the total force of the cloud. There are two forces acting on the cloud, the gravitational force from the galaxy it falls toward and the drag force it encounters as it passes through the galaxy's 'atmosphere'.
Dividing the total force by the mass gives the total acceleration of the cloud, the sum of the gravitational acceleration and deceleration due to drag..
Deceleration due to drag can be shown by the equation below. A drag constant multiplied by the square of the velocity, all divided by the column density of the cloud. To get a better feel for what the equation means, think of column density as a mass per area ratio. An object with a higher mass per area ratio will have a smaller deceleration due to drag. For example, a hammer (high mass to area ratio) will be less affected by drag than a feather (low mass to area ratio).
So with the given initial x, y, and z positions, and an initial velocity (split into x, y, and z vectors), a six dimensional vector q is formed. To find the vector after a certain amount of time you add the original vector and the product of its derivative and a time difference. The derivative of the original vector is the change of the vector over a change in time. Multiplying by the change in time solves for the change in the vector and is equal to the second half of the sum shown below. To get an accurate solution there are two choices. The first is to use very short time steps, a lot happens over any change in time and smaller steps means less is skipped over. The other option is to choose velocities and accelerations that are weighted by both the initial and 'expected' values using the RungeKutta method.
Each parameter set was labeled according to the parameters that were altered for each run. Here is an example:
Ab3009
So, Ab3009 represents a parameter where the clouds have a density of 5E19 cm^2, and are approaching the galaxy with a velocity of 300 km/s and an inclination of 90 degrees from the top of the galaxy.
A : The capitalized initial letter corresponds to the shape of the galaxy.
A = radius: 2 kpc / height: .5 kpc B = radius: 2 kpc / height: 1.5 kpc C = radius: 2 kpc / height: 2.5 kpc D = radius: 6 kpc / height: .5 kpc
b : The lower case letter represents the density of the clouds that are falling into the galaxy.
 a = 1E19 cm^2
 b = 5E19 cm^2
 c = 1E20 cm^2
 d = 5E20 cm^2
30 = The first two digits represent the initial velocity at which the clouds are heading towards the galaxy.
 00 = 0 km/s
 10 = 100 km/s
 20 = 200 km/s
 30 = 300 km/s
14 = The last two digits represent the initial angle from where the clouds head toward the galaxy.
 00 = 0 degrees
 05 = 45 degrees
 09 = 90 degrees
 14 = 135 degrees
 18 = 180 degrees
Figure 1: An illustration of different parameters
From the data sets that were created via Swarm5, I used an IDL program to plot the X, Y, and Z coordinates on 2D diagrams to get our first looks at what these simulations looked like from each of the dimensional planes.
Notable observations from these intial animations were seen when comparing a cloud set of a certain density versus a cloud set of a much higher density. Here are some examples:
 Aa0000 vs Ad0000 : Here, two sets of clouds are allowed to fall without an initial velocity (a rare case indeed) onto the galaxy from above. By comparing the last frames of the data set animations, we can note that the clouds in Ad0000 form a disk with half the diameter of the Aa0000 cloud disk. Since the only variance between the two data sets is the density, we can assume that the clouds of Ad0000 tend to settle closer to the galactic center due to having a density 50 times greater than the clouds of Aa0000.
 Aa1009 vs Ad1009 : These two sets of clouds are fired toward the galaxy at a velocity of 100 km/s from 90 degrees to the Z axis of the galaxy. Though not as clean as the previous set of animations, the more dense set of clouds in Ad1009 again settled into a disk with roughly half the diameter of the disk formed by the Aa0000 clouds.
Fig. 2: Sample frame taken from the Bc2009 space diagram animation.
This new group of diagrams was made by altering the types of information each graph displayed. While the overhead space diagram using x, y, and z coordinates was kept intact, the other three graphs used additional sets of the collected data (energy, angular momentum, etc.) to provide more types of information what we couldn't see from the first diagram. Example: Ca2009
Radius vs Height  Turning the cartesian coordinates into spherical coordinates showed a new animation of the cross section for each parameter set. This provides a visual on when clouds get caught in the galaxy they are falling toward.
Angular Momentum vs Phi  This plot shows the magnitude of the angular momentum of each cloud versus 'phi', the angle between the angular momentum vector and the Z axis. As shown in figure 3, the angle phi will be 180 degrees when the cloud is caught inside the spinning galaxy because the angular momentum of the cloud (L) will be perpendicular to the plane formed by the clouds radius and momentum in the downward direction is determined using the right hand rule.
KE/PE vs TE  This graph displays the ratio of the kinetic energy versus the gravitational potential energy of each cloud plotted against the total energy of each cloud. For a cloud to be bound into orbit around the galaxy, it must have a negative total energy. If the cloud has an energy greater than or equal to zero, it will move infinitly far away from the galaxy! So by monitoring the position of the clouds on the TE axis, we were able to see whether or not any clouds escaped from the galaxy's gravitational pull.
Fig. 3: Sample frame taken from the Bc2009 phase diagram animation.
Fig. 4: The angle 'phi' between a cloud's angular momentum vector and the Z axis is 180 degrees when it is bound in our clockwise spinning galaxy.


By having the clouds not change color as a function of a parameter but instead keep the color they are given in a certain time step (such as the first or last time step), we can see where certain clouds go. In the simulated diagram on the right, the final frame of a certain parameter set results in the clouds forming the disk at the center. By assigning those clouds a permanent color and then 'rewinding' what happened in a way, we are able to see what clouds end up closer and farther away from the center of the galaxy. Here is the below frame in motion Ca1009.

Fig. 6: Sample frame taken from the Ca1009 3D space diagram animation. 
3D Binding Space Diagrams (7/21/06)


Kitt Peak National Observatory