Travis Fischer's REU homepage (Summer 2006)

Introduction

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.

The beginning: Swarm5

Everything I have done this summer has sprouted from one program, Swarm5, which created data that simulated gaseous high-velocity 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 Runge-Kutta method.

Each parameter set was labeled according to the parameters that were altered for each run. Here is an example:

Ab3009

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

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.

Space Diagrams (5/28/06)

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.

Phase Space Plot (6/7/06)

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.

Variable Polychromatic 3D Space Diagrams (6/14/06, 6/26/06-color)

Our next diagram was partially built using a 3d-plot program found in an IDL library maintained by the University of Washington. Using this program, we were able to plot each of the clouds in a three dimensional environment instead of looking at the simulation from three separate directions as in the original space diagram. It also allowed for a larger spacial picture with longer axies since the screen is no longer cut into four separate graphs. Besides, everything looks cooler in color!

Plots can be colored to represent any physical parameter in the system. By entering a maximum and minimum value, the graph colors each cloud depending on its specific parameter value. The graph shown to the right has colored the clouds using a gradient to represent radial distance from the center of the galaxy, purple representing the center and red representing clouds 50 kpc away or farther. The resultant from this graph shows clouds with a small radial distance being shot into the galaxy from below toward the end of the animation, as shown here Cb2000.

Fig. 5: Sample frame taken from the Cb2000 3D space diagram animation.

Fixed Polychromatic 3D Space Diagrams (7/5/06)

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.

The initial disk of clouds is nearly symmetric around the XY plane. This is because the disk is being shot into the galaxy from the side and what happens above the galaxy is duplicated below it.

The center of the cloud disk is orange, that means its clouds will end up farther away from the center than most clouds. This is because as the disk is shot towards the galaxy it is the first part that hits and is immeadiatly swept into orbit, no longer traveling toward the galaxy's center.

To the right of the center clouds, there are deep red clouds. These clouds are described as 'pro-grade' for as they approach the galaxy they are swept up and sent into orbit in the same direction they were traveling. Thus, these clouds retain most of their energy and maintain a farther radius that the rest of the clouds in the disk.

On the opposite end of the disk are greener clouds, which portray 'retro-grade' clouds that more energy than the center or pro-grade clouds as they hit the galaxy against it's spin.

The blue spots above and below the XY plane are clouds that fall down toward the galaxy and are lucky enough to hit near the center. This basically stops them in their tracks and keeps them in orbit extremely close to the center of the galaxy.

Fig. 6: Sample frame taken from the Ca1009 3D space diagram animation.

3D Binding Space Diagrams (7/21/06)

The final diagram reflects on the total energy diagram from the phase space plot. Using the idea from the last 3d diagram, all the clouds are colored by the last time step to determine which clouds will be bound and which will be unbound. By assigning all the clouds with a total energy less than zero to blue and all the clouds with a total energy greater than or equal to zero to yellow, we can a defined boundry in the initial disk of clouds. We can use this information in the future to help create a graph that will plot initial velocity versus impact parameter that shows whether clouds will be bound or not. Within the bound clouds, we also hope to create a contour that shows the final radius of the captured clouds.

Fig. 7: Sample frame taken from a high velocity 3D space diagram animation.

A higher column density will cause clouds to settle closer to the galactic center.
Pro-grade clouds will settle farther from the galactic center than retro-grade clouds.
We know whether a cloud is captured or not by its initial velocity and impact parameter.
If a cloud is captured, we can also determine the final radius of its orbit.

MODELS OF GASEOUS ACCRETION IN GALAXIES