Kyle Barbary

Astro 120

15 April 2002

Homework 4

 

 

I) Spiral Galaxies

a) One of the ways in which the model is inaccurate is that the stars essentially live forever. In the actual density wave theory, stars supernova and new stars are formed from the remnant of the old stars. In the applet, the stars basically just go around in circles, slowing down when they get into a density wave area.

            Another thing that I’m not entirely sure about is that the galaxy in the applet has 4 spirals. For m > 2 (m = number of arms), there should be an inner radius below which there is no spiral structure. However, the applet model seems to have the spirals going all the way to r = 0.

 

b) i)

Formation of O/B star: 10 Myr

Lifetime of O/B star: 20 Myr

Speed of SNR: 200km/sec

Distance to next formation: 100pc

 

[Time to move 100pc] = (100 pc) (200km/sec)-1  +10 Myr +20 Myr = .5 Myr +10 Myr +20 Myr = 30.5 Myr

 

Thus, the speed of motion is about (100 pc)/(30.5 Myr) = 3.28 pc/Myr

 

ii)

Assuming that the above is at a radius of 5kpc, the rotation period is

 

            2p (5,000 pc)/ (3.28pc/Myr) = 9.6 Gyr

 

For a star like the sun, the orbital period is

 

2p (8,000 pc)/ (200km/sec) = 247 Myr

 

This seems to be more or less consistent with the applet and with the discussion in the book. The rotation speed is about 40 times slower than the orbital speed, which is a bit high compared to what the applet looks like, but given the large number of assumptions made, this is a reasonable number.

 

c) After trying various configurations, I found that the ones that worked the best were an exponential disk with a little spin (+) with a plummer sphere orbiting it:

 

Galaxy 1: expdisk (1000 particles) (z spin = .3)

Galaxy 2: Plummer (200 particles) (position = 0,3,0) (velocity = .93,0,0)

 

You can see that by the 10-12th frame, we have a definite spiral structure. However, after that, it tends to dissipate. If the integration continues to run, you get:

The spiral structure is essentially gone, but the remaining central galaxy seems to have a bit of a bar structure. The spiral only persists for about a ¼ of an orbital period, or 50Myr (using the MW as an example). The main reason the spiral is short-lived is that there is no gas in this model. Stellar random speeds grow and the motion becomes randomized. For an accurate depiction of spiral structure, there must be star birth and death.

 

 

II) Dynamical Friction and “Viriality”

a) So to test for the dependence of DV on V and M, I set up a model with a constant b (distance between plummer spheres at closest original encounter) and a constant-mass non-moving galaxy, and varied first the mass M of the moving galaxy and then its velocity V. We are testing the law:

So we should find that DV varies linearly with M and varies nonlinearly with V (inverse cubically to be precise). Also, we note that this formula requires that b be much larger than the core radius of M.

 

(note: To find the initial and final velocities of the passing galaxy, I plugged the ASCII initial and final integration data into MS Excel and took the Vy column and averaged it to find the average velocity of all the particles in the galaxy. Since the ASCII data is space delimited, to get the data into columns you can use the “Text to Columns…” feature under the “Data” menu on the menubar in Excel 2000.)

 

Here is an example of the first run:

 

 

Here is the data for this run and for subsequent runs:

Galaxy 1

Galaxy 2

particles

mass(each)

particles

mass(each)

start pos.

initial velocity

final velocity

dV

300

0.00333

30

0.0033

5,-5

0.909090517

0.80300163

-0.10609

300

0.00333

30

0.0033

5,-5

1.818181567

1.76017333

-0.05801

300

0.00333

30

0.0033

5,-5

2.727272333

2.68584233

-0.04143

300

0.00333

30

0.0033

5,-5

3.636363333

3.604707

-0.03166

300

0.00333

30

0.0033

5,-5

4.545455333

4.519699

-0.02576

 

 

 

 

 

 

 

0

300

0.00333

30

0.0033

5,-10

3.636364

3.61801867

-0.01835

300

0.00333

15

0.0033

5,-10

3.809524667

3.79158933

-0.01794

300

0.00333

60

0.0033

5,-10

3.291626333

3.28907767

-0.00255

 

 

 

 

 

 

 

0

15

0.06666

15

0.06666

5,-10

1.999999333

1.98040933

-0.01959

30

0.03333

30

0.03333

5,-10

2

1.98066467

-0.01934

 

 

 

 

 

 

 

0

300

0.00333

30

0.0667

5,-10

1.333333967

1.31274237

-0.02059

300

0.00333

30

0.000833

5,-10

3.902439333

3.882074

-0.02037

 

As you can see, the data for the V dependence is in the first 5 lines. The graph of it looks like this:

This looks much like we would expect it to look. To get the exact dependence, we do a log-log plot:

This, unfortunately, shows that dV is proportional to V-.88 instead of V-3, which is what the equation predicts. While this is pretty far off, we must remember that there are a lot of factors that effect this value, as well as assumptions in the model that we may not have satisfied the conditions for.

 

The dependence of dV upon mass was a curious thing. I played around with the simulator for quite awhile, but I couldn’t get it to output reasonable values. As you can see from the data in the table, lines 6-12, I tried various combinations of different masses and different scale factors. I think the problem is that if you have scale factor = none, the galaxies total masses are set equal to 1 no matter what the number of particles is. However, if you set the scale factor to some other value, the velocity is scaled with the mass, and this creates problems. [A note for Vicki: This is something about the testbed that could be improved. A more intuitive way to have different mass galaxies, or at least an explanation of how the simulator scales the masses of the individual particles would be very helpful.]

 

 

 

b) I ran a plummer sphere for 1500 iterations to test if it is stable. It certainly appears to be:

However, it was interesting to note that if dt is set to high in the simulator (~.1 or so), the integration is to imprecise and the plummer sphere blows apart somewhat. So dt should always be set small to get accurate results.

 

Next, I tested the dependence of velocity dispersion on mass. Again, I ran into the problem of the scaling factor. I found that with no scaling, the velocity dispersion didn’t really change with increasing number of particles. This makes sense because since the mass is constant, all you are really doing is splitting up the same galaxy into smaller and smaller pieces, which should have about the same dynamics as if they were stuck together in bigger pieces. With scaling, I got the following results:

 

600 particles scale factor .7, 1,1.5,2

As it turns out, the velocity dispersion didn’t actual increase or decrease at all, in the center of the galaxy anyway. It stayed between .47 and .50 for all scale factors. The velocity dispersion did however increase in the outer parts of the galaxy as the scale factor increased. So my conclusion (since I’m not sure exactly what the scale factor does to mass and velocity) is that either I don’t know how to increase the mass of the system properly using this model, or the central velocity dispersion doesn’t depend on mass.

 

c) I played with a number of different spin values on a plummer sphere to see what values gave a stable elliptical galaxy. It seems that values of spin over about 1 cause the system to fly apart. The greater the spin, the more oblate the galaxy became. This seems to be consistent with observations since we see no elliptical galaxies in the sky more flattened than an E7 galaxy: the high angular momentum of such a system would cause it to separate into a disk and a central bulge.

            To create an E3 galaxy we want 3 = 10 (1-b/a), or b/a = .7: one axis is 7/10 the length of the other. Since all the galaxies we create by spinning about the z-axis are of course symmetric about the z axis, we will measure the ratio of the axes in the xz (or yz) plane (the galaxy is oblate spheroidal). We note that in real observations the En type of elliptical is based on the angle at which we observe the galaxy.

 

[Another note for Vicki: no matter what observation plane you select, all the plots are still labeled x-y… it makes it hard to tell which is which axis]

 

I found that the spin value that most closely gave this ratio was .275. (The integration was done using 1000 particles, 1500 iterations, and dt = .005). Here is the final state:

 

To get an E5 galaxy we want a ratio of .5. A spin value of .45 gives roughly this ratio:

Velocity Dispersion:

E0, E3:

E5:

The central velocity dispersion seemed to be about the same for all 3 types of ellipticals. However the averages increased:

Type

Avg. Vel. Disp.

E0

.31

E3

.37

E5

.41

(averages were estimated using graphs)

 

Do these ratios agree with real elliptical galaxies? Well, we know that the effective radius and velocity dispersion have the approximate relation

s µ Re.81

If we take the effective radii of the En galaxies to be based on the ratio of their axes, we get:

Type

Effective Radius

Re.81

.31 Re.81

E0

1

1

.31

E3

1.42

1.33

.41

E5

2

1.75

.54

 

Well, these don’t exactly agree with the table above, but we did make a lot of assumptions, so I feel like we should be happy to be this close.

 

 

III) Creative Simulations

 

I didn’t have a lot of time to work on this, but I figured I’d play around with the normal impact of a plummer sphere on a expdisk galaxy for a little bit at least. After varying many parameters, I used the following:

Plummer Sphere:

200 particles, mass and velocity scale factor .5

initial position: (0,0, .7)   velocity: (0,0,-.05)

 

Expdisk:

500 particles at origin

 

The collision:

I couldn’t see from the pictures whether there was a ring galaxy formed in the final state, but the map of surface density vs radius led me to believe that it was at least somewhat ring-like.(see map below)

 

Map of Surface density vs. Radius