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# Stellar Structure Worksheet

There are four stations to explore each of these four ideas that combine to explain the internal structure of stars.  The four parts of the lab can be done in any order, and are best done in groups of 3. Data Collection should be done as soon as the apparatus is available, data analysis can be done latter.

## Part 1: Equation of State

### Data Collection

In this section, you will compress a gas to determine the change in temperature and density.

1. Get one of the pistons and note the temperature reading on the thermometer.

Troom_initial = _________
2. Use a dry erase marker to mark the initial position of the piston, then measure the distance of the piston to the end of the cylinder (to determine the initial volume of air).

h0 = _____________

Read the next 3 steps before proceeding! Do not hold the cylinder near the thermometer or the heat from your hand will have a bigger effect than the compression of the gas!

1. Have one of the people in your group rapidly compress and hold the piston. The more the cylinder is compressed, the better, but don't push so hard the air leaks out (you may have to make a trial or two to see what the cylinder will tolerate.
2. While the cylinder is compressed, someone else reads the maximum temperature on the thermometer. The temperature will continue to change, first increasing with the compression then decreasing as the heat is transferred to the room.
3. While the cylinder is compressed, someone else marks the end point with the dry erase marker.  Measure the distance from the piston to the end of the cylinder and record this as the height in table 1 (to determine the final volume of air.)
4. Pull the piston back out to the initial position and let the temperature drop down until it stabilizes (it may not come back down to the original room temperature.)
5. Repeat another 3 times until you have the first four rows of table 1 filled in.
6. Note the temperature reading on the thermometer.

Troom_final = _________
7. Average the initial and final room temperature reading to get the average room temperature

Troom = _________
8. Hand the piston off to another group. If you need to collect data for another section, and the equipment is available, go on to that. Otherwise, you can work on the data analysis while you wait.
 Trial Max Temp (ºF) Height 1 2 3 4 Ave

### Data Analysis

1. Find the average temperature and height for your 4 trials.  Enter those values into table 1.
2. The ideal gas law is in Kelvin, but you measured the temperature change in degrees Fahrenheit. To get to Kelvins, you must first convert the temperature to Celsius.  To convert to Celsius, subtract 32, then multiply by 5 and divide by 9.  In equation form, that's TC = (TF – 32)5/9.  Once you have the temperature in degrees Celsius, add 273 to get the temperature in Kelvins.  Show your work and record the temperatures in Kelvins here:

Troom = ________K, TAve = _________K

3. To find the pressure, you also need the density, which is the number of particles per unit volume.  Since the cylinder is sealed, the number of particles remained constant, so the density depends only on the volume.  Luckily, the volume of a cylinder is directly proportional to the height, which is what you changed when you pushed on the plunger.  So the density ends up being proportional to just 1/h!  Use this information and the ideal gas law to figure out by what factor the pressure increased on average when you compressed the cylinder. Show your work.

4. If you assume it started off at 1 atmosphere or 14 lbs/in2, what was the final pressure?

5. Keep in mind that for the solar core, the density is 105 times higher than at sea level on Earth, and the temperature is 15 million K. What is the ratio of the pressure at the center of the Sun to that of the Earth at sea level if we assume the temperature on Earth is about 300 K?

## Part 2: Hydrostatic Equilibrium

How can we actually know that the pressure varies with the force of the weight of water in, say, a glass? We can demonstrate this with a large graduated cylinder.

### Data Collection

1. There are 3 holes in the cylinder.  Make sure they are plugged and place the cylinder in one end of the tray with the holes facing the center so the tray will catch the water.
2. Fill the cylinder up to the 1000 ml level.
3. Remove the top stopper and note how fast (very fast, fast, slow…) the water leaves the cylinder (this is somewhat subjective, but you can sense the speed by placing a finger in the stream just outside the hole).  Note this in table 2.
4. Place a ruler at the level of the middle hole and measure the distance from the center of the cylinder to the stream (see figure).  Record this in table 2 1. Measure the distance of the hole from the ruler and record it in table 2 under distance y.
2. Measure the height of the hole from the bottom of the cylinder and record it in table 2.
3. Plug the hole and refill the cylinder.
4. Unplug the middle hole and note the speed of the water.  Measure the distance from the center of the cylinder to the stream from the level of the bottom hole.  Measure the height of the middle hole from the ruler and above the bottom of the cy lander.  Record these in table 2.
5. Plug the hole, refill the cylinder and repeat the process again for the bottom hole.  When measuring the length of the stream, measure along the bottom of the tray. Measure the height of the hole from the bottom of the tray and enter this value in both the "distance y" column and the height column of table 2.
6. Plug the hole and refill the cylinder again.  This time, unplug all 3 holes.  Draw a sketch of the cylinder and the three streams, and note which stream is the strongest/fastest and which the weakest/slowest.

7. What happens to the streams as the water in the cylinder decreases?

8. The strength/speed of the stream is related to the pressure inside the cylinder: the higher the pressure, the faster the stream.  Based on your observations, how does the pressure change with the height in the cylinder?  How does it change with the amount of water in the cylinder?

9. Hand the equipment off to another group and clean up any spilled water. If you need to collect data for another section, and the equipment is available, go on to that. Otherwise, you can work on the data analysis while you wait.
 Hole Speed (fast, slow…) Length x Distance y Height Speed (calculated) Top middle Bottom

### Data Analysis

1. Plot the horizontal distance, denoted x, that each stream traveled against the height of the hole from the base of the graduated cylinder. Be sure to include the plot when you hand this in.
2. Calculate the speed of each stream as it exited the side of the cylinder where v is the horizontal speed of the stream as it leaves the cylinder, x is the horizontal distance the stream travels, y is the height of the hole from the ruler, and g is the gravitational acceleration at the Earth's surface, or 9.8x102 cm/sec2. Record the calculated speed in the last column of table 2.

1. Plot the resulting velocities against the height of the hole from the base of the graduated cylinder (attach the plot to the back of the lab when you turn it in!)
2. Why do the two plots look different?

3. Based on your graphs, which of these quantities, x or v, is more closely related to the pressure of the water at each hole? Explain.

4. Does this agree with your subjective impression of the pressure as you allowed each stream to hit your finger?

5. Give examples of structures that must be built in such a way that they explicitly take into account the way pressure varies with depth in water or air?

## Part 3: Transfer of Energy

### Data Collection

1. Fill a glass beaker with water and place the beaker on one of the burners of the hot plate.
3. Raise the temperature of the burner until the water begins to boil, then back off until the boiling just stops (about to the MED level on the dials).  Note you will have to give the hot plate and water time to adjust!
4. Observe carefully what is happening in the water itself and to the beads. Add ONE drop of either red or yellow food coloring and observe the movements. Remember, the water is trying to get the energy from the base to escape to the air around it.  Record your observations here.

5. Count how many beads reach the top of the water (n) over 15 seconds.  Do this four times to ensure the temperature is constant (n doesn't vary too much).  If it does vary significantly, you'll have to give it a little more time to adjust.  Once you have 4 fairly stable counts in a row, record the average number here:

n = _______.
6. Add about 10 drops of BLUE or BLACK food coloring to the water until it is opaque. What do you see (or not see)? How is it different from step 4 (other than being opaque)?

7. Count the number of times you see a bead reach the top over 15 seconds again.  Repeat four times. Record the average number here

nblue = _________.
8. Did the number of beads hitting the surface increase or decrease?  By what factor?

9. Describe the changes in motion in the beaker before and after you added the dye. In particular, how did the beads move when the water was boiling the first time? What caused them to rise? What caused them to sink? What happened to the water traveling with the beads during this time?

10. Describe how the medium itself changed: How far through the water could you see before adding the food coloring? After the food coloring was added, how far could you see through the water?

You should collect any data from the other sections before finishing this section.

### Data Analysis

1. Which configuration, clear or colored, do you think is more conducive to transferring energy via radiation? Explain.

2. If radiation is suppressed, how else would the energy escape? Did you see evidence for this after adding the dye?

## Part 4: Energy Generation

To see how the behavior of the cross section matters, we will simulate fusion using clay ‘nuclei' that will be propelled at a target.

### Data Collection

The target consists of a poster board with pieces of Velcro hook tape. There are two targets, a low density board and a high density board, which has twice as many Velcro pieces as the low density board. The "nucleii" are balls designed to stick to Velcro hooks.

1. Count the number of balls in the bag. Roecord this is Table 3.
2. Lay the low density target on the floor. Drop nuclei one at a time onto it from about chest height (roughly 4 feet.) Try to drop them randomly (don't look at where they are already stuck.)
3. Pick up any balls that aren't on a Velcro pad. Gently lift the board to about 45º and remove any balls that roll. Count the balls that are firmly stuck. Record your hits in table 3.
4. Count the balls to make sure you have all of them for the next trial.
5. Repeat the previous 3 steps.
6. Lay the high density target on the floor. Drop nuclei one at a time onto it from about chest height (roughly 4 feet.) Try to drop them randomly (don't look at where they are already stuck.)
7. Pick up any balls that aren't on a Velcro pad. Gently lift the board to about 45º and remove any balls that roll. Count the balls that are firmly stuck to the Velcro. Record your hits in table 3.
8. Count the balls to make sure you have all of them for the next trial.
9. Repeat the previous 3 steps.
10. The number of hits divided by the number of tosses per trial is the interaction probability, which is closely related to the cross section for fusion. Record the probabilities in table 3
11. Average the probabilities for the two low density trials and enter that number into Table 3. Do the same thing for the high density trials.
12. Measure the approximate size of one of the small target squares and calculate the area:

A= __________

 Target drops hits Probability Ave. Probability Low dens. trial 1 Low dens. trial 2 High dens. trail 1 High dens. trial 2

You should collect any data from the other sections before finishing this section.

### Data Analysis

1. Did you ever get 100% fusion rates (that is, did you have any runs where EVERY nuclei you shot at the target stuck to the target)?

2. Let's say you fire a nucleus at your target once every minute. Based on your fusion probabilities, how long do you have to wait for a fusion reaction in even the most likely case?

Note how knowing the fusion probability allows you to calculate the rate of fusion reactions if you know the temperature and density. In a star, we know the latter from the hydrostatic equation and equation of state, so if we can measure the fusion probability, we can determine the typical time a given atom has to wait to fuse with another nucleus. Consequently, experiments like the one done in this demonstration (though a bit more sophisticated and involving actual H atoms and not clay!) are needed to determine the rate that stars form their energy from fusion.

You can roughly calculate the real interaction cross section for H fusion in the sun by noting that every second the sun produces 4 x 10 33 ergs (that's the solar luminosity). Fusing H into He converts 0.8% of the mass of the solar core into energy. The core itself is 10% of the solar mass, so 0.008 * 0.1 * 2 x 10 33 g = 1.6 x 10 30 g gets converted into energy. Putting this mass into Einstein's equation (see intro) tells us that the Sun can eventually produce 1.4 x 10 51 ergs of energy. In one second, 2.9 x 10 -18 of this total energy is emitted (that's the luminosity divided by the total energy), so 2.9x10-18 * the total hydrogen is consumed every second. This is the interaction probability (technically, divided by four since four atoms have to be fused to make one He atom).

To compare this to the experiment you did, you would get a similar interaction rate if the area of one of the little target squares divided by the area of the entire target board was equal to 2.9 x 10 -18!

1. Using the value of the size of the squares, calculate the area of the target board required had we accurately simulated the interaction probability.

2. If the resulting (very large) target board was square, how large would it be one a side? From this you might appreciate that, as far as the Hydrogen atoms are concerned, the center of the Sun is not such a violent place; collisions strong enough for fusion are in fact extremely rare. Good thing there is a LOT of Hydrogen there!