How to use exec to do the math work in


How to use exec to do the math work in this lab?

 

Learning Objective:

To study the life cycles and deaths of stars with a focus on main sequence stars like our own Sun.

Prerequisites: Chapter 17, Chapter 18 section 1. Review Exploration from Module 1: “Math Primer for Astronomy” (note this contains link for a free online scientific calculator).

Materials Required

  • Computer and internet access
  • Textbook
  • Scientific calculator
  • Spreadsheet software like Excel
  • Digital camera 
  • Printer or drawing software
  • Save this worksheet and use it as your report template

Time Required: Between 3-3.5 hours, note that depending if you use Excel (or similar), your time will be shortened.

Introduction

infographic showing many protostars of various sizes and the phases they go through over their life-span.

Figure 1: JP Stellar Revolution

The life cycle of the stars is one of the most fascinating studies of astronomy.Stars are the building blocks of galaxies and by looking at their age, composition and distribution we can learn a great deal about the dynamics and evolution of that galaxy. Stars manufacture the heavier elements including carbon, nitrogen and oxygen which in turn will determine the characteristics of the planetary systems that form around them. It is the mass of the star which will determine its life cycle and this all depends on the amount of matter that is available in its nebula. Each star will begin with a limited amount of hydrogen in their cores. This lifespan is proportional to (f M) / (L), where f is the fraction of the total mass of the star, M, available for nuclear burning in the core and L is the average luminosity of the star during its main sequence lifetime. The larger the mass, the shorter the lifespan ending in a beautiful supernova, the smaller the mass, the longer the lifespan ending as a quiet brown dwarf (Fig. 1).

Main Sequence Stars

an HR diagram showing absolute magnitude, temperature, luminosity and spectral class.

Figure 2: https://imagine.gsfc.nasa.gov/

For this lab we will focus on stars similar to our own Sun (up to 1.4MassSun ), main sequence stars. A star that is similar in size to our Sun will take approximately 50 million years to mature from the beginning of their collapse to becoming an “adult” star. Our Sun, after reaching this mature phase, will stay on the main sequence of the HR-diagram for approximately 10 billion years (Fig. 2). Stars like our Sun are fueled by the nuclear fusion of hydrogen forming into helium at their cores. It is this outflow of energy that provides the outward pressure necessary to keep the star from collapsing under its own weight. And in turn, this energy determines the luminosity of the stars.

Death of Our Sun

shows a telescopic image of NGC 6543

Figure 3. NGC 6543

When a low mass star like our Sun has exhausted its supply of hydrogen in its core, then there will no longer be a source of heat to support the core against the pull of gravity. Hydrogen will continue to burn in a shell around the core and the star will evolve into the phase of a red giant, growing in diameter. The core of the star will collapse under the pull of gravity until it reaches a high enough density, and it will begin to burn helium and make carbon. This phase will last about 100 million years eventually exhausting the helium and then becoming a red supergiant, growing more in diameter. This is a more brief phase and last only a few tens of thousands of years and the star loses mass by expelling a strong wind. The star eventually loses the mass in its envelope, leaving behind a hot core of carbon embedded in a nebula of expelled gas. Because the core is still hot, its radiation will ionize the nebula, which is the planetary nebula phase (Fig. 3). At the end the carbon core will cool and become a white dwarf.

White dwarfs used to be quite a mystery. Astronomers couldn’t figure out why the star didn’t continue to collapse. Quantum mechanics brought about the answer – electron degeneracy pressure. Read through this web material to learn more. In the below table you will find important data that accompanies each phase of a star like our Sun.

Table 1. Stellar Evolution of a Sun-like star. Reminder: When we are examining the physical state of a star, we have to separately consider the core (where temperature and pressure are very high), and the surface (where the temperature and pressure are considerably less). The core is where the fusion occurs and the surface is what we can visually see. Thus, we have to infer what is going on in the core by observing the envelope of the star.

Table 1 Stellar Evolution of a Sun-like Star

Phase

Duration (years)

Diameter (meters)

Density (kg/m3)

Core Temperature

(Kelvin)

Surface Temperature (Kelvin)

1. Interstellar Cloud

2.13×106

6×1017

1.67×10-18

10

10

2. Protostar

(phase 1)

106

1011

.001674

1×106

3,000

3. Protostar

(phase 2)

1×107

1×1010

16.74

5×106

4,000

4. Main Sequence Star

1×1010

1.4×109

1×105

1.5×107

5,770

5. Red Giant

1×108

4.2×109

1×107

5×107

4,000

6. Red Giant (before helium flash)

1×105

1.4×1011

1×108

1×108

4,000

7. Red Giant (after helium flash)

5×107

1.4×1010

1×107

1×108

5,000

8. Super Giant

1×104

  7×1011

1×108

   2.5 x 108

4,000

9. Carbon Core

1×105

1.4×107

1×1010

3×108

1×105

10. White Dwarf

1×10?

1.4×107

1×1010

Starts at 3×108 and cools down

Starts at 1×105 and cools down

1. Activity

Note: Even if you use Excel for your work below, you will still want to show one calculation of each type fully worked out in detail. (typed) Again, it would be helpful to review the Exploration from Module 1: “Math Primer for Astronomy” (note this contains link for a free online scientific calculator). There are also good math examples in the Appendix of our eText.

The evolution of any star is a complex process. In order for us to understand the processes that are taking place and how stars change with time scientists must apply the basic ideas of physics and chemistry to create a mathematical model of a star. By making many observations of many types of stars along with stars at various stages in their lifespan, we can use these observational clues to test these models. By plugging in many variables into sophisticated computer programs we are able to come up with a theory of stellar evolution and this, in turn, can give us the story behind every sort of object in the sky from a main sequence star, supernovae, black hole, to nebulae. Using the information from Part 1 of this lab, let’s see what we can find out about some of the phases that our Sun has and will go through.

A Balancing Act

In looking at Table 1, during the protostar stage our Sun is contracting under its own weight and this results in a rising temperature. Looking back to the ideal gas law, in Chapter 14, we find:

Pressure x Volume = (Number of particles) x (k) x (Temperature of the gas)

or

PV =nkT

where k = 1.38 x 10-23 [joule/K] is Boltzmann’s constant.  This law applies to all gases consisting of simple, freely flying particles, like in our Sun.  We can also relate this formula to the forces being applied.

force divided by area equations

In a star, the pressure will always be changing with the radius, and this keeps the star from collapsing. At each layer, the outward push of the gas is balanced by the inward pull of gravity on the gas. In looking at the above relationship between the variables, if one changes then the others must change to balance the equation out. Thus:

  1. As our Sun is in the phase of Interstellar Cloud, describe what force is acting while the cloud is collapsing.
  2. Describe also what is taking place in terms of conservation of energy, what is happening to the kinetic and potential and thermal energy as the cloud is collapsing?
  3. For each of the Protostar phases calculate the luminosity using: π{“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mi>L</mi><mo>=</mo><mo>(</mo><mi>&#x3C3;</mi><msup><mi>T</mi><mrow><mn>4</mn><mo>&#xA0;</mo></mrow></msup><mo>)</mo><mo>&#xA0;</mo><mo>&#xD7;</mo><mo>(</mo><mn>4</mn><msup><mi>&#x3C0;r</mi><mrow><mn>2</mn><mo>&#xA0;</mo><mo>&#xA0;</mo></mrow></msup><mo>)</mo><mspace linebreak=”newline”></mspace><mspace linebreak=”newline”></mspace></math>”}, which is power (energy per second per unit area) times the surface area.  (L: luminosity in Watts {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mi>&#x3C3;</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>67</mn><mo>&#xD7;</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>8</mn></mrow></msup><mfenced open=”[” close=”]”><mrow><mfenced><mrow><mi>W</mi><mo>/</mo><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced><mo>/</mo><msup><mi>K</mi><mn>4</mn></msup></mrow></mfenced><mspace linebreak=”newline”></mspace></math>”} the Stefan-Boltzmann constant; T: temperature in Kelvin π{“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mi mathvariant=”normal”>&#x3C0;</mi></math>”}= 3.14; r: radius in meters)
    1. Calculate luminosity of  Protostar phase 1:
    2. Calculate luminosity of  Protostar phase 2:
    3. What was the difference in luminosity between the two phases?
    4. What was the surface area 4πr2 for each Protostar phase?  How much did it change?
    5. What variable, surface temperature or radius affected the luminosity the most during the change from Protostar phase 1 to 2? Why? (Hint: think of percent increase or decrease.) Online calculator

Main Sequence Phase

Once our Sun was about 13 million years old and had reached a temperature of approximately {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><msup><mn>10</mn><mn>7</mn></msup></math>”}Kelvin, a special process was about to take place. The luminosity also settles down to around {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>4</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>26</mn></msup></math>”}Watts. It is during this time that the process of fusing hydrogen takes place, two atoms of hydrogen coming together to produce helium.

  1. How many reaction cycles per second was our Sun fusing hydrogen in order to release enough energy to radiate {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>4</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>26</mn></msup></math>”}Watts? (Hints: Each fusion reaction cycle yields {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>4</mn><mo>.</mo><mn>3</mn><mo>&#xD7;</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>12</mn></mrow></msup></math>”}Joules, and Watt is a unit of power = [Joules/second], and you are wanting to find a number of units [1/seconds].)
  2. If each reaction cycle yields {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>4</mn><mo>.</mo><mn>3</mn><mo>&#xD7;</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>12</mn></mrow></msup></math>”}Joules, how much mass per second is the Sun converting into energy? (hint: use Einstein’s equation {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mi>E</mi><mo>=</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></math>”}and solve for mass, E = energy for each reaction cycle, M = mass in kilograms, c = speed of light {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>3</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>8</mn></msup></math>”}[meters/second]; note that units of Joules = {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mfenced open=”[” close=”]”><mfrac><mrow><mi>k</mi><mi>g</mi><mo>&#xB7;</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><msup><mi>s</mi><mn>2</mn></msup></mfrac></mfenced></math>”}, this will give you (mass converted /reaction). Then you can multiply that number by what you found in A., which was (# reactions/second) ending up with (mass/second).
  3. Explain why the Sun is at equilibrium during this phase of main sequence.
  4. Create a table similar to the one at the end of this lab, titled “Luminosity Table” to put in your lab report. Fill in the luminosity stated above for the main sequence line.

Red Giant Phase

In the main sequence phase, the Sun’s diameter is {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>1</mn><mo>.</mo><mn>39</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>9</mn></msup></math>”} meters and has a mass of {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>2</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>30</mn></msup></math>”} kg. This tells us it has a mean density of approximately {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>1400</mn><mo>&#xA0;</mo><mi>k</mi><mi>g</mi><mo>/</mo><msup><mi>m</mi><mn>3</mn></msup></math>”}. We can compare that to the density of water, which is {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>1000</mn><mo>&#xA0;</mo><mi>k</mi><mi>g</mi><mo>/</mo><msup><mi>m</mi><mn>3</mn></msup></math>”}

  • What is the mean density of the Sun as it is in its Red Giant phase? Hint: ({“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mi>d</mi><mi>e</mi><mi>n</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>a</mi><mi>s</mi><mi>s</mi></mrow><mrow><mi>v</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>m</mi><mi>e</mi></mrow></mfrac></math>”} and π{“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mi>v</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>m</mi><mi>e</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><msup><mi>&#x3C0;r</mi><mn>3</mn></msup></math>”}, and assume that the mass remains roughly the same.)
  • If the Sun’s diameter continues to increase, what will happen to the density? Explain your answer.
  • Calculate the luminosity of the Sun at this phase. (Use equation from above and put this number into your Luminosity table for the Red Giant phase.) Hint: You can create a spreadsheet with the luminosity formula and data from Table. 1, this would allow you to calculate the luminosities quite quickly. Make sure to show at least one sample calculation done by hand. How to Create Formulas and Make Calculations.

Red Giant – before helium flash phase

: “triple-alpha process”: two helium nuclei collide and fuse to make beryllium, releasing energy, but before the beryllium can break down another helium collides with it to form carbon, releasing more

Figure4. Institute for Astronomy

Up to the time before the helium flash the core temperature continues to rise and reaches {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><msup><mn>10</mn><mn>8</mn></msup></math>”} Kelvin, and a core density of {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><msup><mn>10</mn><mn>8</mn></msup><mi>k</mi><mi>g</mi><mo>/</mo><msup><mi>m</mi><mn>3</mn></msup><mspace linebreak=”newline”></mspace></math>”}. At this point the helium fuses to ignite a “triple-alpha process”: two helium nuclei collide and fuse to make beryllium, releasing energy, but before the beryllium can break down another helium collides with it to form carbon, releasing more energy. This helium flash releases more energy than had been radiated over 30,000 years while the Sun was in its main sequence phase, all in just a few seconds.

  1. If the helium flash released 30,000 years worth of energy (as in the main sequence phase) in just 10 seconds, what would be the amount of power that was radiated?  Hint: (Remember your units!  How many seconds are in a year?  If the luminosity was {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>4</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>26</mn></msup></math>”}Watts?  Power=Watttime, power has units of [Joules])
  2. Compare that power to what a single hurricane might generate, {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>1</mn><mo>.</mo><mn>3</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>17</mn></msup></math>”}Joules in one day, which is equivalent to about half the world wide electrical generating capacity.  Does that even come close to the number you calculated?
  3. Calculate the luminosity of the Sun at this phase. (Use equation from above and put this number into your Luminosity table.)

Red Giant – helium fusion after helium flash phase

Over the next {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><msup><mn>10</mn><mn>5</mn></msup></math>”} years the core settles into stable helium fusion surrounded by a shell of hydrogen fusion. During the helium flash, this explosive event would produce strong convection currents in the outer envelope of the Sun and perhaps blow out 20-30% of it out into space. In turn, the outer envelope of gas gets hotter. The core will consume the helium quickly because of the high temperature, the triple-alpha fusion lasting maybe on a few million years.

  1. Calculate the luminosity of the Sun at this phase. (Use equation from above and put this number into your Luminosity table.)

Red Giant becomes Super Giant

By this time helium is running out of the core, which is mostly carbon now surrounded by a shell of fusing helium and an outer shell of fusing hydrogen. The core is small and massive and heating up. Eventually, the fusion days are coming to an end. The hydrogen shell dumps helium ash onto the helium fusion shell, then the helium shell dumps its carbon ash into the carbon core. The core continues to contract, which shrink the outer shells. Again temperatures rise and as a result the star bloats up again but even bigger into a super giant.

   A. For the Super Giant phase calculate the average density. (Hint: use the above equation for density, assume the mass is still {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>2</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>30</mn></msup></math>”} kg.)

  1. Calculate the luminosity of the Sun at this phase. (Use equation from above and put this number into your Luminosity table.)

After the Super Giant phase

Finally all of the available gravitational energy is spent. The fusion stops, leaving a carbon core. But just before the core goes out, the outer envelope is transformed. During this period, a number of helium flashes can occur, destabilizing the gas and causing pulsations. The gas would rise and fall a few times until finally it rises fast enough to escape from the core – and we will see a beautiful planetary nebula.

The Carbon Core

By this time our Sun is not shining by fusion, and no longer technically a star, but it is back in equilibrium.  

White Dwarf

At this point our Sun starts to cool off and radiate light.

  1. For the surface temperature of {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><msup><mn>10</mn><mn>5</mn></msup></math>”}Kelvin, what is the initial luminosity of the white dwarf?
  2. How do you think the luminosity will change over time?
  3. Convert your luminosities in the below table to solar units (by dividing each by {“version”:”1.1″,”math”:”<math xmlns=”http://www.w3.org/1998/Math/MathML”><mn>4</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>26</mn></msup></math>”}Watts). Hint: this will be very easy if you have created an Excel spreadsheet with the luminosity formula. Please upload your Excel file to the assignment folder with your lab report.
  4. Print out the below HR Diagram plot and label each of your luminosities in pencil from the Luminosity Table you create. (Note: if you have drawing software, you can use the image and draw the below.)
    1. Draw an arrow in the direction the path will follow from one phase to the next.
    2. Draw a path from the Super Giant phase to where you think the Sun will end up. (Remember, over time how this path would look on the HR Diagram.)

Luminosity Table (to create and place in your lab report)

Phase

Luminosity (in Watts)

4. Main Sequence Star

5. Red Giant

6. Red Giant (before helium flash)

7. Red Giant (after helium flash)

8. Super Giant

HR Diagram (print out, and follow instructions to fill it in, and place in your lab report)

HR Diagram

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