How to use exec to do the math work in this lab?
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).
Time Required: Between 3-3.5 hours, note that depending if you use Excel (or similar), your time will be shortened.
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).
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.
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
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.
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
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.
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:
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>×</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.
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>×</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>×</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> </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> </mo><mi>k</mi><mi>g</mi><mo>/</mo><msup><mi>m</mi><mn>3</mn></msup></math>”}
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.
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.
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>×</mo><msup><mn>10</mn><mn>30</mn></msup></math>”} kg.)
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.
By this time our Sun is not shining by fusion, and no longer technically a star, but it is back in equilibrium.
At this point our Sun starts to cool off and radiate light.
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)
I have attached the instructions in case things go wonky.
