course material

1. [From 10$/Pg] 2 Mn
[From 10$/Pg] 2 MnHow 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 RequiredComputer and internet accessTextbookScientific
calculatorSpreadsheet software like ExcelDigital camera Printer or drawing softwareSave
this worksheet and use it as your report templateTime Required: Between 3-3.5 hours, note
that depending if you use Excel (or similar), your time will be shortened.IntroductionFigure
1: JP Stellar RevolutionThe 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 StarsFigure 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 SunFigure 3. NGC
6543When 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

2. 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 StarPhaseDuration (years)Diameter
(meters)Density (kg/m3)Core Temperature(Kelvin)Surface Temperature (Kelvin)1.
Interstellar Cloud2.13×1066×10171.67×10-1810102. Protostar(phase
1)1061011.0016741×1063,0003. Protostar(phase 2)1×1071×101016.745×1064,0004.
Main Sequence Star1×10101.4×1091×1051.5×1075,7705. Red
Giant1×1084.2×1091×1075×1074,0006. Red Giant (before helium
flash)1×1051.4×10111×1081×1084,0007. Red Giant (after helium
flash)5×1071.4×10101×1071×1085,0008. Super Giant1×104 7×10111×108 2.5 x
1084,0009. Carbon Core1×1051.4×1071×10103×1081×10510. White
Dwarf1×10?1.4×1071×1010Starts at 3×108 and cools downStarts at 1×105 and cools
down1. ActivityNote: 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 ActIn 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)orwhere 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

3. 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:As our Sun is in the phase of Interstellar Cloud, describe
what force is acting while the cloud is collapsing.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?For each of the Protostar phases calculate the luminosity
using: π{“version”:”1.1″,”math”:”L=(σT4 ) ×(4πr2 )”}, which is power (energy per second
per unit area) times the surface area. (L: luminosity in Watts
{“version”:”1.1″,”math”:”σ=5.67×10-8W/m2/K4”} the Stefan-Boltzmann constant; T:
temperature in Kelvin π{“version”:”1.1″,”math”:”π”}= 3.14; r: radius in meters)Calculate
luminosity of Protostar phase 1:Calculate luminosity of Protostar phase 2:What was the
difference in luminosity between the two phases?What was the surface area 4πr2 for each
Protostar phase? How much did it change?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 calculatorMain Sequence PhaseOnce
our Sun was about 13 million years old and had reached a temperature of approximately
{“version”:”1.1″,”math”:”107”}Kelvin, a special process was about to take place. The
luminosity also settles down to around {“version”:”1.1″,”math”:”4×1026”}Watts. It is during
this time that the process of fusing hydrogen takes place, two atoms of hydrogen coming
together to produce helium.How many reaction cycles per second was our Sun fusing
hydrogen in order to release enough energy to radiate
{“version”:”1.1″,”math”:”4×1026”}Watts? (Hints: Each fusion reaction cycle yields
{“version”:”1.1″,”math”:”4.3×10-12”}Joules, and Watt is a unit of power = [Joules/second],
and you are wanting to find a number of units [1/seconds].)If each reaction cycle yields
{“version”:”1.1″,”math”:”4.3×10-12”}Joules, how much mass per second is the Sun
converting into energy? (hint: use Einstein’s equation {“version”:”1.1″,”math”:”E=mc2”}and
solve for mass, E = energy for each reaction cycle, M = mass in kilograms, c = speed of light
{“version”:”1.1″,”math”:”3×108”}[meters/second]; note that units of Joules =
{“version”:”1.1″,”math”:”kg·m2s2”}, 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).Explain why the Sun is at equilibrium during this phase of main
sequence.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 PhaseIn the main sequence phase, the Sun’s diameter is
{“version”:”1.1″,”math”:”1.39×109”} meters and has a mass of
{“version”:”1.1″,”math”:”2×1030”} kg. This tells us it has a mean density of approximately
{“version”:”1.1″,”math”:”1400 kg/m3”}. We can compare that to the density of water, which
is {“version”:”1.1″,”math”:”1000 kg/m3”}What is the mean density of the Sun as it is in its
Red Giant phase? Hint: ({“version”:”1.1″,”math”:”density=massvolume”} and
π{“version”:”1.1″,”math”:”volume=43πr3”}, and assume that the mass remains roughly the
same.)If the Sun’s diameter continues to increase, what will happen to the density? Explain

4. 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 phaseFigure4. Institute for AstronomyUp to the time before the helium flash the core
temperature continues to rise and reaches {“version”:”1.1″,”math”:”108”} Kelvin, and a core
density of {“version”:”1.1″,”math”:”108kg/m3”}. 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.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”:”4×1026”}Watts? Power=Watttime, power has units of
[Joules])Compare that power to what a single hurricane might generate,
{“version”:”1.1″,”math”:”1.3×1017”}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?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
phaseOver the next {“version”:”1.1″,”math”:”105”} 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.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 GiantBy 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”:”2×1030”} kg.)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
phaseFinally 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 CoreBy this time

5. our Sun is not shining by fusion, and no longer technically a star, but it is back in
equilibrium. White DwarfAt this point our Sun starts to cool off and radiate light.For the
surface temperature of {“version”:”1.1″,”math”:”105”}Kelvin, what is the initial luminosity
of the white dwarf?How do you think the luminosity will change over time?Convert your
luminosities in the below table to solar units (by dividing each by
{“version”:”1.1″,”math”:”4×1026”}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.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.)Draw an arrow in the
direction the path will follow from one phase to the next.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)PhaseLuminosity (in Watts)4. Main Sequence Star5. Red Giant6. Red Giant (before
helium flash)7. Red Giant (after helium flash)8. Super GiantHR Diagram (print out, and
follow instructions to fill it in, and place in your lab report)