Week of October 29, 2018

Astronomy

The Life Cycles of Stars: How Supernovae Are Formed

It is very poetic to say that we are made from the dust of the stars. Amazingly, it's also true! Much of our bodies, and our planet, are made of elements that were created in the explosions of massive stars. Let's examine exactly how this can be.

Life Cycles of Stars

A star's life cycle is determined by its mass. The larger its mass, the shorter its life cycle. A star's mass is determined by the amount of matter that is available in its nebula, the giant cloud of gas and dust from which it was born. Over time, the hydrogen gas in the nebula is pulled together by gravity and it begins to spin. As the gas spins faster, it heats up and becomes as a protostar. Eventually the temperature reaches 15,000,000 degrees and nuclear fusion occurs in the cloud's core. The cloud begins to glow brightly, contracts a little, and becomes stable. It is now a main sequence star and will remain in this stage, shining for millions to billions of years to come. This is the stage our Sun is at right now.

As the main sequence star glows, hydrogen in its core is converted into helium by nuclear fusion. When the hydrogen supply in the core begins to run out, and the star is no longer generating heat by nuclear fusion, the core becomes unstable and contracts. The outer shell of the star, which is still mostly hydrogen, starts to expand. As it expands, it cools and glows red. The star has now reached the red giant phase. It is red because it is cooler than it was in the main sequence star stage and it is a giant because the outer shell has expanded outward. In the core of the red giant, helium fuses into carbon. All stars evolve the same way up to the red giant phase. The amount of mass a star has determines which of the following life cycle paths it will take from there.

The life cycle of a low mass star (left oval) and a high mass star (right oval).

The illustration above compares the different evolutionary paths low-mass stars (like our Sun) and high-mass stars take after the red giant phase. For low-mass stars (left hand side), after the helium has fused into carbon, the core collapses again. As the core collapses, the outer layers of the star are expelled. A planetary nebula is formed by the outer layers. The core remains as awhite dwarf and eventually cools to become a black dwarf.

On the right of the illustration is the life cycle of a massive star (10 times or more the size of our Sun). Like low-mass stars, high-mass stars are born in nebulae and evolve and live in the Main Sequence. However, their life cycles start to differ after the red giant phase. A massive star will undergo a supernova explosion. If the remnant of the explosion is 1.4 to about 3 times as massive as our Sun, it will become a neutron star. The core of a massive star that has more than roughly 3 times the mass of our Sun after the explosion will do something quite different. The force of gravity overcomes the nuclear forces which keep protons and neutrons from combining. The core is thus swallowed by its own gravity. It has now become a black hole which readily attracts any matter and energy that comes near it. What happens between the red giant phase and the supernova explosion is described below.

From Red Giant to Supernova: The Evolutionary Path of High Mass Stars

Once stars that are 5 times or more massive than our Sun reach the red giant phase, their core temperature increases as carbon atoms are formed from the fusion of helium atoms. Gravity continues to pull carbon atoms together as the temperature increases and additional fusion processes proceed, forming oxygen, nitrogen, and eventually iron.

The two supernovae, one reddish yellow and one blue, form a close pair just below the image center (to the right of the galaxy nucleus) Image Credit: C. Hergenrother, Whipple Observatory, P. Garnavich, P.Berlind, R.Kirshner (CFA).

When the core contains essentially just iron, fusion in the core ceases. This is because iron is the most compact and stable of all the elements. It takes more energy to break up the iron nucleus than that of any other element. Creating heavier elements through fusing of iron thus requires an input of energy rather than the release of energy. Since energy is no longer being radiated from the core, in less than a second, the star begins the final phase of gravitational collapse. The core temperature rises to over 100 billion degrees as the iron atoms are crushed together. The repulsive force between the nuclei overcomes the force of gravity, and the core recoils out from the heart of the star in a shock wave, which we see as a supernova explosion.

As the shock encounters material in the star's outer layers, the material is heated, fusing to form new elements and radioactive isotopes. While many of the more common elements are made through nuclear fusion in the cores of stars, it takes the unstable conditions of the supernova explosion to form many of the heavier elements. The shock wave propels this material out into space. The material that is exploded away from the star is now known as a supernova remnant.

The hot material, the radioactive isotopes, as well as the leftover core of the exploded star, produce X-rays and gamma-rays.

For the Student Using the above background information, (and additional sources of information from the library or the web), make your own diagram of the life cycle of a high-mass star.

For the Student Using the text, and any external printed references, define the following terms: protostar, life cycle, main sequence star, red giant, white dwarf, black dwarf, supernova, neutron star, pulsar, black hole, fusion, element, isotope, X-ray, gamma-ray.

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Reference URLs:

Supernovae
http://imagine.gsfc.nasa.gov/science/objects/supernovae1.html
http://imagine.gsfc.nasa.gov/science/objects/supernovae2.html

Life Cycles of Stars
http://imagine.gsfc.nasa.gov/educators/lifecycles/stars.html

Functions: Domain and Range

Functions v. Relations

Purplemath

Let's return to the subject of domains and ranges.

When functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. These won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. Small sets of points are generally the simplest sorts of relations, so your book starts with those.

For instance:

 

MathHelp.com

Domain and Range

• State the domain and range of the following relation. Is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

The above list of points, being a relationship between certain x's and certain y's, is a relation. The domain is all the x-values, and the range is all the y-values. To give the domain and the range, I just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(It is customary to list these values in numerical order, but it is not required. Sets are called "unordered lists", so you can list the numbers in any order you feel like. Just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

While the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). Since x = 2gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

Note that all I had to do to check whether the relation was a function was to look for duplicate x-values. If you find any duplicate x-values, then the different y-values mean that you do not have a function. Remember: For a relation to be a function, each x-value has to go to one, and only one, y-value.

• State the domain and range of the following relation. Is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

I'll just list the x-values for the domain and the y-values for the range:

domain: {–3, –2, –1, 0, 1, 2}

range: {5}

This is another example of a "boring" function, just like the example on the previous page: every last x-value goes to the exact same y-value. But each x-value is different, so, while boring,

this relation is indeed a function.

In point of fact, these points lie on the horizontal line y = 5.

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By the way, the name for a set with only one element in it, like the "range" set above, is "singleton". So the range could also be stated as "the singleton of 5"

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There is one other case for finding the domain and range of functions. They will give you a function and ask you to find the domain (and maybe the range, too). I have only ever seen (or can even think of) two things at this stage in your mathematical career that you'll have to check in order to determine the domain of the function they'll give you, and those two things are denominators and square roots.

• Determine the domain and range of the given function:

$\mathbf{\color{green}{\mathit{y} = \dfrac{\mathit{x}^2 + \mathit{x} - 2}{\mathit{x}^2 - \mathit{x} - 2}}}$

The domain is all the values that x is allowed to take on. The only problem I have with this function is that I need to be careful not to divide by zero. So the only values that x can not take on are those which would cause division by zero. So I'll set the denominator equal to zero and solve; my domain will be everything else.

x² – x – 2 = 0

(x – 2)(x + 1) = 0

x = 2 or x = –1

Then the domain is "all x not equal to –1 or 2".

The range is a bit trickier, which is why they may not ask for it. In general, though, they'll want you to graph the function and find the range from the picture. In this case:

As you can see from my picture, the graph "covers" all y-values; that is, the graph will go as low as I like, and will also go as high as I like. For any point on the y-axis, no matter how high up or low down, I can go from that point either to the right or to the left and, eventually, I'll cross the graph. Since the graph will eventually cover all possible values of y, then:

the range is "all real numbers".

• Determine the domain and range of the given function:

$\mathbf{\color{green}{\mathit{y} = -\sqrt{-2\mathit{x} + 3}}}$

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The domain is all values that x can take on. The only problem I have with this function is that I cannot have a negative inside the square root. So I'll set the insides greater-than-or-equal-to zero, and solve. The result will be my domain:

–2x + 3 ≥ 0

–2x ≥ –3

2x ≤ 3

x ≤ 3/2 = 1.5

Then the domain is "all x ≤ 3/2".

The range requires a graph. I need to be careful when graphing radicals:

The graph starts at y = 0 and goes down (heading to the left) from there. While the graph goes down very slowly, I know that, eventually, I can go as low as I like (by picking an x that is sufficiently big). Also, from my experience with graphing, I know that the graph will never start coming back up. Then:

the range is "all y ≤ 0".

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• Determine the domain and range of the given function:

y = –x⁴ + 4

This is just a garden-variety polynomial. There are no denominators (so no division-by-zero problems) and no radicals (so no square-root-of-a-negative problems). There are no problems with a polynomial. There are no values that I can't plug in for x. When I have a polynomial, the answer for the domain is always:

the domain is "all x".

The range will vary from polynomial to polynomial, and they probably won't even ask, but when they do, I look at the picture:

The graph goes only as high as y = 4, but it will go as low as I like. Then:

The range is "all y ≤ 4"

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