Here is a demonstration of the use of group theory that I thought was quite neat. Fermat's little theorem states that if p is a prime number and a is a natural number that is not divisible by p (i.e. a is not congruent to 0 (mod p)) then
$$a^{p-1}\equiv 1 \;\mbox{(mod p) ...(1) }$$
Now recall from M208 that if p is prime then
$$(Z^{*}_{p},\times_{p})$$
is a group of order p-1 (see the Handbook for M208 on page 30). Now we can use Lagrange's Theorem (Handbook page 35) to prove Fermat's little theorem as follows. Let a be a natural number that is not divisible by a prime number p, then a will be congruent to an element of the above group, i.e. to one of the set
$$\left\{1,2,3,..,(p-1)\right\}$$
Let this element be g so that
$$a\equiv g \;\mbox{(mod p)}$$
Now g will generate a cyclic subgroup of order n (Handbook page 28) and by Lagrange's Theorem (page 35) n will divide p-1, the order of the group. Hence
$$p-1=mn$$
for some natural number m and thus
$$a^{p-1}\equiv g^{p-1}\equiv g^{mn}\equiv(g^{n})^{m}\;\mbox{(mod p)}$$
Now by definition
$$g^{n}\equiv 1 \;\mbox{(mod p)}$$
and so
$$a^{p-1}\equiv 1^{m}\equiv 1\;\mbox{(mod p)}$$
Hence (1) is proven.
Friday 15 February 2013
Sunday 3 February 2013
Why is the sun hot?
This is a slight departure from my usual posts here but I wanted to counter something that appears in the course notes for S382 Astrophysics and in the book "Stellar Evolution and Nucleosynthesis" which I think is misleading. Dan has raised the subject of 'Why is the sun hot' in his blog and I want to discuss some of the issues that have been raised here rather than trashing his blog with lots of comments.
First I am going to quote some bits and pieces from "Stellar Evolution and Nucleosynthesis" by Ryan and Norton. In chapter 2 of their book on page 46 they have a section 2.6 entitled "Why are stars hot? Putting fusion in its place". They say:-
"In Chapter 1 you saw that hydrogen fusion provides the power that is radiated by the Sun and other main-sequence stars. In this Chapter, you have seen the importance of the gravitation energy released by a collapsing cloud of gas. In order to understand the roles of these two sources of energy, before we go any further, we pause to consider exactly what fusion is and is not responsible for."
They go on to discuss how much energy is released per cubic metre in the core of the Sun by nuclear fusion and this turns out to be about 300 Watts.
They go on to say:-
"Think about this number: 300 Wm-3. Imagine putting three 100 W lightbulbs in a broom cupboard whose volume is about 1 cubic metre. Would that make the cupboard as bright and hot as the Sun?"
"No, clearly it would not. In fact, 300 W seems like a pathetically tiny power output for a volume as large as 1m3, especially in something as hot as the core of the Sun! Is the calculation grossly wrong? No, the power output per cubic metre is that small. Clearly hydrogen burning by the proton-proton chain is not much of a powerhouse! (But it is a big reservoir!)"
After some further number crunching the books says:
"You might be tempted at this stage to think that, even though the power released in fusion is very small, the energy released heats up the core of a star, and this is why stars are hot. The energy released is indeed very important, but it is not the reason that stars heat up. In fact the opposite is true; fusion prevents a star from getting hotter!"
"In fact, a star's temperature and luminosity are not determined by nuclear burning. However, they are maintained by nuclear burning. Nuclear fusion replenishes the slow leakage of energy from the star's core and eventually from its surface. As you saw in the last Section, the release of gravitational potential energy can do exactly the same thing, but the nuclear fusion energy source has the advantage that it lasts for much longer. That is, nuclear fusion greatly delays the gravitational collapse of the star."
"If someone had asked you at the beginning of this book, 'What makes stars hot?', you could have been forgiven for answering 'It is because they have thermonuclear reaction in their cores.' That answer sounds plausible, but hopefully now you can see that it is incorrect. Stars are hot because they have collapsed from large diffuse clouds and gravitational potential energy has been converted to kinetic energy. That is why stars are hot."
I think that the above discussion and conclusion are very misleading. You could come away with the idea that nuclear fusion in the Sun is not an important source of its energy output ("Clearly hydrogen burning by the proton-proton chain is not much of a powerhouse") but that release of energy from gravitational potential energy is ("the release of gravitational potential energy can do exactly the same thing"). This is of course, incorrect.
The question that needs to be asked is this. Would the Sun be hot now after having existed for approximately 4.6 billion years if nuclear reactions weren't taking place in its core? The answer to this is no. If there was no other energy source in the Sun apart from the release of gravitational potential energy, then at its current luminosity it would only shine for approximately 18 million years , a time-scale that is about 256 times too short. It follows then that the Sun would have cooled to well below its current temperature by now if heating by contraction was its only source of energy. We know, anyway, that the Sun is in equilibrium, which means that it is neither contracting nor expanding. You can't have energy released by gravitational potential if the Sun isn't contracting.
The argument about the three 100 Watt light bulbs in a cupboard is also misleading ("Would that make the cupboard as bright and hot as the Sun? No clearly, it would not!"). The cupboard could get as bright as the surface of the Sun and even as high as the core of the Sun, if the cupboard was perfectly insulated. This is how an ideal oven would work. In fact, I find the langauge used by the authors ("300 W seems like a pathetically tiny power output for a volume as large as 1m3") emotive and inappropriate for a science text.
So is nuclear fusion in the Sun much of a powerhouse? Yes, it is. Although of 300 Watts per metre cubed sounds like a small number it is sufficient to power the output of the Sun for billions of years and maintain the temperature in its core for the duration.
So why is the Sun hot? I would argue it is hot for both of the reasons discussed, neither of which can be disentagled or discounted. The collapse of the gas cloud that led to the formation of the Sun was responsible for getting the core of the Sun up to a temperature where nuclear fusion could take place and nuclear fusion is responsible for maintaining that temperature of the Sun, in both the core and on its surface, until now. To overemphasise the role of gravitational energy in the way that these authors have done in their book is, in my opinion, very misleading and it is disappointing if this is also true of the course notes for the OU Astrophysics course.
First I am going to quote some bits and pieces from "Stellar Evolution and Nucleosynthesis" by Ryan and Norton. In chapter 2 of their book on page 46 they have a section 2.6 entitled "Why are stars hot? Putting fusion in its place". They say:-
"In Chapter 1 you saw that hydrogen fusion provides the power that is radiated by the Sun and other main-sequence stars. In this Chapter, you have seen the importance of the gravitation energy released by a collapsing cloud of gas. In order to understand the roles of these two sources of energy, before we go any further, we pause to consider exactly what fusion is and is not responsible for."
They go on to discuss how much energy is released per cubic metre in the core of the Sun by nuclear fusion and this turns out to be about 300 Watts.
They go on to say:-
"Think about this number: 300 Wm-3. Imagine putting three 100 W lightbulbs in a broom cupboard whose volume is about 1 cubic metre. Would that make the cupboard as bright and hot as the Sun?"
"No, clearly it would not. In fact, 300 W seems like a pathetically tiny power output for a volume as large as 1m3, especially in something as hot as the core of the Sun! Is the calculation grossly wrong? No, the power output per cubic metre is that small. Clearly hydrogen burning by the proton-proton chain is not much of a powerhouse! (But it is a big reservoir!)"
After some further number crunching the books says:
"You might be tempted at this stage to think that, even though the power released in fusion is very small, the energy released heats up the core of a star, and this is why stars are hot. The energy released is indeed very important, but it is not the reason that stars heat up. In fact the opposite is true; fusion prevents a star from getting hotter!"
"In fact, a star's temperature and luminosity are not determined by nuclear burning. However, they are maintained by nuclear burning. Nuclear fusion replenishes the slow leakage of energy from the star's core and eventually from its surface. As you saw in the last Section, the release of gravitational potential energy can do exactly the same thing, but the nuclear fusion energy source has the advantage that it lasts for much longer. That is, nuclear fusion greatly delays the gravitational collapse of the star."
"If someone had asked you at the beginning of this book, 'What makes stars hot?', you could have been forgiven for answering 'It is because they have thermonuclear reaction in their cores.' That answer sounds plausible, but hopefully now you can see that it is incorrect. Stars are hot because they have collapsed from large diffuse clouds and gravitational potential energy has been converted to kinetic energy. That is why stars are hot."
I think that the above discussion and conclusion are very misleading. You could come away with the idea that nuclear fusion in the Sun is not an important source of its energy output ("Clearly hydrogen burning by the proton-proton chain is not much of a powerhouse") but that release of energy from gravitational potential energy is ("the release of gravitational potential energy can do exactly the same thing"). This is of course, incorrect.
The question that needs to be asked is this. Would the Sun be hot now after having existed for approximately 4.6 billion years if nuclear reactions weren't taking place in its core? The answer to this is no. If there was no other energy source in the Sun apart from the release of gravitational potential energy, then at its current luminosity it would only shine for approximately 18 million years , a time-scale that is about 256 times too short. It follows then that the Sun would have cooled to well below its current temperature by now if heating by contraction was its only source of energy. We know, anyway, that the Sun is in equilibrium, which means that it is neither contracting nor expanding. You can't have energy released by gravitational potential if the Sun isn't contracting.
The argument about the three 100 Watt light bulbs in a cupboard is also misleading ("Would that make the cupboard as bright and hot as the Sun? No clearly, it would not!"). The cupboard could get as bright as the surface of the Sun and even as high as the core of the Sun, if the cupboard was perfectly insulated. This is how an ideal oven would work. In fact, I find the langauge used by the authors ("300 W seems like a pathetically tiny power output for a volume as large as 1m3") emotive and inappropriate for a science text.
So is nuclear fusion in the Sun much of a powerhouse? Yes, it is. Although of 300 Watts per metre cubed sounds like a small number it is sufficient to power the output of the Sun for billions of years and maintain the temperature in its core for the duration.
So why is the Sun hot? I would argue it is hot for both of the reasons discussed, neither of which can be disentagled or discounted. The collapse of the gas cloud that led to the formation of the Sun was responsible for getting the core of the Sun up to a temperature where nuclear fusion could take place and nuclear fusion is responsible for maintaining that temperature of the Sun, in both the core and on its surface, until now. To overemphasise the role of gravitational energy in the way that these authors have done in their book is, in my opinion, very misleading and it is disappointing if this is also true of the course notes for the OU Astrophysics course.
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