Nuclear reactions in the Sun and other topics on stars

[Hak]

Arguing with a crackpot is futile. Arguing with a crackpot is more futile. But I understand now that we are arguing with an imaginary crackpot. That's triply futile.

I apologise if I have offended your sensibilities, that was not my intention. I just wanted to create some sort of constructive dialogue and understand all the evidence supporting the 'nuclear reactions inside the Sun' argument that refutes the 'gravitational contraction' argument. Apparently there is more than one. Do you confirm?

That said, it would be good to make up your minds about whether you are discussing main sequence stars or white dwarfs, which behave very differently.

In reference to what? Anyway, what do you think of the ionisation paradox and the white dwarf? I would also like to know your opinion. Thank you very much.

 

[Hak]

Doing it through a third party is actually more important. If you don´t, you are letting the crackpot (who may be invested in his opinions) persuade the third party (who does not have the same investment and who might be open to persuasion).

But now about the paradox of ionization...
Cool metals in solid and liquid form are actually "ionized" in that the outer electrons are shared by neighbouring atoms and free to jump from one atom to another.
At ambient pressure and temperature, no metals share all their electrons. Even Li and Be have two inner electrons that are confined to their respective nuclei. But their outer electrons are shared.
That the electrons in metal are moving around the metal does not mean that the metal is "hot". You can cool metals to absolute zero, and they remain conductive - with one illustrative exception.
Just because electrons stuck in atoms do not move all around the substance does not mean they do not have kinetic energy. Their kinetic energy, confined on their closed and small orbits, is actually quite big!
In many insulating solids and liquids, and also many metals, some electrons are confined not to specific atoms but shared by a few (usually two) atoms, confined at a covalent bond. Again, they do have kinetic energy.
The illustrative exception for metal which does not remain conductive on cooling is tin. At room temperature, or what you want in your room, it is a conductive (and tough/plastic) metal. Cooling to just +13 Celsius, it turns into an insulator (actually semiconductor). It also expands a lot and becomes brittle.
At low pressures, the substances convert in various ways regarding conductivity. A typical example is mercury. It is a conductive, metallic liquid. Boil it at 360 Celsius. Mercury vapour consists almost exclusively of lone, neutral atoms. It therefore is a very poor conductor of electricity. When you heat mercury vapour further, the atoms are gradually ionized and the vapour slowly becomes a better conductor again.

Now, if we apply high pressure on mercury, what happens to conductivity? Well, mercury will stay a liquid, conductive metal to higher temperatures than at atmospheric pressure. So when it eventually boils, it does so at a higher temperature and turns into a hotter, more conductive/plasmalike vapour. We can extrapolate that at sufficiently high pressure, liquid metal conductive mercury would turn directly into hot conductive plasma without passing through insulating gas.

What about tin? Well, since tin expands on converting to insulator, this process could be hindered or reversed by applying sufficient pressure. You could apply pressure to turn cold tin (normally semiconductor) into conductive metal.
It is predicted, and confirmed by experiment in several other cases beside tin, that insulators do tend to convert into conductors at high pressures. But this does not stop them from being cold!

I am not an expert, but it seems to me to be a quite complete and profound description. Thank you very much.

 

[phinds]

Doing it through a third party is actually more important. If you don´t, you are letting the crackpot (who may be invested in his opinions) persuade the third party (who does not have the same investment and who might be open to persuasion).

Then the third party should make his OWN argument, not bring up some random crackpot that we have never heard of or from.

 

[Ken G]

The real question for me that is irresolvable, and very difficult to answer, is the whole issue put forward by the professor.

"The fact that matter is ionized allows ions and electrons to be much closer together than they are in atoms (Bohr radius
), and the result is that some stars, including white dwarfs, have a very small radius and a very high density that could not be had if matter were not ionized.

I think the better way to say this is that if you produce such a small radius and high density, you can be sure that the electrons will ionize. The reason is because they must have very significant kinetic energy, a distinction also pointed to by @snorkack. This is the key point, the electrons can have very high kinetic energy and still be "cold", in the sense of low temperature. If you stick your tongue to metal on a freezing day, your tongue can get stuck by how cold that metal is, but the electrons in that metal will have extremely high kinetic energy. This phenomenon is sometimes given the bizarre name "degeneracy."

A better way to think of it is that this is the quantum mechanical effect called a "ground state." A ground state is a state of zero temperature, but when dealing with forces that get extremely strong at small distance, like gravity and electrostatic attraction, this will also be a state of very high electron kinetic energy. So the paradox you refer to, what makes the professor's comments so hard to understand, is the same thing that leads to a "ground state" in quantum mechanics, and this was completely unexpected until the discovery of quantum mechanics.

A white dwarf, however, is a star that has run out of nuclear fuel; therefore, it slowly cools down. At some point the temperature becomes so low that matter would tend to regroup into non-ionized atoms.

This is the problem, right here. You are implicitly equating being "cold" (in the sense of low temperature) with electrons having low kinetic energies so getting trapped by positive charges. That's what is not the case, the electrons in a cooling white dwarf actually gain more kinetic energy, the more the white dwarf cools. They are approaching their ground state, but it is not a ground state like the electrons in an atom (because that relates to the electrostatic forces from the positive charges), it is a ground state that relates to the strong gravity of the white dwarf. (Electrostatic forces play little role in a white dwarf because there are as many positive as negative charges in any given box of white dwarf "fluid", whereas gravity is always an attractive force on any "fluid element.") The main point is, as they approach their ground state, the electrons have higher and higher kinetic energy (so are even harder to trap into atoms), and lower and lower temperature. That this is possible is pure quantum mechanics, so if you have not learned about quantum mechanics, you share the same confusion that every single physicist living before 1920 would have felt!

But to do that it would have to increase its radius and decrease its density, and this it cannot do because the change in gravitational energy it would take is greater than all the energy it has left.

True, and that's just another way of saying the electrons have way too much kinetic energy to be trapped into atoms, which is correct. They just accomplish this feat at much lower temperature than you would think possible, without quantum mechanics.

So, in a sense, the star would get to a situation where it does not have enough energy to cool down again. In other words, the temperature would stop decreasing without the star being in equilibrium with the part of the universe that surrounds it, and that is obviously not possible."

I'm not sure what the professor is trying to say here, hopefully they know quantum mechanics so are just using this paradox to introduce the fact that there is something missing in the argument. It is true that it would be impossible for the white dwarf to not reach equilibrium with its environment, so what is missing that seems to create this paradox is quantum mechanics.

Sounds like a dilemma to me. A question that is very difficult and complex to answer. Even the professor himself thought so.

It is one of the greatest dilemmas in the history of physics. The fact that it was resolved in the 1920s is also one of the greatest accomplishments in that history.

 

[Hak]

I think the better way to say this is that if you produce such a small radius and high density, you can be sure that the electrons will ionize. The reason is because they must have very significant kinetic energy, a distinction also pointed to by @snorkack. This is the key point, the electrons can have very high kinetic energy and still be "cold", in the sense of low temperature. If you stick your tongue to metal on a freezing day, your tongue can get stuck by how cold that metal is, but the electrons in that metal will have extremely high kinetic energy. This phenomenon is sometimes given the bizarre name "degeneracy."

A better way to think of it is that this is the quantum mechanical effect called a "ground state." A ground state is a state of zero temperature, but when dealing with forces that get extremely strong at small distance, like gravity and electrostatic attraction, this will also be a state of very high electron kinetic energy. So the paradox you refer to, what makes the professor's comments so hard to understand, is the same thing that leads to a "ground state" in quantum mechanics, and this was completely unexpected until the discovery of quantum mechanics.

This is the problem, right here. You are implicitly equating being "cold" (in the sense of low temperature) with electrons having low kinetic energies so getting trapped by positive charges. That's what is not the case, the electrons in a cooling white dwarf actually gain more kinetic energy, the more the white dwarf cools. They are approaching their ground state, but it is not a ground state like the electrons in an atom (because that relates to the electrostatic forces from the positive charges), it is a ground state that relates to the strong gravity of the white dwarf. (Electrostatic forces play little role in a white dwarf because there are as many positive as negative charges in any given box of white dwarf "fluid", whereas gravity is always an attractive force on any "fluid element.") The main point is, as they approach their ground state, the electrons have higher and higher kinetic energy (so are even harder to trap into atoms), and lower and lower temperature. That this is possible is pure quantum mechanics, so if you have not learned about quantum mechanics, you share the same confusion that every single physicist living before 1920 would have felt!True, and that's just another way of saying the electrons have way too much kinetic energy to be trapped into atoms, which is correct. They just accomplish this feat at much lower temperature than you would think possible, without quantum mechanics.

I'm not sure what the professor is trying to say here, hopefully they know quantum mechanics so are just using this paradox to introduce the fact that there is something missing in the argument. It is true that it would be impossible for the white dwarf to not reach equilibrium with its environment, so what is missing that seems to create this paradox is quantum mechanics.It is one of the greatest dilemmas in the history of physics. The fact that it was resolved in the 1920s is also one of the greatest accomplishments in that history.

Thank you very much. Why did you mention such a precise date, i.e. 1920? Excuse my lack of knowledge on the subject...

 

[Ken G]

Thank you very much. Why did you mention such a precise date, i.e. 1920? Excuse my lack of knowledge on the subject...

Just a round number, there was no one single event. Still, it was remarkable how fast physicists went from having no idea about the "dilemma" of which you speak, to having a pretty complete understanding. Perhaps the most important single idea behind what we are talking about is the "deBroglie wavelength" (which essentially gives the smallest length scale an electron of a given momentum can be confined to), and that was proposed in 1924, whereas the seminal paper that set the stage for our current understanding of white dwarfs appeared in 1952 (by Leon Mestel). So it only took 28 years to go from a crazy new idea that related momentum and distance in a way Newton would never have dreamed of, to a rather complete answer to the mystery that motivated your post.

 

[Hak]

Just a round number, there was no one single event. Still, it was remarkable how fast physicists went from having no idea about the "dilemma" of which you speak, to having a pretty complete understanding. Perhaps the most important single idea behind what we are talking about is the "deBroglie wavelength" (which essentially gives the smallest length scale an electron of a given momentum can be confined to), and that was proposed in 1924, whereas the seminal paper that set the stage for our current understanding of white dwarfs appeared in 1952 (by Leon Mestel). So it only took 28 years to go from a crazy new idea that related momentum and distance in a way Newton would never have dreamed of, to a rather complete answer to the mystery that motivated your post.

Crazy. The more I think about it, the more I wonder how such a thing is possible. Anyway, thank you very much for your response.

 

[Ken G]

To expound a little more on the mystery, the real question is not how the electrons stay ionized as the white dwarf "cools", it is why the white dwarf cools at all! The normal thing, classically, is that a radiating ball of self gravitating gas doesn't cool, in the sense of temperature dropping, it gets hotter. That's how the Sun was able to start fusing hydrogen in the first place, after all. The mystery is, why does a white dwarf cool under the same conditions that made the Sun so hot? So no one should be surprised that a white dwarf can lose heat and still keep the electrons ionized, they should be surprised that the temperature drops during the process. That's the only part of all this that requires quantum mechanics to understand, it is because of the presence of a ground state, it means that the system temperature drops as the ground state is approached, even if the electron kinetic energy is getting very high indeed.

 

[Hak]

To expound a little more on the mystery, the real question is not how the electrons stay ionized as the white dwarf "cools", it is why the white dwarf cools at all! The normal thing, classically, is that a radiating ball of self gravitating gas doesn't cool, in the sense of temperature dropping, it gets hotter. That's how the Sun was able to start fusing hydrogen in the first place, after all. The mystery is, why does a white dwarf cool under the same conditions that made the Sun so hot? So no one should be surprised that a white dwarf can lose heat and still keep the electrons ionized, they should be surprised that the temperature drops during the process. That's the only part of all this that requires quantum mechanics to understand, it is because of the presence of a ground state, it means that the system temperature drops as the ground state is approached, even if the electron kinetic energy is getting very high indeed.

Thank you very much. Could you provide more information or insights about "deBroglie wavelength"? Thank you again.

 

[Ken G]

Sure, the key relation is that this is the wavelength associated with the quantum mechanical "wave function," which means that particles always move according to the rules of waves (waves can do even the things baseballs do, but they can also do other things, like interfere with themselves and make some behaviors possible and others impossible). For our present purposes, it means the interparticle spacing between electrons is impossible to be less than that wavelength (approximately, this is essentially the "Pauli exclusion principle", simplified). So if you have a white dwarf with very very small distances between electrons, it means their deBroglie wavelengths have to be equally small.

Now that by itself doesn't say much, until you understand deBroglie's key relation, which says the particle momentum is Planck's constant (h) divided by the deBroglie wavelength. So what matters now is that, since we know the deBroglie wavelengths will need to be very very small in a white dwarf, this implies the electron momentum must be very very big, which also means their kinetic energy must be very big. This is not surprising, because the kinetic energy of all the electrons in a white dwarf has to add up to half the total gravitational energy of the white dwarf in order for there to be force balance (which is called the "nonrelativistic virial theorem"), and we know the gravitational energy is very large because the radius is very small for all that mass. As the white dwarf is forming and its radius is contracting, the total gravitational energy scales like the reciprocal of the radius, so the total kinetic energy of the electrons also scales like the reciprocal of the radius. That means their momentum scales like the reciprocal square root of the radius (since momentum scales like the square root of the kinetic energy, nonrelativistically).

Now here's the surprise: the distance between particles scales like the radius (all the internal distances are shrinking like that), and the deBroglie wavelength has to always be less than that or else there would be destructive interference in the wave function that would make the situation impossible. So imagine a graph of two curves, the actual average momentum of each electron (which scales like one over the square root of the radius) and the minimum possible average momentum (which scales like one over the radius itself), and note what happens as the radius gets very small: at some point the average momentum tries to drop below the minimum possible momentum allowed because the former is dropping faster than the latter as the radius shrinks. What happens when this limit is reached?

This is the approach to the "ground state", which means that it becomes simply impossible for the white dwarf to shrink any more. You can also say it cannot lose any more heat because a ground state is a state of zero temperature, and zero temperature states cannot lose heat, and heat loss is why it was shrinking in the first place. Physically, what makes it impossible to shrink any more is the Pauli exclusion principle, which is essentially the statement that destructive interference in the wave function makes it simply impossible to lose any more heat because to do so would require shrinking that there is not available energy to allow. (That's also how the ground state in a single atom behaves, so the white dwarf is starting to act like a very giant atom, in a sense.)

So that's how a white dwarf reaches a state of very high kinetic energy (the most the star has ever had in its life), and very low temperature. It should be noted that the temperature is generally not so low for our everyday concept of low temperature, but it is low in comparison to the extremely high kinetic energy. By that I mean, it would take infinite time for the white dwarf to actually reach its ground state at zero temperature, but it becomes a reasonable approximation to imagine it is in its ground state if you want to understand why the electrons have the kinetic energy they do, and why the white dwarf has the radius it has. Its actual nonzero temperature is not an important correction for these purposes. So you end up with a state where the electron deBroglie wavelengths (based on their momenta) roughly equal their interparticle spacing (based on the number of electrons and the radius of the white dwarf), and the kinetic energy roughly equals the gravitational potential energy, and that's a white dwarf.

Thus there is no problem with this object being in thermal equilibrium with its surroundings, though it takes a long time to cool that much, because it can cool to any arbitrarily low temperature, but it will still be similar to the object I just described. Only quantum mechanics, and the ground state concept, makes that possible, there is no classical way to understand why the temperature can be going down instead of up. But never is there any surprise that the electrons are ionized, that would hold either classically or quantum mechanically, they just have a huge kinetic energy either way.

 

[Hak]

Sure, the key relation is that this is the wavelength associated with the quantum mechanical "wave function," which means that particles always move according to the rules of waves (waves can do even the things baseballs do, but they can also do other things, like interfere with themselves and make some behaviors possible and others impossible). For our present purposes, it means the interparticle spacing between electrons is impossible to be less than that wavelength (approximately, this is essentially the "Pauli exclusion principle", simplified). So if you have a white dwarf with very very small distances between electrons, it means their deBroglie wavelengths have to be equally small.

Now that by itself doesn't say much, until you understand deBroglie's key relation, which says the particle momentum is Planck's constant (h) divided by the deBroglie wavelength. So what matters now is that, since we know the deBroglie wavelengths will need to be very very small in a white dwarf, this implies the electron momentum must be very very big, which also means their kinetic energy must be very big. This is not surprising, because the kinetic energy of all the electrons in a white dwarf has to add up to half the total gravitational energy of the white dwarf in order for there to be force balance (which is called the "nonrelativistic virial theorem"), and we know the gravitational energy is very large because the radius is very small for all that mass. As the white dwarf is forming and its radius is contracting, the total gravitational energy scales like the reciprocal of the radius, so the total kinetic energy of the electrons also scales like the reciprocal of the radius. That means their momentum scales like the reciprocal square root of the radius (since momentum scales like the square root of the kinetic energy, nonrelativistically).

Now here's the surprise: the distance between particles scales like the radius (all the internal distances are shrinking like that), and the deBroglie wavelength has to always be less than that or else there would be destructive interference in the wave function that would make the situation impossible. So imagine a graph of two curves, the actual average momentum of each electron (which scales like one over the square root of the radius) and the minimum possible average momentum (which scales like one over the radius itself), and note what happens as the radius gets very small: at some point the average momentum tries to drop below the minimum possible momentum allowed because the former is dropping faster than the latter as the radius shrinks. What happens when this limit is reached?

This is the approach to the "ground state", which means that it becomes simply impossible for the white dwarf to shrink any more. You can also say it cannot lose any more heat because a ground state is a state of zero temperature, and zero temperature states cannot lose heat, and heat loss is why it was shrinking in the first place. Physically, what makes it impossible to shrink any more is the Pauli exclusion principle, which is essentially the statement that destructive interference in the wave function makes it simply impossible to lose any more heat because to do so would require shrinking that there is not available energy to allow. (That's also how the ground state in a single atom behaves, so the white dwarf is starting to act like a very giant atom, in a sense.)

So that's how a white dwarf reaches a state of very high kinetic energy (the most the star has ever had in its life), and very low temperature. It should be noted that the temperature is generally not so low for our everyday concept of low temperature, but it is low in comparison to the extremely high kinetic energy. By that I mean, it would take infinite time for the white dwarf to actually reach its ground state at zero temperature, but it becomes a reasonable approximation to imagine it is in its ground state if you want to understand why the electrons have the kinetic energy they do, and why the white dwarf has the radius it has. Its actual nonzero temperature is not an important correction for these purposes. So you end up with a state where the electron deBroglie wavelengths (based on their momenta) roughly equal their interparticle spacing (based on the number of electrons and the radius of the white dwarf), and the kinetic energy roughly equals the gravitational potential energy, and that's a white dwarf.

Thus there is no problem with this object being in thermal equilibrium with its surroundings, though it takes a long time to cool that much, because it can cool to any arbitrarily low temperature, but it will still be similar to the object I just described. Only quantum mechanics, and the ground state concept, makes that possible, there is no classical way to understand why the temperature can be going down instead of up. But never is there any surprise that the electrons are ionized, that would hold either classically or quantum mechanically, they just have a huge kinetic energy either way.

Thank you very much, if you still have insights you'd like to share, don't hesitate to write to me!

 

[snorkack]

Oh, there is obvious classical or "classical" explanation why white dwarfs cool!
First have a look at why stars heat when they lose energy...
Because of PV=nRT.
If you have a thin atmosphere on top of a solid or liquid, incompressible planet then it cools on cooling all right. Because the thickness of atmosphere is from the beginning small compared to the radius of the planet, the gravitational acceleration of the atmosphere is unchanged, its weight and pressure on the ground is unchanged - the atmosphere gets thinner, denser and colder.
But if a star consists of gas which is attracted by its own gravity and is still ideal gas?
If you take a self-gravitating ball and decrease its radius r (R is unavailable, it is the gas constant...) in half, then:

• surface gravity a increases 4 times
• density ρ increases 8 times
• thus the pressure of any column of given length increases 32 times
• though the length of the column shrinks 2 times
• thus the central pressure increases 16 times
• but since the density increased only 8 times and the gas is assumed to be still ideal, the temperature must increase 2 times.

But this assumes that the star consists of ideal gas.
Take any common liquid or solid. Double its pressure. Its volume does NOT halve!
What happens when you double its temperature varies. Some substances boil, and then their volume far more than doubles. Others don´t, and then although they usually expand, their volume expands only slightly. Like, take a litre of water at 20 Celsius. Its density is 998 g/l. If you heat it to 99 Celsius - increasing the temperature about 27 % - it expands only 1,04 times, to 958 g/l. If you heat it to 101 Celsius, it expands about 1700 times, to 0,6 g/l.

The thing is, white dwarfs unlike stars and like planets behave like they consist of condensed matter - solids or liquids - not ideal gas. Low compressibility, low thermal expansion. Of course they cool when they cool!
And being cool can still be burning hot. Water boils to steam at 100 Celsius, but hot iron is still solid at 1500 Celsius. And molten iron is still a condensed substance - low compressibility, low thermal expansion. Incidentally, a decent conductor of electricity too. Boils only over 2800 Celsius at 1 bar. Boiling temperature expected to rise with pressure, as usual for liquids.

 

[Hak]

If you take a self-gravitating ball and decrease its radius r (R is unavailable, it is the gas constant...) in half, then:

• surface gravity a increases 4 times
• density ρ increases 8 times
• thus the pressure of any column of given length increases 32 times
• though the length of the column shrinks 2 times
• thus the central pressure increases 16 times
• but since the density increased only 8 times and the gas is assumed to be still ideal, the temperature must increase 2 times.

Thank you very much. Based on what did you infer these estimates? Thank you.

 

[Ken G]

Oh, there is obvious classical or "classical" explanation why white dwarfs cool!

Only because of quantum mechanics! So what you mean by "classical" really means "with quantum mechanics but in a way we did not recognize before we had quantum mechanics."

My point here is, with no Pauli exclusion principle, it will remain a gas of independent charged particles, and the white dwarf will never cool. But it's true that you could have classical interparticle forces that break the gas/plasma behavior and turn it into some kind of crystal lattice. That would allow cooling because of the huge difference between a force (like gravity) that always increases with reduced interparticle distances, and a force (like molecular binding forces) which has an equilibrium where the force is zero at finite interparticle distance (acting like springs attached between the particles). Usually with white dwarfs, we do not need to consider forces like that (and we don't, in any of the common formulas you will see used to describe white dwarfs). They don't come up until the white dwarf is quite old, i.e., after quantum mechanics has already made its presence felt. Without quantum mechanics that just never happens, so the white dwarf never cools.

First have a look at why stars heat when they lose energy...
Because of PV=nRT.

Yes, and that is the thing that quantum mechanics breaks.

If you have a thin atmosphere on top of a solid or liquid, incompressible planet then it cools on cooling all right.

Certainly, that's because the incompressibility of the planet makes implicit assumptions about interparticle forces of the second type I mentioned above. That's what quantum mechanics is allowing to happen. But it's true that those kinds of forces will eventually matter, as even a white dwarf will eventually crystallize its ions. In a normal picture of a white dwarf, we look at it before that happens, and get things like the white dwarf mass/radius relation. Once the ions crystallize, new physics is needed, but that takes a long time to appear (it's why Jupiter is not a white dwarf).

It's interesting to consider how the physics of interparticle forces other than gravity come into play. I think your perspective here could be framed as the question, is quantum mechanics needed for those forces to come into play, or would they never matter without it? That's what I'm saying the answer is the latter: with no quantum mechanics, interparticle forces other than gravity will never come into play, because one crucial thing that the PEP does is allow the electrons to steal most of the kinetic energy from the ions. It does that because thermodynamics says that what will be the same between the electrons and the ions is their temperature, but the PEP is causing the electrons to have way more kinetic energy than their kT would suggest classically. Hence, classically, the electrons and ions have similar kinetic energy, which is then very high for both, and nothing ever crystallizes nor forms any molecules of any kind. So classically, a white dwarf stays a plasma, and never cools at all.

Take any common liquid or solid. Double its pressure. Its volume does NOT halve!

Yes, because of those molecular type forces that have a zero at a finite equilibrium interparticle spacing at low enough kinetic energy to "trap" the particles in those potential wells. But that is what doesn't happen when both electrons and ions have roughly equal kinetic energy, i.e., that's what doesn't happen without quantum mechanically altered thermodynamics.

The thing is, white dwarfs unlike stars and like planets behave like they consist of condensed matter - solids or liquids - not ideal gas.

Beware, this is one of the common yet misleading things you will hear about white dwarfs. It comes about because the temperature is very low (due to quantum mechanics), compared to the kinetic energy, and that means the ions (which are behaving classically) will eventually crystallize. But most white dwarfs have not had that happen yet, they are still just pure gas, with no important interparticle forces except gravity (and the electrostatic forces that assure charge neutrality, which we safely sweep under the rug by considering "fluid parcels" that are electrically neutral). All of their solidlike behavior comes from their inability to lose heat easily, nothing else. In particular, if you squeeze a highly degenerate gas using some external pressure, it contracts exactly the same as a thermally insulated ideal gas would, i.e., adiabatically. That's because it is exactly a gas that cannot lose heat, just like an adiabatic ideal gas, but with a much different temperature than you would imagine.

 

[snorkack]

Thank you very much. Based on what did you infer these estimates? Thank you.

I thought they were fairly obvious, which is why I did not bother to show the details.
For example the first is the law of gravity.
F=G*m₁*m₂/r²
generalizing away m₂, you get
a=G*m₁/r²
So when you leave m₁ unchanged and decrease r 2 times (halve r), you increase a 2²=4 times.
Now do you see the next estimate?