Undergrad Calculating the Sun's core temperature?

[bbbl67]

Knowing the following information: Sun surface temperature of 5788 K, Sun's core is between 20-25% of radius. I tried to (perhaps naively) calculate the temperature to the edge of its core from the surface temperature, using an inverse square law. Depending on which value you use for the core radius, 0.2^2 or 0.25^2, you get a temperature between 92,608 to 144,700 K, which is well below the known value of 15 million K. So what went wrong?

 

[phinds]

What's going on at the surface and what's going on in the core are WAY different things. You can't compute the temperature of one based on the other, or at least certainly not in the straightforward way you are trying to do.

 

[bbbl67]

What's going on at the surface and what's going on in the core are WAY different things. You can't compute the temperature of one based on the other, or at least certainly not in the straightforward way you are trying to do.

Well obviously, that's why I'm asking here.

 

[Borg]

Maybe some of these links will help to answer your question. The first one is a very simple rough estimate that is still pretty close.

http://burro.case.edu/Academics/Astr221/StarPhys/estcent.html
https://astronomy.stackexchange.com/questions/253/how-was-the-core-temperature-of-the-sun-estimated
https://en.wikipedia.org/wiki/Solar_core

 

[Matterwave]

What went wrong was the entire approach used to try to find the temperature at the core. There is no reason to expect, as far as I can tell, that the internal temperature of the Sun should follow an inverse square dependency on radius.

 

[bbbl67]

What went wrong was the entire approach used to try to find the temperature at the core. There is no reason to expect, as far as I can tell, that the internal temperature of the Sun should follow an inverse square dependency on radius.

Yeah, well somehow there are people who know all of the internal temperatures of the Sun, so I'd like to find out how they calculate it. For example, my first thought to correct this is that the convective zone doesn't follow the inverse square law, but perhaps the core and radiative zones may follow it? Need know if that's the case or not.

 

[mfb]

Why do you expect an inverse square law anywhere? You need constant energy flow in radiative zones. The blackbody radiation scales with the temperature to the 4th power, the heat flow is proportional to its derivative - but it also depends on how opaque the region is. The inner regions have a high density, radiation is scattered quickly, you need large temperature gradients to maintain the heat flow. This doesn't lead to any easy relation between temperature and radius.

 

[stefan r]

Yeah, well somehow there are people who know all of the internal temperatures of the Sun, so I'd like to find out how they calculate it. For example, my first thought to correct this is that the convective zone doesn't follow the inverse square law, but perhaps the core and radiative zones may follow it? Need know if that's the case or not.

I thought it was the fusion people. They can test how often hydrogen atoms will fuse if you accelerate protons and smash them into each other. Temperature is equivalent to particle velocity. We know how much fusion is taking place because we know how much energy the Sun radiates. We also know how much total gas there was when it started fusing. The temperature is "hot enough". You know that a glass of soda is 0C because it has ice cubes floating in it. You know that a pot of boiling soup is 100C because it has steam bubbles. The Sun's core has to be hot enough for fusion to be happening at the rate that fusion happens.

 

[bbbl67]

Why do you expect an inverse square law anywhere? You need constant energy flow in radiative zones. The blackbody radiation scales with the temperature to the 4th power, the heat flow is proportional to its derivative - but it also depends on how opaque the region is. The inner regions have a high density, radiation is scattered quickly, you need large temperature gradients to maintain the heat flow. This doesn't lead to any easy relation between temperature and radius.

Well mainly I expected it to conform to an inverse square law because that's what happens outside the surface of the Sun. All of the energy that flows from the Sun to Earth is through radiation. So I expected that the inner radiative zone would be the same behaviour, since it's also done through radiation.

I knew that the convective zone would be a different rule, but it was only going to be my initial first-cut approximation, ignoring the convective zone and perhaps getting close to the answer.

I thought it was the fusion people. They can test how often hydrogen atoms will fuse if you accelerate protons and smash them into each other. Temperature is equivalent to particle velocity. We know how much fusion is taking place because we know how much energy the Sun radiates. We also know how much total gas there was when it started fusing. The temperature is "hot enough". You know that a glass of soda is 0C because it has ice cubes floating in it. You know that a pot of boiling soup is 100C because it has steam bubbles. The Sun's core has to be hot enough for fusion to be happening at the rate that fusion happens.

Well, the 0C and 100C examples are special cases, as those are the regions where a phase transition happens. So the temperature stays constant even if the actual heat content is changing. In the case of fusion, it's not a phase transition situation, so temperatures and pressures can vary greatly to enable fusion to take place. For example the inside of the Sun would be 15 million K, but inside a tokamak you'd need 100 million K+ to get fusion. The Sun has a lot more pressure than a Tokamak, so the Tokamak has to compensate with higher temperature.

 

[mfb]

Outside the surface of the Sun you have the corona which is much hotter than the surface.
The equilibrium temperature for objects outside drops with the inverse square root, but that only applies because you have a vacuum.

 

[Vanadium 50]

Well mainly I expected it to conform to an inverse square law because that's what happens outside the surface of the Sun.

And why would you think that? That might be appropriate if the surface weren't there, but it is.

 

[jonesdave]

This page may explain how a rough calculation can be made. It results in an overestimate by ~ 50%;

http://www.jgiesen.de/astro/temperature.htm

 

[stefan r]

...
...Well, the 0C and 100C examples are special cases, as those are the regions where a phase transition happens. So the temperature stays constant even if the actual heat content is changing. In the case of fusion, it's not a phase transition situation, so temperatures and pressures can vary greatly to enable fusion to take place. For example the inside of the Sun would be 15 million K, but inside a tokamak you'd need 100 million K+ to get fusion. The Sun has a lot more pressure than a Tokamak, so the Tokamak has to compensate with higher temperature.

The temperature of boiling water varies with pressure too. At 1% of an atmosphere you can boil water at 10C.

 

[Nik_2213]

When the first neutrino detectors came up short, there was much controversy as this could imply the sun's inner processes didn't match the predictions from nuclear theory...
Turned out that due to those slippery little whatsits' tiny but non-zero mass, they were changing into forms those first detectors missed.
So, solar processes' corner of nuclear theory intact, but another having to digest neutrinos with mass...
At least our sun wasn't about to go out !