Radius of the Sun: Can We Accurately Predict It Through Theoretical Models?

[thepolishman]

TL;DR Summary

Are there any theoretical methods that can be used to approximate the radius of the sun?

Basically, I'm wondering if there have been any attempts to calculate/model what the radius of the sun should be based on gravitational, thermal, and electromagnetic pressures. If there has, where can I find the calculation/model, and how closely does it match the actual radius of the sun?

Similarly, are there any calculations/models that accurately predict the expected changes in density as a function of solar radius?

 

[Vanadium 50]

Um...sort of. You can look at a text of stellar modeling (e.g. Clayton) but these models were created looking at the sun as input. So of course it all matches.

 

[sophiecentaur]

Summary:: Are there any theoretical methods that can be used to approximate the radius of the sun?

are there any calculations/models that accurately predict the expected changes in density as a function of solar radius?

Do you want to include the photosphere, chromosphere and corona? Probably as hard to define as giving a 'hard' value for the height of Earth's atmosphere.

 

[phyzguy]

The sun has been modeled in great detail, and the results match the measurements very well. You can even download your own copy of the simple 1D Mesa stellar modeling code and run the simulations yourself. If you download the movie of the sun's evolution ("Evolve a 1 MSun star"), and look at the current age of the sun (about 4.5E9 years), you will see that it predicts a radius of 0.995 Rsun, so it is within about 0.5%. The movie also shows density, temperature, pressure, and composition as a function of radius, as well as many other things. It probably has more details than you want to know.

 

[sophiecentaur]

The sun has been modeled in great detail,

But, as has already been noted, the model was based on measurements of the Sun. I imagine the measurements of other stars would not be precise enough to spot any deviation of the model that we use for the Sun.
I just typed the expression "Near enough is good enough" in another thread and I think that must apply here - until they send a probe to Proxima Cenuri for a closer look.

 

[phyzguy]

But, as has already been noted, the model was based on measurements of the Sun. I imagine the measurements of other stars would not be precise enough to spot any deviation of the model that we use for the Sun.

I imagine that the people that spend their careers building these models and tying them back to first principles would be offended by your suggestion that they have simply empirically tweaked it to match the Sun and left it at that. In fact the models are carefully built using laboratory measurements and tested against existing stellar data across a wide range of masses , compositions, and stellar ages. Why would you think that "measurements of other stars would not be precise enough to spot any deviation of the model that we use for the Sun"? In fact there is a huge database of data on stars: masses, radii, temperatures, compositions, etc. In addition, pulsating stars and "starquakes" give us information about the interior structure of stars, just like earthquakes give us information about the interior structure of the Earth.

 

[sophiecentaur]

I imagine that the people that spend their careers building these models and tying them back to first principles would be offended by your suggestion that they have simply empirically tweaked it to match the Sun and left it at that. In fact the models are carefully built using laboratory measurements and tested against existing stellar data across a wide range of masses , compositions, and stellar ages. Why would you think that "measurements of other stars would not be precise enough to spot any deviation of the model that we use for the Sun"? In fact there is a huge database of data on stars: masses, radii, temperatures, compositions, etc. In addition, pulsating stars and "starquakes" give us information about the interior structure of stars, just like earthquakes give us information about the interior structure of the Earth.

Do these guys really need to be defended? I have in no way attacked them (that would be ad hominem). They will have done the best they can with the available data and there must be a lot more detailed data about the Sun than any other star. Is the measurement error in the diameter of the nearest few stars as good as that of the Sun?

 

[phyzguy]

The point is that the inputs to the model are not measurements of the sun. The inputs to the model are more basic physics things like (1) nuclear fusion cross-sections as a function of energy, (2) measurements of plasma opacity as a function of density and temperature, etc. After these types of measurements and building the model, the results are compared to measurements of the sun and other stars. The agreement is good, but I don't have the information on how accurately it fits stars other than the sun.

 

[sophiecentaur]

The point is that the inputs to the model are not measurements of the sun. The inputs to the model are more basic physics things like (1) nuclear fusion cross-sections as a function of energy, (2) measurements of plasma opacity as a function of density and temperature, etc. After these types of measurements and building the model, the results are compared to measurements of the sun and other stars. The agreement is good, but I don't have the information on how accurately it fits stars other than the sun.

Are there no other factors to bring the model into agreement? There is such a lot that's not actually known about the Sun and a lot has been found out about it so it would be surprising if there were enough data available without Solar data. (I base that on very limited info about the actual topic but I do know that tweaking and more than tweaking is quite acceptable in these matters.)

 

[Vanadium 50]

So let me give a simple example. We know that stars are in hydrostatic equilibrium, which means they follow the Lane-Emden equation. The solutions are $P=k\rho^{\frac{(n+1)}{n}}$.

Can n be infinite? No - that would be appropriate for a ball of gas before it starts to collapse.
What about n = 5? No - that would be a star of infinite extent.
Maybe n = 0? No - that's appropriate for solid bodies (and incompressible fluids). We know the sun cannot be solid because of its rotation.

So, even by the simplest observations, we know that n has to be between zero and 5. (It's actually around 3) And n sets the mass-radius relationship.

 

[Hornbein]

The Sun doesn't really have a surface. I read in one of the Feynman books that the apparent surface has to do with the mean free path of a photon. But I don't understand that.

 

[sophiecentaur]

It's actually around 3

"Around" doesn't lead to the quoted accuracy of 0.5%, does it?

So let me give a simple example. We know that stars are in hydrostatic equilibrium, which means they follow the Land-Emden equation. The solutions are P=kρ(n+1)n.

That appears to assume uniform density, for a start and that is unlikely to apply to the Sun.

There is no shame in the expression "the best we can do at present"; it's the way Physics works for everything. If it was all done and dusted, there would be no need for continuing observations.

 

[Vanadium 50]

"Around" doesn't lead to the quoted accuracy of 0.5%, does it?

An accuracy of x% in polytriopic index ("n") doesn't necessarily translate to an accuracy of x% in radius. Indeed, my example where going from 3 to 5 (66% change) makes the radius go from finite to infinite demonstrates this.

That appears to assume uniform density, for a start and that is unlikely to apply to the Sun.

No, this is an equation that is true at any point, not when the variables are averaged over.

It's also not exact. It doesn't consider, for example, a variation in composition with radius, or the convective region. But the point is that a model of stars that fails to reproduce the sun will be immediately discarded. As such, any "prediction"of solar properties needs to be taken with a grain of salt.

 

[sophiecentaur]

As such, any "prediction"of solar properties needs to be taken with a grain of salt.

That was rather my original point.

 

[phyzguy]

The Lane-Emden equation is an extremely simple model of a star, vastly simpler than modern stellar modeling codes. It's a bit like asking how well Hooke's law fits an apartment building. At some level it fits, but apartment buildings aren't designed using Hooke's law, they are designed with much more sophisticated 3D finite element modeling codes.

 

[Vanadium 50]

The Lane-Emden equation is an extremely simple model of a star

Yes it is. I think I said as much.

But more sophisticated models require more observational input, not less.