The more our gas cloud was compressed, the more pressure there was on the core, and therefore the temperature got higher untill it couldn't compress itself any further. At this point the core is so hot that a process called fusion, where hydrogen can become helium, has began to happen. This creates energy, as the mass of hydrogen is not the same as the mass of helium. You may have heard of a little formula called
\(E=mc^2\)
This formula is famous from films such as "Movie that needed a smart formula on a board
while a scientist talks about something that has nothing to do with it." It allows mass to be turned into energy, creating a pressure that pushes against the gravity (and makes it shiny!), which eventually creates hydrostatic equilibrium and stops the compression of the star. The star will remain in equilibrium untill all the Hydrogen in it's core has turned into Helium, at which point the core isn't hot enough to continue fusion of Helium, and the star moves into its next phase - but more on that later.
Our star has grown (shrank, really) from a GMC, into a young star on the main sequence! But how well do we know this young soul? In order to be able to say anything about our star, we should investigate its core. The processes in the core are essential knowledge, but in order to find them, we need to make a few estimations and assumptions:
- We assume the density of the star is uniform
- We approximate the pressure inside the star as with the gaseous pressure of an ideal gas, \(P=\frac{nRT}{V}\). We ignore pressure from radiation and such things.
- We assume hydrostatic equilibrium when we model the star
- We assume our star consists entirely of protons with a mass \(m_H=1.673\cdot 10^{-27}\) kg.
The temperature of the core is very relevant to what processes are happening there, and will depend on the mass, so our first step will be to develop an expression for the spherical mass profile, \(M(r)\), of our star.
We've talked about hydrostatic equilibrium, which is what we call the case when the force pushing somthing down is cancelled out by pressure pushing back. If we assume that the density, \(\rho\), is constant, we can make a model for \(M(r)\). We've then assumed that the mass is distributed spherically such that any given point a distance \(r\) on a shell from the center, can say something about how much mass is within this shell:
\(M(r)=\int_v\rho dV = \rho V = \rho\frac{3}{4}\pi r^3\)
where \(V\) is the volume of a sphere and \(r\) is the distance to a point within the star. This integral was made a lot easier by keeping the density constant!
Let's now use this spherical mass profile in order to find an equation for the temperature of the star, depending on the distance to the core.
We again use the equation for hydrostatic equilibrium,
\(\frac{\partial P}{\partial r} = - \rho g(r)\)
as we have approximated the pressure nside the star as with the gaseous pressure of an ideal gas, we can rewrite the pressure as \(P=\frac{nRT}{V}=\frac{\rho k T(r)}{\mu m_H}\).
\(\frac{\partial }{\partial r} \frac{\rho k T(r)}{\mu m_H}= - \rho g(r)\)
Using Newtons law of gravitation, we can insert an expression for \(g = G\frac{M(r)\mu m_H}{r^2}\), and for \(M(r)= \rho\frac{3}{4}\pi r^3\) we use the expression found earlier. We then solve with respect to the temperature:
\(\frac{\partial}{\partial r}T(r) = -\frac{4\pi}{3} \frac{G \rho_0\mu m_H}{k}r\)
And we can use this equation to find the temperature in the core- again by integrating! We integrate from the center, \(r=0\), to the surface, \(r=R\), and get:
\(T_{core} = T(R) + \frac{2\pi}{3} \frac{G \rho_0\mu m_H}{k}R^2=27.7\cdot 10^{6}K\)
The temperature of our core is at 27.7 million Kelvin! This is quite a bit hotter than the Sun's core, which is 15 million Kelvin, and makes sense with our stars greater size. The bigger the star, the higher the core temperature needs to be, in order to keep the star stable. If you carry something heavy, you need to spend more energy than if you carry something light, so it makes sense that the star needs more energy in order to 'push back' the gravitational force it exherts on itself.
Now that we know the core temperature, we can make a few assumptions about the what is happening in the core, which we will assume is within a sphere of radius \(0.2R\). Our temperature is under \(90\cdot 10^6 \) K, which means that we can assume that energy production in the core occurs via the pp-chain and CNO-cycle. We can also assume that the core consists of 74.5% Hydrogen, 25.3% Helium, and 0.2& Carbon, Oxygen and Nitrogen. Finally, we assume that the density of the star is uniform, and that the core temperature we calculated is the same throughout the core.
So what happens in the core? Well, there are two processes keeping our star going, as mentioned above:
Writing nuclei: \(X^A_Z\), where \(X\) is the symbol for the element, \(Z\) is the total number of protons in the atom, and \(A\) is the total number of particles in the core.
The PP-chain: This is one of the two most important fusion reaction for main sequence stars, where four \(^1_1H\) are fused to \(^4_2He\). It's most effective around 15 million Kelvin, where 0.7% of the mass in each reaction is converted to energy.
The CNO-cycle: This is the other important fusion reaction where four \(^1_1H\) are fused to \(^4_2He\). C(arbon), N(itrogen) and O(xygen) are only catalysts, and do not change in the reaction. This is valid for much higher temperatures and, and is very sensitive to temperature changes.
\(3\alpha\) reaction (Only in stars with a core temperature \(t_c>90\cdot 10^6\) K) : Helium is fused into even more heavy elements, Beryllium and Carbon. This process is valid for elements as eavy as Iron, but after this stage it would take more energy for fusion to happen than is created. This process is probably not present in our star, but is important to know about.
We can make an estimate of the luminosity based on the energy produced in the core of the star, as luminosity is energy over time. We assume only the PP-chain and the CNO-cycle are happening in the core, each producing a specific amount of energy: \(\epsilon_{PP} \approx \epsilon_{0,PP}X^2_H\rho T^4_6\) and \(\epsilon_{CNO} \approx \epsilon_{0,CNO}X_HX_{CNO}\rho T^{20}_6\). These equations give us the amount of energy produced per mass over time. We can find the amount of the core that is an element, \(X_{element}=M_{element}/M_{core}\), and we have the usual density \(\rho\), and a constant \(\epsilon_0\). The temperature may look unusual to you, but it is simply a way of writing \(T_6 = \frac{T}{10^6}\)K.
Inserting the values into each equation, adding them up and multiplying by the mass of the core, we get that our star should have a luminosity of \(4.74\cdot 10^{26}W\). Comparing this to the number we got using Stephen-Boltzmann's law, \(2.39\cdot 10^{28}W\), we see a really big difference.
We see that the two numbers are very different. While this could be due to mistakes, we should keep in mind that we've made many assumptions, such as assuming that the density of the core is constant. It really isn't. The temperature also changes a lot over the core, and maybe some day we'll model a star that takes this into account, but today is not that day. Note how the formulas for the energy produced are very dependent on temperature? When we get this number wrong, even by just a little bit, they make a huge difference in the output, and are probably the reason for this large difference between the numbers. There are probably also additional reactions happening in the core, not just the PP-chain and CNO-cycle.