The Origin of Stars

A (fictional) man named Javert once said (sang) about stars:

"And each [star] in your season returns and returns, and is always the same."

Stars sort of seem like true constants on our night sky. Sure, they move around, but they always come back, and they don't change. At least not to the untrained eye. But for those who look up at the sky every night, mapping each little detail that they can, they might notice some changes.

Occasionaly, through history, extremely bright stars have been reported, shining for a while before dissapearing forever. Now, we can say that at least some of these were supernovas, the bright death of a supergiant star. And there's a lot more than that to the life of a star, they really arent always the same. You probably knew that though.

We look at the star in our system, and wonder, what did it come from? And what will become of it?

Currently, our star is a main sequence star, meaning that it is currently fusing hydrogen atoms to form helium atoms in its core. Most stars in the universe, including the Sun, are main sequence stars, and they can range from one tenth the mass of the sun, up to 100 times as massive.

In astronomy, the main sequence is a continuous band of stars that appear on plots of the luminosity versus the temperature (alternatively the B-V color index or the specrtal classes) of a star. These plots are known as Hertzburg-Russell diagrams after theit co-developers, Ejnar Hertzsprung and Henry Norris Russel.

The Sun has a luminosity of \(3.85\cdot 10^{26}\), making it a main sequence star, and with a surface temperature of 5778 K it lies on the lower-middle end. We want to have a look at where our star lies, and so we find the luminosity (this has been done before, but we'll show you the formula again):

\(L_* = \sigma T_*^4 \cdot 4\pi r_*^2 = \sigma\cdot (10240.67K)^4 \cdot 4\pi (1746710.68 km)^2 = 2.39\cdot 10^{28}\)

We live in the same system as this star, and our lives are very dependent on it's behaviour. Is it nice to us? Or is it a bastard star, intent on making life as hard as possible? Or, more scientific: Does the star follow the proportions and relations we assume it does? For any star, we assume its mass is proportional to its radius

\(M_*\propto R_* \longleftrightarrow \frac{M_*}{R_*}=C\)

where C is a constant. A star's luminosity is strongly related to it's temperature (we use the same surface temperature as previously),

\(M_*\propto T^2_*\)

We can combine these two expressions for proportionality and see that

\(\frac{M_*}{T_*^2}=C\)

Let's check what this constant is for the Sun, which we know is very well behaved (and well loved), \(M_\bigodot/T_\bigodot^2=5.96\cdot 10^{22}kg/K^2\). If we get something similar for our own star, it points to a well behaved, main sequence star:

\(\frac{M_*}{T_*^2}= 6.55\cdot 10^{22}\)

Not exactly the same, but not too far off! It deviates with a factor of \( 0.09836\% \approx 0.09\%\), which we say is good enough!

We can double check with another relation, this time between luminosity and mass, just to be certain

\(L_*\propto M^4_* \longleftrightarrow \frac{L_*}{M_*^4}\)

Same as before, we check the eqation for the Sun first, \(L_\bigodot/M_\bigodot^2=2.4452\cdot 10^{-95}\), and compare to our star

\(\frac{L_*}{M_*^4} = 1.0764\cdot 10^{-95}\)

Again, the number is a bit different, and we have a relative error of \(0.55978\%\approx0.56\%\), which is bigger than the last one, but still good. We conclude with that we have a well-behaved star on the main sequence. From this we assume that we should be placed in the middle of the main sequence, not at the edges, somewhat higher up than the Sun.

We can plot our star into an HR-diagram with the Sun:

The long diagonal line consists of stars on the main sequence, with our star as a blue dot almost in the middle. The red dot represents the Sun. Our star is pretty far up, either due to its large size, or to a high temperature. As our star is 3.5 times as massive as the Sun, with a radius twice of the Sun's, it makes sense that it should have a higher temperature and luminosity, therefore lying further up on the HR-diagram. Main sequence stars range from 0.1 to 10 times the radius of the Sun.

The temperature of a star is directly related to the color, and we know that our star is a bright, pale shade of yellow, almost blue.

The three other concentrations of stars are the sparse supergiants at the top right (more than 100 times the mass of the Sun), the giants beneath (10-100 times the Sun's mass), and the bottom line which are white dwarves (less than 0.1 times the Sun's mass).

We believe our star started out on the far right side of the HR-diagram, with a very low temperature, as a huge cloud of gas that contracted due to its own gravity. As it contracted, the radius decreased and the temperature increased as the star moved leftwards and up, settling on the main sequence. It won't remain here forever, some time in the future, when the hydrogen in the core has been exhausted, its radius will increase several times, transforming it into a giant (or a supergiant). How soon this happens depends on how quickly it expels energy. We recall how the amount of energy radiated by a star over an amount of time is its luminosity. We can therefore find out how long a star is expected to live if we know the amount of energy the star had to begin with, and its luminosity.

Because the luminosity of a star is proportional to its mass, \(L \propto M^4\), we can say that the lifetime of a star on the main sequence (when its luminosity is almost constant) is:

\(t_{life}\propto \frac{T_{tot}}{L} = \frac{1}{M^4}\)

If we compare our star to the Sun, \(t_{life}M^4 = t_\bigodot M_\bigodot\), we see that our star will remain on the main sequence for \(3.88\cdot 10^7\) years. The Sun moved onto the main sequence about 4.5 billion years ago, and is expected to remain there for another 5.4 billion years, after which it will begin to expand. As our star is more massive, it will burn out quicker, eventually expanding, and then leaving behind a white dwarf star (its too small to become a neutron star or black hole).

That's still in the future, though. Let's stick to exploring the birth of a star, for now.

We begin by assuming that our star started out as a shperically symmetric Giant Molecular Cloud (GMC), with a temperature \(T=10K\), that consisted of 75% Hydrogen and 25% helium atoms. To make the math simpler, we also assume that it began collapsing on itself withour the help of shock waves from supernovae. It's hard to say the exact size our cloud was, it's not like we can ask for it's baby-photos. We can, however, say something about the maximum size it must could have been, because if it's larger than a certain size, it wouldn't have enough gravitational force to begin collapsing on itself.

In order to understand what the maximum size our gas cloud could have been, we will utelize the virial theorem:

\(\langle K\rangle=-\frac{1}{2}\langle U\rangle\)

where \(\langle K \rangle\) is the mean kinetic energy and \(\langle U \rangle\) is the mean potential energy of a system. This theorem is a relation between the total kinetic energy and the total potential energy of a system in equilibrium. The virial theorem tells us that the condition for stability is \(2K+U=0\). If the kinetic energy is large compared to the potential energy (compare 2K to U), the system does not stabilize as the gas pressure is larger than the gravitational forces and the cloud will expand. However, if the potential energy is larger, the cloud will be gravitationally bound and undergo a collapse! So, the condition for our gas cloud to collapse into a star is

\(2K < \langle U \rangle\)

What is the total potential energy of the gas cloud, though?

We can find it by setting up \(E= K + U = -\frac{1}{2}U + U = \frac{1}{2}U\) ( by using the virial theorem to set \(K = -\frac{1}{2}U\)), and considering the gravitational attraction on a small mass dm from the sphere of matter inside the position of the mass. We need only consider the sphere inside as the gravitational forces from aspherical shell of matter cancel each other out inside this shell. This will be a shpere of radius with mass \(M(r)\), which we can apply Newton's law of gravitation to, as if it were a point mass located at the center with mass \(M(r)\). The potential energy between the particle dm and the rest of the cloud becomes

\(du = -G\frac{M(r)dm}{r}\)

By integrating this over all masses in the shell at a distance \(r\) from the center, and assuming that the mass density is given by a constant with a value equal to the mean density of the cloud, and again from the radii \(r\) to the end of the cloud, we obtain the expression

\(U = -\frac{3GM^2}{5R}\)

where \(M\) is the mass of the cloud and \(R\) is the radius. In order to find an expression for the kinetic energy we use the expression (from thermodynamics) of kinetic energy of a gas

\(K=\frac{3}{2}NkT\)

where the number of particles in the gas is \(N=\frac{M}{\mu m_H}\), k is the Boltzmann constant and T is the temperature. Inserting these expressions for U and K into \(2K < \langle U \rangle\), we get

\(\frac{3MkT}{\mu m_H} < \frac{3GM^2}{5R}\)

which we can rewrite to find the minimum mass

\(M > \frac{5kT}{G\mu m_H}R\)

where \(\mu = \frac{\overline{m}}{m_H}\) is the mean mass per particle measured in units of hydrogen mass. Clouds with a larger mass than \(M\), known as the Jean Mass M_J, are large enough to start a gravitational time collapse. We're interested in the radius, though, so we rewrite the formula in terms of the mean density of the cloud, and solve for R:

\(R_J>\left(\frac{15kT_{GMC}}{4\pi G\mu m_H\rho}\right)^{1/2}\)

Alright, that was a lot of things at once, but we have our expression, and we can use this to figure out the upper limit radius of our cloud! We make a little adjustment, though, and assume that the density is constant over the cloud (it isn't, but the approximation isn't too far off):

\(\rho = \frac{M}{V}=\frac{M}{4\pi R_J^3 / 3}\)

We insert this into the expression for the Jean Radius, stack it around a bit, and get

\(R_J=\frac{G\mu m_HM_s}{5kT_{GMC}}=1.94\cdot 10^{15} m\)

Where we've simply used the values for the gravitational constant \(G\), the Boltzmann constant \(k\), the temperature of the gas \(T_{GMC}=10K\), the mass of our star \(M_s\), and the mean molecular weight, found by multiplying \(\mu=\sum\limits^N_{i=1}f_i\frac{m_{i}}{m_H}=\frac{3}{4}\frac{m_{H}}{m_H}+\frac{1}{4}\frac{m_{He}}{m_H}\) with the weight of hydrogen \(m_H\).

This upper limit is equal to \(R_J = 1.3\cdot 10^4\) AU, where 1 AU is the distance from the Sun to the Earth, so it's pretty far! Our calculations do not account for any other forces in space, nor for the rotation of the cloud or any magnetic fields, so it won't be an entierly accurate size, but this is still a good approximation.

The process of collapsing this cloud can take many millions of years, as the cloud becomes smaller and the density gets bigger! The smaller the cloud gets, the more heat it also generates (from the collisions of particles), and once the temperature in the core of the collapsing star is high enough, nuclear fusion begins and a star is born.

If we set the radius of our GMC to what it needs to be in order to collapse, we can use the formula for luminosity in order to find the luminosity the cloud might have had:

\(L_{GMC} = \sigma T_{GMC}^4 \cdot 4\pi R_{J}^2 = 2.67 \cdot 10^{28}\)

The luminosity of our star is currently \(L_* = 2.39\cdot 10^{28}\), meaning that the gas cloud was a bit more luminous. We add it to our HR-diagram:

It's pretty far of the scale, but that's where it should be!