Stellar Physics, The Physics of a Star

Written by: Rahul Raturi (https://raturi2611.blogspot.com/)

We all are made up of Star Dust.

Carl Sagan (1934-1996)

~ Up there in the Night sky you all know what those mysterious twinkling objects are. Yes !! Those are stars. But for a long time, people didn’t wonder and know about stars. And I am not talking about Ancient Greeks or Ancient Mesopotamians but even as recently as Galileo and Kepler. They didn’t know what stars were.

•What is a Star?

Stars are huge celestial bodies mostly made up of Hydrogen (H) and Helium (He), which produces light and heat from the churning nuclear forces inside the core. The closest star to our Earth is Sun.

•Physics of a Star

A star must follow the condition of Hydrostatic Equilibrium to be stable. In a star, inside its core, nuclear fusion Is going on converting H to He 90-95% of the lifetime. So this nuclear fusion creates an outward pressure.

Stars are globes of a perfect ideal gas in which this outward pressure is balanced by the inward gravity and this outward pressure is a combined effect of Pressure due to Ideal Gas (given by Boyles Law) and the Radiation Pressure (Stephen’s Law).

Total Pressure = P₁ + P₂

= nKT + (1/3)aT⁴

P₁= Pressure due to Ideal Gas

P₂= Radiation Pressure

K=Boltzman Constant

But when we talk about the Neutron stars and White dwarfs we have to change these equations as the cores of these types of stars are no more Ideal Gas but made of fermions that obey the Fermi Dirac Laws and there we account for the Degeneracy pressure (It is due to Pauli Exclusion Principle, which states that no two fermions can occupy the same quantum state).

•Why Do Stars Shine?

Arthur Eddington was the first to give the formula for Luminosity of star which shows the relation between Luminosity and Mass of star

L ∝ M³

In General, we think that, if a given mass is in less radius volume then it has a high temperature so it should be more luminous. But here we can notice an astonishing result that luminosity is not depending on the radius. Eddington compared this result with the observations and the theory perfectly fitted the observations. Radius doesn’t come in the picture at all, luminosity only depending on the mass of the star.

But there’s a question, Why do stars shine at all? The Nuclear Reaction releases an enormous amount of energy in the form of gamma rays. These gamma rays are trapped inside the star. The gamma rays jump around in the star, trying to get out. They’re absorbed by one atom and then emitted again. This can happen many times a second, and a single photon can take 100,000 years to get from the core of the star to its surface. When the photons have reached the surface, they’ve lost some of their energy, becoming visible light photons, and not the gamma rays they started as. These photons leap off the surface of the Sun and head out in a straight line into space. They can travel forever if they don’t run into anything.

Luminosity is due to photons you’re seeing that left the surface of the star years ago and traveled through space, without running into anything.

•Quantum Tunneling in Stars.

In Nuclear fusion inside stars, we see two H nuclei (proton) coming closer and fusing.

Figure 2.Fusion Reaction

But we all know there is a Barrier of Coulombian repulsion force between proton-proton. So how these protons overcome this force and fuse? Some of us may think that it is because the Star’s core has such a high temp. which provides the protons enough high energy so they can fuse.

But the repulsive force between proton-proton is about 1 Mev and the temperature of stars are around 10⁷ degrees which are millions of degree but it can only overcome the repulsive force of 1000 eV.

So it’s virtually impossible to think of protons coming close and fuse. It requires billions of degrees temperature to fuse protons.

So, if the temperature is not the only factor for the fusion so then what?

Now, Quantum mechanics comes into the picture. According to De Broglie Hypothesis atoms also possess a wave nature. And these waves are probability waves. Quantum mechanically, these low energy protons have a very small but Non-zero probability of tunneling through the barrier. The probability of tunneling depends on the energy of the particles, their mass, and charge.

Figure 3. Quantum Tunneling of a wave

Although the probability of tunneling is very very low i.e. 10^-20, there are approx 10⁵⁷ protons inside a star. So Statistically fusion can happen even with such a low probability just at a very slow rate.

•LifeTime of a Star

The lifetime of a star mainly depends on Mass. Lower the mass (up to 2 solar masses) will have more life because they have less temperature Inside their core and burn the fuel slower. Another reason for the star’s long life is the weak interaction inside the star where proton converting to neutron which is a very slow process.

A star having around 15 solar masses will live only a few million years. But low mass stars like our Sun can live Billions of years.

•Death of a Star

In a self-gravitating star, the fusion starts with Hydrogen converting into He. Once all Hydrogen is fused, the core starts contracting because Helium is formed, which doesn’t fuse at the temp of 10⁷ degree(stars core temp.) due to which the outward pressure reduces and now gravity starts to dominate, results in contracting the core.

Once core contracted up to a size at which the temp reached up to 10⁸ degrees, Helium starts to fuse into Carbon. And this drama goes on until the core becomes the Iron which requires 10¹⁰ degrees temp.

Fusion is halted at 56Fe, In the graph above we can see that Iron is having the highest Binding energy i.e. 9 Mev per nucleon. Since it is so tightly bound that no energy can be extracted by Fusion. Iron can fuse, but it will absorb energy in the process, and the core temperature drops. So the star stops fusing more and the core stops at Iron nuclei.

After this point Gravity wins the fight and the star collapses within a second, the outer layer bouncing off the core triggering an explosion thus ejecting all the other heavy nuclei into space. This is the most energetic and violent yet beautiful event in the universe called Supernova.

After Supernova, the Main sequence stars of less than 8 sollar masses will become White Dwarf. They will get the ultimate peace because it doesn’t have enough mass to produce gravity which can now overcome the degeneracy pressure. The Chandrasekhar Limit is now accepted to be approximately 1.4 times the mass of the sun; any white dwarf with less than this mass will stay a white dwarf forever.

But if the mass is in between 1.4 to 3 solar masses, it will collapse so much that protons and electrons combine to form neutrons and end up in a Neutron Star. Neutron stars are the more dense objects in the universe. A neutron star is so dense that one teaspoon (5 milliliters) of its material would have a mass over millions of tonnes. A Neutron star is about 20 km in diameter and has a mass of about 1.4 times that of our Sun.

And if mass exceeds more than 3 Solar masses then gravity will become so immense that the whole mass collapses into a singular point of infinite density and it becomes a Black Hole. It also forms an Event Horizon around it. The ‘Event horizon’ is the boundary defining the region of space around a black hole from which nothing (not even light) can escape. In other words, the escape velocity for an object within the event horizon exceeds the speed of light.