Supernova

Supernovae – An Introduction

The following is an excerpt from my book “Introduction to Stars: Spectra, Formation, Evolution, Collapse”, available here for Kindle. Enjoy!

A vast number of written recordings show that people watching the night sky on 4 July 1054 had the opportunity to witness the sudden appearance of a new light source in the constellation Taurus that shone four times brighter than our neighbor Venus. The “guest star”, as Chinese astronomers called the mysterious light source, shone so bright that it even remained visible in daylight for over three weeks. After around twenty-one months the guest star disappeared from the night sky, but the mystery remained. What caused the sudden appearance of the light source? Thanks to the steady growth of knowledge and the existence of powerful telescopes, astronomers are now able to answer this question.

The spectacular event of 4 July 1054 was the result of a supernova (plural: supernovae), an enormous explosion that marks the irreversible death of a massive star. Pointing a modern telescope in the direction of the 1054 supernova, one can see a fascinatingly beautiful and still rapidly expanding gas cloud called the Crab Nebula at a distance of roughly 2200 parsecs from Earth. At the center of the Crab Nebula cloud lies a pulsar (Crab Pulsar or PSR B0531+21) having a radius of 10 km and rotating at a frequency of roughly 30 Hz.

CrabNebulaHubble

Image of the Crab Nebula, taken by the Hubble Space Telescope.

Let’s take a look at the mechanisms that lead to the occurrence of a supernova. At the end of chapter two we noted that a star having an initial mass of eight solar masses or more will form an iron core via a lengthy fusion chain. We might expect that the star could continue its existence by initiating the fusion of iron atoms. But unlike the fusion of lighter elements, the merging of two iron atoms does not produce energy and thus the star cannot fall back on yet another fusion cycle to stabilize itself. Even worse, there are several mechanisms within the core that drain it of much needed energy, the most important being photodisintegration (high-energy photons smash the iron atoms apart) and neutronization (protons and electrons combine to form neutrons). Both of these processes actually speed up the inevitable collapse of the iron core.

Calculations show that the collapse happens at a speed of roughly one-fourth the speed of light, meaning that the core that is initially ten thousand kilometers in diameter will collapse to a neutron star having a radius of only fifteen kilometers in a fraction of a second – literally the blink of an eye. As the core hits the sudden resistance of the degenerate neutrons, the rapid collapse is stopped almost immediately. Because of the high impact speed, the core overshoots its equilibrium position by a bit, springs back, overshoots it again, springs back again, and so on. In short: the core bounces. This process creates massive shock waves that propagate radially outwards, blasting into the outer layers of the star, creating the powerful explosion known as the (collapsing core) supernova.

Within just a few minutes the dying star increases its luminosity to roughly one billion Suns, easily outshining smaller galaxies in the vicinity. It’s no wonder then that this spectacular event can be seen in daylight even without the help of a telescope. The outer layers are explosively ejected with speeds in the order of 50,000 kilometers per second or 30,000 miles per second. As time goes on, the ejected layers slow down and cool off during the expansion and the luminosity of the supernova steadily decreases. Once the envelope has faded into deep space, all that remains of the former star is a compact neutron star or an even more exotic remnant we will discuss in the next section. Supernovas are relatively rare events, it is estimated that they only occur once every fifty years in a galaxy the size of the Milky Way.

At first sight, supernovae may seem like a rather destructive force in the universe. However, this is far from the truth for several reasons, one of which is nucleosynthesis, the creation of elements. Scientists assume that the two lightest elements, hydrogen and helium, were formed during the Big Bang and accordingly, these elements can be found in vast amounts in any part of the universe. Other elements up to iron are formed by cosmic rays (in particular lithium, beryllium and boron) or fusion reactions within a star.

But the creation of elements heavier than iron requires additional mechanisms. Observations indicate that such elements are produced mainly by neutron capture, existing atoms capture a free neutron and transform into a heavier element, either within the envelope of giant stars or supernovae. So supernovae play an important role in providing the universe with many of the heavy elements. Another productive aspect of supernovae is their ability to trigger star formation. When the enormous shock wave emitted from a supernova encounters a giant molecular cloud, it can trigger the collapse of the cloud and thus initiate the formation of a new cluster of stars. Far from destructive indeed.

The rapid collapse of a stellar core is not the only source of supernovae in the universe. A supernova can also occur when a white dwarf locked in a binary system keeps pulling in mass from a partner star. At a critical mass of roughly 1.38 times the mass of the Sun, just slightly below the Chandrasekhar limit, the temperature within the white dwarf would become high enough to re-ignite the carbon. Due to its exotic equilibrium state, the white dwarf cannot make use of the self-regulating mechanism that normally keeps the temperature in check in main sequence stars. The result is thus a runaway fusion reaction that consumes all the carbon and oxygen in the white dwarf within a few seconds and raises the temperature to many billion K, equipping the individual atoms with sufficient kinetic energy to fly off at incredible speeds. This violent explosion lets the remnant glow with around five billion solar luminosities for a very brief time.

Type_Ia_supernova_simulation_-_Argonne_National_Laboratory

Simulation of the runaway nuclear fusion reaction within a white dwarf that became hot enough to re-ignite its carbon content. The result is a violent ejection of its mass.

Mathematics of Explosions

When a strong explosion takes place, a shock wave forms that propagates in a spherical manner away from the source of the explosion. The shock front separates the air mass that is heated and compressed due to the explosion from the undisturbed air. In the picture below you can see the shock sphere that resulted from the explosion of Trinity, the first atomic bomb ever detonated.

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Using the concept of similarity solutions, the physicists Taylor and Sedov derived a simple formula that describes how the radius r (in m) of such a shock sphere grows with time t (in s). To apply it, we need to know two additional quantities: the energy of the explosion E (in J) and the density of the surrounding air D (in kg/m3). Here’s the formula:

r = 0.93 · (E / D)0.2 · t0.4

Let’s apply this formula for the Trinity blast.

———————-

In the explosion of the Trinity the amount of energy that was released was about 20 kilotons of TNT or:

E = 84 TJ = 84,000,000,000,000 J

Just to put that into perspective: in 2007 all of the households in Canada combined used about 1.4 TJ in energy. If you were able to convert the energy released in the Trinity explosion one-to-one into useable energy, you could power Canada for 60 years.

But back to the formula. The density of air at sea-level and lower heights is about D = 1.25 kg/m3. So the radius of the sphere approximately followed this law:

r = 542 · t0.4

After one second (t = 1), the shock front traveled 542 m. So the initial velocity was 542 m/s ≈ 1950 km/h ≈ 1210 mph. After ten seconds (t = 10), the shock front already covered a distance of about 1360 m ≈ 0.85 miles.

How long did it take the shock front to reach people two miles from the detonation? Two miles are approximately 3200 m. So we can set up this equation:

3200 = 542 · t0.4

We divide by 542:

5.90 t0.4

Then take both sides to the power of 2.5:

t 85 s ≈ 1 and 1/2 minutes

———————-

Let’s look at how the different parameters in the formula impact the radius of the shock sphere:

  • If you increase the time sixfold, the radius of the sphere doubles. So if it reached 0.85 miles after ten seconds, it will have reached 1.7 miles after 60 seconds. Note that this means that the speed of the shock front continuously decreases.

For the other two parameters, it will be more informative to look at the initial speed v (in m/s) rather the radius of the sphere at a certain time. As you noticed in the example, we get the initial speed by setting t = 1, leading to this formula:

v = 0.93 · (E / D)0.2

  • If you increase the energy of the detonation 35-fold, the initial speed of the shock front doubles. So for an atomic blast of 20 kt · 35 = 700 kt, the initial speed would be approximately 542 m /s · 2 = 1084 m/s.

  • The density behaves in the exact opposite way. If you increase it 35-fold, the initial speed halves. So if the test were conducted at an altitude of about 20 miles (where the density is only one thirty-fifth of its value on the ground), the shock wave would propagate at 1084 m/s

Another field in which the Taylor-Sedov formula is commonly applied is astrophysics, where it is used to model Supernova explosions. Since the energy released in such explosions dwarfs all atomic blasts and the surrounding density in space is very low, the initial expansion rate is extremely high.

This was an excerpt from the ebook “Great Formulas Explained – Physics, Mathematics, Economics”, released yesterday and available here: http://www.amazon.com/dp/B00G807Y00. You can take another quick look at the physics of shock waves here: Mach Cone.