Stars

Stellar Physics – Gaps in the Spectrum

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

When you look at the spectrum of the heat radiation coming from glowing metal, you will find a continuous spectrum, that is, one that does not have any gaps. You would expect to see the same when looking at light coming from a star. However, there is one significant difference: all stellar spectra have gaps (dark lines) in them. In other words: photons of certain wavelengths seem to be either completely missing or at least arriving in much smaller numbers. Aside from these gaps though, the spectrum is just what you’d expect to see when looking at a heat radiator. So what’s going on here? What can we learn from these gaps?

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Spectrum of the Sun. Note the pronounced gaps.

To understand this, we need to delve into atomic physics, or to be more specific, look at how atoms interact with photons. Every atom can only absorb or emit photons of specific wavelengths. Hydrogen atoms for example will absorb photons having the wavelength 4102 A, but do not care about photons having a wavelength 4000 A or 4200 A. Those photons just pass through it without any interaction taking place. Sodium atoms prefer photons with a wavelength 5890 A, when a photon of wavelength 5800 A or 6000 A comes by, the sodium atom is not at all impressed. This is a property you need to keep in mind: every atom absorbs or emits only photons of specific wavelengths.

Suppose a 4102 A photon hits a hydrogen atom. The atom will then absorb the photon, which in crude terms means that the photon “vanishes” and its energy is transferred to one of the atom’s electrons. The electron is now at a higher energy level. However, this state is unstable. After a very short time, the electron returns to a lower energy level and during this process a new photon appears, again having the wavelength 4102 A. So it seems like nothing was gained or lost. Photon comes in, vanishes, electron gains energy, electron loses energy again, photon of same wavelength appears. This seems pointless, why bother mentioning it? Here’s why. The photon that is created when the electron returns to the lower energy level is emitted in a random direction and not the direction the initial photon came from. This is an important point! We can understand the gaps in a spectrum by pondering the consequences of this fact.

Suppose both of us observe a certain heat source. The light from this source reaches me directly while you see the light through a cloud of hydrogen. Both of us are equipped with a device that generates the spectrum of the incoming light. Comparing the resulting spectra, we would see that they are for the most part the same. This is because most photons pass through the hydrogen cloud without any interaction. Consider for example photons of wavelength 5000 A. Hydrogen does not absorb or emit photons of this wavelength, so we will both record the same light intensity at 5000 A. But what about the photons with a 4102 A wavelength?

Imagine a directed stream of these particular photons passing through the hydrogen cloud. As they get absorbed and re-emitted, they get thrown into random directions. Only those photons which do not encounter a hydrogen atom and those which randomly get thrown in your direction will reach your position. Unless the hydrogen cloud is very thin and has a low density, that’s only a very small part of the initial stream. Hence, your spectrum will show a pronounced gap, a line of low light intensity, at λ = 4102 A while in my spectrum no such gap exists.

What if it were a sodium instead of hydrogen cloud? Using the same logic, we can see that now your spectrum should show a gap at λ = 5890 A since this is the characteristic wavelength at which sodium atoms absorb and emit photons. And if it were a mix of hydrogen and sodium, you’d see two dark lines, one at λ = 4102 A due to the presence of hydrogen atoms and another one at λ = 5890 A due to the sodium atoms. Of course, and here comes the fun part, we can reverse this logic. If you record a spectrum and you see gaps at λ = 4102 A and λ = 5890 A, you know for sure that the light must have passed through a gas that contains hydrogen and sodium. So the seemingly unremarkable gaps in a spectrum are actually a neat way of determining what elements sit in a star’s atmosphere! This means that by just looking at a star’s spectrum we can not only determine its temperature, but also its chemical composition at the surface. Here are the results for the Sun:

– Hydrogen 73.5 %

– Helium 24.9 %

– Oxygen 0.8 %

– Carbon 0.3 %

– Iron 0.2 %

– Neon 0.1 %

There are also traces (< 0.1 %) of nitrogen, silicon, magnesium and sulfur. This composition is quite typical for other stars and the universe as a whole: lots of hydrogen (the lightest element), a bit of helium (the second lightest element) and very little of everything else. Mathematical models suggest that even though the interior composition changes significantly over the life time of a star (the reason being fusion, in particular the transformation of hydrogen into helium), its surface composition remains relatively constant in this time.

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New Release for Kindle: Introduction to Stars – Spectra, Formation, Evolution, Collapse

I’m happy to announce my new e-book release “Introduction to Stars – Spectra, Formation, Evolution, Collapse” (126 pages, $ 2.99). It contains the basics of how stars are born, what mechanisms power them, how they evolve and why they often die a spectacular death, leaving only a remnant of highly exotic matter. The book also delves into the methods used by astronomers to gather information from the light reaching us through the depth of space. No prior knowledge is required to follow the text and no mathematics beyond the very basics of algebra is used.

If you are interested in learning more, click the cover to get to the Amazon product page:

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Here’s the table of contents:

Gathering Information
Introduction
Spectrum and Temperature
Gaps in the Spectrum
Doppler Shift

The Life of a Star
Introduction
Stellar Factories
From Protostar to Star
Main Sequence Stars
Giant Space Onions

The Death of a Star
Introduction
Slicing the Space Onion
Electron Degeneracy
Extreme Matter
Supernovae
Black Holes

Appendix
Answers
Excerpt
Sources and Further Reading

Enjoy the reading experience!

The Jeans Mass, or: How are stars born?

No, this has nothing to do with pants. The Jeans mass is a concept used in astrophysics and its unlikely name comes from the British physicist Sir James Jeans, who researched the conditions of star formation. The question at the core is: under what circumstances will a dark and lonely gas cloud floating somewhere in the depth of space turn into a shining star? To answer this, we have to understand what forces are at work.

One obvious factor is gravitation. It will always work towards contracting the gas cloud. If no other forces were present, it would lead the cloud to collapse into a single point. The temperature of the cloud however provides an opposite push. It “equips” the molecules of the cloud with kinetic energy (energy of motion) and given a high enough temperature, the kinetic energy would be sufficient for the molecules to simply fly off into space, never to be seen again.

It is clear that no star will form if the cloud expands and falls apart. Only when gravity wins this battle of inward and outward push can a stable star result. Sir James Jeans did the math and found that it all boils down to one parameter, the Jeans mass. If the actual mass of the interstellar cloud is larger than this critical mass, it will contract and stellar formation occurs. If on the other hand the actual mass is smaller, the cloud will simply dissipate.

The Jeans mass depends mainly on the temperature T (in K) and density D (in kg/m³) of the cloud. The higher the temperature, the larger the Jeans mass will be. This is in line with our previous discussion. When the temperature is high, a larger amount of mass is necessary to overcome the thermal outward push. The value of the Jeans mass M (in kg) can be estimated from this equation:

M ≈ 1020 · sqrt(T³ / D)

Typical values for the temperature and density of interstellar clouds are T = 10 K and D = 10-22 kg/m³. This leads to a Jeans mass of M = 1.4 · 1032 kg. Note that the critical mass turns out to be much greater than the mass of a typical star, indicating that stars generally form in clusters. Rather than the cloud contracting into a single star, which is the picture you probably had in your mind during this discussion, it will fragment at some point during the contraction and form multiple stars. So stars always have brothers and sisters.

(This was an excerpt from the Kindle book Physics! In Quantities and Examples)

Length – From Electrons to Galaxies

Even compared to a hydrogen atom (with its diameter of 1 A = 1 angstrom), the electron is microscopic. It is about 0.00006 A in diameter, or in other words: you would need 16,700 electrons to fill the length of a hydrogen atom. A water molecule, which consists of two hydrogen atoms and one oxygen atom, measures roughly 3 A. The largest naturally occurring atom, uranium, goes up to 4 A and a common glucose molecule is around 9 A.

This is where we leave the realm of atoms and molecules, that can only be penetrated by hi-tech electron microscopes. With these, resolutions below 1 A can be achieved, making images of individual atoms possible. The good old light microscope can go as low as 2000 A and no further. But this is enough to observe individual bacteria (10,000 A) and human cells (100,000 A). The latter is already about one-tenth the width of a human hair (1,000,000 A = 0.0001 m). This means that now we are nearing the length scales we are familiar with.

The thickness of a credit card is around 0.0008 m, the average red ant is about 0.005 m long and one inch measures 0.025 m. From the length of a cigarette (0.1 m) over the height of a person (1.7 m) and the wingspan of the Boeing 747 (64 m), we quickly approach the high end of the length scale.

The tallest man-made structure is Burj Khalifa, a 830 m tall skyscraper in Dubai. While already mind-boggling, it dwarfs in comparison to the highest mountain on Earth, the mighty Mount Everest, with its height of 8848 m. From this height, it would take about two minutes to reach sea level in free fall. From the International Space Station (400,000 m) however, the Mount Everest is just a small bump in an enormous sphere of diameter 12,700,000 m.

Going into space, the distances quickly grow beyond our comprehension. The Apollo astronauts had to travel 380,000,000 m = 1.3 light-seconds to get to the Moon. Any mission to Mars has to travel one-hundred eighty times that (4 light-minutes). Multiply that by another factor of ninety, and you get to the former planet Pluto (350 light-minutes).

This is where things get crazy. To reach the next star, Alpha Centauri, you’d have to travel 4.2 light-years or about 550,000 times the distance Earth-Mars. The center of our home galaxy is roughly 10,000 light-years away, the nearest galaxy Canis Major Dwarf adds another factor four to that (42,000 ly). In the grand scheme of things though, even this is not that much. The light we we observe coming from the nearest spiral galaxy, Andromeda, has been traveling for a mind-blowing 2.5 million years.

Where does it all end? Nobody knows for sure. The farthest galaxy is z8_GND_5296, discovered 2013 by the Hubble telescope and Keck Observatory in Hawaii. It is 13.1 billion light-years away. This means the light we see has been sent into space long before Earth came to be. Maybe the galaxy does not even exist anymore, maybe all the stars within it are dead by now. We’ll have to wait another 13.1 billion years to see if that’s the case. I’ll update the post then.