Stellar Fingerprints: How Do We Know What Stars Are Made Of? Part 2 by Lanz Lagman


The second part of this article will bring us towards a more in-depth exploration of how light interacts with matter in the world of quantum mechanics.

Absorption Lines and Emission Lines: The Fingerprint of Elements

Absorption and emission lines of hydrogen in visible light.
Image credit: Kahn Academy
Spectroscopy is the study of how light interacts with matter. Matter is composed of a set of elements and an element is represented by an atom. 

When a light of a particular wavelength hits an element, the spectrum detected will have a set of dark lines separated by certain distances called the absorption lines. Each element (and therefore, molecule) has a different absorption line, making this the fingerprint of an element.

This means that by detecting the EM waves of stars, analyzing their spectra, and hence their absorption lines, we can determine their composition. We don't need to actually get star stuff after all! 

Additionally, when chemists use spectroscopy, they use various light-generating devices used to project light through their samples. Astronomers don't need to use these devices because the stars themselves already generate light.

How are these absorption lines made anyway? 

The Bohr Model of the Hydrogen Atom

Some of you may find his world boring, but his atomic model is not!
Image credit: The Simpsons
In order to understand how emission lines are made, let's study Niels Bohr's atomic model. Bohr described the atom as having a dense nucleus in the center, just like how the Sun looks like the dense center of our Solar System. The electrons could only be found at the orbitals, the circular path where they orbit. 

An orbital could contain more than one electron. When an electron gets hit by an EM wave depending on its wavelength, it will teleport (yes, teleport) to an orbital much farther from the nucleus. In order to "step down" a few orbitals, the electron must spit back an EM wave of a specific wavelength.

Bohr's Mini Solar System. Image credit: NASA
The most important aspect of his model is that the angular momentum of the electron is quantized. Quantized means that it is determined by a positive integer greater than 0 (1, 2, 3, 4, 5...), also known as the numbers we learned during our first days in school.

Angular momentum is in this sense, what determines how hard it is to make the electron orbit around the nucleus, and how hard it is to make it slow down once it orbits. Expressed as an equation, it looks like this: 

By looking at this equation, it is clear that the values of the resulting angular momentum would just be h/2π times n, which could only take up a value of 0, or 1, or 2, and so on. This quantization part is one of the principal concepts that would give birth to quantum mechanics, or what I prefer to call, the weird physics.

Now what does it mean to have a quantized angular momentum? The resulting orbital radii are quantized also:

In this equation, k is the Coulomb's constant, Z is the atomic number (number of protons), m is the mass of the electron, and e is the magnitude of the electron's charge (protons and electrons have the same magnitude charge, but electrons have a negative sign before the value). 

Radii being quantized means rcould only take the following values: r122 r32 r, and so on. To make it simple, no orbital radii will exist if its value is 1.52 r, 2.012 r, or even 3.00000000012 r. That's the essence of integers.

Another important thing to remember is that if the electron is at the orbital n=1, it's at ground state. If it goes higher, such that it becomes n=2, it's in an excited state, or in other words, ionized. To make things simpler, let's just say from 1 to 2 instead. 

Transfer from n=1 to n=2. Image credit: NASA
Consequentially, the energy involved in transferring from one lower orbital to the next consecutive orbital is also dependent on the allowed orbital radii:

To move from energy level 1 to 2, an electron must absorb the energy of an EM wave that passes through it. However, transferring from 2 to 1 involves the electron releasing an energy with a corresponding wavelength. In short, you release light.

The electron could only exit its present orbital if the wavelength has just the right length to have the required energy for the electron to transfer.

This is where quantization is very important. Think of how quantized our currency is today - there are no products in grocery stores, for example, that cost 1 centavo. 

You won't see a bar of chocolate that costs 99.9276 pesos. All products' costs are multiples of five centavos and even these coins are almost never seen, because a 1 peso coin is the most practical coin with the lowest value.

An atom jumping from one lower orbital level to a higher one produces an emission line. Jumping from a higher one to a lower one produces an absorption line.

Since each element contains a different number of electrons and orbitals occupied, the number and arrangement of lines recorded by spectroscopes would give away what substances are made of.

In the case of stars, it turns out they're primarily made of hydrogen, then helium, and lastly, traces of other heavier elements (elements containing more than two electrons). 

Bohr's model works best for hydrogen atoms and is only used as an approximation for heavier elements. For other elements, scientists use databases containing the observed spectral lines of individual elements. Now that we've explained how a single emission and absorption line is made, we'll introduce Rydberg's formula and its known offspring: the Balmer series.

Rydberg Formula and the Balmer Series

From his atomic model, Bohr has given an empirical or in simpler terms, a mathematical solution to the spectral lines of hydrogen.
Lambda (λ) is again the wavelength, R is the Rydberg Constant, and Z is the number of electrons. The letter i is the initial orbital of an electron and f is the final orbital of the electron. This is the number of the orbital where the electron would transport to. For the hydrogen atom, Z = 1. 

Hydrogen spectra has a lot of so-called series, distinguished by the value of i, for i=2, the Balmer series is produced:
The other series are the following: Lyman (i=1), Paschen (i=3), Brackett (i=4), Pfund (i=5), Humphreys (i=6), and Further (i>6).

This is how the lines of these series look like for the spectrum of the hydrogen atom.
Hydrogen spectral series. Image credit: Wikipedia.
Of all the hydrogen spectral series, the Balmer series is the most useful. This is because its lines lie on the visible range. Ground-based optical telescopes are frequently able to see on the visible range, so it would be easier for them to observe these lines.

What could we do with all this knowledge?

Observing how these lines move in the spectrum is also a useful indicator on how the star being observed moves with respect to Earth's position. Since stars and therefore galaxies, are mostly made out of hydrogen, we could look at its lines on how they move across the spectrum. 

If the lines move toward the blue side, that means that the object, star, or galaxy is moving towards us. If it moves to the red side, that means it's moving away from us. 

Redshift. Image credit: UCLA
The wavelength of light emitted by a star is also correlated with its surface temperature and brightness. By grouping them based on these criteria, we could classify stars better and this leads to understanding how stars of various sizes evolve over time. The result of this classification is the Hertzsprung-Russell diagram.

An H-R diagram. Image credit: Wikipedia.
Our Sun is a yellow main-sequence star. This means that it is at the middle of its lifespan. Some stars are very big, but their temperature is lower than the Sun's, while some are medium-sized but a lot brighter and hotter than our Sun. Big stars have very short lives (just some millions of years), while smaller stars have longer life spans.

Aside from that, spectroscopy is also an important tool in searching for Earth-like planets. Since the glow of a planet is just the reflection of their parent stars, we could determine the composition of the atmosphere of exoplanets if we have very powerful instruments.

A planet with the right distance from its parent star/s and an atmosphere which is quite similar to ours is generally considered to be habitable. Hopefully, when we have better means of space travel, we could send probes to a potentially habitable planet, or even visit it ourselves!

Learning about the jewels in our sky is just a step towards exploring our universe.

Postscript: For the Advanced Reader - 

 - here is a much more detailed derivation of the previous equations mentioned. 


Bohr's model was made primarily for the hydrogen atom. In this model, the electron orbits the nucleus just like how a planet orbits around its parent star. The electrostatic force is equal to the centripetal force, and we express this as equation (1): 
Simplifying equation (1) leads to:
Bohr has proposed that the angular momentum of the electron is quantized. This is expressed as: 

Substituting the v of equation (2) to equation (4) then isolating r to the left side, we get an expression for the radius of an orbital. If n=1 or the radius of the ground state orbital, then this expression becomes:

We could generalize then that for any n, the resulting radius will be: 


In order to express the total energy of the electron, we must express it first as the sum of its kinetic energy Eand potential energy Ep

Since Ek is just one half of the electron's mass multiplied by the square of its velocity, we could use equation (2) to find an electrostatic expression of its kinetic energy. This results in:


The electric potential energy is expressed as: 


Now that we have equations (9) and (10), we could add them just like in equation (7) to solve for Eand this gives us the equation for an energy level: 



Expressing equation (11) in terms of equation (6) gives us a more quantized form: 



This means that n=1 is the ground state energy level and n>1 are the energy levels for excited states. 

The Rydberg formula tells us how much energy is absorbed or released depending on what orbital the electron goes to: if it goes from a higher orbital or a lower orbital.

As our solutions go by, this will be expressed in terms of the inverse of the wavelength or wave number (1/λ). Nonetheless, the change of energy is the difference between the final energy level Ef and the initial energy level Ei.



We could express the left-hand side of equation (13) in terms of wavelengths. After all, what happens to the electron, whether it goes up or down, depends on the wavelength it absorbs or releases.



Using equations (14) and (11), we could now express equation (13) as: 



In order to make further calculations easier, we must express the Rydberg formula in terms of only the orbital numbers. We do this by expanding equation (15) as: 



The final orbital number then becomes f and the initial becomes i. Using integers is much easier than solving first for the orbital radii of the two required orbits. We further simplify this equation by factoring out all constants from i and j.



Now that we got equation (17), we divide its left side by hc so that the left side contains the wavenumber solely. 



We're almost done. We could combine the fundamental constants and call them the Rydberg constant. Reducing this earlier known constant into more fundamental components is a known feat of the Bohr model. This results to:



For the Balmer series, Z is just equal to 1 and I is equal to 2. Finally, the Balmer series expressed by the Rydberg formula is: 



REFERENCES: 

1. PhysicsLAB. (n.d.). Retrieved September 28, 2016 from http://dev.physicslab.org/document.aspx?doctype=3&filename=atomicnuclear_bohrmodelderivation.xml
2. Niels Bohr. (1913). On the Constitution of Atoms and Molecules, Part 1. Philosophical Magazine 26:1--25.


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