Spectra

Spectra (plural of spectrum) were mentioned throughout this blog, and they will be mentioned many more times.  It is the best kind of spectrum, the electromagnetic.  They can be used to study stars, DSOs, and even the planets and moons!  Since stars can emit all sorts of light they are also split into classes based on their spectra.  Let’s get right into this extremely useful tool, but be warned that there will be some math.

Atoms have quite the history and must be understood to understand those lovely spectra since all atoms produce different spectra.  The atom below is the Bohr model, named after Niels Bohr, of a Hydrogen atom.  Also, the electron paths are called orbitals or shells.

Simply this (astronomyonline.org physics concepts-the atom).

There are essentially seven energy levels, numbered one to seven, represented as n=# (n is known as a quantum number) and referring to the orbitals, such that one is the closest to the nucleus and seven is the furthest.  A relationship exists showing that as an electron gets further from the nucleus it gains energy, usually coming from a photon or from heat.  When an atom is in the ground state all of its electrons are in the lowest possible energy level, while when it is in an excited state at least one electron is in a higher energy level from the lowest possible.  This is known as jumping energy levels, but please don’t think that this involves a child that drinks caffeine and gets all energetic and jumpy.  (In fact, it is probably recommended to not give children caffeine but that’s another story.)  Moving on, ionization can be more specifically defined as when the amount of energy the electron absorbs is greater than the allowed energy of all orbits.  Atoms want to be fairly stable, unlike jumpy little kids, which is why all sorts of things form, from salt to water.  So, an atom that is excited ultimately will have the electrons jump back down to the ground state.  But to have gone to a higher energy level the electron had to gain energy, meaning that for it to drop to a lower energy level, it has to emit energy.  That is where spectra come from.

Basically this. Sadly if the atoms produce gamma rays it won’t turn people into the Hulk. In fact, if atoms produce gamma rays it is recommended to go far far away and to hide (astronomyonline.org physics concepts-the atom).

Spectral lines can be produced at specific wavelengths from atoms.  Electrons that absorb energy must either absorb all or none of it since this energy comes in packets called photons.  Therefore, it is true that each and every single atom that has a different structure MUST absorb and emit different amounts of energy.  In essence, this process starts with absorption of energy and ends with emission of energy.  Atoms have certain lines which show optimally in certain temperature ranges, so this is yet another hint they can show about stars.  Spectra can be split into three types: continuous spectrum, emission lines, and absorption lines.

A continuous spectrum is basically a rainbow, all colors of light.  This is the prime, beautiful example of a spectrum that anyone has seen in nature.  We noted in the history section that it took many people to explain the effects seen in the continuous spectrum as it is normally seen and the dark lines seen by Fraunhofer. Notably Kirchoff did this, and the rules are:

1. A luminous solid or liquid emits a continuous spectrum of all wavelengths.  It has no lines in it.

2. A rarefied luminous gas emits light whose spectrum shows bright lines.  These lines are called emission lines.

3. If the light from a luminous source passes through a gas, the gas may extract certain specific energies from the continuous spectrum.  We then see dark lines where the energy has been removed.  These dark lines are called absorption lines.

Three types of spectra

Very colorful (NASA spectra introduction from Imagine the universe)!

To explain its usefulness in stars, certain elements will show prominently in the spectral lines since stars are insanely hot.  In general, certain objects show more emission lines or absorption lines depending on their composition, and every single aspect to spectral lines can be studied to learn immense amounts of information about stars.  They are made of plasma, but since that is ionized gas the 2nd and 3rd rules apply.

Since Hydrogen is the most abundant element in the universe, a variety of people developed series to explain its various spectral lines.  It was understood that series existed to show different energy wavelengths produced as emission lines by Hydrogen.  The major series consist of the Paschen series in the infrared, the Balmer series in the visible, and the Lyman series in the ultraviolet.  For the Lyman series the shortest transition seen is called Lyman-alpha (n=2 to n=1) at the 122 nm line.  For the 103 nm line (n=3 to n=1) it is called Lyman-beta.  For the Balmer series the 656 nm line is Balmer-alpha or more commonly H-alpha (Hα).  After this the 486 nm line (n=4 to n=2) is H-beta.  Hopefully a trend is shown that the highest nm line transition is named alpha, and then after it goes beta, gamma, etc.

It is from Wikipedia, but do use other sources with Wikipedia when doing research.

Just to make this colorful, we will also show what the Balmer series would look like to us, but it wouldn’t be fun to just have Hydrogen.  Let’s show all the elements!

The wavelengths are measured in Angstrom which is 10^-10 meters, but besides that it’s colorful (ucolick.org under spectra lab)!

From here we must depart to the land of math as the mathematical definition can better show our knowledge of spectra.  If a letter is used once it will not be repeated unless to make a note about it or to change units, so if it isn’t mentioned look to one of the last equations mentioned.  Remember UNITS MUST ALWAYS MATCH.

Going from the Hydrogen series there is a formula known as the Rydberg formula.  It was discovered around the same time the Balmer series was first discovered.  It is highly important because it can be used to calculate the wavelength for energy transitions from emission spectra of Hydrogen, the most abundant element in the universe.  We will show how the formula works.

First, it is important to know the two major equations relating to light.  One is λν = c.  Where:

  • Lambda or λ is wavelength in meters
  • ν or nu (pronounced “new”) is frequency in wave crests or cycles per second
  • c is the speed of light in a vacuum or 2.998 x 10^8 m/s

The other equation is the de Broglie equation E=hν.  Where:

  • E is energy in joules
  • h is Planck’s constant as 6.626 x 10^-34 Joule*seconds.

Using the above equation and substituting for frequency it can be represented as E=c*h/λ.  Both the speed of light and Planck’s constant can be found to have more significant figures, but

For calculating the energy change of an electron:

\begin{displaymath}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
{\mit\Delta} E = E_0\left(\frac{1}{n_f^{\,2}}-\frac{1}{n_i^{\,2}}\right).<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\end{displaymath}

Where:

  • $n_i$ is the initial quantum number
  • $n_f$ is the final quantum number
  •  Eo is the initial energy or the energy of the groundstate

For the energy change of the electron if it is negative then the electron should emit a photon of frequency$\nu=- {\mit\Delta}E/h$.  If the energy is positive then the electron should absorb a photon of frequency $\nu={\mit\Delta}E/h$.  This is exactly what leads into the Rydberg formula.  Since $\lambda^{-1}=\nu/c$, the Rydberg formula when substituting is:

\begin{displaymath}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\frac{1}{\lambda} = R\left(\frac{1}{n_f^{\,2}}-\frac{1}{n_i^{\,2}}\right),<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\end{displaymath}

Where:

    Lambda is in nanometers

   $n_i$ and $n_f$ are the same, but it is assumed in the Rydberg formula that $n_f<n_i$ since it is used for emission lines of Hydrogen.  This gives the definition that the Lyman series is a transition to the ground state ($n_f=1$) releasing UV light, the Balmer series is a transition to the first excited state ($n_f=2$) releasing visible light, and the Paschen series is a transition to the second excited state ($n_f=3$) releasing infrared light.

    And the Rydberg constant R:

With this now even you can relate energy, wavelength, and frequency all from spectra.  In fact, combined with the Doppler effect these equations can be used to analyze and see the composition, temperature, or presence of certain objects.

To finish, some jokes that hopefully will be amusing after we have shed light on yet another subject:

An electron sitting in a prison asked a second electron cellmate,”What are you in for?”

To which the latter replied, “For attempting a forbidden transition.”

Q: Why does hamburger have lower energy than steak?

A: Because it’s in the ground state.

Q: What happens when electrons lose their energy?

A: They get Bohr’d.

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TL;DR: Spectra are up there with one of the best tools to use to analyze objects in Astronomy.  Atoms can absorb specific wavelengths of light or emit specific wavelengths of light depending on their size and structure.  This produces spectral lines which appear on a spectrum of light, and this will be different for all atoms of different structure.  Since most of the objects being studied are made of atoms, we find that they produce spectral lines. Hydrogen has been studied extensively because it makes up so much of the universe, and the Rydberg formula helps us calculate hydrogen’s spectral lines.

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Sources and further reading:

 

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