# Spectroscopy

Recessional velocities and compositions of stars

For the purposes of this discussion we can regard light as a wave without consideration of its particle character. Since it is a wave it has a wavelength $$λ$$ which is defined as the distance between two crests or troughs of the light wave as shown in figure #. (More generally speaking, it is defined as the distance between two points on the wave with the same phase.) If we know what the wavelength $$λ$$ of light is, then we can determine a great deal of information about that light such as its color (or, more precisely, which region of the EM spectrum the light is in), energy, temperature, and so on. Each particular color of light has a specific wavelength. For example red light has a wavelength of roughly $$λ=700nm$$whereas violet light has a wavelength of roughly $$λ=400nm$$. There are also forms of light whose wavelength corresponds to regions in the EM spectrum which are imperceptible to human vision (but, nonetheless, can still be “seen” by our detectors) such as infrared light, microwaves, etc. Given the wavelength then using Plank’s relationship $$E=hc/λ$$, we can determine the energy of the light. Furthermore from the Boltzmann relationship $$E=kT$$, we can also relate the energy of this light to the temperature of the matter which emitted it. On average the atoms composing matter of temperature $$T$$ will emit photons with an energy $$E=kT$$ where $$T=k/λ$$. Thus given $$λ$$ we can determine the temperature $$T$$ of the matter. The value $$λ$$ of radiation emitted by the human body with temperature $$T≈98\text{ degrees F}$$ corresponds to infrared radiation. If you consider hotter matter such as a piece of metal heated to $$500\text{ degrees C}$$, it will emit wavelengths of light which correspond to the red region of the EM spectrum and the object will be glowing red. If you consider still hotter object such as the filament in a light bulb with a temperature of $$3,000\text{ degrees C}$$, it will emit white light because $$λ$$ will have shifted to the middle of the visible part of the EM spectrum.

When light is emitted by a distant stellar object (i.e. star or galaxy) with an initial wavelength $$λe$$, by the time it reaches us its wavelength becomes “stretched” and increases by an amount $$∆λ$$. We say that the light was redshifted by an amount $$∆λ$$. In practice the radial velocity $$V$$ of recession of a distant stellar object a distance $$D$$ away from us can be calculated using Doppler’s equation if we know what the redshift (∆λ\) is. During the 1920s the astronomer Edwin Hubble performed this calculation for many different stellar objects and plotted the velocity $$V$$ as a function of distance $$D$$ on a graph. He then drew a line of regression through the data points (which was a straight line) and concluded that $$V∝D$$. Then by calculating the slope he was able to determine Hubble’s constant $$Ho$$. (Hubble’s initial calculation of $$Ho$$ was off by a factor of 10 but later on this error was corrected.) It is very important to emphasize that these stellar objects are not moving through space; rather it is space itself that is “moving” and expanding. Therefore the redshift of these objects is due not to their motion relative to us but rather to the expansion of space (we will talk about this in detail later on). The redshift $$∆λ$$ of such objects is measured using a device called a spectroscope which is an instrument that contains a glass (or some other refractive material) prism. When light passes through the prism it “spreads out”; this makes it easier to distinguish between the different wavelengths of light. (For example when white light (which is composed of all visible wavelengths) passes through a prism a rainbow is produced and the different wavelengths are easier to distinguish.) This refracted light is then shone on a photographic plate (the detector) which records the “brightness” of each wavelength of light. The intensity as a function of wavelength, $$I(λr)$$, is obtained from the detector and, roughly speaking, is a measure of the “brightness” or “dimness” of each wavelength $$λr$$.

Figure 2

According to the laws of atomic physics (which are derived from quantum mechanics), a particular atom can emit and absorb only certain wavelengths of light. Take for example sodium atoms which compose salt. Sodium atoms emit or absorb only two different wavelengths (which correspond to the “orange region” of the EM spectrum) of photons. If a light source shined light composed of all wavelengths in the EM spectrum through a bunch of sodium atoms all wavelengths of light would pass right through it except for two specific wavelengths (which are in the “yellow region” of the EM spectrum). If this emitted light, after passing through the sodium atoms, is shined through a spectroscope and onto a detector, two distinct bands (which lie in the yellow region) will appear on the detector as shown in Figure 3. Each different kind of atom leaves its own signature and produces different bands on the detector as shown in Figure 3. If light consisting of all wavelengths is shined through some object (that is composed of unknown kinds of atoms) and then through a spectroscope and onto a detector, we can determine what kinds of atoms it is made of by inspecting the dark lines (called spectral lines) on the detector. Throughout the early 20th century astronomers used this technique (which is called spectroscopy) to determine the composition of the Sun, other stars, our Milky Way galaxy, and other galaxies.

A star has an atmosphere made up of certain kinds of atoms each of which absorb only particular wavelengths. Stars generate their light through a process called nuclear fusion at its center. When this light passes through the star’s atmosphere most wavelengths of light will go straight through it but certain wavelengths of light will get absorbed. After the light passes through the star’s atmosphere some of it will passes through an astronomer’s telescope and spectroscope and onto the detector. All wavelengths of light will appear on the detector except for the ones that got absorbed—these will appear as black spectral lines. Suppose that we had two identical stars made of the same atoms: the first is stationary relative to us, but the second is moving away from us due to the expansion of space. The spectral lines will look exactly the same for both stars with one exception: the spectral lines corresponding to the star moving away from us will be slightly redshifted. This measured redshift $$∆λ$$ is what astronomers use to calculate the recessional velocity $$V$$ of the star. The same exact argument applies to entire galaxies: most of a galaxy’s light is generated in the interiors of stars which then pass through the stars atmospheres (some wavelengths getting absorbed) and some of that light reaches the detectors of humanities most powerful telescopes. By examining the redshift in the spectral lines astronomers like Vesto Slipher and Edwin Hubble were able to determine what the galaxies recessional velocities were.

Find massive exoplanets

Spectroscopy also plays an important role in the discovery of massive exoplanets which are several times as massive as the Earth. To understand how this works, it is very useful to start out by thinking about the Jupiter-Sun system. As Jupiter orbits around the Sun, it exerts gravitational forces causing the Sun to oscillate about its equilibrium position with an amplitude of hundreds of miles. If we were living on a distant exoplanet "watching" the Sun with out telescopes, what would we see? Well, if we were at the appropriate point in our orbit we would be able to see the Sun either moving towards us or away from us. This relative motion will cause the light emitted from the Sun to be either blueshifted or redshifted. From there, we could just use Doppler's equations to determine the Sun's relative velocity. After determining the relative velocity, we could then work out the mass of Jupiter, its orbital period, and lastly how far Jupiter is away from the Sun.

The entire aforementioned discussion we just had would also apply to an Eath-based observer watching distant stars with their telscopes. When astronomers see a star wobbling, this gives them a pretty good hunch that a massive exoplanet must be orbiting around it. Using the aforementioned techniques, astronomers ccould deduce the mass of the exoplanet, its orbital period, and its distance away from its parent star. Unfortunately, for smaller planets, this strategy of looking at the star's wobble doesn't work quite so well since small exoplanets (ones which are 0.1 to a few times the size of the Earth) exert very small tugs on their parent star making it difficult to notice any wobbles. To find smaller exoplanets, astronomers use a different strategy: namely, they try to spot a "twinkle" as the exoplanet passes by its neighboring star.