# Astro-Spectroscopy

By

Dr Lim Zhi Han

,

Koh Zhi Yong

,

Gerald Kang Joon Kiat

●

Published 2022-09-10

Let us now discuss one of the most valuable tools in Science applied to the study of the Universe. This section introduces how astronomers determine the chemical compositions and physical properties of celestial objects. We will also look at how velocities of stars and galaxies in the radial direction are deduced by examining their spectra.

Have you seen a piece of material start to glow as it is heated to higher temperatures? Consider an iron rod being heated up. At first you will feel the heat radiating out of the iron. We are feeling the infra-red light that the iron emits. The iron will begin to glow in dull red at temperatures above 500^{\circ}\text{C}. At even higher temperature, it changes to bright red to orange, then yellow and progressively white above 1300^{\circ}\text{C}. From such observations, it seems that light is somewhat associated with temperatures. Indeed it does! In fact, light that are emitted from the iron are not limited to just visible or infrared but covers the entire electromagnetic spectrum.

Consider a hypothetical body that absorbs all incident electromagnetic radiation and reflects none. We termed this perfect absorber as a blackbody, since it will appear black when it does not reflect visible light. While such a body do not reflect light, it is capable of emitting (blackbody) radiation in all directions. The body emits electromagnetic energy as a result of the thermal agitation of electrons in its surface. When the body is in thermal equilibrium, it emits as much energy as it absorbs, thus making a blackbody both a perfect absorber and emitter.

The intensity of the electromagnetic energy emitted by a blackbody depends on the wavelengths (frequencies) and the body's thermal equilibrium temperature; the radiation it emits covers the entire range of the electromagnetic spectrum. The electromagnetic energy per unit volume per unit wavelength emitted by a blackbody at thermal equilibrium temperature T is given by

u(\lambda,T)=\frac{8\pi hc}{\lambda^{5}}\frac{1}{\exp\left(\frac{hc}{k_{B}T}\right)-1}

where h is the Planck's constant, k_{B} is the Boltzmann's constant and c is the speed of light in vacuum.

The figure above shows the radiation emitted at thermal equilibrium by a blackbody at various temperatures. It has a well-defined continuous energy distribution over the entire range of wavelengths of the electromagnetic spectrum. For any specific wavelength, there is a corresponding energy density that depends neither on the chemical composition of the body nor its shape, but only on its surface temperature.

At a certain temperature T, the energy distribution peaks at a particular wavelength, which can be approximated from the Wien's Displacement Law

\lambda_{\text{peak}}=\frac{2.898\times10^{-3}\text{mK}}{T}.

Real materials emit electromagnetic energy at only a fraction of a blackbody. Like any material, gases when heated and at thermal equilibrium also emit electromagnetic radiation. Stars, which are huge dense masses of gases at their core, are therefore no exception. In fact, the interior of stars are good approximations of blackbodies because their hot gases are very opaque, that is, stellar material is nearly a perfect absorber of radiation. At thermal equilibrium, the interior of a star emits electromagnetic energy that has a continuous spectral energy density nearly similar to that of a blackbody. However, when the emitted radiation of a star's interior passes through the star's atmosphere, the relatively cooler less dense gases absorb parts of the wavelengths that are emitted. By studying these wavelengths, we can determine the chemical composition of the stellar atmosphere. Let us take a look at our closest example: The Sun.

The first astro-spectroscopy was carried out more than two hundred years ago by William Hyde Wollaston (1766 - 1828), who noted dark lines in the spectrum of sunlight after passing through a prism. About a decade later, Joseph von Fraunhofer (1787 - 1826) repeated the experiment using a diffraction grating, and recorded about 600 discrete dark lines. Atomic physics at that time was not yet developed, so the nature of the lines remained a mystery.

Modern technologies have allowed us to collect the solar spectrum above Earth's atmosphere (which also causes absorption of wavelengths). Even so, some of the notable discrete dark lines still persist in the solar spectrum. This indicates that these lines are caused by the Sun's atmosphere. The Sun's interior emits a continuous blackbody spectrum of radiation. As the light passes through its cooler outer atmosphere, the atoms and molecules absorb photons at specific wavelengths to undergo electronic transitions. The absorbed wavelengths coincide with the emission spectrum of some heated elements, providing strong evidence that these elements are present in the solar atmosphere of our Sun.

The Fraunhofer absorption lines provide an insight into the chemical composition of our Sun's atmosphere. Extending to other stars, the absorption lines in a stellar spectrum serves as a fingerprint identification of elements present in the stellar atmosphere.

It was found that all stars have roughly the same chemical composition (therefore similar discrete dark lines in their stellar spectrum). However, the relative strength of the absorption lines varies between stars. Stars are therefore classified based on the relative strength of their absorption lines and the line strength mostly traces out the surface temperature of the star.

To see why, consider two extreme cases where the surface temperature of a star is too low or too high. A star with a low surface temperature will have few electrons in the n=2 energy level. As a result the cool star will exhibit weak Balmer lines. On the other hand, a very hot star will mean that many atoms are fully ionised. With few bound-state electrons available, the absorption lines will be weak too. In the intermediary of the two extreme cases, the intensity of the spectral lines is determined by the ratio of the population of electrons in each state.

Therefore, by carefully analysing the intensity of the absorption lines in a stellar spectra, one can in principle determine the surface temperature of the star or at least narrow down its temperature range. The surface temperature of stars are commonly compared against other stellar properties, such as luminosity.

Most stars are generally classified under the Morgan Keenan (MK) system using the letters O, B, A, F, G, K, and M, with O type being the hottest to M type being the coolest. Each spectral type is also further subdivided using a numeric digit with 0 being the hottest and 9 being the coolest.

Table 5.2: Spectral classification of stars.

Go to Activity 2: Doppler Rocket

The Doppler effect is commonly associated with astro-spectroscopy. Let's start off the discussion with an interesting phenomenon you have probably encountered. The pitch of an ambulance siren increases as it moves towards us while it decreases as it moves away from us. Such a phenomenon was first described by Christian Doppler and was eventually termed as the Doppler effect. When a source of sound and a listener is in relative motion relative to each other, the frequency of the sound heard is not the same as the source frequency.

To see why, let's first consider an observer and a source of sound. The source emits a sound wave with frequency, f ,and wavelength \lambda=c/f, where c is the speed of sound in air. To make things easier to visualise, consider the observer's motion in the horizontal direction.

Now suppose the source moves, say at a constant velocity of v_{s} towards the observer. The frequency and wavelength of the sound wave as perceived by the observer is now different. In one period T=1/f, the source moves by a distance of v_{s}T=v_{s}/f. The wavefronts towards to observer are compressed by this amount. The observer thus perceives a shorter wavelength, and experiences a higher frequency of recieving the wavefronts. Let \lambda' and f' be the wavelength and frequency measured by the observer. It is conventional in astronomy to take velocities as "recession velocities". This means sources that move away from us take on positive velocities. Here since the source moves towards us, v_{s}<0. Hence the compressed wavelength is

\begin{align}
\lambda' & =\lambda+\frac{v_{s}}{f}\nonumber \\
 & =\lambda+v_{s}\frac{\lambda}{c}\nonumber \\
\lambda'-\lambda & =\frac{v_{s}}{c}\lambda\nonumber \\
\frac{\Delta\lambda}{\lambda} & =\frac{v_{s}}{c}
\end{align}

where \Delta\lambda is the shift in wavelength. Here the source is approaching the observer, v_{s}<0, \Delta\lambda is negative, meaning that the wavelengths received are shorter than the originally emitted. If the source is receeding away the observer, v_{s}>0,\Delta\lambda>0, the wavelengths received will be longer. The frequency will shift in a similar way:

\begin{equation}
\frac{\Delta f}{f}=-\frac{v_{s}}{c}
\end{equation}

The shifts in wavelengths and frequencies are called the Doppler shifts.

The Doppler effect of sound waves require the speed of sound wave (c), velocities of the source with respect to the observer (v_{s}) to be measured with respect to whatever medium they are travelling within. Electromagnetic waves, such as light, require no medium to propagate. Nonetheless, there is still a Doppler relation for electromagnetic waves (e.g. light) just by having knowledge of the relative velocity between the observer and the source. Here, we quote the result and shall not derive the relation:

f'=\sqrt{\frac{c-v_{s}}{c+v_{s}}}f

where c here is the speed of light. In the low (source) velocity limit where v_{s}\ll c, the above equation reduces to Eq.(2). In practical astronomy, this is almost always the case and hence the simpler formulae Eqs.(1) and (2) are usually used.

Generally all celestial objects are in relative motion with us, hence all astronomical spectral lines are Doppler shifted.

Astronomers define the redshift of a spectral line by

\begin{equation}
z=\frac{\Delta\lambda}{\lambda}
\end{equation}

This term gives a intuitive picture of shifted spectral lines shfted towards higher wavelengths due to recession velocities. Together with Eq.(1) we see that the higher the (recession) velocity, the greater the redshift.

The use of astro-spectroscopy has led to many interesting astronomical discoveries. Here, we briefly discuss how spectroscopy is used to detect binary stars and exoplanets.

A binary star system is a system of two stars that are gravitationally bound and orbit around each other about a common center of mass. While some binary star systems are resolvable by using a telescope, more often than not either the resolving power of the telescope presents itself as a limiting factor or the relative brightness between the stars is too large, resulting in a situation where the glare from the brighter star makes it difficult to observe the fainter star. In such situations, astro-spectroscopy is a useful tool in helping to detect binary star systems through the analysis of the Doppler effect on their stellar spectra.

The absorption lines in the stellar spectra of each star are first blueshifted and then redshifted, as each star moves towards us and then away from us about their common center of mass, with the period of their common orbit. In some spectroscopic binaries, the spectral lines of both stars are visible and lines alternate between single and double. In other systems, only the spectrum of one star is observed and the lines periodically blueshift and redshift. By carefully analysing the Doppler shift of the spectral lines, it is possible to determine the radial velocity of the system and even the shape of the binary orbit.

Astro-spectroscopy also offers a useful indirect method of detecting exoplanets. If the exoplanet is gravitationally bound to a host star, both of them will orbit about a common center of mass. Again, we search for the Doppler shift (periodic blueshift and redshift) in the stellar spectrum of the host star as the star moves towards and away from us. However, compared to binary star systems, the spectral shift brought about by an exoplanet can be much less prominent and harder to detect if the planet is much smaller in size and orbits far from its host star, causing the radial velocity of the star-exoplanet system to be relatively much smaller compared to a binary star system. Therefore, this method is best used for detecting massive exoplanets orbiting close to their host stars, since this will allow for a much larger radial velocity of the system and thus a more observable Doppler shift in the spectral lines of the stellar spectra.

The discovery of the first exoplanet was made in 1995 by Swiss astronomers Michael Mayor and Didier Queloz using astro-spectroscopy, who were both subsequently awarded the 2019 Nobel Prize in Physics. Since then, there have been a few different methods aside from spectroscopy used in detecting exoplanets and the number of verified exoplanets has surpassed 5000.

In 1912, Vesto Slipher examined the spectrum of our neighbour galaxy, Andromeda and reported a shift of the spectral lines. He argued that this (blue)shift of Andromeda's spectrum is most likely due to Andromeda moving towards us with a radial velocity of 300 \text{km/s}. He went on to analyse 15 other galaxies and found that some are approaching us and receding from us. Edwin Hubble and Milton Humason extended Slipher's work in the 1920-30s and found that most galaxies are moving aways from us. In addition to finding the redshifts and radial velocities, they used Cepheids in the galaxies they examine to determine the distance of these galaxies.

In 1929, Hubble published a result that changed the view of the Universe. He found that with the exception of a few nearby galaxies whose spectra are blueshifted and thus movng towards us, all other galaxies have redshifted spectra and move away from us. Even more surprisingly, the further away the galaxy, the faster the receeding velocity. The proportional relationship between distance and recession velocity of a galaxy, now known as the Hubble's law

\begin{equation}
v=H_{0}d
\end{equation}

What kind of force is pulling the Universe apart? Gravity is an attractive force, it will not do this. Electric forces cannot be at play as that will require the galaxies to be highly charged. Nuclear forces do not work in long range. What could be the cause of this phenomenon?

Even now we do not truly know the "cause" of the phenomenon. The best "explanation" is to assume that the Universe is expanding, and model the expansion based on our existing theories. What is the meaning of an expanding Universe? We will dig into the this in the second half of the course.