Lecture 7: Spectra

Don't Panic

Sections

1. Heat Energy and Temperature
2. Blackbody Radiation
3. Wien's Law
4. The Stefan-Boltzmann Law (Omit)
5. Line Spectra
6. Star Spectra
7. The Doppler Effect

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Heat Energy and Temperature

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To understand light spectra, we have to review a bit of relevant thermodynamics oddly enough.

HEAT ENERGY (or just heat or thermal energy) is statistically distributed microscopic energy of various forms:

1. The kinetic and potential energy of atoms.

2. An electromagnetic radiation field in thermodynamic equilibrium with matter. This means the radiation field is being emitted and absorbed by matter without either radiation field or matter changing state---there is an equilibrium.

   Such radiation field has a blackbody spectrum as is discussed below.

   [A thermodynamic equilibrium electromagnetic radiation field can exist without interacting with matter if it has been set up somehow. For example it can be set up by matter interaction and then isolated from the matter: the thermodynamic equilibrium condition of the EMR is maintained.]

3. Other forms we won't go into here: e.g., nuclei and neutrino energies in equilibrated situations.

Recall thermodynamic equilibrium is a changeless, timeless state at the macro-level. The body doesn't change color, size, phase, composition. It just stays as it is.

Heat an extensive quantity: i.e., one just adds up bits of heat to get the total amount in a sample.

TEMPERATURE is an intensive quantity: this means that in physics it is a ratio of extensive quantities. Because it is a ratio, any size sample can have any size temperature.

The actual quantities in the ratio for temperature are beyond the scope of this class, but heat per unit mass serves as proxy in vague way: it's not exactly right, but if heat per unit mass goes up, so does temperature ordinarily.

TEMPERATURE is a measure of the thermodynamic equilibrium state of body. This state is independent of size.

Recall from Intro Astro Lecture 1: Scientific Notation, Units, Math, Angles, Motion, Orbits that in this course we usually only use the ABSOLUTE TEMPERATURE SCALE or, as it is often called, the Kelvin Scale: the unit is the kelvin (equal in size to the Celsius degree).

     Recall

     T_K=T_C+273.15 

     Absolute zero is T_K = 0 kelvin or T_C = -273.15 C. 

Question: We put a cup of hot coffee and a hot vat of water in thermal contact and isolate them from the rest of the world and NOTHING happens to either.
1. The cup and vat had different temperatures (i.e., different states of thermodynamic equilibrium), but then heat energy flowed from COLD TO HOT and the two came into a mutual state of thermodynamic equilibrium.

2. The cup and vat had different temperatures (i.e., different states of thermodynamic equilibrium), but then heat energy flowed from HOT TO COLD and the two came into a mutual state of thermodynamic equilibrium.

3. The cup and vat had the same temperature originally (i.e., were in thermodynamic equilibrium with each other before they were put in contact) and no net heat flows occurred.



Answer 3 is right.

There are no spontaneous heat flows at the macroscopic level between objects of the same temperature: they are in thermodynamic equilibrium with respect to each other.

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Blackbody Radiation

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All matter can emit and absorb electromagnetic radiation (EMR) from its own heat energy by various microscopic processes.

By ``from its own heat energy'' we mean that the radiation is not reflection EMR: reflection is another process in which the EMR does not strongly interact with the body, but just ``bounces off.''

If a body has temperature above absolute zero (i.e., 0 kelvin), then it will radiate EMR.

Dense materials (i.e., solids, liquids, and dense gases) have a CONTINUUM of emission and absorption channels that extend effectively over all wavelengths.

[Recall CONTINUUM means there are no breaks. The real numbers are a continuum of numbers: between any two no matter how close there is another one. Whole numbers are not a continuum: they are a discrete, but infinite, set of numbers.

Actually crystaline solids like rocks can have broad emission and absorption bands (HI-92): but we won't go into that detail here.]

In fact, a dense body at a SINGLE temperature will emit---not counting any reflection---a BLACKBODY (or Planck) spectrum of EMR.

The neglect of reflection explains the word blackbody---only a pure black body doesn't reflect: pure blackbody emitters can look quite bright, but they are NOT reflecting.

A blackbody spectrum is a smooth spectrum that reaches from zero to infinite wavelength whose shape varies depending on only one parameter: the TEMPERATURE of the emitter.

[Dense bodies with varying temperature will emit spectra that are mixtures of the blackbody spectra at different temperatures.]

Example blackbody spectra are given below.

Note in astro-jargon, blueward means toward shorter wavelength and redward means toward longer wavelength: this is without regard to where the visible colors blue and red are in the spectrum.

Blackbody spectra.

Most bodies in nature are not pure blackbody emitters, but many bodies are close to being blackbodies or at least can be approximated that way for many purposes: e.g.,

1. The filament of an incandescent light bulb which is at about 2800 K ( HOW THINGS WORK). It looks white-hot, but in fact has blackbody spectrum peak at about 1 micron which is in the infrared.

   Most metals melt well below 2800 K, but tungsten has melting point of 3653 K (CAC-51).

2. A candle flame with a temperature of about 2000 K ( HOW THINGS WORK). But a candle flame is probably too dilute to radiate like a blackbody---but I don't really know.

3. A stove grill that is red-hot. It's actual temperature must be much lower than 1000 K---just guessing. The peak of it's blackbody spectrum must be fairly far into the IR (i.e., the infrared).

4. A useful experimental setup that is nearly a perfect blackbody emitter is a completely closed oven.

   The walls of the oven are held to a particular temperature. The walls emit and absorb and reflect radiation, but all those processes cancel each other. The blackbody radiation field inside the oven stays unchanging in time and is in thermodynamic equilibrium with the walls.

   A small, negligibly perturbing aperture allows one to study the radiation field.

5. Stars, including the Sun, approximate blackbody radiators which is a key reason for studying them here.

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Wien's Law

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The wavelength peak of a blackbody spectrum is determined by Wien's law.

Wien's law.

Examples of the use of Wien's law:

_____________________________________________________________________________

 Object         Approx. T (K)   Approx. Wavelength of Maximum (microns) 

_____________________________________________________________________________

 hot star            40000         0.075      far in UV
   photosphere
 
 Sun                  6000           0.5      middle of visible (green light),
   photosphere                                  but white light is a mixture
                                                wavelengths. 

 human body            309          10        in IR
  -so we-all
  radiate mainly
  in the IR, but
  reflect mainly
  in the visible.

 Pluto's surface        37          80        far in IR 
 
_____________________________________________________________________________

From Wien's law you can determine the wavelength of maximum of EMR emission for any body at a given temperature---insofar as that body approximates a blackbody.

Wien's law can be inverted and thus if you know the shape of spectrum that approximates a blackbody, you can estimate the temperature of the emitting surface.

This is a crude, but very useful, way of finding star surface temperatures.

Question: The background EMR field that permeates all space has a blackbody form and a peak at about 0.1 cm = 1000 microns in the microwave band of the electromagnetic spectrum (Se-387).

Recall Wien's law:

    lambda_max = approximately 3000 micron-K / T .
    

What approximately is the temperature of the background EMR field

1. 5800 K.
2. 310 K.
3. 3 K.



Answer 3 is right.

    T = 3000 micron-K / 1000 microns = 3 K .
    

A very exact determination gives 2.726 K (HI-481).

This background EMR field is called the cosmic microwave background (CMB). It is a relic of the big bang.

The discovery in 1965 of the CMB was the crucial evidence that convinced most people that something like a big bang had happened.

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The Stefan-Boltzmann Law (Omit)

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Just for completeness we should give the Stefan-Boltzmann law.

This law gives the total power per unit area emitted by a blackbody radiator.

This is the power summed over all wavelengths.

Stefan-Boltzmann law.

For us, the importance of the Stefan-Boltzmann law is that it shows that the power of a radiator increases strongly with increasing temperature.

Question: Say one increases temperature of a blackbody radiator by a factor of 2.

The thermal EMR emitted increases by a factor of:

1. 2.
2. 4.
3. 8.
4. 16.



Answer 4 is right.

If T_1 is the initial temperature and T_2=2T_1 is the raised temperature, then

  F_2 = sigma x T_2**4 = sigma x (2T_1)**4 = sigma x 2**4 x T_1**4

      =16 x F_1 .
      
    

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Line Spectra

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Dense materials have effectively a continuum of emission and absorption channels with wavelength, and so they can emit or absorb a wavelength continuum of EMR which in the case that the material is at one temperature is a blackbody spectrum.
[Recall crystaline solids like rocks can have broad emission and absorption bands (HI-92): but we won't go into that detail here.]

But atoms, ions, and molecules in dilute gases do not have such strong continuum of emission and absorption channels. They have some, but we won't worry about them here.
[Ions are non-neutral atoms. They have lost or gained electrons relative to the neutral version of the atom.]

Such dilute gases tend to have their strongest emission/absorption in a discrete set of narrow wavelength bands that for spectroscopic reasons are called LINES or LINE TRANSITIONS.

The centers of the narrow bands are called the LINE WAVELENGTHS.

Quantum mechanics explains why atoms, ions, and molecules have LINE TRANSITIONS.

A transition is a change in the internal energy state of the atom, ion, or molecule.

These energy states have only certain discrete energy values: they are QUANTIZED.

To conserve energy, energy must be emitted or absorbed during transition.

The energy change can only come in set of quantized values depending on the quantized states.

Emitting or absorbing EMR in photons is one possible way of conserving energy.

If you just have a hot, dilute gas then usually you will see an emission line spectrum from photons from the LINE TRANSITIONS of atoms, ions, and/or molecules.

Cartoon of continuum and emission line spectra.

Each atom, ion, and molecule has a virtually unique set of lines and therefore it's emission (or absorption) line spectrum is virtually its FINGERPRINT.

In many cases, just by looking at the line spectrum from a gas many of the constituent atoms, ions, and molecules can be identified BY EYE by experienced spectroscopists.

A line spectrum can be created using a spectroscope: just a device that disperses light. But for an emission line spectrum you don't see a long band of colors gradually changing; you see just isolated strips (i.e., lines) of colors surrounded by relative darkness.

Thanks to NASA we can look at a few atomic emission line spectra and some star absorption line spectra: we explain absorption line spectra below.

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Atomic spectra for stellar spectroscopic analysis.

An Angstrom is 10**(-10) meters. It is a non-standard unit, but often used by spectroscopists.

The upper spectra are spectra for actual stars. The lower spectra are just the spectra of particular atoms. One uses the lower spectra to identify the atoms in the upper spectra.

Hydrogen offers the simplest emission line spectrum: a red line, a blue-green line, a blue line, and a violet line.

The hydrogen red line or Halpha at 6563 Angstroms (656 nm) is usually the strongest visible line of hydrogen. Hot interstellar gas often emits this line which gives true color pictures of such gas a reddish or pinkish color.

Halpha and the sodium doublet can both be identified in the star A spectrum.

Credit: NASA: Imagine the Universe.

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Question: Emission line spectra are called line spectra because:
1. that is just what they are called.

2. when you disperse the emission from a source with an emission line spectrum through some sort of device with a slit aperture (which is usually what you do), the spectrum consists of lines instead of a continuous band of color. The lines are images of the slit.



Answer 2 is right.

LINE TRANSITIONS are so called because the EMR they emit when dispersed through a device with a slit aperture, gives rise to spectral lines.

As well as identifying the atoms, ions, and molecules, one can also usually learn something about their abundances, and the temperature and density of a gas from spectroscopy plus modeling.
[Also one can determine the source's velocity relative to the observer along the line of sight. This makes use of the Doppler effect which is discussed in the section The Doppler Effect below.]

It is true to say that spectroscopy is the most important scientific chemical analysis technique of all.

You don't need a sample in your hand. You just need light from a source. The light could be from all across the observable universe.

[In 1835, the positivist philosopher Auguste Comte (1798--1857) ( August Comte) composition of stars as an example of something humans can never know ( Studying the Sun and Stars Spectra by Francis Berthomieu; Professor Korista).

Within a few decades this composition was being identified from starlight.

It turns out that stars are mainly hydrogen just as our Sun is.]

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Star Spectra

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What kind of spectra do we get from stars (e.g., the Sun).

The EMR in the deep interior of a star is in thermodynamic equilibrium with very hot matter with temperatures of tens of millions of degrees and typically has a blackbody spectrum with maximum in the X-ray range (HRW-802).

But EMR from the deep interior does NOT escape the star. The photons that are emitted are absorbed after a short distance by matter.

As discussed in IAWL Lecture 6: Light and Electromagnetic Radiation (EMR) can picture the EMR as photons on a random walk from the interior to the outside. (Heat energy flowing from hot to cold.)

In a random walk, photons are emitted in random directions from matter, travel some distance, and then absorbed.

Their energy is then re-emitted as other photons, but at temperature corresponding to the matter where they are re-emitted from.

The temperature, density, and opaqueness of a star decreases going outward. This gives a bias for longer flights in the outward direction. Near the surface the photons come from relatively low temperatures compared to the interior.

Finally there is layer of the star where the density is so low that outward going photons can escape to infinity at least about half the time.

This layer we call the photosphere. It is the layer from which we see most EMR coming from the star.

Cartoon of photons escaping from the photosphere.

The photosphere often called the surface of the star.

But, in fact stars have no definite surface. Their density decreases going outward and probably becomes negligibly small at some point, but without a sharp cut-off.

Many stars are probably like the Sun whose atmosphere extends into a stellar wind where matter is blowing off the Sun. This wind extends outward until it merges with the interstellar medium.

The photosphere is layer not exactly at a single temperature.

Thus you will get a continuum spectrum from the photosphere that is a mixture of blackbody spectra at slightly different temperatures.

Nevertheless, you can often fit the photosphere spectrum by a blackbody spectrum at a single temperature to high accuracy.

But although the photons can escape at most wavelengths from the photosphere, the lines of atoms, ions, and molecules are particularly strong absorbers even when continuum absorption has grown weak because of low density.

So above the photosphere, the lines will absorb some of the continuum spectrum in the narrow wavelength bands corresponding to the lines.

They absorb because the atoms, etc., are colder than the EMR from the photosphere.

This line-absorbed EMR is reprocessed into some other form and probably mostly escapes the star.

The absorption by the lines creates an ABSORPTION LINE SPECTRUM: a bright continuum with superimposed dark absorption lines.

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Making an absorption line spectrum.

The atoms, ions, and molecules immediately above the photosphere are colder than the photosphere. They will absorb the photospheric emission in their lines.

Credit: NASA: Imagine the Universe.

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Cartoon of an absorption line spectrum.

Of course, an absorption line spectrum plotted in colors rather than intensity is our old friend the solar spectrum.

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The solar spectrum from the Earth's surface as the eye sees it strongly dispersed.

The spectrum is spread out on a wrap around line.

The spectrum is basically a Planck spectrum of temperature 5800 K (Se-147) with absorption lines from the Sun and Earth's atmosphere superimposed.

Credit: N.A.Sharp, NOAO/NSO/Kitt Peak FTS/AURA/NSF.

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The solar spectrum log-log plot.

Here we have a blackbody fit to the solar spectrum (dashed line), the solar spectrum above the atmosphere (dark blue), and the relative solar spectrum at the Earth's surface in cloud-free conditions (light blue). The wavelength range extends from far UV to far IR.

Because of the small scale, absorption lines in the spectrum have been mostly smoothed out. A larger scale would show many them.

Note that the Sun spectrum peaks in the visible.

Note also that the Earth's atmosphere is opaque in the UV and in many broad bands in the IR, but the visible is pretty transparent.

The transparent bands are sometimes called WINDOWS in astro jargon.

Credit: US Naval Research Laboratory, Judith Lean; download site NASA.

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The Doppler Effect

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The Doppler effect is a shift in the frequency or wavelength of a wave phenomenon caused by the relative motion of the source and receiver.

It was first discovered by Austrian Johann Christian Doppler (1803--1853) in 1842 for light. The application to sound came only a little later in about 1845.

The Doppler effects for light and sound are, in fact, a bit different because electromagnetic radiation (i.e., light) which needs no medium and always moves at the vacuum speed of light in vacuum: sound, of course, moves much more slowly and requires a medium

The Doppler effect for sound is in fact an everyday phenomenon. The pitch (frequency) of sound from a siren or plane changes noticeably as the siren or plane passes you.

Bats make use of the Doppler effect for sound in navigating: they emit high-frequency sound and judge relative speeds based on the fequency of the echo---and they don't need any formulas (HRW-417).

Police radar also relies on emission and reflected Doppler shifted signals to judge relative speeds: in this case it is the Doppler effect for electromagnetic radiation that is used (KB-185).

Here we only need to concern ourselves with the Doppler effect for light.

It is quite easy to understand qualitatively.

Doppler effect for light.

In astronomy it is conventional to call a Doppler shift that increases wavelength a redshift because red is at the long wavelength end of the visual band.

A Doppler shift that decreases wavelength is blueshift because blue is at the short wavelength end of the visual band.

Understanding and deriving the Doppler effect for light is complicated by the fact that the vacuum speed of light is the same for all observers and other effects of special relativity.

But the formula is simple enough (HRW-947):

    lambda_receiver = lambda_source * sqrt[(1+v/c)/(1-v/c)]  ,

       where c = about 2.998*10**8 m/s is the vacuum speed of light,

       v is the relative velocity component along the line of sight
       from source to receiver.

       v is positive for recession and negative for approach.
   
  Example 1:  say v = c/2.

    lambda_receiver = lambda_source * sqrt[(3/2)/(1/2)]

                    = lambda_source * sqrt(3)

                    = about lambda_source * 1.7321 , and so

                       the wavelength is increased:  this is redshift.


  Example 2:  say v = -c/2.

    lambda_receiver = lambda_source * sqrt[(1/2)/(3/2)]

                    = lambda_source * sqrt(1/3)

                    = about lambda_source * 0.57735 , and so

                       the wavelength is decreased:  this is blueshift.

   There is a simplification for low relative velocities.

   If v << c , then 

   Delta lambda     v
   ___________  =  ___      ,

     lambda         c


       where Delta lambda is the change in wavelength
      
       and lambda is the rest frame line wavelength.

       We will give an example using this formula below.

There is in fact a small relativistic Doppler effect for motion perpendicular to the line of sight (HRW-947), but it doesn't seem to important in astrophysics.
Question: Why is the Doppler effect of interest in astronomy?
1. It allows us to determine the line-of-sight velocities of astro-bodies.
2. It allows us to determine the velocity perpendicular to the line of sight of astro-bodies.
3. It is NOT important. This whole section is a pointless digression.



Answer 1 is right.

Question: The Doppler effect in astronomy is most useful for:
1. continuum spectra since you DO NOT know directly the shape of emitted spectrum if it has been Doppler shifted.

2. continuum spectra since you DO know directly the shape of emitted spectrum if it has been Doppler shifted.

3. line spectra since you know the position of the emitted spectrum lines since those are the positions one determines in the laboratory. In fact, since one can measure lab and astronomically-observed line positions very exactly, one can determine the relative velocity to an astro-body to very high accuracy often.



Answer 3 is right.

You can't know the shift of an observed continuum spectrum unless you know what the emitted spectrum was like in its own rest frame and usually you don't know that until after you have determined the relative velocity.

Determining the line-of-sight velocity of astro-bodies from the Doppler effect.

There are two similar wavelength shifting effects, that are often mistakenly called Doppler shifts, that turn up in astronomy:

1. The gravitational redshift---a negative gravitational redshift is a blueshift---which is discussed in IAWL Lecture 25: Black Holes.

2. The cosmological redshift which is discussed in IAWL Lecture 31: Cosmology.