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9.2 Space
9.3 Motors & Generators
9.4 Ideas-Implementation
9.7 Astrophysics
9.8 Quanta to Quarks

9.7 Astrophysics Page 2
9.7 Astrophysics Page 3
9.7 Astrophysics Page 4

 

NOTE: This page is a continuation of the notes and worksheets for topic 9.7 Astrophysics.  Four separate pages were used for this topic because of the large volume of material in the topic.  This will keep download time within acceptable limits.

 

9.7 OPTION - ASTROPHYSICS CONTINUED PAGE 2

PREPARED NOTES

 

SPECTROSCOPY – A VITAL TOOL FOR ASTRONOMERS

NOTE: Numbers appearing in parentheses at the end of sentences or paragraphs refer to the references provided in the Bibliography at the end of these notes.

Spectroscopy is the systematic study of spectra and spectral lines.  Spectral lines are extremely important in astronomy, because they provide very reliable evidence about the chemical composition of distant objects, the temperature of objects, the density of objects and the motion through space of objects.  Let us now examine the nature of spectra and how they can be used to obtain the information mentioned above.

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SPECTRA

When a beam of sunlight is shone through a triangular glass prism, the white light is dispersed, producing a rainbow of colours, which can be displayed on a screen.  The rainbow of colours is called a spectrum.  In 1814 the German optician Joseph von Fraunhofer discovered that the spectrum of sunlight contains hundreds of fine dark lines, now called spectral lines.  Fifty years later, chemists found that they could produce spectral lines in the laboratory and use these lines to analyze the kinds of atoms of which substances were made.

There are three basic types of spectrum: a continuous spectrum, an emission spectrum and an absorption spectrum.  Let us now examine each in turn.

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CONTINUOUS SPECTRUM

A hot, glowing solid or liquid or a hot, glowing, dense gas produces a spectrum consisting of a continuous series of coloured bands ranging from violet on one end to red on the other (1).  In fact, this is just the visible section of the continuous spectrum of blackbody radiation, which we have previously studied in the Cosmic Engine & From Ideas to Implementation topics.   Examples of objects that produce continuous spectra include: an incandescent light globe, the inner layers of a star and galaxies.

Recall that a blackbody is a hypothetical body that is a perfect absorber and emitter of electromagnetic radiation.  At any temperature above absolute zero a blackbody emits as much energy as it absorbs.  The emitted radiation has a continuous distribution of wavelengths and the intensity of the spectrum at any given wavelength depends only on the temperature of the surface of the body (3).  A perfect blackbody does not reflect any light at all.  This is the reason why any radiation that it emits is entirely due to its temperature (3 & 8).

The intensity (energy density) versus wavelength curves for blackbody radiation should by now be very familiar to you, so they will not be repeated here.  See the curves on the Cosmic Engine page to refresh your memory if necessary.

The point that does need to be made however is that there is a very close correlation between the theoretical blackbody curves and the observed intensity curves for most stars (3).  Since the physics of blackbody radiation is well understood, this correlation allows a great deal of information about stars to be determined by studying their intensity versus wavelength curves.  More on this later.

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EMISSION SPECTRUM

This is a series of bright, coloured lines on a black background, produced by a hot, glowing, diffuse gas.  For example, the spectrum of hydrogen when heated to incandescence by passing an electric discharge through the gas is a series of four spectral lines (violet, blue, green & red) seen on a black background.  See diagram below.  Other examples of objects producing emission spectra include: Wolf-Rayet stars, emission nebulae such as planetary nebulae and quasars (5).

           

 

Note that the emission lines in the above diagram are not representative of the exact colour or line thickness of the real hydrogen emission lines.  The line thickness varies, with red normally being the broadest line and violet the thinnest.

To account for the production of emission spectra it is convenient to refer to the model of the atom, proposed by Niels Bohr in 1913.  In very brief summary Bohr’s model says that: (1) Electrons orbit a central positive, nucleus in certain allowed, circular, orbits called stationary states from which they do not radiate energy; and (2) Electrons only move from one state (orbit) to another by absorbing or emitting exactly the right amount of energy in the form of a photon, whose energy is equal to the difference in energy between the initial & final states, DE = hf (Planck’s formula).

It is the second point that concerns us here.  This explains how atoms emit and absorb specific frequencies of electromagnetic radiation.  An electron in its lowest energy state (called the ground state) can only jump to a higher energy state within the atom when it is given exactly the right amount of energy to do so by absorbing that energy from a photon of EM radiation of the right energy.  Once the electron has jumped to the higher level, it will remain there only briefly.  As it returns to its original lower energy level, it emits the energy that it originally absorbed in the form of a photon of EM radiation.  The frequency of the energy emitted will have a particular value and will therefore be measured as a single emission line of particular frequency and therefore of particular colour if it is in the visible region of the EM spectrum.

In summary then, emission spectra are produced when electrons within an atom make downward transitions from higher to lower orbits.  The energy lost by each electron is carried away by a single photon of set frequency and corresponds to a single spectral line in the emission spectrum (1).  The visible Balmer series of hydrogen emission lines shown above is produced by electrons dropping from higher orbits to orbit n = 2.

 

             

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ABSORPTION SPECTRUM

This is a series of dark spectral lines among the colours of the continuous spectrum.  These dark spectral lines represent wavelengths that are missing from an otherwise continuous spectrum.  Absorption spectra are produced when light from a hot source of continuous spectrum passes through a cooler, non-luminous, diffuse gas.  The spectra of normal cool stars such as the Sun fall into this category (5).  Note that the dark lines in the absorption spectrum of a particular gas occur at exactly the same wavelengths as the bright lines in the emission spectrum of the same gas.

See the diagram of the hydrogen absorption spectrum shown below.  Note that actual colours used in this diagram do not match exactly the colours in the real spectrum.  Also, the transitions between colours in the real spectrum are nowhere near as sharp as those depicted in the diagram.  The line thickness also varies in the real hydrogen absorption spectrum, with the alpha-line normally being the broadest and the delta-line being the thinnest.

           

 

To account for the production of absorption spectra it is again convenient to refer to the Bohr Model of the Atom.  Photons from the source of continuous spectrum (eg the Sun) pass through the cool diffuse gas (eg hydrogen) lying between the source and the observer on Earth.  The atoms of the gas will absorb those photons whose energies match allowed energy transitions within the atom.  The absorbed photons raise electrons within the atom from lower to higher orbits.  When these electrons drop back to their original orbits, they emit the energy they originally absorbed but in all directions not just in the original direction of motion of the photons.  Therefore these energies with their corresponding frequencies are effectively removed from the continuous spectrum observed on Earth and a series of black lines is seen in their place.  The Balmer series of hydrogen absorption lines shown above is produced by atoms absorbing photons that cause electrons to jump from orbit n = 2 to higher orbits.

Try the following website.  It allows you to call up the real emission and absorption spectra for all the elements in the Periodic Table.

http://jersey.uoregon.edu/vlab/elements/Elements.html

It is also worth noting that although the Bohr Model of the Atom gives a good description of the emission and absorption spectra of hydrogen, a more advanced model is needed to do the same for other elements.

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COMPARISON OF EMISSION & ABSORPTION SPECTRA WITH A CONTINUOUS BLACKBODY SPECTRUM

An emission spectrum is clearly very different to a blackbody spectrum.  An emission spectrum is a line spectrum with each individual coloured line corresponding to a particular wavelength (or frequency) of emission.  Between the lines there is darkness.  The blackbody spectrum is a continuous spectrum of colours from red to violet in the visible range.

An absorption spectrum is much closer in appearance to a blackbody spectrum than is an emission spectrum.  An absorption spectrum consists of an almost continuous background of colours from red to violet, with the continuity broken by black absorption lines representing wavelengths of radiation that have been absorbed somewhere between the source of radiation and the observer.

As mentioned earlier, a great deal of information about stars and other celestial objects can be obtained by comparing their spectra to blackbody spectra.  For example by comparing the intensity versus wavelength curves for real stars with those of a blackbody we can determine the surface temperature of stars.  By comparing a star’s absorption spectrum with a blackbody spectrum and noting the differences we can determine the chemical composition of the star.

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MEASURING ASTRONOMICAL SPECTRA

The technology required to measure astronomical spectra is an instrument called the spectrograph.  This is an optical device that is mounted at the focus of the telescope (3).  Its purpose is to diffract light into a spectrum so that the intensity at each wavelength can be recorded by a detector.  Spectrographs have been designed for use with various regions of the spectrum, with particular emphasis on the UV, visible and IR regions of the EM spectrum (5).

In a modern spectrograph, light from the telescope objective is focussed onto the entrance slits of the spectrograph.  It is then collimated by a mirror to produce a parallel beam and directed onto a diffraction grating.  A diffraction grating is a piece of glass onto which thousands of parallel, evenly spaced lines per millimetre have been ruled.  The diffraction grating causes the light from the telescope objective to be diffracted into a spectrum by the way in which light waves leaving different parts of the grating interfere with one another.  This spectrum is then passed through a corrector lens and focussed onto a charge-coupled device (CCD) that records the image.  A CCD is a silicon chip containing an array of light sensitive diodes used for capturing images.  The recorded spectrum is called a spectrogram.  (1 & 5)

Older spectrographs used a prism to disperse the light and produce a spectrum in place of the diffraction grating.  They also used photographic plates rather than CCD’s to record the spectrum (3).

If a photographic plate is used then the result is a spectrum similar to the examples of the hydrogen emission and absorption spectra shown earlier.  If a CCD is used the spectrum produced is in the form of a graph of intensity versus wavelength, in which absorption lines appear as depressions on the graph and emission lines appear as peaks.  (3)

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STELLAR SPECTRA AND THEIR CLASSIFICATION

As astronomers made more and more observations of the absorption spectra of stars during the nineteenth century, they discovered a bewildering variety of different spectra.  We know today that this is due to the fact that a star’s spectrum is profoundly affected by its surface temperature (3).  Every element has a characteristic temperature range over which it produces prominent absorption lines in the observable part of the spectrum (3).  For example, for the Balmer hydrogen lines to be prominent in a star’s spectrum, the star must be hot enough to excite the electrons out of the ground state but not so hot that all the hydrogen atoms become ionized.  A stellar surface temperature of around 9000 K produces the strongest hydrogen lines.  Different temperatures produce different degrees of excitation or ionization of the various atoms and molecules present in a star.  This in turn produces different strength absorption lines (3).

To bring order to the huge variety of different spectra they had found, astronomers grouped stars into a number of spectral classes that summarize features such as colour, surface temperature and chemical composition.  The order of these classes from highest to lowest temperature can be remembered using the mnemonic “Oh, Be A Fine Girl (or Guy), Kiss Me”.  See the Table below which has been taken from Kaufmann & Freedman p.470 (3).

Spectral

Class

Colour

Temperature

Spectral Line Features

Examples

O

Blue-violet

28000-50000

Ionized atoms especially helium

Mintaka

(d Orionis)

B

Blue-white

10000-28000

Neutral helium, some hydrogen

Rigel

(b Orionis)

A

White

7500-10000

Strong hydrogen, some ionized metals

Sirius

(a Canis Majoris)

F

Yellow-white

6000-7500

Hydrogen, ionized metals (Ca, Fe)

Canopus

(a Carinae)

G

Yellow

5000-6000

Both neutral & ionized metals, especially ionized Ca

Sun

K

Orange

3500-5000

Neutral metals

Aldebaran

(a Tauri)

M

Red-orange

2500-3500

Strong titanium oxide & some neutral Ca

Antares

(a Scorpii)

 

Note that astronomers use the term “metals” to refer to any element above helium in the Periodic Table (3).  Clearly, this is different to the term’s meaning in chemistry.

Each spectral class is further sub-divided into finer steps called spectral types.  These are indicated by adding a number from 0 to 9 after the appropriate letter for the spectral class.  The “0” is the hottest spectral type and the “9” is the coldest.  So we have for example, the spectral class F, which includes spectral types F0, F1, F2, ….. F9.

The modern stellar classification system also includes a luminosity class, which indicates for example, whether a star is a supergiant, giant or dwarf.  This is useful, since luminosity can vary widely within a spectral type.  See the H-R plot on the Cosmic Engine page and Ref.(1) pp.246-250 for more detail.

Luminosity classes Ia and Ib are composed of Bright Super Giant stars and Super Giant stars respectively.  Luminosity class V includes all main sequence stars.  The classes in between provide a useful means of distinguishing giant stars of various luminosities – Class II Bright Giant stars, Class III Giant stars and Class IV Subgiant stars.  Class VI consists of Subdwarf stars.  There is no luminosity class assigned to white dwarfs, since they represent a final stage in stellar evolution in which no thermonuclear reactions are taking place.  Consequently, white dwarfs are referred to only by the letter D (for dwarf).  (1 & 3)

On an H-R diagram astronomers describe stars by giving both a spectral class and a luminosity class.  The spectral class indicates the star’s surface temperature and the luminosity class its luminosity.  For instance, Aldebaran is a K5 III star, which means that it is a red giant with a luminosity around 500 times that of the sun and a surface temperature of about 4000 K.  The sun is a G2 V star, which means that it is a main sequence star of luminosity equal to the sun (obviously) and surface temperature of about 5800 K.  (3)

This two-dimensional classification scheme enables astronomers to locate a star’s position on the H-R diagram based entirely on the appearance of its spectrum.  This is very useful, since once the star’s absolute magnitude has been read from the vertical axis of the H-R diagram, the distance to the star can be calculated using a method called spectroscopic parallax as we shall see later (1).

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APPLICATIONS OF STELLAR SPECTRA

Stellar spectra can be used to deduce information about the temperature, chemical composition, density and rotational and translational velocity of stars.

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TEMPERATURE

Using a modern spectrograph to record the absorption spectrum of the star in question, a plot of intensity versus wavelength can be obtained for the star.  The wavelength, lmax, at which the energy output of the star is a maximum, can then be determined and Wien’s Displacement Law used to calculate the temperature of the star.

                                  

This measurement would also suggest the spectral class and composition of the star – see the Table above.

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CHEMICAL COMPOSITION

Each element produces its own unique pattern of spectral lines (3).  To identify the elements present in the outer layers of a star astronomers scan the absorption spectrum for that star for the unique patterns of spectral lines that correspond to particular elements. Thus, by analysing a star’s spectrum an astronomer can determine the star’s chemical composition.  (3)

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DENSITY

To determine the abundance or number densities of the atoms of elements present in a star, astronomers examine the line strengths of each spectral line.  This is done from an intensity versus wavelength plot of the star’s absorption spectrum.  Such a plot allows astronomers to study the shapes of individual spectral lines.  Basically, the line strength of a spectral line depends on the number of atoms in the star’s atmosphere capable of absorbing the wavelength in question.  For a given temperature, the more atoms there are, the stronger and broader the spectral line appears.  This effect is called pressure broadening – the greater the atmospheric density and pressure, the greater the broadening of the spectral lines.  So, by careful analysis of the star’s spectrum, an astronomer can determine the number density of atoms of a particular element in the star’s atmosphere.  (1 & 3)

Note that pressure broadening is responsible for the differences observed in spectral line widths between supergiant stars and main sequence stars like the Sun.  The narrower lines observed for the more luminous supergiant stars are due to lower number densities in their extended atmospheres.  Pressure broadening broadens the lines formed in the denser atmospheres of the main sequence stars, where collisions between atoms occur more frequently.  (1)

More specific detail on determining elemental abundances from stellar spectra is available in Ref (1) pp.293-306 and Ref (3) pp.471-472.  Most of this is way beyond the scope of the current Syllabus.

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THE DOPPLER EFFECT

Before looking at the velocity information that is contained in stellar spectra, it is necessary to understand the Doppler Effect.  This is the apparent change in the wavelength or frequency of a light wave when there is relative motion between the observer and the source of light.  Imagine that a source, S, of light waves is moving to the right as shown in the following diagram:

           

The circles represent the crests of light waves emitted from various positions as the source moves along.  Notice that as the source moves, the light waves emitted become crowded together in front of the source and spread out behind the source.  Hence, Observer 2 sees more wavelengths reach her in a set time period than would be the case if the source were stationary.  So, Observer 2 sees a higher frequency and shorter wavelength than if the source were stationary.  This means that the light seen by Observer 2 is bluer in colour than if the source were stationary.  So, the spectrum of light from an approaching source is blue shifted, that is all lines in the spectrum are shifted towards the short wavelength (blue) end of the spectrum.

By a similar argument Observer 1 sees a lower frequency and longer wavelength for the light than would be the case if the source were stationary.  This means that the light seen by Observer 1 is redder in colour than if the source were stationary.  So, the spectrum of light from a receding source is red shifted, that is all lines in the spectrum are shifted towards the long wavelength (red) end of the spectrum.

Note that motion perpendicular to the observer’s line of sight does not affect the wavelength.  Also, be aware that there is a mathematical formalism associated with the Doppler Effect that enables velocities of recession or approach to be calculated for sources of light from the observed wavelength shifts.  This is beyond the scope of the current syllabus.  Details can be found in many texts – eg Ref (3) pp.124-125.

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TRANSLATIONAL VELOCITY

Stars can move through space in any direction.  To calculate the translational velocity of a star through space (called its space velocity) astronomers measure two quantities.  Consider the following diagram as you read on. 

           

Firstly, astronomers determine the star’s radial velocity vr parallel to our line of sight.  This is calculated from measurements of the Doppler shifts of the star’s spectral lines.  Secondly, astronomers determine the star’s tangential velocity vt perpendicular to our line of sight – that is across the plane of the sky.  This is calculated from knowledge of the distance, d, to the star and the star’s proper motion, which is the number of arcseconds the star appears to move per year on the celestial sphere.  (3) 

Once the radial and tangential components of the star’s motion are known, the star’s space velocity v can be calculated by simple addition of vectors (3).  Note that the space velocity of a star is by definition its velocity relative to the Sun (5).  To calculate an accurate value for this, the component of the Earth’s velocity around the Sun that is parallel to our line of sight to the star must be subtracted from the star’s measured radial velocity (1).

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ROTATIONAL VELOCITY

To measure the rotational velocity of a star it is first necessary to obtain an intensity versus wavelength plot of the star’s spectrum.  As mentioned previously, this enables astronomers to study the shapes of individual spectral lines.

If a star is rotating, light from the side approaching us is slightly blue shifted, while light from the receding side is slightly red shifted.  As a result, the star’s spectral lines are broadened in a characteristic fashion.  By measuring the shape of the spectral lines astronomers can calculate the speed of rotation of the star.

A different case of rotation involves binary star systems.  Where two stars in a binary system have their orbital plane edge on to our line of sight, their speeds of rotation about the centre of mass of the system can be determined from their spectra using the Doppler Effect.  As the two stars move around they periodically approach and recede from us.  Hence, the spectral lines of the two stars will be alternately blue shifted and red shifted.  From the Doppler shifts, the velocities of approach and recession can be calculated and then used to calculate the orbital period and velocities of rotation of the two stars about the centre of mass of the system.

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PHOTOMETRY & PHOTOMETRIC METHODS

A photometer is an instrument capable of measuring the brightness of a light source.  In astronomy, photometry is the measurement of the apparent brightness of stars and other astronomical objects (3).  From such measurements distances to astronomical objects and the luminosities of those objects relative to some standard object (such as the Sun) can be determined.  We will focus on the use of photometry to determine the distance to astronomical objects.

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APPARENT MAGNITUDE

In Astronomy the magnitude scale is used to denote brightness.  Hipparchus invented the scale in the second century BC and modifications over time, especially in the nineteenth century AD, have produced what we now call the Apparent Magnitude Scale.  Apparent magnitude is a measure of the light arriving at earth and is directly related to apparent brightness.  Apparent magnitude describes how bright an object appears to be when seen by an observer on earth (3).  It is dependant on the distance to the object and the luminosity of the object, as well as on the presence of interstellar dust etc, which can make an object appear dimmer than it otherwise would be.  Apparent magnitude can be measured either photographically or photoelectrically. 

Hipparchus originally defined the brightest stars he could see with his naked eye as first magnitude stars.  Stars about half as bright were called second magnitude and son down to sixth magnitude stars, which were the dimmest stars Hipparchus could see with his naked eye.  With the invention of telescopes the scale had to be extended to take account of the much dimmer stars that could then be seen.  Also, as time went by stars even brighter than first magnitude were discovered.  So, Sirius, the brightest star in the night sky, ended up with an apparent magnitude of minus 1.43 (– 1.43). 

Be aware that when reading apparent magnitudes, the greater the apparent magnitude, the dimmer the star.  A star of apparent magnitude +2 (a second magnitude star) is dimmer than a star of apparent magnitude +1 (a first magnitude star).  The limit of vision with the naked eye is about apparent magnitude +6.  The Hubble Space Telescope can photograph stars of apparent magnitude +27 (more than 1020 times fainter than the sun) using very long exposure photography.  The Sun has an apparent magnitude of minus 26.8 (–26.8). 

In the nineteenth century more accurate measurements of the light energy arriving from stars revealed that a first magnitude star is actually about 100 times brighter than a sixth magnitude star.  In 1856, this led the English astronomer, Norman Pogson (1829-91), to define the apparent magnitude scale more precisely (5). 

Pogson defined that a magnitude difference of 5 corresponds exactly to a factor of 100 in brightness.  Therefore, a magnitude difference of 1 will correspond to a factor of (100)1/5 = 2.512 in brightness, since each change by 1 in magnitude must correspond to the same relative change in observed brightness (5).  So, the factor by which brightness changes each time the magnitude changes by 1 must by definition multiply itself 5 times and give a result of 100.  That is 2.512 x 2.512 x 2.512 x 2.512 x 2.512 = 100. 

A useful formula for calculating the brightness ratio of any two stars A and B is:

 

                                   

where mA = apparent magnitude of Star A (the brighter star), mB = apparent magnitude of Star B (the dimmer star) and (IA /  IB) = brightness ratio of the two stars. 

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Example Questions

1.      The Sun is the brightest object in our sky with an apparent magnitude of –26.8.  The giant star Canopus (Alpha Carinae) lying 310 light years (ly) from earth is the second brightest star in the night sky.  Canopus has an apparent magnitude of –0.74.  How much brighter does the Sun appear than Canopus?  (Answer: The Sun appears 2.7 x 1010 times brighter than Canopus.)

2.      Sirius (Alpha Ursae Majoris) lying 8.6 ly from earth is the brightest star in the night sky with an apparent magnitude of –1.43.  How much brighter than Canopus is Sirius?  (Answer: Sirius is about 1.9 times brighter than Canopus.)

3.      Algol is a star with apparent magnitude 2.1.  Given that the brightness ratio of Algol compared to Proxima Centauri, the closest star to earth, is 3600, determine the apparent magnitude of Proxima Centauri.  (Hint: You may need some help with logs from your Teacher.)  (Answer: 11)

 

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ABSOLUTE MAGNITUDE

The absolute magnitude of a star is defined as the apparent magnitude the star would have if it were located 10 parsecs (32.6 ly) from earth (3).  This quantity measures a star’s true energy output – its luminosity.  Again, the greater the absolute magnitude figure, the less luminous is the star.

Absolute magnitude removes the effect of different distances and allows us to compare the luminosities of stars with each other.  For instance, the sun appears to be the brightest object in the sky only because it is so close to us compared to all other stars.  As we have seen, it has an apparent magnitude of –26.8.  If the sun were placed 10 pc from earth, its apparent magnitude would be +4.8.  So, the sun’s absolute magnitude is +4.8.  The most luminous objects have absolute magnitudes around –10 and the least luminous around +15. 

An important equation that relates a star’s apparent and absolute magnitudes and its distance from us is:

 

                                   

where M = absolute magnitude of star, m = apparent magnitude of star and d = distance of star from Earth in parsecs.  The logarithm function used here is logs base 10.  This equation is often re-arranged using the basic rules of logarithms to give:

                                   

where (m – M) is called the distance modulus.  Clearly, if (m – M) is negative, then M > m and the star lies closer to us than 10 pc.  If (m – M) is positive, M < m and the star lies beyond 10 pc from Earth.

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Example Questions

1.      The star Epsilon Indi has an apparent magnitude of +4.7.  It is 3.6 pc from Earth.  Determine the absolute magnitude of this star.  (Answer: +6.9)

2.      The red supergiant Betelgeuse (Alpha Orionis) has an apparent magnitude of 0.5 and an absolute magnitude of –5.2.  Calculate the distance from Earth to Betelgeuse.  (Answer: 138 pc)

3.      Given that Betelgeuse has a parallax of 0.0076” re-calculate the distance from Earth to Betelgeuse using the relationship between parallax angle and distance.  (Answer: 131.6 pc – which is closer to the accepted value of 131 pc – see p.135 ref.11 or a similar reference.)  Note the difference with the answer to question (2).  The accuracy of the distance values calculated using the distance modulus formula depend on the accuracies of the apparent and absolute magnitudes substituted into the formula.

4.      Procyon (Alpha Canis Minoris) has an apparent magnitude of 0.38 and an absolute magnitude of 2.7.  Determine the distance from Earth to Procyon.  (Answer: 3.44 pc)

5.      The star Wolf 359 has an absolute magnitude of 16.55 and is located a distance of 2.39 pc from Earth.  Determine the apparent magnitude of this star.  (Answer: +13.44)

 

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SPECTROSCOPIC PARALLAX 

Spectroscopic Parallax is a very powerful technique that uses the Hertzsprung-Russell (H-R) Diagram and the distance modulus formula to determine the approximate distance to a star (3).

 

Spectroscopic parallax is performed as follows:

 

1.      Measure the apparent magnitude of the star using photometric methods (photographic or photoelectric).

2.      Determine the spectral type and luminosity class of the star from its spectrum.

3.      On an H-R Diagram locate the appropriate spectral type and luminosity class and then read off the absolute magnitude value for the star from the vertical scale.  Click on this link for an example of an appropriate H-R Diagram located on the Cosmic Engine page.  Use the back arrow of your Browser to get back here.

4.      Substitute the apparent and absolute magnitude values into the distance modulus equation and calculate the distance to the star.

 

Note that although this method is extremely useful, since it can determine the distance to a star no matter how remote that star happens to be, the distances obtained are only approximate.  At best they are accurate to ± 10% (3).  This is due to the fact that the luminosity classes on an H-R Diagram are not thin lines but are moderately broad bands.  This can lead to large percentage errors in the absolute magnitudes read from the H-R Diagram and corresponding large percentage errors in the distances calculated by this method.  It is worth noting, however, that sometimes this is the only method available to astronomers to give a ballpark figure for the distance to a remote star. (3)

 

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Example Question 

Use the method of spectroscopic parallax to determine the distance to the star Regulus (Alpha Leonis).  Regulus is a B7 V star (a hot, blue Main Sequence star).  Regulus has an apparent magnitude of 1.36.

SOLUTION: 

Using the H-R Diagram given earlier in these notes, we can determine that the absolute magnitude of Regulus is –0.2.  Thus, substituting the apparent & absolute magnitude values into the distance modulus equation, we obtain a distance to Regulus of 20.5 pc.  The accepted value for the distance to Regulus is 23.8 pc (p.135 of Ref. 11).  So, in this case, the method yields a reasonable approximation (about 14% error).  We could have been even more accurate by obtaining a more accurately drawn H-R Diagram.

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MEASURING A STAR’S COLOUR 

One factor that we have not yet mentioned that does have a bearing on how bright we perceive a particular star to be is the star’s colour.  Stars vary greatly in colour.  Betelgeuse is clearly pinkish-red to the naked eye while Rigel is clearly bluish.  These colours are due to the surface temperatures of the stars.  The intensity of light from a cooler star peaks at longer wavelengths, making the star appear red in colour.  The intensity of light from a hotter star peaks at shorter wavelengths, making the star appear blue in colour. 

Different detectors vary in their sensitivity to the coloured light emitted by stars.  The human eye is most sensitive to light in the yellow-green section of the visible EM spectrum.  Photographic film is most sensitive to light in the blue-violet region of the spectrum.  Thus, a blue star like Rigel for instance does not appear as bright to the human eye as it would appear as an image on photographic film.  Historically, the term “visual magnitude” came to be associated with magnitudes of stars determined by the naked eye, while the term “photographic magnitude” came to be associated with magnitudes determined using photographic emulsions. 

Eventually, photoelectric photometers, such as CCD’s (charge coupled devices), became available.  These are equally sensitive to all wavelengths of the spectrum.  A standardised set of coloured filters is used with these photometers.  The most commonly used filters are the UBV filters and the technique is called UBV Photometry.  This technique is extremely useful for providing an accurate measurement of star colours and a corresponding accurate determination of the surface temperatures of stars.  (3) 

Each of the three UBV filters is transparent in one of three broad wavelength bands: the ultra-violet (U) centred at 365nm wavelength, the blue (B) centred at 440nm and the central yellow (V, for visual) centred at 550nm in the visible spectrum.  The transparency of the V filter mimics the sensitivity of the human eye.  The B filter mimics the sensitivity of photographic emulsions.  The U filter makes the most of the extra sensitivity of available from modern photometers.  (1 & 3) 

In UBV photometry the astronomer measures the intensity of the starlight that passes through each of the filters individually.  This procedure produces three apparent magnitude values for the star, U, B and V as measured through each of the filters.  These apparent magnitude values are called the colour magnitudes of the star.  The astronomer then often subtracts one colour magnitude from another to produce what is called a numerical two-colour value that expresses the colour of the star.  The most common two-colour value is called the (B – V) colour index.  The (B – V) colour index of a star is the difference between the photographic magnitude of the star and the visual magnitude (3).

                                    Colour index = B – V

 

Such two-colour values are very useful because they produce a numerical scale that expresses the colour of stars in a precise manner.  By definition, stars that are of spectral type A0 and luminosity class V (such as Vega) have a colour index of zero (5).  These stars have surface temperatures of around 10 000K and are blue-white in colour (3).  Red stars, such as Betelgeuse, are brighter through the V filter than through the B filter, so their V magnitude is lower than their B magnitude.  Therefore, B – V for a red star is positive.  Blue stars, like Rigel, are brighter through a B filter than through a V filter, so their B magnitude is lower than their V magnitude.  Thus, B – V for a blue star is negative. 

For completeness, realise that there is also the (U – B) colour index, which is the difference between the ultra-violet and photographic magnitudes.  The details of this are not required by the current syllabus.

Once an accurate and precise determination of colour is made for a star, an accurate determination of surface temperature can be made. 

The following Table gives some idea of the range of the colour index scale and how it correlates with colour, spectral class and temperature.  This Table was compiled from Tables in Reference 3 p.470 & Reference 12 p.311.

Colour Index

Colour

Temperature

Spectral

Class

- 0.6

Blue-violet

28000-50000

O

 

 

Blue-white

10000-28000

B

 

0

White

7500-10000

A

 

 

Yellow-white

6000-7500

F

 

+ 0.6

Yellow

5000-6000

G

 

 

Orange

3500-5000

K

 

+ 2.0

Red-orange

2500-3500

M

 

 

The Table below gives some UBV magnitudes and colour indices for a number of stars.  This Table was taken from Reference 10 p.17. 

 

Magnitudes

Colour Index

STAR NAME

V

B

U

B – V

U - B

Altair

+ 0.77

+ 0.99

+ 1.07

+ 0.22

+ 0.08

Arcturus

- 0.06

+ 1.17

+ 2.44

+ 1.23

+ 1.27

Bellatrix

+ 1.64

+ 1.41

+ 0.54

- 0.23

- 0.87

Capella

+ 0.08

+ 0.87

+ 1.32

+ 0.79

+ 0.45

Regulus

+ 1.36

+ 1.25

+ 0.89

- 0.11

- 0.36

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Example Questions

1.      The stars Sirius and Betelgeuse have (B-V) colour indices of 0.0 and +0.86 respectively.  Explain which star has the lower surface temperature.

2.      Stars X, Y and Z have (B-V) colour indices of + 1.23, 0.0 and – 0.11 respectively.  Arrange the stars in order of increasing surface temperature and state which star could have a surface temperature of around 10 000 K.  Give reasons for your answers.

 

Answers 

1.      Lower temperature stars tend to be redder in colour than higher temperature stars. Red stars have (B – V) colour indices that are positive.  Betelgeuse has a more positive colour index than Sirius and is therefore redder in colour and lower in surface temperature.

2.      (B – V) colour indices range from negative values for hotter stars through to positive values for cooler stars.  The more positive the value, the cooler is the star.  Therefore the stars in order of increasing temperature are: X, Y and Z.  Star Y must have a surface temperature around 10 000K, since by definition stars with a (B – V) colour index of zero have surface temperatures of 10 000K.

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ADVANTAGES OF PHOTOELECTRIC TECHNOLOGIES 

As mentioned previously, photometry can be performed photographically or photoelectrically.  Photographic photometry uses photographic film to record the section of sky under study.  Photoelectric photometry uses filters in combination with CCD’s (charge coupled devices) or photomultiplier tubes to detect light signals and convert these into electrical signals that can be multiplied, digitised, analysed and stored electronically.

 

The advantages of photoelectric devices in photometry are:

1.      A greater range of wavelengths to which they are sensitive than is the case for photographic film;

2.      When used with appropriate filters they can detect intensities over broad wavelength bands, as in UBV photometry, or over very narrow bands, as when searching for a particular element in an astronomical object; and

3.      They are more sensitive to faint light sources than is photographic film;

4.      They can be controlled remotely;

5.      They can feed the data obtained straight into a computer for immediate viewing and analysis.  (3)

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BIBLIOGRAPHY

 

1.      Carroll, B.W., & Ostlie, D.A. (1996).  "An Introduction To Modern Astrophysics", New York, Addison-Wesley Publishing Company Inc.

2.      Gondhalekar P. (2001). "The Grip of Gravity", Cambridge, Cambridge University Press

3.      Kaufmann, W.J. III, & Freedman, R.A. (1999).  "Universe", (5th Edition), New York, W.H. Freeman & Company

4.      "Sidereus Nuncius or the Sidereal Messenger". Translated with introduction, conclusion, and notes by Albert Van Helden. Chicago: University of Chicago Press, 1989

5.      Ridpath, I. (Ed.) (1997).  "Oxford Dictionary of Astronomy", Oxford, Oxford University Press

6.      Hollow, R.  "Why Build Big Telescopes?", paper presented at Science Teachers Workshop 2002

7.      http://www.eso.org/projects/aot/introduction.html and http://www.ls.eso.org/lasilla/sciops/ntt/telescope/esontt.html

8.      Eisberg, R. & Resnick, R. (1974).  "Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles", Canada, Wiley

9.      Playoust, D.F., & Shanny, G.R. (1991).  "An Introduction to Stellar Astronomy", Queensland, The Jacaranda Press

10. Bhathal R., (1993).  "Astronomy for the Higher School Certificate", Kenthurst, Kangaroo Press Pty Ltd

11.  Dawes, G., Northfield, P. & Wallace, K. (2003).  "Astronomy 2004 Australia – A Practical Guide to the Night Sky", Australia, Quasar Publishing

12.  Andriessen, M., Pentland, P., Gaut, R. & McKay, B. (2001). "Physics 2 HSC Course", Australia, Wiley

13. Schilling, G. (2004). "Evolving Cosmos", Cambridge, Cambridge University Press

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