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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. 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.
CONTINUOUS SPECTRUMA 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.
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.
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. 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.
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) 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).
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).
APPLICATIONS OF STELLAR SPECTRAStellar spectra can be used to deduce information about the temperature, chemical composition, density and rotational and translational velocity of stars.
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. 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) 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. 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. 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). 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.
PHOTOMETRY & PHOTOMETRIC METHODSA 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.
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.
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)
ABSOLUTE MAGNITUDEThe 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. 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)
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)
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.
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.
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