parallax method / brightness / spectroscopic method / cepheid variables
Parallax Method
The parallax method can be used for the nearest stars to about 100 parsec (see below for definition of parsec). The parallax method relies on the apparent movement of the stars against the background of further stars as the earth orbits the sun.
The parallax movement are very small and are measured in seconds (3600 seconds = 60 minutes = 1 degree = 1/360 of a full circle).
The apparent movement of the star relative to the background of further stars being measured in seconds of an arc leads to the unit Parsec (or per second).
The parallax angle θ is observed and measured as the star position changse over the period of a year. From basic trigonometry and knowing the distance of Earth to the sun, we can work out the distance from the Sun using the following formula:
where θ is the parallax angle
As for small angles tanθ = sin θ = θ in radians
Therefore in the case of using parallax to measure stellar distances,
where d is distance to the Star
This would mathematical be equivalent to
The constant in the equation can be made equal to one by a careful adoption of an appropriate distance unit so that
Calculations show that this will be the case if d is measured in units equivalent to 3.08 x 10^{16} m and this defines the distance unit as one parsec (parallax angle of one second).
In other words:
Therefore to measure the distance to the star, observation will be made of the star “movement” against fixed back of further stars over the period of a year. The movement measurement in seconds of arc can them be applied to the simple formula as described above of
and that produces the distance as measured in parsecs.
Due to the difficulty of measuring small parallax angles, this method is not reliable to stellar distance above 100 parsecs. Parsecs, however, continue to be used for other techniques for measuring greater distances with kpc and Mpc being used for 10^{3} and 10^{6} parsecs respectively. These other techniques are dealt with below.
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Brightness
The apparent magnitude (m) of a star is a measure of its apparent brightness as seen by an observer on Earth. The brighter the object appears, the lower the numerical value of its magnitude. Absolute magnitude is a measure of luminosity, how much light a star radiates into space. Absolute magnitude can be defined as apparent magnitude a star would have 10 pc away from Earth. A star that looks just as bright as one close to us, but being further away has a greater absolute magnitude. The apparent magnitude represents the apparent brightness and the absolute magnitude the absolute luminosity. The absolute magnitude of a star is the apparent magnitude that it would have if it were observed from a distance of 10 parsecs.
The magnitude scale has been used by astronomers for more than 2000 years to classify stars into 6 categories of brightness as it appears to the human naked eye, with magnitude 1 being the brightest and magnitude 6 being the faintest. With the use of telescope and more sensitive instruments, stars beyond magnitude 6 can now be more accurately measured.
The magnitude scale is now defined as magnitude 1 being 100 times brighter than magnitude 6, with the scale being logarithmic. Negative values for stars brighter than magnitude 1 is also allowed. With the difference of magnitude 1 and 6 being defined a 100, this means the each unit decrease of the magnitude scales corresponds to 2.512 times brighter (as 2.512^{5}=100).
For example, to compare the power received from the two stars Sirius, with apparent magnitude 1.46, and Betelgeuse, with apparent magnitude 0.5, raise 2.512 to the difference of the apparent magnitude.
Difference in apparent magnitude of the two stars = 0.5 – (1.46) = 1.96
Therefore power received from Sirius/power received from Betelgeuse = 2.512 x 1.96
While the apparent magnitude described above is the brightness apparent to an observer on Earth, absolute Magnitude is defined as the apparent magnitude of a star at a fixed distance of 10 parsec. Since the apparent brightness of a star depends on the absolute brightness and distance of the star, the relationship is given by the following formula known as the distance modulus.
where M is the absolute magnitude
and m is the apparent magnitude
and d is distance measured in parsec
If the luminosity of a star is known, the inverse square law relating apparent brightness, luminosity and distance can be applied to estimate the distance to the star. The formula being:
b is apparent brightness (m)
L is luminosity (W m^{2})
d is distance (m)
r is radius of star (m)
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Spectroscopic Method
The term spectroscopic parallax is a misnomer as it actually has nothing
to do with parallax. It is, however, a way to find the distance to stars.
Most stars are too far away to have their distance measured directly using
trigonometric parallax but by utilising spectroscopy an approximate
distance to them can be determined. Let us see how this works.

If we take a spectrum of a star we can determine its spectral class.

Knowing either the star's spectral class allows us to
place the star on a vertical line or band along a HertzsprungRussell
Diagram. If we also know its luminosity class we can further constrain
its position along this line, that is we can distinguish between a red
supergiant, giant or main sequence star for example.

Once we know its position on the HR diagram we can infer what its absolute
magnitude, M should be by either reading off across to the vertical
scale of the HR diagram or looking it up from a reference table. A main
sequence (luminosity class V) star with a colour index of 0.0
has an absolute magnitude of +0.9 for example.

Now knowing m from measurement and inferring M we can use the distance
modulus equation:
m  M = 5 log(d/10) to find the distance to the star, d, in parsecs.
The assumption made is that the spectra from distant stars are the same as spectra from nearby stars. The spectra enables us to put the star into a spectral class. If this is the case, the HR diagram can be used to estimate the luminosity of a star that is far away.
In practice this technique is not very precise in determining the distance
to an individual star. Uncertainties in the absolute magnitude of stars of
specific spectral and luminosity class range from about 0.7 up to 1.25 magnitudes.
These then give a factor of 1.4 to 1.8 × variation in the resultant
distance. Nonetheless it is still an important methods for estimating distance
to stars beyond direct trigonometric parallax measurement.
Example of Spectroscopic Parallax Calculation:
γ Crucis is an M3 III star with a measured value of m_{V}
= 1.63 and a colour index of +1.60. This means that it is a red giant. Plotting
its position on the HR diagram below we can estimate its absolute magnitude
to be about 0.8. In fact if we look up a standard reference table we find
the the absolute magnitude for a III luminosity class star with a colour index
of +1.60 is 0.60.
Gamma Crucis is an M3.5 III star, a red giant. Using
its spectral and luminosity classes we can place it where the red circle
is on the HR diagram. Reading across to the vertical axis this corresponds
to an absolute magnitude of about 0.8.
Now if we use the tabulated value of M = 0.60 with the distance modulus
equation (4.2) we have:
M = m  5 log(d/10) so;
5 log (d/10) = m  M
log (d/10) = (m  M)/5
d/10 = 10 ^{(m  M)/5}
d/10 = 10 ^{(m  M)/5}
d = 10*10 ^{(m  M)/5}
which can be written as:
d = 10 ^{(m  M + 5)/5}
now substituting in:
d = 10 ^{(1.63  (0.60) + 5)/5}
d = 10 ^{7.23/5}
d = 10 ^{1.446}
d = 27.9 parsecs
so γ Crucis is about 28 pc distant which is within 1 pc of the published
Hipparcos value. If we used the graphically obtained estimate value of M ≈
0.8 then:
d = 10 ^{(1.63  (0.8) + 5)/5}
d = 10 ^{7.43/5}
d = 10 ^{1.486}
d = 30.6 parsecs
so γ Crucis would have a value of about 31 pc distance, about a 15% error.
Although this method is not accurate for individual stars, if carried out for
many stars it can yield statistically useful values.
This method involves quite a lot of uncertainty. Matter between the star and the observer (for example, dust) can affect the light that is received. It would absorb some of the light and make the star's apparent brightness less than it should be. In addition, dust can scatter the different frequencies in different ways, amnking the identification of spectral class harder.
As the stellar distance increases, the uncertainty in the luminosity becomes greater and so the uncertainty in the distance calcualtion becomes greater. Forthis reason, spectroscopic parallax is limited to measuring stellar distances up to about 10 Mpc.
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Cepheid Variables
Some types of pulsating variable stars such as Cepheids exhibit a definite
relationship between their period and their intrinsic luminosity. Such periodluminosity
relationships are invaluable to astronomers as they are a vital method in
calculating distances within and beyond our galaxy. Cepheid variables are quite rare stars whose outer layers undergo periodic compression and contraction and this produces a periodic variation in its luminosity.
Discovery of the PeriodLuminosity Relationship
During the first decade of the 1900s Henrietta Leavitt (1868  1921), working
at the Harvard College Observatory, studying photographic plates of the Large
(LMC) and Small (SMC) Magellanic Clouds, compiled a list of 1,777 periodic
variables. Eventually she classified 47 of these in the two clouds as Cepheid
variables and noticed that those with longer periods were brighter than the
shorterperiod ones. She correctly inferred that as the stars were in the
same distant clouds they were all at much the same relative distance from
us. Any difference in apparent magnitude was therefore related to a difference
in absolute magnitude. When she plotted her results for the two clouds she
noted that they formed distinct relationships between brightness and period.
Her plot showed what is now known as the periodluminosity relationship;
cepheids with longer periods are intrinsically more luminous than those with
shorter periods.
The Danish astronomer, Ejnar Hertzsprung (18731967) quickly realised the significance
of this discovery. By measuring the period of a Cepheid from its light curve,
the distance to that Cepheid could be determined. He used his data on nearby
Cepheids to calculate the distance to the Cepheids in the SMC as 37,000 light
years away.
Harlow Shapley, an American astronomer using a larger number of Cepheids, recalibrated
the absolute magnitude scale for Cepheids and revised the value of the distance
to the SMC to 95,000 light years. He also studied Cepheids in 86 globular
clusters and found that the few dozen brightest nonvariable stars in each
cluster was about 10 × brighter than the average Cepheid. From this
he could infer the distance to globular cluster too distant to have visible
Cepheids and realised that these clusters were all essentially the same size
and luminosity. By mapping the distribution and distance of globular clusters
he was able to deduce the size of our galaxy, the Milky Way.
In 1924 Edwin Hubble detected Cepheids in the Andromeda nebula, M31 and the
Triangulum nebula M33. Using these he determined that their distances were
900,000 and 850,000 light years respectively. He thus established conclusively
that these "spiral nebulae" were in fact other galaxies and not
part of our Milky Way. This was a momentous discovery and dramatically expanded
the scale of he known Universe. Hubble later went on to observe the redshift
of galaxies and propose that this was due to their recession velocity, with
more distant galaxies moving away at a higher speed than nearby ones. This
relationship is now called Hubble's Law and is interpreted to mean
that the Universe is expanding.
Calculating Distances Using Cepheids
Both types of Cepheids and RR Lyrae stars all exhibit distinct periodluminosity
relationships as shown below.
Periodluminosity relationship for Cepheids and RR Lyrae stars.
Let us now see how this relationship can be used to determine the distance
to a Cepheid. For this procedure we will assume that we are dealing with a
Type I, Classical Cepheid but the same method applies for W Virginis
and RR Lyraetype stars.

Photometric observations, be they nakedeye estimates, photographic plates,
or photoelectric CCD images provide the apparent magnitude values for
the Cepheid.

Plotting apparent magnitude values from observations at different times
results in a light curve such as that below for a Cepheid in the LMC.

From the light curve and the photometric data, two values can be determined;
the average apparent magnitude, m, of the star and its period
in days. In the example above the Cepheid has a mean apparent magnitude
of 15.56 and a period of 4.76 days.

Knowing the period of the Cepheid we can now determine its mean absolute
magnitude, M, by interpolating on the periodluminosity plot.
The one shown below is based on Cepheids within the Milky Way. The vertical
axis shows absolute magnitude whilst period is displayed as a log value
on the horizontal axes.
The log of 4.76 days = 0.68. When this is plotted a value of about 3.6
results for absolute magnitude.

Once both apparent magnitude, m, and absolute magnitude, M
are known we can simply substitute in to the distancemodulus
formula and rework it to give a value for d, the distance
to the Cepheid.
as you should recall,
this can be rewritten as:
now substituting in:

In practice astronomers would try and observe as many Cepheids as possible
in another galaxy in order to determine a more accurate distance. As the
number of stars observed go up the uncertainties involved in calculations
for individual stars can be statistically reduced.
The basic steps, therefore for distance calculation using pulsating variables
are straightforward though the detail makes it harder in practice.
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