An Introduction to Astronomy
Part X: Properties of Stars
Lambert E. Murray, Ph.D.
Professor of Physics
Starlight: The Key to the Universe
 Everything
we can know about a distant
star, we determine from the light that
reaches us from that star.
 By analyzing the light from distant stars,
astronomers can determine:
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Distances to Stars
Surface Temperatures and Luminosities
The Motions of Stars
Stellar Masses
Stellar Chemistries
Time and Distance
Light travels at a rate of 3 x 108 m/sec, so when
we see a star, we are actually seeing the light that
was emitted from that star some time in the past.
 The nearest star, Proxima Centauri, is about 4.3
light years (or 4.08 x 1013 m) away from our Sun,
which means it takes light 4.3 years to reach us
from that star. Thus, if the star were to explode
today, we would not be aware of it for 4.3 years!
 Similarly, when we observe the light from a
distant star we are looking at that stars
characteristics some time in the distant past – that
star may not ever exist today!

Stellar Spectra
 By
studying the spectrum of starlight we
can determine several things about a star:
– The stars surface temperature
– The rate at which energy is being emitted from
the star per surface area
– The chemical make-up of the star (at least its
outer atmosphere)
– The velocity of the star toward or away from us
based upon the Doppler shift.
Color and Temperature
We have already discussed the relationship
between the color of a star and the stars
temperature.
 The blackbody spectrum of a star’s continuous
spectrum can be used to determine surface
temperature of the star.
 In addition, once the surface temperature is
known, the rate at which energy is emitted from
the star can be determined, using the StephanBoltzmann equation, if we know the size of the
star.

The energy radiated by a
blackbody takes a special
form when the relative
brightness (or strength) of
the radiation is graphed as
a function of the
wavelength. Hotter objects
emit more light (the total
area under the curve) and
this light is peaked at
shorter wavelengths.
From a Star’s spectrum, we can determine the star’s
chemical make-up.
The Doppler Effect
The Doppler Effect can be used to obtain a
star’s radial velocity by measuring the shift
in wavelengths of certain spectral lines.
v 

c 
 is the reference wavelength in the laboratory, c
is the speed of light, and  is the measured shift
in the spectral line from the star.
Doppler Shifts can also be used to determine the rate of rotation of a few
nearby, large stars.
Stellar Parallax: A Direct Method
to Determine the Distance to Stars
This method, also known as triangulation, uses simple
mathematical ratios to determine the distances to stars
(see next slide).
 It is the only direct method of determining stellar
distances and can be used only for stars within about
500-1000 light years of our Sun.
 The ancients were not able to see any stellar parallax
because the stars are so far away.
 To determine the distance to more distant stars,
indirect methods must be used.

Parallax Formula
 Distance
in Parsecs (pc) = 1/parallax angle
measured in seconds of arc (arc seconds)
 1 Parsec (pc) = 3.3 light years (ly)
 The parallax angle shift of the nearest star is
the angular size of a dime measured at a
distance of 2.4 km – or about 0.77 arc
seconds. This corresponds to about 4.3 ly.
Limitation of the Parallax
Method
From the Earth’s surface, parallax measurements
can be made accurately only for stars within about
100 parsecs. This is because turbulence in the
atmosphere interferes with the accuracy of small
angle measurements.
 From above the atmosphere (using satellites such
as Hipparcos) we have been able to accurately
determine the distance to stars out to about 250
parsecs. Distances to an accuracy of better than
10% have been obtained for stars out to 150
parsecs (approx. 500 ly).

Proper Motion of Stars
Some stars are observed to move relative to the
background stars over a period of time (this is in
addition to the apparent back-and-forth motion
due to the Earth’s orbit about the Sun).
 This motion is called proper motion, m, and is
measured in arcseconds/year.
 Barnard’s Star exhibits the largest proper motion
of any star – 10.25 arcseconds/year.

Proper Motion for Barnard’s Star – The Largest Observed
Proper Motion for any Star – is 10.25 arcsec/yr
The actual motion of a star
is a combination of the
proper motion and the
radial motion.
Stellar Magnitudes
In the second century B.C., Hipparchus created the
first star catalog in which he tabulated the location
and apparent magnitude of a large number of
stars.
 He designated the brightest stars (the first order
stars) as having a magnitude of +1, while the next
brightest stars (the second order stars) had a
magnitude of +2, etc.
 The dimmest stars visible to the naked eye are
approximately +6 magnitude stars.
 Modern telescopes can image stars as faint as
+25th magnitude.
 Using this scale, brighter objects, like the Sun,
have a negative magnitude.

The Luminosity of a Star
The luminosity is the total amount of radiation
emitted from the star.
 The luminosity depends upon the temperature and
the size of the star according to the equation:

L = T4A =4pR2T4
L is the Luminosity (total energy radiated/sec)
T is the temperature in Kelvin
 is a constant (5.67 x 10-8 watts/m2K4)
A is the surface area of the radiating body
Brightness of Stars
 The
apparent magnitude (or brightness) or a
star depends upon two things:
– The actual brightness of the star – its
luminosity, and
– The distance to the star
 The
Luminosity of a star depends upon two
things:
– The surface temperature of the star, and
– The size of the star
Relationship between Luminosity
and Apparent Magnitude
Since stars are at different distances from the
Earth, the apparent magnitude depends upon that
distance and the actual luminosity, L, of the star.
 The apparent magnitude M (sometimes called the
brightness) is given by:
1 L
M
4p R 2
where R is the distance from the Sun to the Star.

Absolute Magnitude
 The
absolute magnitude Mo of a star is
defined as the apparent magnitude of a star
if that star were a standard distance (10
parsecs) from our Sun.
 If we can determine a stars distance (by
parallax) and the stars apparent magnitude,
we can then determine by using ratios the
absolute magnitude of that star.
Spectral Classification of Stars
Another method of classifying stars is based upon
the absorption lines observed in the spectra of the
stars. A classification scheme was developed
using the letters:
O B A F G K M
 It turns out that this spectral classification is
essentially equivalent to the temperature or color
classifications, the O stars being the hottest
(bluest), and the M stars being the coldest
(reddest). This classification scheme is further
subdivided into subunits (e.g., B0 – B9, etc.)

Spectral Classification of Stars
Hertzsprung-Russel Diagrams
Around 1910, attempting to gain a better
understanding of the different types of stars, Ejnar
Hertzsprung, from Denmark, plotted the star’s
temperature (or color) versus the star’s absolute
magnitude.
 At about the same time, Norris Russell (USA)
made a similar plot of spectral types versus
absolute magnitude.
 These plots, which are essentially identical,
showed a surprising relationship between the
temperature (color or spectral type) of a star, and
the star’s absolute magnitude (or luminosity).
 Sample HR diagrams are shown on the next few
slides.

Observations from the HR Diagram

The stars are not found to be randomly distributed
across this diagram – on the contrary, they seem to
fall into groups:
– A majority of the stars (about 90%) fall along a
diagonal; from hot, luminous stars to cool, dim stars.
This diagonal is called the main sequence, and stars
falling along this line are often referred to as dwarf
stars.
– Another group of very luminous, yet cool (red) stars
occurs in the upper right corner – these stars must be
very large, and are known as red giants.
– Yet another grouping of stars occurs in the lower left
corner – these are hot, dim stars, known as white
dwarfs.

This observation turns out to be significant in
understanding the life cycle of stars.
Stellar Radii
 Since
stars act like blackbodies, we can use
our equation for the luminosity of a star to
determine the radii of different Stars.
L = T4A =4pR2T4
 Using
this equation, we can plot stellar radii
on the HR diagram on the next slide.
Hertzsprung-Russell Diagram with Stellar Radii
Dwarfs, Giants, and SuperGiants
Notice that many of the main sequence stars have
a radius about equivalent to the Sun’s radius. (The
cooler main sequence stars are all somewhat
smaller.)
 White dwarfs are all about 100 time smaller that
our Sun (i.e., about the size of the Earth).
 The giants and super-giants are from 10 to 100
times larger that our Sun. Betelgeuse, for
example, is about 370 times larger that our Sun,
with a radius of about 2 A.U. (larger than the orbit
of Mars).

Luminosity Classes
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In addition to the spectral classification OBAFGKM, which is
related closely to temperature, astronomers have subdivide
stellar spectra into luminosity classes which are related more to
the size (or surface density) of a star.
For example, a B8 supergiant spectra exhibits narrower spectral
lines than a main sequence B8 star.
This is because the density at the surface of the supergiant star is
less than the main sequence star, and the spectral line exhibits
less pressure broadening.
Astonomers have developed 5 luminosity classes, in order of
decreasing luminosity: I, II, III, IV, and V, with I signifying
supergiants and V main sequence stars.
Thus, the complete spectrascopic designation of our Sun is that
it is a G2V star, designating the Sun’s surface temperature and
luminosity.
Luminosity
Classes
The luminosity classification
is based upon the absorption
line spectrum of the star.
Spectroscopic Parallax
The HR diagram can actually help us to determine
the distance to stars whose actual parallax cannot
be measured.
 If the stars temperature (color) is known, and if its
spectral class can be determined from its spectra,
the stars luminosity (or absolute magnitude) can
be determined.
 Knowing the absolute magnitude (the maganitude
at 10 parsecs) and its apparent magnitude (as seen
from the Earth) one can calculate, based on the
1/R2 law, the distance to the star.
 This technique is known as spectroscopic
parallax.

Spectroscopic
Parallax
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By looking at the
continuous spectrum, we
can determine the “color”
of the star.
By looking at the
absorption line spectrum,
we can determine the
spectral class of the star.
From these two, we can
determine the absolute
magnitude of the star –
provided it is properly
placed on the HR diagram.
Binary Stars and Stellar Masses
The masses of individual or bachelor stars cannot
be measured by any current techniques.
 However, more than 50% of all stars are part of
multiple star systems (the majority are binaries).
 This means we can make use of Newton’s law of
motion and gravity to determine the masses of the
stars based upon the period of revolution and the
the distance between the stars.

Multiple Star Systems
 Optical
doubles are two stars that have
small angular separation as seen from Earth
but are not gravitationally linked.
 A binary star system is a system of two
stars that are gravitationally linked so that
they orbit one another.
 A visual binary is an orbiting pair of stars
that can be resolved (normally with a
telescope) as two stars.
 If one uses large telescopes, about 10% of
the stars in the sky are visual binaries.
Visual Binary
The Orbit of 70
Ophiuchi, a faint
double star in
constellation
Ophiuchus.
The orbit is plotted
with the more
massive star held
fixed.
Recall that both stars
actually orbit the
common center of
mass.
Mass of Binary Stars

This last equation determines the total mass of the
binary system in terms of the period of revolution
and the distance between the stars (which can be
determined only if we know the distance to the
binary system). Once the total mass of the binary
system is determined, the mass of each individual
star can be determined by observing the motion of
each star relative to the center of mass, since:
M1R1  M 2 R2
Spectroscopic Binaries
 Many
binary (or other multiple) star
systems are too close together or too far
from the earth for the individual stars to be
resolved.
 Fortunately, many binaries that are visually
unresolved, can be detected from the
Doppler shift of their spectra. Such binaries
are called spectroscopic binaries.
A Double-Line Binary
This is the spectra of the same star at two different
times. Notice the simultaneous red and blue shifts!
Blue Shifted Lines
Red Shifted Lines
Non-Shifted Lines
Careful measurements
of the Doppler shifts
gives a radial velocity
curve from which one
can determine the
periods of the orbits.
Eclipsing Binaries
 Algol,
discovered by Goodricke in 1783, is
an eclipsing binary, in which one star moves
in front of the other as viewed from Earth.
 Algol’s light curve—a graph of the
numerical measure of the light received from
a star versus time—shows peaks and dips that
indicate an unseen companion.
Mass –
Luminosity
Curve
Data from binary stars
enables us to determine
the mass of various stars.
If we plot the mass of
these stars against their
luminosity we find
another interesting
pattern.
Mass - Luminosity Data Plotted
on an HR Diagram
This plot looks almost
like a plot of main sequence
stars. We will notice this
again when we talk about
the birth of stars.
End of Part X