Chapter 13
Taking the Measure of Stars
The material of Chapter 13 introduces the fundamental
ideas about how stars are studied, in particular, how it
is possible to learn incredibly detailed information about
individual stars simply from the light they emit.
1. Examine how one determines the distance, luminosity,
size, mass, temperature, and chemical composition of
individual stars.
2. Organize the properties of stars using the classical
Hertzsprung-Russell (H-R) diagram, and explore the
large range of properties that differentiate one star from
another.
Determining distance using triangulation.
The concept of stellar parallax.
The angle p is smaller as distance becomes larger.
How astronomers
measure parallax
angle. They
observe star
fields 6 months
apart and
compare the
positions of stars
with each other.
All stars close enough
display parallax.
Distance is measured
by inverting the
parallax angle.
Distance = 1/angle.
A new measure of distance is
introduced: the parsec. A
parsec is the distance at which
1 Astronomical Unit subtends
an angle of 1 second of arc
(arcsecond).
1 parsec = 3.26 light years
The image at right
demonstrates parallax.
The Sun is visible
above the streetlight.
The reflection in the
water shows a virtual
image of the Sun and
the streetlight. The
location of the virtual
image is below the
surface of the water
and thus offers a
different vantage point
of the streetlight,
which appears to be
shifted relative to the
stationary, background
Sun.
The Hipparcos telescope of ESA.
The nearest star (beyond the Sun !) has a parallax
of only 0.769 arcsecond. Parallaxes smaller than
0.010 arcsecond become increasingly difficult to
measure and those smaller than 0.001 arcsecond
are often erroneous. Other techniques are used to
derive distances for more distant objects.
Other techniques:
Proper motion (rate stars move across the sky)
Cluster parallax (main-sequence fitting)
Spectroscopic parallax (spectrum gives L, T)
are some of the few.
The proper motion of a star refers to its annual
displacement in the sky relative to a fixed coordinate grid. Proper motion angles are much
larger than parallax angles.
Barnard’s Star, an
example of a star
that has a very large
proper motion
across the sky. The
images were taken a
few years apart.
Measuring stellar luminosity using a star’s
spectrum. For hot stars the hydrogen lines are
broad (wide) in dwarfs but much narrower in
giants and supergiants.
The magnitude system.
As invented by the astronomer Hipparchus 2200
years ago, it was simply a way to “rank” the stars
visible at night. The brightest were ranked as 1st
magnitude, the faintest visible were ranked as 6th
magnitude. In other words, the brightest stars
were assigned the smallest number, the faintest
the largest number. And 6 divisions were used
because of the mysticism about 6, which is the
first “perfect number.”
The scheme was only later put on a formal basis
in which a brightness difference of 5 magnitudes
corresponds to a brightness ratio of 100. Note
that the scheme is “inverse logarithmic.”
Magnitudes relate to the logarithm of star
brightness, and go the “wrong way.”
The magnitude system.
Magnitudes for Pleiades stars.
A chart for M67 used by astronomers to establish
the limiting magnitude for their telescope system.
If brightness (in magnitudes) and distance are
known, it is straightforward to establish the
intrinsic brightness of a star. Astronomers use
another magnitude scale for that, called absolute
magnitude. Apparent magnitude is designated as
m, absolute magnitude as M.
Absolute magnitude is defined as the brightness
at which an object would appear if placed at a
distance of 10 parsecs. As such, M is a measure of
a star’s luminosity.
Another magnitude used is colour index, the
difference between a star’s brightness (in
magnitudes) in different colour bands, usually
blue and yellow, for example b–y (on the Johnson
system it is written B–V).
Recall from Chapter 4 the sequence of spectral types
used to establish the surface temperatures of stars.
Stellar spectra are also used to measure
abundances for the different elements in stars, at
least in their outer regions.
Results:
Element
Hydrogen
Helium
Oxygen
Carbon
Neon
Nitrogen
Iron
All others
No. Percentage
92.5%
7.4%
0.064%
0.039%
0.012%
0.008%
0.003%
0.001%
Mass Percentage
74.5%
23.7%
0.82%
0.37%
0.19%
0.09%
0.16%
0.03%
Summary on a logarithmic scale.
The elements lithium, beryllium, and boron are
of extremely low abundance because they are
destroyed in star interiors.
Even-numbered nuclides come from collisions
with helium nuclei (He), so are more common.
Astronomers normally state for simplicity that
the universe is 75% hydrogen (H) by mass, 24%
helium (He), and 1% everything else. They also
designate that “everything else” as “metals,” even
though only a portion of the elements involved
are metallic.
Another way of stating that is:
X = 0.75, Y = 0.24, Z = 0.01.
If number abundance is considered instead, the
composition of stars and the universe is 92½%
hydrogen (H), 7% helium (He), and ~½% metals
(everything else).
Astronomical Terminology (so far)
Parallax. The half-yearly angular displacement of a star
in the sky created by our orbit about the Sun =
similar to effects of stereoscopic vision in humans.
Parsec. A measure of distance for an object that has a
parallax of 1 arcsecond = 3.26 light years.
Magnitude, m. A scale developed by Hipparchus to rank
the naked-eye stars in terms of brightness.
Luminosity. The rate at which a star emits light, often
measured using absolute magnitude.
Absolute Magnitude, M. A measure of luminosity,
equivalent to the magnitude a star would have if it
were 10 parsecs distant.
Spectral Sequence. The scheme developed to label stars
according to decreasing surface temperature:
OBAFGKM (LT).
Composition. A measurement of the abundances of the
different elements in stars and galaxies.
Sample Questions
1. What would happen to our ability to measure
stellar parallax if we were on the planet Mars?
What about Venus or Jupiter?
Answer: As distance from the Sun increases, a
parallax angle of one arcsecond corresponds to a
greater and greater distance. From Mars we
would be able to measure distance to even
further stars, and from Jupiter even more distant
stars would be detectable by their parallax. From
Venus, however, only closer stars could be
measured.
18. The stars Betelgeuse and Rigel are both
in the constellation Orion. Betelgeuse
appears red in colour and Rigel bluishwhite. To the eye, the two stars appear
equally bright. Can you compare the
temperature, luminosity, and size of the two
stars from just that information? If not,
why?
Answer. Betelgeuse is cooler than Rigel, but
nothing else can be said about the
luminosity or size of the two stars without
knowing their distance.
The Stefan-Boltzmann Law:
Star Sizes
The luminosity of a star is given by the energy radiated
per second from every square meter of surface of the star
multiplied by the surface area of the star.
The surface area of a sphere = 4πR2
(R is the stellar radius)
The radiance of a hot gas = σT4
(T is temperature measured in K)
where σ = Stefan-Boltzmann constant.
So the luminosity, L, of a star is often written:
L = 4πR2σT4
The Stefan-Boltzmann Law:
Star Sizes (2)
But the luminosity of a star can also be found from its
distance and apparent brightness. Namely, distance +
apparent brightness = luminosity. That is how absolute
magnitude is calculated
So, the distance and apparent brightness of a star can be
used to infer its size, i.e. its radius, since the star’s
temperature can be established from its spectrum.
Distance + apparent brightness = luminosity
Spectral Type = temperature
Luminosity + temperature = size
That reasoning is the
basis for the
Hertzsprung-Russell
diagram, H-R
diagram for short,
which plots absolute
magnitude (or
luminosity) of a star
versus its spectral
type (or
temperature).
Position in the H-R Diagram is also related to size.
MK scheme of
using Roman
numerals to
designate stellar
luminosities: I =
supergiant, III =
giant, V = dwarf
(like the Sun),
corresponding to
the main
sequence in the
H-R diagram.
The H-R Diagram simplified.
The colour-magnitude
diagram (CMD) for the
open star cluster NGC
6087 is identical to the
H-R diagram for stars
in general. Note that
most stars lie on the
main sequence, very few
have evolved “away”
from the main
sequence. S Nor is a
yellow supergiant
member of the cluster.
Actually it is a pulsating
Cepheid variable star.
The distances to star
clusters like NGC
6087 and NGC 6830
can be measured,
often to better than
5%, by fitting a
standard mainsequence to the
obvious dwarf stars
in the cluster.
Other examples: P Cygni cluster and NGC 2264 (young).
The masses of stars are measured through the analysis of
binary star orbits. Radial velocity information is
particularly useful for that. Image of a visual binary:
Albireo.
A multiple system: Mizar (R) and Alcor (L).
Orbital motion in a visual binary system: ζ UMa.
The orbit of the visual binary 70 Ophiuchi revealed by
regular monitoring.
Eclipsing binaries are common. In such systems the
orbital plane is close to the line of sight and the orbital
period is short, so eclipses (drops in brightness) can be
detected regularly.
An eclipsing binary system, where one star totally eclipses
the other at regular intervals.
An eclipsing binary system in which only partial eclipses
occur.
Intensity measures of an eclipsing binary system during
total eclipse of the primary star.
Light curves of eclipsing binaries may also reveal tidal
distortions of the two stars caused by their mutual
attraction.
Or reveal the presence of hot spots on the cooler star in
the system.
Spectroscopic binaries are among the most useful of
binary systems. Binarity is revealed in such systems by
the regular variations in the measured radial velocity of
the star. The spectral lines of one (or both) stars are
revealed through the regular red or blue shifts of their
spectral lines. Examples are shown below.
Analysis of the radial velocity curve(s) for a
spectroscopic binary provides information on the
orbital shape and orientation, but not the orbital
inclination to the line of sight.
The “best” systems for giving information on
ALL orbital parameters are
eclipsing/spectroscopic binaries since orbital
inclination is known. From them we obtain
information on the masses and luminosities of the
two stars in the system. The light curve solution
gives the radius and temperature of each star,
from which luminosities are derived.
Recall that the sum of the masses of the two stars
can be found from the semi-major axis of the
orbit and the orbital period. Individual masses
are found from the orbits of each star relative to
the other, and luminosities for visual binaries are
derived once the distance is known.
Results: the
mass-luminosity
relation.
The luminosity
of a star
depends directly
on its mass:
L ≈ M4
The relationship
applies only to
main sequence
stars.
The masses of stars
along the main
sequence in the H-R
diagram.
Stars of large mass
(50-100 M) are very
hot and luminous:
spectral type O. Stars
of very small mass
(0.01-0.1 M) are very
cool and faint:
spectral types M, L, T.
The H-R diagram
for bright stars
(from the Hipparcos
mission). Note that
the diagram
contains a very large
proportion of the
most luminous
classes of stars.
The proportions of stars of different types.
Astronomical Terminology
H-R Diagram. The figure used by Ejnar Hertzsprung
and Henry Norris Russell to demonstrate how
luminosity relates to temperature in stars.
Stefan-Boltzmann Law. The relation describing how a
star’s luminosity depends on its surface area and
temperature.
Dwarf. The term used to describe stars populating the
main sequence in a H-R diagram.
Giant. A star of 10-100 R that is more luminous than
main-sequence stars.
Supergiant. A star of 100-1000 R that is much more
luminous than main-sequence stars.
Binary Star. A system of two stars in orbit about each
other, designated according to how the orbital
motion is detected: visual binary (observed orbital
motion), spectroscopic binary (periodic radial
velocity shifts), eclipsing binary (mutual eclipses).
Sample Questions
1. Although we think of the Sun as an “average”
main-sequence star, it is actually hotter and more
luminous than average. Explain.
Answer: Pick almost any stellar property to
measure (mass, luminosity, size, temperature),
and the Sun resides near the geometric mean of
the range. However, in terms of absolute
numbers of stars, small stars are much more
numerous
than
large
stars.
In
fact,
approximately 90 percent of all stars are cooler
and less luminous than the Sun.
18. Two stars have the same luminosity but
one is larger. Compare their temperatures.
Now suppose that the two stars have the
same size but one is more luminous. Again,
compare their temperatures.
Answer. If two stars have the same
luminosity but one is larger, then the larger
star must also be cooler. If the two stars
have the same size but one is more
luminous, then the more luminous star (the
brighter one) must be hotter.