CHAPTER 12
Measuring the Properties of Stars
CHAPTER OUTLINE
12-1 Stellar Luminosity
1. When we discuss a star’s brightness we must be careful to distinguish between how
bright the star looks to us from how bright it really is.
2. Two stars may differ in brightness due to any combination of the following:
(a) One star is inherently brighter than the other (it has a greater luminosity).
(b) One star is much closer to us than the other, making it appear brighter.
(c) There is more interstellar material absorbing light from one star than from the other.
3. When we refer to a star’s brightness, we will refer to its apparent brightness as seen
from Earth. When we refer to a star’s luminosity, we will refer to the star’s inherent
brightness (the rate at which it emits energy).
Apparent Magnitude
1. Hipparchus created the first star catalog with corresponding brightness determinations
(apparent magnitudes) in the second century B.C.
2. Apparent magnitude is a measure of the amount of light received from a celestial object. Hipparchus assigned an apparent magnitude of 1 for the brightest stars and 6 for the
dimmest.
3. The modern magnitude scale is set up so that a 5-magnitude difference corresponds to
a ratio of 100 in the amount of light received. Also, a one magnitude difference corresponds to a ratio of 1001/5  2.512 in the amount of light received.
4. Modern measuring devices allow astronomers to determine magnitudes to an accuracy
of 0.001 or better.
5. A few stars (e.g., Sirius) are so bright that they have negative magnitudes. Sirius’s apparent magnitude is –1.43 (it is about 10 times brighter than an average first-magnitude
star).
6. Modern, large telescopes equipped with CCD devices can image objects as dim as 25th
magnitude or better.
12-2 Measuring Distances to Stars
1. The absence of observable stellar parallax was an argument against the heliocentric
system. Stellar parallaxes were finally observed in 1838.
2. Parallax angle is half the maximum angle that a star appears to be displaced due to the
Earth’s motion around the Sun. The maximum parallax angle of the nearest star is only
0.76 arcseconds.
3. A parsec is the distance from the Sun to an object that has a parallax angle of one
arcsecond. One parsec corresponds to 3.26 ly or 206,265 AU.
4. Parallax distance formula: Distance to star (in parsecs) = 1/parallax angle (in arcseconds).
5. Only stars within about 120 parsecs (400 light-years) have parallax angles great
enough to allow accurate calculations of their distances from Earth.
6. The satellite Hipparchos measured positions and parallaxes to an accuracy of 0.001
arcsecond. Its successor, Gaia (to be launched in 2011), will give us precise information
for the billion brightest objects in the sky and create a 3-D map of the Milky Way.
7. Accurate stellar distances help to determine other quantities about celestial objects.
They also help to determine the distance scale of the universe more accurately.
Absolute Magnitude
1. The intrinsic luminosity of a star is usually given as its absolute magnitude, which is
defined as the apparent magnitude a star would have if it were at a distance of 10 parsecs.
2. Sirius’ apparent brightness (–1.43) is due to its closeness (2.6 parsecs from Earth). Its
absolute magnitude is +1.47.
Tools of Astronomy: Calculating Absolute Magnitude
1 The difference between a star’s apparent (m) and absolute (M) magnitudes is called the
distance modulus: m  M = 5  log(d)  5, where d is the star’s distance from us in parsecs.
12-3 Motions of Stars
1. In 1718 Halley discovered that stars do move with respect to one another. Thus, constellations do gradually change shapes.
2. A star’s proper motion is expressed as the angle through which a star moves each
year—its angular velocity—as seen from the Sun.
3. Barnard’s star, the second closest star to the Sun, exhibits the greatest proper motion of
any star—10.3 arcseconds/year.
4. A star’s tangential velocity—speed across our line of sight—can be calculated using its
proper motion and distance. This is difficult to determine for stars. It is easier to detect a
star’s radial velocity —its velocity toward or away from Earth, using the Doppler effect.
5. A star’s space velocity relative to the Sun corresponds to its actual motion relative to
the Sun and combines its radial and tangential velocities.
12-4 Spectral Types
1. A star’s color is determined by its temperature. A star’s blackbody curve can be used
to determine its temperature.
2. A star’s absorption spectrum—the absorption of radiation at various wavelengths—can
also be used to determine its temperature.
3. Annie Jump Cannon devised a system for classifying stellar spectra and applied this
system to several hundred thousand stars. The different spectral classes were arranged
alphabetically, based on the strength of the hydrogen lines.
4. The work of Cecelia Payne-Gaposchkin resulted in a reclassification based on temperature. The spectral classes used today (from hottest to coolest) are designated as O B
A F G K M.
5. The temperature range of O stars is 30,000 – 60,000 K. The temperature of M stars is
less than 3,500 K.
6. Within each spectral class, stars are subdivided into 10 categories (spectral types) by
number. The Sun is listed as a G2 star.
The Hertzsprung-Russell Diagram
1. The Hertzsprung-Russell (H-R) diagram is a plot of luminosity (absolute magnitude)
versus temperature (or spectral class) for stars.
2. About 90% of all stars fall into a group running diagonally across the diagram; this diagonal band is called the main sequence.
3. Stars on the H-R diagram fall into categories such as main sequence stars, white
dwarfs, giants, and supergiants.
Spectroscopic Parallax
1. Spectroscopic parallax is the method of measuring the distance to a star by comparing its absolute magnitude to its apparent magnitude.
2. Because of limits in the technique, spectroscopic parallax is most useful when applied
to groups of stars that are nearly the same distance from Earth.
Luminosity Classes
1. In the 1880s Antonia Maury discovered that absorption lines are subject to a smearing
effect, which has become valuable in classifying stars.
2. The greater the density in a star’s atmosphere, the more frequent the collisions among
atoms, the broader the corresponding absorption lines
3. The extent of line broadening allows us to classify stars in different luminosity classes, denoted by roman numerals I, II, III, IV and V; supergiants (I), bright giants (II), giants (III), subgiants (IV), and main sequence stars (V).
4. The classification of stars in luminosity classes allows spectroscopic parallax to be
used with any star.
Analyzing the Spectroscopic Parallax Procedure
1. Knowing the luminosity class of a star and its temperature allows us to determine its
absolute magnitude (and thus luminosity). Knowing its absolute magnitude and its apparent magnitude allows us to find the star’s distance.
2. This procedure can be used to find distances to stars that are too far for parallax measurements.
Luminosity and the Sizes of Stars
1. White dwarfs are hot but dim stars because they are small. They may be 0.01 the Sun’s
size.
2. A giant star is one of great luminosity and radius (10 to 100 times the Sun’s radius). A
supergiant is a star of very great luminosity and radius (more than 100 times the Sun’s
radius).
3. The sizes of a few very large stars have been measured directly by interferometry.
4. Knowing the temperature of a star gives us its energy emitted per square meter. Knowing the total energy emitted (from the absolute magnitude or luminosity) we can then calculate the surface area of the star. From that the diameter of the star can be determined.
12-5 Multiple Star Systems
1. More than half of what appear as single stars are, in fact, multiple star systems.
2. Optical doubles are two stars that have small angular separation as seen from Earth
but are not gravitationally linked.
3. A binary star system is a system of two stars that are gravitationally linked so that
they orbit their common center of mass.
4. In 1802 Herschel obtained the first evidence that some double stars orbit one another.
5. Binary star systems are classified into several categories according to how they are detected.
Visual Binaries
1. A visual binary is an orbiting pair of stars that can be resolved (normally with a telescope) as two stars.
2. If one uses large telescopes, about 10% of the stars in the sky are visual binaries.
3. Visual binaries can be confirmed by observing the system over time and looking for
signs of revolution.
Spectroscopic Binaries
1. A spectroscopic binary is an orbiting pair of stars that can be distinguished as two due
to the changing Doppler shifts in their spectra.
Eclipsing Binaries
1. Algol, discovered by John Goodricke in 1783, is an eclipsing binary, in which one
star moves in front of the other as viewed from Earth.
2. 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.
Other Binary Classifications
1. An astrometric binary is an orbiting pair of stars in which the motion of one of the
stars reveals the presence of the other.
2. A composite spectrum binary is a binary star system with stars having spectra different enough to distinguish them from one another.
12-6 Stellar Masses and Sizes from Binary Star Data
1. Binary stars are important because they allow us to measure stellar masses using Kepler’s third law as modified by Newton.
2. Knowledge of the size of one of the star’s ellipses, along with knowledge of the period
of its motion, permits calculation of the total mass of the two stars.
3. To determine how the system’s total mass is distributed between the two stars, we need
only consider the ratio of the two stars’ distances to the center of mass.
4. Because the inclinations of spectroscopic binary orbits are usually not known, exact
mass calculations cannot be done. However, assuming an average inclination we can obtain valuable information about average masses of spectroscopic binary stars.
5. Eclipsing binaries that are also spectroscopic binaries provide us with a way of measuring not only the masses of the two stars but also their sizes. We derive this information
using measurements of their Doppler shifts.
12-7 The Mass-Luminosity Relationship
1. The mass-luminosity diagram shows the relationship between the luminosity and mass
of main-sequence stars. More massive stars are more luminous (luminosity  mass3.5).
2. The mass-luminosity relationship is valuable in investigating less accessible stars and
in constructing and evaluating hypotheses on the life cycle of stars.
12-8 Cepheid Variables as Distance Indicators
1. In 1784, John Goodricke discovered that the star  Cephei varies in luminosity in a
regular way. Soon after, other stars were seen that exhibit delta Cephei’s characteristic
light curve—rapid brightening followed by slow dimming.
2. Doppler-effect data show that these stars are pulsating in rhythm with their changes in
luminosity.
3. It is fairly easy to identify this class of stars, now named Cepheid variables or Cepheids. Each Cepheid has a very constant period of variation, ranging from about 1 day to
about 3 months for different Cepheids.
4. In 1908 Henrietta Leavitt discovered that for Cepheids in the Magellanic Clouds, the
brighter variables have the longer periods. Even though the stars in these clouds do not
have the same intrinsic luminosity, their apparent magnitudes are directly related to their
absolute magnitudes because they are all at about the same distance from us.
5. Cepheids are important because a Cepheid’s period, which is easy to determine, allows
us to determine its absolute magnitude. This in turn allows us to find its distance by using
the distance modulus.
6. Cepheids could be valuable distance indicators if the distance to one could be determined. None are close enough to be measured by parallax, but beginning in 1917, Shapley worked out a complex statistical method to determine distances to Cepheids in our
Galaxy.
7. Shapley’s work led to a period-luminosity diagram for Cepheid variables.
8. It took until the late 1950s to determine the correct relationship between Cepheids’ periods and absolute magnitudes accurately because there are actually two different types of
Cepheid variables. Cepheid I stars are brighter than Cepheid II stars (for the same period)
by about a factor of 4 (or 1.5 magnitudes). Cepheid I stars are younger and metal richer
than Cepheid II stars.
9. Another group of variables, the RR Lyrae stars, are also used as distance candles.
Their periods range from a few hours to one day and their luminosity is approximately
constant. They are less luminous than the Cepheids and thus can only be seen to smaller
distances.
10. Though only a few thousand Cepheids have been detected, they are extremely useful
in giving us a method of measuring distances not only to distant parts in our Galaxy but
also to faraway galaxies.