Measuring the Properties of Stars
© Sierra College Astronomy Department
Measuring the Properties of Stars
Stellar Brightness and Luminosity



Power is the rate at which energy is transferred,
or the amount of energy transferred per unit time.
Luminosity is the rate at which electromagnetic
energy is being emitted - the total amount of
power emitted by a star over all wavelengths.
Brightness (or apparent brightness) refers to
the luminosity/area of a star as seen at the Earth
and is related to the star’s luminosity through the
inverse square law.
© Sierra College Astronomy Department
2
Measuring the Properties of Stars
Stellar Brightness and Magnitude




In the second century B.C., Hipparchus created the
first star catalog with corresponding brightnesses
determined visually.
Hipparchus quantified each star’s brightness with a
“magnitude”, an integer from 1 (brightest) to 6
(dimmest).
Ptolemy (second century A.D.) expanded the number
of stars with measured magnitudes and popularized
this system of measurement.
Today this magnitude is formally known as
apparent magnitude and is designated by the letter
m.
© Sierra College Astronomy Department
3
Measuring the Properties of Stars
Stellar Brightness and Magnitude



The modern apparent magnitude scale is set up so
that a 5-magnitude difference (say between two
stars) is equal to a brightness change of 100 times
(so as to closely match Hipparchus’s original
data).
Consequently, a one-magnitude difference is
equal to a brightness change of 2.512 times
(2.5125 = 100).
The magnitude system is useful for its historical
connections (for comparisons) and its manageable
range (brightness itself covers a much larger
range).
© Sierra College Astronomy Department
4
Measuring the Properties of Stars
Stellar Brightness and Magnitude



Modern measuring devices allow astronomers to
determine magnitudes to an accuracy of 0.001 or
better.
Modern, large telescopes equipped with CCD
devices can image objects as dim as 25th magnitude
or better.
A few stars (e.g., Sirius) are so bright that they
have negative magnitudes. Sirius’s apparent
magnitude is –1.47.
© Sierra College Astronomy Department
5
Measuring the Properties of Stars
Distances to Stars - Parallax



Stellar parallaxes were not observed until the
mid-1800s.
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 angle of the nearest star is only
about 1.52 seconds of arc, but astronomers define
the parallax angle as half that value, or 0.76
seconds.
© Sierra College Astronomy Department
6
Measuring the Properties of Stars
Distances to Stars - Parallax



Parallax distance formula (in light-years):
Distance to star (ly) =
3.26 ly/parallax angle in arcsec
Parallax distance formula (in parsecs):
Distance to star (pc) =
1/parallax angle in arcsec
A parsec is the distance of a 1 AU object has a
parallax angle of one arcsecond. One parsec is
equal to 3.26 ly or 206,265 AU.
© Sierra College Astronomy Department
7
Measuring the Properties of Stars
Distances to Stars - Parallax
 The
satellite Hipparchos (1989-1993)
measured parallax angles with very high
precision (milli-arcsecond) for over
100,000 stars establishing highly accurate
distance measurements out to about 1000
light-years.
 Accurate stellar distances help to determine
other quantities about celestial objects.
© Sierra College Astronomy Department
8
Measuring the Properties of Stars
Absolute Magnitude & Luminosity



The intrinsic luminosity of a star is usually given
as its absolute magnitude and designated with a
capital M.
M is defined as the apparent magnitude a star
would have if it were at a distance of 10 parsecs.
Sirius’s apparent brightness (–1.47) is due to its
closeness (2.7 parsecs from Earth). Its absolute
magnitude is +1.45 (determined by using inverse
square law).
© Sierra College Astronomy Department
9
Measuring the Properties of Stars
Absolute Magnitude & Luminosity

Given the brightness-magnitude relationship:
b1
 1001/ 5
b2


m2  m1

And the brightness-luminosity relationship:

It is possible to show derive the distance
modulus relationship (d is in parsecs):
d  10x10mM  / 5
L
b
4 d 2
© Sierra College Astronomy Department
10
Measuring the Properties of Stars
Temperature and Spectral Classes




A star’s color is determined by its temperature.
An absorption spectrum - the absorption of radiation
at various wavelengths - can be used to determine a
star’s temperature.
Harvard astronomers, lead by Edward Pickering and
his women “computers” developed the first stellar
classification system using letters A-O, in
alphabetical order.
In particular, Williamina Fleming based the system
on the strength of the stars’ hydrogen absorption lines
(A strong, O weak)
© Sierra College Astronomy Department
11
Measuring the Properties of Stars
Temperature and Spectral Classes



The A-O scheme was eventually found to be
inadequate.
Another “computer”, Annie Jump Cannon, discovered
that a reordering and elimination of some of the letters
gave a better scheme, which is still used today.
Cannon’s system was thought to reflect stellar
composition, but “computer” Cecilia PayneGaposchkin showed that the system was a
consequence of the stars’ surface temperatures.
© Sierra College Astronomy Department
12
Measuring the Properties of Stars
Temperature and Spectral Classes



The principal spectral types used today (from hottest to
coolest) are designated as O B A F G K M.
O stars range in temperature from 30,000 K to 60,000 K.
M stars have temperatures less than 3,500 K.
Within each spectral class, stars are subdivided into 10
categories by number (0 to 9).


For example, the Sun is a G2 star.
There are also other spectral types which are not quite as
well known as the original seven (L and T types are two
new ones used to classify very cool stars which form dust
grains in their atmospheres).
© Sierra College Astronomy Department
13
Measuring the Properties of Stars
The Hertzsprung-Russell Diagram



Hertzsprung-Russell diagram is a plot of
absolute magnitude (or luminosity) versus
temperature (or spectral class) for stars.
About 90% of all stars fall into a group
running diagonally across the diagram
called main-sequence stars.
Stars on the H-R diagram fall into
categories such as main-sequence stars,
white dwarfs, red giants, and supergiants.
© Sierra College Astronomy Department
14
Measuring the Properties of Stars
Luminosity Classes




In the 1880s Antonia Maury and Ejnar Hertzsprung
discovered that the width of a star’s absorption lines
was directly related to the star’s luminosity (which in
turn is related to a star’s surface density).
Luminosity classes are one of several groups into
which stars can be classified according to the
characteristic widths of their spectra.
The luminosity classes are: Ia (supergiants), Ib (dimmer
supergiants), II (bright giants), III (ordinary giants), IV
(subgiants), and V (main-sequence).
Complete Stellar Classification: A star is fully
classified if its spectral class and luminosity class are
specified (e.g., the Sun is designated as a G2 V star)
© Sierra College Astronomy Department
15
Measuring the Properties of Stars
Towards a “Distance Ladder”
Spectroscopic Parallax
 Knowing a star’s luminosity class and temperature
(spectral class) gives its absolute magnitude.
 Knowing a star’s absolute magnitude and apparent
magnitude gives its distance.
 The distances to stars too far away for parallax
measurements can be determined using this procedure.
 Spectroscopic parallax represents the second rung
(geometric parallax being the first) in the distance ladder
created and used to scale the Universe.
© Sierra College Astronomy Department
16
Measuring the Properties of Stars
Star Sizes from Temperature and Luminosity
The Sizes of Stars
 The sizes of a few very large stars have been
measured directly by interferometry.
 Knowing the temperature of a star gives its
energy emitted per square meter.
 Knowing the total energy emitted (from the
absolute magnitude) one can then calculate the
surface area of the star.
 From that the diameter of the star can be
determined.
© Sierra College Astronomy Department
17
Measuring the Properties of Stars
Multiple Star Systems and Binaries
Multiple Star Systems and Binaries
 More than half of what appear as single stars
are in fact multiple star systems.
 Optical doubles are two stars that have small
angular separation as seen from Earth but are
not gravitationally linked.
 Binary star system is a system of two stars that
are gravitationally linked so that they orbit one
another.
© Sierra College Astronomy Department
18
Measuring the Properties of Stars
Multiple Star Systems and Binaries




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.
Binaries can be confirmed by observing the system
over time and looking for signs of revolution.
Spectroscopic binary is an orbiting pair of stars that
can be distinguished as two due to the changing
Doppler shifts in their spectra.
© Sierra College Astronomy Department
19
Measuring the Properties of Stars
Multiple Star Systems and 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.
© Sierra College Astronomy Department
20
Measuring the Properties of Stars
Masses and Sizes from Binary Stars



Binary stars are important because they allow one to
measure masses of stars using Newton’s version of
Kepler’s laws.
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.
To determine how the system’s total mass is
distributed between the two stars, one need only
consider the ratio of the two stars’ distances to the
center of mass.
© Sierra College Astronomy Department
21
Measuring the Properties of Stars
Masses and Sizes from Binary Stars




Because the inclination of spectroscopic binary
orbits are usually not known, exact mass
calculations cannot be done.
However, assuming an average inclination can
provide information about average masses of
spectroscopic binary stars.
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.
© Sierra College Astronomy Department
22
Measuring the Properties of Stars
The Mass-Luminosity Relationship



Mass-luminosity diagram plots the mass versus the
luminosity of a number of stars.
More massive stars are more luminous.
The mass-luminosity relationship holds only for mainsequence stars.
L / L  M / M


p
where p has a value between 3.5 and 3.9
The mass-luminosity relationship is valuable in
investigating less accessible stars and in constructing
and evaluating hypotheses on the life cycle of stars.
© Sierra College Astronomy Department
23
Measuring the Properties of Stars
Main-Sequence Lifetimes


The lifetime on the main-sequence depends on how much fuel
(hydrogen) the star has and how fast the star is consuming it.
This lifetime can be expressed as:
t

M
L
Using the main-sequence mass-luminosity relation, we have:
1 p
t  M 


t  M 


where t⊙ is for the Sun and is ~ 10 billion years
Examples: A 10 M⊙ will last about 10 million years, whereas a
0.3 M⊙ star will last 300 billion years
© Sierra College Astronomy Department
24
Measuring the Properties of Stars
Star Clusters and Aging





Open (galactic) cluster is a group of stars that share a
common origin and are located relatively close to one another.
Globular cluster is a spherical group of up to hundreds of
thousands of stars found primarily in the halo of the Galaxy.
Clusters are important for two reasons:
1. All stars in a cluster are at about the same distance from us,
so their apparent magnitude is a direct indication of their
absolute magnitude.
2. All the stars within a cluster formed at about the same time
(more or less).
Age of cluster determined from main-sequence turnoff
Much of our knowledge of star formation has come from
examination of clusters
© Sierra College Astronomy Department
25
Measuring the Properties of Stars
Variable Stars as Distance Indicators





Not all stars shine steadily like the Sun.
Stars that vary significantly over time are called variable
stars.
A certain sub-class of variable stars are called pulsating
variable stars (based on how the star is pulsates in size)
Most pulsating variable stars occupy the instability strip
on the H-R diagram.
A special class of very luminous pulsating variable stars
called Cepheid variable stars have a well established
period-luminosity relation that provides a powerful
means for determining cosmic distances.
© Sierra College Astronomy Department
26