CHAPTER 8: PROPERTIES OF STARS
Instructor: M. Azad Islam

CHAPTER BRIEF:

8-1 Measuring the distances to the stars
Surveyor's methgod
Astronomer's method
8-2 Intrinsic brightness
Brightness and distance
Absolute visual magnitude
Luminosity
8-3 The diameters of stars
Luminosity, radius and temperature
Dwarfs, Gianta and Supergiants
Luminosity classification
Spectroscopic parallel
8-4 The masses of stars
Binary stars in general
Visual binaries
Spectroscopic binaries
Eclipsing binaries
8-5 The family of stars
Mass, luminosity and density
Surveying the stars
The frequency of stellar types
 
IMPORTANT NOTE: Science is based on observations and measurements. Measurements in astronmy are very difficult. Even, something as simple as measuring diameter is not possible for stars. Observations together with basic laws of physics give us insights into understanding a star. Billions of stars are spread across the sky. They seemed to be the building blocks of the universe, just as atoms are building blocks of a solid or a substance. It might be that understanding the stars will lead us to understanding the earth and us.
 

KEY CONCEPTS 

In this chapter we shall learn about how we know and what we know about the stars. In this chapter we shall study six fundamental properties of all stars,namely the size (diameter), mass, luminosity, temperature, composition and age. At first, luminosity, radius and mass of the stars are determined. Later, some general conclusions are determined from them. Luminosity of a star tells us how much energy (all wavelengths included) the star produces each second. This provides some information about what is inside the star. Determining luminosity requires finding the distance to the star. Method of PARALLAX is used to find distance. Another method of finding distance of a star is spectroscopic parallax.The relations among luminosity, temperature and radius show that stars of similar temperatures can have very different radii. This leads to the luminosity classification.

The mass of an individual star can only be determined from the binary stars. The masses and radii of the individual stars in Eclipsing binary can be determined from the light curve and the radial velocity.The Visual binaries allow for an accurate determination of the mass of each of the stars in the system. The spectroscopic binaries exist in abundance. But, they permit only the determination of a lower limit on the mass of the stars.

The mass-luminosity relation described in the chapter is useful in the determination of age of the star clusters to be studied in the later chapter.The relative abndance of stars of various spectral types and luminosity classes are discussed. The discussion is enlightening, because it points out what the problems are in finding the relative number of main sequence O stars to that of main sequence M stars. The importance of H-R digram is made clear in the chapter. H-R diagram allows us to compare star properties of diameters, temperatures and luminosity among various classes of stars.

We shall classify the stars according to luminosity as in H-R diagram (the Hertzsprung-Russell diagram).
 
In order to grasp the materials in this chapter, we need to know the following:
 
Stars--What Are They Like?
o Distances
o Inverse Square Law of Light
o Magnitude System
o Star Colors (Temperatures)
o Stellar Composition
o Velocity of Stars from Doppler Effect
o Stellar Masses
o Stellar Radius
o Temperature dependence
o Spectral Types
o Hertzsprung-Russell diagram and Color-Magnitude diagram
o Density of stars
o Frequency of stellar types
 
 
MEASURING DISTANCES TO STARS: Surveyor's Method:
 

 

ANGULAR MEASUREMENT:
60 minutes = 1 degree, 60 seconds = 1 minute
206,265 seconds = 1 radian
Angle = (diameter/distance)x206,265 seconds
 
SMALL ANGLE FORMULA:
The angle subtended by opposite ends of an object with respect to an observer's eye, can be used as a measure of distance.
 
DEFINITION: The angle measured in radian is,
Angle (radian) = (Arc length)/(radius of the arc)
2p radians =360 degrees = 4 right angles
 

In the formula, Arc length Diameter, Radius distance to the object

Astronomers use trigonometric parallax to measure the distances of the nearby stars. If we look at a nearby object from different vantage points, it will appear to shift against more distant objects in the background. The farther apart the vantage points are, the greater the shift. The farther away the object is, the less it appears to shift. Since the shifts are so small for the stars, we use arc second as the unit of their angular shift. The distance is quoted in parsec (abbreviated as "pc"). One parsec = 206,265 Astronomical Units
(A.U.= mean distance between the Earth and the Sun or [1.496*10^8] km).
In terms of light years, one parsec = 3.26 light years.

We can measure shifts upto 1/50 arc seconds of arc [or] 50 parsecs in distance.Stars at distance greater than 50 parsec cannot be measured from earth due to uncertainty in measurements originating from blurry images produced by turbulent earth's atmosphere. Space based telescope can measure upto 500 parsec distance by parallax.

 
DISTANCE AND BRIGHTNESS
 
Inverse Square Law of Light Brightness: Energy from any point on the light source radiates out in radial direction. So concentric spheres (centered on the point of light source) have the same amount of energy pass through them every second. As light moves outward it spreads out to pass through each square centimeter of those spheres.
Amount of energy passing through surface of sphere #1 = amount of energy passing through surface of sphere #2.
Surface area of sphere = 4pr2, where [r] is the radius of the sphere. 
Amount of energy passing through sphere surface = (flux) x (surface area). 
The flux is the amount of energy reaching each square meter area (of a detector, CCD) every second.
 
Magnitude System
Astronomers specify the brightness of stars with the magnitude system. The apparent visual magnitude is based on how the human eye perceives differences in brightness. This dependence is logarithmic. Also fainter stars have larger and more positive magnitudes. It's a bit clumsy, but it's the tradition!
So, the apparent brightness of a star observed from the Earth is called the apparent (visual) magnitude. The apparent magnitude is a measure of the star's flux in the visible range of wavelengths. In earlier chapter, we studied the Apparent visual magnitude of stars. Now, we need to understand the Absolute Visual Magnitude of a star. It is measured by placing a star at a hypothetical distance of 10 parsecs.
 
If the star is placed at 10 pc distance from us, then its apparent magnitude is equal to its absolute magnitude. The absolute magnitude is a measure of the star's luminosity . That is, the amount of energy in the visible range (of wavelengths) radiated by the star every second. If we know the apparent magnitude and absolute magnitude, we can find the star's distance. Also, if we know the apparent magnitude and distance, we can find the star's luminosity.

Some apparent magnitudes we studied before:

Sun = -26.8, Moon = -12.6, Venus = -4.4,
Sirius = -1.4, Vega = 0.00, faintest naked eye star =+6.5, brightest quasar = +12.8, faintest object = about +28
 
Most famous apparently bright stars are also intrinsically bright (luminous). Most nearby stars are intrinsically faint. Assume we live in a typical patch of galaxy (Copernican principle) [which means that] most stars are puny emitters of light.
 
Faintest stars have absolute magnitudes = +19, brightest stars have absolute magnitudes = -8 [and that means a] huge range in luminosity!
 
Let us define,
"Distance modulus" = apparent magnitude - absolute magnitude = [5 log(distance in pc) - 5]. That is,
 
Distance Modulus = m - Mv = 5 log10(d) - 5
 
or, d = 10[(5+m - Mv)/5]
 
Now actual total radiation takes place from all over the stellar surface. Total radiation output is Area x Intensity. That is,
Total radiation output = Luminosity (L) = Area x sT4 = 4pR2 x sT4
 
So the ratio of the luminosity of two stars is,
[L1/L2] = [(R1/R2)2 x (T1/T2)4]
 
Also the ratio of the luminosity of two stars is related to distances by the following,
 
L1/L2 = 10{0.4x(abs. mag of #1 - abs. mag. of #2)}
 

MAGNITUDE AND LUMINOSITY CHART

Name of the Star

Apparent Magnitude

Distance

Absolute Magnitude

Luminosity (rel. to Sun)

Sirius

-1.4

2.7

1.4

23

Arcturus

0.0

11.0

-0.2

100

Vega

o.o

8.1

0.5

52

Spica

1.0

8.0

-3.4

1900

Barnard's Star

9.5

1.8

13.3

1/2500

Proxima Centauri

11.0

1.3

15.5

1/19000

Star Colors (Temperatures) 

The color of stars depends on their temperatures. Hotter stars are bluer and cooler stars are redder. Use of different filters allows only a narrow range of wavelengths (colors) to pass through. By sampling the star's spectrum at two different wavelength ranges (``bands''), we can determine if the spectrum is that of a hot, warm, cool, or cold star. Hot stars have temperatures around 40,000 K while cold stars have temperatures around 2,000 K.
 
Stellar Composition
 
We determine the composition of stars by spectroscopic studies. The method breaks the starlight into individual colors and noting the absorption (or sometimes, emission) lines present. From these absorption lines we learn some important things beside the star's composition.
 
1. Structure of stars: hot dense body producing continuous spectrum, covered by cooler thin gas.
2. The physics we use on Earth works everywhere else in the Universe! Hydrogen spectrum is same in Sun, stars, distant galaxies and quasars. All absorption lines seen in celestial objects can be seen in laboratories on Earth. Charge and mass of electron and proton are the same as electrons and protons everywhere in the universe. Laws of Physics are the same everywhere!
3. Since light has a finite speed, the light we receive from far away sources tells us how they were long time ago. Physical laws are the same for all times!
 
Types of Stars and the HR diagram
 
Temperature dependence:
 
Strength and wavelength of absorption lines vary between stars. Some stars have thick or strong (dark) hydrogen lines, other stars have no Hydrogen lines but strong Calcium and Sodium lines. Are their abundances different? No. temperature affects, in particular, the temperature of the photospheres.

1. Example: Hydrogen atom with orbiting electron in different energy levels (remember Bohr model for atom?). To absorb photon of certain energy, electron needs to be at right energy level. The hydrogen atoms at high temperatures are ionized by their own atomic collisions.Which means no absorption lines in the spectrum. If the star's temperature is too low, then there are not many electrons present in 2nd energy level. These atoms will be present in ground state because there are not many atomic collisions. To produce absorption lines in the visible spectrum as Balmer lines, electrons must be excited to 2nd energy level first by collisions.

2. Hydrogen Balmer lines are strong for temperature range of 4,000 -12,000 K. Helium lines are strong for 15,000-30,000 K. Calcium lines are strong for 3000-6000 K.
 
3. The strength of the lines is sensitive to temperature. Comparison of atomic line strengths gives accurate temperature (within 20-50 K). Some stars have peaks of continuous spectrum outside the visible range so use spectral lines. Stars not perfect thermal radiators so continuum spectrum gives only a rough temperature (within a few hundred Kelvin).
 
Spectral Types
 
The spectral types were based on Hydrogen (absorption) line strength. A-type of stars is strongest, B-type next strongest, F-type next, etc. Originally there was the whole alphabet of types, based on Hydrogen line strengths, but then astronomers discovered that the lines depended on temperature. After some rearranging and merging of some classes we now have OBAFGKM classes ordered by temperature. Each class is subdivided into 10 intervals, e.g., G2 or F5, with 0 hotter than F and, 1 hotter than 2 for the same spectral type of star.
 

Color

Class

Temperature (K)

Prominent Lines

bluest

O

40,000

ionized Helium

bluish

B

18,000

neutral Hydrogen

blue-white

A

10,000

neutral Hydrogen

white

F

7,000

neutral Hydrogen, ionized Calcium

yellow-white

G

5,500

neutral Hydrogen, strongest ionized Calcium

orange

K

4,000

neutral metals (Calcium, iron), ionized Calcium

red
M
3,000
remolecules and neutral metals
 
Hertzsprung-Russell diagram:
 
Hertzsprung-Russell diagram looks for correlations between stellar properties. Hertzsprung and Russell independently found a surprising correlation between temperature and luminosity for more than 90% of the stars. Also called a color-magnitude diagram. Diagonal strip is main-sequence. Luminous ones are easier to observe but rarer in existence, faint ones are harder to see but are more common.
 
Also, when the Luminosity versus mass are plotted in a graph, there seems to be a correlation between the two. We have the following:
Luminosity, L = M3.5 where M = mass of the star.

SPECTROSCOPIC PARALLAX
Spectroscopic parallax does not involve measuring parallax angles. But, this method is used to determine the distance of a star. The measurement of spectrum of a star tells us about its temperature, hence the spectral type is known. The width and strength of the spectral lines tell us about the luminosity (class) of the star whether a giant, supergiant or main-sequence star. By plotting this position of the star in the H-R diagram, we know about the absolute visual magnitude. From the difference value between absolute and apparent manitudes, we can find the distance to the star.
 
Spectral Type to Distance
 
Steps:
1. Determine temperature to find spectral type (from spectroscopy).
2. Measure star's flux to find absolute magnitude or luminosity
3. Use Inverse Square Law for Brightness to get distance.
 
BINARY SYSTEM OF STARS

Visual Binary: A system of two stars moving in a closed orbit about their center of mass that are resolved as two stars by a telescope. Ratio of their masses is inversely proportional to the ratio of their orbit radii. In general both components are visible and have long orbital periods (hundreds of years). Example is Sirius and its companion. Large orbit and long peroid of cycle. Another example is 61 Cygni that consists of two stars separated by approximately 30" and has an orbital period of 722 years. Kepler's third law is used to find the mass or ratio of the two masses.

Spectroscopic Binary: A system of two stars that appears as a single star but shows variations in their spectrum. This can usually be attributed to the presence of more than one component. The two stars are too close together and the telescopic image cannot separate the two. The two stars alternately approach and recede from the earth, producing Doppler effect. Doppler effect in their spectra is used to find the radial velocity. From the period and velocity of the star, one can find the circumference and hence the radius of the orbit. Kepler's 3rd law then gives the mass. The inclination of the orbital plane with respect to the earth is a major problem. More than half of the stars are binary systems and most of them are spectroscopic binaries.
 

 
Eclipsing Binaries: The orbital plane of the two stars fully or slightly tipped so that one of them can cross in front of the other in a periodic motion. Usually the system may have one small and the other a large star. The light output from the two stars are detected as a function of time to find the time period. Knowledge of Doppler shift and size of the orbit are used together to find the mass. Diameter of the star is measured from the time of eclipsing. Algol is an example of eclipsing binaries.
 
The study of binary star systems remains an integral part of fundamental stellar astronomy. By applying Kepler's and Newton's laws to the analysis of binary star orbits, it is possible to determine the mass and basic dimensions of the stars. It is the most direct and accurate way of determining stellar mass. Recall Kepler's 3rd law in Newtonian form:

where m1 and m2 are the masses of the components measured in solar masses, P is the orbital time period in years, r is the average orbital separation between the centers of the stars measured in AU.

Astrometric Binaries

In this system of two stars, only one component is visible, and detected due to the"wobble" in the proper motion of the visible component.
 Example: Barnard's Star


Seeing Double! Some stars may appear to be binary from the earth due to small angle of separation. Whereas they may be located at quite different disnces. These are called optical double and are NOT really a system of binaries.

More than half of the stars that we see are members of multiple star systems. An inspection of any field of stars with even a modest telescope will reveal numerous "close pairs" of stars. It was once thought that this closeness was the result of coincident alignment. While some are no more than "chance alignments" (example Albireo (beta Cygni)) it is now understood that stars can orbit each other as part of a common system. Indeed, the observations of binary stars over decades help us confirm in our minds that the laws of physics operating on earth also operate on distant star systems.

How much can be learnt from Hertzsprung-Russell Diagram?
History of the H-R Diagram
1911-13 A.D., Ejnar Hertzsprung and Henry Norris Russell independently developed Hertzsprung-Russell diagram (H-R diagram)
Horizontal axis is used for spectral type (or, equivalently, color index or surface temperature)
Vertical axis is used for absolute magnitude (or luminosity)
Data for individual stars are plotted, are found to be clustered in some definite ways
Data points define definite regions, suggesting common relationship exists for stars composing a particular region similar to Priodic Table of elements
Each region also represents common time in evolution of stars. Consequently, "common relationship" is set of physical processes applying to all stars of a particular region
 
Spectral Classification
Spectral classification is the scheme for grouping the stars according to temperatures or similarities in color as in violet, blue, and green portions of the visible spectrum.
Seven spectral classes are O, B, A, F, G, K, M
Temperature for O stars is the highest (40,000 K) and for M stars the lowest (2500 K)
Each spectral class subdivided into 10 spectral types
Spectral classification is a grading of stars according to their surface temperature (66 bins) and not chemical composition

Region

Spectral Type

Description of stars

Percent of stars

main sequence

O to M

bright-hot to faint-cool

90

red giants

F to M

bright-cool

<0.5

white dwarfs

B to F

faint-hot

10

blue supergiants

O to A

very bright-hot

0

red supergiants

G to M

very bright-cool

0

Main Sequence in Class V:
 
Main sequence is the most prominent region in a band that runs from upper-left corner (extremely bright hot stars) to lower-right corner (faint, cool stars). It contains most of the stars that can be plotted. Sun is G2 main-sequence star appearing iclose to the middle. Temperatures for main-sequence stars varies from approximately 3000 degrees Kelvin for M stars to approximately 40,000 degrees Kelvin for O stars. Main-sequence stars are all members of one luminosity class, luminosity class V. However, they vary from extremely luminous O stars to very faint M stars (dwarfs) with a range of about a billion in luminosity and a range of 100 in size. These stars are numerous and have a common internal structure. Hence, these group of stars form a single class called main sequence. If we observe the number of stars in each spectral type, the number of M-type stars far exceeds the number of A or F-type stars. Also, the number of M-type stars far exceeds number of K-type stars and so on up the main sequence. The most numerous common type of star in our Galaxy is faint, cool, M-type star. It is known as red dwarf. Astronomers think this to be true for other galaxies, but too faint to observe all over the universe.
 
Mass-Luminosity Relation for Main-Sequence Stars
L = M3.5
 
From orbital motion of binary stars, astronomers are able to estimate masses of component stars in binary system. It is observed that masses of main-sequence stars increases from spectral class M up main sequence to spectral class O.
 
Other Properties of Main-Sequence Stars
 
Radii of stars on main sequence increases from small radii M dwarfs to large radii O stars
Luminosity, temperature, radius or mass of a main sequence increases from M to O stars
 
Bright Giants and Giants (also called Red Giants) in Classes II and III
 
The two classes of giants make up the second most prominent region in H-R diagram. It is composed of bright and cooler stars
Giants are luminous stars in spectral classes F, G, K, and M lying above main sequence in region that angles up toward right cool stars in upper right-hand corner of H-R diagram.
Despite being members of one family of stars, these stars vary by at least a factor of 100 in luminosity
100 times more luminous than Sun on the average
Surface temperature varies 3000 K to 7000 K
No relationship exists between mass and temperature (spectral type) on this branch
No mass-luminosity relation has been found for red-giant stars
Radii of these stars increase progressively upward toward upper right-hand corner of H-R diagram
 
Bright Supergiants and Supergiants (also called Blue and Red Supergiants) in Classes Ia and Ib
 
Blue supergiants of classes O and B are early-type stars
Red supergiants of classes G, K, and M are late-type stars
Blue and red supergiants can be hundreds or thousands of times more luminous than Sun
Blue and red supergiants also do not possess definite relation between mass and luminosity in the H-R diagram
Radii do increase toward upper right-hand corner
Although supergiants can be seen at tremendous distances because of their enormous luminosity, they appear to be very rare in the known universe
Certainly, there are far more giant stars in our Galaxy than supergiants.
 
White Dwarfs:
White dwarfs span spectral classes of B, A, and F. This class of stars is composed of faint stars lying below main sequence
Note: when star referred to as being "on" or "off main sequence," reference is to position in H-R diagram and not to its actual position in space
White dwarfs appear to be second most populous region in H-R diagram
While the supergiants are seen at great distance across Galaxy, white dwarfs though more numerous are far less visible at great distances
White dwarfs are typically thousand times less luminous than Sun
But, surface temperature are far greater than that of Sun
White dwarf stars all possess Masses less than about 1.4 mass of Sun
Their radii are much smaller than Sun (700,000 km) making them about the size of Earth (7,000 km)
Consequently, white dwarfs must have mean densities of the order of millions of g/cm3 and a spoonful of the material would be 15 tons.
This suggests white dwarfs composed of matter in state unlike anything we possess on Earth
 
Bright Stars vs. Nearby Stars
 
Stars that are among the brightest in night sky are in general intrinsically brighter than Sun
About 70% of these bright stars are either giants or supergiants.
Some are early-type stars on upper end of main sequence, about 30%
Stars that are among closest to Solar System are in general intrinsically fainter than Sun.
Most numerous are the red dwarf stars, i.e., spectral class M stars, hard to see even with large telescopes
The second most numerous are the white dwarfs
Stars within about 15 ly of Solar System number about only 50.
About 0.004 star/ly3 or about 1.0 star/300 ly3
 
For every 1000 small, red, class M stars, there are 350 class A to K main-sequence stars, and 1 star in 4 million is an O star
Star formation clearly favors formation of small stars
Brightness of stars,
 
luminosity of Rigel (B8 Ia) = 1000[ luminosity of Vega (A0 V) ] = 50,000[ luminosity of Sun (G2 V) ] = 2.5 x 109[ luminosity of Wolf 359 (M8 V) ]
 
Range of Stellar Properties

Stellar Property

Range*

mass

10-2 to 102 Msun

radius

10-2 to 103 Rsun

mean density

10-7 to 107 rsun

luminosity

10-5 to 105 Lsun

surface temperature

103 to 105 K

heavy-element mass abundance**

0.05 to 2.0 Zsun

age

104 to 1010 y

* Solar units: Msun = 2 x 1033 g, Rsun = 7 x 10+5 km, Lsun = 4 x 1033 erg/s, Zsun = 0.02
** Fraction of all elements heavier than hydrogen and helium
 


Estimated Numbers of Stars in Our Galaxy:

 
The following is a census of stars in our Galaxy. This estimate is constructed assuming that total number of stars in the Galaxy is 400 billion. With assumed masses for stars, their contribution to mass of the Galaxy is approximately 175 x 109 Msun, and with their assumed luminosities, their contribution to luminosity of the Galaxy is approximately 20 x 109 Lsun or about 8 x 1041 ergs/s.
 
CUMULATIVE PERCENT

Luminosity Class

Spectral Class

Typical Mass

[Msun]*

Typical Luminosity

[Lsun]*

Number

of Stars

Number

Mass

Luminosity

Supergiant (I & II)

O-M

?

50,000

~105

~0

~0

~3

Red Giant (III)

F-M

~1.2

40

~2 x 109

0.5

0.6

~41

Main Sequence (V)

O

~25

80,000

~104

0.5

0.6

~42

.

B

5

200

300 x 106

0.6

1.6

~70

.

A

1.7

6

3 x 109

1.2

4.6

~79

.

F

1.2

1.4

12 x 109

4.2

13.6

~87

.

G

0.9

0.6

26 x 109

11.2

~27

~94

.

K

0.5

0.2

52 x 109

~24

~42

~99

.

M

0.25

0.005

270 x 109

~91

~80

~100

White

Dwarf

B-F

~1.0

0.005

35 x 109

~100

~100

~100

.

.

.

Total

400 x 109

100

100

100

* Sun's mass: Msun = 2 x 1030 kg; Sun's luminosity: Lsun = 4 x 1026 Joules/sec


 
Hertzsprung-Russell diagram:
Hertzsprung-Russell diagram looks for correlations between stellar properties. Hertzsprung and Russell independently found a surprising
correlation between temperature and luminosity for 90% of the stars. Also called a color-magnitude diagram. Diagonal strip is main-sequence.
Luminous ones are easier to observe but rarer to find; faint ones are harder to see but more common. H-R diagram found a correlation between mass and luminosity:
Luminosity = [Mass^3.5"]. The H-R diagram is for all stars visible to the naked eye (down to apparent magnitude = +5) plus all stars within 25 parsecs.
 

Considerable materials adopted above is from Prof. Martin's web page. 

A NICE WEB PAGE FOR PROPERTIES OF STARS: Prof. Martin's Astronomy Page

 
 
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UPDATED 04/18/00