Module 14
H. R. Diagram
TABLE OF CONTENTS
1. Learning Outcomes 2. Introduction 3. H. R. Diagram 3.1. Coordinates of H. R. Diagram 3.2. Stellar Families 3.3. Hertzsprung Gap 3.4. H. R. Diagram and Stellar Radii 4. Summary
1. Learning Outcomes
After studying this module, you should be able to
recognize an H. R. Diagram explain the coordinates of an H. R. Diagram appreciate the shape of an H. R. Diagram understand that in this diagram stars appear in distinct families recount the characteristics of these families recognize that there is a real gap in the horizontal branch of the diagram, called the Hertzsprung gap explain that during their evolution stars pass very rapidly through this region and therefore there is a real paucity of stars here derive the shape of the 퐥퐨퐠 푻 − 퐥퐨퐠 푳 plot at a given stellar radius explain that the stellar radius (and mass) increases upwards in the H. R. Diagram
2. Introduction
In the last few modules we have discussed the stellar spectra and spectral classification based on stellar spectra. The Harvard system of spectral classification categorized stellar spectra in 7 major classes, from simple spectra containing only a few lines to spectra containing a huge number of lines and molecular bands. The major classes were named O, B, A, F, G, K and M. Each major class was further subdivided into 10 subclasses, running from 0 to 9. Considering the bewildering variety of stellar spectra, classes Q, P and Wolf-Rayet had to be introduced at the top of the classification and classes R and N were introduced at the bottom of the classification scheme. Some prefixes and suffixes were also suggested to be used with the spectral classes to take account of special features of their spectra.
Earlier it was thought that the difference in spectra was due to the evolving chemical composition of the stars. However, in 1920 Saha showed that the progression of spectra from class O to M could be explained in terms of the decreasing surface temperature of stars. Saha likened the process of ionization of atoms to a chemical process, and derived what we now call Saha’s ionization formula. This formula gives the fraction of ionized atoms as a function of temperature and pressure in the stellar surface layers. Taking the examples of atoms of hydrogen, helium and calcium, we showed in the last module how the intensities of lines of these atoms vary with temperature, in agreement with the spectral classification. In this module we discuss a type of plot which has become an important diagnostic tool for the astronomers.
3. H. R. Diagram
We have already seen that astronomers are always on the lookout for relations which help them to study the objects of their interest. One such relation is the one between some measure of the luminosity of a star and some measure of its surface temperature. The relation, found independently by Hertzsprung and Russell, is named after these astronomers and goes by the name of Hertzsprung – Russell (H. R.) diagram. It is one of the most useful tools for the study of stars and their physical properties.
3.1. Coordinates of an H. R. Diagram
Besides luminosity itself, absolute magnitude is measure of luminosity. The measures of surface temperature are spectral class and colour index. Therefore, the coordinates of an H.R. diagram
are those shown in Fig. 14.1. Note that the surface temperature increases towards the left while the colour index (퐵 – 푉) increases towards the right. Similarly, the absolute magnitude, being anti-correlated with luminosity, increases downwards.
Surface Temperature AbsoluteMagnitude
y Luminosit
O B A F G K M (퐵 − 푉)
Fig. 14.1. The coordinates of an H. R. Diagram. Notice that temperature decreases towards right and absolute magnitude decreases towards the top.
Fig. 14.2 shows a schematic H. R. diagram. On the 푦-axis are the absolute magnitudes and luminosities; on the 푥-axis are the temperature and spectral class. Separation of stars in neat groups is immediately noticed.
Fig. 14.3 shows the Hertzsprung–Russell diagram with 22,000 stars from the Hipparcos Catalogue and 1,000 from the Gliese Catalogue of nearby stars.
Fig. 14.2. Schematic H. R. Diagram. Notice that the temperature and luminosity scales are not linear. The density on the Main Sequence is indicative of the actual stellar numbers of stars on it. (Source: http://chandra.harvard.edu/edu/formal/variable_stars/HR_student.html)
3.2. Stellar Families
In the H. R. diagrams, the group of stars running from the top left to the bottom right is the most populous group. This group is called the Main Sequence. On the Main Sequence the luminosity steadily decreases as we go from the early to the late spectral classes. The lower region of the Main Sequence is more crowded than the upper region. The stars at the lower end are red in colour (because of the low surface temperature) and are very small in size. These stars are therefore called Red Dwarf stars. These are the most abundant stars in the Galaxy. Another group of stars is situated below the Main Sequence, occupying the left bottom corner. The stars in this group have luminosities like those of the Red Dwarfs, but their surface temperatures are
much higher. These stars are known as White Dwarf stars. In abundance, they are next only to the red dwarfs. That their name is descriptive of their size is clear from the fact that the white dwarfs are about 10 magnitudes fainter than the Main Sequence stars of the same surface temperature, and so must have very small surface area (and radius).
Fig. 14.3. H–R diagram with 22,000 stars plotted from the Hipparcos Catalogue and 1,000 from the Gliese Catalogue of nearby stars. Notice the use of colour index (퐵 − 푉), surface temperature and the spectral class along the 푥- axis. (Source:
https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram. Diagram made by Richard Powell.)
Next in abundance are the stars called Giant Stars, lying above the Main Sequence. These stars are much brighter than their surface temperatures would suggest, indicating that they are large in size. The giant branch is almost horizontal, the luminosity of giants changing little with their surface temperature. These stars belong mostly to spectral classes F, G, K and M. Brighter than the giant stars by about 5 magnitudes (a factor of hundred in luminosity) are the Supergiant
Stars having radii of about 100 푅⨀. In this group, too, luminosity does not change much with spectral class. These stars are the least abundant stars. Between giants and the Main Sequence is a group of stars called the Subgiant stars. These belong mostly to the latter spectral classes. Finally, between the Main Sequence and the White Dwarf stars are the Subdwarf stars.
If we plot apparent magnitude instead of the absolute magnitude, no such correlation with the surface temperature is found (Fig. 14.4). There is no separation into families either.
Fig. 14.4. Lack of correlation between the apparent magnitude and the spectral
class. (Source: http://spiff.rit.edu/classes/phys230/lectures/hr/hr.html)
3.3. Hertzsprung Gap
4. Following comments are called for about the H. R. diagram:
Fig. 14.5. An HR diagram with the instability strip and its components
highlighted. (Source: Wikipedia)
1. The stars are found all over the diagram; the groups simply define the locations where the stars tend to congregate. 2. There is a real paucity of supergiant stars of spectral classes A, F and G. This defines a real gap in the supergiant branch, called the Hertzsprung gap (Fig. 14.5). The reason for the presence of this gap is that during their evolution stars stay at this location for a very, very short time. 3. The H. R. diagrams shown above feature the stars found in the spiral arms of our galaxy and other galaxies. These stars are the so-called Population I stars. These are young stars. The stars found in the globular clusters and in the central bulge of the Galaxy and other galaxies are generally old stars (Fig. 14.6). These are known as Population II stars. The H. R. diagram of the Population II stars is quite different from that of the Population I stars (Fig. 14.7).
Fig. 14.6. The Messier 80 globular cluster contains hundreds of thousands of stars (Source: NASA/ESA). The density of stars in a globular cluster is so large that the cluster appears almost spherical. All these stars are old Population II stars. Since the stars in globular clusters can be assumed to be of the same age (born at the same time), though different in masses, H.R. diagrams of these clusters throw a lot of light on stellar evolution.
Typical H. R. Diagram of a Globular Cluster
Horizontal Branch
) 4 Hertzsprung Gap
10
⨀ 퐿
1
Luminosity ( Luminosity Main −4 Sequence 10
40000 20000 10000 5000 2500 Temperature (K)
Fig. 14.7. A typical globular cluster H. R. Diagram.
H. R. diagram of globular cluster M3 is shown in Fig. 14.8.
3.4. Importance of H. R. Diagram
The importance of H. R. diagram to astronomers can hardly be overstated. It is an important tool for them since all physical properties of a star can be read from its location on the H. R. diagram. The diagram has been called the horoscope of stars, because astrologers claim that they can read the events of the entire life of a person from birth to death from her horoscope. Luminosity and temperature of stars are, of course, the coordinates of the diagram and can obviously be read. As we shall just see, the radius of a star, the mass of a star, its spectral type, its colour, the processes of energy production in its core, its age and the stage of its evolution, can all be inferred from the location of the star
on the H. R. diagram. Whenever a new object is discovered, its location on the H. R. diagram is of prime importance, because it gives clue to all its physical properties.
Fig. 14.8. H. R. diagram of globular cluster M3. (Source:
https://commons.wikimedia.org/wiki/File:M3_color_magnitude_diagram.jpg)
3.5. H. R. Diagram and Stellar Radii
We know that the luminosity of a stars is given by the Stefan-Boltzmann law:
퐿 = 4휋푅2휎푇4. (14.1)
For the Sun, we have
2 4 퐿⨀ = 4휋푅⨀ 휎푇⨀ . (14.2)
If we express luminosity, radius and surface temperature of a star in solar units, then its luminosity can be written as
퐿 = 푅2푇4. (14.3)
In terms of logarithms, we can write
log 퐿 = 2 log 푅 + 4 log 푇. (14.4)
For a fixed 푅, this equation represents a straight line in the log 푇 − log 퐿 plane. Since
푅1 log 퐿
푅2
푅1 > 푅2
log 푇
Fig. 14.9. Showing the line of Equation (14.4) for two values of the stellar radius.
temperature increases to the left; the straight line is as shown in Fig. 14.9. Fig. 14.10 shows an H.R. diagram with lines for various stellar radii drawn in its plane. We note the increase of stellar radius going upwards. Since mass of a star is also determined by its radius, it also increases upwards in the diagram.
Fig. 14.10. An H.R. diagram with stellar radii drawn on it. The stellar radius increases upwards. (Source: https://www.eso.org/public/images/eso0728c/)
4. Summary
A plot of some measure of the luminosity of stars against some measure of their surface temperature is called an H. R. diagram. Stars arrange themselves into neat families on the H. R. diagram. There is no such correlation between the apparent magnitude and the surface temperature. The major families are: Supergiant Stars, Giant Stars, Main Sequence stars, White Dwarf stars. The Red Dwarf stars on the lower reaches of the Main Sequence are the most abundant stars in the Galaxy. Next in abundance are the White Dwarf stars which lie about 10 magnitudes below the Main Sequence. These stars are very small in size. Next in abundance are the stars called Giant stars, which are really huge in size, lying above the Main Sequence. Very bright Supergiant stars are rather rare. They lie about 5 magnitudes above the Giant stars. The radii of Supergiant stars may be as large as 100
푹⨀. There is a real paucity of Supergiant stars of spectral classes A, F and G. This defines a real gap in the Supergiant branch, called the Hertzsprung gap. H. R. diagram of old stars (Pop II stars) found in the globular clusters is different from the H. R. diagram of young (Pop I) stars. In the H. R. diagram the radius and mass of stars increase as we go up.