Magnitudes and the H-R Diagram

The Inverse Square Law

We are all familiar with the fact that at night the headlights of an approaching car appear brighter as they get nearer. From experience, we feel that fainter corresponds to farther. How bright a light source seems, the apparent brightness, does indeed depend on the distance. However, it also depends on how bright the source actually is. This actual brightness is related to the amount of energy being emitted. It is the wattage of the light source. The apparent brightness may also depend on the presence of intervening material. For example, approaching headlights seem fainter when there is fog.

Disregarding the latter complication, the inverse square law of light states that the apparent brightness, b, is directly proportional to the actual brightness, B, and inversely proportional to the distance, D, squared,

b ~ B/D²

To illuminate this law, consider 60 and 120 watt light bulbs placed at the same distance. The 120 watt bulb is actually twice as bright as the 60 watt bulb and will therefore appear to be twice as bright. If the distance to the bulbs is doubled, both will appear only one-fourth as bright. Removed to three times the original distance their apparent brightness becomes only one-ninth as great. On the other hand, if the 120 watt bulb were placed twice as far away as the 60 watt bulb, it would appear to be one-half (2/4) as bright as the 60 watt bulb. As another example, the Sun is the brightest star in our sky. But if we could view the Sun from 30 times farther out, it would appear to be only onenine-hundredth (1/302 =1/900) as bright. The Sun appears bright not because it is actually brighter than other stars but because it is so very much closer.

The Magnitude System

In the second century BC the Greek astronomer Hipparchus grouped the stars visible to the naked eye into six brightness categories. The apparently brightest stars he called first magnitude stars, the next, second magnitude stars, and so on to the just barely visible sixth magnitude stars. The numbers assigned to stars based on their apparent brightness are known as apparent magnitudes. Astronomers today use an extended version of Hipparchus’ system. With telescopes it is possible to detect very faint stars having magnitudes of around +27. At the bright end of the magnitude scale is the Sun with an apparent magnitude of about -27. Notice that very bright stars have large negative magnitudes while very faint stars have large positive magnitudes. One of the closer stars, Alpha Centauri, has a magnitude of 0 right at the center of the magnitude scale. The apparent brightness of other objects such as planets and comets can also be expressed in terms of magnitudes.

Since apparent magnitudes depend on distance, they cannot be used to compare the actual brightness of objects unless their distances are known. Sometimes it is convenient to imagine all stars being at the same distance. If this were the case, apparent brightness would be a true indication of an object’s actual brightness. Astronomers define absolute magnitude as the apparent magnitude that an object would have if it were at a distance of 10 parsecs. A parsec (pc) is 3.26 light years (l.yr.) or approximately 19.6 trillion miles.

The human eye perceives brightness in such a way that a difference of one magnitude corresponds to a factor of 2.512 in brightness. That is, a first magnitude object is 2.512 times brighter than a second magnitude object, and a second magnitude object is 2.512 time brighter than a third. A difference of two magnitudes, however, corresponds to a factor of 2.512 times 2.512, or as it is often expressed, 2.512². Thus, a first magnitude star is 2.512² or 6.31 times brighter than a third magnitude star. A difference of three magnitudes is a factor of 2.512³.

The relationship between magnitude difference and apparent brightness can be expressed as a brightness ratio:

b_n/b_m = 2.512^(m-n)

where bn and bm are the apparent brightness of objects having apparent magnitudes n and m, respectively. For example, the brightness ratio for a first and third magnitude object is b1/b3 = 2.512^(3-1) = 2.512². The magnitude difference between the apparently brightest object, the Sun, and the faintest observable object is 54 magnitudes. The corresponding brightness ratio is

b_-27/b₂₇ = 2.51254

This is truly an astronomically large range in the brightness of observable objects.

The actual brightnesses of two objects can also be compared using the above formula. In this case the brightness ratio would be written as

B_N/B_M = 2.512^(M-N)

where BM and BN are the actual brightnesses of objects having absolute magnitudes M and N. If one of the two objects is the Sun,

where 4.8 is the Sun’s absolute magnitude.

You intuitively judge the distance of approaching headlights at night by comparing how bright they appear with your knowledge of how bright the headlights should be. That is, you compare the apparent brightness and the actual brightness of the headlights. Astronomers do the same sort of thing using the inverse square law of light. The absolute magnitude, M, is a measure of the actual brightness while the apparent magnitude, m, is related to the apparent brightness. Using the inverse square law and the expression for brightness ratio, it is possible to show that the distance of an object is

D = 10 x 2.512 ^0.5(m-M) parsecs

The difference between apparent and absolute magnitudes, m - M, is called the distance modulus. If the distance modulus is known, the distance to the star can be computed.

Spectral Classification

Light can be thought of as energy in the form of a wave. Each wave can be characterized by giving its wavelength, which is the distance between two peaks or troughs (see Figure 10.1).

Figure 10.1 Wavelength

The energy associated with the wave is inversely proportional to the wavelength. So short waves have the largest energy and long waves the smallest energy. The entire range of wavelengths that light can assume ranges from short wavelength, high energy gamma rays to long wavelength, low energy radio waves. This range of energies is referred to as the electromagnetic spectrum (see Figure10.2).

Figure 10.2 The Electromagnetic Spectrum

When sunlight is passed through a prism a band of colors ranging from violet to red is produced. These colors correspond to wavelengths of light between 4000 and 7000 angstroms, a unit equal to 10-8 centimeters. This very small part of the entire electromagnetic spectrum is called visible light because our eyes perceive these energies as color.

The visible spectrum of the Sun and stars can be photographed with a device known as a spectrograph. The resulting picture is called a spectrogram.

The spectra of most stars are absorption spectra, composed of discrete dark lines produced by various atoms and ions (see Figure 10.3).

Figure 10.3 Absorption Line Spectrum

Some spectra also have broader bands produced by molecules. Since each atom, ion, or molecule has a unique and characteristic set of lines or bands, the chemical composition of stars can be determined.

Stars can be classified using criteria based on the appearance of their spectra. The Draper catalog, published in 1890, was the initial effort of Harvard University astronomers to classify stars based on the varied patterns of lines and bands in stellar spectra. One of the greatest women astronomers of the time, Williamina Fleming, made monumental contributions to the development of the Draper catalog, which contained spectral classifications for 10,351 stars.

Most stars have very similar chemical compositions. They are composed primarily of hydrogen with some helium and very little of the remaining elements. The variation in the appearance of stellar spectra is therefore not due to large differences in chemical composition but rather results from a variation in the temperatures of stellar atmospheres, where the absorption spectrum is formed.

The horizontal axis of the graph in Figure 10.4 is labeled with the letters astronomers call spectral classes. Each class is subdivided into 10 subclasses designated by a number. For example, A0, A1, A2, . . ., A9. Not all spectral classes and subclasses are shown in the figure. The vertical axis of the graph indicates the temperature in degrees Kelvin (deg K). It is obvious that the atmospheres of B0 class stars are considerably hotter than those of G2 stars such as the Sun. The coolest stars are those of spectral class M. Since the color of a star depends on its temperature, the spectral sequence, as the arrangement is called, is also a color sequence.

Figure 10.4 The Spectral Sequence

The hottest stars, O (not shown) and B type, are blue-white, while G type stars are yellow like the Sun, and the coolest M type stars are red. Notice that the relationship between temperature and color is just the opposite of what everyday usage suggests, namely, blue with cold and red hot!

The Hertzsprung-Russell diagram

Early in the twentieth century the Danish astronomer Ejnar Hertzsprung and the American astronomer Henry Norris Russell independently developed a diagram that is now named after them and is often referred to simply as the H-R diagram. This diagram, which is one of the most important in modern astronomy, is a plot of the absolute magnitude of stars versus their spectral class. Since absolute magnitude is a measure of actual brightness and spectral class is related to temperature, the H-R diagram is really a plot of actual brightness versus temperature. In Figure 10.5 the diagram is shown in both ways. Note that stars are located in the four areas labeled Supergiants, Giants, Main Sequence, and White Dwarfs.

Figure 10.5 The Hertzsprung-Russell Diagram

Besides its spectral class a star is also assigned a brightness, or as it is more often called, a luminosity class. For example, main sequence stars belong to luminosity class V, giants to class III, supergiants to class I, and white dwarfs to Wd.

Recall that the relationship between magnitude difference and brightness can be expressed as a brightness ratio,

where BN and BM are the brightness of objects having absolute magnitudes N and M, respectively. The actual brightness of the Sun and a star can be compared using this formula. In this case the formula is written as

where BSun and BN are the actual brightness of Sun and star, 4.8 is the absolute magnitude of the Sun, and N is the absolute magnitude of the star.

In addition, it should be noted the actual brightness, B, of a star is determined by its radius, R, and its temperature, T. The relationship between these quantities is given by the expression

B/BSun = (R/RSun)2 x (T/TSun)4

where BSun, RSun, and TSun are, respectively, the intrinsic brightness, radius, and temperature of the Sun.