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Module 09 Colour Index TABLE of CONTENTS 1. Learning Outcomes Module 09 Colour Index TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Dependence of magnitudes on wavelength 3.1. UBV System of Johnson and Morgan 3.2. Colour Index 3.3. Colour-Colour Diagrams 3.4. Colour Excess 3.5. Reddening and Redshift 4. Summary 1. Learning Outcomes After studying this module, you should be able to Appreciate that the magnitude depends on the wavelength at which it is measured Understand the need of UBV photometric system Realize that the UBV system at present measures magnitudes at various wavelength bands other than the visual region Understand why magnitudes of stars at various wavelength bands of their spectra are not equal Grasp the concept of colour index Understand that the colour index follows the surface temperature of the star Realize the importance of colour-colour diagrams in describing stars 2. Introduction In the last module we introduced the concept of magnitudes. Apparent magnitude is a number assigned to a star (or any other celestial object) to indicate its brightness. Brighter stars are assigned lower magnitudes and fainter stars are characterized by higher magnitudes. Only stars brighter than 6th magnitude are visible to the unaided eye. Since brightness is a function of distance and the intrinsic brightness of a star, apparent magnitude does not allow us to compare stars with respect to their intrinsic brightness. Therefore, to compare brightness all stars are thought to be at a standard distance of 10 pc. The apparent magnitude at this distance is called the absolute magnitude of the star. It turns out that the difference between the absolute magnitudes of two stars is proportional to the ratio of their luminosities. So, in absolute magnitude, we have now a handle to estimate the luminosity of as star. We also found that the difference between the apparent and absolute magnitudes of a star, the distance modulus, allows us to determine the distance of the star. It was pointed out that overwhelming majority of stars in the sky are faint stars; there is real paucity of bright stars. The number of stars brighter than 6th magnitude, those visible to the naked eye, is only about 5000. Also most stars are less luminous than the Sun; very few stars have luminosity greater than the Sun. We know that stars radiate in many wavelengths. Therefore, we now turn to the dependence of magnitudes on wavelength and introduce an important quantity: colour index. 3. Dependence of magnitudes on wavelength If the radiation of stars were monochromatic, magnitude differences would be entirely independent of the instruments used to determine them. But stellar radiation is not monochromatic. Moreover, the measuring instruments select a band of wavelengths rather than a single wavelength (Fig. 9.1). Therefore, magnitude becomes instrument dependent. However, this is not the case with the instruments which are nonselective, such as radiometers, bolometers and thermocouples. These instruments measure the true luminosities of stars once they have been corrected for absorption of radiation in the atmosphere of the earth. Unfortunately, these instruments are rather insensitive and, therefore, are not used for determining stellar luminosities. Instead one uses an indirect method which employs the idea of bolometric correction, as we shall see later. 3.1. UBV System of Johnson and Morgan Magnitudes are measured these days with special filters attached to the telescopes. The filters belong to what is called the UBV Johnson Morgan photometric system. The V band was provided to approximate to the visual magnitude system. The B band approximated to the photographic magnitude. The peak sensitivity of the filter in the visual region (denoted by V), for example, centres around a wavelength to which the eye is most sensitive, 훌545 nm in the yellow-green region of the spectrum. Similarly, filters for blue (B) and ultraviolet (U) have their peak sensitivities centred around 훌435 nm and 훌350 nm, respectively (Fig. 9.1). A filter in the red region (푹) has its peak sensitivity around 훌640 nm, and the one in the infrared region has peak around 훌800 nm (Fig. 9.2). ) 훌 Sensitivity S( Fig. 9.1. Sensitivity functions of the U, B and V filters (Johnson and Morgan, 1953). The magnitudes determined with these filters are denoted as 푚푉, 푚퐵, and 푚푈, or simply as 푉, 퐵, and 푈. Notice that these are all apparent magnitudes. The corresponding absolute magnitudes are denoted by 푀푉, 푀퐵, and 푀푈. Fig. 9.2. Modern system of filters. (Source: Bessell, Annu. Rev. Astron. Astrophys., 2005). All the magnitudes of a star in various wavelength bands are not equal, because each filter samples a different region of the blackbody spectrum corresponding to the surface temperature Blue Filter Yellow Green Filter Black Body Curve at Star’s Temperature Higher Intensity Temperature Lower Temperature Wavelength Fig. 9.3. Blue band and yellow-green band filters sample different regions of the spectrum. The blue magnitude of the star at higher temperature is lower than the yellow-green magnitude. Opposite is the case with the star at lower temperature. of the star. Referring to Fig. 9.3, the blue filter samples higher intensity region of the spectrum at higher temperature while yellow-green filter samples region of lower intensity of the same curve. At lower temperature the blue filter samples the lower intensity and the yellow-green filter samples the higher intensity. Since higher brightness is associated with a lower magnitude, and vice versa, the blue magnitude of the star at higher temperature in Fig. 9.3 is lower than its visual magnitude. On the other hand, for the star at lower temperature in the same figure, the blue magnitude will be slightly higher than the yellow-green magnitude. 3.2. Colour Index The difference between the magnitudes of a star at different wavelengths is called its colour index (C.I.), or just the colour, of the star. It is obvious that the colour index is a measure of the surface temperature of the star. That is why instead of the temperature of the star, many times its colour index is stated. We shall state the precise relationship between the temperature and the colour index a little later. One kind of colour index is defined as C. I. = 푚퐵 − 푚푉 = 퐵 − 푉. (9.1) The colour index is stated in units of magnitude. Another colour index is 푈 − 퐵: 푚푈 − 푚퐵 = 푈 − 퐵. (9.2) The reference star for measuring colour indices is star Vega (Surface temperature, 9602 K; Apparent magnitude, 0.03; Luminosity, 50 푳ʘ; Distance, 7.68 pc). For this star, by definition, all the three indices are equal, that is, 푼 = 푩 = 푽, which implies that 푩 − 푽 = ퟎ. ퟎ and 푼 − 푩 = ퟎ. ퟎ. So, star Vega is a white star. A star with 푩 − 푽 < ퟎ has higher intensity in the blue region than Vega. We usually say that the star is bluer than Vega; a star with 푩 − 푽 > ퟎ is redder than Vega. Table 9.1 shows the sample calibration of various colour indices for stars of various surface temperatures, 푻풆풇풇. The first column in the Table shows the spectral class of the star. We shall study spectral classification of stars later. For the present, you can ignore the first column. Table 9.1. Sample Calibration Colour Indices of Stars of various Surface Temperatures Class 푩– 푽 푼– 푩 푽– 푹 푹– 푰 푻풆풇풇 (K) O5V –0.33 –1.19 –0.15 –0.32 42,000 B0V –0.30 –1.08 –0.13 –0.29 30,000 A0V –0.02 –0.02 0.02 –0.02 9,790 F0V 0.30 0.03 0.30 0.17 7,300 G0V 0.58 0.06 0.50 0.31 5,940 K0V 0.81 0.45 0.64 0.42 5,150 M0V 1.40 1.22 1.28 0.91 3,840 Source of data: Zombeck, Martin V. (1990). "Calibration of MK spectral types". Handbook of Space Astronomy and Astrophysics (2nd ed.). Cambridge University Press. p. 105. ISBN 0-521-34787-4. Look at the 퐵 − 푉 colour index of stars of temperatures 42000 K and 30000 K. They have very high intensity in the blue colour of their spectra. These are called blue stars. Stars of temperature ~ 1000, of the type of star Vega, are white. Stars of lower temperatures are redder than Vega, redness increasing as the temperature decreases. Sun, with a surface temperature ~ 6000 K is yellowish in colour, while stars with temperatures ~ 4000 K are red stars. Colour index (푩 − 푽) for the Sun has been estimated as 0.656. A category of dwarf stars are red dwarf stars. These stars actually outnumber all other categories of stars. Their surface temperatures are low and their masses are a fraction of solar mass while -4 their luminosities are typically 10 L⨀. There are also stars which are many times larger in size than the Sun. These, the so-called red giants and red supergiants are also red because of their low surface temperatures. Another collection of red stars is a globular cluster. Globular clusters are very old clusters, 10 billion years or older. They contain very old, therefore, red stars. There are about 150 of such clusters in the Milky way galaxy. The new-born stars are blue because of their high surface temperatures. Galaxies in which there is a burst of new stars, star burst galaxies, appear blue, because of preponderance of blue stars in them. An example of a blue star is the star Rigel, the brightest star in Orion constellation. Just like red giants, there exist also blue giant stars, stars of huge sizes and high surface temperatures.
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