Depending on the temperature of an object, it thermally radiates a unique spectral flux. A black body is an object that absorbs all the electromagnetic energies incident on it and does not reflect anything. On the other hand, if the temperature of the black body is non-zero it radiates energy governed by Planck’s law in Eq. (5.1).
I(ν, T) = 2 2 3 1
The quantity I is called the spectral radiance or spectral intensity, which is the energy per unit time per unit surface area per unit solid angle per unit frequency and is measured in W m–2 sr–2 Hz–1, ν is the frequency of radiation measured in hertz, T is the temperature of the black body in K, h is the Planck’s constant (6.6260693
× 10–34 Joules/Hz), c is the velocity of light (2.998 × 108 m/sec), e is the base of natural logarithm (2.718282), and k is the Boltzmann constant (1.3806505 × 10–23 joule/kelvin). If the black body is in thermal equilibrium with the surroundings, then Kirchoff’s law states that the emissivity of the body is equal to its absorptivity.
The radiated energy increases rapidly with increases in temperature and can be expressed by Steffan-Boltzmann law as
F = σT4 W/m2. (5.2)
Here, F is called the energy flux or power per unit radiated from an object and σ = 5.7 × 10–8 W m–2 K–4 is the Steffan-Boltzmann constant for all wavelengths.
The total energy radiated from an object per unit time is defined as the luminosity.
The surface area of a star can be calculated from its radius R as 4πR2 and thus the luminosity L of the star can be expressed as
L = 4 π R2σ T4 W. (5.3) Eq. (5.3) clearly indicates that the luminosity of a star is directly proportional to its surface area and to the square of the square of its temperature. The wavelength at which the emission spectra of black body is maximum, i.e., λpeak is also dependent on its temperature and is governed by Wien’s displacement law as expressed in Eq. (5.4).
λpeak = 2.9 10 3
T m.
× − (5.4)
When viewed from a distance like from Earth, the observed flux of an object, also known as its apparent brightness, is the power that we actually receive from it. This quantity is generally measured in Wm–2 and is dependent on the distance of the object. If r is the distance between the object and point of measurement, then apparent brightness or observed flux f can be expressed as
f = 2 4
L r ⋅
π (5.5)
The brightness of the star is sometimes expressed logarithmically and called magnitude. These could be (i) apparent or (ii) absolute, respectively, denoted by m and M. The apparent magnitude m of a star is a measure of its apparent brightness as seen by an observer on Earth. Since the amount of light received actually depends on the thickness of the Earth’s atmosphere coming in the line of sight to the object, the apparent magnitudes are normalized to the value it would have outside the atmosphere. The apparent magnitude mx in the spectral band x is defined as
mx = –2.5 log10 (fx) + C (5.6)
Here, fx is the flux at the spectral band x and C is a constant, which depends on the units of the flux and the band. Thus, the dimmer an object appears, the higher its apparent magnitude. On the other hand, absolute magnitude refers to a measurement at a fixed distance of 10 parsecs. That is, if the luminous object is placed at a fixed distance of 10 parsecs from Earth, the magnitude measured would be its absolute magnitude. If the actual distance of the object is known, then the absolute magnitude M is expressed as
Mx = mx – 5(log10 DL – 1). (5.7) Here, DL is the luminosity distance in parsecs. For the nearby objects like
the stars in our galaxy, the luminosity distance DL is more or less identical to the real distance, since the space-time is almost Euclidean. For much more distant objects, this approximation is invalid and the general relativity must be applied for calculating. Table 5.1 lists the apparent and absolute magnitudes of some stars.
Although in a true sense a star is not strictly a black body, it may be approxi-mated with it within some range of frequency. The reason being is that the star contains various elements, which produce absorption lines at different wavelengths depending on the elemental composition of its photosphere and chromospheres.
Thus, if the luminosity of a star is plotted as a function of wavelengths, absorbing type glitches would be found on the curve. Moreover, the star also produces sudden active emissions such as radio bursts, etc., whose intensity are much higher than when the star is quiet. If the glitches in the spectrum produced out of absorption lines in a star are neglected, then within a small range of spectrum the quiet stars may be approximated to a black body as shown in Fig. 5.1.
THE STARSINTHE SKY
51
Figure 5.1: Approximate black body models of stars at different temperatures using Planck’s law. Note that the wavelength at peak emission is highly temperature dependent and is governed by Wien’s displacement law.
A star like our Sun may be approximated to a black body when the glitches due to spectral absorption are neglected and when it is quiet for within a range of 10–7 to 10–4 m wavelengths as shown in Fig. 5.2. The Sun produces an extra amount of electromagnetic radiations when certain activities in it are alive. These
Figure 5.2: The Sun is approximated with a black body at 5800 K. This approximation is reasonable within the range of the wavelength between 10–7 to 10–4 meters.
confer to the large bursts in the gamma-rays and flares in upper ultraviolet and x-rays, microwaves, and radio waves. The shaded regions in the diagram show the most probable ranges of intensity fluctuation of these special radiations, which vary with time and are unpredictable. However, at certain times, the amplitude range could exceed.
Table 5.1: Apparent and absolute magnitudes of some stars.
S.N. Name of the
2. Sirius Canis Major –1.46 +1.45 2.63
3. Canopus Carina –0.72 –5.63 29.14
4. Arcturus Bootes –0.04 –0.30 11.34
5. Rigil Kent A Centaurus –0.01 +4.34 1.35
6. Vega Lyra +0.03 +0.58 7.66
7. Capella Auriga +0.08 –0.48 12.88
8. Rigel Orion +0.12 –6.75 236.08
9. Procyon Canis Minor +0.38 +2.68 3.37
10. Achernar Eridanus +0.46 –2.76 42.92
18. Fomalhaut Piscis Austrinus +1.16 +1.73 7.67
19. Mimosa Crux +1.25 –3.92 107.31