The spectrum of electromagnetic radiation ranges from gamma rays , X-rays, UV, visible light to IR, microwaves, and radiowaves. This is also a sequence of

8 Most cholesterol is synthesized in the body, some is of dietary origin. LDL (low density lipoprotein) is harmful; HDL (high density lipoprotein) is thought to be beneficial.

decreasing energy and increasing wavelength (decreasing frequency) of the radiation. The wavelength X and frequency v are related by c = Xv (9.7)

where c=300,000 km/s, the speed of light. The energy of electromagnetic radiation (Fig. 9.12) depends on its frequency:

h is the Planck constant, h = 6.626 x 10-34 J • s.

Some examples of energy of different types of radiation:

To a good approximation, we can assume that stars radiate like a black body. A black body is a theoretical idealization: an object that absorbs completely all radiation at all wavelengths. The radiation of a black body at temperature TS is given by the Planck law:

Here, Iv is the intensity of radiation at frequency v; h, k, and c are Planck's constant, Boltzmann's constant, and the speed of light, respectively. k = 1.38 x 10-23JK-1.

Increasing Frequency (V)

1034 10" 10™ 10" ¡II"1 I0U Kl" 10'" 10s 10fi W 10* 10° v (Hz)

Increasing Frequency (V)

1034 10" 10™ 10" ¡II"1 I0U Kl" 10'" 10s 10fi W 10* 10° v (Hz)

Y rays

X rays

UV

IR

Mkrowave

FM

AM

1|ri6 1()-I4 |(riz 1 (>-'■> JO"8 \ ¡nr6 10"^ 10"2 10° HI2 IO4 It)'' 10s Mm)

Increasing Wavelength (K) -»

1|ri6 1()-I4 |(riz 1 (>-'■> JO"8 \ ¡nr6 10"^ 10"2 10° HI2 IO4 It)'' 10s Mm)

Increasing Wavelength (K) -»

" Visible spectrum

Increasing Wavelength iKi ill Ttm ~

Fig. 9.12 Electromagnetic radiation. Only the visible part (400-700 nm), some part of the IR, and the radio part is observable from the ground. Wikimedia Cosmos

9.2.2 Spectral Lines

By observing spectral lines, we can deduce many important parameters: the elements that emit these lines, relative velocities between source of emission and observer (Doppler effect), temperature, pressure, etc.

Spectral lines can be in emission or absorption. In atoms, the electrons are found on defined orbital states. To make a transition to a higher state, energy in the form of a photon is required—this causes an absorption line; when the electron jumps back to its original state (which depends on the temperature and pressure), then, energy is emitted in form of an emission line. Electrons can only be found in defined energy states (see textbooks on quantum mechanics).

Let us calculate a very simple example: what is the energy released when in a hydrogen atom, a transition from orbital states n=3 to n=2 occurs? The formula is (RH is the Rydberg constant)

For the wavelength, X = 6,563nm is obtained, which corresponds to 1.9 eV. The energy from n = 1 to ^ corresponds to 91.13 nm or 13.6 eV. This energy is needed to ionize a hydrogen atom from its ground state (Lyman series). The lines of the Balmer series are in the visible part of the spectrum and denoted as Ha (n = 3 to n = 2), H^ (n = 4to n = 2), etc. and they converge to n = 2 ^ ^ and X = 364.51 nm. The electron volt, eV, is given by:

Energy, frequency, and wavelength of a photon are related by hc 1240 nm E = hv = J =—x-eV (9.12)

9.2.3 Stellar Parameters

Stars are gaseous spheres being most of their lifetime in hydrostatic equilibrium. The only information we can directly obtain from a star is its radiation and position.

To understand the physics of stellar structure, stellar birth, and evolution, we have to derive quantities such as stellar radii, stellar masses, composition, rotation, magnetic fields, etc. We will just very briefly discuss how these parameters can be obtained for stars.

• Stellar distances: a fundamental but not an intrinsic parameter. Stellar distances can be measured by determining their annual parallax, that is, the angle the Earth's orbit would have seen from a star. This defines the astrophysical distance unit parsec. A star is at a distance of 1 parsec (pc) if its parallax is 1". 1 pc = 3.26 light years.9

• Stellar temperatures: can be derived from Planck's law (see Sect. 9.2.1).

• Spectral lines indicate which elements are present. By measuring the wavelength X of an observed stellar spectral line and comparing it with the laboratory wavelength Xo, we can determine the radial velocity vr of the star:

If the lines are blueshifted, then, the object approaches the observer; if they are redshifted, the objects moves away. If the star expands, then, the lines will also be shifted.

• Stellar radii: once the apparent diameter of a star is known, its real diameter follows from its distance d. The problem is to measure apparent stellar diameters, which are extremely small because of the large stellar distances. One method is to use interferometers and another method is to use occultation of stars by the moon or mutual occultations of stars in eclipsing binary systems. Consult general textbooks about astronomy.

• Stellar masses: can be determined by using Kepler's third law in case we observe a binary system. Stellar evolution strongly depends on stellar mass, however, we know accurate masses only for some 100 stars.

• Once mass and radius are known, the density and the gravitational acceleration follow. These parameters are important for the stellar structure.

• Stellar rotation: for simplicity, we can assume that a star consists of two halves: one half approaches to the observer and the spectral lines from that region are blueshifted, the other half moves away and the spectral lines from that area are redshifted. The line profile we observe in a spectrum is a superposition of all these blue- and redshifted profiles, therefore, stellar rotation causes a broadening of spectral lines;

• Stellar magnetic fields: magnetically sensitive spectral lines are split into several components under the presence of strong magnetic fields. This is called the Zeeman effect and the amount of splitting AX depends on g... Lande factor, follows from quantum mechanics. Lines with a g = 0 are not split. H... magnetic field strength. Note that the amount of splitting depends on the square of the wavelength, so that this effect is larger in the IR.

c Xo

9 1 Lyr is the distance that light propagates in 1 year at a speed of 300,000 km/s. The distance is thus 365 x number of seconds per day (86,400) x speed of light (300,000 km/s) = 1013 km.

9.2.4 Stellar Spectra, the Hertzsprung-Russell Diagram

The analysis of stellar radiation is fundamental for the derivation of physical quantities describing a star. Putting a prism or a grating inside or in front of a telescope, we obtain a spectrum of a star. Such a spectrum contains many lines, most of them are dark absorption lines. Each chemical element has a characteristic spectrum.

In the HRD, the temperature of stars is plotted versus brightness (Fig. 9.13). The temperature of a star is related to its color (Wien's law): blue stars are hotter than red stars. In the HRD, the hottest stars are on the left side. The temperature increases from right to left. Stellar brightness is given in magnitudes. The magnitude scale of stars was chosen such that a difference of 5 magnitudes corresponds to a factor of a 100 in brightness. The relation between intensity and magnitude is given by

The smaller the number (which can be even negative) the brighter the star. The brightest planet Venus, e.g., has magnitude -4.m5 and the Sun has -26.m5. The faintest stars that are visible to the naked eye have magnitude +6.m0. Since

O5 G

Fig. 9.13 Sketch of the Hertzsprung-Russell diagram with evolutionary path of the Sun

O5 G

Fig. 9.13 Sketch of the Hertzsprung-Russell diagram with evolutionary path of the Sun apparent magnitudes depend on the intrinsic luminosity and the distance of a star, absolute magnitudes (designated by M) are defined as the magnitude a star would have at a distance of 10 pc. In the HRD, often absolute magnitudes are plotted as ordinates instead of luminosity. The relation between m and M is given by m - M = 5logr - 5 (9.16)

r is the distance of the object in pc. The Sun has M = +4.M5; seen from a distance of 10 pc, it would be among the fainter stars but still visible with the naked eye.

How can we determine stellar temperatures? If the Planck equation (9.9) is integrated over all frequencies (wavelengths), we obtain a formula for the total power emitted by a black body, the Stefan-Boltzmann law:

and for the luminosity of a star:

For the Sun, Teff = 5,785K. This formula defines the effective temperature Teff of a star. o = 5.67 x 10-8 Wm-2K-4 is the Stefan-Boltzmann constant.

What is the power emitted per unit area on the Sun's surface? Answer: put T = 6,000 K, we find that the Sun radiates 70 MW per m2 of its surface.10

By taking the derivative with respect to X of Planck's law and setting it equal to zero, one can find the peak wavelength, for which the intensity is at maximum:

This is also called Wien's law.

At about which wavelength can planets be expected to radiate most of their energy? Answer: Let us assume the temperature of the Earth = 300 K. Then,

The Sun has a surface temperature of about 6,000 K. At what wavelength does the Sun's spectrum peak? Answer:

10 The worldwide nuclear energy generation is about 350 GW. Thus, an area of 5,000 m2 on the Sun generates this amount

The temperature derived from the peak wavelength is called Wien Temperature, the temperature derived from the difference of intensity between two wavelengths (=color) is called Color temperature, etc. To measure color, a filter system must be defined. The most commonly used system is the UBV-system (Table 9.3) which has three bands that are located in the UV (U), blue (B), and visual (V) to measure the intensity Iv. The color of a star is measured by comparing its magnitude through one filter (e.g., red) with its magnitude through another (e.g., blue).

Table 9.3 Central wavelength and bandwidth of the UBVRI filter set

Name Meaning CentralX Bandwidth [nm]

Table 9.3 Central wavelength and bandwidth of the UBVRI filter set

Name Meaning CentralX Bandwidth [nm]

 U Ultraviolet 360 66 B Blue 440 98 V Visual (green) 550 87 R Red 700 207 I Infrared 900 231

For example, mV means the magnitude measured with the V filter. Therefore, instead of determining temperatures from the comparison of the spectrum of a star with the Planck law, one can use, e.g., color indices. The value B — V will be (see, e.g., Table 9.4):

• Positive for the cooler star, because it is brighter in V than in B (blue). If the cool star is brighter in V, it means that its magnitude has a lower value and, therefore, B-V is positive.

• Negative for the hotter star. The hotter star is brighter in B than in V, mB < mV and B-V<0.

 Star B-V Effective T Sun +0.6 5,800 K Vega 0.0 10,000 K Spica -0.2 23,000 K Antares +1.8 3,400K

Habitable planetary systems need host stars in the spectral range F, G, or K. By measuring the colors of stars, these spectral types can be determined easily. F and G stars are yellow; K stars are orange.