A spectral line is a dark or bright line over a continuum spectrum. Each chemical element pro- duces its own unique pattern of spectral lines, a unique set of wavelengths characteristic of that
1.2. SPECTRAL LINE FORMATION 19
element alone, allowing us to study the composition of celestial objects with spectral analysis. Figure 1.10 shows the spectra of Sodium in emission and absorption. To understand the differ- ences between these two spectra, we must remember some basic quantum mechanics: given the Plank-Einstein relation:
E=hc
λ (1.20)
where h is the Planck’s constanth=6.62606876e34 J s (Carrol & Ostlie, 2007). This relation shows that the energy of a photonE is inversely proportional to its wavelengthλ. The wavelength is related to the frequency (ν) asλ=c/ν. Given Bohr’s frequency condition:
∆E=hν (1.21)
where∆Eis the energy difference between two energy levels involved in an electronic transition. When the electron jumps from one level to another, the atom emits or absorbs a pho- ton of energy equal to the difference in energy between the two levels. From Equation 1.20, if two photons have the same energy, they must also have the same wavelengthλ. Hence, if an atom can emit photons of a certain wavelength, it can also absorb photons of precisely the same wavelength, explaining what we see in Figure 1.10; a hot, dense gas or solid object produces a featureless continuous spectrum, a blackbody. A hot, diffuse gas produces bright emission lines. These are produced when an electron jumps from a higher to a lower orbit. The energy lost is carried away by a single photon that has two ways to decay: to ground state or to cascade one orbital at a time. A cool, diffuse gas in front of a source of continuous spectrum produces dark absorption lines in the continuous spectrum. These are produced when an electron jumps from a lower to a higher energy level, absorbing a photon that carries exactly the amount of energy equal to the difference in energy between the higher orbit and the electron’s initial orbit.
These observations are summarised in Kirchhoff’s laws (Kirchhoff, 1860):
1. A hot opaque body, such as a perfect blackbody, or a hot dense gas produces a continuous spectrum.
2. A hot transparent gas produces an emission line spectrum.
3. A cool, transparent gas in front of a source of a continuous spectrum produces an absorp- tion line spectrum.
These results are used for the identification of chemical elements found in the stars (Fig- ure 1.10), and also to study their velocities. The motion of a light source also affects wavelengths,
20 CHAPTER 1. INTRODUCTION
Figure 1.10: Emission and absorption spectra of Sodium.
permitting us to deduce how fast stars and other objects are approaching or receding. Measur- ing the non-relativistic Doppler shift of spectral lines can be used to determine the star’s radial velocity (vr): λobs−λrest λrest = ∆λ λrest = vr c (1.22)
whereλobsis the observed wavelength,λrestis the rest wavelength andcis the speed of light.
An ideal surface that absorbs all wavelengths of electromagnetic radiation that are inci- dent upon its surface, will also emit at all possible wavelengths. This ideal surface is called a blackbody, and the continuous spectrum radiation that is emitted is known as blackbody radia- tion, which at absolute temperatureT is given by
Bν(ν,T)=2hν 3 c2 1 e hν kB T−1 (1.23)
whereBνis the energy per unit time radiated per unit area of emitting surface by a blackbody of temperatureT,his the Planck constant, c is the speed of light,kB is the Boltzmann constant (1.3806488(13)×10−23J K−1)andνis the frequency. This is the greatest amount of radiation that any body at thermal equilibrium can emit from its surface. Figure 1.11 shows blackbody curves for different temperaturesT, we can see that blackbodies with different temperatures will peak in different colours of the visible spectrum.
Bearing these ideas in mind, it is possible to have a physical understanding of the nature of light in astronomical sources.
1.2.1 Spectral Features in CBs
In the following Sections, we will describe some of the characteristic spectral features of both CVs and LMXB.
1.2. SPECTRAL LINE FORMATION 21
Figure 1.11: Blackbody radiation curves. Image created for educational purposes, courtesy of “Darth Kule”.
Spectral Features of CVs
CVs spectra originate from a range of individual components with different temperatures. If we remember equation 1.23, different temperatures will peak in different regions of the spectrum, making CVs suitable to study from the X-ray to the infra-red (Hellier, 2001). In optical wave- lengths, the spectrum is usually dominated by bright accretion disc lines from low ionisation states of hydrogen, helium, iron, calcium and oxygen: HI, HeI, HeII, Fe II, CaII and OI. Some- times, the Bowen blend and other heavier elements are also present.
White Dwarf: The WD has typical temperatures ofT∼10000Kand peaks near the ultra-violet. The high atmospheric pressure of WDs causes broadening of its spectral lines, most of which are absorption lines of hydrogen. In accreting WDs, some narrow metal lines can also be seen in absorption, due to accreted gas. Detecting any WD absorption line in a CV spectrum can allow us to directly determineK1. Unfortunately, these are only visible when they outshine the accretion
22 CHAPTER 1. INTRODUCTION
Figure 1.12: Accretion disc’s double peaked profile formation. The hatched areas are regions of constant velocity. The corresponding radial velocities of the disc hatched on the left (a) are indicated by the same hatched pattern on the double peaked profiles on the right (b). From Horne & Marsh (1986).
Donor Star: As the donor star is a cool body (T ∼3000−6000K), the spectrum is often sig- nificant only at red and infrared wavelengths. Spectral absorption features caused by titanium oxide (TiO) molecules can be present in M dwarf type secondaries. If enough energy is available to excite the atoms on the irradiated face of the donor, we may also see emission lines.
Accretion Disc: The accretion disc is often the dominant feature in a CV and it is normally modelled as the sum of a series of blackbodies with temperatures appropriate for each radius of the disc (hotter on the inside and cooler on the outside of the disc), although this is a quite simple approximation. It can show either emission or absorption lines, depending on its temperature. In a quiescent CV, it is optically thin, showing double peaked emission lines due to the radial velocities of the rotating disc projected along our line of sight, as shown in Figure 1.12. During outburst, the disc becomes optically thick and the spectral features become often dominated by strong, single peaked absorption lines. As the disc is centred on the WD, it follows its motion about the common centre of mass. When no WD feature is present in the spectrum, we use this fact to indirectly traceK1through the radial velocity of the disc.
Total Spectrum: The total spectrum of a CV will be a superposition of the individual spectra of the WD, the mass donor and the accretion regions. The extent to which each component will be visible depends on the accretion rate. In high accretion rate systems, the disc will dominate. In low accretion rate systems, we may see the WD and donor features on top of the disc’s spectrum, allowing direct measurement of their radial velocities. Figure 1.13 shows an example of CV spec- tra over a wide range of wavelengths. Overlapped in grey are the models of a WD and a late type