Spectral Line Shapes. Line Contributions

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1 Spectral Line Shapes Line Contributions The spectral line is termed optically thin because there is no wavelength at which the radiant flux has been completely blocked. The opacity of the stellar material is greatest at the line s center and decreases moving into the wings. The center of the line is formed at higher (and cooler) regions of the stellar atmosphere. Moving into the wings, the line formation occurs at progressively deeper (and hotter) layers of the atmosphere. 1

2 Line Broadening The three contributors to the broadening, depth, and shape of lines are Natural Broadening Doppler Broadening Collisional or Pressure Broadening 1. Natural Broadening Spectral lines cannot be infinitely sharp, even for motionless, isolated atoms. According to Heisenberg s Uncertainty Principle, as the time available for an energy measurement decreases, the inherent uncertainty of the result increases. Because an electron in an excited state occupies its orbit/level for only a brief instant, t, the orbit s energy, E, cannot have a precise value. Thus the uncertainty in the energy, E, or the orbital is h / 2 E t Electrons can make transitions from and to anywhere within these fuzzy energy levels, producing an vary small uncertainty in the wavelength of the photon absorbed or emitted in a transition. 2

3 Natural Broadening We find the width of a spectral line due to natural broadening is c t i t where t i is the lifetime of the electron in its initial state and t f is the lifetime in the final state. f Example 1 The lifetime of an electron in the first and second excited states of hydrogen is about t = 10-8 sec. The natural broadening of the H line of hydrogen, = 6563 Å, is then Δ 6563 x x x 1014 m4.57x10 5 nm This is a real but tiny effect. 3

4 2. Doppler Broadening In thermal equilibrium, the atoms in a gas, each of mass m, are moving randomly about with a distribution of speeds that is described by the Maxwell-Boltzmann distribution function, with the most probable speed given as v mp 2kT m The wavelengths of light absorbed or emitted by the atoms in the gas are Doppler-shifted according to (nonrelativistic) v r c Doppler Broadening Thus the width of a spectral line due to Doppler broadening is 2 c 2kT m 4

5 Example 2 For H atoms in the Sun s photosphere (T = 5770 K), the Doppler broadening of the H line should be about Δ x x x x x10 11 m nm which is about 1000 times greater than that of natural broadening. 3. Pressure Broadening The orbits/levels of an atom can be perturbed by a collision with a neutral atom or by a close encounter involving the electric field of an ion. The results of individual collisions and closely passing ions can be termed pressure broadening. The outcome depends on the average time between collisions/encounters with other atoms and ions. 5

6 Pressure Broadening Calculating the precise shape of a pressure-broadened line is complicated. Atoms and ions of the same or different elements, as well as free electrons, are involved in these collisions and close encounters. The general shape of the line, however, is like that found for natural broadening. The values for natural and pressure broadening usually prove to be comparable, although the pressure profile can at times be much wider. Pressure Broadening An estimate of pressure broadening due to collisions with atoms of a single element can be obtained by taking the value of t o to be the average time between collisions. This time is approximately equal to the mean free path between collisions divided by the average speed of the atoms. The mean free path is vt 1 l n vt n and the speed is given by v mp 2k T m 6

7 So we find that Pressure Broadening l 1 t o v n 2kT where m is the mass of an atom, is its collision cross section, and n is the number density of the atoms. Thus the width of a spectral line due to pressure broadening is on the order of m 2 c 1 t o 2 c n 2 k T m The width of the line is proportional to the number density n of the atoms. Example 3 The pressure broadening of the H line in the Sun s photosphere (T = 5770 K, n = 1.5 x m -3, and = 3.52 x m -2 ) should be roughly Δ 6563 x x x x x x x m which is comparable to the result for natural broadening found earlier. However, if the number density of the atoms in the atmosphere is larger, the line width will be larger as well. 7

8 The total line profile, called a Voigt Profile, is due to the contributions of both the Doppler and Pressure + Natural (i.e., damping) profiles. Doppler Broadening dominates near the center of the line. Pressure Broadening dominates in the wings. Total Line Profile Rotational Broadening Other sources of broadening means involve coherent mass motions, such as stellar rotation and pulsation. These phenomena can substantially effect the shape of the line profiles, but cannot be combined with the results of the previously described broadenings produced by random thermal motions. Rotational Broadening does not change the line s Equivalent Width. 8

9 Calculating Line Profiles The simplest model used for calculating a line profile assumes that the star s photosphere (top layer) acts as a source of blackbody radiation, and that the atoms above the photosphere remove photons from this continuous spectrum to form absorption lines. The temperature, density, and composition determine the importance of the Doppler and Pressure Broadening and are also used in the Boltzmann and Saha equations. Column Density The calculation of a spectral line depends not only on the abundance of the element forming the line but also on the quantum-mechanical details of how atoms absorb photons. Let N be the number of atoms of a certain element lying above a unit area of the photosphere. N is a column density and has units of cm -2. (In other words, suppose a hollow tube with a cross-section of 1 cm 2 was stretched from the observer to the photosphere; then the tube would have N atoms of this specified type in it, but almost all of the atoms are at the bottom of the tube on the photosphere.) 9

10 Column Density To find the number of absorbing atoms per unit area, N a, that have electrons in the proper orbital for absorbing a photon at the spectral line s wavelength, the temperature and density are used in the Boltzmann and Saha equations to calculate the atomic states of excitation and ionization. Our goal is to determine the Number of specific Atoms (N a ) by comparing the calculated and observed line profiles. For example, say, can we determine the number of Barium atoms in a star s photosphere by using the equivalent widths of one or more lines? The equivalent width, W, of the line varies with N a. The Curve of Growth graph is used to determine the value of N a and thus the abundances of elements in stellar atmospheres. A Curve of Growth is a logarithmic graph of the equivalent width as a function of the number of absorbing atoms, N a. Curve of Growth 10

11 To begin, imagine that a specific element is not present in a stellar atmosphere. As some of that element is introduced, a weak absorption line appears. If the number of the absorbing atoms is doubled, twice as much light is removed and the equivalent width is twice as great. So W is proportional to N a, and the curve of growth is initially linear with N a. Curve of Growth Curve of Growth As the number of absorbing atoms continues to increase, the center of the line becomes optically thick as the maximum amount of flux at the line s center is absorbed. With the addition of still more atoms, the line bottoms out and becomes saturated. The wings of the line, which are still optically thin, continue to deepen. 11

12 Curve of Growth This occurs with little change in the line s equivalent width and produces a plateau on the curve of growth where W is proportional to the sqrt(ln N a ). Curve of Growth Increasing the number of absorbing atoms still further increases the width of the pressure broadening profile, enabling it to contribute to the wings of the line. The equivalent width grows more rapidly, although not as steeply as at first, with W being proportional to the sqrt N a for the total line profile. 12

13 Curve of Growth (1) Using the curve of growth and a measured equivalent width, we can obtain the number of absorbing atoms. (2) The Boltzmann and Saha equations are then used to convert this value into the total number of atoms of that element lying above the photosphere. Thus the number of absorbing atoms can be determined by comparing the equivalent widths measured for different absorption lines produced by atoms or ions initially in the same state (and so having the same column density) with a theoretical curve of growth. Curve of Growth 13

14 Example Two Na I Lines The first three quantities are known/measured. Calculate the last two. Regarding the f parameter, not all transitions between atomic orbitals are equally likely. The relative probabilities of an electron making a transition from the same initial orbital are given by the f-values or oscillator strengths for that orbital. Curve of Growth

15 Example Use log(w/) on the Curve of Growth plot to get the values below: Example (cont.) 15

16 Example (cont.) Higher state, shorter Lower state, longer By using the Curve of Growth, we got the number of atoms in the lower energy level. Using the Boltzman Equation, we see that there are very few atoms in the higher excited states. Now, let s use the Saha Equation for ionizations. Example (cont.) (T = 5800 K P e = 10 dyne/cm 2 ) Even though we used the Curve of Growth for neutral Na, the Saha Equation shows us that most of the Na atoms are singly ionized and we have our total number of Na atoms. 16