Stellar Astrophysics: The Classification of Stellar Spectra

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1 Stellar Astrophysics: The Classification of Stellar Spectra

2 Temperature and Color The intensity of light emitted by three hypothetical stars is plotted against wavelength The range of visible wavelengths is indicated Where the peak of a star s intensity curve lies relative to the visible light band determines the apparent color of its visible light The insets show stars of about these surface temperatures

3 Spectra of Stars Spectra of stars, like the Sun, show the continuous optical spectrum with dark absorption lines The quasi-continuous band of emission frequencies is missing a few frequencies These missing energies have been absorbed in the outer heliosphere by the hydrogen gas to excite the hydrogen atoms into higher energy states (Balmer series) Joseph von Fraunhofer ( )

4 Spectral Types of Stars Classification of stars by the strength of their hydrogen absorption lines began at Harvard in the 1890s Edward Pickering together with several assistants developed the basic scheme still used today Williamina Fleming, Antonia Maury, Annie Jump Cannon Basic sequence O B A F G K M from O (blue) to M (red) Sub-sequences, for example A0 - A9 Edward C. Pickering ( )

5 Spectral Types of Stars

6 Principal Types of Stellar Spectra

7 Spectral Line Strengths For different star temperatures or spectral types, different elements contributions dominate the spectrum Stars at around 10,000 K are dominated by hydrogen lines The Sun at 5,777 K is dominated by calcium and iron lines

8 Maxwell-Boltzmann Velocity Distribution How can we understand the underlying physics of the classification? In what atomic orbitals are electrons most likely to be found? What are the relative numbers of atoms in the various ionization levels? Statistical mechanics describes large ensembles of particles by statistic quantities, like temperature, pressure, For example, the number of gas particles per unit volume with velocities between v and v + dv is given by n v m 3/2 m v dv = n 2 / ( 2 k T ) e 4 π v 2 dv 2 π k T n total number density k Boltzmann constant T temperature of the gas

9 Maxwell-Boltzmann Velocity Distribution n v m 3/2 m v dv = n 2 / ( 2 k T ) e 4 π v 2 dv 2 π k T The exponent is the ratio of the particles kinetic energy to the characteristic thermal energy k T We obtain the most probable velocity v mp = 2 k T m 1/2 And for the root-mean-square velocity v rms = 3 k T m 1/2

10 Boltzmann Equation Colliding atoms in a gas gain and lose energy This produces an equilibrium distribution with higher energy orbitals being less likely occupied The ratio of probabilities for two given states is P (s b ) P (s a ) E b / ( k T ) e = = e E a / ( k T ) e ( E b E a ) / ( k T ) Ludwig Boltzmann ( ) with s a being the state with lower energy E a With temperature decreasing towards 0 : P (s b ) / P (s a ) 0 With temperature increasing towards : P (s b ) / P (s a ) 1

11 Boltzmann Equation Generalize the Boltzmann equation for the case of different (quantum) states with the same energy by introducing g a as the number of states with a given energy E a (degeneracy) g a is also called the statistical weight of the energy level E a P (E b ) P (E a ) E b / ( k T ) g b g = b e = e E a / ( k T ) g a e g a ( E b E a ) / ( k T ) Stellar atmospheres contain very large numbers of atoms leading to an equality between probability and number ratios (Boltzmann equation) N 2 N 1 g 2 ( E 2 E 1 ) / ( k T ) = e g n = 2 n 2 g 1

12 Boltzmann Equation Apply Boltzmann equation to the ground state (1) and first excited state (2) of hydrogen Plot the ratio of excited state to the sum of both states Contradiction to Balmer lines (transitions from n = 2 to higher levels) which reach their maximum intensity at 9520 K

13 Saha Equation Consider also ionization of atoms Ionization energy χ i needed to ionize atom in the ground state (hydrogen χ i = 13.6 ev) Atoms may not be in ground state Average over orbital energies accounts for possible distribution of electrons among orbitals Introduce the weighted sum of the number of ways the electrons can be arranged in an atom of a given energy The weight for higher energy states is lower according to the Boltzmann distribution Meghnad Saha ( ) Z = Σ g j e ( E j E 1 ) / ( k T ) j = 1 Partition function

14 Saha Equation Looking at ratio of atoms in initial (i) and final ionization stage (i+1) N i +1 N i 2 Z i +1 2 π m = e k T 3/2 e n h 2 e Z i χ i / ( k T ) Here n e is the free electron number density, often expressed by the pressure of the free electrons P e = n e k T The factor 2 in front of the final state partition function accounts for the two possible spin states of the electron m s = ± 1/2

15 Saha Equation for Hydrogen Consider the number ratio of ionized atoms H II to neutral atoms H I N II N total = N II N I + N II = N II / N I 1 + N II / N I Graph for P e = 20 N m -2

16 Combining Boltzmann and Saha Equation for Hydrogen Balmer lines depend on the ratio N 2 / N total N 2 N total = N 2 N 1 + N 2 N I N total = N 2 / N N 2 / N N II / N I Assuming N 1 + N 2 N I Saha equation only applicable for gas in thermal equilibrium (Boltzmann velocity distribution) Gas density should be less than 1 kg / m -3

17 The Electron in the Hydrogen Atom at Different Temperatures

18 Hertzsprung-Russell Diagram Large number of data on stellar luminosities, absolute magnitudes and masses indicated that O stars are brighter and hotter than M stars O stars are more massive than M stars Ejnar Hertzsprung and Henry Russel independently published a correlation between stellar luminosity and their spectral type

19 Hertzsprung-Russell Diagram Luminosities of stars are plotted against their spectral types Main-sequence stars fall along the red curve Giants are to the right and supergiants are on the top White dwarfs are below the main sequence

20 Hertzsprung-Russell Diagram Sizes of stars form straight lines on log scale presentation of Hertzsprung-Russel diagram

21 The Pleiades The blue glow surrounding the stars of the Pleiades (375 ly from Sun) is a reflection nebula created as radiation scatters off dust Most of the cool, low-mass stars have arrived at the MS, indicating hydrogen fusion in their cores

22 H-R Diagram of Globular Cluster M55 Las Campanas 1 m reflector