Physics 321 

Prof. Dale E. Gary
NJIT 
HertzsprungRussell Diagrams and
Structure of Spectral Lines
HertzsprungRussell Diagram
The first thing we note
is that the stars tend to fall in a fairly narrow band, stretching diagonally
across the diagram. This band is called the main
sequence, because most stars (between 80 and 90% of all stars)
lie along it.
Figure 6.1 HR Diagram of Stars Within 25 pc of the Sun
Clearly the bluer stars (zero or negative BV) tend to have higher absolute magnitudes, while red stars (large positive BV) have lower absolute magnitudes. Remember that we can convert absolute magnitude to luminosity, through the equation
We can ask what causes the relationship we see between luminosity and temperature. We know that these two are related by
In addition to the stars that lie on the main sequence, Figure 6.1 shows a few stars off the main sequence, both above and below. The ones below must have a smaller size than ones on the main sequence at the same color index. The ones above must have a larger size. Note that the ones above tend to be red (cool), while the ones below tend to be bluewhite (hot). These are, not surprisingly, called red giants and white dwarfs, respectively. Note that once we plot a star on an LT diagram, we know its radius. We find
We have used the terms giant, supergiant, dwarf, without defining them. Because of the relationship between size and luminosity, the size categories correspond to luminosity class. The MorganKeenan (MK) system is based on spectra. One can see subtle differences in spectra of stars of otherwise similar spectral type, due to the different sizes of the stars. One of the main effects is the width of the spectral lines, which get narrower for more luminous stars, as may be seen below:The Structure of Spectral Lines
Figure 6.3 Spectra near spectral type F5, for different luminosity classes
Adapted from data in the electronic version of "A Library of Stellar Spectra,"
by Jacoby G.H., Hunter D.A., Christian C.A. Astrophys. J. Suppl. Ser., 56, 257 (1984).The MK system of luminosity classes are shown in the table below, and their position on the HR diagram is shown in Figure 6.4, below.
Figure 6.4 An HR diagram showing the MorganKeenan luminosity classes,
along with locations in the diagram for some nearby or bright stars.
Class Type of Star IaO Extreme, luminous supergiants Ia Luminous supergiants Ib Less luminous supergiants II Bright giants III Normal giants IV Subgiants V Mainsequence (dwarf) stars VI Subdwarfs D White dwarfs The MK classification scheme enables astronomers to place a star on the HR diagram solely on the basis of the star's spectrum. Once the star is placed on the HR diagram, one can simply read off the absolute magnitude M_{V} from the vertical scale, and from its measured apparent magnitude find its distance from
m_{V}  M_{V} = 5  5 log dSuch a distance determination is called spectroscopic parallax, although note that it has nothing to do with parallax. It is simply a distance determination based on its spectral type. Note that because the luminosity classes are of finite extent in magnitude, this method is only good to roughly +/ 1 magnitude, which corresponds to a distance accuracy of about 10^{1/5} = 1.6.Let's compare spectral types from our catalog with the listed absolute magnitudes. Star Gl 3 is listed as a K5 V (a K5 mainsequence, or dwarf star), and looking up that spectral type in Table A4.3 of the text, we find a predicted absolute magnitude of 8.0. The absolute magnitude listed above is 7.1  somewhat brighter than predicted. The star GJ 1003 has a listed spectral type of 'm', which must mean that the spectral type is uncertain, but it is some type of M star. If we assume that it is luminosity class V, then from its measured absolute magnitude of 12.82 we would predict that it should be somewhere between M5 and M6. Its BV color index, however, would place it as about M2 or M3. You can see that Table A4.3 is not exactly precise, but it makes a useful guide.
Now let us focus on individual spectral line shapes and see what more they can tell us about the physical conditions in stars. One simple measurement we can do is the width of the spectral line, but spectral lines can have different shapes. A precise definition of line width that is independent of line shape is given by the equivalent width, which is defined asEquivalent Width Versus Line Strength
F_{c}  F_{l} W = dl F_{c} where F_{c} is the flux of the continuum, and F_{l} is the flux elsewhere in the line. The figure below shows the relationship between the equivalent width and the shape of the line, for a normalized line profile.
Normalized line profile and equivalent width. Note that the blueshaded region above the line has the same area as the blueshaded region below the line, so the equivalent width has the same area as the line itself. The width of the line is contributed to by three main effects:
 Natural Broadening
This is the effect we already discussed, relating the lifetime of an energy state Dt with the uncertainty in energy DE due to the Heisenberg Uncertainty Principle. This uncertainty in energy corresponds to an uncertainty in wavelength:
l^{2} ( 1 1 ) Dl = + 2pc Dt_{i} Dt_{f} The natural broadening of the Halpha line is about 0.46 mA (very small).
 Doppler Broadening
This is due to the thermal and/or turbulent motion of atoms in the atmosphere of the star. A moving atom absorbs a photon that is slightly shifted in wavelength due to the doppler shift Dl/l = +/  v_{r} /c. If the velocity is due to thermal motion, v = [2kT/m]^{1/2}, then we have a width of
2l ( 2kT ) 1/2 Dl = c m where the factor of 2 is due to the motion both toward and away from the observer. Turbulent motion with a velocity v_{turb} can be included as
2l ( 2kT ) 1/2 Dl = + v^{2}_{turb} c m Doppler, or thermal, broadening of the Halpha line for the Sun (T = 5780 K) is about 0.43 A, or 1000 times greater than natural broadening.
 Pressure Broadening
When the stellar atmosphere is very dense (large number n of atoms per cubic meter), an additional source of broadening comes into play. An estimate for the effect of pressure broadening can be obtained by taking the equation above for natural broadening, and replacing the lifetime Dt with the mean time between collisions among atoms. This time is Dt = l/v where l = ns is the meanfreepath and s is the collision crosssection. The collision crosssection is just the "area" of the atom, and these values are tabulated for different species. Here, of course, we will again use the thermal speed v = [2kT/m]^{1/2}. Putting it all together, we have
l^{2 }ns ( 2kT ) 1/2 Dl = pc m Note that this broadening is proportional to the density of atoms. This explains the narrowing line profiles as we go up in the MorganKeenan classification to more luminous stars. The supergiant stars have such large sizes that their outer atmosphere is very low density. The pressure broadening term is absent in such stars. Dwarf stars, on the other hand, have appreciable broadening (in the Sun, the Halpha line is broadened by 0.24 mA, for example).
In addition, the line profile changes depending on the number of atoms per unit area, (column density) N_{a} contributing to it. For the Halpha line, this would be the number of H I atoms along the line of sight that are in the n = 2 state. Here is the behavior as the column density N_{a} grows:
It is possible to draw
a curve of growth that expresses how
the equivalent width log W/l
changes with log f N_{a}(l/5000).
This is shown for the Sun in the figure below. Here, f
is
the oscillator strength for the transition,
and is a value tabulated for each electron transition in an atom that expresses
the relative likelihood of that transition occurring rather than another
from the same initial energy level.
A general curve of growth for the Sun, where N is in atoms/cm^{2}.
(Figure from Aller, Atoms, Stars, and Nebulae,
Revised Edition, Harvard University Press, Cambridge, MA, 1971).
Example:
We will use the above curve
of growth to find the number of sodium atoms above each square centimeter
of the Sun's surface from measurements of two sodium lines given in the
first two columns of the table below. The third column is the oscillator
strength f measured in the laboratory,
and the last two columns are just combinations of the first three columns,
for calculation convenience below.



log (W/l)  log[f (l/5000 A)] 










Values of the Sun's surface temperature T = 5800 K and pressure P = 0.01 N/m^{2} were used to construct the above curve of growth, and will be used in this example.
Both of the lines are produced
when an electron makes an upward transition from the ground state of the
neutral sodium atom (Na I) and so they both have the same value of N_{a},
the number of absorbing sodium atoms per unit area above the surface.
This number can be read from the figure, using the values in the table,
to obtain the value of log[f N_{a}(l/5000
A)] for each line. The results are:
To obtain the value of
the number of absorbing atoms per unit area, N_{a},
we use the measured equivalent widths, in the form of the last column of
the table, log[f (l/5000 A)],
together with
The average value of
log
N_{a} is
thus 15.0, so there are about 10^{15}
Na I atoms in the ground state per cm^{2 }area
of the Sun's surface.
To find the total number
of sodium atoms, we use the Boltzmann and Saha equations from the previous
lecture. The difference in energy between the final and the initial
states, E_{b }E_{a},
is just the energy of the emitted photon, so in the Boltzmann equation
N_{II} 

(  2pm_{e}kT  )  

= 


exp( c_{I}/ kT )  
N_{I} 


Using atomic parameters
for
Na, Z_{I} = 2.4,
Z_{II}=
1.0 (partition functions), and c_{I}=
5.14 eV for the ionization potential, the
above equation evaluates to N_{II}/N_{I}=
2.43 x 10^{3}.
This means that there are about 2430 singlyionized sodium atoms for every
neutral atom, so N_{II} = 2.43
x 10^{3 }N_{I} =
2.43 x 10^{18 }cm^{2}.
Where
to find atomic data (ionization
potentials, oscillator
strengths)