Consider the image on the left. The two blue-green circles represent the Earth at opposing points in its orbit.
The large yellow circle between them of course represents Sol (our Sun). Suppose we want to measure the distances to the red and gold
stars. As the Earth moves over half its orbit, each star appears to change position. This is called
parallax and the change in position is twice the parallax angle. The parallax angle for the red star is
a and that for the gold star is b. Note that b is
less than a because the gold star is farther away.
If we know the radius r of the Earth's orbit we can compute the distances Ri to the stars:
Rred = r cot (a) and Rgold = r cot (b)Since for large distances cot(q) is very close to 1/q (if q is measured in radians!), we have
Rred = r / a and Rgold = r / bA parallax angle of 1 ArcSecond (there are 60 minutes of arc in a degree and 60 seconds of arc in a minute) yields a distance of 1 parsec. Look up the average radius of the Earth's orbit and show that 1 parsec is approximately 3.26 light-years (the distance light travels in one year).
We do not have parallax data for some of the stars; in this case, you will not be able to perform any of the computations which require the distance. In these cases, notate the result as "N/A" (Not Available).
Also compute the relative error in that distance using the parallax error provided for each star. While it is not obvious, the relative error in the distance is equal to the relative error in the parallax.
Our first attempt to classify our stars will be made using a visual inspection of the spectra. The colored image of the spectrum
consists of a continuum background interrupted by dark lines or regions. The continuum background arises from the nearly
black body spectrum of radiation generated in the stellar interior. A black body is by definition a perfect absorber of
radiation. If a black body is exposed to radiation, its temperature increases and to maintain thermal equilibrium it must
also radiate. The end result is that a perfect absorber is also a perfect radiator. The intensity (power per unit area) per unit
wavelength is called the flux, and for such a radiator, depends only on its temperature and the wavelength emitted:
FT (l) =
2 h c2 / (l5 (eh c / (l k T) - 1))
where h is Planck's Constant, c is the speed of light, k is Boltzmann's Constant and T
is in Kelvin. Show that the units of this function are consistent with Watts per meter squared per Angstrom.
The following table [6] will help you to classify each star according to the black body fits. It gives the most likely effective temperature for each class. You should use interpolation to pick each spectral class (ie., a temperature of 46000 K would be an O4), but round to the nearest one half (ie., 45200 would be O4.5). More precision than that is almost certainly unjustified by the data.
The range of temperatures associated with each spectral class depends on what kind of star it is: a main sequence star, which is powered by the fusion of Hydrogen nuclei into Helium in the core, or a giant or supergiant star, which burns heavier nuclei and is much more luminous for its temperature. For the moment, assume that your stars are all main sequence stars (but one or more may not be!).
Are the classes the same for the peak fits versus the tail fits?
Spectral Class T (Main Sequence) T (Giants) T (Supergiants) O3 48000 O5 44000 O6 43000 O8 37000 B0 31000 30000 B1 24100 B2 21080 B3 18000 B4 15870 B5 14720 B8 11950 A0 9572 9550 A2 8985 9000 A5 8306 8500 A7 7935 8300 F0 7178 7178 8030 F2 6909 6909 7780 F5 6528 6528 7020 F8 6160 6160 6080 G0 5943 5943 5450 G2 5811 5811 5080 G5 5657 5657 4850 G8 5486 5486 4700 K0 5282 5282 4500 K2 5055 5055 4400 K3 4973 4973 4230 K5 4623 4623 3900 K7 4380 4380 3870 M0 4212 4212 3850 M2 4076 4076 3800 M5 3923 3923 M8 2400
Note that an absorption line is surrounded on both sides by brighter continuum colors. In a display like ours with discrete color and intensity levels, the border between two continuum colors may appear to be "line-like", but these are not absorption lines. True absorption lines or bands will show a clear dip in the graph with higher fluxes on either side of the line (or band).Which lines appear are largely a function of temperature: lines associated with neutral atoms will disappear when T is high enough to ionize those atoms, and bands associated with molecules will disappear when T is high enough to break their bonds.
Compute the wavelengths associated with electron transitions in Hydrogen from ninitial = 2, to nfinal = 3 through 7. Choose Hydrogen as the reference spectra element from the pop-up menu [5]. Identify the lines corresponding with those wavelengths.
Compare the emission line data from the first class meeting with the data in this table. If there are discrepancies, how might you explain them?
For each specie in the table, choose it from the pop-up menu [5] and see if any of the lines or bands corresponding to these wavelengths are present. If they are, the temperature of the star's atmosphere must be in the given range. If the line is particularly strong, the temperature is likely close to the given peak temperature. Since we are assuming that the temperature is the same for every specie, you should be able to narrow the possible temperature range by a process of elimination.
specie lines (Angstroms) T Range (K) Peak T (K) H (you just computed these in the last step) 5000-40000 9000 He 4471, 4542 10000-50000 29000 Ca 4227 2000-5500 3000 Ca+ 3934, 3968 3000-7000 5000 Na 5890, 5896 2000-5500 3000 Mg+ 2796, 2802, 4481, 5173 8000-30000 9200 Si+ 4128, 4131 8000-20000 9200 Si++ 4552 20000-40000 25000 Si+++ 1394, 1403, 4089, 4116 30000-50000 40000 Fe 4045, 4143, 4299, 4325, 4384, 5270 2000-7000 4500 Fe+ 4173, 5316 4000-8000 5700 CH band 4300-4315 5000-6000 5500 TiO bands 4762, 4955, 5167, 5448, 5862, 6159, 6384 2000-4000 3000 Note that many of the spectra have insufficient range or resolution to see many of these lines. It may not be possible to classify all of your stars by this method.
Classify your stars again using the narrowest consistent temperature range and the previous table linking temperature and spectral class. Again, assume for the moment that all of your stars are main sequences stars. You will only be able to specify a range of classes for each star using this method.
For each range, there is an effective wavelength which is essentially the median wavelength in the range. We will be interested in the b and v ranges, whose effective wavelengths are 4361 and 5448 Angstroms, respectively. The following definitions for these spectral indices are adjusted so that the values are zero for the reference star Vega:
b = -2.5 log (f4361) - 28where the fi denotes the flux at wavelength "i". There is a one to one correspondence between the difference b-v and spectral class, depending again on the type of star [6]:v = -2.5 log (f5448) - 28.6
Use the applet to measure the flux at each wavelength for each star. Compute the spectral indices b and v for each star, and use the difference to classify them. Some of the spectra have insufficient range to use this method.
Spectral Class b - v (Main Sequence) b - v (Giants) b - v (Supergiants) O5 -0.35 O6 -0.32 O8 -0.31 B0 -0.29 -0.25 B1 -0.26 B2 -0.24 B3 -0.21 B4 -0.18 B5 -0.16 B8 -0.10 A0 0.00 0.01 A2 0.06 0.05 A5 0.14 0.10 A7 0.19 0.13 F0 0.31 0.31 0.16 F2 0.36 0.36 0.21 F5 0.44 0.44 0.33 F8 0.53 0.54 0.55 G0 0.59 0.64 0.76 G2 0.63 0.76 0.87 G5 0.68 0.90 1.00 G8 0.74 0.96 1.13 K0 0.82 1.03 1.20 K2 0.92 1.18 1.29 K3 0.96 1.29 1.38 K5 1.15 1.44 1.60 K7 1.30 1.53 1.62 M0 1.41 1.57 1.65 M2 1.50 1.60 1.65 M5 1.60 1.85 M8 1.80 Note that the b-v value is the most likely value for each class.
M = m + 5 - 5 log Dwhere D is the distance in parsecs. Compute m and M for each star. Again, some of the spectra have insufficient range to perform this computation.
A white dwarf is a star which has collapsed to a small, hot cinder after fusion processes have stopped.
The luminosity of Sol is 3.85 * 1026 W, and its radius is 7 * 108 m. Compute the luminosity and radius of each star,
in SI and in Solar units.
The mass of the Sun is 1.989 * 1030 kg.
Since our spectra do not cover the entire electromagnetic spectrum, we expect the flux ratios to be less than 1. For
those stars whose ratio is greater than 1, show that there are at least two possible sources of error:
the parallax and the v spectral index.
Let fi denote the set of values for the flux ratios and pi denote the relative errors in the parallax.
Let F and P denote the averages of those values. The quantity
Make a scatter plot of flux ratio versus parallax error. Then compute the correlation coefficient for the two. For your sample of stars,
is parallax error a likely candidate to explain the flux ratios greater than 1? If the spectral index is a more likely candidate,
how would that affect the values for radius and mass that you calculated?
For each star, choose three narrow absorption lines from one of the atoms in the element menu,
and fit a Gaussian to each line. Be sure to clearly identify
the value of the standard deviation that most exactly fits the tip (and large wavelength edge) of the graph at the absorption line.
This means that you will not be able to perform this computation (and the next) for some stars (particularly those with low resolution spectra).
I will distribute the final classes and absolute magnitudes to everyone at that time.
This is called a "Hertzsprung-Russell Diagram". Notice how the main
sequence is clearly identifiable from upper left to lower right, and how the white dwarfs reside in the lower left. We do not have
many giants and they are close to the main sequence, but in general giants would be in the upper right region.
What can you learn from the patterns in the H-R Diagram?
Use your absolute magnitude and spectral class(es) to identify the luminosity class for each star.
Spectral Class M (Main Sequence) M (Giants) M (Supergiants) M (White Dwarfs) O5 -5.8 O6 -4.8 O8 -4.1 B0 -3.3 -6.4 10.2 B1 -2.9 B2 -2.5 B3 -2.0 B4 -1.5 B5 -1.1 B8 0.0 A0 0.7 -5.0 A2 1.3 -5.0 A5 1.9 -5.0 A7 2.3 -4.9 F0 2.7 1.0 -4.8 12.9 F2 3.0 0.9 -4.8 F5 3.5 0.8 -4.7 F8 4.0 0.7 -4.6 G0 4.4 0.6 -4.6 G2 4.7 0.5 -4.6 G5 5.1 0.4 -4.5 G8 5.6 0.3 -4.5 K0 6.0 0.2 -4.5 K2 6.5 0.1 -4.5 K3 6.8 0.1 -4.5 K5 7.5 0.0 -4.5 K7 8.0 -0.1 -4.5 M0 8.8 -0.2 M2 9.8 -0.2 M5 12.0 -0.2 M8 16.0
This is sometimes
an exercise in finding the most consistent classification based on everything you have done so far.
L = 10(4.83 - M) / 2.5 * LSolar
Given this value and the temperature, we can compute its radius as
r = (L / (4 p s T4))1/2
(where s is the Stefan-Boltzmann constant).
log (r / rSolar) = log (M / MSolar) + 0.10
while for stars with mass greater than 0.4 Solar Masses but less than or equal to 30 Solar Masses,
log (r / rSolar) = 0.73 log (M / MSolar)
Use these relationships to compute the mass (in Solar Masses) of each star. For most of your stars
(except main sequence stars of class M2 or cooler), you should probably use the second equation.
You must submit your final spectral classes, along with your absolute magnitudes, by the end of class during the 6th week.
F = L / (4 p r2)
where r is here the distance to the star. Compute the total flux using this equation, and the ratio of the total flux
reported by the applet to the total calculated from the luminosity, for each star.
(S (fi - F) (pi - P))2 /
((S (fi - F)2) * (S (pi - P)2))
is called the correlation coefficient and measures the extent to which the two sets of data are related, assuming that there is
a linear relation between them. If it is close to 1, there is a nearly linear relation between the two sets of data; if it is close to
0, the two sets of data are not linearly related.
v = c s / l
where c is the speed of light, s is the standard deviation and l
is the wavelength at the central minimum of the flux at that line.
Do not use any Hydrogen or Helium lines. Those lines are broadened by the Stark Effect: since their
nuclear charge is small, they are particularly susceptible to electric fields created by ions in the stellar atmosphere.
These fields perturb their energy levels and are primarily responsible for their line broadening.
Average the 3 velocities for each star to obtain the maximum possible velocity for the absorbing species.
m v2 / 2 = k T / 2
where m is the mass of an atom of the specie you selected in the previous step, in kg. Note that only one component of velocity contributes to the
broadening. For each star, are the velocities comparable? Note
that the maximum velocities in a statistical ensemble of atoms and molecules (such as occur in stellar atmospheres) are far
down the tail of the Gaussian velocity distribution, very far away from the average.
A discussion of this diagram and its implications will be a central part of your project report. Be sure to identify
your stars on the diagram, and explain how their attributes fit (or not!) our expectations regarding mass and
size.
The Roman Numeral designates the star's luminosity class: Ia and Ib are supergiants, II are bright giants, III are
giants, IV are subgiants and V are "dwarfs" (which includes most main sequence stars).
A spectral class beginning with "D" indicates a degenerate star, or white dwarf. Additional designations include
e (has emission lines), n (has diffuse lines), p (has a peculiar spectrum; see, even physicists don't understand
it all yet!), s (has sharp lines), and v (variable star).
Evaluate each of the classification methods we used for both accuracy and precision (yes, these are different!). Defend your evaluations.
With these things in mind, discuss the utility of spectral classes as a way of organizing our knowledge about stars.
Why use classes at all? Why not just temperatures and luminosities?