AS1001 ASTRONOMY & ASTROPHYSICS
LABORATORY PRACTICAL
SOLAR AND STELLAR SPECTRA
I. The Sun's Continuous Spectrum And Energy Output
Introduction
The Sun is of extreme importance to us in providing the heat and light for our planet; we would not exist without it! The energy (light and heat) radiated by the Sun, and our understanding of it, are the focus of this first part of the practical. It is important that we determine how much radiation, at which wavelengths, is emitted by the Sun, our nearest star, for it is typical of billions of stars in our own Milky Way Galaxy and in other galaxies. Study of the Sun allows us to study these more distant objects too.
The visible "surface" of the Sun is a specific layer in its atmosphere from which most of the visible light is emitted, called the PHOTOSPHERE. The Sun emits electromagnetic radiation at all wavelengths. Most of the energy is emitted in the visible part of the spectrum (i.e., the part of the spectrum to which our eyes are most sensitive), but substantial amounts are emitted also in the infrared (which we feel as heat) and in the ultraviolet (hence sunburn). Small amounts are emitted in the X-ray and radio regions.
Table 1 gives measures of the amount of radiation emitted by the Sun at different wavelengths. >From these data, you will be able to determine the average temperature of the solar photosphere and the luminosity of the Sun - the total energy it emits per second over all wavelengths and in all directions. A very large number of careful observations with many different telescopes and detectors are required to obtain these data.
Procedure
This distribution of the solar energy with wavelength is called the continuous spectrum, or the solar continuum. It is a characteristic of the radiation emerging through the photosphere from the solar interior. The lower-density gases in the solar atmosphere then superpose their own signature on this continuum, by producing dark absorption lines and some bright emission lines too. But Table 1 provides you with the continuum values only of measured FLUX DENSITY as a function of wavelength. Flux density is the rate at which energy crosses a unit area of surface, and is the quantity which comes directly from the detector measurements. So the units of flux are W m-2 nm-1.
1)	Using the data from Table 1, plot the flux density f(l) vs. its corresponding wavelength l on a sheet of graph paper, and sketch a curve that passes through ALL the points. Do this carefully, since this graph forms the basis of part I of the practical.
2)	Determine from your graph the wavelength at which f(l) attains its greatest value. This wavelength of maximum light lmax can be related to the temperature TW of the Sun by Wien's Law,
lmax TW = 2:898 x 106 nm K,
for lmax in nanometres (nm) and TW in Kelvin (K). Using this equation and your measured value of lmax, determine the value of TW. 

Table 1
Values of flux density f(l) vs. wavelength l for the Sun, measured at the Earth, and corrected for absorption through the Earth's atmosphere.

l (nm)	f(l) (10-2 W m-2 nm-1)	l (nm)	f(l) (10-2 W m-2 nm-1)	
200	1	460	215	
220	5	480	213	
240	6	500	204	
260	13	550	198	
280	25	600	187	
300	59	650	167	
320	85	700	149	
340	114	750	129	
360	115	800	114	
370	127	900	90	
380	121	1000	74	
390	115	1100	61	
400	160	1200	50	
410	187	1400	33	
420	189	1600	22	
430	183	1800	15	
440	201	2000	10	
450	213	2500	5	

3)	Now find the total area under the f(l) vs. l graph by using two straightforward techniques. Firstly, count all the squares under the curve - it really does take only ten minutes! (Don't forget to include the partial squares through which your curve passes.) Your total will be in units of the area of an individual square on the graph paper and it will be necessary to convert this area into units of Watts per square metre (W m-2). The conversion factor will be determined by the scale used on each axis for your graph.
Estimate the percentage error in the area under the curve due to inaccuracies in counting squares.
Secondly, use a simple but more sophisticated numerical method called Simpson's Rule to assess the area under the curve. The graph is defined between points a = 200 nm and b = 2500 nm. Divide this interval (b – a) into an EVEN number n of subintervals of equal length h = (b – a) / n. A value of n = 10 should be adequate. Then the area E under the curve is given directly in units of W m-2 by the equation:
,
where f(a) and f(b) are the flux densities at a and b respectively, f(a +2h) is the flux density at wavelength (a + 2h), etc.
This wavelength-integrated or BOLOMETRIC flux density E is equal to the total solar energy between 200 nm and 2500 nm striking one square metre of surface just outside the Earth's atmosphere per second. It is referred to as the SOLAR CONSTANT, and is a fundamental quantity for meteorology.
Assuming the percentage error on E is the same as you got from counting squares, give the uncertainty on E. Then compare the values you obtained for the area under the curve using the two different methods. Do they agree within the measuring (counting) error?
Propagate this measuring error through the following three calculations, giving the uncertainty on EQ, Teff and LQ.
4)	We can now calculate the total energy per unit area per second emitted at the surface of the Sun. This quantity is given simply by the inverse-square law of radiation,
,
where aE = 1:498 x 1011 m is the mean Earth - Sun distance and RQ = 6:96 x 108 m is the radius of the Sun.
We can use this rate of energy output to define another temperature for the Sun, called the EFFECTIVE TEMPERATURE, Teff. The effective temperature of a star is the same as the temperature of a black body that emits energy at the same rate per unit area, and is given by
,
where s is the Stefan-Boltzmann constant; s = 5:67 x 10-8 Wm-2K-4. Can you explain why these two temperatures for the Sun, TW and Teff , should be different? Which should be higher, and why?
5)	With this value of the energy emitted by the Sun per second through each square metre of its surface, it is easy to calculate the TOTAL energy per second emitted over all wavelengths in all directions by the Sun. This quantity is called the LUMINOSITY of the Sun and is given by
 .
In astronomy we usually express the luminosities of other stars and even of whole galaxies in units of the solar luminosity, LQ. An accurate determination of this quantity is therefore very important. Your result should be close to 4 x 1026 W.
II. Classification Of Stellar Spectra
The observed spectrum of a star is composed of a continuous spectrum of radiation that is emitted by the photosphere, plus a line spectrum superposed by the outer layers of the stellar atmosphere. This line spectrum may contain absorption lines only (most typical), or, for some stars, a combination of absorption and emission lines.
The spectral sequence from class O to class M is a sequence of decreasing surface temperature. It is well calibrated with other observed quantities like colour indices, and derived quantities like absolute visual magnitudes MV and temperature Teff. The spectral classes in the MK system are defined by a set of standard stars. To say that a star has a spectral type of F5, for example, means its spectrum is very closely similar to the spectrum of the F5 standard star.
Spectral typing is pattern matching, usually by eye, and it takes only a short time to achieve that match. A spectral type may be assigned quickly from one spectrum of a star, is well calibrated, and is independent of complications like interstellar extinction. So a spectral type is important for determining stellar distances, for studies of star clusters, for assessing the stellar content of galaxies, etc. This second part of the practical uses the CLEA - Classification of Stellar Spectra program on the PCs in the Laboratory to introduce you to the method.
Procedure
Ask a demonstrator to start the CLEA - Classification of Stellar Spectra program for you, and then log on. Click on RUN, and then select classify spectra. Click on LOAD, and then select main-sequence standards. Click on LOAD again, and select unknown spectra.
Select any one from the list of 25 unknowns. The spectrum will be plotted in the middle panel of the display, as a plot of flux density vs. wavelength. Each spectrum has a sloping continuum and superposed absorption lines that are due to various atoms or molecules in the atmosphere of that star. Using the spectral line identification feature in the program, spend a little time identifying the major lines in the spectrum - Hydrogen (H), Helium (He), Calcium (Ca), Iron (Fe), ... Table 2 at the end of this document contains a useful summary of the main spectral features for each letter class.
Use particular spectral lines or blends to compare the unknown and the standard spectra, remembering that you are trying to match an overall pattern. You can quickly determine a spectral type to within 5 subclasses e.g., between F0 and F5. It takes a little longer to interpolate between these subclasses. Use the difference command to compare the unknown and a standard spectrum.
By experimenting with the above, you will soon see how to classify the spectrum of an unknown star. Estimate the spectral type to one-tenth of a letter class, and also estimate the luminosity class (Roman numerals V, IV, III, II, I, where 'V' indicates a dwarf like the Sun, and 'I' means the most luminous supergiant). You can do the latter by comparing your unknown spectrum with the luminosity effects lists via the LOAD command and selecting the appropriate atlas of standards. Record also which spectral lines you used in deciding upon the spectral type.
Having familiarised yourself with how spectral classification is done, make your own selection of 10 unknown spectra from the list of 25, and determine full spectral types (e.g. F5 III) for those 10 stars. Ensure that you cover a reasonable range of letter class (directly related to surface temperatures of stars) and luminosity class. As you are classifying each spectrum, make a judgement about how well you can discriminate between spectral subtypes (e.g. one-tenth of a type), and between luminosity classes (e.g. one class).

Provided by I A Bonnell, University of St Andrews, for the LTSN/HEA Physical Sciences Centre 
Table 2
THE SPECTRAL SEQUENCE
Spectral class	Temperature (K)	Colour	Main spectral lines	Example	
O	40000-30000	blue-violet	Ionised atoms, esp. helium (He II)	d Orionis
(Mintaka)	
B	30000-10000	blue-white	Neutral helium (He I),
some hydrogen	b Orionis
(Rigel)	
A	10000-7500	white	Strong hydrogen,
some ionised metals	a Lyrae
(Vega)	
F	7500-6000	yellow-white	Hydrogen & ionised
metals (Ca, Fe)	a Canis Minoris
(Procyon)	
G	6000-5000	yellow	Ionised calcium (Ca II),
neutral & ionised metals	Sun; a Auriga
(Capella)	
K	5000-3500	orange	Neutral metals	a Tauri
(Aldebaran)	
M	3500-2500	red-orange	Strong titanium
oxide (Ti O), Ca I	a Orionis
(Betelgeuse)