How Big Are the Stars? 173183 Luminosity, Radius and Temperature  the Critical Link In an earlier section we introduced the oddlynamed "black body" to describe how hot objects emit light energy. You learned that the energy emitted by a blackbody (i.e. Flux) depends on the "4th power" of the temperature. This is a very strong relationship. For example, if you double the temperature of an object then the flux of energy radiating from the object increases by a factor of 2^{4} or 2x2x2x2 =16 times! As you have just seen, the luminosity of a star also depends on its surface area. So, combining these ideas tells you that knowing the intrinsic brightness of a star also tells you about its size and temperature. This is illustrated for you in the following applet.
Example 8.11 Use the following data to compare the size of the bright star Capella and our sun. Do this without using any formulas but remember that a magnitude difference of 5 is the same as a brightness factor of 100.
Solution: Since Capella and our sun have essentially the same temperature any difference in intrinsic brightness must be due to a difference in their surface areas. The difference in the absolute magnitudes of the two stars is nearly 5, so Capella must be about 100 times brighter. Since we just concluded that this is due to a difference in surface area we see that Capella must have about 100 times the surface area. Surface area depends on the square of radius so Capella must be approximately 10 times the radius of the sun. Example 8.12 The bright star Arcturus has an apparent magnitude of 0.29. Its distinctly orange colour is consistent with a surface temperature of 4300 K. The parallax for this star is 0.089". Use this information to estimate the size of Arcturus. (Hint: use some of the applets given in this and previous sections to assist you.) Solution: This is a multipart problem which combines ideas from the previous 3 sections. Use the following strategy:
The HertzsprungRussell Diagram All of what you have learned in the last 3 sections can be combined with what you learned earlier about stellar spectral classification to construct one of the most significant relationships in astronomy  the HertzsprungRussell or HR diagram. The HR diagram shows a relationship between the intrinsic brightness of a star (expressed either as absolute magnitude M_{v} or as luminosity) versus the star's temperature (or spectral type).
The HR diagram is remarkable. When stars are plotted according to their intrinsic brightness and temperature a number of very distinct patterns emerge. This is shown in the following diagram produced by data from the Hipparchus satellite and represents one of the most complete and accurate HR diagrams produced to date.
Exploring the HR Diagram Use the following applet (HRexplorer) to explore the wealth of information contained within the HR diagram. To assist you, complete the following tasks:
Just What are Luminosity Classes? The division of the HR diagram into luminosity classes is not "accidental". Rather it points to the underlying physics of how stars create energy and how they evolve. Roughly 90% of stars are found in the mainsequence section of the diagram because this is where a stars spends 90% of its lifetime. As a star ages important changes occur to the star's surface temperature and size. This means a specific star's position on the HR diagram changes as the star evolves. Thus, knowing which luminosity class a star is in tells us something about the age of that star. So, how do you know what luminosity class a star belongs to? This is not simple but ultimately rests on looking at the spectrum of light coming from the star. The following figure shows two spectra for two very different stars. Each of the stars is an Atype stars. One, however, has much narrower spectral lines than the other. The narrow line tells the astronomer that the star producing this absorption line ha a very rarefied atmosphere and is a supergiant star. In the other spectrum, the lines are broader suggesting a much denser atmosphere.
Using the HR Diagram  Spectroscopic ParallaxAs we proceed in the course we will discover a number of uses for the HR diagram. Some of the key uses of the HR diagram are:
Example 8.13 Through the telescope you find a faint blue star. Spectroscopic examination reveals that it is a B5 V star and your CCD photometer tells you that it has an apparent magnitude of 6.7. How far away is the star? Solution: Use the applets HRexplorer and distanceModulus to help answer this. The first thing to do is to look at the HRexplorer . Since a B5 V star has a surface temperature of 15 500 K we can judge that its absolute magnitude is 1.1. We now know the distance modulus: If you use distanceModulus you can conclude that the star must be about 360 pcs away or about 1200 ly away. Spectroscopic Parallax for Math Geeks! (or people who just like equations...) There is a relatively straightforward mathematical connection between distance modulus and distance that all starts with the brightnessmagnitude relation that you learned at the beginning of the course. If we let 'r' stand for the distance (in pcs), m the apparent magnitude and M the absolute magnitude then we get two important formulae: Example 8.14 Use the formulae given above to confirm that the star in Example 8.13 is 360 pcs away. In example 5.13 you concluded (by the method of spectroscopic parallax) that the distance modulus of the star is 7.8. Put this into to get The formulae gives a more exact answer of 363.1 pcs. Practice

To determine the size of the stars
Chapter 9.3,9.4
Altair is one of the very few stars that we have been able to resolve. The image shown below is a direct image of Altair produced by the CHARA array at Mount Wilson. Altair is not spherical! Its high rate of rotation squashes into this oblong shape.
The HR diagram is a fundamental diagram in astrophysics that relates stellar luminosity to temperature. Eijnar Hertzsprung (18731967) was a Danish astronomer who was also trained as a chemical engineer. Other achievements include the recognition that the thin shape of spectral lines in supergiant stars indicate the very low density of the atmosphere of these stars. Henry Norris Russel (18771957) was one of the most influential American astronomers of the last centaury. He made fundamental contributions in many parts of astrophysics as well as atomic physics. 