The study of mathematics is apt to commence in disappointment.... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it. Alfred North Whitehead, An Introduction to Mathematics

Assignments: Problem Set #3 Solutions are on the Web Hour Exam #1 on Friday Last Year's First Hour Exam is on the Web Read Chapter 25, Sections 1 through 3 (pp. 399-404) Look at the Special Web Page on The H-R Diagram

In Class: The best summary of what we did in class today can be found in the Special Web Page on the Hertzsprung-Russell Diagram, but here are my class notes just for completeness. Last time, we saw that two properties of stars are often correlated: broadband color and luminosity you all know that "color" is just a codeword for temperature and that "red" means lower temp and "blue" means higher temp so we really have a correlation between surface temperature and luminosity So let's try to see if we can explain it Look at an idealized H-R diagram note first the log-normal scale each division is ten times greater on the y-axis recall that there are an enormous range of stellar luminosities 0.01 -- 100,000 L_o x-axis is scale is "normal" range in temperatures is smaller 3000-30000 K drawn in the Main Sequence where most of the stars are instead of a scatterplot, I've just put a line showing where all the points would lie We want to try to understand why there is a correlation between TEMPERATURE and LUMINOSITY We actually know a relationship between TEMPERATURE and LUMINOSITY for stars: Recall the Stefan-Boltzmann law: I = sigma T^4 intensity is the power per unit surface area emitted by something at a temperature T To get the total power, or LUMINOSITY, emitted by a star P (or LUMINOSITY) = I x S where S is the surface area of the star remember that for spheres, S = 4 pi R^2 so LUMINOSITY = 4 pi R^2 sigma TEMPERATURE^4 This is a nice relationship because it does just what we want - explains LUMINOSITY as a result of TEMPERATURE - higher TEMPERATURE means higher LUMINOSITY good! looks like H-R diagram - double the TEMPERATURE and LUMINOSITY increases by 2x2x2x2 = 16 times Let's see if it works in detail We can put a line on the H-R diagram corresponding to this relationship choose a TEMPERATURE calculate a LUMINOSITY place a point on the H-R diagram corresponding to that TEMPERATURE, LUMINOSITY pair In order to do this, we need a value for R, the radius of the star choose for now the radius of the Sun that is, assume that all stars are the same size as the Sun then there is a direct relationship between the TEMPERATURE and LUMINOSITY for any TEMPERATURE LUMINOSITY = 4 pi sigma (R_sun)^2 TEMPERATURE^4 = 3.45 x 10^11 W/K^4 (TEMPERATURE)^4 if we put in the TEMPERATURE of the Sun, 5800 K, we get LUMINOSITY = 3.45 x10^11 W/K^4 (5800 K)^4 = 3.8 x 10^26 W which is the LUMINOSITY of the Sun good! At least it works for one star what about the rest? If we do this calculation for a large number of TEMPERATURES, we can then plot our model onto the H-R diagram. IF the model values fall in the same place on the diagram as the observed data then our model FITS the data. i. e., we can have some confidence in our explanation of the relationship between TEMPERATURE and LUMINOSITY IF it doesn't fit, then we have not FIT the data and we can't claim to have explained the relationship between TEMPERATURE and LUMINOSITY i. e., we have more work to do Let's try it hmm.... well, we've had some success -- trend is in the right direction -- higher L for higher T -- lower L for lower T -- but the curve definitely doesn't fit the data IF our hypothesis were correct, then the real star data would fall along our model line that would mean real stares behave like our model. Apparently, this is not the case. Are we completely wrong? Is our BB idea wacko? -- possibly, but still the fit seems kind of close -- the line just isn't tilted enough -- and we think we understand how matter produces light -- there aren't a lot of other choices -- maybe our model just isn't complete -- what else to vary? Well, let's examine our model's assumptions In addition to saying that stars emit like blackbodies, we also stipulated that all stars have the same size as the Sun. What if they didn't? What if stars had different sizes? How would that change our model? It would make R and additional variable Now LUMINOSITY depends on TEMPERATURE and R LUMINOSITY = 4 pi sigma RADIUS^2 TEMPERATURE^4 So, for a given TEMPERATURE, if the RADIUS of the star increases, the LUMINOSITY will increase We can create the same type of line we've already done for RADIUS = 1 solar radius for other sized stars try RADIUS = 3 solar radii try RADIUS = 1/3 solar radius Now we can plot three lines: one for model stars of each radius the lines are parallel they increase at the same rate with TEMPERATURE but displaced vertically because each line represents a different RADIUS model it's not obvious that this helps - none of the lines fits the data -> CONCLUSION: stars aren't all of the same size But wait, that's intersting there's a systematic relationship between the size of a star and it's TEMPERATURE e.g., the RADIUS = 1 solar radius lline fits 6000 K stars fine the RADIUS = 3 solar radii line fits 12000 K stars fine the RADIUS = 0.33 solar radii fits 3000 K stars OK --> hotter stars on the Main Sequence are physically bigger as well - that's a conclusion from the data i.e., the position and slope of the Main Sequence - not demanded by our models - we didn't specify how RADIUS changes with TEMPERATURE However, we have discovered from the data that there appears to be a relationship between RADIUS and TEMPERATURE for Main Sequence stars ----------------------------------- We can lay this game further and use it to look at the oddball stars not on the Main Sequence -- not all stars are on the MS; just most of them -- a few others in different parts of the diagram -- upper right -- lower left Now we can characterized them -- upper right -- cool stars, but high L -- they must have really huge sizes -- GIANTS -- lower left -- hot stars, low L -- they must be pretty small -- WHITE DWARFS To make the H-R diagram, we needed to input LUMINOSITY and TEMPERATURE -- we get out RADIUS (or size) -- we also see for for most of the stars, higher TEMPERATURE means larger RADIUS (or size)