## Calculate the energy emitted per second by the entire star if we multiply the Stefan-Boltzmann law by the spherical surface area of a star: 4 pi R2.

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Introduction
• Originators: Ejnar Hertzsprung and Henry Russell developed the diagram independently. It is a plot of the magnitude (absolute magnitude (related to luminosity) or apparent magnitude (related to brightness)) versus spectral type (which is related to temperature and color). This diagram is as important to stellar astronomy as the periodic table is to chemistry. It allows us to identify the stage of stellar evolution a star is in but also to estimate distances to galaxies, and determine the age of stars in star clusters.
• Brightness/Apparent Magnitude and Luminosity/Absolute Magnitude: The brightness of a star is defined as the amount of energy we receive from it here on Earth, how bright a star appears to us in the night sky. Brightness depends on the distance of the star from Earth. A low energy star closer to Earth can appear equally bright as a high energy star further away. Brightness is measured by the apparent magnitude scale. The apparent magnitude is a logarithmic property that decreases as brightness increases. A star that appears 100 times brighter is has an apparent magnitude of 5 less. The brighter the star, the lower its apparent magnitude. Similarly the luminosity of a star is defined as the amount of energy it emits per second. Luminosity does not depend on the distance of a star from Earth and is an intrinsic property of the star itself. Luminosity is measured by the absolute magnitude scale. It is also a logarithmic scale that decreases when luminosity increases. Similar to the apparent magnitude scale, a star that emits 100 times more energy has an absolute magnitude of 5 lower. The absolute magnitude of a star is defined as its apparent magnitude if it were 10pc away from Earth. Basically, this system lines all stars up at the same distance of 10pc from Earth and their differences in brightness are due to their differences in energy output.
• Stefan-Boltzmann Law and Size-Luminosity Relationship: The Stefan-Boltzmann law describes the energy flux emitted from the surface of a star: F = sigma T4, where F is the energy per second per cm2, sigma a constant of proportionality and T the temperature. We can calculate the energy emitted per second by the entire star if we multiply the Stefan-Boltzmann law by the spherical surface area of a star: 4 pi R2. This yields the Size-Luminosity Relationship: L = sigma T4 4 pi R2. The luminosity then, is proportional to the fourth power of the temperature but also the square of the radius of the star. A cool but large star can emit just as much energy as a smaller, hotter one. In the lab we will be comparing one star whose properties we know to another so that we deal with luminosity ratios. The constants sigma and pi will cancel out and we wind up with L1/L2 = (T1/T2)4(R1/R2)2.
◦ Sample Problem: A star has the same temperature as the Sun but is 100x brighter (emits 100x more energy). How much larger or smaller is the star compared to the Sun? So, Tstar = Tsun, that means the temperature ratio in the equation above will be 1 and we won’t have to worry about it in our calculation. That leaves Lstar/Lsun = (Rstar/Rsun)2. The luminosity ratio equals 100, so: 100 = (Rstar/Rsun)2. To solve for Rstar, we need to take the square root of and multiply both sides by Rsun. We get 10 Rsun = Rstar. The answer is that the star is 10 times larger than the sun. If their temperatures had been different we would simply plug in the temperatures, calculate the ratio of temperatures, take that ratio to the fourth power, and divide both sides by that number to help isolate Rstar.
• Interpreting the H-R Diagram: When Hertzsprung and Russell plotted their data they found that stars grouped in well defined regions of the diagram. This points to a relationship between physical stellar properties and brightness and temperature of the stars. The most significant region is a band of stars that spans diagonally across from top left (high energy, high temp.) to bottom right (low energy, low temp). This band is called the main sequence because most stars are found on it. The of stars on the main sequence only varies by a factor of +- 10. Then, there is a region on the bottom left where low energy, hot stars can be found. They are the white dwarfs because to be very hot and only emit very little energy, means that they must be very tiny, not a whole lot of surface area over which the energy is emitted. Yet another region is in the top right corner of the diagram where the high energy, cool stars can be found. These stars are referred to as Red Giants and Super Giants because in order to be as cool as they are and emit a large amount of energy they must have a huge surface over which to emit this energy.
Procedure
• Conversion of logarithm of temperature to temperature: You can use the 10x or INV log button on your calculator to convert the logarithm of a temperature to the actual temperature.
• Some hints for specific questions:
◦ To answer questions 6 and 7 divide the number of stars with the property being asked for by the total number of that data set.
◦ For question 12 use Lstar/Lsun = (Tstar/Tsun)4(Rstar/Rsun)2. You can set Tsun = 5700K, and use Lstar/Lsun = 100.
◦ For question 17 you can start with Dsirius/Dsun = (Msirius/Msun)(Rsun/Rsirius)3. Multiply both sides by Dsun, plug in what is given about Sirius’ mass and your result from question 15 (be careful, you may have to use the inverse of it).

The following resources are available if questions should arise.

YouTube: Live Lab Help Session Recording – H-R Diagram