Wk08-Scaling_Relations_Activity_avaiceman

Student Name: Lab TA: Group Members: Astronomy 1101 Scaling Relations Activity Part 1: Remembering Parallax and the Inverse Square Law of Light As we get deeper into stars it is useful to remember the concepts we have used to get here. These ideas help us to go deeper into the Universe building on themselves. It’s been a while since we have done Parallax but let’s review it and also see how Parallax and the Apparent Brightness of light are different. Parallax Formula : D = 1 / P where P is the parallax angle and D is the distance. If P is measured in arcseconds, the distance D has units of parsecs (pc). Inverse Square Law : Brightness = Luminosity / ( 4 π d 2 ) The star Betelgeuse is 3 times more distant than the star Achernar, but has about the same apparent brightness. 1. Based on the information above, how does the parallax angle of Betelgeuse compare to that of Achernar? Start by figuring out which one should be larger, then figure out by how much. D=1/p Parallax angel of Betelgeuse is 3 times smaller than that of Achernar. 2. How does the luminosity of Betelgeuse compare to the Luminosity of Achernar? (Always show your work or explain your answer!) B=L/ ( 4 π d 2 ) Luminosity of Betelgeuse is 9 times less than Achernar. For the next question you do not have to show your work, but you should still set up the problem and have an answer that makes sense. These are directly related to the idea of having a habitable zone which needs a certain amount of energy from the host star for water to be liquid. 1

3. Compared to its current appearance on Earth, how bright would the Sun appear when viewed from… a. half of Earth’s current distance from the Sun? The sun would be 2 times as bright. b. 5 times Earth’s current distance from the Sun (close to Jupiter’s orbit)? The sun would be 25 times dimmer. c. the (average) orbital distance of Pluto at 40 AU? The sun would be 1600 times dimmer. Part 2: Luminosity, Temperature, and Area Breaking down the Luminosity Equation into its parts, L = 4 π R 2 σ T 4 , We can write in words what each of the following terms are. Look at the term as a whole, not as individual values. When we compare objects relative to the Sun then σ , 4, and  divide out. a. 4 π R 2 – Surface Area (m 2 ) of a sphere b. L – Luminosity (Watts) c. T – Temperature (Kelvin) d. σ – Stefan-Boltzmann constant . A physical constant like G in the gravity equation. 4. Using the Luminosity Equation above, we are going to take the ratio of L star to L sun . Write down the complete equation for the star’s luminosity below and simplify by dividing out all of the common terms (like  ) to get a relation that involves only those quantities that can change, such as L, R, and T. Make sure that subscript labels indicate the object to which a quantity belongs (i.e. R star is the radius of a star, as you can see R sun is the radius of the Sun). Because we’ve made a formula that takes out all of the constants, we can more clearly see how luminosity changes when we change those quantities that can change. We call such formulae scaling relations . DO NOT use numerical values to answer this question. 2

You DO NOT need a calculator for most of the next few problems. If you find yourself needing a calculator, you’re approaching the problem incorrectly. Additional Information: Our Sun’s temperature is about 6,000 Kelvin (K) and its spectrum peaks at roughly 500 nm (roughly the middle of the visible part of the electromagnetic spectrum). 5. Achernar, is a star with a temperature of roughly 18,000 Kelvin. What color will Achernar appear to be? Hint : It is as simple as it seems. The star will appear blue. 6. Will the peak wavelength for Achernar be in the visible, infrared, or ultraviolet region of the electromagnetic spectrum? Explain why you think so, you do not need an equation. It would be in the ultraviolet region because it is so hot, so it’s peak wouldn’t be visible. 7. How much more light does Achernar give off compared to the Sun ONLY due to having a higher surface temperature? (Show or explain your work.) Hint: Same area. (18000)^4/(6000)^4 Achernar gives off 81 times more light. 8. It turns out Achernar has a radius that is 7 times bigger than the Sun. How much more light does Achernar give off ONLY because it is a larger size? 7^2=49 Achernar gives off 49 times more light. 9. Combined with your answer to the previous 2 questions how much more total light does Achernar give off compared to the Sun? (You can use a calculator for this one if you need to, but still show or explain your work!) 81 X 49 = 3969 Achernar gives off 3969 times more light in total. 4

10. Do you think it would be possible to have life (as we know it) if we lived on a planet like Earth orbiting 1AU from Achernar? I don’t think so because it is so much brighter. The UV would probably be more harmful to life so there couldn’t be life on a planet 1 AU from Achernar. Part 3: The Main Sequence Mass – Luminosity Relation Definition: Mass – Luminosity Relation: L star L sun = ( M star M sun ) 4 We know that the luminosity of a main sequence star depends on its mass. This allows us to estimate main sequence luminosities for types of main-sequence stars that are rare. Because they are so rare, there are no nearby examples to show up on the H-R diagram of nearby stars. 11. Calculate the main sequence luminosity (compared to the luminosity of the Sun) for a star that is 10 times the mass of the Sun. (10/1)^4 = 10,000 12. Calculate the main sequence luminosity (compared to the luminosity of the Sun) for a star that is 0.1 times the mass of the Sun. (0.1/1)^4 = 0.0001 Part 4: Main Sequence Mass – Lifetime Relation Before getting to stars, let’s begin with an example from everyday life: How far can a car travel on a full tank before running out of gas? The distance a car can travel depends on the amount of fuel and the rate at which it is consumed: distance = fuel (gallons) / consumption rate (gallons per mile) . Let’s compare a Prius (50 miles/gallon) and a Hummer (5 miles/gallon). 5

17. Using the Prius and Hummer questions at the start of this part, discuss with your lab partner how to come up with a formula to estimate the lifetime of a star (  ). 0.10 mass/ luminosity because the mass can relate to how much gas can be in the tank and the rate of fuel consumption can be related to the luminosity. 18. Using your equation from above, write out the equation for each  star on top and  sun on the bottom of the equation below and simplify to get the lifetime of a star relative to the Sun. τ star τ sun = 0.1 M st ar / Lstar 0.1 Msun / Lsun = (Mstar/Msun)/(Lstar/Lsun) 19. Once you have your formula, combine it with what we know about the Mass – Luminosity relation in Part 3 to create a new formula for the main-sequence lifetime that only has the star’s mass in the equation. Hint : Solve for Luminosity in the Mass–Luminosity equation and replace L star and L sun with those equations in the lifetime equation. (Mstar/Msun)/(Lstar/Lsun) = (Mstar/Msun)/(Mstar/Msun)^4 = (Msun/Mstar)^3 Using your formula for the main-sequence lifetime, answer the following questions: 20. Calculate the main-sequence lifetime for a star that is 10 times the mass of the Sun. (1/10)^3 = 0.001 21. Calculate the main-sequence lifetime for a star that is 0.1 times the mass of the Sun. (1/0.1)^3 = 1,000 7

Part 5: The H-R Diagram and Stellar Data This is a good review of all the concepts we have learned about the H-R Diagram so far. 22. Mark all the boxes that accurately describe where on the H-R Diagram to find these stars: Upper Left Upper Right Lower Right Lower| Left Hot stars X X Dim stars X X Luminous stars X X Cool stars X X Large Blue stars X Small Red stars X Small Blue stars X Large Red stars X 23. Stars with the same color have the same __ Temperature ___ and ____ Lifetime ____ (optional). 24. Stars with the same total energy output have the same __ Luminosity __. 25. Complete these comparisons of main sequence stars: a. As the Mass increases, the Temperature ( increases / decreases) b. As the Mass increases, the Size ( increases / decreases) c. As the Mass increases, the Luminosity ( increases / decreases) d. As the Mass increases, the Lifetime ( increases / decreases) 8