Parallax and Luminosity: Understanding Scaling Relations

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 : 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 : 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. Betelgeuse has parallax angle that is 1/3 times smaller than achernar 2. How does the luminosity of Betelgeuse compare to the Luminosity of Achernar? (Always show your work or explain your answer!) L = B(4pi(3d^2)). The Luminosity of betelgeuse will be greater than luminosity of achernar D = 1/ P Brightness = Luminosity / ( 4 π d 2 ) 1

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. 2

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? It will be 2 times more brighter b. 5 times Earth’s current distance from the Sun (close to Jupiter’s orbit)? It will be 5 times less brighter c. the (average) orbital distance of Pluto at 40 AU? It will appear 40 times less brighter Part 2: Luminosity, Temperature, and Area Breaking down the Luminosity Equation into its parts, , 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 , 4, and π divide out. a. – Surface Area (m 2 ) of a sphere b. L – Luminosity (Watts) c. T – Temperature (Kelvin) d. – 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 π ) 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. L = 4 π R 2 σ T 4 σ 4 π R 2 σ L star L sun = 4 × π × σ × R 2 sun × T 4 sun = 3

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^ 4 x pi x Boltzmann constant x R^2 Star x T^4 star ——-> L star 4

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. It will appear to be a blue color 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 will be in the ultraviolet region since the temp at Achernar holds more energy 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. It will 81 time more light Achernar 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? It will give a 49 time more 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!) 3969 times 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? 5

No because it’s to bright 6

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Part 3: The Main Sequence Mass – Luminosity RelaIon Definition: Mass – Luminosity Relation: 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. (X)^4 - no change 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. It would 0.1 times less than the sun Part 4: Main Sequence Mass – LifeIme RelaIon 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) . L star L sun = ( M star M sun ) 4 7

Let’s compare a Prius (50 miles/gallon) and a Hummer (5 miles/gallon). 13. We typically compute miles/gallon for cars but the rate at which it consumes fuel is in expressed in gallons/mile. What is the rate at which the Prius and Hummer consume fuel? Prius: 0.02 gallons/mile Hummer: 0.2 gallons/mile 14. The Prius has a 10-gallon gas tank and the Hummer has a 30-gallon gas tank. How many miles can a Prius and Hummer travel on one tank of fuel using the equation above? Prius: 500 miles Hummer: 150 miles 15. Using your numbers above, approximately how many times farther can a Prius travel than a Hummer on a single tank of gas? 350 times more Luminosity measures the rate at which a star is losing energy, and because lost energy must be replaced to be in thermal equilibrium, it also measures the rate at which a main- sequence star needs to generate energy by nuclear fusion. We can use this to derive a formula that gives us an estimate of a star’s main-sequence lifetime relative to the lifetime of the Sun. In main sequence stars only about 10% of their mass is available for fusion. The amount of fuel the star has is proportional to the star’s mass . The rate at which it consumes that fuel is proportional to the star’s luminosity . The lifetime of the Sun is approximately 10 billion years. 16. What is the mass of nuclear fuel available for each of these main-sequence stars: (give that mass as a fraction of the mass of the Sun) Amount of fuel for a 0.1 M sun star : 0.01 m Amount of fuel for a 1 M sun star : 0.1 m 8

Amount of fuel for a 10 M sun star : 1 m 9

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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 ( τ ). We can use the info of the amount of fuel to calculate the distances 18. Using your equation from above, write out the equation for each τ star on top and τ sun on the bottom of the equation below and simplify to get the lifetime of a star relative to the Sun. = 10^10 / 10^10 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. 10^10/10^10. ——-> ( m star / m sun )^4 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. It will be 10 times more lifetime than the sun τ star τ sun = 10

21. Calculate the main-sequence lifetime for a star that is 0.1 times the mass of the Sun. It will be 0.1 times less lifetime than the sun 11

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: 23. Stars with the same color have the same temperature and lifetime (optional). 24. Stars with the same total energy output have the same _________temperature___________. 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) Upper Left Upper Right Lower Right Lower| Left Hot stars Dim stars Luminous stars Cool stars Large Blue stars Small Red stars Small Blue stars Large Red stars 12

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d. As the Mass increases, the Lifetime (increases / decreases ) 13

Wk08-Scaling_Relations_Activity

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