Studying the Crab Nebula: Supernova Remnants Exploration

Your Name: TA: Group Members: Astronomy 1101 “Supernova Remnants” Activity Studying Exploding Stars Studied Via Light This activity applies the tools we have gained to study light to a “supernova remnant.” This is the wreckage left by of an exploding star, a supernova. These supernovae are among the most energetic and important phenomena in the universe. In this lab, we will see that we have all the tools to use light to figure out the basic properties of a stellar explosion. We’ll pay particular attention to the Crab Nebula, one of the nearest and newest such stellar explosions in the Milky Way. Part 1: Determining the Age of the Crab Nebula Part A. Measure the Angular Diameter of the Crab N ebula First, we will figure out how big, the physical size, the nebula is from a picture and the known distance. The figure below is an image of the Crab Nebula, the expanding gas left behind by a massive star that exploded as a “core-collapse” type supernova. The explosion itself was seen by Chinese astronomers in the year 1054 AD so we know that the Crab Nebula is just about 1000 years old. The whole image is 0.17 degrees on each side. The picture was taken in 2010 using the Large Binocular Telescope in Arizona (in which OSU is a major partner). 1

Using a ruler or other measurement tool, measure the angular diameter of the Crab Nebula in degrees, first along its longest dimension, then along its shortest dimension (remember that the whole image is 0.17 degrees on a side, so measure the image in inches to figure out the number of degrees per inch, then measure the diameter of the nebular in inches and convert). The Crab isn’t a perfect circle, so measure along the long axis and along the short axis separately. 1. What is the “angular diameter” in degrees of the Crab Nebula along the long dimension: 10.7 degrees 2

2. What is the “angular diameter” in degrees of the Crab Nebula along the short dimension: 7.3 degrees Now compute the average of these two numbers to compute the average angular diameter of the Crab Nebula: 3. The average angular diameter of the Crab Nebula in degrees: 9 degrees Ultimately based on something very similar to parallax (remember parallax!), we have an estimated distance to the Crab Nebula of d = 6,500 light years = 6.2 × 10 19 m Using the formula from the book and lecture of Physical diameter = d × ( θ / 57.3 degrees) where θ is the average angular diameter you found from your measurements above. Calculate the physical diameter of the Crab Nebula, first in light years , then in meters . 4. Physical diameter of Crab Nebula, light years: 6,500 ly (9/57.3 deg) = 1020.94241 5. Physical diameter of Crab Nebula, meters: 6.3 x 10^19 (9/57.3 degrees) = 9.7 x 10^22 M 3

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Part B. Estimate the Expansion Rate of the Crab N ebula Next, we will use the Doppler shift to figure out the rate of expansion of the Crab Nebula. The figure below shows a visible-light spectrum of the Crab Nebula, taken with the MODS1 spectrograph on the Large Binocular Telescope (the same telescope used to take the picture above). The horizontal axis of this “spectrum” plot shows the wavelength of light, measured in nanometers (1 nanometer = 1 nm = 10 − 9 m). The vertical axis shows the brightness of light from the Crab Nebula at that wavelength. The spectrum shows a number of “emission lines” produced by atoms in the hot nebula. Focus on the bright lines in the spectrum near 500 nm. These are emission lines from ionized oxygen known as the “O III lines.” The first of these lines is emitted at 500.7 nm The second of these lines is emitted at 495.9 nm. These lines would be bright green if you looked at them with your eyes. If the Crab Nebula were not moving, we would just see two lines at these two wavelengths, but the Nebula is an explosion, with one side coming towards us and another going away. We see the light emitted from the far side and the near side of the expanding Crab Nebula separately. Each side emits two lines. Thus, what started as two lines becomes four lines, as both O III lines are Doppler shifted to the red (receding side) and both are also Doppler shifted to the blue (approaching side). From this graph, we can calculate the Doppler velocity of the gas on the near side moving toward us ( blueshifted ) and on the far side moving away from us ( redshifted ) to solve for the rate of expansion of the Crab Nebula. 4

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Use a straight line to measure the wavelengths (x-axis) of the peaks for the Doppler shifted 500.7 nm oxygen lines to the nearest 0.1 nm. The wavelengths of the Doppler shifted 495.9 line has been measured for you. Hint: the measured 495.9 lines are marked on the spectrum so you know which ones are the 500.7 lines. • O III 495.9 nm (redshifted) ____497.9___ nm • O III 495.9 nm (blueshifted) ____493.6___ nm 6. O III 500.7 nm (redshifted) _____502.7_______ nm 7. O III 500.7 nm (blueshifted) _____498.4_______ nm Compute the Doppler shift-based velocity for the expansion of the Crab Nebula using each of the observed lines of the Crab Nebula in meters/sec. The Doppler shift formula is v = 3.0 × 10 8 m/sec × (( λ obs / λ emit ) − 1) where c = 3 × 10 8 m/sec is the speed of light, λ obs is the observed wavelength of light and λ emit is the emitted wavelength of light, so 495.9 nm or 500.7nm here. Do this calculation separately for both the redshifted and blueshifted components. Show your calculations (i.e., write down the numbers you are multiplying and dividing). V = 30 x 10^8 ((497.9/495.9) 8. O III 495.9 nm (redshifted) _____1209920_______ m/s V = 30 x 10^8 ((493.6/495.9)-1) 9. O III 495.9 nm (blueshifted) _____-1391410_______ m/s Finally, combine the blue and redshifted estimates to compute the average speed (average of the absolute values of the two Doppler velocities) from your measurements to get your estimate of the expansion rate or expansion speed of the Crab Nebula: (1209920+-1391410)/2 = -90745 6

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10. Average Expansion Rate: ____-90745________ m/s This is the speed at which the Crab Nebula is expanding in ALL directions. 11. Now draw a simple schematic sketch of the Crab Nebula with the expansion motion and size that you have estimated labeled. Part C. Estimate the Age of the Crab N ebula Now we will combine our size and expansion rates to figure out how old the Crab Nebula appears to be. If the Crab Nebula expanded at the speed you calculated for a time t = its age , it would grow to a diameter of Diameter = 2 × (Average Expansion Rate) × t This equation can be rewritten to solve for the age of the nebula if we assume that the Crab Nebula has had the same expansion speed for its whole lifetime: t = Radius / (Average Expansion Rate) Using your numbers from parts A and B, compute the age of the Crab Nebula in seconds. Show your calculation: 12. Age of the Crab Nebula in seconds: T = 9.7 x 10^23 /-90745, t= 1.06893 x 10^18 Divide the age in seconds by (3.16 × 10 7 ) seconds/year. This number just 365.25 days × 24 hours/ day × 3600 seconds/hour and will allow you to convert the age to years. 13. Age of the Crab Nebula in years: T/3.16 x 10^7 = 3.3 x 10^11 years 7

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You made assumptions to determine the size of the Crab Nebula. Combined with measurement error (i.e., your numbers above aren’t perfect) there are a range of possible values for its age. One of these assumptions was taking the average of the longest and shortest sides to determine the average size of the Crab, which assumes it is a spherical object. 14. What is the ratio of size of the longest side divided by the shortest side of the Crab Nebula? 10.7 /7.3 This assumption is most likely your largest source of uncertainty. Given this information, how accurately do you think you know the age of the Crab Nebula? A. Accurate to 10% (i.e., true age is within 10% of your estimate) B. Accurate within a factor of two (i.e., true age could be twice your estimate or half your estimate, but probably not further off than that) C. Accurate within a factor of ten (i.e., true age could up to ten times smaller or ten times larger than your estimate) 15. Circle the best choice for your numbers and then explain why it is the most likely uncertainty: It’s plausible that my numbers will be 1/2 or twice what they should be. Chinese astronomers observed a “guest star” that appeared suddenly in the year 1054 AD in the constellation Taurus, near the present location of the Crab Nebula, and eventually faded away. 16. Based on your result for the age and your answer to the previous question about the uncertainty of your measurements, could this “guest star” have been the supernova that we see today as the Crab Nebula? Why or why not? Talk this out with your group. No it couldn’t since when the nebula was formed there would have been observers who should have seen it due to it’s brightness. Part 2: The Energy of a Supernova Explosion Next, we will use some of what we have learned about energy and power to calculate the total energy involved in one of these stellar explosions. Note that this is about the most complex calculation we will do in the class. 9

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The graph below shows a plot called a “light curve.” This is a plot of luminosity (light power) vs. time and the plot shows data for a supernova produced by an exploding white dwarf star (data are from 2002 observations by astronomer Nick Suntzeff). 10

The vertical axis is marked in billions of solar luminosities. A “solar luminosity” is the light power of the Sun (given below). So the plot says that for the few weeks near maximum luminosity, a supernova shines with more energy than a billion suns. The horizontal axis is time since the explosion occurred, in days. To get the total energy emitted as light by the supernova explosion, we need to add up the energy emitted each day. This is equivalent to calculating the area under the curve that connects the points in this figure. To make our lives easier, we’ll calculate the area of the dotted box, which is about the same as the area under the curve. Thus, an exploding white dwarf supernova emits about the same total amount of energy as if it shone at 3 billion solar luminosities for 30 days. First convert 30 days into seconds, using 60 seconds/minute × 60minutes/hour × 24 hours/day = 86,400 sec/day. 17. 30 days × 86,400 sec/day = ______2.512 x 10^6__________ secs Now calculate the luminosity of the explosion in joules/sec. A “joule” is a standard unit of energy, equal to about the energy needed to lift an apple one meter. One solar luminosity is (approximately) 4 × 10 26 watts = 4 × 10 26 joules/sec. How many joules/sec is 3 billion (3 × 10 9 ) solar luminosities? 11

18. 3 billion solar luminosities = ___1.2 x 10^26_____________ joules/sec Now compute the total energy emitted by the supernova, in joules, by multiplying the time (in secs) with the luminosity (in joules/sec) (1.2 x 10^41)(2.5432 x 10) 19. Supernova energy = ______3.0504 x 10^42_________ joules. This is a big number, but what can we compare it to? Let’s compare it to the total amount of energy that our Sun has emitted during its entire 4.5 billion years since it began its life. 4.5 billion years is equal to 1.4 × 10 17 sec. The Sun is emitting energy at a rate of 4 × 10 26 joules/sec. How much energy has the Sun emitted thus far? (1.4 x 10^12)( 4 x 10^26) 20. Energy emitted by the Sun in 4.5 × 10 9 years = ______5.6 x 10^43___________ joules. Now, compute the ratio of these two numbers: 5.6 x 10^43 / 3.0504 x 10^42 21. Energy emitted by Sun in 4.5 × 10 9 years / Energy emitted by supernova explosion = _______18.358 _____________ 12

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This is a comparison of how much energy the Sun has produced so far vs how much energy comes out in a supernova explosion. 13

Part 3 – Extra Credit: The Energy of the Death Star Let’s compare the energy of the supernova explosion to that of another memorable, and seemingly big, explosion. In the first Star Wars movie (Episode IV, A New Hope , 1977), the Galactic Empire’s Death Star blows up the planet Alderaan. If you somehow missed this bit of educational pop culture, google “Destruction of Alderaan” and watch any of the YouTube videos that pop up. Using Newtonian gravity, it’s fairly easy to calculate the minimum energy required to destroy a planet. Expressed as a formula, the answer is: Minimum destruction energy = G × (Mass of planet) 2 / (Radius of Planet) In essence, this formula tells you how much energy is needed to accelerate every atom in the planet over the escape velocity so that it flies apart. Let’s assume that Alderaan is a planet like the Earth, with mass of 6 × 10 24 kg and radius of 6.4 × 10 6 meters. If you plug these numbers into the above formula with the value G = 6.7 × 10 − 11 m 3 / kg / s 2 , you will get out an energy in joules. Do this calculation on your calculator twice and check that you get consistent answers, and check consistency with your lab partner’s calculation, since it’s easy to enter the numbers wrong and you probably have no idea what kind of number the answer “should” be. When you’re confident of your answer, write it down below. Show your calculation. 22. Energy Required to Destroy Alderaan = _______________ joules. Now refer back to your answer from Part 2. If all of the energy of a supernova could be channeled into destroying Earth-like planets… 23. How many planets could one supernova destroy? Show your calculation. Concluding remarks: 14

1) A note for those of you who like to track the units in your calculations, a very good way to catch errors: G can also be written with the units G = 6.67 × 10 − 11 joule m/kg 2 from which you can see that multiplying by (mass) 2 and dividing by radius will give you an answer in joules. 2) The explosive deaths of stars, whether caused by the gravitational collapse of the core of a massive star or the thermonuclear explosion of a white dwarf, are fantastically energetic, far beyond anything in our everyday experience. 15

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Wk09-Light and Crab_Age_Activity (2)

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