Name: Partner(s): Lab #8 The Crab Nebula Introduction One of the most fascinating objects of the winter night sky is the famous Crab nebula, located near the tip of one of Taurus the Bull s horns. The nebula was discovered by the well known French astronomer, Charles Messier, in 1758. It is the first object in his catalog of nebulous objects of the night sky that he began compiling in 1764. The Crab nebula is in fact the remnants of the bright supernova of 1054. This supernova was recorded by Chinese astronomers to have been visible during the day for 23 days and in the nighttime sky for two years. In 1968, radio astronomers Staelin and Reifenstein found the stellar remnant at the core of the nebula - a neutron star! This neutron star spins on its axis 30 times a second. The star s magnetic field causes it to emit beams of light from its magnetic polls. These twin spotlight beams sweep by the Earth, causing the neutron star to appear to blink on and off. Because of this flickering, the neutron star is also called a pulsar. The purpose of this lab is to learn about a number of fascinating properties of the Crab Nebula, including its appearance, radiation mechanisms, expansion rate, age, distance, and some of its spectral properties. Finding the Crab Nebula s Age For this part of the lab, you will need the photographs taken of the Crab nebula in 1973 and 2000 so that you can find the rate of expansion (both of these photographs are negatives so the brightest spots appear dark). The location of the pulsar is indicated in the following image: Astronomy 101 8 1 Introduction to Astronomy
1 (2 pts). To estimate how long the Crab Nebula has been expanding, you must first obtain the scale for each photograph (at the end of the lab). In both cases, measure the distance between the two marked stars in millimeters, estimating to the nearest 0.1mm (in other words try to judge the distance in between each millimeter tick mark). Knowing that the angular distance between the stars is 385 arcseconds, find the scale of each photo in units of [arcseconds/mm]. Date 1973 2000 Distance Between Marked Stars (mm) Photographic Scale (arcseconds/mm) 2 (2 pts). (a) Carefully locate the pulsar as indicated in the diagram above. (b) Identify 10 relatively well-defined knots in the filaments around the periphery of the Crab on both photos. Be sure to distribute your selections around the nebula as much as possible, and select at least four knots near the edges of the minor axis of the nebula. The term minor axis is used to refer to the shortest dimension across the nebula. Clearly number the knots you select on both photos so you don t confuse them. 3 (5 pts). Now use a millimeter ruler to measure the distance of each knot, to the nearest 0.1 mm, from the pulsar on both photos. Note: knots are fainter and fuzzier than stars which are darker and circular. Write your results in Table 2 under the columns labeled r 1973 and r 2000 (r stands for radius). 4 (2 pts). Use the correct scale from step (1) to obtain the angular distances of the knots from the pulsar, q, by converting r into q and fill in the corresponding spaces in Table 2. 5 (2 pts). We will now calculate the average speed of the ejected material in the knots relative to the central pulsar. The angular velocity of any knot, w, is given by the expression w = dq/dt (1) where dq is the angular change in position of a knot, and dt is the interval in time between the two photos (i.e. 27 years). Using this formula, calculate w for each knot in units of [arcseconds/year] and enter the results in Table 2. 6 (2 pts). Knowing the angular speed, w, and the angular position, q, of each knot in 1973, we can solve for the total time, T since the explosion using the simple relation T = q/w (2) Find the estimated time since the explosion for each knot and place the results in Table 2. 7 (2 pts). The mean scatter in dq, where dq is the change in the angular distance of each knot from the pulsar, gives an indication of the random errors in your distance measurements. Indicate the mean error in the space provided below Table 2. Astronomy 101 8 2 Introduction to Astronomy
Knot # r 1973 (mm) 1 2 3 4 5 6 7 8 9 10 q 1973 ( ) r 2000 (mm) q 2000 ( ) dq ( ) w ( /yr) T (yr) 8 (3 pts). Calculate the mean of the 10 T values you obtained, then use this to calculate the date of the supernova occurred. (a) The mean T is (b) Thus, the date of the explosion, according to your results, was 9 (2 pts). Use your spread in values of T (and the following equation) to estimate you random uncertainty: largest time shortest time T = (3) 2 The uncertainty in T is. 10 (2 pts). When calculating the date of the supernova explosion, what have you assumed about the velocity of the gaseous knots? 11 (3 pts). Compare your value for the date of the supernova event to the accepted value of year 1054. What does this suggest about the expansion velocity of the nebula? Explain. Astronomy 101 8 3 Introduction to Astronomy
Finding the Distance to the Nebula In terms of the velocity of expansion, v [km/sec], and the expansion rate, w [arcseconds/yr], measured between 1973 and 2000, we can compute the distance to the Crab nebula. To do this, recognize that in its actual motion, v, across the plane of the sky, a knot can be considered as having traversed a tiny fraction of the circumference of the celestial sphere. (The total circumference is 2πd, where d is the distance from the observer to the nebula.) This fraction is just a portion of a complete 360 angle that has been swept out by the angular motion of the knot. Thus we can set up a relation between the angular and spacial velocities: w/365 = v/2π (4) From the above formula, we can solve for the distance, d, in units of light years. Modifying the expression appropriately so that v has units of [km/sec] and w has units of [arcseconds/year] (along with knowing that one light year is equal to 9.46 10 12 kilometers), the distance of the nebula is given by d = 0.69v/w (5) So far we have found the angular rate of expansion, w, of the Crab Nebula. To obtain its distance, the equation above shows that we need to measure the linear velocity, v, by some other method. To accomplish this, you will learn some of the spectral properties of a supernova remnant, and use the same technique that astronomers use to measure velocities from spectral lines. Look at the spectrum of the Crab Nebula on the following page. In this negative image, the bright emission lines of the nebula and laboratory comparison spectra above and below show as dark lines. The spectrum was taken by aligning the slit of the spectragraph along the major axis of the nebula. The brighter spots along each spectral line occur where a bright filament crossed the slit. Notice that each of the filaments is either redshifted or blueshifted, with nothing in between. This occurs because we are seeing material that is either at the very nearside of the nebula, rushing towards us, or material at the backside of the nebula, rushing away. The filaments are on the outer edges of the nebula. Astronomy 101 8 4 Introduction to Astronomy
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12 (2 pts). Examine carefully the region around the [OII] 3727 line on the Crab Nebula Spectra. First calculate the spectral scale in this region as described below: The distance between the 3690A and the 3719A palladium lines is and is measured to be mm. Angstroms The spectral scale is thus Angstroms/mm. 13 (2 pts). Now use a millimeter ruler and the scale you found in step (12) to find the maximum Doppler shift between the blueshifted and redshifted branches of the [OII] 3727 necklace. maximum Doppler shift = maximum Doppler shift = mm. Angstroms. 14 (2 pts). Using the Doppler formula to calculate the relative velocity between the approaching and receding filaments. Show your calculation in the space provided. Recall the Doppler formula is: change in wavelength wavelength = velocity speed of light The change in the wavelength is the difference in wavelength between the red and blueshifted sides (use your answer from above) and the wavelength is the unshifted line wavelength, i.e. 3727 Angstroms. The speed of light, c = 300,000 km/s. (6) 15 (2 pts). We wish to measure the true expansion of the nebula. Explain, using a drawing, what velocity was found in question 14. Determine what velocity we are really interested in and calculate its value. The spatial velocity of the filament along the line of sight with respect to the center of the Crab Nebula is km/sec. 16 (2 pts). We now have the data needed to establish the distance of the Crab Nebula using the formula developed above. Note however, that the nebula is not spherically symmetrical. Does the radial velocity of a filament rushing toward us from the center of the nebula correspond to the average angular velocity calculated from all of the motion of the knots? Maybe it corresponds to the angular velocity at the ends of the major axis, or maybe the Astronomy 101 8 6 Introduction to Astronomy
ends of the minor axis. The answer depends on the shape of the nebula. Is it an oblate spheroid (like a slightly squashed orange) or is it a prolate spheroid (like an elongated lemon)? Probably the latter, which means that the extension of the nebula toward us is about the same as the shorter dimension in the plane of the sky. Therefore, you should use an average of the proper motions near the end of the minor axis of the nebula for the calculation. Find the average angular velocity (proper motion) of the knots near the edge of the minor axis of the nebula. average w = arcseconds/year Indicate the number of knots on your photograph that you used to derive this value: 16 (3 pts). Finally, calculate the distance to the Crab Nebula using the distance formula given at the beginning of this section. A drawing in which the distance from observer to nebula is one leg of a triangle and the expansion rate in km/sec is another leg may help you make sense of this. Show your work. 17 (2 pts). Find the percent error in your result compared to the accepted value of about 6300 light years. Relative error = d 6300 ly /6300 ly 100 = 18 (2 pts). When calculating the spatial velocity of expansion in km/s and the distance to the Crab Nebula, what have you assumed about the shape of the nebula? 19 (2 pts). List here what you consider to be the primary sources of error in your distance estimate. Astronomy 101 8 7 Introduction to Astronomy
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Astronomy 101 8 9 Introduction to Astronomy