University of Michigan - Department of Astronomy

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updated: 03/09/2003

The Crab Nebula

Before you come to lab

Print out the following images:

Crab Nebula in 1973
Crab Nebula in 2000
Spectra of 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 well known French astronomer Charles Messier in 1758. It is the first object in his catalog of nebulous object 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 it's 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.

Part I: Finding the Crab'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. Find the location of the pulsar on the photos using the image to the right:
  1. To estimate how long the Crab Nebula has been expanding, you must first obtain the scale for each photograph. In both cases, measure the distance between the two marked stars in millimeters, estimating to the nearest 0.1mm. Knowing that the angular distance between the stars is 385 arcseconds, find the scale of each photo in units of [arseconds/mm].

    Date Distance Between Marked Stars (mm) Photographic Scale (arcseconds/mm)
    1973  
    2000  

  2. 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 distribue 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.
    c) Now use a millimeter ruler to measure the distance of each knot, to the nearest 0.1 mm, from the pulsar on both photos. Write your results in Table 2.
    d) Use the correct scale from step (1) to obtain the angular distances of the knots from the pulsar, and fill in the corresponding spaces in Table 2.
    e) The mean scatter in Dq, where q is the angular distance of each star from the pulsar, gives an indication of the random errors in your distance measurements. Indicate the mean error in the space provided below Table 3.

  3. 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

    where dq is the angular change in position of a knot, and dt is the interval in time between the two photos. Using this formula, calculate w for each knot in units of [arcseconds/year] and enter the reults in Table 2.

  4. Knowing the angular speed, w, and the angular position, q, of each knot in 1973, we can solve for the total time since the explosion using the simple relation

    Find the estimated expansion time for each knot and place the results in Table 2.


Table 2: The Expansion of the Crab Nebula

Knot # r
1973
q
1973
r
2000
q
2000
dq
(")
w
("/year)
DT
(years)
1              
2              
3              
4              
5              
6              
7              
8              
9              
10              

Calculate the mean of the 10 DT values you obtained, then use this to calculate the date of the supernova occured

(a) The mean DT is _______________________ years.

(b) Thus, the date of the explosion, according to your results, was _________________.

(c) When calculating the date of the supernova explosion, what have you assumed about the velocity of the gaseous knots?










(d) Compare your value for the date of the supernova event to the accepted value of 1054. What does this suggest about the expansion velocity of the nebula? Explain.









Finding the Distance to the Nebula

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 2pd, 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 spactial velocities:

From the above formula, we can solve for the distance, d. Modifying the expression appropriately so that v has units of [km/sec] and w has units of [arcsec/year] (along with knowing that one light year is equal to 9.46x10^12 kilometers), the distance of the nebula is given by

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 remnat, and use the same technique that astronomers use to measure velocities from spectral lines.

Look at the spectrum of the Crab Nebula. 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.

  1. 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 ______________ Angstroms and is measured to be _________________mm.

    The spectral scale is thus _________________ A/mm.

  2. Now use a millimeter ruler and the scale you found in step (1) to find the maximum Doppler shift between the blueshifted and redshifted branches of the [OII] 3727 "necklace".


  3. 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 Dl/l = v/c.)









  4. We wish to measure the true expansion of the nebula. Explain, using a drawing, what velocity was found in question 3. Determine what velocity we are really interested in and calculate its value.









    The spatial velocity of the filament along the line of sight with repsect to the center of the Crab Nebula is ______________km/sec.

  5. 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 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 motins 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.



    Indicate the number of knots on your photograph that you used to derive this value: _____________.

    Finally, calculate the distance to the Crab Nebula. Show your work.








  6. Find the percent error in your result compared to the accepted value of about 6300 light years.



  7. 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?









  8. List here what you consider to be the primary sources of error in your distance estimate.









  9. With the high resolution of radio telescopes, namely the Very Long Baseline Interferometer, you can use the method of expansion parallax for supernova remnants that are much farther away from us than the Crab Nebula. Supernova 1987A exploded in the large Magellanic Cloud and was first observed on February 23, 1987. At that point, the remnant had a radius of zero. 5.2 days later, the remnant had a radius of 0.0022 arcseconds. The radial velocity of the nebula was 36,000 km/s. What distance does this suggest to the Large Magellanic Cloud?









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