Lesson 6 Lab - The Cosmic Distance Ladder

Exercises 

The Cosmic Distance Ladder Module consists of material on seven different distance determination techniques. Four of the techniques have external simulators in addition to the background pages. You are encouraged to work through the material for each technique before moving on to the next technique.

Radar Ranging 

Question 1: (2 points) over the last 10 years, a large number of ice balls have been found in the outer solar system out beyond Pluto. These objects are collectively known as the Kuiper Belt. An amateur astronomer suggests using the radar ranging technique to learn the rotation periods of Kuiper Belt Objects. Do you think that this plan would be successful? Explain why or why not? 

The technique can be a challenge to use for an amateur astronomer and might not be successful because of the hardships that the environment has set through the transparency of particular radio waves. Furthermore, no sooner the radio waves are sent than they begin to spread, making it even more challenging to get the return signals as they will be very weak. Thus, for the Kuiper objects which are further beyond Pluto, the technique will be ineffective, and unsuccessful results will be achieved. Since the objects under observation are far placed, then the radio signals sent will continue to be dispersed, and the final signal that will reach the body will be extremely weak, thereby making the echo signals even more weak and hard to detect. But if a very strong signal is sent and extremely sensitive detectors are employed, then there is a possibility for the detection of the echoes. 

Parallax 

In addition to astronomical applications, parallax is used for measuring distances in many other disciplines such as surveying. Open the Parallax Explorer where techniques very similar to those used by surveyors are applied to the problem of finding the distance to a boat out in the middle of a large lake by finding its position on a small scale drawing of the real world. The simulator consists of a map providing a scaled overhead view of the lake and a road along the bottom edge where our surveyor represented by a red X may be located. The surveyor is equipped with a theodolite (a combination of a small telescope and a large protractor so that the angle of the telescope orientation can be precisely measured) mounted on a tripod that can be moved along the road to establishing a baseline. An Observer’s View panel shows the appearance of the boat relative to trees on the far shore through the theodolite.

Configure the simulator to preset A, which allows us to see the location of the boat on the map. (This is a helpful simplification to help us get started with this technique – normally the main goal of the process is to learn the position of the boat on the scaled map.) Drag the position of the surveyor around and note how the apparent position of the boat relative to background objects changes. Position the surveyor to the far left of the road and click take measurement which causes the surveyor to sight the boat through the theodolite and measure the angle between the line of sight to the boat and the road. Now position the surveyor to the far right of the road and click take the measurement again. The distance between these two positions defines the baseline of our observations, and the intersection of the two red lines of sight indicates the position of the boat.

We now need to make a measurement on our scaled map and convert it back to a distance in the real world. Check show ruler and use this ruler to measure the distance from the baseline to the boat in an arbitrary unit. Then use the map scale factor to calculate the perpendicular distance from the baseline to the boat.

Question 2: (2 points) enter your perpendicular distance to the boat in map units. ___7.5___________ Show your calculation of the distance to the boat in meters in the box below. 

1 map scale = 20m,  Therefore,  7.5 map scales = ?  The distance = 7.5 x 20 = 150m

Configure the simulator to preset B . The parallax explorer now assumes that our surveyor can make angular observations with a typical error of 3 ° . Due to this error, we will now describe an area where the boat must be located as the overlap of two cones as opposed to a definite location that was the intersection of two lines. This preset is more realistic in that it does not illustrate the position of the boat on the map.

Question 3: (2 points) repeat the process of applying triangulation to determine the distance to the boat and then answer the following: 

What is your best estimate for the perpendicular distance to the boat?	6.8 map scales which is equivalent to 136m
What is the greatest distance to the boat that is still consistent with your observations?	The greatest distance is 7.3 map scales or 146m.
What is the smallest distance to the boat that is still consistent with your observations?	The smallest distance is 6.6 map scales or 132m

Configure the simulator to preset C, which limits the size of the baseline and has an error of 5° in each angular measurement.

Question 4: (2 points) repeat the process of applying triangulation to determine the distance to the boat and then explain how accurately you can determine this distance and the factors contributing to that accuracy. 

The distance of the boat is 2.8 ruler units or 56m. This method can accurately determine the accuracy of the objects that are at a close range due to the fact that the angle of triangulation is very small. Thus, the factors that contribute to this accuracy are the angle of triangulation, the smaller the angle, the more accurate the method is in determining objects that are at a closer range. 

Distance Modulus 

Question 5: (2 points) complete the following table concerning the distance modulus for several objects. 

Object	Apparent  Magnitude m	Absolute  Magnitude M	Distance  Modulus m-M	Distance (pc)
Star A	2.4	2.4	0	10
Star B	-3.98	5	1.02	16
Star C	10	8.01	1.99	25
Star D	8.5	0.5	8	398.1

Question 6: (2 points) could one of the stars listed in the table above be an RR Lyrae star? Explain why or why not. 

Star D is most likely the RR Lyrae star as it has an absolute magnitude of 0.5 a feature that is common in most RR Lyrae stars. It is also the farthest placed star of the four identified. 

Spectroscopic Parallax 

Open up the Spectroscopic Parallax Simulator . There is a panel in the upper left entitled Absorption Line Intensities – this is where we can use the information on the types of lines in a star’s spectrum to determine its spectral type. There is a panel in the lower right entitled Star Attributes where one can enter the luminosity class based upon information on the thickness of the line in a star’s spectrum. This is enough information to position the star on the HR Diagram in the upper right and read off its absolute magnitude.

Let’s work through an example. Imagine that an astronomer observes a star to have an apparent magnitude of 4.2 and collects a spectrum that has very strong helium and moderately strong ionized helium lines – all very thick. Find the distance to the star using spectroscopic parallax.

Let’s first find the spectral type. We can see in the Absorption Line Intensities panel that for the star to have any helium lines it must be a very hot blue star. By dragging the vertical cursor, we can see that for the star to have very strong helium and moderate ionized helium lines it must either be O6 or O7. Since the spectral lines are all very thick, we can assume that it is a main sequence star. Setting the star to luminosity.

Class V in the Star Attributes panel then determines its position on the HR Diagram and identifies its absolute magnitude as -4.1. We can complete the distance modulus calculation by setting the apparent magnitude slider to 4.2 in the Star Attributes panel. The distance modulus is 8.3, corresponding to a distance of 449 pc. Students should keep in mind that spectroscopic parallax is not a particularly precise technique even for professional astronomers. In reality, the luminosity classes are much wider than they are shown in this simulation and distances determined by this technique are probably have uncertainties of about 20%.

Question 7: (2 points) Complete the table below by applying the technique of spectroscopic parallax. 

	Observational Data			Analysis
m	Description of spectral lines	Description of line thickness	M	m-M	d  (pc)
6.2	strong hydrogen lines  moderate helium lines	very thin	0.9	5.3	117
13.1	strong molecular lines	very thick	16	-2.9	2.62
7.2	strong ionized metal lines  moderate hydrogen lines	very thick	3.1	4.1	65

Main Sequence Fitting 

Open up the Cluster Fitting Explorer. Note that the main sequence data for nearby stars whose distances are known are plotted by absolute magnitude in red on the HR Diagram. In the Cluster Selection Panel, choose the Pleiades cluster. The Pleiades data are then added in apparent magnitude in blue. Note that the two y-axes are aligned, but the two main sequences don’t overlap due to the distance of the Pleiades (since it is not 10 parsecs away).

If you move the cursor into the HR diagram, the cursor will change to a handle, and you can shift the apparent magnitude scale by clicking and dragging. Grab the cluster data and drag it until the two main sequences are best overlapped as shown to the right.

We can now relate the two Y-axes. Check show horizontal bar which will automate the process of determining the offset between the m and M-axes. Note that it doesn’t matter where you compare the m and M values; at all points, they will give the proper distance modulus. One set of values gives m –

M = 1.6 – (-4.0) = 5.6 which corresponds to a distance of 132 pc.

Question 8: (2 points) Note that there are several stars that are above the main sequence in the upper left. Can you explain why these stars are not on the main sequence? 

The stars, referred to as “blue stragglers” are above the main sequence in the upper left as they are constituents of binary systems and their companions have passed mass to them. It is the mass added to them that places them on the left of the cluster turnoff point. 

Question 9: (2 points) Note that there are several stars below the main sequence especially near temperatures of about 5000K. Can you explain why these stars are not on the main sequence? 

These stars are most probably non-member stars found in the same direction as the cluster, and have been added in the cluster photometry. 

Question 10: (2 points) determine the distance to the Hyades cluster. 

Apparent magnitude m	Absolute Magnitude M	Distance (pc)
6.4	3.5	38
Question 11: (2 points) Determine the distance to the M67 cluster.
Apparent magnitude m	Absolute Magnitude M	Distance (pc)
12.3	3.5	575

Cepheids 

Question 12: (1 point) A type II Cepheid has an apparent magnitude of 12 and a pulsation period of 3 days. Determine the distance to the Cepheid variable and explain your method in the box below? 

From graphical representation in http://astro.unl.edu/naap/distance/cepheids.html , a period of 3 days matches to an absolute magnitude of -3.  Hence,  Modulus distance = m –M  = 12- (-3)  = 15  Which is an equivalent to a distance of 10,000pc  Note: These estimates do not factor in the numerous subtleties considered by astronomic researchers.

Supernovae 

Open up the Supernovae Light Curve Explorer . It functions similarly to the Cluster Fitting Explorer. The red line illustrates the expected profile for a Type I supernovae in terms of Absolute Magnitude. Data from various supernovae can be graphed in terms of apparent magnitude. If the data represents a Type I Supernovae, it should be possible to fit the data to the Type I profile with the appropriate shifts in time and magnitude. Once the data fit the profile, then the difference between apparent and absolute magnitude again gives the distance modulus.

As an example, load the data for 1995D. Grab and drag the data until it best matches the Type I profile as shown. One can then use the show horizontal bar option to help calculate the distance modulus. One pair of values is m – M = 13-(-20) = 33, which corresponds to a distance of 40 Mpc.

Question 13: (1 point) determines the distance to Supernovae 1994ae and explains your method in the box below? 

From the simulator of the 1Supernova Light Curve Fitting Explorer,  The apparent magnitude is 16.5, and the absolute magnitude is -16  Thus,  the distance modulus (m – M) = 16.5 – (-16)  = 32.5  This corresponds to a distance of 31.6Mpc

Question 14: (1 point) Load the data for Supernova 1987A. Explain why it is not possible to determine the distance to this supernova? 

The distance cannot be determined as they are only visible after several years, and when they do appear, they are visible for only a few days before their brightness fades away, and they become invisible. The Supernova is believed to appear only twice or maybe once a century hence, making it hard to determine the distance as within the short time they are visible, scientists have to make lots of efforts to detect them. The type 1 constellation is also uniform, making it difficult to determine its distance.