Overview
In this activity, you will explore the relationship between the period of pulsation of Cepheid variable stars and their intrinsic luminosity, and how to utilize that relation for distance measurement.
Objectives
After completing this activity students will be able to:
· Calculate the absolute magnitude for a Cepheid variable with a known distance
· Plot a period-luminosity diagram for classical Cepheids
· Use Leavitt’s Law to determine the distance to another galaxy
Definitions
Here are some terms from lecture that we will be using today in lab:
· Cepheid – a star that varies in brightness with a recurring cycle of pulsations in brightness and size
· Period – P; the time required to complete one cycle; in the case of Cepheid variable stars, the time between cycles of varying brightness
· Leavitt’s Law – also known as the Period-Luminosity Relation; a relationship discovered by Henrietta Leavitt which relates a Cepheid variable star’s period of pulsation to its absolute magnitude.
· Apparent magnitude – m; the apparent brightness of a celestial object as measured from Earth.
· Parsec – equal to 3.26 light-years; the distance an object must be to have a parallax angle of 1 arcsecond.
· Absolute magnitude – M; the brightness of a celestial object as measured from 10 parsecs.
· Distance Modulus – m-M; the difference between an objects apparent and absolute magnitudes.
To begin, download and open the Excel file named ‘LeavittsLaw.xlsx’.
After you open your spreadsheet, you will notice there are two tabs at the bottom left (seen in the screen capture). The first tab, ‘Calibrating Leavitt’s Law’, will be used for Part 1 of the lab and the second tab, ‘Determining Distances’, will be used for Part 2.
Part 1. Calibrating and Plotting Leavitt’s Law
In your spreadsheet, make sure you are on the tab named ‘Calibrating and Plotting Leavitt’s Law’!
Let’s begin with some observations of Cepheids in the star cluster χ Persei. The technique of main sequence fitting gives a distance to χ Persei of about 2600 parsecs. THE DISTANCE TO ALL STARS IN THIS CLUSTER IS 2600 PARSECS. We want to use the distance modulus equation (Equation 1) to calculate the absolute magnitude of these stars:
where M is the absolute magnitude, m is the apparent magnitude, and D is 2600 parsecs.
1. In your spreadsheet, under Question 1, using the apparent magnitude values and Equation 1 to calculate the absolute magnitude in column 5 (labeled ‘M’).
In your spreadsheet, under Question 2, there are data on 8 additional Cepheids which are found in 6 other star clusters. Their absolute magnitudes have been determined by the same method you just used for Question 1. The spreadsheet should automatically fill in the absolute magnitude values for VY Persei, V Persei, VX Persei, and SZ Cassiopeiae from Question 1, but if not, COPY AND PASTE THE ABSOLUTE MAGNITUDE VALUES FROM QUESTION 1 TO THE MATCHING STARS UNDER QUESTION 2!
2. Using the log(P) and M columns for the 11 stars in your spreadsheet under Question 2, you are going use Excel’s graphing abilities to plot these two columns to make your period-luminosity relation.
a. Select the log( P) and M columns
b. Click on ‘Insert’ tab
c. Click on ‘Recommended Charts’ (just ‘Charts’ in Numbers) and select ‘Scatter’ (or ‘2D Scatter’ in Numbers).
i. Note: Make sure none of the columns are in the first column on the left if using Numbers
d. Click on your plot and then click on ‘Chart Design’ Tab in the upper left (if using Numbers, click on plot and the menu bar to the right will list plot options)
e. Need to label axes. Click on ‘Add Chart Element’. Select ‘Axis Titles’. Select both ‘Primary Horizontal’ and ‘Primary Vertical’. This will create textboxes next to each axis. (If using Numbers, select ‘Axis’ in the right menu bar, click the box ‘Axis Name’ under both ‘Value (X)’ and ‘Value (Y)’). Label the x-axis as log( P) and the y-axis as M.
f. Finally, to best represent the period-luminosity relation, we will draw a trendline that best fits the data. In Excel, select your chart, click ‘Add Chart Element’, ‘Trendline,’ ‘More Trendline Options…’ then select ‘Linear’ and check the box labeled ‘Display Equation on chart’ (in Numbers, select your chart, under the ‘Series’ tab in the menu bar on the right, click the bar under ‘Trendlines,’ then select ‘Linear,’ and check the box labeled ‘Show Equation’).
LEAVE THE GRAPH ON THE SPREADSHEET! WHEN YOU SUBMIT YOUR EXCEL FILE!
Part 2. Distance Determination Using Leavitt’s Law
Switch to the ‘Determining Distances’ tab of your spreadsheet!
The data here are from a Cepheid in the Large Magellanic Cloud (LMC), a nearby irregular galaxy. The first column, labeled Time, lists the time in fractional days of each observation. The second column, labeled m, gives the apparent magnitude measured at each time. Using these data, you are going to make a light curve!
3. Plot the light curve for the LMC Cepheid data in Excel in the same method you used to graph Leavitt’s Law in Question 2. Label the x-axis as ‘Days’ and the y-axis as ‘m.’
LEAVE THE GRAPH ON THE SPREADSHEET! WHEN YOU SUBMIT YOUR EXCEL FILE!
From your graph, determine the time for each maximum brightness, i.e. the times of the lowest m (remember the magnitude system is reversed, so m=3 is brighter than m=4). ( Note. If using Excel, simply hover your curser over the peaks, and the x,y coordinates will pop up, telling you the times of the peaks (If you are using Numbers, click on your chart, go to ‘Series’ in the menu tab on the right, under ‘Value Labels’ click on ‘Values:,’ then select ‘X’ as its option. Now every m point will have its corresponding time labeled).
The difference in time is the pulsation period ( P). Record your value of P below
4. P =
Use a calculator to determine the value of log( P):
5. log(P) =
6. Look up your value for log( P) on your first plot in your Excel sheet. From your trendline equation, calculate the Cepheid’s M (plug in log( P) for x in your equation, and solve for y, that will give you M). Record your answer below.
M =
7. From Table 3, calculate the average m, and record your results below.
m =
8. Use m and M to compute the distance modulus, m-M. Record your answer below.
m – M =
9. Use the distance modulus equation (Equation 1) to determine the distance to this Cepheid. This is the distance to the Large Magellanic Cloud. Currently, the distance to the LMC is believed to be between 57000 and 65000 parsecs. Record your answer below.
D = pc
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