Blog Post 15, Worksheet 5.2, Cepheid Relations

As we previously learned, Cepheid variables are a special class of stars that radially pulsate in a predictable way. In 1908, Henrietta Swan Leavitt discovered that there is a distinct relationship between a Cepheid's luminosity and pulsation period by examining many stars in the Magellanic Clouds. Henrietta was a member of "Harvard's computers," a group of women hired by Edward Pickering to analyze stellar spectra and light curves. In this worksheet, we will use Henrietta's original data set to find our own Period-Luminosity relation for Cepheid variables. 

1. The data file contains data for 25 Cepheid variables located in the Small Magellanic Cloud (SMC). Each lines contains a specific Cepheid's: (1) ID number, (2) Maximum apparent magnitude, (3) Minimum apparent magnitude, and (4) Period. Calculate the mean apparent magnitude for each Cepheid. 

Below are the data for the 25 Cepheids:

ID number 1505	Max 14.8	Min 16.1	Period(days) 1.25336
1436	14.8	16.4	1.6637
1446	14.8	16.4	1.762
1506	15.1	16.3	1.87502
1413	14.7	15.6	2.17352
1460	14.4	15.7	2.913
1422	14.7	15.9	3.501
842	14.6	16.1	4.297
1425	14.3	15.3	4.547
1742	14.3	15.5	4.9866
1646	14.4	15.4	5.311
1649	14.3	15.2	5.323
1492	13.8	14.8	6.2926
1400	14.1	14.8	6.65
1355	14	14.8	7.483
1374	13.9	15.2	8.397
818	13.6	14.7	10.336
1610	13.4	14.6	11.645
1365	13.8	14.8	12.417
1351	13.4	14.4	13.08
827	13.4	14.3	13.47
822	13	14.6	16.75
823	12.2	14.1	31.94
824	11.4	12.8	65.8
821	11.2	12.1	127

To calculate the mean apparent magnitude for each Cepheid, we add the maximum and minimum apparent magnitudes for each Cepheid, and then divide by two to get: 

ID number 1505	Mean 15.45
1436	15.6
1446	15.6
1506	15.7
1413	15.15
1460	15.05
1422	15.3
842	15.35
1425	14.8
1742	14.9
1646	14.9
1649	14.75
1492	14.3
1400	14.45
1355	14.4
1374	14.55
818	14.15
1610	14
1365	14.3
1351	13.9
827	13.85
822	13.8
823	13.15
824	12.1
821	11.65

2. The distance to the SMC is about 60 kpc, where kpc = 1000 pc. Convert your mean apparent magnitudes into mean absolute magnitudes. Plot the Cepheid mean absolute magnitudes as a function of period. This plot should look exponential. 

To determine the mean absolute magnitude, we use the equation from Worksheet 5.1, Problem 2: \[m = 5(\text{log}(d) - 1) + M\]

\[M =  5(\text{log}(d) - 1) - m,\] where d = 60,000 pc. If we plug in each mean apparent magnitude at a distance of 60,000 pc, we get the following graph relating mean absolute magnitudes to period: 

3. It is often handy to plot exponential (or power-law) functions with one or more logarithmic axes, which "straightens out" the data. Magnitudes are already exponential, we we don't need to adjust that axis. Plot the Cepheid mean absolute magnitudes as a function of \(\text{log}_{10}\)(Period). Verify that the plot now looks linear. 

If we plot the logarithm of period along the x-axis, the plot does indeed look linear:

4. Now that the data looks linear, we can estimate the parameters of a linear relation, \(M_V(P) = A\text{ log}_{10}(\text{Period}) + B\). A and B are "free parameters" that allow the function to match the data. 

Using Python's linear regression function, we can determine that A, the slope of the line, is -2.033, and B, the y-intercept, is -2.726. We get a final function of: \[M_V(P) = -2.033 \text{ log}_{10}(P) - 2.726\]

We can plot this on the same graph as before to see the linear approximation: