MEASURING DISTANCE WITH CEPHEID VARIABLES

Name Date Partner(s) Grade / MEASURING DISTANCE WITH CEPHEID VARIABLES Written by T. Jaeger INTRODUCTION Cepheid stars (named after the class prototype star, DELTA CEPHEI) are of great interest because their intensity varies over periods as short as a day to as long as 50-100 days. Furthermore, the observed variation period is linked to the star s luminosity, allowing astronomers to estimate the ABSOLUTE MAGNITUDE of the star. This unique property means Cepheid variables can be used as STANDARD-CANDLES, providing the ability to accurately measure the distance to the host galaxy or cluster. The science behind a Cepheid s variability is quite interesting. The intensity of the star varies due to a repetitive process in the stars atmosphere. In this process, the helium in the atmosphere ionizes and expands, then deionizes and contracts. As the gas expands, it becomes more opaque to light, making the star appear dimmer. This process repeats on regular intervals. The period of this process and the change in magnitude provide valuable information about the luminosity and density of the star. There are two types of Cepheid variables, classified by their change in luminosity and period. 1. DCEP - classical Cepheids, Delta Cephei-type variables. These stars are young, POPULATION I (like the sun) stars that have left the main sequence and are in the instability strip of the Hertzsprung-Russell diagram. This class of Cepheid Variables obeys a predictable PERIOD-LUMINOSITY RELATIONSHIP which can be used to estimate the distances to far away objects. 2. CW - variables of W VIRGINIS type. These Cepheids are POPULATION II (metal rich) stars and can also be used to estimate distances, but their periods and luminosities are much smaller than classical Cepheids. They too obey a well-known Period-Luminosity relationship, but its form is different than DECP Cepheids. In today s exercise, you will study the LIGHT CURVES (plot of star intensity versus time) of various Type I Cepheids located in distant open clusters. Your goal will be to determine the distance to each cluster by measuring the

period of the Cepheid and comparing its apparent magnitude to the absolute magnitude predicted by the Period- Luminosity relationship. Take note of the unusual shape to the Cepheid light curve. The variability is not always symmetric like you might see with other variable stars, but more SAW TOOTH, with a fast increase in intensity and slow decrease. PERIOD-LUMINOSITY RELATIONSHIP The relationship between a Cepheid s period and its luminosity was first discovered by HENRIETTA SWAN LEAVITT in 1912. Henrietta worked as a woman-computer at the Harvard College Observatory, hired to measure and catalog the brightest stars on the observatory PHOTOGRAPHIC PLATES. She noticed that some variable stars observed in the MAGELLANIC CLOUDS displayed a peculiar property. These stars not only have a periodic variation in their intensity, but the brighter stars also appeared to have the longest periods. Through further study, she worked out an empirical relationship (based on observations) which is now referred to as the PERIOD- LUMINOSITY RELATIONSHIP. This equation has been refined slightly by observations over the last century and in its current form is the following: Here, Mv is the absolute magnitude and P is the period in days. FIGURE 1 - PERIOD-LUMINOSITY RELATIONSHIP FOR TYPE I CEPHEIDS

STANDARD CANDLE This relationship above for Type I Cepheids is quite accurate and has been verified by observations of nearby Cepheids. In these cases, the distance and magnitude can be calculated through other means, such as parallax. The Period-Luminosity relationship however is much more useful when extrapolated to distant sources where tools such as parallax are inadequate. For these distant objects, Cepheids act as Standard Candles (a star that has a known Luminosity) and are invaluable for measuring distances. Once the absolute magnitude of a star is known, it can be used along with the APPARENT MAGNITUDE (how dim/bright the star appears to the observer) to calculate the stellar distance. The relationship between distance, absolute magnitude and apparent magnitude is called the MAGNITUDE-DISTANCE FORMULA. It is shown below. Here, d is the distance in parsecs, Mv is the absolute magnitude and mv is the apparent magnitude. MEASURING THE DISTANCE TO DISTANT CEPHEIDS You are provided with five observations of Cepheid Variables located in distant clusters to implement the concepts discussed earlier in the lab. Your final goal is to measure the distance to each cluster. To accomplish this, follow the steps below for each dataset. Make sure to enter your results in the table located at the end of the lab. 1. Each observation folder (S###) contains 40 FITS images and one JPEG image. The JPEG image is called a FINDING CHART and can be used to locate the various stars you will need for this lab. An example Finding Chart is shown to the right. Each chart will contain the location of the three stars you will need to locate for each observation (two cluster stars and the Cepheid). 2. Open the FITS images in MaxIM DL. This may take a minute, so don t be alarmed if the screen freezes for a short time. 3. Now open the JPEG Finding Chart in MaxIM DL. Try and locate the three stars indicated on the chart. You will need each star for the next step. 4. You will use the PHOTOMETRY TOOL in MaxIM DL to generate a light curve of the Cepheid Variable. While Henrietta Swan Leavitt was working at the Harvard College Observatory, this would have been done by hand using a photometer to measure how much light FIGURE 2 - SAMPLE FINDING CHART could still pass through the region of photographic plate or film exposed by the target star. For comparison, this measurement would be made for a known, non-varying star called a Reference Star and multiple other stars called Check Stars. These plates could be many feet across and would have carefully made this measurement for each image and

recorded her results in a table, noting the time each image was taken. To generate a light curve, the final step would be to plot the measurements from the photometer (compared to the Reference Star) versus the observation time. For a skilled worker such as Henrietta, this process may have taken days to hours. Today, computers can do this calculation in a few seconds, but the basic principles are still the same. 5. The photometry tool in MaxIM DL can be found under Analyze -> Photometry. First mark the Cepheid (indicated as S### on the image). Do this by setting the MOUSE CLICK TAG AS: option to NEW OBJECT and clicking on the target star. The green Photometry Tool should just fit around the star. If the tool is too large, you can change the size of the tool components by right-clicking inside the image. A typical setup is to set the APERTURE RADIUS, GAP WIDTH and ANNULUS THICKNESS each to a value of 5. 6. Now mark the Reference Star by changing the MOUSE CLICK TAG AS: option to NEW REFERENCE STAR and clicking on the Reference Star. Set the REF MAG value to the magnitude indicated in your results table (Table 1) at the end of the lab write-up. This will properly scale your magnitudes for later. 7. Finally, mark the Check Star by changing the MOUSE CLICK TAG AS: option to NEW CHECK STAR and clicking on the indicated Check Star. This step is important because it allows you to verify that the variation in the Object is due to the Object itself, and not due to variation in the Reference Star. If the Check Star has a flat light curve, then the Reference Star too must be non-varying. FIGURE 3 - USING PHOTOMETRY IN MAXIM DL

8. To see the light curve for the three selected stars, press the VIEW PLOT button. MaxIM DL will perform all the needed calculations (along with finding the stars in all 40 images automatically) and generate a light curve. 9. Save your light curve data to the desktop by pressing the SAVE DATA button. Make sure to give your file a name that will help you remember which object you are analyzing, as you will save a light curve for each cluster. 10. Repeat the above steps and create/save light curves for each Cepheid and Open Cluster. When complete, you will have 5 files in all. 11. Now that you have saved each light curve has been calculated and saved, you can use LOGGER PRO to examine the data closer and estimate the Cepheid period. To open your data, select FILE -> IMPORT FROM -> TEXT FILE. The time axis of your photometry data is plotted as Julian Date (JD, decimal days since noon, Jan 1st, 4713 BC). The Julian Date for midnight on the first observing day is 2454542. Optional: To scale the time axis in Logger Pro, do the following: a. Navigate to DATA -> NEW CALCULATED COLUMN. b. Type a Name and Short Name of MJD, and for Equation, type "Column" - 2454411 c. Change x axis to use MJD by clicking on axis name d. Rescale x axis. This can be done by pressing CTRL-J or right clicking on axis for menu and selecting AUTOSCALE. 12. Estimate the period for each Cepheid and record your results in Table 1 at the end of the lab exercise. To do this, use Logger Pro to fit the light curve using a SINE function. A Sine function is desirable because it is a periodic function and has a shape close to that of the saw-tooth shaped Cepheid data. The general form of a sine function is as follows: Don t worry if you don t understand all the particulars. Logger Pro will do all the heavy lifting. Navigate in Logger Pro to Analyze -> Curve Fit. This will open a new window where you can fit your Cepheid data to the function above. Select Sine from the General Equations list and press Try Fit. This should close the window and display the results along with the original data. The Fit Parameters (A,B,C,D) are now listed in a box attached to Logger Pro s solution. Write down the following information: a. The Fit Parameter B represents the frequency. You may recall that period is inversely related to the frequency. The period of the Cepheid is then: b. The Fit Parameter D represents the average magnitude of your data. It you entered your reference magnitude correctly, D will equal the apparent magnitude of the Cepheid. 13. Once the Cepheid periods have been estimated, use the PERIOD-LUMINOSITY RELATIONSHIP (or corresponding graph) to find the absolute magnitude of the variable star. Record you results as well.

14. Finally, use the absolute magnitude and apparent magnitude of the Cepheid to calculate the distance to each cluster. Use the MAGNITUDE-DISTANCE FORMULA to calculate the distance. Move on to the final section. HUBBLE LAW AND THE CURRENT HUBBLE CONSTANT The rate at which the universe is expanding is called the Hubble Constant, named after the astronomer Edwin Hubble. Hubble (along with Milton Humason) realized through observations of distant objects that the observed velocity of objects (seen through the red shifting of light) were proportional to their distance. This proportionality is denoted as H and is typically measured in km/sec/mpc (10 6 pc). Scientists soon realized that the magnitude of this expansion must have changed over the life of the universe, so we denote the present-day value of the Hubble constant as H o. We can estimate H o by using the following formula (where v is the velocity (or redshift) in km/s and d is the distance in Mpc): Below is an observation of a M100 Cepheid Variable taken by the Hubble Space Telescope. The background light from the galaxy was subtracted from each image so the star could be seen more easily. Notice the star change intensity. This Cepheid has a period of approximately one month. FIGURE 4 - CEPHEID VARIABLE IN M100 OBSERVED BY THE HUBBLE SPACE TELESCOPE

Cepheids observations provide a valuable tool for measuring the Hubble constant. The clusters analyzed in this lab are all inside the Milky Way Galaxy (distances less than 10-20 kpc), but Cepheids are also be observed in distant Galaxies (distances greater than 10 s of Mpc). At these large distances, Standard Candles like Cepheid Variables are crucial for estimating distances and the expansion rate of the Universe. Table 2 contains a list of famous Galaxies and Cepheid Variables, along with a few helpful parameters. Often, dozens of Cepheid are observed in a Galaxy, but for ease, only one is listed. Use the given velocity and period for each Galaxy/Cepheid listed to estimate the Hubble constant in km/sec/mpc. To get a more accurate value, plot the given velocities versus calculated distance and use a ruler to draw a best fit line to your data. The slope of this line will be your estimate of the current Hubble Constant. The Hubble Constant also indicates the age of the Universe. If the number of kilometers in a parsec (or in this case, Mega-parsec) is known, then the age of the universe is simply n (km/mpc) divided by H o. Once you have completed your graph, use the current value of n = 3.0857 10 19 km/mpc to make an estimate of the Universe s age. Your answer will be in seconds, so use 1 yr = 3.156 x 10 7 sec to express your answer in years.

TABLE 1 OPEN CLUSTER RESULTS TABLE Cepheid Ref P (days) m v M v d (kpc) S011 9.56 S204 7.69 S366 8.23 S435 8.10 S797 7.54 Work Space

TABLE 2 - HUBBLE CONSTANT Galaxy Observed Velocity (km/s) Cepheid Period (days) Apparent Magnitude M100 1571 2.7 28.4 M94 308 72.1 21.8 NGC 7331 816 22.6 25.2 M82 203 43.1 21.7 M49 997 38.4 25.0 Absolute Magnitude Distance (Mpc) Hubble Constant Age of Universe (km/sec/mpc) (years)