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Measuring Distances to Cepheids


Discover the Cepheids stars, the standard candles to extragalactic distances


How to measure the distance to the stars? Or to the galaxies? Brightness in not a reliable measure, because two very different objects, one faint but close, and the other bright but very distant, may look the same on our sky. For many centuries astronomers did not even realise that each star is at a different distance from the Earth - they assumed that all stars were located on a single, revolving sphere. In the 19th century distances to a few stars were measured by geometrical methods, and this allowed us to determine the shape and size of our Galaxy - The Milky Way. Yet immediately a new question arose: are the objects, seen in the night sky as nebulae, located inside our Galaxy, or perhaps they are also separate galaxies, similar to ours?
This question was resolved in 1912 by American astronomer, Henrietta Leavitt, who investigated the Cepheids - unusual, pulsating stars. After her discovery the determination of distances to the Andromeda Galaxy and the Large Magellanic Cloud was possible. They turned out to be separate galaxies, different from our Galaxy. Using the data of the stars from Large Magellanic Cloud, we can actually perform similar analysis.
minervg.gif "Measuring the Universe: distances to the Cepheids" (W. Śliwa)

 

Cepheids - the unusual stars


Mesuring Distance to Cepheids - description of the exercise (zip rtf  file 0.3MB)

Measuring Distance to Cepheids - description of the exersise (pdf file 330 KB)

Mesuring Distance to Cepheids - data (rar file 1.5MB)

Measuring distance to Cepheids - data (zip file 1.68 MB)

If you have questions about this excersise, please contact This email address is being protected from spambots. You need JavaScript enabled to view it.

Data needed:

 

1)      Cepheid images - zip they are here  

2)      Information on the relation between periods of Cepheids and their luminosity (amount of energy radiated by a star in 1 second), Figure 4.

3)      Luminosity of the Sun (3.85´1026 W)

4)      SalsaJ image processing software.

 

Cepheids - the cosmic candles

 

Cepheids are very bright. Their brightness exceeds the brightness of the Sun by a thousand or even ten thousand times. They regularly change their brightness. Each Cepheid is a pulsating star: it periodically changes its size and the surface temperature. The periods are in the range of a few days up to several months. An extremely interesting and useful property of such stars is the relation between their average brightness and pulsation period. Brighter Cepheids pulsate slower than the faint ones. A 3 days period Cepheid emits in a second 800 times more energy than the Sun. A 30 days period Cepheid, is brighter than the Sun by 10 000 times! Measuring the pulsation period of a particular Cepheid, we can determine the amount of emitted energy. When we compare this with the energy reaching our Earth, we can determine the distance of this star. Thus the method of distance determination resembles the method used by someone who can measure the distance to some distant person (in the night!) evaluating the brightness of his torch, provided he knows the real brightness of the torch. Interestingly, Polaris is also a Cepheid, albeit an unusual one (thus not usable for such method of distance determination).

 

 

Necessary formulae

 

The amount of energy radiated in a unit time by a star is called luminosity, denoted by L. At a distance r, a unit surface perpendicular to the direction of a star receives a flux of energy F:

One of the data files with images

For example, the luminosity of the Sun is Ls = 3.85´1026 W, and the flux of solar energy reaching a unit surface on Earth is Fsun = 1370 W/m2. Thus if we know the luminosity of a star, and measure the energy flux, we can determine the distance r.

 

 

Measurements

 

In order to measure the distance to the Cepheids in the Large Magellanic Cloud a data file for these Cepheids is needed. You can find them in the Cepheids directory. The data were obtained by the OGLE experiment (http://www.astrouw.edu.pl/~ogle/index.html).

 

The name of each file contains the date of observation. Thus the file CEP-43522-1999-10-24-03-25.FTS is the observation made on 24 October 1999 at 3:23 (you can check the date of observation opening the window Image: show information [HOU-IP: Data Tools/image info]). On all images there is the same part of the sky (see Fig. 1). The location of the  Cepheid and comparison stars of constant brightness is shown in Fig. 2. All observations were performed in red and near infrared wavelength.

 

 

 


 

One of the data files with images

 

Fig.1. One of the data files with star pictures.

 

 

 

 

Fig.2 The location of the Cepheids and stars used for comparison.

 

 

Observational conditions may be different. The sky could be more or less clear, and the sensitivity of the detectors could vary. The best way to eliminate these unwanted factors in determination of the brightness changes of the Cepheid is comparison of its flux with fluxes of stars of constant luminosity. Different observations of two stars of constant brightness (say, one twice as bright as the other) could give different fluxes, but each time the ratio of their fluxes should be the same (equal two). In the case of an observation of a Cepheid and a constant comparison star, the change of the ratio of their fluxes is the effect of the change of the  brightness of a Cepheid only.

 

Let us start the analysis.

 

1.      Evaluate the brightness of a chosen Cepheid and comparison star in consecutive observations. To do this

 

a.       Open the data file. Note the date of observation.

b.      Adjust the image in such a way that you can clearly distinguish the stars [Image: Adjust Brightness/Contrast Auto]. Find the Cepheid and the comparison star.

c.       Using Auto aperture tool [Analysis: Photometry; find the star and click on it] measure the brightness of both stars.

d.      Repeat the procedure for all data files.

 

2.      Put the results in a table. Subsequent columns should contain day and hour of each observation, time difference (hours or days) between each measurement and the first one Dt, measured brightness of the Cepheid Lc and of the comparison star Lg , and their ratio Lc/Lg.

 

time

Dt (days)

Lc

Lg

Lc/Lg

26-09 18:01:00

0.000

168086

357753

0.4698

28-09

19:36:00

2.066

179784

340024

 

0.5287

...

 

 

 

 

 

3.      Calculate average brightness of the Cepheid compared to the comparison star. Using the value of the flux of the comparison star (presented in Fig. 2) and this ratio, calculate the average flux Fav from the Cepheid. You can use Excel spreadsheet (appendix 2).

 

4.      Plot the ratio of the fluxes of the Cepheid and the comparison star as a function of time (such a plot is called a lightcurve). Try to fit a sinusoid to the observed changes. Determine the period.

  light curve

 

Fig.3 Example of a lightcurve of a Cepheid star

 

 

 

Using the relation between pulsation period and the average luminosity of the Cepheid, presented in Fig.4, find the ratio between the luminosity of the Cepheid and of the Sun. Determine the luminosity of the Cepheid.

 

 

 

Fig.4 Cepheid luminosity (in solar units) as a function of pulsation period

 

 

5.      Using the formula given at the beginning of this text, calculate the distance of the Cepheid from the Sun. This is the distance between the Sun and the Large  Magellanic Cloud (where the Cepheid is located).

 

6.      Problem: What are the sources of uncertainty of your result? Which of them is most significant? Remember, the results depend on the (unknown) location of the Cepheid within the Magellanic Cloud. Still, as the size of the Large Magellanic Cloud is much smaller than the distance between the Sun and the Magellanic Cloud, this uncertainty does not influence the result very much. There are also other sources of uncertainty. Space between Large Magellanic Cloud and the Earth is filled with small size particles of dust, partially absorbing the radiation. How does this affect the evaluation of the distances?

 

You can learn more about the Cepheids at those web pages:

 

http://sswdob.republika.pl/cefeidy.htm

http://orion.pta.edu.pl/astroex/ex2/cefeidy.html

http://www.uni-sw.gwdg.de/~hessman/MONET/AstroKiste/Sterne/Cepheiden/

 

 

Acknowledgments: We are grateful to Fabrice Mottez (CETP) for providing us with the French version of the exercise and to Bohdan Paczyñski for continuous encouragement and help with the selection of appropriate data. We thank OGLE team for permission to use their data in the exercise.