The Distance to M100 using Cepheid Variables
Summary Determine the distance to the galaxy Messier 100 using Hubble Space Telescope Observations of Cepheid variables.
Background and Theory
Among the main scientific reasons for building the Hubble Space Telescope was to determine the value of the Hubble Constant, the rate of expansion of the Universe, from observations of Cepheid variables in galaxies belonging to the great Virgo Cluster of Galaxies. The Virgo Cluster contains some 2500 galaxies, among them the magnificent spiral Messier 100.
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Figure 1: The spiral galaxy Messier 100 (Hubble Space Telescope
rom repeated observations of M100, astronomers were able to identify more than a dozen Cepheid variables, and measure their brightness as a function of time. Cepheid variables are pulsating stars with regular periods of a few weeks. Henrietta Leavitt, working at the Harvard College Observatory early in the 20th century, discovered that the period of a Cepheid variable is related to its brightness. Brighter Cepheids pulsate with longer periods. From the Hubble measurements of Cepheid variables in M100, the periods of the variables, and hence the absolute magnitudes of the variables, could be determined. Once both the absolute magnitudes and the apparent magnitudes are known, the distance to M100 can be calculated.
The relation between a Cepheid's period and its absolute magnitude is given by the following expression:
M_{v} = 2.8 log(P) 1.4
Where M_{v} is the absolute magnitude and P is he period in days.
For convenience, this expression is shown in graphical form in the plot below.
The accompanying materials from the European Southern Observatory (Exercise Two) contain background information about Cepheid variables, as well as light curves for a dozen Cepheid variables in M100.
Determining the distance to M100  Use the light curves given in the ESO materials to determine the periods of each of the Cepheids, as well as the average apparent magnitude of each one.

The period is the time between successive maxima of the Cepheid variable.

The absolute magnitude can be determined from the period using either the chart above or the expression on the previous page.

The average apparent magnitude is half way between the maximum and minimum brightness in the light curve.
Enter the period, absolute magnitude, and average apparent magnitude in the table on the next page. The measurements for Cepheid No. 1 have already been entered.
Be sure to use appropriate significant figures!
Cepheid Number

Period (days)

Absolute Magnitude (M)

Maximum App. Mag. (m_{max})

Minimum App. Mag (m_{min})

Average App. Mag. (m_{ave})

Distance Modulus
(m_{ave}M)

Distance (Mpc)

1

53.4

6.2

24.5

25.3

24.9

24.9(6.2)=31.1

16.6

2








3








4








5








6








7








8








9








10








11








12








Average distance to M100 in megaparsecs:


The distance to each Cepheid can be calculated from the inverse square law that relates a change in brightness to a change in distance. An object twice as far away will appear one quarter as bright as one of the same intrinsic brightness.
In astronomy, the "absolute magnitude" of a star is the apparent magnitude that star would have if it were at a distance of 10 parsecs (a parsec is 3.26 light years). Thus the difference between the absolute magnitude and the apparent magnitude is related to the distance. The relation can be written mathematically as:
Distance in parsecs = 10^{(m}^{ave}^{ M+5)/5}
where m_{ave} is the average apparent magnitude and M is the absolute magnitude. The derivation of this relation is left as an exercise for the reader!
For Further Discussion…
The Cepheid variables in M100 give a wide range of distances. What are some possible reasons why the distances might vary so much from star to star?
Compare the size of M100 to its distance (M100 is similar in size to our own Milky Way, with a diameter of about 30,000 parsecs). Could the variation in distance result from the difference in location of the Cepheid variables within M100?
In the original scientific paper reporting on the distance to M100 using the Hubble observations, the distance was calculated to be 17.1 ± 1.8 Mpc (about 56 million light years). The presence of interstellar dust within M100 causes the apparent magnitudes of the Cepheids to be dimmer than they should be, so that they appear to be more distant than they really are.
This exercise is based on an ESA/ESO Astronomy Exercise, Series 2, and has been adapted for use in the Workshop "Exploring the Dark Universe" at Indiana University. The original data on which the exercise is based were taken from Freedman et al. 1994, Nature, 371, 757. 