ESA/ESO Astronomy Exercise Series 2
The Distance to Messier 100 as Determined by Cepheid Variable Stars
Arntraud Bacher, Lars Lindberg Christensen
In this exercise we measure the period and apparent magnitudes of Cepheid variables in the galaxy M100. The absolute magnitude is derived using the Period-Luminosity relation and the distance to M100 can then be determined using the distance relation. Finally we calculate a value for the Hubble constant (using a value for the recession velocity of M100 observed by other scientists) and estimate the age of the Universe.
Measuring distances with Cepheids
Measuring the distance to an astronomical object is a difficult task and is one of the greatest challenges facing astronomers. Over the years a number of different distance estimators have been found. One of these is a class of special stars known as Cepheid variables.
Cepheids are rare and very luminous stars that have a very regularly varying luminosity. They are named after the star d-Cephei in the constellation of Cepheus, which was the first known example of this particular type of variable star and is an easy naked eye object.
In 1912 the astronomer Henrietta Leavitt (see Fig. 1) observed 20 Cepheid variable stars in the Small Magellanic Cloud (SMC). The small variations in distance to the individual Cepheid variable stars in the Cloud are negligible compared with the much larger distance to the SMC. The brighter stars in this group are indeed intrinsically brighter and not just apparently brighter, because they are closer. Henrietta Leavitt uncovered a relation between the intrinsic brightness and the pulsation period of Cepheid variable stars and showed that intrinsically brighter Cepheids have longer periods. By observing the period of any Cepheid, one can deduce its intrinsic brightness and so, by observing its apparent brightness calculate its distance. In this way Cepheid variable stars can be used as one of the 'standard candles' in the Universe that act either as distance indicators themselves or can be used to calibrate (or set the zero point for) other distance indicators. Cepheid variables can be distinguished from other variable stars by their characteristic light curves (see Fig. 2).
Figure 1: Henrietta Leavitt
The understanding of the relative brightness and variability of stars was revolutionised by the work of Henrietta Swan Leavitt (1868-1921). Working at Harvard College Observatory, Leavitt calibrated the photographic magnitudes of 47 stars precisely to act as standard references or 'candles' for the magnitudes of all other stars. Leavitt discovered and catalogued over 1500 variable stars in the nearby Magellanic Clouds. From this catalogue, she discovered that brighter Cepheid variable stars take longer to vary, a fact used today to calibrate the distance scale of our Universe (Courtesy of AAVSO).
Figure 2: Typical Cepheid light curve
The light curve for a Cepheid variable star has a characteristic shape, with the brightness rising sharply, and then falling off much more gently. The amplitude of the variations is typically 1-2 magnitudes.
The most accurate measurements of both velocity and distance are naturally obtained for objects that are relatively close to the Milky Way. Before the NASA/ESA Hubble Space Telescope was available, ground-based observatories had detected Cepheid variables in galaxies with distances up to 3.5 Megaparsecs from our own Sun.
However, at this sort of distance, another velocity effect also comes into play. Galaxies attract each other gravitationally and this introduces a non-uniform component to the motion that affects our measurements of the uniform part of the velocity arising from the expansion of the Universe. This non-uniform part of the velocity is known as the peculiar velocity and its effect is comparable with the expansion velocity in our local part of the Universe. In order to study the overall expansion of the Universe, it is necessary to make reliable distance measurements of more distant galaxies where the expansion velocity is significantly higher than the peculiar velocity. Hubble has measured Cepheid variables in galaxies with distances of up to ~20 Megaparsecs.
Before Hubble made these measurements astronomers argued whether the Universe was 10 or 20 billion years old. Now the agreement is generally much better the age of the Universe is believed to be somewhere between 12 and 14 billion years.
One of the Hubble's Key Projects had as a longterm goal a more accurate value for the Hubble constant and the age of the Universe. Eighteen galaxies located at different distances have been monitored to reveal any Cepheid variables. One of these galaxies is M100 (see Fig. 3).
Figure 3: Hubble
tracks down Cepheid variable stars in M100
Hubble's high-resolution camera detected and picked out one of the Cepheid variable stars used in this exercise. The star is located in a star-forming region in one of the galaxy's spiral arms (the star is at the centre of the box).
Measurements and calculations
The Period-Luminosity relation for Cepheid variables has been revised many times since Henrietta Leavitt's first measurements. Today the best estimate of the relation is:
M = 2.78 log (P) 1.35
where M is the absolute magnitude of the star and P is the period measured in days. Light curves for the 12 Cepheids in M100 that have been measured with Hubble are shown in Figure 4 and 4a.
Calculating the absolute magnitude
Using the information in these curves, calculate the absolute magnitude M for the 12 stars.
Figure 4 and 4a: Cepheid light curves (click on images for larger versions)
Light curves for the twelve Cepheid variables in M100 that have been observed with Hubble. The absolute magnitude, M, is determined from the period of the Cepheids. Adapted from Freedman et al. (1994).
Calculating the apparent magnitude
Students are first asked to think of a way to determine the apparent magnitude. Either they use their method or they use the one, described by us: At the beginning of the 20th century astronomers measured the minimum apparent magnitude (mmin) and the maximum apparent magnitude (mmax) and then took the average (<m>) of the two.
Calculating the distance to each Cepheid and to M100
For this task the distance equation is used:
m-M = 5 log (D/10) = 5 log(D) 5,
where D is in parsecs (1 parsec (pc) = 3.086 _ 1013 km = 3.26 light-years).
The distances for the twelve Cepheids are not all the same, although the measured stars are located in the same galaxy. The students are asked to find reasons for the differences.
The distance found by scientists is given in the exercises. They took the interstellar dust into account for determining the value and their result is therefore more precise. By comparing the calculated value with the scientists' value, students will see how interstellar matter affects measurements of distances in space.
As a final task the students calculate the Hubble constant and estimate the age of the Universe.
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