The ESA/ESO Astronomy Exercise Series 2

The Distance to M100 as Determined by Cepheid Variable Stars

Astronomy is an accessible and visual science, making it ideal for educational purposes. Over the last few years the NASA/ESA Hubble Space Telescope and the ESO telescopes at the La Silla and Paranal Observatories in Chile have presented ever deeper and more spectacular views of the Universe. How- ever, Hubble and the ESO telescopes have not just provided stunning new images, they are also inva- luable tools for astronomers. The telescopes have excellent spatial/angular resolution (image sharp- ness) and allow astronomers to peer further out into the Universe than ever before and answer long- standing unsolved questions.

The analysis of such observations, while often highly sophisticated in detail, is at times sufficiently simple in principle to give secondary-level students the opportunity to repeat it for themselves.

This series of exercises has been produced by the European partner in the Hubble project, ESA (the Eu- ropean Space Agency), which has access to 15% of the observing time with Hubble, together with ESO (the European Southern Observatory).

Figure 1 : The NASA / ESA Hubble Space Telescope

The NASA/ESA Hubble Space Telescope has presented spectacular views of the Universe from its orbit above the Earth.

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Preface

Introduction

Cosmology and distance measurements

How old is the Universe? How fast is it expan- ding? Will it one day start to contract? These are fundamental cosmological questions that have long awaited satisfactory answers.

The fate of the Universe is closely linked with the future behaviour/evolution of its expansion rate. If the expansion slows down sufficiently then the Universe may one day start to recon- tract. Observations currently suggest that it is

more likely that the Universe will continue to expand forever.

The expansion makes all galaxies recede from a given observer (e.g. on Earth) and the further away they are, the faster they recede. The ex- pression known as Hubbles law (formulated by Edwin Hubble in 1929) describes the relation between the distance of a given object and its recession velocity, V. Hubbles law is:

v = H0D It states that the galaxies in our Universe are

Figure 2: The Fate of the Universe

This graph relates the size of the Universe with time in other words it shows how it expands and/or contracts with time. The different lines in the future (to the right in the diagram) show different models for the fate of the Universe an ever-expanding Universe or a contracting Universe.

Figure 3: Receding Galaxies

This diagram illustrates how the galaxies recede from each other due to the expansion of the Universe.

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Introduction

Introduction

Figure 4: Remote Galaxies with High Redshifts

This image, taken by the Wide-Field and Planetary Camera (WFPC2) of the Hubble Space Telescope, shows many galaxies, billions of light years away. Most of the fuzzy patches are galaxies containing billions of stars. The galaxies in this image are receding from us at high velocities.

flying away from each other with a velocity, v, proportional to the distance, D, between them. H0 is a fundamental property of the Universe the Hubble constant important in many cos- mological questions and is a measure of how fast the Universe is expanding today.

The age of the Universe, t, can be approximated by the inverse (or reciprocal) of the Hubble con- stant H0:

t = 1/H0 The value of H0 has enormous significance for 4

estimates of the age of our Universe. But how do we measure it? To determine H0, we simply need to measure both the recession velocity, v, and the distance, D, for an object, usually a ga- laxy, or, even better, for many galaxies and find the average measurement.

The recession velocity is relatively easy to deter- mine: we can measure the so-called redshift of the light from the galaxy. Redshift is a direct consequence of an objects motion away from us. It is a Doppler-shift of the light from the in- dividual galaxies, resulting in a shift of the

Introduction

Introduction

Figure 5: 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

brighter stars in this group are indeed intrinsi- cally brighter and not just apparently brighter, because they are closer. Henrietta Leavitt un- covered a relation between the intrinsic bright- ness and the pulsation period of Cepheid varia- ble stars and showed that intrinsically brighter Cepheids have longer periods. By observing the period of any Cepheid, one can deduce its in- trinsic brightness and so, by observing its ap- parent 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. 6).

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).

wavelength of the light from the galaxies to- wards the red end of the spectrum. As the wave- length of red light is longer than blue light, the wavelength of the light from the galaxies has increased during its journey to the Earth. The fractional change in wavelength due to the Dop- pler-shift is called the redshift and galaxies with a high redshift have high recession veloci- ties.

Using Cepheids as distance estimators

Measuring the distance to an astronomical ob- ject is much more difficult 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 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 -Cephei in the con- stellation 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. 5) observed 20 Cepheid variable stars in the Small Magellanic Cloud (SMC). The small varia- tions in distance to the individual Cepheid vari- able stars in the Cloud are negligible compared with the much larger distance to the SMC. The

Figure 6: 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 veloci- ty and distance are naturally obtained for ob- jects 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 dis- tances up to 3.5 Megaparsecs (see the defini- tion of Megaparsecs in the Mathematical Toolkit) from our own Sun. However, at this sort of distance, another velocity effect also comes into play. Galaxies attract each other gravita- tionally and this introduces a non-uniform com- ponent to the motion that affects our measure- ments of the uniform part of the velocity arising

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Introduction

Introduction

from the expansion of the Universe. This non- uniform part of the velocity is known as the pe- culiar velocity and its effect is comparable with the expansion velocity in our local part of the Universe. In order to study the overall expan- sion of the Universe, it is necessary to make re- liable distance measurements of more distant galaxies where the expansion velocity is signifi- cantly higher than the peculiar velocity. Hubble has measured Cepheid variables in galaxies with distances of up to ~20 Megaparsecs.

Before Hubble made these measurements astron- omers argued whether the Universe was 10 or

20 billion years old. Now the agreement is gen- erally much better the age of the Universe is believed to be somewhere between 12 and 14 billion years.

One of the Hubbles Key Projects had as a long- term 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.

Figure 7: The Spiral Galaxy M100

If we could observe our galaxy, the Milky Way, face on from an extragalactic space vessel, its shape would be similar to the Spiral Galaxy M100.

Spiral galaxies are rich in dust and gas. The dust shows up in this image as dark lanes running between the majestic spiral arms.

M100 is a popular target for amateur astronomers and is located in the spring sky in the direction of

the constellation of Coma Berenices. The image was taken with Hubbles Wide Field and Planetary Camera 2. Blue colours correspond to regions with young hot stars.

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Introduction

Introduction

Figure 8: Hubble tracks down Cepheid variable stars in M100

Hubbles 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 galaxys spiral arms (the star is at the centre of the box).

M100 a Grand Spiral

The galaxy M100 is a magnificent spiral galaxy in the large Virgo cluster of galaxies. The Virgo cluster contains 2,500 galaxies. M100 is a rota- ting system of gas, dust and stars similar to the Milky Way and is viewed face on. The name M100 stems from the fact that it is number 100 in the Messier catalogue of non-stellar objects.

M100 is one of the more distant galaxies where accurate measurements of Cepheid variables have been made. This exercise is based on Hubbles images and data from this galaxy.

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Introduction

Tasks

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Figure 9: A Cepheid variable in M100

Six images taken at different times depicting one of the Cepheid variable stars in the galaxy M100 are shown. The Cepheid is in the centre of each frame. It is clear that the Cepheid varies in brightness over time.

Measurements and calculations

The Period-Luminosity relation for Cepheid vari- ables has been revised many times since Henri- etta Leavitts 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 on pages 9 and 10.

Task 1

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? Using the information in these curves, cal- culate the absolute magnitude M for the 12 stars.

Our goal is to calculate the distance to M100. If you remember the distance equation, you will know that the absolute magnitude alone is not enough to calculate the distance you also need the apparent magnitude.

Apart from problems in measuring the amount of light received accurately and calibrating the magnitudes that were measured, for a hundred

Tasks

Tasks

years astronomers have discussed which appar- ent magnitude, m, to use in the distance equa- tion for a Cepheid that is actually varying in magnitude.

Task 2

? Think about a method to estimate the ap- parent magnitude, m, using the curves.

At the beginning of the 20th century astrono- mers measured the minimum apparent magni- tude (mmin) and the maximum apparent magni- tude (mmax) and then took the average () of the two.

If you do that or use your own method you now have all the information you need to calcu- late the distance to M100.

Figure 10: Cepheid light curves

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).

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Tasks

Tasks

Figure 10 (continued): Cepheid light curves

Task 3

? Calculate and D (in Mpc) for each Ce- pheid.

You could, of course, just do the same calcula- tion twelve times, but you can reduce the amount of work by, for example, writing a small program for a calculator or using a spreadsheet.

Task 4

? Consider the probable reasons why you do not find exactly the same distances for the different Cepheid variables.

Task 5

? Now you have calculated the distance to the twelve different Cepheid variable stars in M100. Does that give you the distance to M100?

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Tasks

Tasks

• ? Could the fact that the twelve stars have different positions in M100 be the reason for the variation in the distances of the twelve stars?

• ? Find out how large the Milky Way is (you can, for instance, look in an astronomy book or on the Internet). Assume that the size of M100 is of the same order. Now think about the previous question once again.

  Task 6

• ? Calculate the mean value of the distances to the twelve Cepheid stars and consider this as the distance to M100.

• ? In the original scientific paper using the Hubble measurements, the distance to M100 was calculated as 17.1 1.8 Megaparsecs. The presence of interstellar dust was taken into account in determining this value. Compare your results with that distance.

Task 7

As you may remember from the introduction (p. 5), the recession velocity, v, of a galaxy like M100, together with information about its dis- tance can give you a value for the general ex- pansion velocity of the Universe as described by Hubbles constant, H0. H0 is expressed in units of km/s/Mpc. The recession velocity of the Virgo Cluster, of which M100 is a member, has been measured earlier to be 1400 km/s (Freedman et al., 1994).

? Calculate Hubbles constant using this v, and the average of your distance measure- ments as D.

Task 8

? Assuming that the age of the Universe, t, is given by t = 1/H0, calculate a value for the age of the Universe. Remember to convert to the correct units. How much older is this than the age of the Earth?