AST 114 – Spring 2005 Variable Stars

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VARIABLE STARS

What will you learn in this Lab?

This lab will cover three of the main types of variable stars that we observe in our nightsky: RR Lyrae stars, Cepheid variable stars and Eclipsing Binary stars. After discussinghow these stars are manifest in terms of their light curves, you will be asked to take aseries of provided data and determine for yourselves which stars are which types andfrom there derive their particular properties.

What do I need to bring to the Class with me to do this Lab?

For this lab you will need:

• A copy of this lab script• A pencil• A scientific calculator

Introduction:

Not all the stars in the heavens have a constant appearance. Many of them change theirapparent brightness over time. The nature of these variations can tell us much about thestar itself, how far away it is and some other physical properties of the object. Thisexercise is billed under the Braeside Observatory banner, but for this semester will usedata acquired elsewhere.

RR Lyrae Stars

RR Lyrae stars are a special type of star that physically pulsates to cause the apparentchange in brightness that we see from Earth. The pulsation comes from the fact that thestar itself has entered an internally unstable state in its evolutionary history. This phasecauses the star to “bounce” or oscillate back and forth across a stable size and brightnessbecoming alternately too large and fainter, and then too small and brighter.

Since the conditions for this type of behavior are so explicit, the absolute magnitude, M, ofthese stars does not vary much and is typically right around +0.5 magnitude. Such starscan be used as “standard candles” – if you know how bright it is intrinsically, then bymeasuring its brightness (its apparent magnitude, m) you can determine its distance.

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The equation for this calculation is:

m −M = 5log10d10

= 5logd − 5

where d is the distance of the starmeasured in parsecs. 1 parsec = 3.26light years = 3.09x1016 m.

When the brightness of a star istracked over a period of time, a plot ofthe changes in brightness on a graphis called a light-curve. In the case ofRR Lyrae stars, there are three typesof light curve, shown here, that aregiven the names RRa, RRb and RRc.Each has a unique shaped curve.However, the designations are definedin terms of different types of pulsation.a and b oscillate by the whole starswelling and contracting; c oscillatesby alternate hemispheres swelling andcontracting generating a sloshingmotion in the surface of the star.

Cepheid Stars

These are similar to RR Lyrae stars in that they oscillate in size and shape to provide thechanges in brightness we see. However, Cepheids (named after the first one found: δCepheus) are much bigger and brighter stars than RR Lyrae stars. Cepheid stars alsoobey a period-luminosity relation that predicts that a star of a given period of oscillationhas a certain intrinsic brightness, but that stars of different period have differingbrightnesses. This is a remarkable result – it means that how fast a star changes in sizeand brightness is affected by how absolutely bright that star is. There are actually twotypes of Cepheids, but we will deal with only the brighter ones – Type I’s.

The period of a variable star is found simply by measuring the time between the sameparts of the light curve:

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The final piece to this puzzle is the nature of the period-luminosity relation, shown below:

Therefore, if you can measure the period of a Cepheid variable star, you can determinehow intrinsically bright it is, and thus the distance to the star. Note that the maindifference between Cepheids and RR Lyrae stars is how long the period is – with the latteralways having a period less than a day, and the former always being more than a day.

Why they pulsate – the Instability Strip

Both RR Lyrae stars and Cepheids pulsate because of a change in the internal structureof the star as it evolves. The state is semi-stable, but oscillates to either side of a stableconfiguration to give the alternate brightening and dimming of the stars. This state isactually well-defined on an H-R diagram and is called the “instability strip” as shownbelow. Also shown are the various locations you will find the types of variables stars thatexist, and not just the ones we deal with in this lab. (Note: W Virginis stars are Type 2Cepheids that obey the period-luminosity relation, they’re just a lot fainter – hence theparallel second track on the diagram above.)

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Eclipsing Binaries

The final category of commonly observed variable stars is eclipsing binaries. These arepairs of stars that orbit very closely with each other, in some cases sharing a commonouter envelope of gas. What makes them unique from our point of view is that their orbitalplane is seen almost edge on, so the two stars take turns eclipsing the light from the otherstar. As such the light curve seen from these objects obeys certain rules that tell usinformation about the stars involved.

Similarly, when the small star is obscured, the light level drops to that of the larger staralone. Thus, the light curve tells us how bright each star is, and how fast they’re orbitingeach other. The amount of time each trough lasts also tells us something about therelative sizes of the two stars.

An eclipsing binary usuallyconsists of two stars: one largewith a low surface brightness,and another small but verybright. As the stars orbit oneanother, the bright star isalternately visible and invisible.When both stars are visible thebrightest light level is seen. Asthe small bright star moves infront of the larger dimmer star,the light level drops to just thatof the smaller star alone.

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The light curve shown is somewhat idealized, but you can see that it looks very differentfrom the other ones you’ve seen. Again, the period of the system is measured betweenthe peaks (or troughs) of the light curve.

Procedure

Now you are going to put all this knowledge into play. We are going to give you lightcurve data acquired from a variety of sources, and you are going to use the informationabove to figure out what type of variable star each one is, and derive whatever propertiesfor each object you can. Don’t be intimidated – this is exactly what astronomers whostudy variable stars do all the time. Believe it or not, variable stars like these are beingdiscovered every day, and in each case the astronomers have to employ information likethis to figure out what each system is really like.

For each of the following light curves, identify the type of variable star and in each casewhat physical properties of the star you can determine from the information presented.TAKE YOUR TIME. Study each one carefully – some of the details are not as obvious asyou might think. Write down and justify your answer next to the light curve.

1.

1.

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1.

1.

1.

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2.

3.

4.

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1. Consider the following properties: period, absolute magnitude, average apparentmagnitude of an individual star, distance to the object, relative size of star

a. Which of the above properties can be determined for an RR Lyrae star?

b. Which of the above properties can be determined for a Cepheid variablestar?

c. Which of the above properties can be determined for an eclipsing binarysystem?

2. Using your answers to the previous questions, fill out the following table. Note: theperiod – luminosity relationship for variable stars indicates their average luminosity⇒ average absolute magnitude ⇒ average apparent magnitude. If there is onlyone star, refer to it as “star A.”

Lightcurve

Variabletype

Period(days)

Absolutemagnitude

Apparentmagnitudeof star A

Apparentmagnitudeof star B

Relativesize ofstars

Distance(parsecs)

1

2

3

4

5

6

7

8

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Questions

1. What is the main difference between RR Lyrae stars and Type 1 Cepheids? What arethe other differences? What makes these stars particularly useful in astronomy today?

2. What is special about eclipsing binaries? What can we determine from these starsfrom the length of time for each dip in brightness? Which of the two stars is thebrighter – the larger or smaller one? Why?

3. Based on the accuracy with which you measured your data from the graphs provided,how accurately do you think you know the derived distances (for those stars it appliesto)?

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4. Which can you see farther away, RR Lyrae or Cepheid variable stars?

Conclusion: