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# How Big Is That Star?

## Objectives

1. Students will be able to explain the relationship between radius and mass among a list of stars.

2. Students will understand how a binary star system's orbit can cause changes in the observed brightness of the system.

3. Students will determine the diameters of stars by analyzing data and manipulating equations.

## Prerequisites

If students have participated in and completed the lesson titled "Time that Period!", they have a good foundation in astrophysical data analysis and binary star systems. But if "Time that Period!" cannot be completed, please refer to the following:

#### Math

Students should have learned some pre-algebraic concepts such pattern recognition, ratios, proportions, decimals, and percents.

#### Science

Students should have had an introduction to the concepts of stars and space astronomy.

## Time Requirements

For each class of students, you will probably need at least 2 periods/mods.

## Warm - Up

Here are some examples of what you may want students to do. Substituting your own is, of course, always acceptable!
1. To refresh the student's memory, instruct students to write the ratio of girls to boys, people in the class over 5 ft. tall vs. people under 5 ft. tall, etc.

2. Again, to remind students of previously learned concepts, instruct students to convert a set of given ratios into decimals and percents.

## Introduction

It is important for students to generally comprehend the size, mass, and density of stars. The following activity should establish this general understanding in order for students to be prepared for both the lab and the analysis of actual data from stars within this lesson.

Students should be instructed to examine the table "Dimensions of Typical Stars" for this lesson and complete the following activities.

1. In a column format, write a list of the stars from least to greatest in terms of their radius (measured in Suns).

2. In another column next to that list, write the ratio of radius to mass (measured in Suns).

3. Locate some of these stars on a constellation map.

4. Explain the (very general) relationship between radius and mass among this list of stars.

Students should also examine the colorful illustration of the sizes of various stars just below in this lesson. You'll see that some of the stars depicted in the artist's rendition are the same stars listed in the table of data. This type of representation should assist visual learners with the concept of the size of stars.

# Day 1

## Materials

Teacher
flash light
pen light
two pieces of paper
red cellophane or filter paper
overhead to emphasize points in introduction and throughout the lesson

## Guided Practice or Developmental Procedure

1. Explain that all types of electromagnetic radiation are emitted in variable amounts from binary systems of stars. Also inform students that high-energy satellites have detected X-rays from these sources, which is the type of data retrieved and used on the second day of this lesson.

2. Next, roll one piece of paper like a tube and tape it around the light end of the flash light. Point it at a blank screen. This demonstrates the "brightness" or "magnitude" of a larger star in a binary system. The paper tube will enable to students to see a well defined circumference around the light.

3. Next, roll one piece of paper like a tube. Tape the red cellophane or filter paper to one end of it. Tape the other end around the light end of the pen light. Be sure the pen light shines through the tube! Now point it at a blank screen. This demonstrates the magnitude of a smaller star in a binary system.

4. Now hold the pen light next to the flash light, pointing both light sources toward a blank wall or screen.

5. Move the pen light so that is "orbits" the flash light. Direct students to notice the change in overall "magnitude" or "brightness" as the pen light is behind, next to, and in front of the flash light. This should be somewhat noticeable since the pen light will show on the wall or screen as red.

6. Prompt the students with questions like, "do you see how the binary flash light system is brightest when the flash lights are next to each other?" and "do you see how the binary flash light system is dimmest when the pen light is behind the flash light?" and "do you see how the magnitude is somewhere in between brightest and dimmest when the pen light is in front of the flash light?"

Below, there is a representation of various light curves as they would appear from plotting data from eclipsing binaries. Notice how their "brightness" or "magnitude" changes as the smaller star is behind, next to, and in front of the larger star. Finally, direct students to see how the flash light model and the light curves of binary stars are related. A light curve is simply a plot of some measure of a source's intensity or brightness versus time.

Example of Light Curves and Model of Orbital Periods of Binary Systems

# Day 2

## Materials

TeacherFor each student or pair of students
overhead to emphasize the points of the guided practice calculator
Plot of HT Cas data metric ruler
Plot of X0748-676 data Plot of HT Cas data
Plot of Vela X-1 data Plot of X0748-676 data
Plot of Vela X-1 data
Few sheets of graph paper

## Guided Practice or Developmental Procedure

If you look at the illustration below, you can see how the diameter of the smaller star (d1) relates to t1 on the light curve. You can also see how the diameter of the larger star (d2) relates to t2 on the light curve.

In this lesson, we will examine data from from sources: HT Cas, X0748-676, and Vela X-1.

It is known that distance = velocity x time. In our first binary star system example, HT Cas, the velocity is known. It is 390 km/sec. We can now combine that information with the equations d2 = v t2 and d1 = v t1 to determine the diameters of the stars.

In order to determine the diameter (d2) of the larger star, or the diameter (d1) of the smaller star, we need to look at the light curve, determine both t1 and t2, and calculate. If we indeed look at the light curve of HT Cas, t1 is equal to 0.4 minutes and t2 is equal to 5.1 minutes. Once we substitute these values, we get:

```
d2 = v t2
=  390 km/sec x 4.8 minutes
=  390 km/sec x 4.8 x 60 (secs/min.)
=  112,320 km

d1 = v t1
=  390 km/sec x 0.4 minutes
=  390 km/sec x 0.4 x 60 (secs/min.)
=  9,360 km
```
How close are these to the values determined using other data? Using optical data, we obtain values of d2 to be about 214,300 km and d1 about 16,700 km. So we have underestimated both. The reason is because we are using the x-ray light curve. For d1, did you notice that it was somewhat difficult to determine t1? Astromers' measurements of t1 from this data range from 0.08-0.88 minutes. Also, we are measuring the size of the x-ray emitting region. We find that it is smaller than the size of the white dwarf itself. Both of these contribute to underestimating d1. For d2 remember that the smaller star does not cross the full diameter of the larger star. This will underestimate the size of the companion star, d2. In addition, studies suggest that the x-ray emitting region is slightly above the plane of orbit of the two stars. This would also cause an additional underestimation of d2.

## Independent Practice

The students are now ready to determine the sizes of the larger stars found in the binary systems X0748-676 and Vela X-1. The students can work independently or in pairs for the completion of this task.

## Assessment

Formative assessment and observation should be evident throughout lesson. The worksheet, final questions during closure, or a future quiz may serve as summative assessment.

## Closure

Direct students to write for ten minutes in their journals summarizing the lab and all procedures in this lesson. Encourage students to then share their findings and what they might have written in their journals.

## References

1. Giommi, P., Gottwald, M., Parmar, A. N., and White, N. E., "The Discovery of 3.8 Hour Intensity Dips and Eclipses from the Transient Low-Mass X-ray Binary EXO 0748-676", The Astrophysical Journal, 308:199-212, 1986, September 1.

2. Horne, K., Steining, R. F., & Wood, J. H., "Eclipse Studies of the Dwarf Nova HT Cassiopeiae", The Astrophysical Journal, 378:271-280, 1991 September 1.

3. Kaufmann, William J. III, Universe, Freeman and Company, 1994, pgs. 336-340

4. Kerrod, Robin, Encyclopedia of Science: The Heavens Stars, Galaxies, and the Solar System, Macmillan Publishing Company, 1991

5. Kondo, Herbert, The New Book of Popular Science Vol. 1, Grolier Incorporated, 1982, pgs. 174-190

6. Seward, Frederick D. and Charles, Philip A., Exploring the X-ray Universe, Cambridge University Press, 1995

7. Sato, N., et al., Publications of the Astronomical Society of Japan, 1986, Vol. 38, pg. 731

8. Parmar, A. N., et al., Astrophysical Journal, 1991, Vol.366, pg. 253

The graphics and other information found within this lesson can also be found on Imagine the Universe! which is located on the World Wide Web. The URL for this site is http://imagine.gsfc.nasa.gov

Some of the data was retrieved within The HEASARC Data Archive using W3Browse which is located on the World Wide Web. The URL for this site is http://heasarc.gsfc.nasa.gov/