How Big Is That Star?
- Students will be able to explain the relationship between radius
and mass among a list of stars.
- Students will understand how a binary star system's orbit can cause
changes in the observed brightness of the system.
- Students will determine the diameters of stars by analyzing
data and manipulating equations.
6th through 9th grades
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:
Students should have learned some pre-algebraic concepts such
pattern recognition, ratios, proportions, decimals, and percents.
Students should have had an introduction to the concepts of stars and
For each class of students, you will probably need at least
Warm - Up
Here are some examples of what you may want students to do. Substituting
your own is, of course, always acceptable!
- 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.
- Again, to remind students of previously learned concepts, instruct students
to convert a set of given ratios into decimals and percents.
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.
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.
- In a column format, write a list of the stars from least to greatest in
terms of their radius (measured in Suns).
- In another column next to that list, write the ratio of radius to mass
(measured in Suns).
- Locate some of these stars on a constellation map.
- Explain the (very general) relationship between radius and mass among this
list of stars.
|two pieces of paper|
|red cellophane or filter paper|
|overhead to emphasize points in introduction and throughout the lesson|
Guided Practice or Developmental Procedure
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.
- 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.
- 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.
- 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.
- Now hold the pen light next to the flash light, pointing both light
sources toward a blank wall or screen.
- 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.
- 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?"
|Example of Light Curves and Model of Orbital Periods of Binary Systems|
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:
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.
| 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
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.
Formative assessment and observation should be
evident throughout lesson. The worksheet, final questions
during closure, or a future quiz may serve as summative
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.
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
- 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.
- 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.
- Kaufmann, William J. III, Universe, Freeman and Company,
1994, pgs. 336-340
- Kerrod, Robin, Encyclopedia of Science: The Heavens Stars,
Galaxies, and the Solar System, Macmillan Publishing
- Kondo, Herbert, The New Book of Popular Science Vol. 1,
Grolier Incorporated, 1982, pgs. 174-190
- Seward, Frederick D. and Charles, Philip A., Exploring the
X-ray Universe, Cambridge University Press, 1995
- Sato, N., et al., Publications of the Astronomical Society of Japan,
1986, Vol. 38, pg. 731
- Parmar, A. N., et al., Astrophysical Journal, 1991, Vol.366,
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