How Much do Stars "Weigh"?

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As surprising as it may seem, we can determine the mass ("weight") of stars. The underlying physics of orbital motion which embraces Newton's law of gravity expressed through Kepler's laws of orbital motion will help us complete our goals for this chapter. The key to this comes through the fact that roughly half of the stars you see at night are not "alone" but are in fact members of double or multiple star systems orbiting around one another.

Seeing Double!

Roughly half of the stars that we see are members of multiple star systems. An inspection of any field of stars with even a modest telescope will reveal numerous "close pairs" of stars. It was once thought that this closeness was the result of coincident alignment. If you look carefully at the Big Dipper you will notice that the second star in the handle is actually two stars - Mizar and Alcor. In fact, these stars are about 3 light years from each other and though they are moving together as part of a common group do not orbit each other.

	While some are no more than "chance alignments" it is now understood that stars can orbit each other as part of a common system. Indeed, the observations over decades of binary stars helps confirm in our minds an assumption that we made earlier - that the laws of physics operating on earth also operate for distant star systems.
Figure 8.6 Mizar is the second star in the handle of the Big Dipper If you look carefully you see it has a companion - Alcor.

What Good are Binary Stars?

Binary stars are our "laboratories" in which we can learn about:

• the masses of stars
• the sizes of stars
• the subtleties of stellar surfaces

Types of Binary Systems

1. Visual Binaries both components visible long orbital periods (typically hundreds of years) 61 Cygni for example consists of two stars separated by approximately 30" and has an orbital period of 722 years.
	Figure 8.7 Hubble image of the binary system Gliese 623

2. Astrometric Binaries

• only one component visible
• detected by "wobble" in the proper motion of the visible component. The Flash movie shown below greatly exaggerates such motion. The curved line along which the same stellar image has been placed simulates the view an astronomer might put together of this star by observing it for many decades.
• example: Barnard's Star

  Figure 8.8 Animation of an astrometric binary system in which the "wobble" in the motion of the major star betrays the presence of a companion.

3. Spectroscopic Binaries

3a. Double-Line Spectroscopic Binaries

• These are systems in which the spectrum of the "star" actually appears as two spectra "sandwiched" together. As the individual stars move to and away from the earth the astronomer sees the lines of each star shifted in opposite directions. This allows one to determine such things as the eccentricity of the orbits and the mass ratio of the stars.

3b. Single-Line Spectroscopic Binaries

• Often only one spectrum is visible. This is especially so when the component stars are of much different luminosity. Still, in a manner analogous to that of the astrometric binary, the nature of these systems is revealed by the periodic back-and-forth shifting of the spectral lines in the stellar spectrum.

4. Eclipsing Binaries

• Occasionally, if the orbit is aligned properly with respect to earth, an eclipse will occur. When this happens the binary system is considered to be an eclipsing binary. All of the previous cases could also be eclipsing binaries.

What Binary Stars Can Tell Us:

The study of binary star systems remains an integral part of fundamental stellar astronomy. By applying Kepler's and Newton's laws to the analysis of binary star orbits it is possible to determine the mass and basic dimensions of stars. This is our most direct and accurate way of determining stellar masses.

Use the Eclipsing Binary Simulator applet show below to investigate the following:

1. How mass of the stars affects the period of their orbits. (Do this by adjusting the masses using the scrollbars on the right side of the applet.)
2. How the separation between the stars affects the period of their orbits. (Do this by adjusting the separation scroll bar on the right side of the applet.) (Follow the take a closer look link to learn how to use this applet)

To show you how binary stars can tell us some of the basic properties of stars consider the following worked example in which the mass of a binary system will be determined.

Example 8.14 Use the following data for the star 61 Cygni to determine the mass of the system

You may recall from section 8.1 that 61 Cygni was the first star for which parallax had been detected. 61 Cygni is also a visual binary with both stars separated by about 30 seconds of arc. Since we know the star's parallax we can determine its distance. This data was critical in the determination of the separation of the two stars. Also, the because both stars are visible it is possible over the course of many decades to observe the stars orbiting each other. From this the period was determined. (Note - you don't need to watch for an entire 722 years to determine the period! Only a "few decades" will suffice!)

Recall that Kepler's 3rd Law provides a crucial link between the size of orbits and the period of the orbits. If we use units of years for period and AU for star separation Kepler's 3rd Law assumes the following form:

where m₁ and m₂ are the masses of the stars measured in solar units, "a" is the orbital separation in AU and P is the orbital period in years.

This leads to

This tells us that the total mass of the 61 Cygni system is 2.1 solar masses but it does not tell us the individual masses of the stars. In order to do this you must have more data which normally is provided by a spectroscopic analysis of the light coming from the binary system.

Unlocking the Secrets of the Stars - Looking at Spectra

Figure 8.8 shows the periodic variation in shape of the spectral lines that occurs when two stars in a single-lined spectroscopic binary orbit each other. These periodic variations contain a wealth of information and an important part of an astrophysicist's training is devoted to learning to tease out this information. By studying the way in which the lines vary (and if two sets of spectra are visible) it is possible to identify the two stars and to determine the mass of each star. (In Example 8.14 the best we could do was determine the total mass). In addition to mass spectral line variation can tell us surface temperatures, composition and even if the surface is vibrating.

If you return to Figure 8.6 and study the combined spectrum you will see two distinct sets of lines always shifting in opposite directions. Sometimes the lines from one star shift toward the blue end of the spectrum - this indicates that the star is moving toward you. At the same time the other star's lines are shifting to the red and you can conclude that the star is moving away from you. By measuring the amount of spectral line shifting, and by using the Doppler effect formula, you can deduce the speed of each star.

Example 8.15 In a double-lined spectroscopic binary system the a spectral line whose rest-frame wavelength is 434.1 nm is observed to be split into two lines, one at 433.9 nm and the other at 434.4 nm. Determine the velocity of each star with respect to your line of sight.

Solution: You need to determine how much each line has shifted by subtracting its wavelength from the rest wavelength of 434.1 nm. This gives: Blue-Shift: Dl = (433.9 nm - 434.1 nm) = -0.2 nm. Use the Doppler formula Inserting numbers gives Multiply both sides by "c" - the speed of light to get v = -138 km/s (-ve means the star is moving toward you).
	Figure 8.12 Spectrum of a double-lined spectroscopic binary show blue and red-shifted components.

You can do the same calculation for the red-shifted component to conclude that the other star is moving away from your line of sight with a velocity of +207 km/s

Practice

1. Use the applet Eclipsing Binary Simulator to investigate the concept of centre of mass how the mass of the stars in a binary influences their motion around the centre of mass. (You may find it helpful to se the inclination equal to zero - the centre of mass is indicated by a green cross.)
2. In a double-lined specroscopic binary one star you determine that one star is orbitin at 100 km/s while the other star orbits at 220 km/s. Which of the stars is the most massive?
3. Two stars in a visual binary orbit each other with a period of 125 years. Telescopic observation indicates that their separation is 75 AU. What is the total mass of both stars in the system?
4. Spectroscopic analysis reveals that one of the stars in question 3 above is twice as massive as the other. What are the masses of the stars in this binary system?
5. Match the following light curves on the right with the correct star positions as shown on the left. (The red vertical line on each light curve corresponds to of the orientations.)

1		a
2		b
3		c
4		d

To determine the mass and size of the stars

Chapter 9.5