Properties of Stars

The Composition of Stars

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The compositions of stars are determined through spectroscopy. Spectroscopy is the study of something using spectra. Recall from the electromagnetic radiation chapter that a spectrum is what results when you spread starlight out into its individual colors. By noting what absorption lines (or sometimes, emission lines) are present and their strengths, you can find out a tremendous amount of information. Stars have absorption lines patterns similar to the Sun. This means that they are composed mostly of hydrogen and helium with traces of other elements.

From these absorption lines you learn some important things beside the star's composition:

Structure of stars: From the simple fact that you see absorption lines in most stellar spectra, you know that the stars must have a hot dense part that produces a continuous spectrum and an outer layer, or atmosphere, made of cooler, low density gas. The general trend is density and temperature of stars decreases as the distance from the star's center increases. The hot dense part is also gaseous because of the extreme temperatures. Stars have no molten rock in them like the interiors of some of the planets.
Universality of physical laws: The same pattern of hydrogen lines are seen in the in spectra of the Sun, stars, distant galaxies, and quasars (active galaxies at very great distances from us). This is a sensitive test of whether or not the laws of physics used in the structure of atoms works everywhere in the universe. Because the same patterns are seen in the spectra, regardless of where the light comes from, the physics used on Earth must work everywhere else in the universe! All of the absorption lines seen in celestial objects can be seen in laboratories on Earth. The charge and mass of the electron and proton are the same everywhere you look. Physical laws are the same everywhere!
Permanence of physical laws: Since light has a finite speed and the distances are vast, the light received from far away sources has been travelling for billions of years. The light from those remote regions tells us about the physical laws way back then. The spectra seen can be explained with the same physical laws in operation here on Earth at the present time. Physical laws are the same throughout time!

The Velocities of Stars

The velocity of stars are determined by using the doppler effect (see the doppler effect section). The motion of a star causes a shift in the wavelengths received. The star emits the same spectrum it would emit if it was at rest. The pattern of lines depends on the internal physics of the star itself. The doppler effect depends on the external property of the star's motion through space relative to us. The wavelength shift

_new -

_rest, where

_new is the observed wavelength and

_rest is the wavelength you would observe if the star was not moving relative to us. The amount of the shift depends on the velocity of the star in this way:

_rest = (radial velocity) / c, where c is the speed of light and the radial velocity is the component of the star's motion that is along the line of sight. Rearranging the terms you find the radial velocity = (

_rest) × c.

doppler effect tells you the amount of radial
velocity

If the star is moving away from you, then you see the wavelengths stretched out and is a positive number. The spectral lines appear shifted to the red end of the spectrum, so the shift is called a redshift. If the star is moving toward you, then the wavelengths appear to the compressed and is a negative number. The spectral lines are shifted to the blue end of the spectrum, so the shift is called a blueshift. The doppler effect tells you only about the motion relative to you. If you move toward the star or it moves toward you, the doppler effect will be the same. The Earth is in motion relative to the star because it are orbiting the Sun, so you have to take the Earth's orbital motion into account when figuring the star's velocity from the doppler effect.

Also, the doppler effect gives you only the speed along the line of sight. Most stars move at an angle to our line of sight. The part of a star's total velocity that is perpendicular to the line of sight is called the tangential velocity.

To get the tangential velocity, you need to first measure the angular velocity of the star across the sky (d/dt). This is how many degrees on the sky the star moves in a given amount of time and is called the proper motion by astronomers. If you determine the star's distance from its trigonometric parallax or the inverse square law method, you can convert the angular velocity (proper motion) to tangential velocity in physical units such as kilometers/second. The tangential velocity = k × the star's distance × the proper motion, where k is a conversion factor that will take care of the conversion from arc seconds and parsecs and years to kilometers/second. Using the Pythagorean theorem for right triangles, you find that the star's total velocity = Sqrt[(radial velocity)² + (tangential velocity)²].

tangential velocity is the speed across the
sky

Vocabulary

blueshift	doppler effect	proper motion
radial velocity	redshift	spectroscopy
tangential velocity

Formulae

redshift: > 0. Object moving away from you.
blueshift: < 0. Object moving toward you.
Radial velocity = (/ _rest) × c, where is the doppler shift, _rest is the wavelength you would see if the star was at rest. The rest wavelength is also the wavelength of light emitted by the star. Larger doppler shift means greater radial velocity.
Tangential velocity = k × (the star's distance) × (d/dt), where k is a conversion factor and (d/dt) is the angular velocity or proper motion of the star.
Total velocity = Sqrt[(radial velocity)² + (tangential velocity)²]

Review Questions

How is spectroscopy used to find the composition of stars?
How do scientists know that the laws of physics are the same everywhere in the Galaxy and have been the same for billions of years?
How does the amount of doppler shift in the spectral lines depend on the relative speed? How does the direction of shift of the spectral lines depend on the direction of motion?
Given three stars moving with the same total speed of 40 kilometers/second but in three different directions: (A) moving away at an angle=45° from our line of sight, (B) moving directly away from us, (C) moving perpendicular to our line of sight, put the stars in order of increasing amount of shift you see in their spectral lines (the smallest shift first).
The H-alpha spectral line is at 6563Å. Star (A) has that line at 6568Å, star (B) has that line at 6560Å, star (C) has that line at 6563Å. Which star is moving the fastest (along the line of sight) and what is the three stars' directions of motion?
What must you first know about a star before you can convert its angular velocity to a tangential velocity in kilometers/second?
Two stars have proper motions of 0.5 arc seconds/year. Star (A) is 20 parsecs away and star (B) is 30 parsecs away. Which one is moving faster in space?
Two stars orbit about a common point in a plane that is oriented parallel to our line of sight. What will you see happening to the spectral lines of each star? If there is a change, will both stars' spectra be changing in the same direction at the same time?

The Masses of Stars

To determine the masses of stars, Kepler's third law is applied to the motions of binary stars and Kepler's third law (see the gravity chapter). Usually the binary stars are spectroscopic binary stars. A spectroscopic binary system is two stars orbiting a common point at too great a distance away from us to resolve the two stars individually, but whose binary nature is indicated in the periodic shift of their spectral lines as they orbit around each other. Spectroscopic binary stars are used because (a) there are a lot more far away stars than nearby ones and (b) more importantly, you can easily measure their speeds from the doppler shifted lines.

Newton's form of Kepler's third law gives the combined mass of the two stars: (mass 1 + mass 2) = (separation distance)³/(orbital period)² if you use solar mass units, the A.U. for the distance unit between the stars, and the time unit of years for the orbital period. The total distance between the two stars is used in Kepler's third law, but their individual distances from the common point they orbit is used to determine the stars' individual masses.

ratio of distances from center of mass is the
inverse of the mass ratio

Since stars have about the same mass (within a factor of 20), they both orbit around a common point, called the center of mass, that is significantly different from one of the star's center. The center of mass (C.M.) is the point where (mass star 1) × (C.M. distance 1) = (mass star 2) × (C.M. distance 2), or the point they would be balanced upon if the stars were on a stellar seesaw (it is the ``x'' in the figure below). The massive star is closer to the center of mass than the low-mass star and the massive star also moves proportionally slower than the low-mass star so its spectral lines have a smaller doppler shift.

binary stars orbit the center of mass and
the effect on their spectra

Newton's Law of Gravity explains why this is. The gravitational force exerted by the massive star causes a large acceleration in the motion of the low-mass star, so the low-mass star moves faster and has a larger orbit. The weaker gravity exerted by the low-mass star produces a smaller acceleration on the massive star, so the massive star's orbital speed is less and its orbit is smaller. Think about how you could also explain this using Newton's second and third laws of motion.

The distance travelled by an object = velocity × the time it takes. The distance travelled by the star is just the circumference of the orbit = 2p × the radius of a circular orbit and something similar for an elliptical orbit. Therefore, each star's C.M.-distance r = the star's velocity × the star's orbital period / (2p). This allows you to use the easily measured velocity in Kepler's third law and in the center of mass relations. The doppler shifts of the spectral lines are used to construct a radial velocity curve---a plot of the radial velocity (line of sight velocity) vs. time. The low-mass star will move proportionally faster than the massive star.

Uncertainty arises, though, if the binary orbital plane is inclined to our line of sight. In that very common case, the radial velocity = total velocity × Sin(inclination angle). The inclination angle i ranges from i=0° for a face-on orbit (viewing the orbit from directly above the system) to i=90° for an edge-on orbit (viewing the orbit along its plane). The inclination angle can be approximately determined from the plot of radial velocity vs. time. If the binary is an eclipsing binary, then you know that i=90° because you see them periodically pass in front of each other. Eclipsing binaries also allow us to accurately determine the diameters of stars (discussed in the next section). The radial velocity measurement technique has also been used to find planets around other stars and to locate black holes from the doppler shifts they produce in the visible stars they orbit around.

Remember these rules:

Stars stay on the opposite side of the center of mass from each other.
The massive star moves slower than the low-mass star.
The center of mass is also the point where mass1 × velocity1 = mass2 × velocity2

Using the distance of the center of mass from each star, you can proportion out the total mass to each star. Here are the steps to figure out each star's mass:

Find the total mass (mass star A + mass star B) from Kepler's 3rd law.
Find the proportion of each star's mass to the total mass from the center of mass: (mass star A)/(mass star B) = (C.M. distance B)/(C.M. distance A) or (mass star A)/(mass star B) = (velocity star B)/(velocity star A). Note which star's values are on top of the fraction and which are on the bottom! Simplify the fraction down as far as possible.
Set the mass of star A = (mass of star B)×(the fraction of the previous step) and substitute this for the mass of star A in the first step (Kepler's 3rd law step).
You will find star B's mass = the total mass/(1 + the fraction from step 2).
Star A's mass = star B's mass × (the fraction from step 2).
Check that the proportions add up to the total mass!

(By the way, you can use this proportion idea in cooking if you need to have a 32-ounce mixture and the recipe calls for 3 parts sugar to 2 parts flour or if a recipe is for 6 people but you need to serve 8 people.) The masses of different types of stars are summarized in the Main Sequence Star Properties table below.

How do you do that?

Use the observed velocities in the figure below to find the individual masses of the stars. The stars have a measured period of 4/3 years and a separation distance of 4 A.U.

binary system seen edge-on by observer and the
resulting radial velocity curve

Step 1:: Kepler's third law says the total mass = 4³/(4/3)² = 64/(16/9) = 36 solar masses.
Step 2:: I will let the massive star be ``star A''. So (mass star A)/(mass star B) = 400/100 = 4.
Steps 3 and 4:: mass star B = 36 solar masses/(1 + 4) = 7.2 solar masses.
Step 5:: mass star A = mass star B × 4 = 28.8 solar masses.
Step 6: Check:: 28.8 solar masses + 7.2 solar masses does equal 36 solar masses. This step makes sure you did not make an arithmetic error in the previous steps. If the sum does not equal the value in step 1, then re-check your math!

derived masses and distances from the given
velocities