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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 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](radvel.gif)
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.
![a star's total velocity in 3D space](totalvel.gif)
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](tangvel.gif)
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]
- 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=45ƒ 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 6563‰. Star (A) has that line at
6568‰, star (B) has that line at 6560‰, star (C) has that line at
6563‰. 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?
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.
![spectroscopic binary](specbina.gif)
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](cm-anim.gif)
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](specbin-anim.gif)
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=0ƒ for a
face-on orbit (viewing the orbit from directly above the system) to i=90ƒ
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=90ƒ 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](massexb.gif)
- 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!
|
Vocabulary
center of mass | radial velocity curve | spectroscopic binary |
Formulae
- Center of mass distance: mass star A / mass star B =
distance B / distance A, where the distances are each
measured from the center of mass. Notice which star's distance is in the
top of the fraction!
- separation distance = distance A + distance B, where the
distances are each measured from the center of mass.
- Center of mass velocity: mass star A / mass star B =
velocity star B / velocity star A. Notice which star's velocity is in
the top of the fraction!
- Kepler's 3rd law: (mass star A + mass star B) =
(separation distance)3 / (orbital period)2 if use solar mass
units, A.U. for the distance unit between the stars, and the time unit of years for the
orbital period.
- How the masses of stars found? What kind of star systems are used
and which famous law of orbital motion is used?
- How many times closer to the center of
mass is the massive star than the low-mass star?
- How do you use the radial velocity curve to find the
mass proportions and separation distance? How much faster or slower does the low-mass
star move than the massive star?
- Three binary systems with a separation of 2 A.U. between the two
stars in each system. System (1) has the two stars orbiting the center of mass
in 1 year, system (2) has the two stars orbiting the center of mass in 5
months, and system (3) has the two stars orbiting the center of mass in 2
years. Put the binary systems in the correct order by increasing
total mass (least massive first and ignore the inclination angle i).
- Star A is 0.2 A.U. from the center of mass and its companion star B is
0.6 A.U. from the center of mass. Which star is more massive?
- If the two stars in the previous question have orbital periods of 0.35777
years, what are the individual masses of the two stars? (Hint: find their
combined mass from Kepler's third law and then use
their relative center of mass distances to find how many times more massive one
star is than the other.)
- Use the radial velocity curve graph in the text above. Assume that
star A reaches a velocity of 90 kilometers/second and star B reaches only 10
kilometers/second. If the separation distance = 10 A.U., and the orbital period
= 10 years, what is the combined mass of the two
stars? From the center of mass relation also find (star A mass)/(star B mass) and
their individual masses.
- Which star system(s) would you be able to measure the radial velocity:
(a) stars orbiting in a plane that is along our line of sight (i=0ƒ);
(b) stars orbiting in a plane that is perpendicular to our line of sight
(face-on, i=90ƒ); (c) stars orbiting in a plane with i=30ƒ.
Eclipsing binaries are used to determine the diameters of stars. These
are two stars orbiting each other in a plane that is parallel to your line of
sight so you see their orbits edge-on. This means that one star will periodically
cover up the other star. During the eclipse the total brightness measured
for the binary will decrease. The amount of the dip in brightness depends on
the luminosity and relative size of the two stars.
A star's
diameter is found from speed = (distance travelled)/(time it takes). The
speed comes from the
doppler shift and the time is the length of the eclipse. The distance travelled
during the eclipse is equal to the
diameter of the star = 2 × radius. The light
curve---plot of brightness vs. time---is used to derive the star
diameters. Here is an example of two stars orbiting each other in circular orbits
seen edge-on with one star small and hot and the other large and cool:
![deriving the diameter of
stars in an eclipsing binary system](eclipse-size.gif)
When the small star moves from position 1 to position 2 (or from position 3 to
position 4), it has moved a distance equal to its diameter. When the small
star moves from position 1 to position 3 (or from position 2 to position 4), it
has moved a distance equal to the diameter of the large star.
Star sizes can also be found (less accurately) from the luminosity and the
flux. Recall from the magnitude section above that
the luminosity = [4p×(star radius)2] ×
[
×(star's surface temperature)4],
where
is the Stefan-Boltzmann constant. If you
compare the star with the Sun, you can cancel out the constants to get
(star's radius)/(Sun's radius) =
(Sun's temperature/star's temperature)2 ×
Sqrt[star's luminosity/Sun's luminosity]. See the ``How do you do that?'' box
below for an example. The sizes of different types of stars are summarized in the
Main Sequence Star Properties table below.
How do you do that?
Antares is 9120 times more luminous than the Sun (Antares' luminosity/Sun's luminosity)=
9120) and has a temperature of only
3340 K and the Sun's temperature is 5840 K. Antares' size/Sun's size =
(5840/3340)2 ×
Sqrt[9120] = 3.057 × 95.5 = 292. Antares is almost 300
times the size of the Sun! If the Sun were replaced by Antares, the inner planets
Mercury, Venus, and Earth would be inside Antares! It is a red giant star---a
star close to death.
|
Vocabulary
light curve | eclipsing binary |
Formulae
- Eclipsing binary: diameter = speed × time of eclipse.
- Size from luminosity: star's radius/Sun's radius = (Sun's temperature/star's
temperature)2 Sqrt[star's luminosity/Sun's luminosity].
- How do you use the light curve to find the
diameters of stars?
- What special type of binary star system is used to find the diameters of
stars?
- Use the light curve in the figure in the section above. Assume that when
star A is behind star B, the small dip in brightness is seen. When star B is
behind star A, the big dip in brightness is seen. Which star is more luminous?
- From the previous problem, if t1 = 45 minutes, t2 = 60 minutes, t3 = 105 minutes,
t4 = 120 minutes, what is (star A diameter)/(star B diameter)? [Hint: find which
star is brighter and in this circular orbit system (t8 - t6) = (t4 - t2).]
- From the previous problem, if the velocity is 750 kilometers/second, what is the
diameter of the larger star?
- The white dwarf Sirius B has a temperature of 14,000 K and a luminosity
only 0.00794 times the Sun's luminosity. What is the diameter of Sirius
B in kilometers? (The Sun's radius = 696,000 kilometers.)
Go to
Magnitude System section
Go to Stellar Types and HR diagram section
last update: 02 March 1999
Nick Strobel --
Email:
strobel@lightspeed.net
(661) 395-4526
Bakersfield College
Physical Science Dept.
1801 Panorama Drive
Bakersfield, CA 93305-1219