Astronomy Assignment #11 Solutions
1. How is spectroscopy used to find the composition of stars?
2. 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?
The proper motion is a measure of the angular shift of the star per year. If two stars have the same
proper motion then they have the same angular shift in their position per year. A star that is more
distant must move faster in space to have the same proper motion as a closer star. See figure below.

, proper motion
Tangential velocity
of the close star
Tangential velocity
of the far star
3. How the masses of stars found? What kind of star systems are used and which famous law of orbital
motion is used?
Star masses are found by applying Kepler’s famous 3rd law to binary star systems. The combined mass
of the binary star system can be determined from the orbital period of the binary star system and the
a3
separation of the two stars in the system: m1  m2 Solar Masses  AU
. When many pairs of binary star
2
PYr
combined masses where the spectral type and luminosity class of each member of the pair is known, then
individual masses of the members of the binary stars can be determined.
4. How many times closer to the center of mass is the massive star than the low-mass star?
The distance of each star from their center of mass is very similar to the distance that two people of
different masses must sit on a teeter-totter to achieve balance. The more massive person must sit closer
to the fulcrum (i.e. so the center of mass is over the fulcrum). In binary stars, the distance of a star from
m2
 R , where R is the separation between the two stars. If m1 is
the center of mass is given by R1 
m1  m2
the more massive star then you can see that R1 is going to less than the half the separation of the two
stars consistent with the notion that the more massive star must be closer to the center of mass.
5. 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?
The distance of each star from their center of mass is very similar to the distance that two people of
different masses must sit on a teeter-totter to achieve balance. The more massive person must sit closer
to the fulcrum (i.e. so the center of mass is over the fulcrum). In binary stars, the distance of a star from
m2
 R , where R is the separation between the two stars. In this
the center of mass is given by R1 
m1  m2
problem the more massive star is the star that is 0.2 AU from the center of mass. Additionally, we can
say that the ration of masses of the stars is equal to the inverse of the ration of their distances from the
center of mass. Therefore the ratio of the masses of the stars is 3; the massive star is three times more
massive than the less massive star.
6. What special type of binary star system is used to find the diameters of stars?
The special type of binary star system that is used to find the diameters of stars is the eclipsing binary
star. These binary stars have their orbital plane exactly aligned with the Earth so that one star passes
in front of and behind the other star. This exact geometry creates a distinctive light curve as one star
first passes in front of the other and then behind it. If the separation of the stars can be measured, then
the velocities of the stars can be determined from Kepler’s 3rd Law and the orbital velocity relation.
With these parameters known the dips in brightness in the light curve can be used to determine the
diameter of each star.
7. 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.)
This is a Stefan-Boltzmann Law problem. To solve this problem use a ratio to solve for the radius of
Sirius B compared to the Sun and then convert that radius into a diameter in kilometers.
2
4
2 2
4
LSirius B  4 2 RSirius
B  TSirius B and LSun  4 RSun  TSun
2
4
2
4
R

L
4 2 RSirius
RSirius
B  TSirius B
B  TSirius B
 Sirius B 

  Sirius B 
2 2
4
2
4
LSun
4 RSun  TSun
RSun  TSun
 RSun 
2
T

  Sirius B 
 TSun 
4
Filling in temperature and luminosity values from the problem yields
2
R
  14,000 K 
0.00794   Sirius B   

 RSun   5,800 K 
4
2
R

4
0.00794   Sirius B   2.42 
 RSun 
2
R

0.00794   Sirius B   33.9 
 RSun 

RSirius B

RSun
0.00794
 0.0153
33.9
Thus Sirius B has a radius that is just 0.0153 times the radius of the Sun. This corresponds to a
diameter for Sirius B of 21,303 km. Note that the diameter of the Earth is about 13,000 km. So Sirius B
has a diameter that is just under twice the radius of the Earth.
8. Why do very hot stars have no hydrogen lines?
Very hot stars have no (visible) hydrogen lines because the hydrogen atom is ionized and there are not
electronic transitions at the lower energy levels.
9. Why do very cool stars have no hydrogen lines?
Very cool stars have no (visible) hydrogen lines because their temperatures are insufficient to excite
ground state hydrogen electrons up to the n=2 energy level. Only electrons that can be excited to the
n=2 energy level in hydrogen and then further excited to higher energy levels will absorb visible
photons creating spectral absorption lines.
10. What is the grouping of stars by spectral type based on?
The grouping of stars by spectral type is based on the stars temperature.
11. What are the 7 basic spectral types in order of temperature (hottest to coldest)?
The seven basic spectral types (hottest to coolest) are designated as O, B, A, F, G, K and M. The
mnemonic is “Oh Be A Fine Girl/Guy, Kiss Me.”
12. If our Sun has a surface temperature of 5840 K, how many times hotter than the Sun is the hottest Otype star? How many times cooler than the Sun is the coolest M-type star?
The hottest O-type star has a temperature of about 50,000 K and this is approximately 10 times hotter
than the Sun (5,800 K). The coolest M-type star has a temperature of about 2,000 K and this is
approximately 3 times cooler than the Sun (5,800 K).
13. What fraction of the stars are main sequence stars?
An overwhelming percentage of stars are main sequence stars – about 99% of all stars are main
sequence dwarf stars.
14. What is the range of temperatures found on the surface of main sequence stars?
The range of temperatures found on the surface of main sequence star range from about 50,000 K for O
stars to about 2,000 K for M stars. Thus the range of temperatures of main sequence stars is a factor of
about 25 from the coolest to the hottest.
15. What is the range of luminosities produced by main sequence stars? Compare them to the Sun (Watts
are ridiculously small energy units to use).
The range of luminosities produced by main sequence stars is huge. The most luminous main sequence
star may be 1 million times as luminous as the Sun and the least luminous main sequence star is about
one millionth the luminosity of the Sun. Thus the range of luminosity of main sequence stars is a factor
of 1012 from the least to greatest luminosity.
16. What is the range of diameters for main sequence stars? Compare them to the Sun (miles and kilometers
are ridiculously small length units to use). Red giants, supergiants, white dwarfs are not main sequence
stars.
Main sequence stars have arrange of diameters of between about 10 solar diameters to 1/10’th a solar
diameter.
17. What is the trend in the stellar diameters vs. temperature for main sequence stars? (As temperature
increases, the diameter increases.)
18. What is the trend in the stellar luminosities vs. temperature for main sequence stars? (As temperature
increases, the luminosity increases.