Astronomy Assignment #7: Stellar Nomenclature I
Review Questions
1. Describe the procedure used to find distances to the nearby stars.
To measure the distances to nearby stars using stellar parallax you need to take at least two photographs
of the star from opposite sides of the Sun as illustrated in the figure below. From the change in position
of the nearby star in each photogr5aph you can measure the parallax angle, p. Once the parallax angle p
is known and using the baseline of 1 AU, you use simple right triangle trigonometry:
tan( p ) 
1 AU
D
Solving for D yields
D
1 AU
tan( p)
Since parallax angles in astronomy are always exceedingly small, you may use the small angle
approximation that states if p is much less than1, then tan(p) =p. So the distance relation becomes
1 AU
p
In the relation above, p must be expressed in radians, an angular unit used by physicists and engineers.
1
A slightly simpler form of this relation used by astronomers is Dpc 
, where the parallax angle
parc -seconds
is expressed in arc-seconds and the distance is then given in parsecs.
D
2p
p
2. What do you need to know in order to get the scale of interstellar space in terms of kilometers or meters?
You need know the relationship between the parsec and meters…in other words the number of meters in
AU. Unless you know the precise length of the baseline in stellar parallax, you cannot express the
distance to a star in absolute units.
3. If the star-Sun distance = 30 parsecs, how far is the star from the Earth?
Since 1 parsec is the same as 3.26 light years, then a star that is 30 parsecs from Earth is 97.8 or about
100 light years from Earth.
4. If you measure the parallax of a star to be 0.1 arc second on Earth, how big would the parallax of the
same star be for an observer on Mars (Mars-Sun distance = 1.5 A.U.)?
Referring to question 2 above, you should be able to see that if you were measuring the parallax angle of
a stars from Mars, where the baseline of observation is 1.5 AU, then the distance relation would be
1 .5
Dpc 
. So a star with a parallax angle of 0.1 arc-seconds as measured from Earth would be 10
parc -seconds
parsecs away. This same star would have a parallax angle from Mars given by
1.5
parc-seconds 
 0.15 arc - seconds .
Dpc
5. If you measure the parallax of a star to be 0.5 arc-second on Earth and an observer in a space station in
orbit around the Sun measures a parallax for the same star of 1 arc-second, how far is the space station
from the Sun?
In this question to you need to determine the baseline b of the parallax observation from the space
station. Since the distance to the stars is the same from the Earth and the space station, we can write
1 
b 
Dpc   
 
 0.5  Earth 1.0  space station
Clearly b must equal 2 for the equality to be maintained. So the space station is 2 AU from the Sun.
6. If you can measure angles as small as 1/50 arc second, how far out can you measure star distances from
the Earth using the trigonometric parallax method? How long do you have to wait between
observations?
The distance formula for stellar parallax is Dpc 
have a distance of Dpc 
1
. So a star with a parallax of 1/50 arc second will
p"
1
 50 
 1     50 pc .
 1 
 1 
 
 50 
Since you have to wait to take two photographs from opposite sides of the Sun for a parallax
measurement, you would have to wait for 6 months for the Earth to carry you halfway around the Sun.
7. If you can measure angles as small as 1/50 arc second, how far out can you measure star distances from
Jupiter (Jupiter-Sun distance = 5.2 A.U.) using the trigonometric parallax method? However, how long
do you have to wait between observations? (Use Kepler's third law to find Jupiter's orbital period and
divide by two.)
The distance formula for stellar parallax from Jupiter would be Dpc 
1/50 arc second will have a distance of Dpc 
5.2
. So a star with a parallax of
p"
5.2
 50 
 5.2     260 pc .
 1
 1
 
 50 
Since you have to wait to take two photographs from opposite sides of the Sun for a parallax
measurement and Jupiter takes about 12 years to complete one orbit (use Kepler’s 3rd Law), you would
have to wait for 6 years for Jupiter to carry you halfway around the Sun.
8. What does a magnitude interval of 5 correspond to in brightness? How about an interval of 1? How
about an interval of 3?
A magnitude interval of 5 corresponds to difference in brightness of 100 exactly. For example a star of
apparent magnitude +5 is 100 times dimmer than a star with apparent magnitude of 0.
A magnitude interval of 1 corresponds to difference in brightness of about 2.5. For example a star of
apparent magnitude +3 is about 2.5 times dimmer than a star with apparent magnitude of +2.
A magnitude interval of 3 corresponds to difference in brightness of about 2.53 ≈ 16. For example a star
of apparent magnitude +5 is about 16 times dimmer than a star with apparent magnitude of +2.
9. Do bright things have larger or smaller magnitudes than fainter things?
The magnitude scale (both apparent and absolute) are “backwards” in that the lower the magnitude
numerically the greater the brightness or luminosity of the star. For example a star of apparent
magnitude +5 is much dimmer (100 times dimmer actually) than a star with apparent magnitude of 0.
10. How is apparent magnitude different from absolute magnitude?
The apparent magnitude of a star is a numerical code that represents how bright a star appears. The
absolute magnitude of a star is a numerical code that represents how bright a star appears if it were
placed at a standard distance of 10 parsecs. Within the absolute magnitude concept, since all stars are
considered to be at the same distance of 10 parsecs, the absolute magnitude is also a measure of the
luminosity of the star.
11. Put the following objects (given with their apparent magnitudes) in order of brightness as seen from
Earth (faintest first): Sun (-26.7), Venus (-4.4), Barnard's Star (9.5), Sirius (-1.4), Proxima Centauri
(11.0).
Faintest
Brightest
Apparent
Magnitude, m
Name
11.0
9.5
-1.4
-4.4
-26.7
Proxima Centauri
Barnard’s Star
Venus
Sun
Comments
Invisible to the
naked eye
Invisible to the
naked eye
Sirius
Brightest star in
the sky excluding
the Sun
Brightest4 object in
the sky
12. What two things does luminosity depend on?
Luminosity of an object depends on the surface temperature of the object and its surface area. As
expressed by the Stefan-Boltzmann Law, the luminosity of a spherical star is L  4R 2  T 4 , where
4R 2 is the surface area of the spherical star and T 4 represents the luminosity per unit surface area of
the star.
A star can be very luminous if it is either very hot, so that the luminosity per unit surface area creates the
high total luminosity, or if it is very large in radius, so that the large surface area creates the high
luminosity.
13. 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?
Proper motion μ is the motion of a stars across the sky measured in milli-arcseconds per year. If two
stars have the same proper motion, but one star is farther away, then the more distant star must be
moving faster in space. See the figure below. The more distant star must move farther in the same time
period as the nearer star to have the same proper motion μ.
14. 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?
50,000 K
 8.6 times the temperature of the hottest O-type star.
5,800 K
2,500 K
 .43 times the temperature of the coolest M-type star.
The Sun is about
5,800 K
The Sun is about
15. 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 stars are between approximately
50,000 K for O stars and 2,500 K for M stars.
16. What is the range of luminosities produced by main sequence stars? Compare them to the Sun (Watts
are ridiculously small energy units to use).
Stars can range in absolute magnitude from the most luminous at -10 to the least luminous at +20.
These absolute magnitude limits translate to luminosity limits of 1 million solar luminosities at most to 1
millionth of a solar luminosity at the least.