Astronomy Assignment #8 Solutions
Unit Number
54
55
56
Problems
14, 15, 16
11, 12, 13, 14, 15
13, 14, 15, 16
Problems in Unit 54
14. Sirius has a parallax of 0.377 arsec. How far away is it?
This problem requires a straight forward application of the stellar parallax relation d Parsecs 
d Parsecs 
1
p"
1
1

 2.65 Parsecs
p" 0.377"
The star is 2.65 parsecs away according to the method of stellar parallax.
15. A dim star is believed to be 5,000 pc away. What should its parallax be?
This problem requires another straight forward application of the stellar parallax relation
1
d Parsecs 
, but we must solve for p" .
p"
1
d Parsecs 
p"
1
1
 p" 

 2  104 "
d Parsecs 5,000 Parsecs
The star at 5,000 pc distance would have a parallax of 0.0002 arcsec.
16. If a star’s parallax is 0.1 arcsec on Earth, what would it be if measured from Mars
This problem requires understanding the parallax principles. Recall from your class notes and
from the reading in the text that the formula for the parallax angle of a star was
1 AU
tan( p" ) 
d
Using the small angle approximation, we can drop the tangent function and write
p"
1 AU
d
If we were measuring parallax angels from Mars we would replace the 1 AU in the formula
above with the semi-major axis of Mars’ orbit, 1.5 AU.
"
pfrom
Mars 
1.5 AU
d
Since the distance to the star will be essentially the same as measured from Earth or Mars, the
parallax measured from Mars will simply be 1.5 times that measured from Earth. Thus, the
parallax of the star measured from Mars will be 0.15 arcseconds.
Problems in Unit 55
11. Suppose star A is 36 times brighter than star B, but we have reason to believe that they have
the same luminosity.
a. Which is more distant?
Since the two stars are assumed to have the same luminosity, they the brighter star must be
closer. So star A is the closer star.
b. How many times more distant is it than the other star?
The brightness of a star varies as an inverse square law (page 445 & 446). So for a star to appear
36 times brighter than another star of the same luminosity, the brighter star must be six times
closer. Mathematically we could express the solution as follows:
L
Let the subscript A stand for star A, then the brightness of star A is B A  A 2 . Similarly, the
4d A
brightness of star B can be written as
BB 
LB
4d B2
(from page 446- in text). If the ratio
of these two equations is taken we see the following:
BA 
LA
4d 2A
BB 
LB
4d B2
LA
2
2
B A 4d A
L A 4d B2 L A d B2  L A   d B 





  
LB
BB
LB d 2A  LB   d A 
4d 2A LB
4d B2
Since the luminosities are equal, the ratio of luminosities id just 0ne. So the relationship
2
d 
B
simplifies to A   B  . The left hand side of the equation is 36 (ratio of their brightness’s).
BB  d A 
So the number on the left that is squared must be 6 because 62 =36. So the ratio of the distances
must be six. As before, Star A is six times closer than star B.
12. Star C and star D are at the same distance from us, but star D is 10,000 times more luminous
than star C. How do their brightness levels compare?
Star D will appear brighter by a factor of 10,000 over star C.
13. How do the magnitudes of stars C and D in problem 4 compare? (Problem 4 synopsis: Stars
C and D are the same distance from us, but star D is 10,000 times more luminous.)
To answer this problem you need to use the rule regarding magnitudes and brightness’s: if two
stars differ in magnitude by 5 magnitudes then they differ in brightness by a factor of 100
exactly. Since 10,000 = 100 x 100, then the brightness difference of a factor of 10,000 is
equivalent to two differences in magnitude of 5. In other words, star D will be 10 magnitudes
brighter than star C. If star C were a 9TH magnitude star, then star D would have magnitude -1.
14. The star Deneb has an apparent magnitude of 1.25 and an absolute magnitude of -8.5. What
two statements can you make about it, based on this data?
Several statements can be made from these data. First, Deneb is easily visible by the naked eye
since its apparent magnitude is much brighter than the limit of naked eye visibility of m=6.
Second, Deneb is much more luminous than the Sun. The Sun’s absolute magnitude is about +5.
So Deneb is 13.5 (= 5 + 5 + 1 + 1 + 1 + ½ ) magnitudes more luminous than the Sun. This
difference in absolute magnitude corresponds to a difference in luminosity of a factor of 100 x
100 x 2.5 x2.5 x 2.5 plus “change”, equaling about 160,000 solar luminosities. We can also say
that Deneb cannot have much time left before it explodes since this high luminosity means that it
is consuming its mass very rapidly. Further, if Deneb were a main sequence star (which it is
not), its lifetime would be only about 100,,000 years or so.
15. Star A has a magnitude of 3.5. Star B has a magnitude of -1.5. Which star is brightest, and by
what brightest ratio?
Star B is brightest by 5 magnitudes (3.5 - -1.5). Thus star B is 100 times brighter than star A
(Magnitude Rule #2 from class notes).
Problems in Unit 56
13. Suppose a star radiates most strongly at about 200 nm. How hot is it?
All these problems in Unit 55 deal with Wien’s Law T 
2.9  10 6 K  nm
MAX
, where T is the
temperature of the star in Kelvin and MAX is the wavelength at which the stars radiates most
strongly expressed in nanometers.
T
2.9  10 6 K  nm
MAX

2.9  10 6 K  nm
 14,500 K
200 nm
The temperature of the stars is 14,500 K, which is much hotter than the Sun (TSun=5,800 K)
14. The bright southern star Alpha Centauri radiates most strongly at about 500 nm. What is its
temperature? How does this compare to the Sun’s?
T
2.9  10 6 K  nm
MAX
2.9  10 6 K  nm

 5,800 K
500 nm
The temperature of the stars is 14,500 K, which is the same temperature as the Sun.
15. A star radiates most strongly at 850 nm. What spectral type is it?
T
2.9  10 6 K  nm
MAX

2.9  10 6 K  nm
 3,412 K
850 nm
The temperature of the stars is 3,412 K, which is cooler than the Sun (TSun=5,800 K). Following
Table 55.1, this star would have a spectral type of an early M star, perhaps an M1.
16. If star T has a surface temperature of 1,000 K, at what wavelength should it be brightest?
T
2.9  10 6 K  nm
 MAX 
MAX
2.9  10 6 K  nm 2.9  10 6 K  nm

 2,900 nm
T
1,000 K
This star, which is much cooler than the Sun, will radiate most strongly at 2,900 nm which is far
into the infrared portion of the electromagnetic spectrum.