Download Astronomy Assignment #1

Astronomy Assignment #11
Text Problems:
Unit Number
57
58
59
Questions for Review
4, 5, 6, 7, 9
7, 8, 9
4, 5, 6, 8
Problems
10
12, 13, 14, 17, 18
11, 12, 13, 16
Problems in Unit 58
11. Alpha Centauri is actually two stars that orbit each other. Recent measurements show that Alpha
Centauri A has an angular diameter of 0.0085 arcsec, while Alpha Centauri B has an angular diameter of
0.0060 arcsec. The pairs are at a distance of 4.4 lyr. What are their diameters compared to the Sun.
This is a fairly messy problem. There is one short cut we can use. After we calculate the radius of Alpha
Centauri A, we can simply calculate the radius of Alpha Centauri B using the ratio of their angular sizes, Alpha
Centauri B will be a factor of (60/85) smaller than Alpha Centauri A.
A
L
,


2D
360
where A is the angular diameter of the star, L is its true diameter, and D is its distance. We need to solve for
L. We will have to change 4.4 light years into meters and 0.0085 arcsec into degrees to get a meaningful
answer.
To determine the angular size of Alpha Centauri A we need to use the angular size formula
A
L


2D
360

 1  
 .0085"

 3,600"  





A
m



 L  2D 
 2   4.4 ly   9.46  1015    


ly
360
360




L  2  4.16  10
16
2.36  10 6
m
 1.71  10 9 m

360
The diameter of Alpha Centauri A is 1.71 x 109 meters. The Sun’s diameter is 1.39 x 109 meters as
determined from the table in the text’s appendix. Thus, Alpha Centauri A is slightly larger than the Sun
with a diameter of 1.23 solar diameters.
Alpha Centauri B is (60/85) = 0.706 times smaller than Alpha Centauri A. based on the ratio of their
angular sizes (and the fact that they are at the same distance). So Alpha Centauri B is slightly smaller
than the Sun with a diameter of 0.867 solar diameters.
12. A star with the same color as the Sun is found to produce a luminosity 81 times larger. What is its
radius, compared to the Sun’s?
The following three problems all use the Stefan-Boltzmann Law L  4R 2  T 4 . We can avoid painful
and error prone calculations, by using a ratio technique as illustrated below.
2
4
2
4
LStar  4RStar
 TStar
and LSun  4RSun
 TSun
Take the ratio of the two equations above…
2
4
R
LStar 4RStar
 TStar

  Star
2
4
LSun 4RSun  TSun  RSun
L
  Star
 LSun
  RStar
  
  RSun



2
T
  Star
 TSun






2
T
  Star
 TSun



4
4
Now we have an expression where only ratios are used…much simpler that working with the full
equation. This problem gives us the ratio of temperatures (1, since the stars are the same color they are
also the same temperature) and the ratio of the luminosities (81). We need to solve for the ratio of radii.
L
  Star
 LSun
  RStar
  
  RSun
2
  TStar
  
  TSun



4
 LStar 


L
 RStar 
Sun
  81  81

  
4
14
 TStar 
 RSun 


 TSun 
The ratio of the radii squared is 81, so the ratio of the radii must be 81  9 . The radius of the star in
question is 9 times the radius of the Sun. A note is worthy here: We expected this star to be larger
because it was the same temperature as the Sun, by quite a bit more luminous. What this problem is
trying to illustrate, is the luminosity depends on the square of the stellar radius, not just on the radius
alone.
2
13. A star with the same radius as the Sun is found to produce a luminosity 81 times larger. What is its
surface temperature compared to the Sun?
As in the preceding question
L
  Star
 LSun
 TStar

 TSun
  RStar
  
  RSun



2
T
  Star
 TSun



4
 LStar 


L

81
   Sun 2  2  81
1
 RStar 



 RSun 
4
TStar 4
 81  3
TSun
This star is only three times hotter than the Sun. Notice how important temperature is to luminosity. A
factor of three increase in temperature increases the luminosity by a factor or 81!

14. The surface temperature of Arcturus is about half as hot as the Sun’s, but Arcturus is about 100
times more luminous that the Sun. What is its radius, compared to the Sun’s?
As before
2
L
  Star
 LSun
  RStar
  
  RSun
2
  TStar
  
  TSun



4

100  100  1,600
 
4
0.0625
1

 
2
The ratio of the radii squared is 1600, so the ratio of the radii must be 1,600  40 . The radius of
Arcturus is 40 times the radius of the Sun. A note is worthy here.
 RStar

 RSun
2
17. A star is five times as luminous as the Sun and has a surface temperature of 9,000 K. What is its
radius compared to that of the Sun?
This is a classic Stefan-Boltzmann Problem.
2
4
2
4
LStar  4RStar
 TStar
and LSun  4RSun
 TSun
Take the ratio of the two equations above…
2
4
R
LStar 4RStar
 TStar

  Star
2
4
LSun 4RSun  TSun  RSun
L
  Star
 LSun
  RStar
  
  RSun
2
  TStar
  
  TSun
2
R
5   Star
 RSun
  9,000 K 
  

  5,800 K 
R
5   Star
 RSun

  1.554

R
5   Star
 RSun

  5.80


2
  TStar
  
  TSun






4
4
4
2
2
RStar
5

 .928
RSun
5.80
The star has a radius just slightly smaller than the Sun at 0.928 solar radii.
18. Using the information that Betelgeuse has an angular size of 0.05 arcsec and is 130 pc from Earth,
show how you can calculate its diameter in km.
A
L
, where A = 0.05” = 1.39x10-5, and


2D
360
D = 130 pc = 130 x (3.08x1016m) = 4.00x1018m.
This is an angular size problem. So we use
3
A
L


2D
360



A
1.38  10 5
L

2

D

 2  (4.00  1018 m)  9.63  1011 m
360 
360 
The diameter of Betelgeuse is 9.63 x 1011 m or 692 solar diameters.
4
Instructor Assigned Topic:
Compare and contrast two sets of stars in two paragraphs. Data for those sets appears on the following
pages. Also a blank HR diagram follows so you can plot the stars from both sets on it to aid in your
comparison.
First paragraph: Summarize the properties of the twenty brightest stars and compare the twenty brightest
stars to the Sun. Is the Sun like or unlike the brightest stars in the sky? Be detailed in your description.
Second paragraph: Summarize the properties of the twenty nearest star systems and compare them to the
Sun. Is the Sun like or unlike the nearest stars in the sky? Be detailed in your description.
Third Paragraph: Does the Sun appear to be a common star. Comment on your feelings or the
implications regarding your discovery.
Include the completed HR diagram for both sets of stars with your paragraphs. Use easily distinguished
symbols for the Bright Stars and the Nearest Stars.
5
The Twenty Brightest Stars
Multiple Star
System
Spectral
Type
Luminosity
Class
Apparent Magnitude,
m
Absolute Magnitude,
M
Distance,
lyrs
Yes
A1
V
-1.46
1.4
8.8
F0
Ib/II
-0.72
-3.1
98
G2
V
-0.01
4.4
4.3
Arcturus
K2
III
-0.06
-0.3
35.9
Vega
A0
V
0.04
0.5
26.1
Star
Sirius
Canopus
Alpha Centauri
Yes
Capella
Yes
G0
III
0.05
-0.7
46
Rigel
Yes
B8
Ia
0.14
-6.8
815
Procyon
Yes
F5
IV
0.37
2.6
11.4
Betelgeuse
M2
Ia/Ib
0.41
-5.5
490
Achernar
B5
V
0.51
-1
65.2
B1
III
0.63
-4.1
295
A7
IV
0.77
2.2
16.6
Beta Centauri
Yes
Altair
Alpha Crucis
Yes
B1
IV
1.39
-4
390
Aldebaran
Yes
K5
III
0.86
-0.2
52.2
B1
V
0.91
-3.6
260
M1
IB
0.92
-4.5
390
K0
III
1.16
0.8
39.1
A3
V
1.19
2
22.8
Deneb
A2
Ia
1.26
-6.9
2,600
Beta Crucis
B0
IV
1.28
-4.6
490
Spica
Antares
Yes
Pollux
Fomalhaut
Yes
6
The Twenty Nearest Star Systems
Common Name
Catalog
Name
Multiple Star
System
Spectral
Type
Luminosity
Class
Apparent
Magnitude, m
Absolute
Magnitude, M
Distance,
lyrs
G2
V
0.01
4.38
4.36
K0
V
1.34
5.71
4.40
Alpha Centauri A
Gl 559
A
Alpha Centauri B
Gl 559
B
Proxima Centauri
Gl 551
M5.5
V
11.09
15.53
4.22
Barnard's Star
Gl 699
M4.0
V
9.53
13.22
5.96
Wolf 359
Gl 406
M6.0
V
13.44
16.55
7.78
Lalande 21185
Gl 411
M2.0
V
7.47
10.44
8.29
Sirius
Gl 244
A
Sirius B
Gl 244
B
Gl 65
A
UV Ceti
Gl 65
B
Ross 154
Yes
A1
V
-1.43
1.47
8.58
White Dwarf
V
8.44
11.34
8.58
M5.5
V
12.54
15.4
8.72
M6.0
V
12.99
15.85
8.72
Gl 729
M3.5
V
10.43
13.07
9.68
Ross 248
Gl 905
M5.5
V
12.29
14.79
10.32
epsilon Eridani
Gl 144
K2
V
3.73
6.19
10.52
Lacaille 9352
Gl 887
M1.5
V
7.34
9.75
10.74
Ross 128
Gl 447
M4.0
V
11.13
13.51
10.91
EZ Aquarii
Gl 866
M5.0
V
13.33
15.64
11.26
----
--
13.27
15.58
11.26
----
--
14.03
16.34
11.26
F5 I
IV
0.38
2.66
11.40
10.7
12.98
11.40
Procyon
61 Cygni
Yes
Yes
A
Gl 866
B
Gl 866
C
Gl 280
A
Gl 280
B
Gl 820
A
Gl 820
B
Gl 725
A
Gl 725
B
Gl 15
A
Gl 15
B
Yes
Yes
White Dwarf
Yes
Yes
Yes
K5.0
V
5.21
7.49
11.40
K7.0
V
6.03
8.31
11.40
M3.0
V
8.9
11.16
11.52
M3.5
V
9.69
11.95
11.52
M1.5
V
8.08
10.32
11.62
M3.5
V
11.06
13.3
11.62
epsilon Indi
Gl 845
K5
Ve
4.69
6.89
11.82
DX Cancri
GJ 1111
M6.5
V
14.78
16.98
11.82
tau Ceti
Gl 71
G8
Vp
3.49
5.68
11.88
RECONS 1
GJ 1061
M5.5
V
13.03
15.21
11.92
7
HR Diagram
Main Sequence Stars
-10
Absolute Magnitude
-5
0
5
10
15
20
O0 O5 B0 B5 A0 A5 F0 F5 G0 G5 K0 K5 M0 M5
Spectral Type
8