Download Problem 4: magnitude of the star?

Problem 4: An A0 main sequence star is observed at a distance of 100 pc through an interstellar
dust cloud. Furthermore, it is observed with a color index B-V = 1.5. What is the apparent visual
magnitude of the star?
The distance modulus in the presence of absorption is given by
m – M = 5 log d – 5 + A
From Appendix 4 Table 3, we have, for an A0V star,
M = 0.6
B–V = 0
where the magnitudes and absorption are in the visual (V) band. Therefore, the
color excess is 1.5−0=1.5, and so A=3×1.5=4.5. This gives
m = 0.6 + 5 log 100 – 5 + 4.5 = 10.1
for the apparent magnitude.
Problem 3: Imagine that all of the Sun’s mass is concentrated in a thin spherical shell at the Sun’s
radius. Imagine further that the Sun is powered by this mass slowly falling piece by piece into a
black hole at the center of the sphere. If 100% of this energy is radiated away from the surface of
the Sun, calculate the lifetime of the Sun, given its observed luminosity. Comment on your
answer.
If a small bit of mass ∆m falls from a radius r onto a radius R, then the energy
released is given by
GM∆m
GM∆m
GM∆m
∆E = – ------------------ –  – ------------------ ≈ -----------------
R
r
R 
where, in this case, the black hole radius R is much smaller than the Sun’s radius r.
Using the Schwarzschild radius R=2GM/c2 for the black hole, we calculate the
luminosity as
GM
∆m
1 ∆m
∆E
L = ------- = --------------------------- -------- = --- -------- c 2
2 ∆t
∆t
( 2GM ) ⁄ c 2 ∆t
This is precisely the same relationship used when we studied Cygnus X-1 and also
appeared on the second class exam. The rate at which mass is used is therefore
9
– 13
∆m
2L
solar mass/year
-------- = ------ = 8.7 ×10 kg/sec = 1.4 ×10
∆t
c2
Therefore, the lifetime of the sun is 7.3×1012 years. This is nearly a thousand times
longer than the age of the solar system. In other words, we cannot exclude this
model of powering stars on the basis of lifetime.
Problem 2: In the present day universe the size scale R grows according to R ∝ t2/3. Show that in
the very early universe, the scale R grew according to R ∝ t1/2.
In class, we came up with the following expression (or something like it), based on
the concept of escape velocity and the size scale of the universe:
 dR
-------
 dt 
2
8πG
2GM
= ------------- = ----------- ρR 2
3
R
NOTE: In clasa and for homework, you worked in the present day universe, which is
dominated by matter, so that ρ=A/R3 for some constant A. This leads directly to
R 1 ⁄ 2 dR ∝ dt
so
R ∝ t2 ⁄ 3
In the very early universe, radiation dominated and ρ=A/R4 for some constant A:
 dR
-------
 dt 
∫
2
8πG A
= ----------- -----3 R2
8πGA 1 ⁄ 2 t
R dR =  ---------------
dt
 3 
0
0
R
∫
8πGA 1 ⁄ 2
1 2
--- R =  ---------------
t
 3 
2
and this directly imples that R ∝ t1/2.
A different way of doing this problem uses the fact that, as demonstrated on a
homework problem, T ∝ 1/R (at all times) and, as demonstrated in class, for very
early times we have T ∝ 1/t1/2. Putting these two statements together says R ∝ t1/2.
Part II Answer the following four short problems. Each is worth 10 points.
Problem 1: Sketch an HR diagram, plotted as absolute magnitude versus temperature, for the
stars in M5, displayed in image E (labeled AAT 70). Include at least two labeled tick marks on
each axis. Label the primary features of the diagram. Also,indicate the position on the diagram
of a star like our sun.
M5 is a globular cluster. These are collections of very old stars, so the brightest are
yellow and red giants, as opposed to blue main sequence stars. You can see that
clearly by comparing it with image D, a very young star cluster (the Pleiades).
The HR diagrams of globular clusters all look the same and reflect the fact that the
more massive stars have evolved off the main sequence. Figure 16-10A in your
textbook sketches several gobular clusters, and 16-10B in fact displays it for M5
itself. However, the vertical scale must be converted to absolute magnitude (as is
done in 16-10A) and the color index horizontal scale must be converted to temperature, using, for example, equation 11-11b. The sun lies on the main sequence, just
near the “turnoff”, at a temperature near 6000K.
The minimal acceptable sketch for full credit looks something like the following:
Horizontal branch
0
Giant branch
4
Sun
Main sequence
10,000 K
6000 K
__E___23. The object which is farthest from the Earth is shown in
A. Image A
B. Image B
C. Image C
D. Image E
E. Image F
__B___24. The red glow in Image A is most likely
A. Hydrogen Lyman emission
B. Hydrogen Balmer emission
C. Hydrogen 21cm emission
D. Synchrotron radiation
E. Blackbody radiation
__E___25. The oldest collection of stars is shown in
A. Image A
B. Image B
C. Image C
D. Image D
E. Image E
__A___26. New stars are presently being formed in
A. Image A
B. Image B
C. Image C
D. Image D
E. Image E
__D___27. Which of the following is located at the center of Image B:
A. Hot, young star
B. Typical solar mass star
C. White dwarf star
D. Neutron star
E. Black hole
__A___28. The dark patch in image G is
A. An interstellar dust cloud
B. A supermassive black hole
C. Caused by cold interstellar gas
D. Very close to the solar system
E. A region where stars have died
__B___18. In the neighborhood of our solar system, the stellar density is ≈0.1 stars per cubic parsec. If we take this to be a measure of the matter density in the universe, the universe would be
A. Open
B. Closed
C. Flat
D. Spherical
E. We don’t know the Hubble constant well enough to tell
__C___19. The full disk of a typical spiral galaxy covers 15 arcmin on the sky. A reasonable estimate for its distance might be
A. 100 kpc
B. 500 kpc
C. 10 Mpc
D. 100 Mpc
E. 1000 Mpc
__A___20. The goalposts on a football field are separated by about 100 m, but this distance is
constantly expanding according to Hubble’s Law. After one year, this distance increases by about
A. 10 nm
B. 10 µm
C. 10 mm
D. 10 cm
E. 10 m
__C___21. The intrinsic color index B-V of the Sun is approximately
A. -0.2
B.
0.2
C.
0.6
D.
1.0
E.
1.4
The remaining questions in Part I all refer to the sheet of color optical images.
22. Indicate the image (A through G) which best typifies the following classes of objects.
One point for each answer. Put down only one answer on each line, even if you think there
may be more than one possibility.
Globular Cluster
____E_________
HII Region
____A or G_____
Planetary Nebula
____C__________
Spiral Galaxy
____F__________
Supernova Remnant
____B__________
Young star cluster
____D or A______
__A___12. Extremely strong hydrogen lines, surface temperature close to 10,000K, and color
index B-V=0 are all characteristics of stars with spectral type
A. A0
B. B0
C. F0
D. G0
E. K0
__C___13. A star forms on the main sequence with twice the mass of our sun. It’s lifetime on the
main sequence, relative to that for a solar mass star, is
A. The same
B. Half as long
C. 1/4 as long
D. 1/9 as long
E. 1/16 as long
__D___14. The evolutionary path of a star takes it horizontally and to the left on an HR diagram.
Which of the following is true:
A. The radius and temperature are both decreasing
B. The radius and temperature are both increasing
C. The radius is increasing and the temperature is decreasing
D. The radius is decreasing and the temperature is increasing
E. The change in size of the star cannot be determined
__C___15. The Milky Way galaxy has a bulge and disk characterstic of spiral galaxies. These
features are most plainly seen in images taken in
A. Visible light
B. Ultraviolet light
C. Infrared light
D. 21cm radio observations
E. X-rays
__D___16. Compared with population I stars, population II stars of the same mass
A. Are smaller
B. Are more blue
C. Are more luminous
D. Have longer total lifetimes
E. Can generally be found in star forming regions
__E___17. It makes sense for Dallas Cowboys fans to study Astronomy because
A. The Cowboys make their opponents see stars
B. The Cowboys have a lot of stars on their team
C. The Cowboys wear stars on their helmets
D. All of the above
E. It doesn’t make sense for anyone to be a Dallas Cowboys fan
__A___6. Which of the following is not considered to be an active galaxy:
A. M31
B. M87
C. Cygnus A
D. Centaurus A
E. 3C273
__C___7. The final stage in the evolution of our Sun will most likely be a
A. Black hole
B. Neutron star
C. White dwarf
D. Planetary nebula
E. Red giant
__C___8. A blackbody radiates with a temperature T=3K. The signal is strongest for a wavelength near
A. 1 nm
B. 1 µm
C. 1 mm
D. 1 cm
E. 1 m
__E___9. A patch of sky shows a dark region nearly devoid of stars when viewed in visible light.
However, an infrared image shows a small area within the region that is more than ten times as
bright as the Sun. You are most likely observing
A. A nova
B. A pulsar
C. A black hole
D. A planetary nebula
E. A collapsing cloud that will eventually form a star
__E___10. Star #1 is 100 times more luminous than star #2. Star #1 is also 100 times farther
away than star #2. The difference in apparent magnitudes m1-m2 is
A. -5
B. -2.5
C.
0
D. 2.5
E.
5
__A___11. Which of the following is not known to be associated with astrophysical jets:
A. Supernovae
B. Star formation
C. Active galaxies
D. Quasars
E. SS433
Part I: Multiple choice worth two points each.
Give the best choice for each question.
__D__1. Which of the following were the last to appear in the early universe:
A. Protons
B. Neutrons
C. Electrons
D. Hydrogen atoms
E. Helium nuclei
__E___2. Hydrogen Balmer Hα emission is observed at a wavelength of 754.8 nm in a distant
galaxy. The distance to the galaxy is independently found to be 450 Mpc. From this data, we
determine the Hubble constant to be
A. 50 km/sec•Mpc
B. 62 km/sec•Mpc
C. 75 km/sec•Mpc
D. 87 km/sec•Mpc
E. 100 km/sec•Mpc
__A___3. The Hubble Space Telescope has made important recent observational contributions to
our best value for the Hubble constant. These have mainly been from
A. Cepheids measured in galaxies closer that 25 Mpc
B. Cepheids measured in galaxies farther than 25 Mpc
C. Supernova observations in very distant galaxies
D. Verification of the Tully-Fisher relationship
E. Establishing quasars as Active Galactic Nuclei
__C___4. Most of the galaxies in the local group
A. Are similar to the Milky Way
B. Are spiral galaxies
C. Display negative redshift values
D. Are within 100 kpc of each other
E. Are more massive than the Milky Way
__B__5. Saturn’s orbital distance from the Sun is about 10 times that for the Earth. Saturn’s
orbital velocity around the Sun (in km/sec for example) is which of the following, relative to the
orbital velocity for the Earth:
A. 1/10
B. 1/√10
C. The same
D. √10
E. 10
Final Exam
79205 Astronomy
Fall 1996
NAME:
Solution Key
You have three hours to complete this exam. Part I has 27 multiple choice questions, each
worth 2 points, and one one six-point question. Part II has four short problems, each worth 10
points. You are to answer all questions on both parts.
You may use your textbook (Zeilik), workbook (Hoff), and class notes and handouts. You
may not share these resources with another student during the test.
GOOD LUCK!
Score
Part I:
Part II:
________
1.
2.
3.
4.
Total Score: