Foundations of Astronomy (MindTap Course List)
14th Edition
ISBN: 9781337399920
Author: Michael A. Seeds, Dana Backman
Publisher: Cengage Learning
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Textbook Question
Chapter 3, Problem 22RQ
How do we know? Some people think science is like a grinder that cranks data into hypotheses. What would you tell them about the need for scientists to be creative and imaginative?
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
Chapter 3 Solutions
Foundations of Astronomy (MindTap Course List)
Ch. 3 - Why are most of the constellations that were...Ch. 3 - Prob. 2RQCh. 3 - Which is the asterism and which is the...Ch. 3 - Prob. 4RQCh. 3 - Prob. 5RQCh. 3 - What does the word apparent mean in apparent...Ch. 3 - Prob. 7RQCh. 3 - Prob. 8RQCh. 3 - Prob. 9RQCh. 3 - Could a solar powered spacecraft generate any...
Ch. 3 - If a lunar eclipse occurred at midnight, where in...Ch. 3 - If Earth had no atmosphere, what color would the...Ch. 3 - If the Moon orbited Earth from North Pole to South...Ch. 3 - Why do solar eclipses happen only at new moon? Why...Ch. 3 - Why isnt the corona visible during partial or...Ch. 3 - Which has the larger angular diameter in the...Ch. 3 - What is the angular diameter of the Moon in the...Ch. 3 - Why cant the Moon be eclipsed when it is halfway...Ch. 3 - Why are solar eclipses separated by one Saros...Ch. 3 - How could Thales of Miletus have predicted the...Ch. 3 - Will an eclipse occur in February 2025? In July...Ch. 3 - How do we know? Some people think science is like...Ch. 3 - Pretend the Moons orbit around Earth is a perfect...Ch. 3 - Identify the phases of the Moon if on March 20 the...Ch. 3 - Identify the phases of the Moon if at sunset in...Ch. 3 - What fraction of the Moons surface area is the far...Ch. 3 - About how many days must elapse between...Ch. 3 - Tonight you see a waning crescent in the night...Ch. 3 - If on March 1 the Moon is full and is near Famous...Ch. 3 - How many times larger than the Moon is the...Ch. 3 - Use the small-angle formula to calculate the...Ch. 3 - Prob. 10PCh. 3 - At perigee, the Moon is closer than average by...Ch. 3 - Examine the list of upcoming lunar eclipses in...Ch. 3 - Prob. 13PCh. 3 - If a solar eclipse occurs on October 3: (a) Why...Ch. 3 - A total eclipse of the Sun was visible from Canada...Ch. 3 - Prob. 16PCh. 3 - When will the eclipse seasons occur during the...Ch. 3 - Examine Figure 3-16. List the letter S for each...Ch. 3 - Arrange the following in order of increasing...Ch. 3 - Prob. 2SOPCh. 3 - Look at the Chapter 2 Concept Art: The Sky Around...Ch. 3 - To take the photos that are combined on the...Ch. 3 - Look at the Chapter 3 Concept Art: The Phases of...Ch. 3 - Look at the Chapter 3 Concept Art: The Phases of...Ch. 3 - Use the photos in Figure 3-1 as evidence to show...Ch. 3 - Prob. 6LTLCh. 3 - Prob. 7LTLCh. 3 - Prob. 8LTLCh. 3 - What evidence of the Saros cycle can you see in...Ch. 3 - The accompanying cartoon shows a crescent moon....Ch. 3 - This photo shows the annular eclipse of May 30,...
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