Astronomy
1st Edition
ISBN: 9781938168284
Author: Andrew Fraknoi; David Morrison; Sidney C. Wolff
Publisher: OpenStax
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Chapter 3, Problem 14E
To calculate the
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ROTATIONAL DYNAMICS
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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]
Chapter 3 Solutions
Astronomy
Ch. 3 - State Kepler’s three laws in your own words.Ch. 3 - Why did Kepler need Tycho Brahe’s data to...Ch. 3 - Which has more mass: an armful of feathers or an...Ch. 3 - Explain how Kepler was able to find a relationship...Ch. 3 - Write out Newton’s three laws of motion in terms...Ch. 3 - Which major planet has the largest . . . A....Ch. 3 - Why do we say that Neptune was the first planet to...Ch. 3 - Why was Brahe reluctant to provide Kepler with all...Ch. 3 - According to Kepler’s second law, where in a...Ch. 3 - The gas pedal, the brakes, and the steering wheel...
Ch. 3 - Explain how a rocket can propel itself using...Ch. 3 - A certain material has a mass of 565 g while...Ch. 3 - To calculate the momentum of an object, which...Ch. 3 - To calculate the angular momentum of an object,...Ch. 3 - What was the great insight Newton had regarding...Ch. 3 - Which of these properties of an object best...Ch. 3 - Pluto’s orbit is more eccentric than any of the...Ch. 3 - Why is Tycho Brahe often called “the greatest...Ch. 3 - Is it possible to escape the force of gravity by...Ch. 3 - What is the momentum of an object whose velocity...Ch. 3 - Evil space aliens drop you and your fellow...Ch. 3 - A body moves in a perfectly circular path at...Ch. 3 - As friction with our atmosphere causes a satellite...Ch. 3 - Use a history book, an encyclopedia, or the...Ch. 3 - Two asteroids begin to gravitationally attract one...Ch. 3 - How does the mass of an astronaut change when she...Ch. 3 - If there is gravity where the International Space...Ch. 3 - Compare the density, weight, mass, and volume of a...Ch. 3 - If identical spacecraft were orbiting Mars and...Ch. 3 - By what factor would a person’s weight be...Ch. 3 - Suppose astronomers find an earthlike planet that...Ch. 3 - What is the semimajor axis of a circle of diameter...Ch. 3 - If 24 g of material fills a cube 2 cm on a side,...Ch. 3 - If 128 g of material is in the shape of a brick 2...Ch. 3 - If the major axis of an ellipse is 16 cm, what is...Ch. 3 - What is the average distance from the Sun (in...Ch. 3 - What is the average distance from the Sun (in...Ch. 3 - In 1996, astronomers discovered an icy object...
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