Download Homework 6

ASTR100 – Fall 2009 (McGaugh)
Homework #6
(due Thursday, December 10th)
***SOLUTIONS***
Problems: Ch. 15 #36, 59; Ch. 16: #33; Ch. 17: #30; Ch. 18: #47, 48; EC: Ch. 16, #49
Chapter 15
36. (Quick Quiz: Choose the best answer to each of the following. Explain your reasoning with
one or more complete sentences.) Which kind of object is the best standard candle for
measuring distances to extremely distant galaxies?
(a) white dwarf
(b) Cepheid variable star
(c) white dwarf supernova (type 1a SN)
The first choice, the white dwarf, is much to small to be visible in a nearby galaxy, let alone a
distant galaxy. The second, the Cepheid variable, is a standard candle, but only works for
nearby galaxies. White dwarf supernovas are extremely bright and their luminosity is almost
always the same (it occurs when the white dwarf reaches the Chandrasekhar limit), so they are
an ideal standard candle for distant galaxies.
59. (Quantitative Problem: Be sure to show all calculations clearly and state your final answers
in complete sentences.) Distances from Hubble's Law. Imagine that you have obtained spectra
for several galaxies and have measured the redshift of each galaxy to determine its speed away
from us. Here are your results:
● Galaxy 1: Speed away from us is 15,000 km/s
● Galaxy 2: Speed away from us is 20,000 km/s
● Galaxy 3: Speed away from us is 25,000 km/s
Estimate the distance to each galaxy from Hubble's law. Assume that H0 = 22 km/s/Mly.
V = H0 * D, therefore D = V / H0 .
D1 = (15000 km/s) / (22 km/s/Mly) = 681.8 Mly
D2 = (20000 km/s) / (22 km/s/Mly) = 901.1 Mly
D3 = (25000 km/s) / (22 km/s/Mly) = 1136 Mly = 1.14 * 109 lightyears
The first galaxy is almost 700 million lightyears away, the second galaxy is just over 900
million lightyears away, and the third galaxy is over one billion lightyears away. Note that the
further a galaxy is from us, the faster it is travelling away from us.
1
Chapter 16
33. (Quick Quiz: Choose the best answer to each of the following. Explain your reasoning with
one or more complete sentences.) Based on current evidence, which of the following is
considered a likely candidate for the majority of the dark matter in galaxies?
(a) subatomic particles that we have not yet detected
(b) swarms of dim, red stars
(c) supermassive black holes
Scientists have determined that non-baryonic matter must dominate the dark matter in the
galaxy. The second choice is an example of baryonic dark matter called MACHOs. They
contribute to the total dark matter but are not the majority. The first choice, however, gives an
example of WIMPs, which are non-baryonic and the answer to this question. Supermassive
black holes are too rare to be found in enough quantity to be the majority of dark matter.
Chapter 17
30. (Quick Quiz: Choose the best answer to each of the following. Explain your reasoning with
one or more complete sentences.) Which of the following does not provide strong evidence for
the Big Bang theory?
(a) observations of the cosmic microwave background
(b) observations of the amount of Hydrogen in the universe
(c) observations of the ratio of helium to hydrogen in the universe
The first choice is the leftover radiation from the event itself, so it provides good evidence for
the Big Bang. The second choice alone is not good evidence because there is nothing to
compare it to. The last choice, however, is good evidence because it is comparing two separate
amounts. Therefore, the amount of hydrogen is just a number and provides no insight.
Chapter 18
47. (Quantitative Problem: Be sure to show all calculations clearly and state your final answers
in complete sentences.) Cruise Ship Energy. Suppose we have a spaceship about the size of a
typical ocean cruise ship today, which means it has a mass of about 100 million kg, and we
want to accelerate the ship to 10% of the speed of light.
a. How much energy would be required? (Hint: you can find the answer simply by
calculating the kinetic energy of the ship when it reaches its cruising speed; because 10% of the
speed of light is still small compared to the speed of light, you can use the formula kinetic
energy = (½) * m * v2.)
b. How does your answer compare to total worldwide energy use at present, which is 5 *
1022 joules per year?
c. The typical cost of energy today is roughly 5 cents per 1 million joules. Using this
price, how much would it cost to generate the energy needed by this spaceship?
2
a. KE = (½)mv2 = (½)(108 kg)(0.1*3*108 m/s)2 = 4.5 * 1022 Joules
b. world energy = WE = 5 * 1022 Joules/year
Ratio: KE/WE = (4.5*1022 J)/(5*1022 J/yr) = 0.9 years
Just from looking, we can see that the two values are about the same. In fact,
it would take 0.9 years' worth of the world's energy to accelerate the spaceship.
c. The cost is 5 cents / 106 J.
To accelerate the spaceship to 0.1c:
(4.5*1022 J)*(5 cents/106 J) = 2.25 * 1017 cents.
It would cost 2.25*1015 dollars!
48. (Quantitative Problem: Be sure to show all calculations clearly and state your final answers
in complete sentences.) Matter-Antimatter Engine. Consider the spaceship from Problem 47.
Suppose you want to generate the energy to get it to cruising speed using matter-antimatter
annihilation. How much antimatter would you need to produce and take on the ship? (Hint,
remember that when matter and antimatter meet, they turn all their mass into energy equivalent
to mc2.)
E = mc2, where m = matter + antimatter, i.e. amount of antimatter = m/2
m/2 = E/(2c2) = (4.5 * 1022 J) / [2*(3*108 m/s)2] = 2.5 * 105 kg
You would need 2.5 * 105 kg of antimatter to generate enough energy to bring the ship to its
intended cruising speed. This is one quarter of a million kilograms (250 metric tons)!
102
Extra Credit: Chapter 16
49. Mass from rotation curve. Study the rotation curve for the spiral galaxy NGC 7541, which
is shown in Figure 16.4.
a. Use the orbital velocity law from cosmic calculations 14.1 to determine the mass (in
solar masses) of NGC 7541 enclosed within a radius of 30,000 lightyears from its center (hint:
1 lightyear = 9.461 * 1015 m).
b. Use the orbital velocity law from cosmic calculations 14.1 to determine the mass of
NGC 7541 enclosed within a radius of 60,000 lightyears from its center.
c. Based on your answers from parts (a) and (b), what can you conclude about the
distribution of mass in this galaxy?
Orbital velocity law is Mr = (rv2)/G
a. ra = 30000 ly * 9.461*1015 m/ly = 2.84 * 1020 m and va ≅ 200 km/s = 2.0 * 105 m/s
Mr = (rava2)/G = (2.84 * 1020 m)(2 * 105 m/s)2/(6.67*10-11 m3/s2kg) = 1.7 * 1041 kg
b. ra = 60000 ly * 9.461*1015 m/ly = 5.68 * 1020 m and va ≅ 230 km/s = 2.3 * 105 m/s
Mr = (rava2)/G = (5.68 * 1020 m)(2.3*105 m/s)2/(6.67*10-11 m3/s2kg) = 4.5 * 1041 kg
c. Because at twice the radius, there is more than twice the mass, the mass distribution in
the galaxy is not concentrated in the central bulge as we might assume, but is rather
spread out equally at all radii. This must be attributed to dark matter.
3