Astronomy 101 Problem Set #6 Solutions |

** This Problem Set is due by 1:00 pm on Thursday, 13 October**

**Problem #1:** Pluto's moon Charon has a
radius of 635 km, and a mass of 1.8 x 10^{21} kg. What is
its average density?

Based on this result, what kind of material do you think Charon is
made of? (*Hint: You can find the densities of various materials on
the 30 September web page.*)

**Solution:** OK, we've got mass and radius,
which is pretty much all we'll need to a density calculation. Recall
that:

density = mass/volume

The volume of Charon is:

volume = 4/3 pi r^{3} = 4/3 x 3.14 x (635 km)^{3} =
1.07 x 10^{9} km^{3}

Note that the units of the radius were cubed along with the value, so our volume has units of cubic kilometers.

Now we can calculate the density:

density = mass/volume

density = 1.8 x 10^{21} kg / 1.07 x 10^{9}
km^{3}

density = **1.67 x 10 ^{12} kg/km^{3}**

Now this is a perfectly valid answer with perfectly valid units. Unfortunately, it's not in the same units as the table I gave you on the 30 September web page, nor is it in the units given in the back of the book. If you're interested in comparing this density to the ones I gave you, you'll need to convert:

- 1.67 x 10
^{12}kg/km^{3}x (1 km/1000 m)^{3} - = 1.67 x 10
^{12}x 1/10^{9}kg/m^{3} - =
**1.67 x 10**^{3}kg/m^{3}, or 1670 kg/m^{3}

Now this density lies between that of rock and water, so it might be reasonable to assume that Charon is made from a mixture of these. Really, what the average density tells you is that it's pretty unlikely that Charon has a large iron core, or that it's entirely rock and denser materials.

Charon is in fact an amalgam of rock and ice (it's really far away from the Sun, so the water is in the form of ice).

**Problem #2:** A 60-watt light bulb emits 60
joules/sec of energy ( 1 Joule/sec == 1 watt). Pretend for a moment
that all of this energy is emitted in the form of photons with
wavelength of 600 nm (this isn't true, of course -- as a blackbody, a
light bulb emits photons of a wide range in wavelengths). Calculate
how many photons per second are emitted by such a light bulb.

**Solution:** This problem asks you to
calculate how many photons per second are emitted from a 60-watt
lightbulb. Since 60 watts is 60 Joules per second, we know that we
need enough photons to carry 60 Joules of energy each second.

So how much energy does one 600nm photon carry? We'll need to use the relation between energy and wavelength

- E
_{photon}= h c / lambda - = (6.63 x 10
^{-34}J s) x (3.00 x 10^{8}m/s)/ 600 x 10^{-9}m - = 3.32 x 10
^{-19}J

- 60 Joules = N x 3.32 x 10
^{-19}J

- N = 60 J / 3.32 x 10
^{-19}J - = 1.8 x 10
^{20}

**Problem #3:** Read the *Executive
Summary* of the Department of Energy's Report, Emissions
of Greenhouse Gases in the United States 2003 and answer the
following questions:

**a)** How many metric tons of anthropogenic carbon dioxide were emitted
from US sources in 2003?

**b)** What is the percentage increase in the US-generated carbon
dioxide emissions since 1990?

**c)** Which sector of the US economy (e.g., residential,
commercial, industrial, or transportation) is responsible for the
largest fraction of 2003 carbon dioxide emissions?

**d)** US emissions of which greenhouse gases actually
*decreased* between 2002 and 2003?

**e)** Based on the information in **Table 3** of Chapter
1 of this report, by what amount would
we have to reduce yearly anthropogenic carbon dioxide emissions to
result in no increase in carbon dioxide in our atmosphere?

**f)** By what fraction of current worldwide human-made emissions
levels would we have to reduce to achieve this goal?

**Solution:** This problem should have been
fairly straightforward, provided that you spent the time reading the
report. Skimming, on the other hand, likely didn't work.

**a) 5870.2 million metric tons of gas** (from the first sentence
in the "Carbon Dioxide" section). Note that 6935.7 million metric tons
listed in the first sentence of the Executive Summary was the "carbon
dioxide equivalent," which is a quantity that also includes the
effects of emissions from other greenhouse gases such as methane,
nitrous oxide, and others.

**b) a 17.6 percent increase since 1990** (from the second
paragraph of the "Carbon Dioxide" section).

**c) commercial** (from Figure ES3). I was surprised to learn that
industrial emissions contributed so little in relative terms.

**d) nitrous oxide, hydrofluorocarbons (HFCs), perfluorocarbons
(PFCs), and Sulfur Hexafluoride(SF _{6})** (from first
paragraph in the "overview").

**e)** Table 3 of Chapter 1 (*not* Table 3 of the Executive
Summary) shows that the total human made carbon dioxide
emissions total 23100 million metric tons. It also shows that the
total natural emissions produce 770000 million metric tons and that
natural absorption accounts for 781400 million metric tons. Given the
imbalance between natural emission and absorption, we can add 781400
- 770000 = 11400 million metric tons into the atmosphere and it will
be absorbed by the ecosystem. Thus, if we were to reduce our emissions
to that level, the carbon dioxide concentration would remain constant.

How much would we need to reduce? We're currently emitting 23100
million metric tons, and the ecosystem can absorb 11400 million metric
tons, so it looks like we'll have to reduce by 23100 - 11400 = **11700
million metric tons of carbon dioxide.**

**f)** The fraction is just the amount we have to reduce by over
the total amount we're currently emitting:

11700/23100 = **0.506, or 51%**.

That means we would have to halve our carbon dioxide emissions to
reach the goal of not increasing the CO_{2} in the
atmosphere. Perhaps this gives you a sense of the magnitude of this
problem. It won't be solved with energy efficient cars and lower
winter thermostat settings. To really halve our CO_{2}
production will require a massive change in how we live.
(One last note: the data in Table 3 you used for parts **e)** and
**f)** is actually based on emissions in the 1990s; today, nearly a
decade later, the problem is even more severe because we've actually
increased the production of anthropogenic carbon dioxide. This means
that the numbers you calculated for this part of the problem
*underestimate* the amount by which we would need to cut back now.)