Homework 2, due in class, Monday, October 11

To receive full credit, you must give the correct
answer *and* show that you understand it. This requires writing your
explanations in full, complete English sentences, clearly labeling all
figures and graphs, showing us how you did arithmetic, and being
explicit about the units of all numbers given. All relevant
mathematical symbols should be explicitly defined. And please use your
best handwriting; if I can't read it, I can't give you credit for it!

In this class you'll see some problems to which
a good answer is basically verbal, some for which a fairly precise
calculation is needed, and some for which ``order-of-magnitude"
accuracy is all that's appropriate. It's usually up to you to figure
out which is called for. As you will see, astronomers and
mathematicians have rather different approaches to numbers. In
particular, it is *wrong* to simply write down all the digits that
your calculator spits out; if the numbers going into a given
calculation are imprecise, the result will have a similar
imprecision. Indeed, do the problems without a calculator whenever
possible.

Feel free to work with your classmates on this homework, but your write-up should be your own. Answer all questions.

**100 points total, plus 10 points extra credit**

**1. The age of the Sun** (40 points)

We saw in class that the Sun shines by burning hydrogen into helium by
the process of thermonuclear fusion. We're going to explore this in
some detail in this problem. First, some useful numbers and facts:

- When 4 protons fuse to make a helium nucleus, 0.7% of the original
mass of the protons is converted to energy via Einstein's famous
equation, (
*c*is of course the speed of light). - The mass of the Sun is grams, and its luminosity is ergs per second. From the fact that the Earth's oceans have remained liquid for roughly 4 billion years, we are confident that the Sun's luminosity has remained close to constant that entire time. The radius of the Sun is cm. The Sun is 75% hydrogen by mass (the rest is helium).
- When a gravitating object shrinks, it releases gravitational
potential energy, just in the same way that when I drop a ball in the
Earth's gravitational field, its gravitational potential energy is
converted to kinetic energy (and then to heat when it hits the ground
and comes to rest). The gravitational potential energy released in
forming a uniform spherical body of mass
*M*and radius*R*is:

where*G*is Newton's Gravitational Constant, .

**a.** (10 points) First, suppose you didn't know about thermonuclear fusion,
and hypothesized that the Sun shone from the gravitational potential
energy of its collapse. Approximating the Sun as a uniform sphere,
calculate how long it could shine at its present rate from this source
of energy; express your answer in years. Compare with the current
lifetime of the Sun, and comment.

**b.** (5 points) The Sun really isn't a uniform sphere; the mass is quite
concentrated to its center. Does this mean that more or less
gravitational potential energy is available? Explain your answer.

**c.** (10 points) Calculate the amount of energy that one gram of hydrogen
would release if it all fused to Helium via thermonuclear fusion.
Express your answer in ergs (1 erg is 1 gram cm^{2}
sec^{-2}). Then
calculate how much mass is converted into energy every second in the
Sun. Express your answer in tons (one metric ton is grams) and
blue whales (a big blue whale weighs about 100 tons).

**d.** (10 points) Only the central 10% of the mass of the Sun is close enough
to the core, i.e., hot and dense enough, for thermonuclear fusion to
take place. With this in mind (and remembering that only 75% of the
mass of the Sun is hydrogen) calculate how long the Sun can shine with
its current luminosity. Express your answer in years. Compare with
the known age of the Sun, and comment.

**e.** (5 points) Finally, something to ponder: it turns out in fact that the
Sun has not stayed at exactly constant luminosity, but has rather
increased in luminosity by something like 30% over the past four
billion years, as the amount of helium in its core has gradually
increased. Speculate on what effect, if any, the smaller luminosity
in the past
might have had on the early Earth, especially on the presence of
liquid water.

**2. Temperatures of planets** (35 points)

In class, we found that the equilibrium temperature of
the Earth can be calculated by assuming that it absorbs all the energy
it receives from the Sun, and re-radiates it as a blackbody. In this
problem, you will derive the surface temperature of 7 of the 136
planets that have been discovered to date around other stars.
The data for the planets can be found on the web site: `
http://exoplanets.org/planet_table.txt` (and check out the home
page from which this comes, `http://exoplanets.org`, for all the
latest on these discoveries). The planets you should consider are:

- 51 Pegasus (called 51 Peg in the table; this was the first extrasolar planet discovered).
- HD 209458 (this planet passes directly in front its parent star as seen by us, and thus briefly eclipses it once per orbit).
- 55 Cancri b, c, and d (three planets around a single star).
- HD 28185.
- GJ 486 (this one is so new as to not be on their table: the star has a mass of and the planet orbits at 0.028 AU, with an eccentricity of 0.12. It is one of the lowest mass planets yet found).

**a.** (20 points) The radius of a star on the main sequence is
approximately proportional to its mass, while its surface
temperature is approximately proportional to the square root of its
mass. Thus in each case, scaling from what you know about the Sun,
you can calculate the surface temperature and radius of the parent
star. Using the data on the web site above, calculate the
equilibrium surface temperature for
each of the planets listed above. You may ignore the effects of
albedo and greenhouse effect on these planets (after all, we know
absolutely nothing about their atmospheres!).

**b.** (5 points) Is it possible that liquid water can exist on the surface of any of
these planets, assuming that they have an atmospheric pressure similar
to that of the surface of the Earth? Discuss.

**c.** (10 points) Some of the planets have very eccentric orbits (i.e., with
eccentricity more than 0.2). Discuss what effect this might have on
any life that might exist on them, and the presence of liquid
water. *Hint: The eccentricity of an orbit refers to how squashed
it is. A perfect circle has an eccentricity of 0; the most flattened
ellipse has an eccentricity of 1.*

**3. Lifetimes of stars** (25 points)

In this problem, you will use concepts from both of the previous two
problems, to calculate how the lifetimes of stars depend on their
masses.

**a.** (10 points) We saw in Problem 2 how the surface temperature and radii of
main sequence stars depend on their mass. Assuming that the stars
radiate as black bodies, calculate how the luminosities of stars
depend on their mass. Your answer will have the form:

you have to determine .

**b.** (5 points) In Problem 1, the lifetime of the Sun was determined by its
mass and luminosity. Calculate how the lifetime of a star of mass *M*
depends on its mass.

**c.** (10 points) Now let's plug in numbers. You know the lifetime of the Sun
(you calculated it in Problem 1!). Determine the luminosity (relative
to that of the Sun) and lifetime (expressed in years) for:

- an O star, 60
- a B star, 8
- an A star, 2
- a G star, 1
- an early M star, 0.5
- a late M star, 0.1

**4. Reflection vs. Black-body radiation** (10 points extra credit)

Go back to Homework 1, problem 3c. Consider that the planet in
question there has an albedo of 60%, like the Earth. Calculate the
ratio of the brightness it has because it is radiating like a
blackbody, to the brightness it has because it is reflecting light
from its parent star. Describe your analysis carefully. *Hint:
This problem is made more complicated by the fact that the reflected
light is not sent uniformly in all directions. So you will need to
consider from which direction you are viewing the planet. You need
not solve the general problem, but choose a direction for viewing that
makes the problem easy, and use this to calculate.*

Mon Sep 27 09:42:18 EDT 2004