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 cm² 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.