Phys 312 - Assignment 11

Due Tuesday, April 27, 1999.

1.

Solar temperatures. Here we'll try to estimate some relevant temperatures in the sun. Assume you start with an initially diffuse cloud of hydrogen and helium atoms (initially at rest), which subsequently collapses under its gravitational attraction.

(a)

Use dimensional analysis to show that the total energy released in the gravitational collapse is about

$$E_{\odot }={\frac{GM_{\odot }^{2}}{R_{\odot }}}.$$ (1)

Calculate this energy.

(b)

How does this energy compare to the total energy that could be released by (hypothetical) chemical reactions in the sun? To the total energy that could be released by converting all the hydrogen into helium? (You can assume that the cloud has the "primitive" abundance of helium--it does not make much difference as long as you have mostly hydrogen).

(c)

Assuming that all of the energy (1) is converted into heat (a rather dubious assumption), estimate the interior temperature of the sun. Compare with the accepted value of the temperature at the center of the sun (look it up).

(d)

The intensity of solar radiation has a peak at a wavelength of 490 nm. What is the surface temperature of the sun? As you go from the surface of the sun toward to center, what is the approximate temperature gradient dT/dx? How can this temperature gradient be maintained?

2.

List three radioactive nuclides that naturally occur (in easily detectable amounts) on Earth today. What are the half -lives of these nuclides? How did they originate? At least one of them should have a half-life of less than a thousand years. Explain how is it possible to find nuclides with a half-life so much shorter than the age of the Earth.

3.

Particle decays.

(a)

Name the particles involved in the process $n\rightarrow p+e+\bar{\nu}_{e}.$ Which are hadrons, mesons, baryons, leptons? Complete the following table

$$\begin{array} {lllll} & n & p & e & \bar{\nu}_{e} \\ & &... ...ber} & & & & \\ {\rm Electron\;number} & & & & \end{array}$$

(b)

Some of the particles in this process are made out of quarks. Describe the process in terms of the transformation of the constituent quarks.

(c)

Why are these decays forbidden?

i.

$n\rightarrow \ \pi ^{+}+e^{-}+\bar{\nu}_{e}$

ii.

$n\rightarrow e^{+}+e^{-}$

iii.

$n\rightarrow p+\mu ^{-}+\bar{\nu}_{\mu }$

iv.

$n\rightarrow \bar{p}+e^{+}+\nu _{e}$

4.

Smoke detectors.

(a)

Read about them in Bloomfield. The one we looked at in class uses an ²⁴¹Am source with an activity of 1 $\mu$Ci = 37kBq. Find out something more about ²⁴¹Am. How is it manufactured? Is it cheap and plentiful? How does it decay, mostly?

(b)

How many decays per second is 1 $\mu$Ci? How much ²⁴¹Am, in grams, gives out 1 $\mu$Ci? After one year, how much is the activity of the decay product(s), compared to that of the remaining americium? (Is it significant?) Why is it important that the isotope used in a smoke detector not decay into a radioactive gas such as radon?

(c)

If you kept 1 $\mu$Ci of ²⁴¹Am in your mouth for an hour, would you get a bad dose of radiation? You can find out in Melissinos what is the permissible quantity of an ingested isotope, but here we have to worry about a local burn, rather than a whole-body effect.

5.

Expansion of the universe

(a)

Does the expansion of the universe imply that the earth is getting bigger? The solar system? Our galaxy? The distance from our galaxy to the Virgo cluster?

(b)

If Hubble's constant is 70 Km/sec/Mpc, how far are galaxies that are moving away from us at one third of the speed of light? At even greater distances, the galaxies eventually move away from us at a speed exceeding the speed of light, if Hubble's law is valid. Is this possible?

(c)

If Hubble's constant is 70 Km/sec/Mpc, how old is the universe in the simplest model in which the "radius" increases in proportion to t^2/3? Justify your answer.