Derivation of Radiation Laws

I will be deriving the Stefan-Boltzmann Law of Radiation:

as well as the Planck Law of Radiation:

I've already derived the equation of the thermal average energy in a mode (see previous post), so now to find a description of the mode with which we are going to determine the energy,  let us take a look at these energy levels within a cavity in the form of a hollow conducting cube of edge L. From E&M, we know that there are three field components, (x,y,z), and the corresponding fields within the cavity are of the form:

We know that the divergence of the field must be zero, and as such the fields are dependent upon each other:

Defining the polarization direction to be the direction of E-naught, we can choose two directions that are both perpendicular to n-hat to satisfy the above equation. 

Now substituting one of our field vectors into the laplacian wave equation we can find the frequency of the mode in terms of our triplet of integers within the n-hat vector (n-sub x,y,z)

and if we define a n to be:

then the frequencies of the modes are of the form:

and thus our total energy of all the photons within the cavity is just a sum of the average energy per mode across all modes:

Positive values for the x,y,z components of n are sufficient to describe all modes of the field vectors, therefore if we replace the sum by an integral over the volume dn, we would only be interested in the positive octant of the resulting volume, multiplied by a factor of two from our previously identified polarization direction; since there are two sets of possible modes for each nsub(xyz):

The last equation is an expression of the Stefan-Boltzmann law of radiation.

Now in order to derive the Planck radiation law, it is necessary to define the term "spectral density." Spectral density ( u sub omega) is the energy per unit volume per unit frequency range. We can find it using one of the above equations:

This law gives the frequency distribution of thermal radiation. Now it is useful to introduce the term "black body." An object is said to radiate as a black body if the radiation emission is characteristic of a thermal equilibrium distribution. Going back to our cavity problem, if this cavity were to have a small hole in it, that hole would radiate as a black body. 

In effort to manipulate the previous expression of the Stefan-Boltzman law, it is necessary now to introduce the energy flux density term J sub U. Energy flux density is defined as the rate of energy emission per unit area.

where the x(g) is a geometrical factor that arises form the angular distribution of the radiant energy flux.  The ratio of the amount of flux through the solid angle dΩ is the ratio dΩ/4π, where 4π is the total solid angle Ω. Also, since blackbody objects follow Lambert's cosine law in that the radiant intensity is directly proportional to the cosine of the angle theta away from the solid angle, we can find the geometrical factor and derive the Stefan-Boltzman constant:

Sources: Thermal Physics 2nd edition by Charles Kittel and Herbert Kroemer.

Derivation of the Planck Distribution function

It's about to get real, time for some fun ha ha.
The Planck distribution function: 

This function can be used to describe the emission spectra of stars and is used to derive Planck's Law and Stefan-Boltzmann law, which I will be doing in a later blog post. It was the first application of Quantum Thermal physics. It is very elegant in that it arose from the simple idea of the quantization of energy.

A "mode" is a particular state in which a system can exist. For now, it will describe a particular oscillation pattern in a cavity, or frequency 

As we know now, the energy of the mode is given be the quantized number of photons (s) within the mode 

It is useful now to introduce the Partition function, which is the sum over all states of the Boltzmann factor:

Where tau is the fundamental temperature of the system and is equivalent to the Boltzmann constant times the Absolute temperature in Kelvin.

However, this summation is of the form:

where x is

 and so since we know that this summation converges, we can find that the value of Z is:

The probability of finding the system in a given state is:

With this we can find <s>, the thermal average value of the number of photons in a given state.

Using the substitution of:

We can find that:

and using the same trick we did with the partition function this is then:

Plugging this summation back into our thermal average value equation:

Which is equivalent to:

This is the more conventional form of the Planck Distribution Function and is applicable to any wave field with energy of the form :

Since we now know that the thermal average value of photons in a single mode of a system is given by Planck's distribution function, we can now find the thermal average energy in each mode:

Sources: Thermal Physics 2nd edition by Charles Kittel and Herbert Kroemer.

Here's a fun app.

For those of you with Droid phones, there's a cool app that lets you see the sky through your phone. Who needs eyes anymore?

Link: Google Sky Map

It's a pretty neat app to fiddle with if you can't see the stars directly. It allows you to see comets as they make their way through space, approaching meteoroids and meteor shower events, and it shows you the location of the planets as well as many other stellar bodies. The coolest thing though is the time travel function that the app has that lets you set a date in the past or future and you can see what the sky looked like or will look like at that time.

Unfortunately, this app is only on for Android Phones. "Sky Walk" seems to be similar app for the iPhone and costs about $3.00.
Here's a link to the iTunes app store: Sky Walk

What does an Astronomer do?

They study the universe, or so I'm led to believe by the lasers and telescopes pointed skyward.

The question posed is one with many answers. Not being one who has looked very hard at the cosmos, I can't really answer this to the satisfaction of someone seriously asking such a question. However, I can indeed answer one very similar: what is there for an astronomer to do?

An Astronomer can do many things, things such as
Study the inner machinery of the stars, namely fusion and the creation of atoms heavier than hydrogen and helium.
Study the surface of these stars, the fields created by the swirling masses, and the particles ejected out and away from the star.
Look for new planets, new asteroids, new comets, new star systems, new anything really. Space whales. I'm sure they're out there somewhere.
Destroy years worth of Mn(D)emonic indoctrination by tyrannical K-12 teachers. No Mother, I don't care how Educated you are, we're apparently not allowed to have any Pizza anymore.
Study the age of the Universe.
Look for residue from the big bang
Examine the expansion of space-time and the universe.
Study the weird things that light up our sky with light we can't even see like quasars and gamma-ray bursts from supernova.
Try to look at black holes and ponder what is really going on inside the event horizon.

Astronomers can also try to answer interesting questions:

Why the night sky is dark in a universe filled with trillions and trillions of stars, also known as Olber's paradox, illustrated below.

http://abyss.uoregon.edu/~js/ast123/lectures/lec15.html

also:
Why comets have tails.
Whether or not we are alone in the universe.
What causes the Northern lights.
What shapes solar systems and galaxies.

They can do all theses things by making observations of the night sky. Using extremely powerful telescopes both on the earth and above it, they can observe slight movements of stellar objects, perhaps allowing them to determine the gravitational pull or other effects of the neighbors of the objects. Using lasers they can determine atmospheric distortions and adjust their telescopes to account for them, effectively increasing the clarity of the image. Or they could use telescopes in space that don't have to deal with problems involving atmospheric distortion.

There is quite a lot for an astronomer do to. What they actually depends on the person, I'd imagine.

Welcome!

Welcome to my first blog post, it's really just a test of the structure of this site.