NuSTAR

The Nuclear Spectroscopic Telescope Array is the first telescope of its kind to be sent into space. It is unique in that it has actual focusing optics to observe the high energy x-ray (6-79 KeV) spectrum.


NuSTAR overview

The purpose of this mission is to basically allow for a wider and clearer range of detection. It give us a wide range of information, such as the distribution of black holes throughout the universe, data on the evolution of various structures in the universe, and information about super-massive black-holes and supernovae. The number of possible uses of this data is only limited by the time and interest of astronomers.

The resolution difference is quite profound as seen below





The Rocket was set to launch sometime this month, but was delayed so that software could be fine tuned to ensure smooth communication between the rocket and the launch vehicle's flight computer. The launch vehicle is actually a pegasus rocket, brought to about 40k feet altitude by an airplane, specifically an L-1011 aircraft named Stargazer. The rocket is then deployed and activates, bringing the array to its desired orbit before separating.





When the array reaches the desired orbit it unfolds a 33 foot extension arm, separating the optics from the detectors.

Let's get to leaping.

How long has it been since we've personally explored the stars? Too long I say. Too long have we been bogged down by other "important" projects that "need" funding: Green technology, feeding the poor, curing cancer, childhood diabetes...we need to focus on what's really important: a moon base. How are we to know whether or not the decepticons are going to start invading us tomorrow while we try to search for some guy named Kony today? How are we supposed to make nachos without all that Cheese? I ask what is more important to your daily life, the price of tea in china or being a member of United States of the MOON?

Finally showing some initiative, NASA has started humanity along the path that we fell off some 50 years ago: colonizing the stars. With the launch of the Orion Exploration First Flight-1 in 2014, NASA will obtain crucial data to this end, mainly testing the technology necessary for further manned space-flights into deep space. Here is a video showing the entirety of the mission that will take place.

Why aren't we giving them more money yet?

Why are we allowing our representatives to cut funding of our future. We all saw the graphs on sustainability in class. We don't have much time left to get off of this dying planet. Why clean up a trash bin I always say, even if you're the most conservative person in the world some uncaring, cancer-ridden, poor, diabetic is gonna come along and steal all your money and throw his trash all over your house. Do YOU want a messy house or a brand new house with a nice view of the world?

The Equipartition Theorem

I'll be deriving this with a little help from the partition function and a particle in a box.
First let's find out what the partition function is for this box of volume:





The wave function for this configuration is of the form:

To find the eigenstates of the Free-Particle Hamiltonian, that is the allowed energy levels of this configuration, we simply use the Free-particle wave equation:

The partition function is given by:

Where the spacing between modes is relatively small compared to tau, this can be written in integral form and calculated:

Now we define the quantum concentration, the concentration associated with one atom in a cube of side equal to the thermal average de Broglie wavelength:

Where n_c is just the concentration of particles. As ideal gas is defined as a gas of noninteracting atoms in the classical regime, that is Z1 >> 1.

Now that we have the partition function, we can go about calculating our thermal average energy similarly to how we did in a previous post.

It is useful now to define the Helmholtz free energy function, which is just how much energy is free to use from a closed system at constant temperature and volume; how obtainable energy is with regards to entropy:







Where sigma is the entropy of the system, which is just the logarithm of the number of states accessible to the system.

From differential relations of F and the thermodynamic identities relating entropy, thermal average energy, pressure, and volume it can be shown that:











Using logarithmic rules and our latest formula for Z in terms of tau, we can note that Log(Z) has only one term involving tau, namely: -3/2log(1/tau). We have thus found the equipartition theorem, the energy per atom of an ideal gas:

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

The Blue Marble

New images have been recently produced from Nasa's Visible Infrared Imaging Radiometer Suite (VIIRS) housed by the Suomi National Polar-orbiting Partnership satellite (Suomi NPP, named after Verner Suomi)  of the Earth as seen from space. These two images are the highest resolution pictures of the Earth to date and detail both the western and eastern hemispheres.

VIIRS Eastern Hemisphere Image
Unfortunately, the Pyramids are covered by clouds.

 Information on how this picture was constructed is in the link.

Credit Nasa/Noaa

Full Size Image

VIIRS Western Hemishphere Image,
If you look closely, you can see the Grand Canyon and effects of the Hoover dam

Credit: NASA/NOAA/GSFC/SuomiNPP/VIIRS/Norman Kuring

Full Size Image

The purpose of Suomi isn't just to take pretty pictures though; it is to mainly understand, measure, and predict long-term climate change and short-term weather patterns. It does this by collecting measurements of clouds, ocean color, land and sea temperatures, and the Earth's reflection coefficient or Albedo. The amount of data that this satellite provides to the environmental sciences is vast. The information can be used to model vegetation dynamics, to model regional to global climate, to understand the cryosphere, and to study the energy and water balance of the earth. More information can be found at Nasa's page: Suomi NPP.

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