In my hunt for a grand equation that would describe the motion of a satellite, I realize that I have a big project ahead. It’s bigger, still, than I would have imagined before starting this internship; Newton’s equations of gravity work beautifully for all objects we can see that are not molecules or atoms or quarks… but there is something big that makes analyzing a satellite’s motion more difficult than entering in a few numbers into this equation. While the force of gravity from the earth is influencing the motion of any satellite in orbit, there are additional forces that greatly alter the satellite’s motion. It is because of the satellite’s path through the Earth’s upper atmosphere that I must bring into play “the drag force.”

There are many different equations that can describe drag, derived from greatly simplified empirical observations that are still not completely understood. Some equations are for objects moving slowly within a fluid, such as water, where the liquid exerts influence on the object in the direction opposite of motion. The faster an object moves through a liquid, the greater the drag force against it.

If the object moves even faster, there becomes a point where that equation is much less accurate, and we must rely on another to take its place. This new equation is a drag equation that is much more accurate for the high-speed situation that we are studying:

Well, that’s pretty self explanatory, isn’t it? Let’s move on to something else…

Actually, this equation is relatively simple to understand, once the meanings of these symbols are given.

is the magnitude of the drag force on the object. This is the output variable, and it is pronounced “F sub D.”

is the density of the fluid we are moving the object through. For water, this is roughly one thousand kilograms per cubic meter, or about 8 pounds per gallon. This value, however changes drastically with different temperatures, as do ALL liquids. This means that air at 40 degrees Fahrenheit will naturally be more dense then air at 70 degrees. This number is a Greek letter, pronounced “rho.”

is simpler to understand, simply being the magnitude of velocity times itself, or velocity squared. This part tells us that the drag force increases with the increase of the square of the velocity, which means that the drag force will not only increase with greater velocity, but will do so at an astounding rate. For instance, the drag force against an object moving at 200 miles per hour will be four times as large than for an object moving at 100 miles per hour. Similarly, an object increasing its velocity by four times will feel an increase in drag by sixteen times.

A is probably the most abstract number in the equation, because of our situation we must model. Simply put, it is an area of the object. Not the surface area, but the cross-sectional area. Where the section is cut depends on the direction of velocity for the object; it is sliced along a plane perpendicular to the direction of motion. Hypothetically, let’s imagine a cube roaring through the water in the Pacific Ocean. Directly behind the cube as its speeding along is a large flashlight, pointed in the direction of the cube’s motion, shining light that cuts perfectly through the water. Suddenly, the cube races towards a large undersea wall directly ahead! In the moments before the cube smashes against it, we can observe its shadow on the wall’s surface, where the flashlight’s output cannot reach. The area of the shadow on the wall is identical to the cross-sectional area of the cube.

The area of the shadow could change, given different orientations of the cube: it could be a square, it could be an elongated rectangle, it could be a diamond, it could be a hexagon… for other, more complex shapes, the possible cross-sectional areas are infinite in variety. The larger the cross-sectional area of an object, the fluid will oppose its motion more intensely because there is more of the object the fluid can hit and oppose. This is why a knife slicing though water moves much easier than a knife swung sideways, with the flat of the blade against the liquid.

, pronounced “C sub D” is a drag constant that is dependent on the object’s orientation and other characteristics, such as the roughness of its surface. These numbers can be found in the back of textbooks for simple objects, such as cubes and spheres, and have been determined for objects like cars and plane wings in wind tunnels. Most of the time, this constant is simply guessed, but for our purposes, we may need to spend more time finding the perfect values.

is the very last bit of the equation. This number is a vector, pronounced “V hat.” It is simply an indicator of the direction of the object’s velocity, and tells us a very important thing: the direction of the drag force always opposes the direction of motion. The “opposition” to velocity is shown by the negative sign at the beginning of the drag equation.

There you have it… drag equations made easy. The drag equation is just one tiny part of the whole process, however. Remember the density of the air? This is no longer a simple constant, as far as we are concerned. It is barely even an equation, and more like a computer program that is dependent on the exact location of the object over the earth, the time of day, the amount of humidity, and the sunspot activity… these programs are created by NASA researchers that take fifty years of atmospheric data and incorporate it into highly theoretical but highly accurate models, from sea-level to 600 miles up. Yes, there is still air where satellites will fly, and it makes a difference. It is a main cause of orbital decay, where satellite loses energy and falls back to earth (a process that could take a few weeks or decades).

My next post will discuss more about the cross-sectional area of the object. If you didn’t quite get what it was all about, or why it matters in our drag equation, you certainly will next time. We will find out that the cross-sectional area of our satellite is not a constant, but another equation! This is because the orientation of a satellite changes with respect to its direction of motion, depending on where it is during its orbit. Why? All in due time.

First things first, we need to test my camera out. Just to make sure we are all working:

I don’t think Scott was expecting anyone to take his picture… he was a bit puzzled. Can’t you tell?

By the way, there are several pictures coming up soon that you are more than welcome to click on to enlarge, if you are so inclined.

It’s just a short walk down to the student union building where we can eat a bit, microwave the heck out of our food, eat a bit more, buy a beverage (if we feel like it), and converse with the fellow nerds and nerdettes. Our boss Keith, a grad student that has been with SSEL for a long time, keeps all of us entertained during lunch with his great stories.

The very first day, he gave Scott and I a grand tour, the size of which was rivaled only by his gigantic frame. He is about 6’5′, has a bit of a southern drawl, and keeps himself very… imposing, but approachable.

Here is a single solar cell. This cell, when properly illuminated (notice the light on the table) has a voltage of 2.0 to 2.5 volts across it. Little tiny thing. This is one of over a hundred cells that have been donated over the years, and most of them are not in as good of a condition as you see here. Some of us are currently on the hunt for additional cells from potential discounters/ donors… you can NEVER have too many solar cells. As a general rule: all will be tested, many will break, and few will be utilized in the final product. Such is life.

Behind the notice hanging clear from the ceiling is a large vacuum chamber. This is one of the many instruments that will be used in the testing (and hopefully not the breaking) of the $40,000 satellite. Other such tests include: ultra-heating and ultra-cooling the satellite in a vacuum, placing the satellite on a table and shaking it as violently as possible, and subjecting the satellite’s structure and antennae to strong, controlled bursts of low-frequency vibrations.

This thing HAS to hold… I mean, during the accent of the rocket, it will have be perched right next to the thrusters (there is no way NASA will risk our little satellite damaging their multimillion-dollar gargantuan weather satellite, which is placed as far away from ours as possible). A coworker named Rubin, who uses a cool program “SolidWorks” to model hardware components, is already designing new protective panels to shield a male connector on the satellite. The circuitry could be exposed to heated plasma from the thrusters, and that potentially mess up a lot of electrical equipment. I have already seen the prototype shielding cut from plastic… (I guess that it’s easier to design and cut some metal shielding than to switch a male connector to a female connector… oh well).

All the last minute designs explain these next pictures: the satellite is apart, in pieces… and stored across several bags.

So… there are many more things to add (such battery brackets)…

…and more pages documentation to write up (two interns, on the task of going over a few steps in the construction of the satellite).

That’s just about it for tonight… I am tired. I know this is not much, but here are a few more pictures to make up for the short post today:

A significantly older and larger satellite; entered into a contest in an attempt to get more funding… and it lost. Oh well. It wins in my book.

A simple circuit on a breadboard. Someone is tinkering with the design while simultaneously modeling it on the computer… I hope no one accidentally moves a wire or two.

Breadboards are designed to put together a relatively simple prototype circuit very quickly. The design is then shipped off to a professional circuit manufacturer that can fit every component very neatly on a printed circuit board with a few chips. Fits well in an enclosed space after that! (Don’t make it too complex, though!)

More to follow.

This is perfect. Made from a composite carbon material six times stronger than steel, and acheives energy efficiency equivalent to 300 miles per gallon.

Watch it. LISTEN to it. In HQ, if you please!

The image enlarges for a better view of the details in the skin and hair.
For the curious, here is a brief pictorial explanation of a but of the process behind something like this. Also, here’s a link to the ZBrush forum thread featuring this project, with links to many other impressive images.

This is a great example of the extreme realism in graphics that many are unaware exists. The implications of ultrarealism in fabricated graphics are astounding, and in some cases, very dangerous. For example, if placed in the wrong hands and coupled with the best voice replication technology available, think about how much damage could potentially be caused with this.

Proof that the internets are insane.

“Passion Pit – The Reeling”

Edit:

Ok, so this one is really great, too…

“Passion Pit – Sleepyhead”