One of the big things that I will be doing this summer is trying to mathematically describe the motion of a satellite in its orbit. The fine details of the project are not all there yet, but I have a good idea of what I need to know in order to do something like this.
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.
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