Politics, Revolutions, and Orbital Mechanics ... online!

The word revolution has a multitude of meanings to fit a multitude of situations and academic contexts. In the study of history, the most cataclysmic, earth-shattering events – the ones that make the discipline so interesting and exciting, “like sex for politics,” as an astute observer once put it – are referred to by this term. Synonyms of the word that invoke the same meaning, such as cycle, are found to have significance in many more fields. Students of the biological sciences are familiar with the cycles of nature – the water cycle, carbon cycle, energy cycle – and even those with a merely casual interest in the purpose of the discipline (preserving the environment) contribute to the cycle that takes place whenever an item is taken to a factory to be broken down and used for the construction of another: recycling. Those involved with classic literature have even learned over time to look upon the archetypical storyline that takes place when the hero of a story follows a certain, predetermined path in the journey towards a foregone conclusion – the hero’s journey – as a cycle of sorts. The idea, in general, of a defined entity progressing from a certain point in the universe appropriate to the aforementioned entity through many different, yet predictably reached, points, only to return to the same point, is a universal one.

As such, the term has an application in the study of the wide range of motions and artificial processes that take place in the world, known as physics. In the realm of physical science, revolutions fall into a vast and far-encompassing category known as orbital mechanics. This “branch of physics,” as a lecturer explained once, “…allows us to place probes anywhere in the solar system but also makes us only another cog in the celestial clockwork…”(No Calculus Lecture One). Also known as flight mechanics, it is, more specifically (and applicably), the study of the motion of artificial satellites and other vehicles of space travel under the influence of such forces as gravity, atmospheric drag, and thrust, to name a few. Admittedly, this definition is based primarily on one side of the equation, so to speak. Its applications, in general, are most useful and thus most commonly found in the field of engineering, in complex calculations such as those for ascent trajectories, re-entries and landings, lunar and interplanetary trajectories, and “rendezvous computations” (Braeunig, 1997).

Yet, as Goodstein put it (with reference to his own work):

This is…about a single fact, although certainly not a small one. When a planet, or a comet, or any other body arcs through space under the influence of gravity, it traces out one of a very special set of mathematical curves – either a circle or an ellipse or a hyperbola…[This] turns out to be not only of profound scientific and philosophical consequence but on immense historical importance as well (7).

And as the current field of orbital mechanics originates from the ancient field of celestial mechanics, in which the study of the motion of “natural celestial bodies” such as the moon, planets, and outlying stars was conducted (using Newton’s laws of universal gravitation and of motion, as we shall see later in this introduction), the principles of orbital mechanics can and will be used for the purpose of this study – provided, not the branch that has the engineering applications discussed above, but that which can be applied in ways that relate to my experiment, as will be seen (Braeunig).

1.2 History of related concepts

If I have seen farther, it is by standing on the shoulders of giants.”

– Newton, in a letter to Robert Hooke, 1676

To approach the specific problem that will be addressed through my research, I must first look to what the “giants” have theorized and discovered on the topic of orbital mechanics, over the centuries of formation of this field that we now call “physics.” The first of them was Aristotle, who believed in a world of four major elements – earth, water, air and fire – that took the forms of four interlocking spheres, surrounding each other in that order. Motion, he believed, was the result of elements acting according to the position of their spheres in the model. A body made of mostly “earthly” material, then, would be pulled down, into its earth sphere, the innermost of the four (Goodstein, 1996). Circular motion, according to this model, was the result of the influence of a “prime mover,” a sort of spiritual force emanating from deep within the universe (Hawking, 2002). The planets also followed this spherical model, he claimed, with their motions as a whole being “serene, harmonious, and eternal” – consisting of perfect circular motion. Ptolemy saw things differently. To him, motion was far from perfect – especially in the planets’ case. Their inconsistencies could be explained by the idea that they moved in circles called epicycles, which were at once controlled by and centered upon other circles called deferents (Goodstein). Conveniently for him, he could manipulate these circles, changing their sizes and adding or taking them away to fit the predetermined paths of motion. His work Almagest laid out these general principles (Hawking).

For over one thousand years, then, the picture of the universe most people held in their minds was a combination of those which Aristotle and Ptolemy, respectively, had laid out. When Copernicus came along, he simplified this collective view by adding one previously unconsidered detail. As many are aware, his major contribution to scientific thought was the demolition of the popular notion that the Earth was at the center of the universe. A modest man in this sense, as compared to the millions of his time and before who saw themselves as the cosmos’ center of attention, he stated in the first chapter of his life’s work, On the Revolutions of the Celestial Spheres, that the great bulk of knowledge referred to by scientists in their attempts to explain the universe could be eliminated – the whole flux of machinery of astronomical data could be simplified – by adopting a heliocentric view of the universe. As many scholars have noted in retrospect over the centuries, this simple correction in scientists’ and laymen’s philosophies had at the time, and still has to this day, an impact on our view of the cosmos that cannot be overemphasized. “Today, when we refer to political and other upheavals as revolutions,” one such scholar has written, “we are paying homage to Copernicus, whose book about revolutions started the first revolution.” Copernicus then went on to corroborate Ptolemy’s work by supplementing charts of astronomical data, based on his system of epicycles and deferents (Goodstein).

Next came Tycho Brahe, a man to whom my classmates and I were introduced in our Physics I class as one who, as Mr. Murray referred to him, “had no nose! He had no nose!!!” Interestingly enough, as I later learned, it had been lost in a duel, after which a replacement organ had been constructed out of silver, wax, and gold. His contributions to scientific knowledge, while not quite as entertaining to learn about, are no less important. Like Copernicus, he developed his view of the universe by taking from the accomplishments of his predecessors and adding his own. His, as laid out in On the New Star, which owed its name to his discovery of a new star in Cassiopeia, was “ a cosmos of his very own,” different from – in what some might say a compromise of – the perspectives of Copernicus and Ptolemy (Goodstein). The planets still revolved around the sun, he said, but at the same time, the Sun – along with the planets around it – revolved around the Earth. That is, the Earth was still the same old center of the universe.

Brahe’s legacy, it seems, would be that of a scientist clinging on to an anachronistic philosophy, were it not for his relationship as a mentor of sorts to another great scientific mind. Johannes Kepler was invited to be Brahe’s apprentice in the latter’s later years, assisting him in the gathering of data to confirm his theories on the dynamics of the cosmos. They didn’t have the warmest of relationships – the young man failed to agree with the old astronomer on a variety of subjects. And so when Brahe died of a urinary infection, after having asked his apprentice to carry on his life’s work, Kepler took all of his data and used it to confirm his own ideas. As it was once described, “A few hundred years ago, a dead Mathematician, by the name of Johan Kepper, stole some data from a dead Astronomer, named Tycho Brahe. The Mathematician worked for the Astronomer. The Astronomer was a jerk” (NCLO).

Kepler, in general, approached his study of the field from a more mathematics-based perspective, as opposed to the astronomy-based background of his mentor. In fact, he was once a high school geometry teacher. As legend has it, he had discovered, while lecturing to high schoolers one day, that the ratio of the diameters of two circles that, respectively, inscribed and circumscribed an equilateral triangle was essentially that of the diameters of the orbits of Jupiter and Saturn. He devised a model for this property that showed spheres (meant to represent planetary orbits) fitted on both sides with perfect solids (which had equal sides), and then stacked inside one another. All of this he laid out in his book The New Astronomy (Goodstein).

Kepler – as any astronomer has, it seems – borrowed from the work of yet another past figure. Inspired by Copernicus, he later mapped Brahe’s data on the orbital motion of Mars and showed it to be an ellipse, lending credit to the former. “His discovery of elliptical orbits helped usher in a new era in astronomy,” it has been claimed. “The motions of planets could now be predicted” (Hawking). Finally, in what is his most famous contribution to physics, he derived three basic laws that are now collectively know as “Kepler’s laws of planetary motion,” using his former mentor’s aforementioned data. Published in Harmony of the World, they can be summarized as follows:

1. The orbits of all the planets are ellipses, in which the Sun is at one focus.

2. Planets move faster when closer to the Sun and slower when further away, witin their orbits. A line drawn from any planet to the Sun “sweeps out” equal areas in the ellipse over equal amounts of time, regardless of its length – this distance from the planet to the Sun. (This holds true for circular orbits as well as ellipses, with the same equal times sweeping out the same equal times by their radii)

3. Based on this line, the further the orbit is from the Sun (the greater the radius), the slower the planet moves within the orbit. In other words, “the square of the period of any planet about the Sun is proportional to the cube of the planet’s mean distance from the Sun,” by

P^2 ∞ r^3

where P is the period and r the radius (Goodstein).

“These three laws made astronomy a hundred times more accurate than it had ever been before,” a scholar once wrote (Goodstein). These laws, as shall be seen, can be deduced from Newton’s laws of motion and of universal gravitation – in fact, Newton used Kepler’s work as a background for his gravitational theory. But all that is still to come.

Moreover, while all planets move in elliptical orbits (as Kepler pointed out), by simplifying the whole thing and considering only the case of circular orbits we can learn more about these orbits. For instance, in the example of two bodies with masses m1 and m2 that are moving in separate orbits, under each other’s attraction, the center of mass between the two lies at a point along the line joining them, such that the product of the mass and the radius of the first equals the mass-radius product of the second. In other words,

(m1)( r1) = (m2)( r2)

(Hawking).

This applies only if they have the same angular velocity. Also, it should be noted that a gravitational force provides this centripetal acceleration, and because it is a pair of forces, they must be equal in magnitude and opposite in direction. But what is a gravitational force?

The answer is known to use today as the result of the life’s work of a man whose name is quite possibly one of the most recognizable in Western society: Newton. In his great work Mathematical Principles, he “postulated that there is a force of attraction between any two objects, called gravitation.” This force was known as Newton’s law of universal gravitation, and defined by the following formula: for two particles with masses m1 and m2, separated by a distance r, these forces are attracted to each other by “equal and opposite forces” along the line joining them, with the magnitude of each force equal to

F = G ((m1m2) / (r^2))

where G is the gravitational constant or constant of gravitation, equal to 6.673 x 10^-11 Nm^2/kg^2, on Earth (Orbital Equations, 1995).

Newton also proposed three well-known laws – major in his time as well as in ours – which collectively “form the basis for all we know about how an object’s motion is affected by gravity.” The second of these, stating that F = ma (where m is the mass of the object concerned and a is its acceleration), allows us to find another important value in the study of gravity, g. This value, the acceleration of any object towards the center of the earth as the result of gravity, which is determined by substituting the above formula for gravitational force – that of universal gravitation – into Newton’s second law and solving for a (or, in the case of earth, g), resulting in the formula of

g = (Gm) / r^2

ends up being approximately 9.81 m/s^2 (based on the Earth’s 6375-kilometer radius) (OE). He proclaimed it to be the so-called “universal constant,” making the assumption that distant bodies and the objects on them follow the same rules as Earth. Before this, Aristotle had people thinking that the Earth obeyed different rules from the other planets in terms of the dynamics it had with other objects within its systems. Yet here Newton was, as one writer put it, allowing “physics to take the Universe as its laboratory,” and the concept he developed remained both widely accept and rarely challenged for several centuries (Sherwood and Sutton, 1988).

But before pushing on with the rest of the experiment, I would like to step back for a moment and look at the more basic concepts of orbital mechanics, including the all-important concept of uniform circular motion.

“He believed in perfect circular motion, and had good evidence to believe the earth to be at rest.”

– Dr. Stephen Hawking in his On the Shoulders of Giants, 2002 (referring to Copernicus)

When an object is moving in a perfect circle its direction of movement is continually changing, but the magnitude of this movement around the center remains the same. It thus does not accelerate within its circular path, around the center of its orbit. It does accelerate, however, in another sense. By Newton’s fundamental laws (which, it seems, nearly everything boils down to), whenever there is a change in velocity of any sort, there is an acceleration. Due to the fact that velocity is a vector with components pointing both parallel and perpendicular to the direction of movement at a certain instant – both towards the center of that circle and tangentially away from it at that instant – then simply by a change in one component of that vector, forcing a change in its direction, it has an acceleration, nevertheless. And because the vector changing is the one constantly pointing towards the center, then its acceleration is towards the center. This is known as centripetal acceleration, forcing the object towards the center in addition to along the tangent, every instant, and thus giving rise to its circular motion. Its magnitude is given by the formula of

a = (v^2) / r

where v is the speed – not the velocity of the object, which is a vector product, but the tangential speed, the scalar exponent always perpendicular to the radius of the orbit in which it moves (Sherwood and Sutton).

As Newton’s second law states that there is a force behind every acceleration and vice versa, then along with this centripetal acceleration there is a centripetal force, in the same direction as the acceleration – inward, along the radius of the circle, towards the center. The existence of this force contradicts the popular myth of centrifugal force, which states roughly that the force felt when rounding a corner, for instance, is supposedly that which pushes you outward (Sherwood and Sutton). And in affirming that the force is towards the center, it lends itself to an example from modern-day political behavior that illustrates this phenomenon, also outlined to us by Murray last year to help our comprehension – that during an election year, candidates for public office tend to move towards the center of the political spectrum, becoming equaly moderate on both sides of the aisle – or, in this case, both ends of the orbit. But that’s another study altogether.

Back-tracking just a bit, we now look at the specific case of the velocity of this orbital. The expression for centripetal acceleration – as it takes into account this tangential velocity – can be used in order to find a formula for velocity simply by setting it equal to the expression which determines g, the constant we saw earlier. And as this takes place on Earth, where g applies as the constant of gravity, there is no reason why we can’t do this very thing, which leaves us with the formula of

v = √(G m / r)

Now, getting close to the real “meat” of it all, we look at the velocity more in-depth. The velocity we found was a quantity in and of itself, with itself as its own frame of reference. In order to study the velocities of the two orbitals of a binary system, for the purpose of finding the difference between them, we must be able to find a velocity relative to the velocity of another; that is, find the velocity of each orbital using the moving reference frame of the other, thus inherently finding the difference between them. The source I consulted for information on this specific topic dealt with relative velocity in the case of a binary orbiting another binary – a binary system, in other words, where one is stationary and serves as the center of orbit and source of gravity for the other, much like that which exists between the Earth and the moon. For it to move in an orbit around this planet, the probe had to be within a specific region in which the gravitational force exerted by this planet on the latter had to be greater than that which exists between it and the sun. This region is called the sphere of influence (Bogan).

Laplace defined the radius of this region as being determined by the formula of

r = d(M / m)^2/5

, in which r equals the radius of the sphere concerned, d is equal to the distance between the sun and the planet around which the probe moves, and M and m, respectively, stand for the values of the masses of the planet and the sun. While this specific detail is not the most useful in a direct sense – for the purposes of this investigation – it is indirectly useful in that it provides a context in which to look at relative velocity, as each binary in the system, by its definition, must be able to orbit around the other and thus, must be within this region. As it was once put, “In order to properly describe the motion of the probe within the sphere of influence of the planet, we must know its velocity with respect to the planet” (Bogan)

And in order to find its velocity with respect to the planet (and not to the sun, contrary to the frame of reference normally used and the way motion of the planets is generally looked at in the solar system), we simply subtract its velocity, as compared to the sun, from the velocity of the planet to which we want to compare it (meaning we need to know the velocity of that other binary as well:

v = v1 – v2

where v is the velocity with respect to the planet, v1 is that which is compared to the sun, and v2 is that of the planet (according to the sun) (Bogan). In other words, we subtract the velocity of the frame of reference itself from that of the planet, which is moving outside this force of reference, in order to get that which is inside, or viewed based on, this frame.

In order to connect this with what I discussed earlier, when this probe is moving within the planet’s sphere of influence – staying within the field of gravitational pull of the planet – the shape of the orbit it makes around the planet is that of either an ellipse or a circle. It can thus be referred to as having elliptical or circular motion, as opposed to hyperbolic motion, which would be the result of it being flung around the planet, using the force exerted by the planet to accelerate its movement and “shoot” it off further into space (Bogan, 1996). This is done by those responsible for the movement of interplanetary vehicles like the International Space Station, yet I believe I can safely assume that such a process will not have to be undergone in the process of conducting the experiment described in this second-year physics research project.

1.3 Statement of the problem

The purpose of this study is to examine the relationship between the orbital radius (the distance from the center of orbit, the star, around which it rotates) and the characteristics of the binary system that orbits this star, specifically focusing upon the possibility of and investigating through experimentation the existence of a variation in the orbital velocity over time of the innermost of the two binaries, in order to gather results. Results that show such a variation – henceforth to be known as a “velocity variation” – will allow me to see whether the distance at which the system, as a whole, orbits its star (as measured from a point halfway between the binaries in the system) affects the dynamics of the binary therein that is closest to the aforementioned star.

1.4 Review of concepts appropriate to experiment

It is obvious from the laws and formulae discussed above that gravitational force and orbital radius have an inverse relationship. That in any system governed by the rules of physics that man has derived – including every universe that is accessible to man – as the distance between an orbital and the body it is orbiting increases in length, the force of gravity that the body exerts on the orbital, holding it within its gravitational field and within its sphere of influence at the cost of the sphere of a competing star, decreases, and vice versa. This was seen in Kepler’s second law, concerning a body not orbiting another, in Newton’s gravitation formula, concerning the attraction between two bodies, and even, indirectly, through the general equation contained within his second law (F = ma). In any system, then, any decrease of an orbiting radius will cause a proportional increase in the orbiting velocity, acceleration, and period of that orbital.

For the specific case of this study, it is also important to consider the effect that this greater velocity for the orbital might have on the amount that the velocity varies over time – the quantity that I have named the “velocity variation.” But first, it should be realized that if the binaries are allowed to orbit each other long enough, each will have an equal amount of time as the innermost of the two binaries in the system (the closest to the star). This can be applied to this study to mean that the binaries at a certain radius will have equal amounts of variation in their velocity at any given radii of orbit. Therefore I can examine the velocity variation over time of only one of the binaries, and derive a quantity that applies to both of them – to the system, as a whole, at that radius.

It should also be understood that by setting the binaries at a certain distance from one another, they will continue to stay at that separation as they go through their orbit around the star, orbiting each other all the while. At a smaller radius of orbit for the inner binary, this distance between them will have a greater effect on their rotation around the star – the difference between the actual distances at which they will be orbiting the star at the innermost and outermost points within the orbit will be proportionally greater, with respect to the radius at which the inner binary is originally set from the star – than it will at a larger radius, as measured from the inner binary. It is therefore not unreasonable to predict that orbital radius and velocity variance will prove to hold an inverse relationship, for the purposes of the situation created for this study – that as this inner radius continues to increase, this difference between the maximum and minimum velocities experienced by an arbitrary binary in the system will decrease, and vice versa.

1.5 Hypothesis

Considering the problem from the perspective of the basic concept mentioned above, by relating it to the particular circumstances and conditions of my study I can form the hypothesis that as the distance between the binary system as a whole and the star it orbits – the orbiting radius of both of the binaries – is increased or decreased, there will be a corresponding decrease or increase, respectively, in the variation of the velocities of these two binaries. This radius is to be measured from the center of the star to the center of mass between the two binaries of the system.

2 A method by which to conduct the study

2.1 List of equipment required

With the sole exceptions of a simple lab notebook and writing utensil for collecting numerical data and recording observations, the only apparatus that I used in my study – regardless of whether the task at hand was to plan, set-up, run, or analyze the experiment – was a standard personal computer. The most important software program used during my study was one known as Interactive Physics. This program was indescribably crucial for the purposes of conducting the experiment, simply for the reason that I would have been unable to create any form of binary system – one which could be run with the exact properties under which such systems operate in space, that is – without it (or, rather, I should say, without a similar program to allow me to simulate such events in such an easily accessible and alterable setting, at the very least).

Other specific programs used on this computer were the Microsoft Excel spreadsheet program (including its graphing capabilities), Microsoft Word (for the purpose of writing this paper, in which I will eventually display the results of the study), and Microsoft PowerPoint (used in preparing my presentation that was used in defense of my research at the planned “IB Physics II symposium” event).

2.2 Experimental procedure

For the purpose of this study, as I have attempted to make clear already, I have chosen to look at the innermost of the two binaries, as they are originally positioned in the “universe” created by Interactive Physics.

1. Opened the aforementioned program. Created the binary system that I have described, and with which I will be concerned, here. First, made three small spheres. Gave two of the bodies a mass of 5 grams, and one a mass of 1000 grams. Second, arranged them in such a manner that they form a straight line (this can be accomplished using the cartesian coordinates of the program’s grid), and set the two 5-g bodies a distance of 5 meters apart. Positioned the 1000-g body 85 meters from one of the two lighter bodies.[1]

2. Relying on formulae described in introduction, determined the velocities that were required for the binaries to orbit and the system as a whole to remain stable, based on the masses of the bodies and their distances from each other.[2] Used spreadsheet for the sake of convenience, entering in under appropriate labels each of the values referred to within step one of this procedure (measuring the orbital radius of the binary system from the center of mass of the two binaries – the midpoint of the imaginary line running between them, which in this case is 87.5 m from the central star), and once determined, the necessary values for the velocities of the two binaries and the star in the center of the system (in order to maintain the general stability of the system overall and counteract the “downward drift” that results from the orbit, the star must have a slight velocity in the positive direction, in a vertical sense). In order to determine these velocities, created a function in the spaces in which the desired values were to be placed, allowing the program to compute these values.[3]

3. Manually copying and pasting the values into their respective spaces in the “properties” window of the program, entered these velocities as those at which the inner and outer binaries, and central star, would travel.

4. Ran the system[4] for the length of time necessary for the binary pair to make two complete orbits around the star (making several orbits around each other all the while, of course).

5. Selecting the window in which the velocities were displayed in real-time (and recorded as well), manually copied and pasted this data into a second spreadsheet, using an appropriate heading.[5]

6. Looking at the velocities thus recorded for the inner binary over time at that radius, determined the minimum and maximum values of the data set, and their average (this is greatly simplified, again, by making use of the spreadsheet’s features). Used these values to derive the Sparksian coefficient for that specific radius of orbit, and recorded this latter value under another appropriate heading.

7. Increasing the inner binary’s original radius of 85 m, as described in step 1, by one meter, repeated steps 2-6.

8. Repeated step 7, as necessary, until desired results were achieved (the number of radii tested, the number at which the velocities of the inner binary at that radii were eventually recorded and analyzed, and the specific values of the radii used for the experiment, will be described later).

2.3 Conceptual review with formulae

As mentioned, in order to have the binaries orbit each other, it was necessary for me to make use of several key formulae from the topic of orbital mechanics, which I first defined and later referred to earlier within this paper. Specifically, I had to set the two basic expressions for force, as it relates to the situation of an orbit – mv/(r^2) and G(m1)(m2)/(r^2) – equal to one another. By knowing each of the required masses m1 and m2, the radius r, and the gravitational constant g (here, G), I was able to solve for v, the orbital velocity, for each of the three bodies in my system, (each of which I was concerned with, in order to have the system operate as a whole).

2.4 The Sparksian coefficient

The so-called “Sparksian coefficient,” as I refer to it here for the purpose of presenting my data, is simply a fancy name for a rather simple concept. In order to be able to compare the data sets – containing so many velocities at so many points in time – in the context of the amount in which they vary between their extreme upper to lower values, I derived this simple number with the help of the aforementioned physics teacher (who will be referred to in my journal entries simply as “Murray”). It is a measure of the average variation between two consecutive values within the set, and as such, it follows the general principle of “the greater the coefficient, the greater the ________” that is also seen in similar quantities in friction and other physical scenarios. In this case that _____ is velocity variation, so the Sparksian coefficient is, essentially, intended to be used as a direct, arbitrary measure of this quantity.

“Sparksian coefficient” for a defined set of data =

(maximum value in set – minimum value in set) / (average value)

2.5 Diagram of experimental setup

What I have included here is essentially a scaled-down version of the image created by Interactive Physics as that of the system created for this study. The body that I have alternately referred to here as the “central star,” among similar terms, appears on the screen in the form of the blue dot in the approximate middle of the image. This generally corresponds to its placement in the image created by the program itself – at the origin of the Cartesian grid. In addition, the binary pair can be located on the right side of the image, in the two points that are labeled with the letter V to represent the vectors that were displayed as pointing in the direction of their movement at any given point in time to indicate their velocities in the two dimensions.

Figure 1

3 An analysis and discussion of data collected during the study

3.1 Entries from laboratory journal

I conducted this study and made the following entries in my “lab journal” of sorts over the course of two months. This figure is a rather misleading one, however, as one may discover when looking at the specific dates upon which I recorded these thoughts and observations. In reality, the study was completed during what can now be seen as two distinct phases: the first phase, in which the rather-lengthy introduction was written and the setup was constructed, taking place in the form of short periods of work that were isolated from each other by the necessity of completing other tasks; and the second, in which the experiment was actually conducted. The first entry was recorded on November 18, the day that I first began to construct the experimental setup within the program. I had formed my research proposal several weeks earlier, and I had spoken with my teacher about my plans for the experiment several times before this date. I was then unable to complete any significant work on the study until almost two months later, after winter break, at which time I was free to conduct my experiment at last. These two weeks in the middle of January, leading up to the date of my research defense on January 22, are recorded in the form of entries made on a near-daily basis.

My hope is that this brief record of my proceedings throughout this study provides some insight into the individual process through which it was taken, the personal state of mind with which it was all the while conducted, and, finally, the specific circumstances under which it was at last completed.

DAY ONE – 11.18.03

I drop my innocent, blindly euphemistic self into Murray’s room after school in order to set up my experiment on Interactive Physics (and to figure out what exactly I’m doing, at the same time). Jesse Buss comes in to hand him a teacher recommendation form for his application to West Point. Yay Jesse. We get to work.

Murray clears up some of the confusion by explaining that, yes, I really am doing an experiment on binary stars and orbits thereof. I argue; he confirms. Begins to set up the system by getting the stars in orbit around each other (“Stay together, little binaries!”) – uses formulae. My head hurts. He’s obviously enjoying this. Finally the system is set up – it doesn’t work. They’re drifting apart like members of a dysfunctional family. We try slight adjustments – the stars continue to orbit separately around the larger star. Hmm. Murray is frustrated. A junior is doing his plot-matching lab behind us. We change reference frames; no luck. Murray curses. Finally the star is taken away, and we look at the two binaries. They orbit. We increase their masses to see if they’ll orbit the big ol’ star. Nope; alas, we then change their velocities, making them go slower with respect to each other. They begin the orbit, but it is an odd pattern.[6]

We discuss the concept some more, and decide where to go with it: the variation of the velocities at different radii. Murray says he’s going to “make like a baby and head out,” and I do the same. The end of another long day, but the beginning of a true odyssey, I foresee.

DAY TWO – 12.18.03

Hard to believe, but I haven’t done any work on my experiment in the last month – save for the seven-page, ultra comprehensive intro that I wrote up a few weeks ago to know what I was doing, in order to be able to collect data. And I haven’t really collected any data (it was just a wee bit hard to work in time for it alongside that which I spent on my area 4 project – standing wave frequencies at different water depths in a bathtub…yeah).

I present my experimental set-up (which Murray and I did last time, as described in detail above) to the class, and explain the idea of a binary system, binary stars, etc. I apologize for not being able to entertain them with any explosion-creating / shooting / actual-physical-apparatus-including presentations. They look disappointed. Sorry. I explain to them the simple fact that “I’m a lover, not a shooter.” And that it made perfect sense to do something with orbital mechanics, “because I’ve always been and continue to be a little ‘out there’ at times” (well…not really; I made that last part up for fun – but it would have sounded pretty good). My friend Ryan points out that it is really a trinary system that I am trying to replicate, when taking the star in the center into account (shit – I thought I had already figured out that one little detail, at least…however trivial it may end up being to my experiment as a whole). Murray saying he’ll help me after winter break, in the one-and-a-half weeks or so I’ll have to prepare before the big ol’ symposium. For now, though, I have two weeks off.

And then it’s crunch time.

DAY THREE – 1.12.04

I have returned t the world of physics yet again, having “endured” three weeks of not being in this lovely, mind-enriching, life-potential-cultivating building they call Tualatin High School. Today marks the one-week point until my presentation of this project. I’ve done all my background research, written the topic intro and pretty much everything else (including my hypothesis and statement of the problem) – everything, that is, leading up to the experiment itself. I know what I want to do, and Murray is helping me figure out how I need to do it.

He’s basically refreshing my memory after the two-month hiatus in the experiment we’ve had, since that day almost two months ago where it all arose out of nothing. For instance, I had assumed going into this that my subject was the difference of the stars’ two velocities at different radii – only after getting back into the work today did I recall that it was actually variations in one star’s velocity that I was to look at. Big difference.

Our presentations are one week from Thursday. How am I ever going to be able to do this on my own, much less talk about it to a room (aka lecture hall)-full of older, wiser people (aka parents – including among their ranks, possibly, one or two who are phairly phamiliar with physics)? And, in the meantime, how will I be able to dig up enough time / energy / motivation this weekend to write it all up?

Aaahhhhhhh……

DAY FOUR – 1.13.04

I drop in for thirty minutes or so. Murray goes through it again, at my request – and the second he leaves the room, I forget everything he says. I’m not comfortable working with this program; in general, I’m really not comfortable working for any length of time with unfamiliar objects in the first place, and when this contraption has one-thousand-and-one features and six-hundred-and-sixty-six or so buttons to select from, it only adds to my bewilderment.

DAY FIVE – 1.14.04

I tell him this (that which I rambled / ranted on yesterday, that is). I tell him I’m struggling with all this. I tell him I won’t be able to do my presentation a week from tomorrow unless one of the following things happens: 1) Murray shows me how to do it “one more time,” going through each step in even greater detail with me, or 2) he writes everything down. He tells me to come in again tomorrow, “right after school.” I agree.

DAY SIX – 1.15.04

I’m late (again) in making my “appointment” with Murray. But I think it’s finally starting sinking in. He has gone through it that one more time – and this time, I took notes. I think I have a project after all. Yay.

DAY SEVEN – 1.16.04

Workin’ like a dawg. More radii. Ooo.

DAY EIGHT – 1.20.04

Now is the time when everything has to come together into one wonderful, happy product. I have about one-half of the data points I need (or, rather, want to have), and I need/want to do some qualitative observations of the orbits as well. And then there’s that whole business with the presentations (aka “research defense”) forty-eight hours or so away. But I’m determined to get this part of it done before I leave this afternoon. Yar…

DAY NINE – 1.21.04

Worked some more. Got even more data. Found the coefficients with my namesake for each of the points, including the new ones. For aesthetic purposes, eliminated unneeded data on the chart and organized it in a more efficient manner, for the purposes of my presentation and the writing of my paper. Even made a graph from the data set of coefficients over time. A very productive day.

DAY TEN – 1.22.04

With the symposium to be held later tonight, January 22^nd in the year of our lord 2004 thus marks the end of the official research phase for my project. Today I tied up all the loose ends for the project and went over the data again, for the purposes of both being able to make a visual presentation (PowerPoint slides – hey, it’s the “aughts”) and to know what in the world I am talking about when I triumphantly take the stage tonight. “I am physics man, hear me roar…”

It’s the bottom of the ninth, with two men on and a full count as Casey steps to the plate. I’ve done the work, put in the time, and now I can bring it all to a triumphant conclusion with one mighty swing of my bat. The baseball metaphor is complete – I’ve even got my “red ropes” in hand (I love this stuff). Mudville, here I come.

3.2 Environmental conditions specific to study

The following settings for the program had already been selected by the time I started running the program to simulate the orbits and collect data, so I suspect that they might have been default properties for the “universe” created therein. In the chance that they were not, and instead were values that I had inadvertently put in for the simulated environment in which the study was conducted (on the program, remember), I have included them here for future reference:

Gravity = 10 m/(s^2) Charge = 1

Static friction = 3 Density = 159.15

Kinetic friction = 3 kg/(m^2)

Elasticity = 5 Moment = 250 kg(m^2)

3.3 Observations made in collecting data

When I first began to experiment with the program, following the general procedure described above, I planned to gather data from radii of regular intervals, being able to thus determine the average velocity variance for each of the possible values in orbital radius within the arbitrary range of 50 – 100 meters (the distance being simulated by the program, of course). However, when running the system for the first trial thus planned in this range – having positioned the inner binary at the initial distance of 50 meters, as described – I found that the binaries collided while orbiting each other.[7] After trying a few more distances on both ends of the range, under the general method of trial and error (ie. “messing around”), in order to determine what the range of the data that I would record and study would be, I settled upon the 30-m range of 65 – 95 meters. In order to have sufficient data for the purposes of my intended analysis, I decided in addition to record this data at intervals of only one meters.

I then began collecting data a second time, running the system at the inner radius of 65 meters. Similar results were found for this radius, however, and it was later to be found for the distances of 66, 67, 75, 78, 79, and 94 meters, as well, that their positions were such that they again resulted in either a collision or another form of separation during their course of orbit.[8] As I was unable to collect data at these distances in the same manner as that with which I had done so for those that corresponded to trials that “worked”, I decided to instead make qualitative observations of the behavior of the binaries, and the system as a whole, at each of these “unusual” radii. This was in order to replace the normal set of velocities over time that was found for the other distances, and that would have been found for that distance as well under normal circumstances.

In regards to the collection of data that I was thus able to assemble for these 23 distances (at which the system “worked”), it should be noted that the binary pair orbited the star at a greater overall velocity at some radii than others. A greater velocity of orbit, of course, results in a smaller orbital period – in this case, the time required for the system to make one complete orbit around the star. As in my procedure I made sure to run the system for the entire length of time that was required for the binaries to orbit the central star twice, it stands to be recognized that this time was greater for the systems positioned at some distances than it was for those set at others. And as the program recorded velocity values at set time intervals – every one-eighth of a second or .125 seconds, in fact – it therefore stands to be inferred that there was a smaller number of inner binary velocities recorded in the systems that were set at smaller distances.

Under ordinary circumstances, this would certainly have affected the results. I would have had less data points for smaller radii, and more for larger radii, thus making the coefficients calculated for these larger distances more accurate. However, during my reorganization of the data on January 21, as noted above, in order to “clean up” the sheet and thus be able to more easily and conveniently calculate the coefficients from the data sets (by being able to input the coefficient function in one cell and then use the “fill” feature to apply that function to other cells, for example), I deleted the all velocities that were recorded after the duration of 500 seconds, for each orbital radius. 500 seconds had been observed to have been the slowest time required for two complete orbits by the system around the star – therefore making this simple procedure a way of setting all the data sets on an “equal basis” in terms of the number of values from which to derive their averages.

The following are the observations that I made of the behaviors of the orbits that (somehow) gave rise to collisions, separations, or other interactions between the binaries that kept me from collecting valid data for the velocity variation of one binary at that radius.

RADIUS (m) NOTES

65 Sped up, crossed once, eventually spread apart – one orbited in close circle, the other made large

oval around it. It appeared as though they would eventually come back together, but they didn’t.

66 Moved slowly together, crossed twice, spread apart faster after four rotations around each other.

67 Came together quickly, crossed twice – first time, slowly, second time, quickly – and then spread apart

after one half of an orbit.

75 Pulled together slowly, made one revolution, collided and spread apart whicle trying to make a second

revolution, collided and spread apart while trying to make second revolution, at almost a

180-degree angle to each other, after only one-fifth of an orbit.

78 Moved together slowly in order to try to make first rotation around each other, collided and spread apart

at about 30-degree angle (the angle at which they were, in relation to each other, at the time of the

collision).

81 Very odd case. Moved together slowly, collided, yet continued to orbit each other after collision – wasn’t

a strong enough collision to force them out of orbit. Made one small, quick rotation around each

other, and then began a very close, fast orbit, which they continued to make.

94 Separated after two lazy rotations of each other, at slow speeds, having made one and a half orbits in all

during the course of 75 seconds (was therefore able to use the data collected before the collision).

95 Moving fast – drawn together for one slow, and then two fast, rotations – finally separated after

three-fourths of the first orbit (approximately 15 seconds having gone by), with the inner one then

being drawn to the other one more time, to then collide with it and be shot quickly, at a 90-degree

angle, directly towards the star. It then approached the star at a slight angle, being then “shot”

quickly around it and propelled off to begin a comet-like orbit.

3.4 Table of Sparksian coefficients at selected radii of orbit

Figure 2

Radius of orbit for binary system, as measured from inner binary (m)																						68
"Sparksian coefficient" of velocity variance between binaries (arbitrary units)																						0.3944

69			70			71			72			73			74			76			77
0.6223			0.4385			0.4364			0.3885			0.3837			0.5728			0.4219			0.7476

79		80			82			83			84			85			86			87
0.5644		0.1810			0.4804			0.4669			0.3908			0.3359			0.2893			0.2615

88	89			90			91			92			93			94			95
0.2331	0.2059			0.1781			0.1580			0.1707			0.1871			0.2067			0.1640

3.5 Graph of experimental results

Figure 3

(go to data file)

4 A conclusion to the study

4.1 Analysis of experimental results

I have been able to come to many conclusions in this lengthy study, in the course of my exploration and “discovery” of binary systems. In affirmation of my conclusion, I have found that my original hypothesis was essentially correct – that as the radius of a binary system, as measured from the inner binary, increases, the variation over time of the orbital velocity of that binary decreases. The difference in orbital radius experienced by the binary between the innermost and outermost points of orbit (within the binary pair – the orbit that they form around each other) has less significance at the larger orbits, as it is smaller in proportion to the distance from the central star to that binary at that innermost position.

4.2 Consideration of uncertainties, potential for error

Error is a fact of life, and a factor whose existence is always to be considered and taken into account when conducting any form of investigation, whether scientific or non-scientific. In my experiment, there was no one specific area of my procedure that could have served as a source for error and thus affected the results published above (and diminished the credibility thereof – hey, that rhymes…I’m tired). Several factors in my procedure, however, could have produced slight changes from what the results would have been, had they been found through an experiment conducted in perfectly controlled, environmentally ideal conditions (which, of course, are theoretically impossible). One of these was simply that this study required the use of much raw data. For each radius at which I set the binary pair to orbit the star, over five hundred velocities were recorded over time (one for every one-eighth of a second). It is highly likely – in fact, I acknowledge that it most likely came about – that I made an error or two in my raw calculations. Despite the fact that so many were made, automatically, by the programs themselves as I stared blankly at the computer monitor. Despite the fact that I took the time to constantly double-check my work as I went on my merry little way.

One final potential source of error should be mentioned here. In any scientific field of which experimental study is a major component (including physics – that’s why I’m bringing it up!), the concern is raised that a certain expectation – regarding the results that will be found through the experiment, or the like – is always held by the person conducting the study that thus affects, whether by a conscious or non-conscious process, the results themselves that are gathered by this “experimenter.” This so-called “experimenter’s bias” or “expectancy bias” (I can’t remember which exact term is used to describe this phenomenon) results in a shift, however slight it may be, in the results towards those which favor these expectations. I cannot think that the expectation that I might have held during the conduction of this experiment – that the data collected therein would confirm my hypothesis, or whatever – could possibly have had any effect on it as it was being played out. After all, the effort required to complete this study was only about 10 percent mine, and 90 or so percent that of the computer, I acknowledge (thank God for technology!). But it’s something to consider, especially when conducting similar laboratory studies in the future (as I know I will).

4.3 Discussion of significance of results

If the results that I have found through experiment and reported here in this paper are to have any significance, they must be applied to some real-world phenomena. And I want to apologize to you, Mr. Murray, because I can’t. My brain is fried from all these calculations and diagrams, and I cannot for the life of me come up with a single application for this conclusion.

I’m done.

WORKS CITED

Bogan, Larry. Satellite Orbits Gravitational Assist From Planets. <http://www.go.ednet.ns.ca/~larry/orbits/gravasst/gravasst.html>.

Bogan, Larry. The Geometry of Orbits: Ellipses, Parabolas, and Hyperbolas. July 1996. <http://www.go.ednet.ns.ca/~larry/orbits/ellipse.html>.

Brauenig, Robert A. Orbital Mechanics Page 1. 1997. <http://users.commkey.net/Braeunig/space/orbmech.htm>.

Brauenig, Robert A. Orbital Mechanics Page 2. 1997. <http://users.commkey.net/Braeunig/space/orbmech1.htm>.

Goodstein, David L., and Judith R. Goodstein. Feynman’s Last Lecture: The Motion of Planets Around the Sun. New York: W.W. Norton, 1996.

Hawking, Stephen. ed. On the Shoulders of Giants: The Great Works of Physics and Astronomy. Philadephia: Running Press, 2002.

No Calculus Lecture One. University of Arizona. <http://www.seds.org/seds/chapters/UASEDS/o-mech/no_calc.lecture.1.html>.

Orbital Equations. NASA. 19 Sep.1995. <liftoff.msfc.nasa.gov/academy/rocket_sci/orbmech/formulas.html>.

Sherwood, Martin, and Christine Sutton. The Physical World. New York: Oxford, 1988.

RELATED SITES

First, I wish to acknowledge once more my predecessors in the study of orbital mechanics in the Tualatin High Physics II program, Ms. Jody Forness, '03, and the late Mr. Ales Vilimovsky, '01, by providing links to their own research websites from years past. In searching for a topic upon which to conduct research last fall, I looked to their work for ideas and a bit of inspiration for generating my own.

____________________

Jody's website was created in order for her to share the results of her project on "the orbital velocities of a binary star system orbited by a single star." When I was knee-deep in the process of trying to figure out exactly what specific topic of the universal (ha), big ol' subject of o-mech I wanted to explore, I used her particular approach to the subject to help me dig myself out. You will note the similarities between her work and mine, even in such trivial characteristics as the titles we chose. I've always admired Jody, and I thought she did a pretty good job with it and all. She chose to further her studies at Northeastern University, back in Boston, and with my own decision to attend George Mason University, near Washington, D.C., this fall, I guess I'm once more following her lead as she walks with her humble, small stature down the road of life.

Ales made his own site under slightly different circumstances, as I understand it. A foreign exchange student from the Czech Republic, he started out here as a senior in Physics I, moving up to IB II halfway through the year. As a result, his site is essentially one containing links to all the other astrophysics sites out there; hence, one that was impossibly important to me for the purpose of my initial research, and of this last section in my report. It also has some really funny stuff on it.

Sadly, the page now remains on Murray's surver as a memorial of sorts to its very creator. Ales passed away the following summer, in a bicycling accident back home in his mother country. A small picture, and a short biography of the young man, can now be seen taped to the tower of the classroom computer that has since been named in his honor.

____________________

One site that I recommend for a closer look into the wide world of orbital mechanics and introductory astrophysics is that of the NASA Astrophysics Data System, hosted by the Harvard-Smithsonian Center for Asstropissicks (err ... astrophysics). I have to admit that I did not find the information that is included in this site incredibly useful for the purposes of mine, the most basic of high school science research projects. However, I'm sure it could be of use to greater minds who will follow in my quickly-fading footsteps, in the study of physics here at Tualatin. According to the site's creators, it is "The Digital Library for Physics, Astrophysics, and Instrumentation" - as such, it reportedly offers a gateway into the project's database of 3.7 million texts and bibliographic records. To be honest, I haven't taken the time to explore it in very much detail. But I have to take them for their scentifically-trained word - after all, this is Harvard we're talking about ... a great place. And you can take me for my word with that.

I happened to stumble upon it while browsing the astrophysics site created by Vilimovsky in the spring of 2001 (re: site description included above).

Another site I recommend for those who happen to be interested in binary star systems can be found on the physics server of Cornell University. I recall getting this link from Jody's site, although in reality it very well may have come from one of the Google searches on "astrophysics" and/or "orbital mechanics" that I conducted way back in October. And since memory has been proven to be fallible and reconstructive and all, it is highly possible that this is indeed the case.

I'm only human.

At the time, I recorded other web addresses from my search. Referring back to them recently for the purpose of this activity, I found that I was unable to access them again. I figure that either the site has since been taken down by those "powers that be," making the link a broken one, or I simply wrote it down wrong, way back when. At any rate, they're gone forever. Gone. It makes me sad...one of them was to a site created by the physics department of my dear sweet George Mason. Oh well. Time to move on.

____________________

In closing, therefore, I wish to refer the reader to one last source of information - one which, as it turned out, I had never happened to come across in my earlier research. This appears to be a general history of o-mech, from a modern perspective. Its author, a NASA employee of some sort named John F. (Kennedy/Kerry) Graham, begins his narrative with a reference to the 1957 launch of Sputnik. Anyways, it looks fairly interesting, and worth a read - even to hopelessly idealistic/realistic/confused politicos like myself. I printed out a copy to read, if only for the experience of reliving the days of Physics II research. Yay.

So that's that. Nothing more to say here, I guess. I thank you for coming, and for having the intellectual curiosity/determination/patience that is necessary for one to have in order to have read this far, as you have so successfully done.

HOWARD DEAN RULES!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[1] A diagram of the situation is included later in this chapter on page 13, under the title Figure 1.

[2] The specific formulae used for determining velocities, as described earlier in my introduction and now indirectly referred to here, are included on the following page.

[3] For the sake of space, I will let this description of how, using the selected formulae, one can determined the desired values, suffice. For a more in-depth description, consult someone who can go through the steps with you on the spreadsheet itself.

[4] The procedural steps that I have outlined here are not meant as a reference for those who are unfamiliar with the specific program with which I was concerned. It is intended as a guide for those who are reasonably familiar with the features of the Interactive Physics software. Sorry.

[5] In this process it is necessary that the data that is transferred to the spreadsheet (that which is recorded by the program in the first place) is that of the vector product of the horizontal and vertical velocities of the inner binary (the so-called component velocity – hypotenuse of the triangle created by the velocity vectors in the two dimensions). That is, there must be one velocity that is recorded for each arbitrary point in time.

[6] One of the more enjoyable aspects of reflecting on my notes from that time period is the sheer irony of what is recorded in them. The “odd” orbit that is referred to here ended up being the actual form of all of the orbits that I was to later create, study, and present data from during my presentation, and now during this paper. Yay.

[7] I should probably admit that I never was able to come to any form of conclusion on this matter – the system “broke apart” at several radius values, as I will describe here.

[8] This phenomenon described to have occurred at these data points is well established by the fact that I checked my calculations for these trials multiple times before recording it, as it appears here. It should be admitted, however, that this does not fully absolve me from the possibility that this is not exactly what should have happened at these points, and I acknowledge this freely. I further discuss this idea, as it applies to this specific case, in my conclusion.

AND ORBITAL MECHANICS

A study of velocity variations of binary stars at varying radii of orbit

by Casey J. Sparks

IB Physics II

28 January 2004

TABLE OF CONTENTS

2 A method by which to conduct the study

2 A method by which to conduct the study

Figure 1

Figure 2

Figure 3