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Those who like mathematics know perfectly well that the circle, the ellipse, the parabola, the hyperbola and even a straight, can be obtained by convenient sections of the cone. This is precisely why these lines are called conical. ......................................................................................... Those who like physics know perfectly well that the orbit of a satellite may be a circle, an ellipse, a parabola or a hyperbola. It can also be a straight, which is the case when dropping like a rock or on rocket take-off. New is, that these two facts are much more related than we can imagine. Our only difficulty to associate these two facts is that the Cone of Orbits is in Hyperspace. In space, nothing is showy. Much to the contrary, the beauty of this solution lies in its simplicity. Let us have a closer look at the cone. Observe Figure 1 ......................................................................................... I took a glass aquarium and filled it up with water dyed blue. Then I put a mirror in it, in an inclined position. It gives us two views of the partially immersed cone. The mirror shows it from underneath, and from front we see the cone directly. ......................................................................................... Hypergeometry teaches us how to use merely the dimensions required to survey the phenomenon. Observe that orbits are always in a plane; so this plane is called orbit plane. This is quite obvious when only two objects are involved. Earth and a satellite, for example. As we are using one plane only, I may use axis W of the fourth dimension. We shall
place our cone inside this XYW trihedral. To make it
better understood, the surface of the blue liquid represents
the orbit plane and the points in this plane are the only
ones belonging to our three-dimensional space.
What lies beyond, is in the fourth dimension, but that does
not make much difference. In other words, the satellite
moves exclusively on the surface of the liquid.
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The intersection of the cone and the liquid, as shown in Figure 1, is a circle. ......................................................................................... Let us imagine a satellite in an orbit that is exactly the circle of the intersection of the cone and the liquid. ......................................................................................... So far, nothing further. Apparently, the cone has no use whatsoever. Imagine a plane alpha, tangential to the cone. ......................................................................................... Imagine the intersection of plane alpha with the surface of the liquid. The intersection is a straight, tangential to the cone in point P, which is on the surface
of the liquid. Accordingly, point P belongs to the
tangent and to the cone. It is the point of union, or,
mathematically speaking, the intersection point.
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Be it as it may, once the tangent is fixed, we can pivot the cone clockwise or counterclockwise. ......................................................................................... First, another glance at the satellite. The satellite gravitates in a circular orbit, at constant altitude and constant speed. As a matter of fact, this is a very special orbit: an particular case. ......................................................................................... Let us imagine a satellite, exactly in point P,
at speed v.
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We are going to exert a force on this satellite in the direction opposed to its motion, as to brake it. What is going to happen, assuming that this force is strictly constant and applied during a complete revolution? Quite simple; the satellite will describe a spiral in the cone and the cone will protrude, i.e. move upward. After this, the orbit continues circular and speed will drop. The satellite will have been slowed down and come down to a lower circular orbit. ......................................................................................... In the same manner, if the force applied is perfectly constant and remains so for the whole duration of a revolution so that the satellite´s speed increases, the cone will sink into the (hypothetical) liquid while the satellite is going to a higher orbit, with higher tangential speed, but still in a circular orbit. ......................................................................................... This is not how orbit corrections should be carried out. ......................................................................................... Let us imagine our satellite in point P at speed v.
A meteorite collides with the satellite (common occurrence)
and satellite speed changes to v1, lower than v. Speed has
changed almost instantly. What is going to happen? ......................................................................................... Figure 1 shows the cone rotating counterclockwise and the orbit becoming elliptic. Point P has become the apex, the highest point of the orbit. The point on the other side becomes the perigee. The satellite continues crossing point P, but its speed now is v1, that is lower than
the original speed v.
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Let us now imagine the opposite situation. The meteorite hits the satellite and modifies its speed to v2, higher than its initial speed v. What is going to happen now? ......................................................................................... The cone rotates clockwise, in a rather symmetrical manner. Now point P becomes the perigee, the lowest point on
the orbit, while the point on the opposite side
becomes apogee. Yet the satellite continues passing through
point P, but its speed is v2 now, that is higher than
the initial speed v.
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Time has come to set a little rule. ......................................................................................... We are in point P. If the tangential speed of the
satellite were equal to speed v of the circular orbit,
it would obviously be in a circular orbit at constant speed.
If its speed were greater than its speed on circular orbit,
then P would be the perigee and the satellite would
slow down as its climbs to the apogee. If, contrarily, speed
in point P were lower than the speed on the circular
orbit, then P is the apogee and the satellite will
accelerate as it descends to the perigee.
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In the light of all this information I shall ask you a question now: what would you do to move the satellite, on circular orbit, to a lower circular orbit? ......................................................................................... In practice, orbit adjustment thrusters are of the stop and go type. These are good old airborne firework-type rockets fueled with hydrogen peroxide. They are fired inside a small chamber with a catalyzer, where the substance is turned into vapor. Vapor is jetted out and so allows minor orbit adjustments. These little thrusters increase or decrease the satellite´s speed almost instantly. ......................................................................................... The proposed adjustment is done as follows: thrust is applied on point P to lower the speed of the satellite.
Wait till the satellite has completed half a circle. Now
slow down the satellite again until it reaches the speed at
which it is on circular orbit going through the perigee.
Now it is on circular orbit at lower speed. Adjustment is
completed in two steps.
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See Figure 2. ......................................................................................... Orbit adjustment is controlled by powerful software and commensurate computers. The operator is little involved in what is taking place. I do not want to belittle anybody, but I bet that few among those involved in the operation know why this is done in two steps. Now think of a student about to graduate. This mathematical model and the orbital cone are very useful to the student and to all those who wish to better understand the orbits of satellites, comets, the Earth and even the asteroids that are on hyperbolic orbits. Heaven knows where they come from. Their orbit bends, and they go to nowhere, and sure will leave the solar system. All is contained in this model, the study of which is not to be limited to this sole paper. Much to the contrary, I invite all scientists to scrutinize this model with utmost attention, for many questions remain to be answered. I have much to add, which I shall do in ‘Hyperphysics’, the book I am currently working on. This cone is characteristic of these two bodies. The angle of the cone is what causes this characteristic. Which approaches can we make, keeping in mind that the mass of Earth is much larger that the mass of the satellite? And so on... . |