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Orbits with Richard Feynman

     Throughout our daily lives, we focus on our nearby surroundings rather than the bigger picture. When we take a look at the skies above, we start to notice astonishing phenomena such as stars, comets, and other celestial objects. One commonality between all of these objects within space is that they are all affected by each other’s gravitational force. Sometimes, systems of large stars and spherical bodies of mass form recurring patterns due to their force of gravity upon each other. These periodic paths between objects in space due to gravity are known as orbits. Interestingly enough, all of these orbits have one similar aspect: an elliptical shape. Why this is the case is difficult to understand. Luckily, the legendary teacher and physicist Richard Feynman shared his thoughts on the subject for everyday students to understand. His teachings of this problem and its solution are part of a famous session known as Feynman’s Lost Lecture which is also conveyed well through a video on the YouTube channel Minute Physics.

If all orbits follow the path of ellipses, there must be some sort of special reason as to why these ellipses appear. In order to find this reason, the definition of an ellipse can be used. An ellipse can be mathematically defined as the graph of an equation in the form, (x²/a²)+(y²/b²)=1. Ellipses can be constructed with two axes perpendicular to the plane where the shape is to be constructed. You can do so by taking a loop of string, tensioning it with a pencil, and tracing with the pencil, by which an ellipse can be constructed. The two axes used are each called a focus (foci for plural). A fundamental property of an ellipse is that the distance from one focus to the perimeter of the ellipse added to the distance from that point on the perimeter to the other focus is always constant. In regards to the other shape in question, circles are a special type of ellipse where the foci are both in the center, or effectively one focus. Even the mathematical equation for a circle, x²+y²=r², is a variation of the ellipse equation where the constants a and b are equivalent.

Another essential concept about ellipses for understanding orbits is the fact that circles can be used in order to create ellipses. Instructions on how to do so are the following:


1. First, take a circle and find its center. Then, identify another point within the circle. This will be known as the eccentric point.


2. Draw various line segments connecting the eccentric point to various points on the perimeter of the circle. Then, draw perpendicular bisectors to each of these line segments.


3. The shape formed by the many line segments resembles an ellipse. Both the center of the circle and the eccentric point can be considered the foci of this ellipse.


4. In order to prove this, the distance from one focus to the perimeter of the ellipse added to the distance from that point on the perimeter to the other focus must always be constant.


5. By adding the distances from the foci to various points and sign that they are equivalents, it is noticeable that this shape exhibits the properties of an ellipse.


A more specific proof that this shape is an ellipse can be shown with the following demonstration:


1. Draw various radiuses of the circle along with finding the intersection point between this radius and the ellipse; connect this point to the focus that is not the circle’s center. 


2. Create a line tangent to the ellipse-looking shape that includes the point shared by both the radius and ellipse. This tangent line just so happens to be an angle bisector of the triangle that can be formed by the tangent point, the focus that is not the center of the circle, and the point on the circle’s circumference that the radius ends. 


The triangle bisected can be found to be isosceles, implying that the focal sum of the ellipse is always equal to the radius, a constant. Therefore, the focal sum is also a constant meaning that this shape is indeed an ellipse.


Kepler’s Laws of Planetary Motion

Now that some knowledge regarding the topic of circles and ellipses has been established, it is time for the physics portion. When it comes to planets and orbits, an important set of laws that scientists use are Kepler’s Laws of Planetary Motion. Specifically, Kepler’s Second Law of Planetary Motion states that the area a planet sweeps out while rotating the sun in a certain amount of time will always be constant. This means that a comet’s location during the orbit does not matter. For example, let's look at the orbit of a comet around the sun. Since it travels along a skewed orbit, the closer it gets to the Sun, the faster it will travel and the arc length will be longer. As it gets further and further away from the Sun, however, the comet will travel slower and have a smaller arc length, but the radius from the comet to the Sun is longer. In this situation, there is a balance between the arc length and the radius which proves that time it takes the comet to travel along the orbit will be constant. 


Fun fact: An important thing to note is that even if orbits were not elliptical, this law would still be true. Therefore, this in and of itself does not prove that orbits are ellipses. 


In any case, when analyzing the amount of area swept by a planet throughout a time that approaches 0, the shape can be treated as a triangle. Therefore, the area of this triangle can be shown to be the ½ multiplied by the radius (R), treated as the base, and the component of tangential velocity perpendicular to the radius multiplied by the duration of time (vₚ△t). Although mass is a variable within the equation of angular momentum conservation, this should not be much of a problem as the planets’ mass most likely stays constant throughout its orbit. Additionally, the Conservation of Angular Momentum has proven that the equation

((1/2)Rvₚt) rotating around an origin point, in this case, the sun means that Rvₚ will remain constant as long as all the forces are acting on the origin point. This means that the area that the comet is sweeping is only dependent on x.

In practice, the shape of a celestial object’s orbit is not a perfect ellipse, however, we can find the exact shape of their orbit by graphing out the velocity vectors of the orbit about a center point. When done so, the outer ends of these vectors trace out a perfect circle. The proof for this circle can be understood more simply by dividing up this ellipse into sections with equal angle measures about a point within the ellipse, observing how the time it takes the orbiting object to travel across these sections is proportional to the area between the object itself and the object it is orbiting as well as the radius or the distance from the orbiting object squared. This can be written as Time ∝ Area ∝ (Radius)2. n simplest terms, the inverse square law states that intensity is proportional to the reciprocal of squared distance from the source. A good real-life example of this is with light. If a light is turned on and an object is distance 1 from it, then brightness is at 1. If the object is now distance 2 from it then the light will have an intensity of ¼th the original. If 2 units (the distance) is squared then it is now 4. The reciprocal of 4 is ¼ so the brightness would be ¼ that of the original. Now we will replace this with an orbital example. The radius (distance) between the Planet A and the Sun (which is orbiting the Sun) is 2 units. Now the force (intensity) would be ¼ that of the original force.  So, when connecting the velocities from the beginning of the section to the end and measuring the distance between the tips of these velocity vectors, the distance is always the same. Also, the smaller the angle of the sections gets, the more close the measured distance vectors turn into a perfect circle. Now, using this circle formed with the velocity vectors and our unknown ellipse, we can find the shape of this ellipse. Using how the circle was formed in the first place, the orbiting object traveling Θ degrees in its orbit corresponds to Θ degrees about the center of the circle as the velocity is proportional to the change in radius. Now to find the ellipse: first, rotate the circle 90 degrees, then take each velocity line and rotate then 90 degrees. Doing this shows an ellipse formed within the circle housing two points: the center of the circle and the eccentric point from which the velocity vectors all originated from. Although an ellipse has emerged, it is not so obvious that this specific ellipse would translate to the entire orbit being an ellipse as well. This requires us to examine the demonstration regarding ellipses formed from circles. Specifically, we must pay special attention to the perpendicular bisector between the line that connected the circumference of the circle and the eccentric point of the ellipse. This bisector is tangent to the circumference and this tangent point also intersects the line between the center of the circle and the circumference. In the velocity space diagram, the angle between the tangent point and the positive x-axis has a velocity that is equal in magnitude, but perpendicular to the line connecting the eccentric point and the circle’s circumference. If we take this understanding of the planet’s velocity diagram and apply it to the larger system encapsulating the entire orbit, we can start understanding how elliptical orbits are obtained. Since the velocity vectors were already rotated 90 degrees beforehand, the perpendicular lines in the velocity diagram at a certain angle are parallel to those corresponding in the overall diagram. After drawing out the specific details, we can notice that we have finally arrived at our result: a perfect ellipse.

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