Orbital Mechanics

The title should be “Elementary Orbital Mechanics,” but I wanted to keep it short. The story goes back 60 years.

My friend Patrick and I were the class nerds, and this was a great time for us. First there was lots of news about America’s plans for putting satellites into orbit around the earth. Colliers magazine ran a series of articles about space travel based on a book titled Across the Space Frontier, with writings by Cornelius Ryan, Willy LeyWerner von Braun and others and illustrations by Chesley Bonestell Besides that there was constant news of rocket launches with some successes and some spectacular failures. The Soviet Union launch the first Earth satellite. We built our own rockets and shot them off. Rockets and space flight were the thing.

Our school library got in this book by Willy Ley titled Rockets, Missiles and Space Travel. The book told all about the history of rocket development, up through the story of the German V2 missile of World War 2. It also had a lot of useful information on rocket propulsion facts and a smattering of orbital mechanics. We learned a lot and stretched what we could achieve without any knowledge of calculus. This is some interesting stuff, but I need to start with at tale from my college days, because this seems to be the likely place to put it.

It went like this. I went down to the University of Texas at Austin to enroll for a degree in physics. However I realized that after getting said degree I would really be working as an engineer, so I enrolled in the engineering college with a degree plan for aerospace engineering. I figured that if I were going to work with rockets and space travel, that was the degree I would need. It took me two years to realize I was in the wrong plan. There were a couple of incidents that made that determination for me, and here is one of them.

I was in this course in my degree plan, and the instructor was giving us the straight skinny on orbital mechanics. He explained what keeps a satellite in orbit. See the diagram:


That black ball is the satellite, and it’s traveling along a circular path around the earth. Gravity exerts a downward force on the satellite, represented by the down arrow. What keeps the satellite in orbit is the centrifugal force due to the satellite’s curved path. The centrifugal force is represented by the up arrow. Now, as explained to us, if the two forces are exactly equal, then the satellite will not fall but will continue to orbit the earth.

I wasted abut a half second thinking about this before realizing it was absolutely wrong and absurd, besides. Somebody who has made a passing grade in a high school physics class will see that if the two forces (assuming there really are two forces) are equal, then the satellite will not follow a curved path. Since the two forces are equal and opposite they cancel out, and the net force on the satellite is zero. The satellite will follow a straight line, according to a well-known observation of Galileo. There is no actual centrifugal force acting here, only gravity. The person teaching us about orbital mechanics knew absolutely nothing about mechanics in general and about orbital mechanics in particular. Shortly after this I exited the aerospace engineering program and wound up getting a degree in engineering science.

Anyhow, back in high school Pat and I worked through some elementary investigations into orbital mechanics, and here are the high points. See the next figure:


This represents a generalized view of the orbit of a small satellite (Sputnik) around a large object (Earth). Contrary to popular perception, all orbits are not circular. At best, all orbits are elliptical. The figure shows an exaggerated case for illustration. Even this is an idealization, because the large objects and the small objects are not uniform and spherical, so there is some deviation from the idealized case. For the ideal case (perfectly uniform and spherical objects) the statement that orbits are elliptical is always mathematically correct. This is because a circle is the special case of an ellipse. The next figure shows the relationship between ellipses and circles:


On the left is an ellipse as defined mathematically. The ellipse is defined by two points called the focus points or the foci. The ellipse is all points on a plane containing the foci such that a + b is constant. When a = b the foci become a single point, and we have a circle, as shown on the right with radius r. The ellipse has two axes. One is longer and is called the major axis. The other is the minor axis. This is important.

When a very small object orbits around a very large object, then for all practical purposes the orbit is an ellipse, and the center of the large object is one of the focus points of the ellipse. The other focus point is somewhere along the major axis.

Johannes Kepler a long time ago made a number of important observations. One was that the orbits of the planets around the sun are elliptical, and another is that the area swept by the planets position is constant for a constant period of time. See the diagram:


This shows a planet in two positions in its orbit around the sun. The blue wedge is the area swept by the planet’s line to the sun in a certain time interval, a day for example. What Kepler observed is that for any position around the planet’s orbit this blue area is the same for a one-day interval. What Kepler did not state, but what this shows, is that angular momentum is preserved. This would be immediately clear for anybody having an understanding of basic mechanics and the law of gravitation. Following Kepler, Newton described gravity as a force between two masses, acting in a line that connects them. Since the force of gravity has no component perpendicular to the line it cannot contribute to any change in the angular momentum. If gravitational attraction is the only force acting, then angular momentum is conserved, and Kepler’s second law is in agreement.

Now to the (more) real world. Large objects are not absolutely large, and small objects are not absolutely small. There is some equality between the two. When a satellite is orbiting around a large object, the large object does not stand still. It reacts to the gravitational pull of the small object. Never is this illustrated better than the case of the Earth and the Moon. The Earth is about 81 times as massive as the moon. This means that when the Moon goes around the Earth, the Earth moves. The fact is they follow elliptical orbits having a common focus point located 600 miles beneath the surface of the Earth. See the diagram:


The big object is the Earth. The smaller object is the Moon. I did not get these sizes to scale, but the Moon’s orbit is the large ellipse, and the Earth’s orbit is the small ellipse. This is important, because if you are taking observations of objects within or in the vicinity of the Solar System, you need to know exactly where the Earth is at all times.

Using only simple physics my classmate and I determined that we could compute the orbit of an object around the Earth (or the Sun) if we knew two things at once:

  • The location of the orbiting body
  • Its velocity vector

It was amazing to us (to me at least) that this problem was so easily resolved. We went further. There was talk of sending spacecraft to Mars. The problem is, what is the minimum transfer orbit between the Earth and Mars? That is, what transfer orbit requires the least expenditure of energy That was easy to solve, as well. See the next diagram:


If the Earth and Mars orbits are idealized circular orbits, then the transfer orbit is tangent to the Earth orbit and also to the Mars orbit. From that it is easy to compute the major axis of the transfer ellipse and also the transfer time and the initial velocity (relative to the Sun).

All that was a lot of fun in high school, and in college I took a course in celestial mechanics and another one in planetary navigation. The problems got a lot harder, and it wasn’t as much fun. It was more like work. But that’s the way life is.

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