1. Chapter Fourteen Notes: Satellite Motion
Conceptual Physics
Chapter Fourteen Notes:
Satellite Motion

2. 14.1 Earth Satellites Circular Motion Principles for Satellites
A satellite is any object which is orbiting the earth, sun or other massive body. Satellites can be categorized as natural satellites or man-made satellites. The moon, the planets and comets are examples of natural satellites. Accompanying the orbit of natural satellites are a host of satellites launched from earth for purposes of communication, scientific research, weather forecasting, intelligence, etc. Whether a moon, a planet, or some man-made satellite, every satellite's motion is governed by the same physics principles and described by the same mathematical equations.
The fundamental principle to be understood concerning satellites is that a satellite is a projectile. That is to say, a satellite is an object upon which the only force is gravity. Once launched into orbit, the only force governing the motion of a satellite is the force of gravity. Newton was the first to theorize that a projectile launched with sufficient speed would actually orbit the earth. Consider a projectile

3. launched horizontally from the top of the legendary Newton's Mountain - at a location high above the influence of air drag.
As the projectile moves horizontally in a direction tangent to the earth, the force of gravity would pull it downward. And, if the launch speed was too small, it would eventually fall to earth. The diagram at the right resembles that found in Newton's original writings. Paths A and B illustrate the path of a projectile with insufficient launch speed for orbital motion. But if launched with sufficient speed, the projectile would fall towards the earth at the same rate that the earth curves. This would cause the projectile to stay the same height above the earth and to orbit in a circular path (such as path C). And at even greater launch speeds, a cannonball would once more orbit the earth, but now in an elliptical path (as in path D). At every point along its trajectory, a satellite is falling toward the earth. Yet because the earth curves, it never reaches the earth.
So what launch speed does a satellite need in order to orbit the earth? The answer emerges from a basic fact about the curvature of the earth. For every 8000 meters measured along the horizon of the earth, the earth's surface curves downward by approximately 5 meters. So if you were to look out horizontally along the horizon of the Earth for 8000 meters,

4. you would observe that the Earth curves downwards below this straight-line path a distance of 5 meters. For a projectile to orbit the earth, it must travel horizontally a distance of meters for every 5 meters of vertical fall. It so happens that the vertical distance which a horizontally launched projectile would fall in its first second is approximately 5 meters (0.5*g*t2). For this reason, a projectile launched horizontally with a speed of about 8000 m/s will be capable of orbiting the earth in a circular path. This assumes that it is launched above the surface of the earth and encounters negligible atmospheric drag. As the projectile travels tangentially a distance of 8000 meters in 1 second, it will drop approximately 5 meters towards the earth. Yet, the projectile will remain the same distance above the earth due to the fact that the earth curves at the same rate that the projectile falls. If shot with a speed greater than 8000 m/s, it would orbit the earth in an elliptical path. 

5. 14.2 Circular Orbits Velocity, Acceleration and Force Vectors
The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The velocity of the satellite would be directed tangent to the circle at every point along its path. The acceleration of the satellite would be directed towards the center of the circle - towards the central body which it is orbiting. And this acceleration is caused by a net force which is directed inwards in the same direction as the acceleration.
This centripetal force is supplied by gravity - the force which universally acts at a distance between any two objects which have mass. Were it not for this force, the satellite in motion would continue in motion at the same speed and in the same direction. It would follow its inertial, straight-line path. Like any projectile, gravity alone influences the satellite's trajectory such that it always falls below its straight-line, inertial path.

6. This is depicted in the diagram below
This is depicted in the diagram below. Observe that the inward net force pushes (or pulls) the satellite (denoted by blue circle) inwards relative to its straight-line path tangent to the circle. As a result, after the first interval of time, the satellite is positioned at position 1 rather than position 1'. In the next interval of time, the same satellite would travel tangent to the circle in the absence of gravity and be at position 2'; but because of the inward force the satellite has moved to position 2 instead. In the next interval of time, the same satellite has moved inward to position 3 instead of tangentially to position 3'. This same reasoning can be repeated to explain how the inward force causes the satellite to fall towards the earth without actually falling into it.

7. Ellipatical Orbits
Occasionally satellites will orbit in paths which can be described as ellipses. In such cases, the central body is located at one of the foci of the ellipse. Similar motion characteristics apply for satellites moving in elliptical paths. The velocity of the satellite is directed tangent to the ellipse. The acceleration of the satellite is directed towards the focus of the ellipse. And in accord with Newton's second law of motion, the net force acting upon the satellite is directed in the same direction as the acceleration - towards the focus of the ellipse. Once more, this net force is supplied by the force of gravitational attraction between the central body and the orbiting satellite. In the case of elliptical paths, there is a component of force in the same direction as (or opposite direction as) the motion of the object. As discussed in an earlier Lesson, such a component of force can cause the satellite to either speed up or slow down in addition to changing directions. So unlike uniform circular motion, the elliptical motion of satellites is not characterized by a constant speed.

8. In summary, satellites are projectiles which orbit around a central massive body instead of falling into it. Being projectiles, they are acted upon by the force of gravity - a universal force which acts over even large distances between any two masses. The motion of satellites, like any projectile, are governed by Newton's laws of motion. For this reason, the mathematics of these satellites emerges from an application of Newton's universal law of gravitation to the mathematics of circular motion. The mathematical equations governing the motion of satellites will be discussed in a later section.

9. 14.4 Energy Conservation and Satellite Motion
The orbits of satellites about a central massive body can be described as either circular or elliptical. As mentioned earlier in the chapter, a satellite orbiting about the earth in circular motion is moving with a constant speed and remains at the same height above the surface of the earth. It accomplishes this feat by moving with a tangential velocity that allows it to fall at the same rate at which the earth curves. At all instances during its trajectory, the force of gravity acts in a direction perpendicular to the direction which the satellite is moving. Since perpendicular components of motion are independent of each other, the inward force cannot affect the magnitude of the tangential velocity. For this reason, there is no acceleration in the tangential direction and the satellite remains in circular motion at a constant speed. A satellite orbiting the earth in elliptical motion will experience a component of force in the same or the opposite direction as its motion. This force is capable of doing work upon the satellite. Thus, the force is capable of slowing down and speeding up the satellite. When the satellite moves away from

10. the earth, there is a component of force in the opposite direction as its motion. During this portion of the satellite's trajectory, the force does negative work upon the satellite and slows it down. When the satellite moves towards the earth, there is a component of force in the same direction as its motion. During this portion of the satellite's trajectory, the force does positive work upon the satellite and speeds it up. Subsequently, the speed of a satellite in elliptical motion is constantly changing - increasing as it moves closer to the earth and decreasing as it moves further from the earth. These principles are depicted in the diagram below.

11. KEi + PEi + Wext = KEf + PEf
In an earlier chapter, motion was analyzed from an energy perspective. The governing principle which directed our analysis of motion was the work-energy theorem. Simply put, the theorem states that the initial amount of total mechanical energy (TMEi) of a system plus the work done by external forces (Wext) on that system is equal to the final amount of total mechanical energy (TMEf) of the system. The mechanical energy can be either in the form of potential energy (energy of position - usually vertical height) or kinetic energy (energy of motion). The work-energy theorem is expressed in equation form as
KEi + PEi + Wext = KEf + PEf
The Wext term in this equation is representative of the amount of work done by external forces. For satellites, the only force is gravity. Since gravity is considered an internal (conservative) force, the Wext term is zero. The equation can then be simplified to the following form.
KEi + PEi = KEf + PEf

12. In such a situation as this, we often say that the total mechanical energy of the system is conserved. That is, the sum of kinetic and potential energies is unchanging. While energy can be transformed from kinetic energy into potential energy, the total amount remains the same - mechanical energy is conserved. As a satellite orbits earth, its total mechanical energy remains the same. Whether in circular or elliptical motion, there are no external forces capable of altering its total energy.
Energy Analysis of Circular Orbits
Let's consider the circular motion of a satellite first. When in circular motion, a satellite remains the same distance above the surface of the earth; that is, its radius of orbit is fixed. Furthermore, its speed remains constant. The speed at positions A, B, C and D are the same. The heights above the earth's surface at A, B, C and D are also the same. Since kinetic energy is dependent upon the speed of an object, the amount of kinetic energy will be constant throughout the

13. satellite's motion. And since potential energy is dependent upon the height of an object, the amount of potential energy will be constant throughout the satellite's motion. So if the KE and the PE remain constant, it is quite reasonable to believe that the TME remains constant.
One means of representing the amount and the type of energy possessed by an object is a work-energy bar chart. A work-energy bar chart represents the energy of an object by means of a vertical bar. The length of the bar is representative of the amount of energy present - a longer bar representing a greater amount of energy. In a work-energy bar chart, a bar is constructed for each form of energy.

14. A work-energy bar chart is presented below for a satellite in uniform circular motion about the earth. Observe that the bar chart depicts that the potential and kinetic energy of the satellite are the same at all four labeled positions of its trajectory (the diagram on the previous page shows the trajectory).
Energy Analysis of Elliptical Orbits
Like the case of circular motion, the total amount of mechanical energy of a satellite in elliptical motion also remains constant. Since the only force doing work upon the satellite is an internal (conservative) force, the Wext term is

15. zero and mechanical energy is conserved
zero and mechanical energy is conserved. Unlike the case of circular motion, the energy of a satellite in elliptical motion will change forms. As mentioned above, the force of gravity does work upon a satellite to slow it down as it moves away from the earth and to speed it up as it moves towards the earth. So if the speed is changing, the kinetic energy will also be changing. The elliptical trajectory of a satellite is shown below.
The speed of this satellite is greatest at location A (when the satellite is closest to the earth - perigee) and least at location C (when the satellite is furthest from the earth - apogee). So as the satellite moves from A to B to C, it loses kinetic energy and gains potential energy. The gain of potential energy as it

16. moves from A to B to C is consistent with the fact that the satellite moves further from the surface of the earth. As the satellite moves from C to D to E and back to A, it gains speed and loses height; subsequently there is a gain of kinetic energy and a loss of potential energy. Yet throughout the entire elliptical trajectory, the total mechanical energy of the satellite remains constant. The work-energy bar chart below depicts these very principles.
An energy analysis of satellite motion yields the same conclusions as any analysis guided by Newton's laws of motion.

17. Summary:
A satellite orbiting in circular motion maintains a constant radius of orbit and therefore a constant speed and a constant height above the earth. A satellite orbiting in elliptical motion will speed up as its height (or distance from the earth) is decreasing and slow down as its height (or distance from the earth) is increasing. The same principles of motion which apply to objects on earth - Newton's laws and the work- energy theorem - also govern the motion of satellites in the heavens.

18. 14.5 Kepler’s Laws of Planetary Motion
In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor - Tycho Brahe - with three statements which described the motion of planets in a sun- centered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any planet and any satellite.
Kepler's three laws of planetary motion can be described as follows:

19. The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

20. Kepler's first law - sometimes referred to as the law of ellipses - explains that planets are orbiting the sun in a path described as an ellipse. An ellipse can easily be constructed using a pencil, two tacks, a string, a sheet of paper and a piece of cardboard. Tack the sheet of paper to the cardboard using the two tacks. Then tie the string into a loop and wrap the loop around the two tacks. Take your pencil and pull the string until the pencil and two tacks make a triangle (see diagram at the right). Then begin to trace out a path with the pencil, keeping the string wrapped tightly around the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented here by the tack locations) are known as the foci of the ellipse. The closer together which these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple - all planets orbit the sun
in a path which resembles an ellipse, with the sun
being located at one of the foci of that ellipse.

21. Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are longer when the earth is furthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.

22. Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws which describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.

23. Planet Period (s) Average Dist. (m) T2/R3 (s2/m3)
Earth
3.156 x 107 s
x 1011
2.977 x 10-19
Mars
5.93 x 107 s
2.278 x 1011
2.975 x 10-19
Observe that the T2/R3 ratio is the same for Earth as it is for mars. In fact, if the same T2/R3 ratio is computed for the other planets, it can be found that this ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T2/R3 ratio.
Planet
Period
(yr)
Ave.
Dist. (au)
T2/R3
(yr2/au3)
Mercury
0.241
0.39
0.98
Venus
.615
0.72
1.01
Earth
1.00
Mars
1.88
1.52
Jupiter
11.8
5.20
0.99
Saturn
29.5
9.54
Uranus
84.0
19.18
Neptune
165
30.06
Pluto
248
39.44
(NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun x 1011 m. The orbital period is given in units of earth-years where 1 earth year is the time required for the earth to orbit the sun x 107 seconds. )

24. 14.6 Escape Speed
In physics, escape velocity is the speed at which the kinetic energy of an object is equal to its gravitational potential energy. It is commonly described as the speed needed to "break free" from a gravitational field, for example for a satellite or rocket to leave earth. The term escape velocity is actually a misnomer, as the concept refers to a scalar speed which is independent of direction.
In practice the escape velocity sets the bar for any rocket aiming to bring a satellite into earth orbit or beyond. It gives a minimum delta-v budget (See next slide) for rockets when no benefit can be obtained from the speeds of other bodies, for example on interplanetary missions where a gravitational slingshot may be applied.
When escape velocity is calculated by the gravitational potential energy (Ug) equation
atmospheric friction or air drag is neglected.

25. Delta-v budget (or velocity change budget) is an astrogation term used in astrodynamics and aerospace industry for velocity change (or delta-v) requirements for the various propulsive tasks and orbital maneuvers over phases of a space mission.
Sample delta-v budget will enumerate various classes of maneuvers, delta-v per maneuver, number of maneuvers required over the time of the mission.
In the absence of an atmosphere, the delta-v is typically the same for changes in orbit in either direction; in particular, gaining and losing speed cost an equal effort.
Because the delta-v needed to achieve the mission usually varies with the relative position of the gravitating bodies, launch windows are often calculated from porkchop plots that show delta-v plotted against the launch time.
Set equal to Kinetic Energy
and solve for velocity, yields m/s, or 11.2 km/s. Fire anything at a speed greater than this, and it will leave Earth, going more and more slowly, but never
stopping!

26. Table 14.1: Escape Speeds at the Surface of Bodies in the Solar System
Astronomical Body
Mass
(Earth Mass)
Radius
(Earth Radii)
Escape Speed
(km/S)
Sun
333,000
109
620
Sun (at a distance of Earth’s orbit
23,500
42.2
Jupiter
318 11
60.2
Saturn
95.2
9.2
36.0
Neptune
17.3
3.47
24.9
Uranus
14.5
3.7
22.3
Earth
1.00
11.2
Venus
0.82
0.95
10.4
Mars
0.11
0.53
5.0
Mercury
0.055
0.38
4.3
Moon
0.0123
0.28
2.4