# Orbital Mechanics

#### AstronomyMathematicsGeometryForces & MotionNewton's LawsAstrophysicsOrbital Mechanics

Why doesn’t the Moon fall out of the sky, the way the apple fell out of the tree and hit Isaac Newton on the head?  The answer is…the Moon actually is falling!

Imagine firing cannonballs off a hypothetical “extremely high” mountain.  The Earth’s gravity (actually air resistance too, but we will blithely ignore that) would bend the path of the cannonball and cause it to fall back to Earth (1).  The faster the cannonball is fired, the farther it would go before falling back to Earth (2).  If the cannonball was going fast enough, it would not hit the surface because the Earth is curving away from the cannonball at the same rate the cannonball is falling.  Like the Moon, the cannonball would be in orbit (3).

Around 1605, the German astronomer and mathematician Johannes Kepler presented his three laws of planetary motion.  These laws form the basis of our understanding of satellite and planetary orbits.

Kepler’s First Law – The orbit of each planet is an ellipse, with the Sun at one focus.

• Kepler’s Second Law – A line that joins a planet to the Sun sweeps out equal areas in equal times.  For satellites in Earth orbit, Kepler’s Second Law could be restated as “A spacecraft orbits such that the line joining it to the centre of the Earth sweeps over equal areas in equal time intervals.”  In other words, a satellite moves faster when it is close to the Earth and more slowly when it is further away.

• Kepler’s Third Law – The square of the period of a planet’s orbit (P) is proportional to the cube of its mean distance from the Sun (a).  In other words, the further a planet is from the Sun (or the higher a satellite is from the surface of the Earth), the longer it takes to complete an orbit.

While Kepler’s Laws describe orbital motion, it was the English mathematician and physicist Isaac Newton (1642–1727) who would discover the underlying physics behind the observed motion.  Newton’s most famous book was called Philosophiae Naturalis Principia Mathematica, more commonly known simply as the Principia, which was published in 1687 and contained his Three Laws of Motion.  This is the basis of what is now known as “Newtonian” or “classical” mechanics.  Newton’s Laws of Motion are as follows:

• Newton’s First Law – An object at rest will tend to stay at rest, and an object in motion will tend to keep moving in a straight line, unless acted upon by some outside force.
• Newton’s Second Law – The acceleration of an object is proportional to the force applied and is in the same direction as that force.
• Newton’s Third Law – For every action there is an equal and opposite reaction.

The latter is probably the best known of Newton’s three laws, and is really what makes spaceflight possible.  Rocket engines work by propelling exhaust out the back of the spacecraft, and since every action has an equal and opposite reaction, the spacecraft is propelled forward.

According to Newton’s Law of Universal Gravitation, the force of gravitational attraction between any two objects (such as a cannonball and the Earth) is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them.  The more massive the objects and/or the closer they are together, the greater the force of gravitational attraction between them, and vice versa.  This law is called “universal” because as far as we know the same principle applies everywhere in the Universe.  The motion resulting from this gravitational attraction is called two-body motion.

The type of orbit that a satellite is launched into depends on its mission.  Low Earth orbit (LEO) is the simplest orbit to achieve and is also the most extensively used.  Over 90% of artificial objects orbiting the Earth are in the LEO “corridor”, an area bounded on the low end by atmospheric drag factors (at about 200 km altitude) and at the high end by the lower Van Allen radiation belt (at about 1,000 km altitude).

Sun-synchronous orbit (SSO) is a type of polar LEO in which the Earth’s uneven gravitational field “twists” the orbit at a rate of one revolution per year.  The result is that the orbit plane will always maintain the same angle with respect to the Sun, and the satellite crosses the equator at the same local time every orbit.  Keeping the orbit at the same orientation with respect to the Sun and having a fixed equator crossing time means the spacecraft will experience the same lighting conditions every time it passes over a particular point on the surface.  This is why remote sensing, weather and reconnaissance missions use Sun-synchronous orbits (for example, motion can be detected by a change in shadows).

At an altitude of 37,500 km is geostationary orbit (GEO), where a satellite goes around the equator at the same rate that the Earth is rotating.  Therefore, it appears “stationary” as viewed from the Earth.  British science fiction author Arthur C. Clarke first proposed that this orbit could be used by communications satellites in a Wireless World article published in 1945.  This is why GEO is sometimes called the “Clarke orbit”.  The majority of communications satellites and many weather satellites are in GEO orbits.

#### Eric Choi

Eric Choi is an aerospace engineer, writer and editor from Toronto, Ontario.  For more information please visit his website at www.aerospacewriter.ca