Orbital Dynamics

Discovery

Our objective here is to gain a simple understanding of the movements of the planets and see what that tells us about their physical properties.

How has the understanding of celestial mechanics evolved over the centuries?


An ancient Mayan concept of the cosmos.
Throughout human history, people noted that while most celestial objects stayed in one position, a few "wanderers" (planets) moved around. To most, these "wanderers" represented deities wandering around heaven.


The Ptolomeic Solar System from John Wright - Central Michigan University
Theories of classical antiquity are the best known of the early thinkers' because many of their records survived. Although they had trouble separating the physical heavens above our heads with the metaphysical Heaven of religion, Greco-Roman astronomers understood that celestial objects were physical things whose motion could be described analytically. Most believed in the geocentric (Earth-centered) model called the ptolemaic system (after Claudius Ptolemy who best described it). Ptolemy's system emphasized the perfection of "the heavens" by requiring the motion of the planets around the Earth be "perfect" circles.


Mars in retrograde motion from Astronomy Picture of the Day
Geocentric models required some complexities to explain simple observations. Example: They need to explain retrograde motions of planets where a planet appears to stop and reverses direction for some time. Ptolemy's model invoked the concept of smaller epicycles superimposed on an orbit.

Aristarchus (310-230 BC) proposed a heliocentric (sun-centered) model in which retrograde motion was the result of the relative motion of Earth and the planets around the Sun, but didn't convince many people because of the problem of parallax (stars appearing to change position against the background as Earth moves). (In fact, parallax is there, but naked eye observations are too insensitive to detect it.) Ptolemy's codified geocentric system was embraced both by the Church and Islamic scholarship, and Western astronomy changed little until the end of the Middle Ages.


Nicholaus Copernicus from Wikipedia
Nicholaus Copernicus (aka Mikolai Kopernik, 1473-1543) was a Polish cleric and astronomer. He viewed the geocentric system as unreasonably complex, and became the first modern proponent of a heliocentric system, explaining retrograde motions as the Earth either passing an outer planet, or an inner planet passing the Earth (because of different planet velocities).


Tycho Brahe from Wikipedia
Tycho Brahe (1546-1601) was a Danish astronomer-aristocrat who equipped an observatory at Uraniborg from his own fortune. He made superb naked-eye observations of the planets and recorded their movements with unprecedented accuracy. Developed a poorly-received "compromise" system in which the Sun and Moon orbited the Earth but everything else orbited the Sun. His great achievement was the mass of excellent data he assembled - the best ever produced by the pre-telescope era of astronomy.


Johannes Kepler from Wikipedia
Johannes Kepler (1571-1630) worked as an assistant to Tycho Brahe after the latter relocated to Prague. Given the task of calculating the precise orbit of Mars based on Brahe's observations, he came to the startling conclusion that its shape was that of an ellipse, not a circle.
Re ellipses:
Kepler went on to formulate three major laws of planetary motion:


Link to a NASA video review of Kepler's Laws.


Galileo Galilei from NuclearPlanet.com
Galileo Galilei (1564-1642) Italian mathematician and physicist. Began using telescopes around 1609, although he did not invent them. Important discoveries included:

Based on the weight of his evidence, Galileo embraced the heliocentric model. Unlike Copernicus, who cautiously expressed his concept of the heliocentric Solar System as more of a useful mathematical conceit to facilitate calculating the paths of the planets than as literal fact, Galileo bluntly reported his findings as objective facts, bringing him to the attention of church authorities. In 1633, charged with "grave suspicion of heresy" and having been "shown the instruments," he publicly recanted. It is hard to know what the church found more disturbing, the threat that the heliocentric solar system posed to the primacy of humans in the universe, or his inductive (evidence based) mode of argumentation. The charges against him, however, make patently clear that the church viewed the geocentric model as anti-biblical. (Note: In 1822, Galileo's works were removed from the Church's Index of Banned Books, and in 1992, he was fully exonerated.)

In any case, Galileo's ideas were still in circulation when the European Enlightenment began during the century after his death.


Isaac Newton from Wikipedia
Isaac Newton (1643-1727) formulated the laws of motion which provided elegant explanations for celestial movements. So why don't the planets move in straight lines? The force of gravity is acting on them.

Laws of Gravity

Gravitational force F = G M1 M2/R2

Where:

The gravitational force between two objects is:

The effect of any force on an object is to accelerate it.


Acceleration = F / m

where:

Applying this to gravity, the gravitational force at a planet's surface causes objects to fall with a gravitational acceleration g:

g = F / Mobject = G MP / RP 2

Where: MP and RP are the mass and radius of the planet. Examples of g:

Newton's law of gravity, combined with his development of mechanics (the laws of motion), offered an explanation of Kepler's laws, describing the detailed observations of planetary motions:

P2 = (4π2/GMsun)a3

where:

From Newton's laws of motion, one can also calculate the velocity necessary to stay in orbit.

Vo = √(GM/R)

where M is the mass of the body being orbited, and R is distance from the center of the body orbited. Note that the velocity is high if the distance R is small, and it is low if the distance is large.

Calculation of Mass

How do we know the masses of celestial objects? From celestial mechanics, as derived by Newton from Kepler's third law:

P2 = (4π2/GMP)a3

Rearranging this math,

MP = (4π2/GP2)a3

Thus, we need only know the semimajor axis and orbital period of an object to know the mass of the thing it orbits.

What is an orbit

The basic idea was originally put forward by Newton. If you placed a cannon on a mountain top and fired a projectile, its motion would be the vector sum of its sideways motion and gravitational acceleration. What if the projectile were shot so fast that it moved completely past the sphere of the Earth before falling into it? Unless friction slowed it down, it would continue to fall with enough sideways motion to miss the Earth. Instead it would circle it. This, in essence, is an orbit.


Transfer to a higher orbit from Edwin Muth.com
What happens when we accelerate an orbiting object (say we execute an engine burn on a spacecraft)? As long as MP stays constant, Kepler's third law says that we can't make it go faster in its original orbital path. Instead, the energy of acceleration kicks it into a higher orbit.

Tides

Now for a paradox: Suppose two particles orbit a planet in orbits whose semimajor axes differ by several hundred kilometers (1). The outer particle should have:

They therefore conform to Newton's laws of motion. But what if they are physically attached to one another by a cable? (2) In that case, their tendency to move at different rates would apply a force stretching the cable, and the mechanical strength of the cable would exert a direct and opposite force.

Now take away the cable. Instead, substitute a moon exerting its own gravitational force. The particles are rocks lying on its surface on opposite sides. The force holding them together is the moon's gravity. The force resulting from their tendency to orbit at different speeds remains the same. That last force is a tide. Note: Remember that g varies as the inverse square of distance from the primary. If two identical moons orbit a planet at different distances, the inner on will feel the stronger tidal force.

In reality, every particle of the moon experiences the tidal force, so that it is effectively stretched by the planet it orbits.


Consequences of tides:


Libration: Note - to say that an object in a synchronous orbit keeps exactly the same face toward its primary is an oversimplification because the orbiting object's rate of rotation is constant, but its orbit is an ellipse, not a circle. Thus, it will appear to wobble a little, as viewed from the primary. This is libration. (Follow the link and watch cool video with sound ON.)


Key concepts and vocabulary. Understand these or you're toast: