**
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.

The Ptolomeic Solar System from John Wright - Central Michigan University

**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

**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:

- There are two
**foci**(sing.**focus**) - An ellipse has a
**major axis**and a**minor axis**. Half of the major axis is the**semimajor axis**(The measurement typically used to describe the radius of a planet's orbit**abbreviated "a"**.) - The shape of an ellipse can be described by its
**eccentricity**e = c/a

Where c = distance from center to a focus and a = the

**semimajor axis**, the distance from the center to the apex. An ellipse with eccentricity of 0.0 is a circle. As an ellipse becomes more elongate, its eccentricity approaches 1. Orbital eccentricity is simply a special case of this measurement.

Kepler went on to formulate three major

**laws of planetary motion**:

**Kepler's First Law**: Planets move in elliptical orbits about the Sun with the Sun at one focus. In this case, the Sun is the**primary**- the dominant body in the orbital system. Terms for a planet's closest an most distant points from its primary:**Apoapsis**: the point in an orbit of greatest distance between the orbiting body and the primary.**Periapsis**: the point in an orbit of closest approach of the orbiting body to the primary.

**Aphelion**: the point in an orbit of greatest distance between the orbiting body and the Sun.**Perihelion**: the point in an orbit of closest approach of the orbiting body to the Sun.

**Apogee**: the point in an orbit of greatest distance between the orbiting body and the Earth.**Perigee**: the point in an orbit of closest approach of the orbiting body to the Earth.

Confused? Wikipedia will straighten you out.

**Kepler's Second Law**: Imaginary lines drawn between Sun and planet sweep out equal areas in equal time as planets move. Planets travel fastest at periapsis and slowest at apoapsis.

**Kepler's Third Law**: A planet's semimajor axis (a) and orbital period (P) are related by**P**^{2}∝ a^{3}Thus, planets farther from the Sun take longer to complete their orbits not only because they follow longer orbital paths, but because their absolute speed is lower.

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:

- Sunspots
- Craters and mountains on the moon.
- The four largest moons of Jupiter, demonstrating that the Earth was not center of
**all**celestial movement. - The phases of Venus, which were flatly impossible under the Ptolomeic system.

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.

- Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by the action of some outside force (things remain in motion unless friction slows them down).
- The change of motion (acceleration) is proportional to the force acting on the body and inversely proportional to the mass of the body (can calculate the motions from the force exerted on an object, for example a spacecraft).
- To every action there is an equal and opposite reaction (apple attracts Earth as Earth attracts apple).

**Laws of Gravity**

**Gravitational force F = G M _{1} M_{2}/R^{2}**

Where:

- G = the gravitational constant, ~6.67384 * 10
^{-11}m^{3}kg^{-1}s^{-2} - M
_{1, 2}= masses of the bodies - r = distance between masses.

The gravitational force between two objects is:

- Proportional to the product of their masses
- Inversely proportional to the square of the distance between them.

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

**Acceleration = F / m **

where:

- F is force
- m is mass.

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

**g = F / M _{object} = G M_{P} / R_{P} ^{2}**

Where: M_{P} and R_{P} are the mass and radius of the planet. Examples of g:

- Earth: 9.8 m/s
^{2} - Moon: 1.6 m/s
^{2} - Jupiter: 23.1 m/s
^{2}.

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:

- Kepler's first law, that orbits take the form of ellipses, is a mathematical consequence of Newton's gravitational force (proportional to 1/R
^{2}). - Kepler's second law, that orbits sweep out equal areas over equal times, is a consequence of Newton's laws of motion (conservation of angular momentum).
- Kepler's third law follows from the first two, after working out the math:

**P ^{2} = (4π^{2}/GM_{sun})a^{3}**

where:

- p = the orbital period of a satellite
- a = the semimajor axis of the elliptical orbit
- M
_{sun}is the mass of the object (the sun) being orbited.

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

**V _{o} = √(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:

**P**

^{2}= (4π^{2}/GM_{P})a^{3}Rearranging this math,

**M _{P} = (4π^{2}/GP^{2})a^{3}**

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

_{P}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:

- a longer orbital period
- slower orbital velocity

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.

- If the moon orbits its primary synchronously, then the bulges never move, as in Mimas.
- If the moon rotates with respect to the primary, however, the bulges move around its circumference with every rotation.
- Of course, a moon also exerts a tidal force on the planet it orbits. On Earth, we see the effect of the moon's tidal force in the "stretching" of the oceans to produce two ocean tidal bulges, one facing the moon and one facing away. Additionally, Earth experiences a rock tide of up to 0.38 m.
- What if the moon's orbit is highly eccentric? In that case, the tidal force (and the amount of stretching) will vary with each orbit. Io experiences 100 m rock tides even though its orbit is synchronous for this reason.

**Consequences of tides:**

**The Roche limit**: What would happen in scenario 2 above if the tidal force were stronger than the tensile strength of the cable? Obviously it would break and the two fragments would orbit independently. For scenario three we have a similar question: What if the tidal force were stronger than the moon's gravity? In that case, the moon would be pulled apart. For every pair of planet and moon, there is a distance within which the planet's tidal force will overwhelm the moon's gravity and tear it apart. This is the**Roche limit**(after Edouard Roche who first calculated it in 1848.)

Not really like this: Orbit Buddy by Royaba**Barycenters**: But wait! We know that the moon exerts a tidal force on Earth, raising ocean tides, even though the moon orbits Earth (right.) In fact, both Earth and moon orbit a common center of gravity, their**barycenter**. Because Earth is much larger, it lies inside Earth's volume, but not at its center. Thus, the moon's gravitational pull exerts a tidal force on Earth, as well. As Earth rotates, the tidal bulge stay (more or less) aligned with the moon. To an observer on Earth's surface, two tidal bulges pass by every day. We mostly perceive these as ocean tides, because ocean waters deform more readily than rock. (In the Pluto Charon system, the barycenter actually lies outside of Pluto's volume.)

**Tidal retreat**: Tides influence the orbital relationships of planets and their moons in subtle ways. The moon raises oceanic (and modest rock) tidal bulges on Earth. Because Earth rotates faster than the bulge can mechanically subside, the tidal bulge is always slightly ahead of (and out of alignment with) the moon. As a result:- The moon is pulled into a slightly higher orbit by the gravity of the tidal bulge
- The moon's gravitational pull on the bulge slows Earth's rotation.

**synchronous rotation**in which it always keeps one face toward Earth.

Of course, if a moon's orbit is

**retrograde**like that of Triton around Neptune, then its orbit gradually gets lower because the tidal bulge that it raises slows it down. Eventually it reaches**the Roche limit**, the point at which the tidal force stretching it is stronger than its own gravity. At this point it is torn apart. Triton will eventually suffer this fate.

**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:

- Geocentric model of Claudius Ptolomey
- Retrograde motion
- Epicycle
- Heliocentric model of Aristarchus
- parallax
- Nicholaus Copernicus' explanation of retrograde motion
- Tycho Brahe's "compromise"
- Johannes Kepler
- Features of ellipses
- Foci
- Major axis
- Minor axis
- Semimajor axis
- eccentricity formula - e=c/a

- Kepler's laws of planetary motion:
- Kepler's First Law: elliptical orbits with the Sun at one focus
- Kepler's Second Law: imaginary line from planet to Sun sweeps out equal area per unit time
- Kepler's Third Law: Square of orbital period is proportional to cube of semimajor axis

- Orbital extremes
- Apoapsis
- Periapsis

- Galileo Galilei - telescope observations of "the impossible."
- Isaac Newton's laws of motion
- Formulae for laws of gravity
- Calculation of mass
- What happens when you add or subtract energy from an orbiting body
- Tides
- The Roche limit
- Barycenter
- Tidal retreat
- Synchronous rotation
- Libration