Planetary Motion

by Nick D’AnnaEarth Science Teacher

Plainedge Middle School

Planet Names

Tuesday Martedì (Italian) Mars’ dayWednesday Mercoledì (Italian)

Mercury’s dayThursday Giovedì (Italian) Jupiter’s

dayFriday Venerdì (Italian) Venus’ daySaturday Saturn’s day

Sunday Sun’s day (not a planet but still important)

Monday Lunedì (Italian) Moon’s day (also not planet, but also important because it moves differently than the other things in the sky.

The Greeks believed that the planets traveled in circular paths.

Since the acceleration (force of gravity) is perpendicular to the velocity of the body, the torque on the body is zero. Thus, the velocity of the body remains constant.

However, the planets did not move with constant speed.

Planet comes from the Greek word Planētēs:

WANDERER

The planets move differently than all the other celestial

objects.

Retrograde motion explained by Hipparchos & Ptolemy

• Ptolemy believed in a Geocentric (Earth Centered) model of the Solar System

• Ptolemy explained retrograde motion with DEFERENTS & EPICYCLES.

• The math involved for Ptolemy’s model with epicycles became extraordinarily complicated.

Copernicus & the Heliocentric model (Sun-centered) Solar

System Copernicus was able to explain the retrograde motion of the planets just as well as Ptolemy. However, Copernicus’ model still had it’s problems.

Copernicus used perfect circular motion, unlike Ptolemy, who had the Earth offset as an equant (not centered in circular orbits

Ockham’s Razor

Cited from http://sbast3.ess.sunysb.edu/fwalter/AST101/occam.html

The most useful statement of the principle for scientists is: "when you have two competing theories which make exactly the same predictions, the one that is simpler is the better.“

The Copernican system was more elegant and more aesthetic than Ptolemy’s system. Hence, it had favor.

Johannes Kepler (1571 – 1630)

• Believed the Universe was driven by mathematical principals

• There must be a force, propelling planets to move. The force was something like magnetism between the Sun and the planets.

• Devised Three Laws of Planetary Motion

Kepler’s Laws

• Law of Ellipses (1609)

• Law of Equal Areas (1609)

• Harmonic Law (1618)

Kepler’s First Law

An ellipse is a geometric shape somewhere between a circle and a parabola. ECCENTRICITY measures how round or flattened an ellipse.

Ellipses

EccentricityE = distance between the foci ÷ Length of major axis

Effects of elliptical orbits

• Changes in gravitational pull between planet and Sun

• Changes in orbital velocity

• Changes in apparent angular diameter

Kepler’s 2nd Law

Kepler’s 2nd law can be equated to the conservation of angular

momentum.

If the net torque on a body is zero, then the angular momentum will be

conserved

A of ∆AoB closely approximates the area swept out in time (dt) by a line connecting the Sun and the planet

dѲ

The base of ∆AoB = rdѲ and the height is r.

dA/dt = ½(r2)(dѲ/dt) dѲ/dt = ω, where ω is the angular velocity

dA/dt = ½r2ω or r2ω/2

Area of triangle = ½(base x height)

Area = ½(r)(rdѲ) = ½r2dѲ

The angular momentum (L) of a planet around the sun is the

product of the r and the component of the momentum

perpendicular to r. L = rp┴ = (r)(mv┴) = (r)(mωr) = mr2ω

Bringing it all together:

dA/dt = (r2/2)(dѲ/dt) = r2ω/2L = rp┴ = (r)(mv┴) = mr2ω

dA/dt = /2

r2ω = L/m

r2ωdA/dt = L/2m

If angular momentum is conserved, L is constant, then dA/dt must also be constant.

Kepler’s 3rd Law: Harmonic Motion

Galileo• Lived at the same time as Kepler.• Studied falling bodies and the way they accelerate

toward Earth

• Introduced the Law of inertia• Made crucial astronomical observations:

– Moon’s orbiting Jupiter.– The surface of the Moon looks like the surface

of Earth. It has mountains and craters, etc… It is not perfect.

• Dealt the final blow to the Ptolemic system of the Solar system. And also a major problem for the Roman Catholic Church

Isaac Newton (1643 – 1727)Unified Kepler’s and Galileo’s work.

Ode to Newton

“Once in a great while, a few times in history, a human mind produces an observation so acute and unexpected that people can’t quite decide which is the more amazing – the fact or the thinking of it. Principia was one of those moments”

Bill Bryson, A Short History of Nearly Everything.

Newton’s 1st Law: Inertia and Momentum

Inertia: A moving body tends to keep moving, and a stationary body tends to remain at rest.

Momentum: The product of mass and velocity ρ = mv

Newton’s 2nd Law: Force

ƒ = maNewton’s 3rd Law: Reaction

For every applied force there is an equal and opposite reaction force

Derivation of the Universal Law of Gravity from Newton’s Laws of motion and Kepler’s

Laws.ƒ = ma For circular

motion a = v2/r

ƒ = mv2/r

Centripetal Force on a planet

From F = mv2/r, let’s look at v

Velocity is distance over time. For simplicity we’ll use a circular path, so the distance is 2πr (the circumference of a

circle)And the time for a planet to travel in its orbit is called the Period (P)

Therefore,

V = 2πr/P

In the centripetal force equation, F = mv2/r , the

velocity is squaredRecall v = 2πr/P

Square it V2 = 4π2r2/P2

Substituting everything into the centripetal force equation, F =

mv2/r

F =

m • v24π2r2/P2

r

Recall Kepler’s 3rd Law of Harmonic Motion P2 = Ar3

Apply this law to the centripetal force equation:

F = m4π2r2/

P2 rAr3

Simplify the equation to:

F = m4π2/Ar2

F = m4π2/Ar2

Remove the constant value from the above equation

F ά m/r2

The mass of the planet (m) is also constant, therefore,

F ά 1/r2

Where, ά means proportional to

We’re not done.According to Newton’s 3rd law (reaction), if the sun exerts a force on the planet, the planet must exert a force on the Sun.

F ά m/r2 is the force on the planet by the sun, then F ά M/r2 is the force on the Sun exerted by the planet, where M is the mass of the Sun.

Which produces a net force of :

F ά mM/r2

Satellite in Motion

Bibliography• Zielik, Michael. Astronomy, The Evolving Universe 7th Edition.

John Wiley & Sons, Inc., 1994.• Cutnell, John D and Kenneth W. Johnson. Physics, 3rd Edition.

John Wiley & Sons, Inc., 1995.• Abell, George O, David Morrison and Sidney C. Wolf. Exploration

of the Universe, 6th Edition. Saunders College Publishing, 1991.• Halliday, David, Robert Resnick and Jearl Walker. Fundamentals

of Physics, Volume 1, 5th Edition. John Wiley & Sons Inc., 1997.• Epstein, Lewis C. Thinking Physics is Gedanken Physics. Insight

Press, 1983.• Byson, Bill. A Short History of Nearly Everything. New York:

Broadway Books, 2003.• Seifert, Howard S and Mary Harris Seifert. Orbital Space Flight,

The Physics of Satellite Motion. New York: Holt, Rinehart and Winston, Inc., 1964.

• Goldstein, David L and Judith R. Goldstein. Feynman’s Lost Lecture, The Motion of Planets Around the Sun. New York: W.W. Norton & Company, Ltd., 1996.