Astonishing Astronomy 101With Doctor Bones (Don R. Mueller,

Ph.D.)

EducatorEntertainer

JU

G G LE

RPLANETARY

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Chapter 2 - The Copernican RevolutionFrom Ptolemy: to Copernicus, Galileo, Brahe, Kepler and Newton

CopernicusPtolemy Galileo

Brahe Kepler Newton

Because the orbits all lie in more or less the same plane, the paths of the planets through the sky all lie close to the ecliptic, appearing to move through the constellations of the Zodiac.

The Motion of the Planets

Early Models of the Universe

• Aristotle & Ptolemy • Copernicus

Retrograde Motion – Backward moving

• As the Earth catches up to the orbital position of another planet, that planet seems to move backwards through the sky. This is called retrograde motion.

• For example, Mars first moves slowly eastward with respect to the stars. Then this eastward motion falters, stops for a bit and then Mars appears to move westward. In turn, the westward motion wanes, stops and Mars begins its journey eastward. Defined as retrograde motion. Mars shines brightest mid-retrograde.

Geocentric Models • Models in which everything revolves around the Earth are called Geocentric models.

• From earliest Greek times, this kind of model was used to describe the heavens.

• Planets and stars resided on their own spheres, each tipped slightly relative to each other. This reproduced the motion of the planets and Sun through the sky.

Ptolemy and his Epicycles

• Ptolemy introduced epicycles into his model in order to explain retrograde motion:– Planets were attached to small circles

(epicycles) that rotated.– These epicycles were attached to a larger

circle, centered on Earth.

• This can be visualized as a planet attached to a Frisbee, attached to a bicycle wheel with the Earth at the center.

• This model did a fair job of reproducing retrograde motion.

Epicycleshttp://www.lasalle.edu/~smithsc/Astronomy/retrograd.html

• Ptolemy’s (87 - 150 A.D.) solar system was a refinement of Aristotle's (384 - 322 B.C.) universe. The model consisted of a series of concentric spheres, with the Earth at the center (geocentric). To account for the observed retrograde motion of the planets, it was necessary to employ a system of epicycles, wherein the planets moved around smaller circular paths, which in turn moved around larger circular paths around the Earth. This type of retrograde motion is shown in the animation:

Heliocentric Models

• Nicolas Copernicus explored a heliocentric (Sun-centered) model in which everything, including the Earth, revolves around the Sun.

• Retrograde motion is a natural result of these models.

• Copernicus was also able to measure the relative distances between the Sun and the planets.

Copernicus and Planetary Distances

• Copernicus was able to calculate a planet’s distance from the sun by noting the planet’s position at various times– Opposition: When the Earth lies

directly between the Sun and the planet

– Conjunction: When the planet lies directly on the Earth-Sun line• Inferior conjunction: Planet

is between the Earth and the Sun

• Superior conjunction: When the Sun lies directly between the Earth and the planet

– Quadrature: When the planet’s position makes a right angle with the Earth-Sun line

Mercury and Venus

• It was found that Mercury and Venus were closer to the Sun than the Earth, as they were never found very far from the Sun in the sky.

• Mercury’s greatest elongation, or angular separation from the Sun, is never more than 28 degrees.

• Venus’s greatest elongation is never more than 47 degrees.

• Mercury is therefore closer to the Sun than Venus.

Distances to the Other Planets

Copernicus measured the time it took a planet to move from conjunction to quadrature and calculated the planet’s relative distance from the Sun.

Copernicus’ Theorem: Two circles of radii R and R/2 with the smaller one rolling inside the bigger one. A point on the circumference of the small circle traces a straight line segment (i.e., the diameter of the big circle).

http://www.cut-the-knot.org/pythagoras/Copernicus.shtml http://www.youtube.com/watch?v=JuMmFdoMjt0

PlanetCopernicus’s Calculation

(AU)

Actual Distance (AU)

Mercury 0.38 0.39Venus 0.72 0.72Earth 1.00 1.00Mars 1.52 1.52

Jupiter 5.22 5.20Saturn 9.17 9.54

A convenient measure – the Astronomical Unit

• Measuring planetary distances using the Astronomical Unit or AU

• 1 AU = average distance between the Earth and the Sun

• 1 AU ~ 150 million km (1.6 km/mi)

Planetary distances:• Mercury: 0.4 AU• Mars: 1.5 AU• Saturn: 10 AU• Neptune: 30 AU

From the Metric System to Astronomical Numbers

English Units (Distance)12 “lines” = 1 inch12 inches = 1 foot3 feet = 1 yard5.5 yards = 1 rod4 rods = 1 chain10 chains = 1 furlong8 furlongs = 1 mile3 miles = 1 league

Metric Units10 millimeters = 1 centimeter100 cm = 1 meter1000 m = 1 kilometer103 micrograms 103 milligrams

==

1milligram1 gram

1000 g106 milligrams106 micrograms 109 nanograms

====

1 kilogram1 kilogram1 gram1 gram

Scientific Notation

Commonly used prefixes

Number Scientific Notation Prefix Abbreviation

1,000,000,000 1 109 giga G

1,000,000 1 106 mega M

1,000 1 103 kilo k

0.01 1 10-2 centi c

0.001 1 10-3 milli m

0.000001 1 10-6 micro

0.000000001 1 10-9 nano n

Special Units

• The Light Year (ly)Distance light travels in 1

year

Equivalent to a “look-back time” as the light we see from a star left a long time ago.

Example: Proxima Centauri is 4.1 ly away, so the light we see from it today left the star 4.1 years ago!

• The Parsec (pc)

“PARallax SECond” Distance to a body

whose parallax motion covers

1 second of arc

1 pc = 3.26 ly

A Sense of Scale

• Brahe built several instruments to measure the positions of the planets and he did so very accurately :

(~1 arc minute)

• Found that comets moved outside of the Earth’s atmosphere.

• Witnessed the supernova of 1572 and concluded that it was much farther away than any celestial sphere, thus refuting Aristotle's cosmos.

• As he could detect no parallax motion in the stars, he held that the planets go around the Sun, but the Sun, in turn, orbits around the Earth.

Tycho Brahe (1546-1601)

Johannes Kepler (1571-1630)

• Using Tycho Brahe’s data, Kepler discovered that planets follow ellipses with the Sun located at one of the two foci.

• Astronomers use the term eccentricity (e) to describe how round or “stretched out” an ellipse is. For an ellipse, the eccentricity is 0 < e < 1. The closer to e = 1, the flatter the ellipse. In fact, a line segment is a degenerate ellipse with minor axis = 0 and e = 1. When e = 0, the foci coincide with the center point and the figure is a circle.

The Ellipse

The ellipse: a curve that is the locus of all points in the plane, where the sum of the distances from two fixed points (foci F1 and F2) to every point on the ellipse is a constant.

Example of locus: the locus of all points equidistant from a

given point is a circle.http://www.mathopenref.com/ellipse.htmlhttp://mathworld.wolfram.com/Ellipse.html

Draw an ellipsehttp://www.mathopenref.com/constellipse1.html

Kepler’s First Law

• Planets move in elliptical orbits with the Sun at one focus of the ellipse:– Developed a heliocentric

(Sun-centered) model.– Did not agree with the

ancients (or Brahe).– The shape of the ellipse is

described by its semi-major and semi-minor axes.

Aphelion versus Perihelion

Apogee: farthest from Earth Perigee: closest to Earth

Kepler’s Second Law

• The orbital speed of a planet varies so that a line joining the Sun and the planet will sweep out equal areas in equal time intervals.

• Planets move faster when closer to the Sun and slower when farther from the Sun.

Kepler’s Third Law

• The amount of time a planet takes to orbit the Sun (its period) P is related to the size of its orbit a by the equation:

P2 = a3

• Kepler’s Laws describe the shape of a planet’s orbit, its orbital period and how far from the Sun the planet is positioned.

• The closer a planet is from the sun, the faster is its orbital speed. By contrast, the further a planet is from the sun, the slower is its orbital speed.

Copernicus versus Kepler• The model of Copernicus: the eccentricity of a planetary orbit is zero. In

other words, the planetary orbit is a circle.A. The Sun is at the center of the circle.B. The planet’s orbital speed is constantC. The square of the Period is proportional to the cube of the distance from

the Sun.

• The model of Kepler: the eccentricity of a planetary orbit is non-zero. That is, the planetary orbit is an ellipse.

A. The Sun is at a focal point.B. Neither the linear speed nor the angular speed of the orbiting planet is a

constant. However, the area speed is constant (i.e., equal areas in equal times).

C. The square of the Period is proportional to the cube of the average of the maximum and minimum distances from the Sun.

Astronomy (1500 – 1700)

The Science of Astronomy (circa 1500 – 1700)

Carl Sagan on Ptolemy and Copernicushttp://www.youtube.com/watch?v=faqjmAoXpM4&NR=1

Copernicus, Brahe and Keplerhttp://www.youtube.com/watch?v=IBvMhpx8Q0Q

Kepler’s 3rd Law (enter Newton’s law of gravity)http://www.youtube.com/watch?v=ShQXRBDBfaA&NR=1

Carl Sagan on Brahe and Kepler:http://www.youtube.com/watch?v=xsX6BeBOorA&NR=1

Galileo Galilei (1564-1642)

• Using a Dutch-designed telescope that he built, he made several observations that disproved ancient thinking about the Universe:

• Found sunspots. Found craters on the Moon.• Discovered four moons of Jupiter, showing

that not everything revolved around the Sun.• Observed the rings of Saturn.• Observed that Venus passed through all

phases, just as the Moon does. In a geocentric model, the phases of Venus were limited to crescents.

Interestingly, the changing phases of Venus were used by Galileo to refute Ptolemy’s epicycles.

Isaac Newton (1642-1727)• Isaac Newton described his three

laws of moving bodies:

1. 1st Law: Inertia2. 2nd Law: F = ma3. 3rd Law: Action-Reaction

• Co-invented Calculus.

Interestingly, according to Newton's laws of motion, an object undergoes natural motion when it moves at a constant speed along a straight line.

Mass and Inertia

• Mass is described by the amount of matter an object contains.

• This is different from weight as weight requires gravity or some other force to exist: W = mg

• Inertia is simply the tendency of mass to stay at rest or to remain in motion at constant velocity.

• Objects at rest may also possess potential energy Epe

Epe = mgh h - height

The Law of Inertia

• Newton’s First Law is sometimes called the Law of Inertia:– A body continues in a state of

rest or in uniform motion in a straight line at a constant speed, unless made to change by forces acting on it.

– More simply, a body maintains the same velocity unless forces act on it.

• A ball rolling along a flat, frictionless surface will keep going in the same direction at the same speed, unless something pushes or pulls on it.

Another View of Newton’s First Law

• If an object’s velocity is changing, there must be forces present:– Dropping a ball– Applying the brakes in a car

• If an object’s velocity is not changing, either there are no forces acting on it, or the forces are balanced and cancel each other out:– Hold a ball out in your hand,

and note that it is not moving.

– Force of gravity (downward) is balanced by the force your hand applies (upward).

Circular Motion

• Tie a string to a ball and swing it around your head:– The Law of inertia says that the

ball should go in a straight line.– The ball goes in a circle – there

must be forces at work.

• Where’s the force?It’s the tension in the string. The swing

changes the direction of the ball’s velocity vector.

If the string breaks, the ball will move off in a straight line (while falling to the ground).

Force, Work and Energy

Force = (mass)(acceleration) Work = (Force)(distance)

Total Energy (Etotal) = Kinetic (Eke) + Potential (Epe)

• Eke = ½ mv2 Energy is the capacity to do work

• Epe = mgh

Force units: Newton (N) = kg m/s2

Energy units: Joule (J) = Nm = kg m2/s2

Work is measured in Joules

Acceleration

• The term acceleration is used to describe the change in a body’s velocity over time:– Stepping on the gas

pedal of a car accelerates the car – it increases the speed.

– Stepping on the brakes decelerates a car – it decreases the speed.

• A change in an object’s direction of motion is also acceleration:– Turning the steering

wheel of a car makes the car go left or right – this is an acceleration.

– Forces must be present if acceleration is occurring.

timein changevelocity in change

ation acceler

Newton’s Second Law

• The force (F) acting on an object equals the product of its acceleration (a) and its mass (m):

• F = ma• We can rearrange this to be:

• a = F/m• For an object with a large

mass, the acceleration will be small for a given force.

• If the mass is small, the same force will result in a larger acceleration.

• This simple expression can be used to calculate everything from how hard to hit the brakes to how much fuel is needed to go to the Moon!

Newton’s Third Law

• When two bodies interact, they create equal and opposite forces on each other.

• If two skateboarders have the same mass and one pushes on the other, they both move away from the center at the same speed.

• If one skateboarder has more mass than the other, the same push will send the smaller person off at a higher speed and the larger one off in the opposite direction at a smaller speed.

Newton’s Third Law

• The Sun is 300,000 times more massive than Earth– The gravitational force of the

Sun on the body of Earth is equal to the gravitational force of Earth on the body of the Sun.

– The Sun’s acceleration is 300,000 times less than the acceleration of Earth

– A wobble is still present.• Extra-solar planets are often

found using this effect.

Orbital Motion and Gravity

• Astronauts in orbit around the Earth are said to be in free fall, a weightless state.

• Imagine a cannon on top of a mountain that fires a cannonball parallel to the ground.

• The cannonball leaves the cannon and is pulled toward the ground by gravity.

• What if we gave the cannonball a very large velocity, so large that it “misses” the Earth?

• The cannonball would be in orbit around the Earth and it would be falling!

Newton’s Universal Law of Gravitation

• Every mass exerts a force of attraction on every other mass. The strength of the force is proportional to the product of the masses divided by the square of the distance between them– Simply put, everything pulls on

everything else.– Larger masses have a greater

pull.– Objects close together pull more

on each other than objects farther apart.

• M: Mass of one body• m: Mass of the other body• d: Distance between the two

bodies’ centers• G: Gravitational constant

• G = 6.67 10-11 N m2/kg2

Surface Gravity

• Objects on the Moon weigh less than objects on Earth.

• This is because surface gravity is less:– The Moon has less mass

than the Earth, so the gravitational force is less.

• We let the letter g represent surface gravity, or the acceleration of a body due to gravity.

• F = mg = W = weight• On Earth, g = 9.8 m/s2

• g on the Moon is around 1/6 as much as on the Earth.

Centripetal Force• If we swing a mass around in a circle, a

force is required to keep the mass from flying off at a tangent to the circle.

• This is a centripetal force FC, a force directed towards the center of the circle.

• The tension in the string provides this force.

• Newton determined that this force can be described by the following equation:

dmVF

2

C

• For planets, the centripetal force FC keeping the planet moving in an elliptical path is the gravitational force FG.

• Therefore, we set FG = FC and solve for the planet’s mass M.

• If we know the orbital speed of an object orbiting a much larger one and we know the distance between the two objects, we can calculate the larger object’s mass.

Masses from Orbital Speeds

GdVM

2 G2

2C F

dGmM

dmVF

Newton’s Modification of Kepler’s 3rd Law

• Newton applied his ideas to Kepler’s 3rd Law and developed a version that works for any two massive bodies, not just the Sun and its planets:

• Here, MA and MB are the two object’s masses.

• This expression is useful for calculating the mass of binary star systems and other astronomical phenomena.

2

324Pd

GMM BA

• We can find the mass M of a large object by measuring the velocity of a smaller object orbiting it and the distance between the two bodies:

• Rearrange this expression to get something very useful:

Orbital Velocity

dGMVcirc

GdVM

2

Calculating Escape Velocity

• From Newton’s laws of motion and gravity, we can calculate the escape velocity: Vesc

a) G – gravitational constantb) M – mass of planetc) R – radius of planet

• The following equation for the escape velocity Vesc shows that the velocity required to put a satellite into orbit does not depend on the mass of the satellite.

R2GMVesc

Please insert figure 18.2

What Escape Velocity Means

• If a rocket, is launched with a velocity less than the escape velocity, it will eventually return to Earth.

• If a rocket travels at the escape velocity, it will leave Earth following a parabolic trajectory.

• If a rocket achieves a speed greater than the escape velocity, it will leave the Earth along a hyperbolic trajectory, and will not return.

Escape Velocity is for more than just Rockets!

• The concept of escape velocity is useful for more than just rockets.

• It determines which planets have an atmosphere and which don’t:

• Objects with a smaller mass (such as the Moon or Mercury) have a low escape velocity. Gas particles near the planet can escape easily, so these bodies don’t have much of an atmosphere.

• Planets with large mass, such as Jupiter, have very high escape velocities, so gas particles have a difficult time escaping. Massive planets tend to have thick atmospheres.