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1445 Introductory Astronomy I

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1445 Introductory Astronomy I

Chapter 2

Gravity and Planetary Motion

R. S. Rubins Fall, 2009

- Cosmologyis the study of the origin and structure of the Universe.
- In geocentric cosmology, which was the prevailing cosmology up to the 16th century, the Earth was considered to be at the center of the universe.
- In heliocentric cosmology, the planets were assumed to orbit the Sun, which was at the center of the universe.
- Apart from the ideas of Heracleides (c. 350 BCE) and deductions of Aristarchus (c. 270 BCE), geocentric ideas held sway for about1800 years, both in science and religion, until Copernicus (1476 – 1543) reintroduced the heliocentric system.

- Plato (428 - 348 BCE) asserted that heavenly motion must be in perfect circles, which was an idea which hindered science for about 2000 years.
- Aristotle (384 - 322 BCE) argued for a geocentric universe.
- Heraclaides (c. 350 BCE) is credited with suggesting that the “heavenly” objects orbit the Sun and not the Earth, and also that the Earth rotates around its north-south axis.
- However, Aristarchus of Samos (c. 270 BCE), the philosopher most strongly associated with the heliocentric universe, based his deductions on his trigonometric estimates of the relative sizes of the Sun, Moon and Earth.
- Although his book “On the Sizes and Distances of Sun and Moon” became famous, his heliocentric philosophy was harshly criticized.
- His status as a philosopher was insufficient to overcome the prejudice against the heliocentric universe for over 1800 years.

- Aristotle (ca. 350 BCE) deduced that the Earth was spherical from the following observations:
- the curved shadow of the Earth during a lunar eclipse;
- the change in position of the constellations with latitude;
- the disappearance of the hull of a ship first when it crosses the horizon.
- Eratosthenes (ca 200 BCE) determined the Earth’s radius by comparing angles made by the Sun with the vertical at the summer solstice. This angle was about 7o at Alexandria, Egypt and 0o (directlyoverhead) at Syene, about 520 miles to the South.
- Since a circle corresponds to a rotation of 360o, he estimated the Earth’s circumference to be 520 x (360/7) ≈ 27,000 mi; i.e. a diameter of 8600 mi (actual value, 7970 mi).

Vertical

Vertical

- By measuring the Sun-Earth-Moon angle when the Moon is exactly half lit, Aristarchus (ca. 270 BCE) calculated that the Sun is about 20 times further from Earth than is the Moon (correct answer 390).

23½o

By observing the the Earth’s curved shadow on the Moon during a

lunar eclipse, Aristarchus deduced that RM≈ 0.3 RE.

- Knowing the Moon’s diameter DM = 2 RM and the angle it subtends at the eye Φ(in radians), Aristarchus determined the distance of the moon R to be about
R = DM/Φ ≈ 250,000 mi.

- Since the Sun has a much larger diameter than the Earth, Aristarchus proposed that the Earth must orbit the Sun, and not vice-versa.

- Apollonius (240 - 190 BCE) introduced the concept ofepicycles (circles upon circles) to explain the retrograde motions of planets.
- Hipparcus (190 - 120 BCE) developed Apollonius’ theory into a method that could predict planetary positions.
- In medieval times, the triumphs of Greek thought were preserved and enhanced in the Arab world.
- In about 140 CE, Ptolemy used a geocentric model to perfect a method of predicting planetary motions, which remained in use for 1500 years.
- Over 1300 years later, the Polish astronomer , Copernicus (1476 -1543), was able to remove the complexities of the Ptolomaic system, by reviving the heliocentric model.

- Planets (from the Greek for wanderer) were so-called because they changed their positions on a nightly basis with respect to the “fixed stars”.
- Viewed from the northern hemisphere, the usual motion of the planets, known as direct motion, is to the east with respect to the fixed stars.
- Occasionally, retrograde motion - a reverse motion to the west with respect to the fixed stars - is observed.
- Ptolemy (ca. 140 CE) represented planetary motion by superimposing two circles, a large one known as the deferent (guiding circle), and a small one, theepicycle.
- When his book was translatedby Arabic scholars in about 800 CE, it was given the title Almagest, meaning “the greatest compilation”.

- Lie on your back, with your head pointing north.

East

West

Surrounding the Sun,

starting on the outside

are the following:

I. fixed stars;

II. Saturn;

III. Jupiter;

IIII. Mars;

V. Earth and Moon;

VI. Venus;

VII. Mercury.

- Throughout history, there have been thinkers, who have been far ahead of their time, such as Heraclaides and Aristarchus, who introduced the heliocentric universe nearly 1900 years before Copernicus.
- Giordano Bruno was such a thinker; he wrote:
“In space there are countless constellations, suns and planets; we see only the suns because they give light; the planets remain invisible, for they are small and dark. There are also numberless earths circling around their suns, no worse and no less than this globe of ours.”

- Bruno was burnt at the stake in Rome for his views, but over 400 years later, in 1995, the first extrasolar planet was discovered orbiting the star 51 Pegasi by Swiss astronomers.

- The siderial period (the time for one complete circle) of Mercury is 0.24 y.
- The synodic period, which is measured between successive inferior conjunctions, is 0.32 y (or 116 days).

- Tycho Brahe was colorful character and a remarkable observational astronomer.
- Living before the invention of the telescope, Tycho made the most precise naked-eye planetary observations of all time.
- His planetary measurements were accurate to within 1 arcsecond (less than the thickness of a finger nail held at arms length).
- In 1600, he hired Johannes Kepler as his apprentice, and, on his death-bed a year later, begged Kepler to make sense of his observations.
- He moved to Austria, and in 1599, as court astrologer and mathematician to the emperor, he made the observations that lead Kepler to deduce his famous laws of planetary motion.

- Kepler was a mystic who believed that numbers and simple geometrical shapes, in particular circles, governed the structure of the universe.
- He was for many years the imperial court astrologer and mathematician, and since the emperor rarely paid his salary, made money by selling astrological material.
- He spent many years trying to fit Brahe’s observations to the Copernican heliocentric system, using Ptolemy’s idea of circles and epicycles.
- Kepler was a prolific writer, who wrote perhaps the first science fiction novel Somnium, a dream of a journey to the Moon, in which the concept of a gravitational force was anticipated.

- He labored for several years, but could not overcome a difference between theory and observation of up to 8 arc-minutes (one quarter the angular diameter of the Moon) between theory and observation.
- Trusting Tycho’s careful work, Kepler was forced to abandon his deeply-held belief in circular orbits, and work on the hypothesis that planetary orbits were elliptical.
- In 1609, Kepler published his laws of planetary motion.
- Possibly Kepler’s greatest achievement was the major psychological breakthrough he achieved through his rejection of the circular orbit dogma. (He was also excommunicated by the Church.)

- The eccentricity e varies from 0 for a circle to 1 for a straight line.

The orbit of a planet about the Sun is an ellipse, with the Sun at one focus.

- The line joining a planet to the Sun sweeps out equal areas in equal times.
- Thus, a planet moves fastest when closest to the Sun.

- The square of the sidereal periodof a planetis proportional to its (Mean distance from the Sun)3 ; i.e.P2 = a3.

A conic section is a curve obtained by slicing a cone with a plane.

- Galileo is famous for the following contributions to science:
i. his contributions to the physics of falling bodies;

ii. the first known telescopic observations of the night sky;

iii. publicizing the work of Copernicus.

- He observed the following:
i. mountains on the Moon;

ii. dark spots on the Sun (sunspots);

iii. four moons orbiting Jupiter;

iv. the phases of Venus, showing that Venus orbited the Sun;

v. a ring on Saturn;

vi. that the white band in the night sky, known as the milky

way is composed of many stars too close and faint to be

resolved as separate by the unaided eye;

vii. that planets appear as disks, and the stars as points.

- Galileo made important contributions to basic physics, paving the way for Newton’s monumental work.
- From experiments on objects rolling or sliding down slopes, he deduced that, in the absence of air resistance, all objects would fall at the same rate at the Earth’s surface.
- This hypothesis disagreed with Aristotle’s 2000 year-old idea that heavier objects fell faster.
- Galileo realized that an object moving on a frictionless horizontal surface could circle the Earth forever.
- This idea was the basis for Newton’s 1st Law of Motion, also known as the Law of Inertia.

First observed by Galileo in 1610, these drawings of Jupiter

and its 4 largest moons were made by Jesuits in 1620.

- Galileo noted that the apparent size of Venus was always largest at the crescent phase, and smallest at the gibbous phase.
- This observation was a direct verification of the heliocentric model.

- Newton’s greatest work was done at the age of 25, when Cambridge University was closed for 18 months because of the Great Plague, and he was forced to live at home.
- In his major work on physics and mathematics, “The Principia”, published in 1687, Newton introduced the first great laws of physics: the Laws of Motion and the Law of Gravitation. The consequences of these laws are still being calculated today.
- Starting with “common sense” ideas about space and time, his three Laws of Motion deal with the effects of force on physical objects.
- To obtain his Law of Gravitation, Newton applied the Laws of Motion to both a falling object and the Moon’ orbit, realizing that each was due to the same force – gravity.
- Newton’s laws of motion and gravity (1687), the greatest achievement of classical physics, explained all of mechanics, including Kepler’s laws.

Nature and Nature’s laws lay hid in the Night

God said, Let Newton be! and all was Light

Alexander Pope,1727

- Mass is the quantitative measure of inertia, which is the resistance of an object to a change of motion.
- We normally find the mass of an object by measuring its weight, which is the force of the Earth’s gravity on it; i.e.
Fgrav= weight = mg,

where g = 9.8 m/s2 is the gravitational acceleration.

- Weight depends on location, but mass is independent of location.
- On the Moon, gravity is 1/6 th of its value on Earth, so that a “120 lb person” would weigh just 20 lb there.
- In a coasting space vehicle, objects are weightless, but their masses remain unchanged, so that an elephant hitting you in space would have the same effect as on Earth.

- First Law
An object moves with constant velocity unless a net force acts to change either its speed or its direction.

- This law is also known as the Law of Inertia, since an object in motion resists being slowed down or speeded up.
- A coasting spaceship needs no fuel to keep moving.

- Second Law
A net force F gives an object of mass m an acceleration a according to the equation

F = ma.

In the photo, the pitcher’s arm gives the ball its acceleration.

Remember that an acceleration is a change of velocity.

An object moving at

constant speed in a

circle has a centripetal

acceleration, given by

ac = v2/r.

A centripetal force Fc, which is a force towards the center

of the circle, is needed to produce a centripetal acceleration

ac; i.e.

Fc= mac.

For astronomical objects, this force is gravity.

Without gravity, the

satellite would move in

a straight line.

Gravity continuously

forces it from its straight

line, causing it to move

in a circle.

The gravitational force

on the satellite always

points towards the

Earth’s center.

- Third Law To every action there is an equal and opposite reaction.
- The downward force with which the gas is expelled from the rocket is equal in magnitude to the upward force on the rocket.

- Every object in the Universe attracts every other object with a force proportional to the product of the masses divided by the square of their separation d; i.e.
F = G m1m2 /d2,

where G is the (universal )gravitational constant.

- Newton’s realized that an apple falls to the ground for the same reason as the Moon orbits the Earth - gravity.
- Using his Laws of Motion and a little guesswork, Newton compared the centripetal acceleration of the Moon in orbit about the Earth with the downward acceleration (g = 9.8 m/s2) of a falling object near the Earth’s surface.
- Gravity was the first of the fundamental forces to be described mathematically; the others we know about are the electromagnetic force (nineteenth century) and the strong and weak nuclear forces (twentieth century).

- Physics deals with energy and matter.
- Energy, comes in many forms, such as:
kinetic energy (KE), the energy of motion;

potential energy (PE) of many types, so-called

because it can be converted to kinetic energy.

radiative energy, which is carried by EM waves.

- Matter is simply material, characterized by its mass. Einstein’s famous equation, E = mc2, indicates that matter is just a form of energy.

- The KE of a moving object is ½ mv2.
- The thermal energy of a gas is the total KE of its molecules.
- The temperature of a gas (in K) is proportional to the average KE of its molecules.
- Heat is the energy transferred from one object to another because of a difference in temperatures.
- Potential energy has many forms, such as gravitational, elastic, electrical, chemical.
- Energy units are J (joules) and eV (electron-volts).

- The Law of Conservation of Energy states that, although the form of energy may change, the total quantity of energy remains constant.
- Example 1 When you drop a rock , its gravitational potential energy is converted to kinetic energy. When the rock hits the ground, its kinetic energy is transferred largely to thermal energy in the rock and ground.
- Example 2 When a positron meets its antiparticle, an electron, the two particles annihilate each other, converting their mass-energy to electromagnetic energy in the form of gamma rays.

- Newton’s Laws not only explained the motions of the planets, given by Kepler’s Laws, but also the motions of moons and comets
Newton’s Laws indicated that an unknown planet was affecting the orbit of Uranus. Astronomers searched, and found Neptune.

The limitations of Newton’s theories became apparent in the twentieth century, when they were superseded by the revolutionary new theories of:

- i. Special Relativity for objects traveling at very
high speeds;

- ii. Quantum Mechanics for the smallest particles;
- iii. General Relativity for the behaviour of large
masses.

- Newton’s theories are still used widely, for example, in structural design and aerospace engineering.

“…the most impressive fact is that gravity is simple. It is simple to state the principles completely and not have left any vagueness for anybody to change the ideas of the law. It is simple, and therefore it is beautiful. It is simple in its pattern. I do not mean it is simple in its action - …to follow how all those stars in a globular cluster move is quite beyond our ability.”

Richard Feynman in the ‘The Character of Physical Law’, 1965.

“Finally comes the universality of the gravitational law,… that Newton, in his mind, worrying about the solar system, was able to predict what would happen in an experiment of Cavendish, where Cavendish’s little model….of two balls attracting, has to be expanded ten million million times to become the solar system. Then ten million million times larger again we find galaxies attracting each other by exactly the same law.“

Richard Feynman in the ‘The Character of Physical Law’, 1965.