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Gravity . Geocentric vs. Heliocentric Model. The Geocentric Model. Arguments For: Parallax not seen Almagest says so Fits with “heavenly” perfection. Arguments Against: Complicated system -- wheels within (off-centered) wheels Not very accurate No “why”. The Heliocentric Model.

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slide2

Geocentric vs. Heliocentric Model

The Geocentric Model

Arguments For:

  • Parallax not seen
  • Almagest says so
  • Fits with “heavenly” perfection

Arguments Against:

  • Complicated system -- wheels within (off-centered) wheels
  • Not very accurate
  • No “why”

The Heliocentric Model

Arguments For:

  • Simple system
  • Almagest is wrong
  • Not everything revolves about the Earth (Jupiter’s Moons)
  • Heavens aren’t perfect

Arguments Against:

  • Parallax not seen
  • Not very accurate
  • No “why”
  • Earth no longer special
tycho brahe measurements
Tycho Brahe Measurements

Tycho Brahe (without a telescope) made extremely accurate measurements of the positions of the stars and planets.

what tycho brahe observed
What Tycho Brahe Observed
  • A nova was outside the Earth’s atmosphere
what tycho brahe observed5
What Tycho Brahe Observed
  • A nova was outside the Earth’s atmosphere
  • 20 years of extremely accurate measurements of the positions of the planets (~ 2 arcmin precision)
tycho brahe s model
Tycho Brahe’s Model
  • Earth at the center of the “Universe” (because parallax is not seen)
  • The Sun travels about the Earth in a perfect circle
  • The planets move around the Sun in perfect circles
kepler s analysis
Kepler’s Analysis

By working for Tycho Brahe (and then stealing the data), Johannes Kepler had access to the most precise data on planetary positions in history.

He fit the data in every way imaginable.

kepler s laws
Kepler’s Laws

First define the distance from the Earth to the Sun to be 1 Astronomical Unit (1 A.U.)

Kepler’s Laws

  • Planets move in ellipses with the Sun at one focus
kepler s laws9
Kepler’s Laws

First define the distance from the Earth to the Sun to be 1 Astronomical Unit (1 A.U.)

Kepler’s Laws

  • Planets move in ellipses with the Sun at one focus
  • Planets in their orbits sweep out equal areas in equal times
kepler s laws10
Kepler’s Laws

First define the distance from the Earth to the Sun to be 1 Astronomical Unit (1 A.U.)

Kepler’s Laws

  • Planets move in ellipses with the Sun at one focus
  • Planets in their orbits sweep out equal areas in equal times
  • The period, P, of an orbit (in years) squared is equal to a, the semi-major axis of the orbit (in A.U.) cubed. In terms of mathematics,P2 = a3

These laws are empirical – Kepler knew no reason for them

state of physics
State of Physics

By now the world knew:

  • Bodies of different weights fall at the same speed
  • Bodies in motion did not necessarily come to rest
  • Moons could orbit different planets
  • Planets moved around the Sun in ellipses with the Sun at one focus
  • The orbital speeds of the planets obeyed Kepler’s 2nd and 3rd law
  • But why??? Isaac Newton put it all together.
newton s concepts
Newton’s Concepts
  • m(mass): How much stuff something contains
  • v (velocity): A body’s speed and direction
  • a(acceleration): The change in a body’s velocity
  • F (force): What is needed to change a body’s velocity
newton s laws of motion

A body’s velocity will remain constant, unless acted upon by an outside force.

Newton’s Laws of Motion
  • A body’s acceleration depends on the force acting upon it, and will be in the direction of that force. Its resistance to acceleration depends on its mass. In equation form, this is

F = m a

  • For every force, there is an equal and opposite force.
newton s law of gravity
Newton’s Law of Gravity

G M1 M2

F = 

There is an attractive force between two bodies called gravity. The force of gravity depends on the masses of the two bodies, and their separation (squared); the larger the mass, the greater the attraction; the larger the separation, the smaller the attraction.

r 2

Note that the word “separation” means the distance between the centers of the two bodies.

example of gravity a thrown ball
Example of Gravity – a Thrown Ball

When you throw a ball, there are two motions: sideways, and up.

The sideways motion obeys Newton’s first law (bodies in motion will stay in motion). The attractive force of gravity causes the upward motion to decelerate, and then change direction. You see the composite of the two behaviors.

example of gravity a planetary orbit
Example of Gravity – a Planetary Orbit

Imagine a planet moving sideways with respect to the Sun. Newton’s first law says that it will continue to move sideways. But the law of gravity says that it will also be pulled towards the Sun. The result is a combination motion, in which the planet falls towards the Sun, but misses. This is an orbit.

example of gravity a planetary orbit17
Example of Gravity – a Planetary Orbit

Imagine a planet moving sideways with respect to the Sun. Newton’s first law says that it will continue to move sideways. But the law of gravity says that it will also be pulled towards the Sun. The result is a combination motion, in which the planet falls towards the Sun, but misses. This is an orbit.

If the initial sideways motion exactly balances the attractive motion, the orbit will be circular. Otherwise, it will be elliptical.

example of gravity weightlessness
Example of Gravity – Weightlessness

You feel weight because of Newton’s third law. Gravity is pulling you down, but the ground is not allowing you to fall. It must therefore be exerting a force on you to keep you from falling. That force is the weight that you feel.

If you were allowed to fall, you would not feel any weight.

example of gravity binary stars
Example of Gravity – Binary Stars

According to Newton’s third law, just as the Sun exerts a force on the Earth, the Earth exerts a force on the Sun. However, since the Sun is so much more massive, it doesn’t move much. But if the Earth were more massive, the Sun would move. In fact, both the Sun and the Earth circle a common center of mass.

Thus, Kepler’s 3rd law is not quite complete. The true law is

(M1 + M2) P2 = a3

where the orbital period (P) is in years, the semi-major axis (a) is in A.U.s, and the masses (M) are in solar units.

example of gravity binary stars20
Example of Gravity – Binary Stars

According to Newton’s third law, just as the Sun exerts a force on the Earth, the Earth exerts a force on the Sun. However, since the Sun is so much more massive, it doesn’t move much. But if the Earth were more massive, the Sun would move. In fact, both the Sun and the Earth circle a common center of mass.

Thus, Kepler’s 3rd law is not quite complete. The true law is

(M1 + M2) P2 = a3

where the orbital period (P) is in years, the semi-major axis (a) is in A.U.s, and the masses (M) are in solar units.

example of gravity binary stars21
Example of Gravity – Binary Stars

According to Newton’s third law, just as the Sun exerts a force on the Earth, the Earth exerts a force on the Sun. However, since the Sun is so much more massive, it doesn’t move much. But if the Earth were more massive, the Sun would move. In fact, both the Sun and the Earth circle a common center of mass.

Thus, Kepler’s 3rd law is not quite complete. The true law is

(M1 + M2) P2 = a3

where the orbital period (P) is in years, the semi-major axis (a) is in A.U.s, and the masses (M) are in solar units.

example of gravity tides
Example of Gravity – Tides

d

The effects of gravity do not depend on the composition of a body, just its mass and distance. The Moon exerts a force on the Earth, but since the Earth has a finite size, this force is different from one side of the Earth to the other. The side of the Earth near the Moon gets pulled most, the center of the Earth less, and the backside least of all. Since most of the Earth is solid, it doesn’t move much, but water reacts to this difference. So we have tides.

Note that due to the Earth’s rotation, there are 2 high tides and 2 low tides per day.

d

The difference in gravity over on a body with size s is

G M1

F =  s

r 3

example of gravity tides23
Example of Gravity – Tides

The effects of gravity do not depend on the composition of a body, just its mass and distance. The Moon exerts a force on the Earth, but since the Earth has a finite size, this force is different from one side of the Earth to the other. The side of the Earth near the Moon gets pulled most, the center of the Earth less, and the backside least of all. Since most of the Earth is solid, it doesn’t move much, but water reacts to this difference. So we have tides.

Note that due to the Earth’s rotation, there are 2 high tides and 2 low tides per day.

spring and neap tides
Spring and Neap Tides

The Sun also produces tides on the Earth, but due to its much larger distance, the solar tides are only half as strong as the lunar tides. When the Sun, Earth, and Moon are aligned, you get “spring’’ or extra-strong tides. When the Sun, Earth, and Moon are at right angles, the solar tide partially cancels the lunar tide. This produces “neap” tides.

Note that since you can tell time from the Moon, you also can approximate the time of the local high and low tides.

spring and neap tides25
Spring and Neap Tides

The Sun also produces tides on the Earth, but due to its much larger distance, the solar tides are only half as strong as the lunar tides. When the Sun, Earth, and Moon are aligned, you get “spring’’ or extra-strong tides. When the Sun, Earth, and Moon are at right angles, the solar tide partially cancels the lunar tide. This produces “neap” tides.

Note that since you can tell time from the Moon, you also can approximate the time of the local high and low tides.

spring and neap tides26
Spring and Neap Tides

In reality, the times of high and low tides are usually an hour or so off the simple prediction, due to the rotation of the Earth. The water wants to align itself with the Moon, but the rotation of the Earth moves it away.