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Measuring the Universe: Earth

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Eratosthenes (about 200 BC):

- Sun overhead at Syene on summer solstice
- Sun 7º from zenith at Alexandria on same day
SO: Alexandria is 7º in latitude from Syene

In terms of fractions of a circle:

Distances to planets and Sun helped settle whether Earth or Sun was at the center of everything…

Ancient people attempted measurements, but techniques weren’t accurate enough

- planets appear to circle Earth once a day
- BUT planets move compared to stars over many days
- planets stay near ecliptic
ecliptic goes through zodiac constellations

VENUS

VENUS

- always near Sun in sky
- best seen just before sunrise or just after sunset
- never seen at midnight

MERCURY

MERCURY

LOOKING WEST AT SUNSET

LOOKING EAST AT SUNRISE

- sometimes seen high overhead at midnight
- usually move W to E relative to stars: “direct motion”
- sometimes move E to W relative to stars: “retrograde motion”

Mars in direct motion (compared to stars)

E

W

Mars in retrograde motion

At roughly what time would the planet at position 5 be highest above the horizon? (Remember that Earth rotates counterclockwise from this point of view.)

- 3 am
- 9 am
- 3 pm
- 9 pm
- It is not possible to tell from the diagram

Where would planet A be seen in the sky from Earth at sunset?

A

Geocentric Model

(Claudius Ptolemy, around 140 AD)

All planets move on epicycles (circular paths) that circle Earth

- epicycles of inferior planets attach to a line between Earth and Sun
- epicycles of superior planets circle Earth independently

Epicycles are needed to create retrograde motions of planets

happens when Earth catches and passes a superior planet

Where would you look to see a planet rise when it is in retrograde motion?

- near the eastern horizon
- near the western horizon

If you lived on the planet Mars and you monitored Earth’s position in the sky over the course of several years, what would see?

- Earth always moves from east to west relative to the stars.
- Earth always moves from west to east relative to the stars.
- Earth usually moves from west to east relative to the stars, but occasionally undergoes retrograde motion (east to west).
- Earth is always fairly close to the Sun in the sky, and is most easily visible before sunrise or after sunset.

Nicolaus Copernicus (around 1530 AD)

maximum elongation (e): largest angle between planet and Sun

used to determine planet’s distance from Sun:

dp

e

d

Discovered by trial and error…

Kepler’s First Law: (SHAPES OF ORBITS)

All planet orbits are ellipses with Sun at one focus

eccentricity

flashlet

applet

Solar eclipses can be either total or “annular”…

the Moon’s distance from Earth changes…

semi-major axis (a): half length of long side

average distance from Sun

eccentricity (e):

center

Sun

c

a

rP

rA

aphelion: farthest point from Sun

perihelion: closest approach to Sun

SPECIAL CASE: CIRCLE

semi-major axis (a) equals radius

Sun

a

TOP VIEW:

Sun

Orbits are flat (they can fit in a flat plane)

BUT

They are usually tilted relative to each other…

inclination (i): angle between Earth’s and object’s orbit planes

rP

rA

SIDE VIEW:

Sun

Inclination i

Earth orbit

- Kepler’s Second Law: (SPEED DURING ORBIT)
A line connecting Sun and planet sweeps out equal areas in equal times.

A1

flashlet

A2

CLOSER TO SUN GREATER SPEED

applet

Orbit

(COMPARING PLANETS)

- Kepler’s Third Law:
P: orbital period of planet (sidereal period)

a: planet’s average distance from Sun (semi-major axis)

Sun

applet

a

1 AU (Astronomical Unit) is Earth’s average distance from Sun

1 AU = 1.5 1011 m

applet

applet - defunct

NASA wants to launch a spacecraft to go out to the planet Mars (without stopping there), and then come back. If the spacecraft follows the orbit below (dotted line),

- What is the semi-major axis of the orbit?
- How long would it take to get to Mars from Earth?

1 AU

1.5 AU

If it takes Eris 557 years to orbit the Sun, what is its average distance from the Sun?

- Jupiter:
- Eris:

- Sedna:

The asteroid Apophis has an orbit with the following characteristics:

- semi-major axis: a = 0.922 AU
- orbit period: P = 0.89 yr
- eccentricity: e = 0.191
- inclination: i = 3.3º
If Earth’s orbit has a semi-major axis of 1 AU and Venus has a semi-major axis of 0.723 AU, does Apophis’ orbit cross the planet orbits?

- Apophis always stays between Earth’s and Venus’ orbits.
- Apophis goes outside Earth’s orbit and inside Venus’ orbit.
- Apophis goes outside Earth’s orbit.
- Apophis goes inside Venus’ orbit.

- asteroid about 350 m across
- close approach to Earth in 2029

- both Earth and other planets are moving
SO…

- how do you determine when a planet comes back to the same place in its orbit?

Sidereal (orbit) periods (Porb):

time for a planet to make exactly one orbit around Sun

- only Earth’s is directly measurable

SUN

Synodic period (Psyn) :

time between “line-ups” of Earth, Sun, and planet

- measurable from Earth

From Sun’s point of view:

Fast planet orbits in timePfast:

- moves W to E with angular speed:
Slow planet orbits in timePslow:

- moves W to E with angular speed:
LINE-UPS:

- Because both move in same direction, it takes longer for fast planet to “lap” slow one (travels an extra 360º in time Pline-up)
- rate:

It takes Earth, Venus, and Sun 584 days to go between line-ups. What is Venus’ orbit period?

(Hint: Which planet is the fast one, and which is the slow one?)

Two observers on Earth see planets in slightly different positions in sky (compared to stars)

- the bigger the angle, the closer the planet must be

angle a

… this is the same idea behind your two eyes

- spacecraft lands and fires rocket to push asteroid
- if asteroid is spinning, spacecraft must land at a “pole”
- less effort required if done farther in advance
- What direction will the asteroid go?

VELOCITY

- speed: how fast an object’s position changes
measured by speedometers, radar guns

- velocity: speed and direction of travel
measured by weather vanes

- First Law: An object will maintain a constant velocity if there is no net force acting on it.

- acceleration: how fast velocity changes
3 ways to accelerate a car:

- GAS PEDAL (change speed)
- BRAKE PEDAL
- STEERING WHEEL (change direction)
- force: strength and direction of a push or pull
any effort that can cause acceleration

- Second Law: For an unbalanced force,
a: acceleration (units: m / s2)

m: mass(units: kg)

F: force (units: Newton = kg · m / s2)

The pictures below show strobe-light pictures of trucks at different times.

- Which of the trucks is showing acceleration? (There may be more than one.)
- For the accelerating trucks, what direction is the net force on the truck pointing?

POSITION (m)

DISTANCE FALLEN

TIME

SPEED

0 s

0 m/s

0 m

SPEED

1 s

9.8 m/s

4.9 m

2 s

19.6 m/s

19.6 m

TIME

DISTANCE

3 s

29.4 m/s

44.1 m

TIME

Newton’s First Law says: If object is traveling on a curved path, there MUST BE an unbalanced force.

FORCE

TOP VIEW:

VELOCITY

VELOCITY

FORCE

(friction between tires and road)

(gravity)

Newton’s Second Law says: object accelerates (turns) in direction of unbalanced force

force is NOT pushing planet forward

force IS pulling toward inside of orbit (toward Sun)

The picture below shows the velocity of a planet at different times in its orbit (larger arrow means larger speed).

B

Draw the direction of the force on the planet at the different positions shown

C

PLANET’S VELOCITY

SUN’S FORCE TURNS AND SLOWS PLANET

PLANET’S VELOCITY

SUN’S FORCE TURNS AND SPEEDS PLANET

SUN’S FORCE JUST TURNS PLANET

Newton’s Thought Experiment

Fire cannonballs from tall mountain at different speeds:

low speed: crash into surface

medium speed: circular orbit

high speed: ellipse orbit (cannonball gets farther from Earth)

A ball is attached to a string and swung in a circular path above my head. At the point shown below, I suddenly release the string. If this is viewed from directly above, which of the paths below would the ball most closely follow when released?

VIEW FROM ABOVE:

USA

- Third Law: When one object exerts a force on a second one, the second object exerts an equal and opposite force back on the first.

EXAMPLES:

GAS FORCE ON ROCKET

ROCKET’S FORCE ON GAS

SKATER FORCES ON EACH OTHER

ICE

A compact car and a large truck collide head-on and stick together.

- Which one feels the largest force during the collision?
- Which one experiences the largest acceleration?

- The car.
- The truck.
- Both experience the same amount.
- You can’t tell without knowing how fast they were moving before the collision.

- satellite uses rocket to hover near asteroid
- gravity of satellite changes path of asteroid
- less effort required if done farther in advance
- What direction will the satellite pull the asteroid?

- Fg: force
- m1, m2: masses
- d: distance between centers of objects
- G: universal gravitational constant
- attractive force: always pulls masses together
- equal strength forces pull on both masses

Galileo’s Experiment:

Two different masses dropped at same time hit ground at same time…

implies equal acceleration

At Earth’s surface, force is

which creates an acceleration:

that doesn’t depend on the mass of the object!

- Mtotal: total amount of mass involved (example: star and planet)
- applies to any elliptical orbit
- appliesto any masses (Sun and planet; Earth and satellite; Jupiter and a moon;…)
- AN ASTRONOMER’S MAIN WAY TO DETERMINE MASS!
- …just need orbital period and average orbital distance

center of mass

Stars appear to be orbiting something dark…

and about 4 million times the Sun’s mass!

Newton’s Third Law: star moves slightly as planet orbits

If we can figure out roughly how massive the star is, we can figure out how big the planet’s orbit is…

PLANET’S FORCE ON STAR

STAR’S FORCE ON PLANET

Thought Question

- It has a planet orbiting at less than 1 AU.
- It has a planet orbiting at greater than 1 AU.
- It has a planet orbiting at exactly 1 AU.
- It has a planet, but we do not have enough information to know its orbital distance.

A new planet is discovered orbiting a star that is 4 times as massive as the Sun. Astronomers find that it takes 0.5 yr to make one orbit. What does this say about the planet’s orbit?

- The planet orbits 1 AU from its star.
- The planet orbits more than 1 AU from its star.
- The planet orbits less than 1 AU from its star.
- It isn’t possible to tell how big the planet’s orbit is.

Specific combinations of variables have the property of keeping the SAME TOTAL VALUE before and after collisions, etc.

MOMENTUM: mass velocity

Forces change momentum, but “equal and opposite forces” ensures total momentum remains constant

The larger the angular momentum, the harder it is to stop its spinning or revolving

- m: mass of object
- v: speed of rotation or revolution (perpendicular to line from center to object)
- r: distance from center of motion

CONSERVATION OF ANGULAR MOMENTUM: amount of angular momentum does not change unless a twisting force acts on the object

Your daring professor sits in the “CHAIR OF DEATH” and starts rotating with his arms outstretched. If he pulls his arms in toward his body, and then moves them back to their original position, what will happen?

- He will be spinning noticeably slower at the end.
- He will be spinning at about the same speed at the end.
- He will be spinning noticeably faster at the end.
- Something horrible will happen…

If your daring professor sits in the “CHAIR OF DEATH” and starts rotating, what will happen if he moves his arms up and down parallel to his spin axis?

- He will start to spin much slower.
- He will keep spinning at the same speed.
- He will start to spin much faster.
- Something horrible will happen…

Sun

The ability to get masses to move…

- Metric unit: Joule (J)
- English unit: Calorie
- energy to pick a burger off floor (move it up by 1 m): 1 J
- energy from eating candy bar: 106 J
- energy released by H bomb: 5 1015 J
- energy released by Sun each second: 4 1026 J

- kinetic energy: energy of motion
- thermal energy (heat): kinetic energy involving random motions of atoms and other particles
higher temperature more thermal energy

EXAMPLE: GAS IN A BOX

How fast does a 1010 kg asteroid have to be traveling to have a kinetic energy equal to an H bomb (5 1015 J)?

- potential energy: energy of position – a kind of “stored” energy
The farther an object falls, the more “stored” energy is turned into kinetic energy:

POTENTIAL ENERGY DECREASES AND…

BECOMES KINETIC ENERGY

CONSERVATION OF ENERGY: energy can be transferred from object to object, or converted from one form to another, but never destroyed

USA

USA

A spacecraft is moving away from the Sun without firing its engines. Which of the following describes what will happen to…

- the spacecraft’s kinetic energy?
- the spacecraft’s potential energy?
- the spacecraft’s total energy?

- BOUND ORBIT – object does not have enough kinetic energy to escape
- UNBOUND ORBIT – object can eventually reach d = ∞

miniature golf analogy: imagine the Sun at bottom of a well, and object rolling along the sides

Imagine objects rolling on a surface that is dented by the gravity of large masses:

- BOUND ORBIT – object does not have enough kinetic energy to escape
- UNBOUND ORBIT – object can eventually reach d = ∞

Escape Velocity

… minimum speed needed to escape to infinite distance