Measuring the Universe: Earth 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:
Measuring the Universe: Distances to Planets 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
Planet Motions Across Sky • 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
“Inferior Planets”: Mercury, Venus 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
“Superior Planets”: Mars, Jupiter, Saturn • 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
Thought Question: 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
Thought Question: Where would planet A be seen in the sky from Earth at sunset? A
Competing Ideas about the Universe 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
Competing Ideas about the Universe Epicycles are needed to create retrograde motions of planets
Retrograde Motion happens when Earth catches and passes a superior planet
Thought Question: Where would you look to see a planet rise when it is in retrograde motion? • near the eastern horizon • near the western horizon
Thought Question: 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.
Measuring the Universe: Inferior Planets 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
Kepler’s Laws of Planetary Motion 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
Eclipses Solar eclipses can be either total or “annular”… the Moon’s distance from Earth changes…
Ellipses 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: Ellipses 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 Laws of Planetary Motion • 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
Kepler’s Laws of Planetary Motion (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
Thought Questions: 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
Thought Question: If it takes Eris 557 years to orbit the Sun, what is its average distance from the Sun?
Kepler’s Third Law Examples • Jupiter: • Eris: • Sedna:
Thought Question: 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.
Apophis – Killer Asteroid? • asteroid about 350 m across • close approach to Earth in 2029
Measuring the Universe: Problems • both Earth and other planets are moving SO… • how do you determine when a planet comes back to the same place in its orbit?
Orbit and Synodic Periods 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
Periods and Angular Speeds 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:
Thought Question: 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?)
Parallax 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
The “Asteroid Tugboat” • 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 Newton’s Laws of Motion • 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.
Newton’s Laws of Motion • 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)
Thought Question: 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)
Earth Acceleration due to Gravity (g = 9.8 m/s2) 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
Orbits are Curved Paths 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)
Thought Question: 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 Ellipse Orbit: 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)
Thought Question: 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 Newton’s Laws of Motion • 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
Thought Questions: 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.
The “Gravity Tractor” • 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?
Universal Gravitation • 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
Acceleration of Gravity 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!