PES 1110 Fall 2013, Spendier Lecture 38/Page 1

Today: - Orbits (13.8) - Kepler’s Laws (13.7) - Last Quiz 6 this Friday on HW 9! - Wednesday, HW 9 due. - End of class FCQs Newton’s Law of Gravitation

1 22G

GM MFr

(call this an inverse square law)

Gravitational potential energy 1 2

gGM MU

r [J]

Escape velocity 2 planet

planet

GMv

r

Example: (Review) The earth is not flat! It has a curvature of roughly 8000m to 5m (horizontal to vertical). A projectile is fired horizontally at 8000m/s (Mach 23), 20m above the ground on earth. How long does it take it to hit the ground? Ignore air resistance. Projectile: Vertical distance traveled in 1 sec

20 0

12yy y v t gt

20

10 (9.8)(1) 4.92

y y y m

Horizontal distance traveled in 1sec 0 0xx x v t

8000 / 1 8000x m s s m The projectile follows the curvature of the earth and will never hit the ground. Every second, the projectile goes 8000m horizontal and drops 5m vertical. It has become a satellite! Satellites: Any projectile with sufficient horizontal velocity to “miss" the ground. This would not work without gravity. If there were no gravity, the projectile will keep going in a straight line, if fired horizontally. We need gravity in order to pull the satellite down, keep it in orbit. Why do we put satellites into space? Normal people get this wrong sometimes; they say it is because there is no gravity. This is not true as we have just discussed. The reason for it is that in outer space there is no

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atmosphere. 8000 m/s is very fast, about 20 times the speed of sound. Therefore, in real life if we launched a projectile at this speed 20 m above the ground, it would catch on fire because of all the air resistance. That is what meteorites do when they come in through the atmosphere. Astronauts inside the space station are floating around. Hence it looks like there is no gravity acting on them. In fact, astronauts are in free fall. The space station is accelerating (radial direction) which causes a change in direction of the space station. Hence, an astronaut in a space station is also accelerating in order to change his/her direction. They are basically in an elevator that is falling downwards at 9.8 m/s2 near the surface of the earth. In this situation, an objects apparent weight is zero. So the astronauts are apparently weightless inside the space station. Orbits Orbits for something that goes around come in two types: (1) Closed Orbits - Satellite returns to its starting point. (2) Open Orbits - Satellite escapes to infinity. Newton did so much theoretical work, without knowing the magnitude of G. His crowning achievement was to show that if gravity were the only force doing work on a satellite there are only four possible shapes for orbits; the only allowed closed orbits are either circular or elliptical in shape. While the only allowed open orbits are parabolic or hyperbolic. If you are in calculus you might get an idea why this is the case because these four orbits are the four conic sections. If I cut a 3D right circular cone with a plane in the right way, you will get circles, ellipses, parabolas and hyperbolas.

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The initial velocity of the satellite determines whether the orbit is open or closed. (Here we consider no air resistance and gravity is the only force)

Circular orbits: This is a very clear application of what we have already done. We know that gravitational force is towards the center. In circular orbit, gravity creates the centripetal acceleration, arad. Satellites are accelerating radially, which causes a change in direction. Just like a ball on the string going in a circle. Speed of the satellite in a circular orbit:

(M2 is the satellite, the object in orbit, so we need the net force on M2 )

2

2 2 2 2g radvF M a F M a Mr

21 2

22

GM M vMr r

1GMvr

(r measured from the center of the planet, M1 mass of the object being orbited,) This is the speed of the satellite in a circular orbit. The speed is constant, since r is not changing. Since there is only one speed that keeps you in circular orbit, you need to make sure that the satellite ends up at the correct distance from the center of the planet. You need to put it at the right place with the right speed.

When you don’t throw it fast enough (1) and (2), the projectile will just curve and hit the ground. (3) you get circular orbits (4) and (5) elliptical orbits (6) and (7) open orbits (6) – maybe a parabola (7) - maybe a hyperbola

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Period: T (circular orbits) Time for one complete revolution. How long it takes to go around once. For constant speed:

12 GMcircumference rvtotal time T r

p

rearrange 3/ 2

1 1

22 r rT rGM GM

pp (for circular orbits)

Example 1: On July 15, 2004 NASA launched the Aura spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface, and we shall assume a circular orbit. a) How many hours does it take this satellite to make one orbit? b) How fast (in km/s) is the Aura spacecraft moving?

Total energy: E (circular orbits)

2 1 22

12 2

GM MK M vr

1 2g

GM MUr

1 2 1 2 1 2

2 2gGM M GM M GM ME K U

r r r

1 2

2GM ME K

r

For a satellite in a circular orbit, the total energy E is the negative of the kinetic energy K.

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Kepler’s Laws Before Newton, all astronomical work had been observational. Using the data of Danish astronomer Tycho Brahe (1546-1601), the German mathematician Johannes Kepler (1571-1630) was able to deduce by fitting data (but not explain), three statements about planetary motion. Kepler’s Laws: 1) Each planet’s orbit traces out the shape of an ellipse with the sun located at one focus. 2) The imaginary line from the sun to a planet sweeps out equal areas in equal times. 3) The period of the planet’s motion is proportional to the orbit’s semi-major axis to the 3/2 power. Kepler was not able to explain why. He derived these laws from data. Newton was able to explain all of these using his force and gravity. Ellipses Ellipse (oval) is the other kind of closed orbits.

Ellipse centered at origin Foci = plural of focus (two important points) S’P + SP = constant (the sum of the distances from both foci to any point P) a = distance from center out to the long edge (semi-major axis) b = distance from center out to the short edge (semi-minor axis) Algebraic equation for an ellipse

2 2

1x ya b

ea = distance from center out to the foci. This is some fraction of a. e = eccentricity of ellipse (a number between zero and one) The eccentricity gives the amount of “oval-ness" of the ellipse. If Eccentricity = 0, both foci are at the origin, we get a circle. If Eccentricity = 1, we get parabolas. If Eccentricity > 1, we get hyperbolas. And there is this connection to conic sections.

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Kepler’s First Laws 1) Each planet’s orbit traces out the shape of an ellipse with the sun located at one focus.

This means of course that throughout the year the distance between the sun and the planet changes. The place (point) where the planet is closed to the sun = Perihelion. The place where the planet is furthest from the sun = Aphelion. Helios = Greek for sun Peri- = prefix for “I like” A- = prefix for “anti” or “against”

Comets have orbits that are this elliptical. They freeze solid at the Aphelion and burst into flames at the Perihelion. None of these planets have an eccentricity close to this. Many people believe that Earth is closer to the Sun in the summer and that is why it is hotter. And, likewise, they think Earth is farthest from the Sun in the winter. Although this idea makes sense, due to Earth’s elliptical orbit, it is incorrect.

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Earth's axis is an imaginary pole going right through the center of Earth from "top" to "bottom." Earth spins around this pole, making one complete turn each day. That is why we have day and night, and why every part of Earth's surface gets some of each. Earth has seasons because its axis doesn't stand up straight. Long, long ago, when Earth was young, it is thought that something big hit Earth and knocked it off-kilter. So instead of rotating with its axis straight up and down, it leans over a bit. The axis is rotated away from the sun at the Perihelion. As Earth orbits the Sun, its tilted axis always points in the same direction. So, throughout the year, different parts of Earth get the Sun’s direct rays.

Southern Hemisphere: Perihelion = summer Aphelion = winter Northern Hemisphere: Right now, it is getting colder and colder outside. So we, Earth, must be approaching the Perihelion of the Earth’s orbit. Around the winter solstice, we are at the Perihelion. Seasons are a matter of how many hours of daylight. The earth’s axis of rotation is also undergoing a very slow but steady precession. Precession is a change in the orientation of the rotational axis of a rotating body. The period for a complete processional cycle is ~26000 years for the earth. So eventually the Northern Hemisphere will be pointing towards the sun at the Perihelion. So seasons can flip.

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Kepler’s Second Law 2) The imaginary line from the sun to a planet sweeps out equal areas in equal times.

If these areas are equal then the planet takes the same amount of time going from A to B as it does going from C to D. The speed of any object in an elliptical orbit is NOT constant. From A to B the planet travels a further distance than from C to D in the same time. Therefore the speed going from A to B must be faster. In a circular orbit they would be the same. Proof:

You draw a line from the sun to the planet. As the planet moves this line changes. Over two very short time intervals we can approximate the area swept out as

12

A r l

Equal area over equal time is what we are looking for hence 12

A lrt t

To make it precise, as 0t 12

dA dlrdt dt

and 12 t

dA r vdt

(vt = tangential velocity of the planet)

On an ellipse, r

and v are not perpendicular! ( r

perpendicular to v is only true for circular motion). When you go around an ellipse the planet is changing its distance (vr) from the sun and its direction (vt) so the velocity must have two components.

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sintv v f

1 sin2

dA r vdt

f

Angular momentum of a satellite: sinL Mvr f At this scale all our planets look like dots So the change of area over times depends on the angular momentum of the planets

1 sin2 2

dA LMr vdt M M

constant ?f

On satellites, gravity causes no torque

sin(180) 0g gdL r F rFdt

t

Gravity has no perpendicular force component on the planet – only radial.

No external torque, hence the angular momentum of the satellite is constant

tan2

dA L cons tdt M

Hence equal ∆A for equal ∆t! Finally, at these two special places, at perihelion and aphelion the angle ϕ for satellite motion is 90º.

At Perihelion: p pL Mv r At Aphelion: a aL Mv r Conservation of angular momentum

p p a aMv r Mv r So for any satellite

p p a av r v r The planet goes fastest at Perihelion, since here the r is smallest.