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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
GM 1 M 2
FG 
(call this an inverse square law)
r2
Gravitational potential energy
GM 1M 2
Ug  
[J]
r
Escape velocity
2GM planet
v
rplanet
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
1
y  y0  v0 y t  gt 2
2
1
y  y0  y  0  (9.8)(1)2  4.9 m
2
Horizontal distance traveled in 1sec
x  x0  v0 x t
x  8000 m / s 1 s  8000 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
PES 1110 Fall 2013, Spendier
Lecture 38/Page 2
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.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 3
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)
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
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 )


v2
F

M
a

F

M
a

M
 2 2
g
2 rad
2
r
2
GM 1M 2
v
 M2
2
r
r
GM1
v
r
(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.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 4
Period: T (circular orbits)
Time for one complete revolution. How long it takes to go around once.
For constant speed:
circumference 2pr
GM 1
v


total time
T
r
rearrange
r
2pr 3/ 2
T  2pr

(for circular orbits)
GM 1
GM 1
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)
GM 1M 2
1
K  M 2v2 
2
2r
GM 1M 2
Ug  
r
GM 1M 2 GM 1M 2
GM 1M 2
E  K U g 


2r
r
2r
GM 1M 2
E 
 K
2r
For a satellite in a circular orbit, the total energy E is the negative of the kinetic energy K.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 5
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
 x   y 
      1
 a   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.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 6
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.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 7
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.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 8
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
1
A  r l
2
Equal area over equal time is what we are looking for hence
A 1  l 
 r  
t
2  t 
To make it precise, as t  0
dA 1  dl 
 r  
dt
2  dt 
and
dA 1
 r vt
(vt = tangential velocity of the planet)
dt
2




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.
PES 1110 Fall 2013, Spendier
Lecture 38/Page 9
vt  v sin f 
dA 1
 r v sin f 
dt
2
Angular momentum of a satellite: L  Mvr sin 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
dA
1
L

Mr v sin f 
 constant ?
dt
2M
2M
On satellites, gravity causes no torque

 dL  
t
 r  Fg  rFg sin(180)  0
dt
Gravity has no perpendicular force component on the planet – only radial.
No external torque, hence the angular momentum of the satellite is constant
dA
L

 cons tan t
dt
2M
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: L  Mv p rp
At Aphelion: L  Mva ra
Conservation of angular momentum
Mv p rp  Mva ra
So for any satellite
v p rp  va ra The planet goes fastest at Perihelion, since here the r is smallest.