Download Gravity - Indiana University Astronomy

H205
Cosmic Origins
APOD
Making Sense (Ch. 4)
EP2 Due Today
Nicolaus Copernicus (1473-1543)
Astronomy in the
Renaissance
 Could not reconcile
Brahe’s measurements
of the position of the
planets with Ptolemy’s
geocentric model
 Reconsidered Aristarchus’s
heliocentric model with the
Sun at the center of the
Solar system
Johannes Kepler (1571-1630)
Using Tycho’s precise
observations of the
position of Mars in
the sky, Kepler
showed the orbit to
be an ellipse, not a
perfect circle
Three laws of
planetary motion
Kepler’s 1st Law
Planets move in
elliptical orbits with
the Sun at one focus
of the ellipse
Words to remember
Focus vs. Center
Semi-major axis
Semi-minor axis
Perihelion, aphelion
Eccentricity
Definitions
 Planets orbit the Sun
in ellipses, with the
Sun at one focus
 The eccentricity of
the ellipse, e, tells
you how elongated it
is
 e=0 is a circle, e<1
for all ellipses
e=0.02
e=0.4
e=0.7
Eccentricity of Planets
(& Dwarf Planets)
Mercury
0.206
Saturn
0.054
Venus
0.007
Uranus
0.048
Earth
0.017
Neptune
0.007
Mars
0.094
Pluto
0.253
Jupiter
0.048
Ceres
0.079
Which orbit is closest to a circle?
Kepler’s 2nd Law
Kepler’s 2nd Law
Planets don’t move at
constant speeds
The closer a planet is
to the Sun, the
faster it moves
A planet’s orbital speed varies in such a way
that a line joining the Sun and the planet will
sweep out an equal area each month
Each month gets an equal slice of the orbital
pie
Kepler’s 2nd Law:
Same Areas
If the planet sweeps out
equal areas in equal times,
does it travel faster or
slower when far from the
Sun?
Kepler’s 3rd Law
• The amount of time a
planet takes to orbit
the Sun is
mathematically related
to the size of its
orbit
• The square of the
period, P, is
proportional to the
cube of the semimajor
axis, a
2
P
=
3
a
Kepler’s 3rd Law
Third law can be
used to determine
the semimajor axis,
a, if the period, P, is
known, a
measurement that is
not difficult to make
Express the period in
years
Express the semi-major
axis in AU
2
P
=
3
a
Examples of
Kepler’s 3rd Law
Body
Period
(years)
Express the period in
years
Express the semi-major
axis in AU
For Earth:
Mercury 0.24
Venus
0.61
Earth
1.0
Mars
1.88
Jupiter
11.86
Saturn
29.6
Pluto
248
P = 1 year, P2 = 1.0
a = 1 AU, a3 = 1.0
P2 = a3
Examples of
Kepler’s 3rd Law
Body
Period
(years)
Mercury 0.240
9
Venus
0.61
Earth
1.0
Mars
1.88
Jupiter
11.86
Saturn
29.6
Pluto
248
For Mercury:
P = 0.2409 years
P2 = 5.8 x 10-2
a = 0.387 AU
a3 = 5.8 x 10-2
P2 = a3
Examples of
Kepler’s 3rd Law
Body
Period
(years)
Mercury 0.240
9
Venus
0.615
2
Earth
1.0
Mars
1.88
Jupiter
11.86
Saturn
29.6
Pluto
248
For Venus:
P = 0.6152 years
P2 = 3.785 x 10-1
What is the
semi-major axis
of Venus?
P2 = a3
a = 0.723 AU
Examples of
Kepler’s 3rd Law
Body
Period
(years)
Mercury 0.240
9
Venus
0.615
2
Earth
1.0
Mars
1.88
Jupiter
11.86
Saturn
29.6
Pluto
248
For Pluto:
P = 248 years
P2 = 6.15 x 104
What is the
semi-major axis
of Pluto?
P2 = a3
a = 39.5 AU
Examples of Kepler’s
3rd Law
Body
Period
(years)
Mercury
0.2409
Venus
0.6152
Earth
1.0
Mars
1.88
Jupiter
11.86
Saturn
29.6
Pluto
248
The Asteroid
Pilachowski (1999 ES5):
P = 4.11 years
What is the semi-major axis
of Pilachowski?
P2 = a3
a = ??? AU
Comparing Heliocentric Models
Kepler’s 3 Laws of
Planetary Motion
 Planets move in elliptical orbits
with the Sun at one focus of
the ellipse
 A planet’s orbital speed varies
in such a way that a line joining
the Sun and the planet will
sweep out an equal area each
month
 P2 = a3 (the square of the
period of a planet orbiting the
sun is equal to the cube of the
semi-major axis of the planet’s
orbit)
But WHY
????????
The Problem of
Astronomical Motion
Astronomers of
antiquity did not
connect gravity and
astronomical motion
Galileo investigated this connection with
experiments using projectiles and balls rolling
down planks
He put science on a course to determine laws
of motion and to develop the scientific method
inertia!
Galileo
experimented
with inclined
planes
Demonstrated the ideas of inertia and forces
Without friction…
a body at rest tends to remain at rest
a body in motion tends to remain in motion
Isaac Newton, the Laws of Motion,
and the Universal Law of Gravitation
Newton
Born same year Galileo died
Attempts to understand motion of the
Moon
Leads him to deduce the law of gravity (as
we still use it today!)
Requires him to invent new mathematics
Leads him to deduce the general laws of
motion
Galileo’s ideas of inertia became Newton’s
First Law of Motion:
A body continues
in a state of rest
or uniform motion
in a straight line
unless made to
change that state
by forces acting
on it
Newton’s First Law
 Important ideas
What is a force?
A push or a pull
The sum of all the
forces on an object
is the net force
If the forces all
balance, the net
force is zero, and
the object’s motion
will not change
If the speed or
direction of motion of
an object changes,
then a nonzero net
force must be present
Newton’s 2nd Law: Acceleration
 Acceleration
 An object increasing or decreasing in speed along a straight
line is accelerating
 An object changing direction, even with constant speed, is
accelerating
 Acceleration is produced by a force
 Acceleration and force are proportional (double the force,
double the acceleration
Newton’s Second Law: Mass
 Mass is the amount of matter an
object contains
 Technically, mass is a measure of
an object’s inertia
 Mass is generally measured in
kilograms
 Mass should not be confused with
weight, which is a force related
to gravity – weight may change
from place to place, but mass
does not
Newton’s Second Law of Motion
F = ma
The amount of acceleration (a) that
an object undergoes is proportional to
the force applied (F) and inversely
proportional to the mass (m) of the
object
This equation applies for any force,
gravitational or otherwise
Newton's Second Law
When a force acts on a body, the resulting
acceleration is equal to the force divided by
the object's mass
Acceleration – a change in speed or direction
F
a
m
or
F  ma
Notice how this equation works:
The bigger the force, the larger the acceleration
The smaller the mass, the larger the acceleration
F = ma
Newton’s Third Law of Motion
When two objects
interact, they
create equal and
opposite forces on
each other
This is true for
any two objects,
including the Sun
and the Earth!
Newton and the Apple - Gravity
 Newton realized that there
must be some force
governing the motion of the
planets around the Sun
 Amazingly, Newton was
able to connect the motion
of the planets to motions
here on Earth through
gravity
 Gravity is the attractive
force two objects place
upon one another
The Gravitational Force
Gm1m2
Fg 
r2
G is the gravitational constant
G = 6.67 x 10-11 N m2/kg2
m1 and m2 are the masses of the two
bodies in question
r is the distance between the two
bodies
Newton solved the premier scientific problem of his
time --- to explain the motion of the planets.
To explain the motion of the planets, Newton
developed three ideas:
F = ma
1. The laws of motion
Gm1m2
2. The theory of universal gravitation
F
3. Calculus, a new branch of mathematics
r2
“If I have been able to see farther than others
it is because I stood on the shoulders of
giants.”
--- Newton’s letter to Robert Hooke,
probably referring to Galileo and Kepler
 Kepler's Laws were a revolution
in regards to understanding
planetary motion, but there was
no explanation why they worked
 The explanation was provided by
Isaac Newton when formulated
his laws of motion and gravity
 Newton recognized that the same
physical laws applied both on
Earth and in space.
 And don’t forget the calculus!
Isaac Newton
Remember the Definitions
Force: the push or pull on an object that
affects its motion
Weight: the force which pulls you toward the
center of the Earth (or any other body)
Inertia: the tendency of an object to keep
moving at the same speed and in the same
direction
Mass: the amount of matter an object has
Astronomical
Motion
Planets move along
curved (elliptical)
paths, or orbits
Speed and direction
are changing
Must there
be a net
force on
the
planets?
Yes!
Gravity is
that force
Gravity gives the Universe its structure
a universal force that causes all objects to pull on
all other objects everywhere
holds the Earth in orbit around the Sun, the Sun in
orbit around the Milky Way, and the Milky Way in
its path within the Local Group
Everything attracts everything else!!
Newton’s Law of Gravity
Orbital Motion
and Gravity
Newton
Explained the Moon’s
motion with force
that pulls the Moon
from a straight,
inertial trajectory
Showed that the
force must decrease
with distance
 The Moon moves “parallel” to
the Earth’s surface at such a
defined the
speed that its gravitational
properties of gravity
deflection toward the surface
is offset by the surface’s
wrote the equations
curvature away from the
of motion with gravity
projectile
Orbital Motion Using Newton’s First Law
At a sufficiently
high speed, the
cannonball travels
so far that the
ground curves out
from under it.
The cannonball
literally misses
the ground!
Determining the Mass of the Sun
 How do we determine the mass of the Sun?
 Put the Sun on a scale and determine its weight???
 Since gravity depends on the masses of both
objects, we can look at how strongly the Sun
attracts the Earth
 The Sun’s gravitational attraction keeps the Earth
going around the Sun, rather than the Earth going
straight off into space
By looking at how fast the Earth orbits the
Sun at its distance from the Sun, we can
get the mass of the Sun
Measuring Mass with Newton’s Laws Assumptions to Simplify the Calculation
Assume a small
mass object orbits
around a much
more massive
object
The Earth around
the Sun
The Moon around
the Earth
Charon around Pluto
Assume the orbit
of the small mass
is a circle
Measuring the Mass of the Sun
 The Sun’s gravity is the force that acts on the Earth
to keep it moving in a circle
GMSunmEarth
Fg 
r2
 MSun is the mass of the Sun in kilograms
 MEarth is the mass of the Earth in kilograms
 r is the radius of the Earth’s orbit in kilometers
 The acceleration of the Earth in orbit is given by:
a = v2/r
 where v is the Earth’s orbital speed
F  mEarth a  mEarth
v2
r
Measuring the Mass of the Sun
 Set F = mEarthv2/r equal to F = GMSunmEarth/r2
and solve for MSun
MSun = (v2r)/G
 The Earth’s orbital speed (v) can be
expressed as the circumference of the
Earth’s orbit divided by its orbital period:
v = 2pr/P
Measuring the Mass of the Sun
 Combining these last two equations:
MSun = (4p2r3)/(GP2)
 The radius and period of the Earth’s orbit are both
known, G and π are constants, so the Sun’s mass
can be calculated
 This last equation in known as Kepler’s modified
third law and is often used to calculate the mass
of a large celestial object from the orbital period
and radius of a much smaller mass
So what is the Mass
of the Sun?
MSun = (4p2r3)/(GP2)
rEarth = 1.5 x 1011 meters
PEarth = 3.16 x 107 seconds
G = 6.67 x 10-11 m3 kg-1 s-2
Plugging in the numbers gives
the mass of the Sun
MSun = 2 x 1030 kg
What about the Earth?
 The same formula gives the mass of the Earth
MEarth= (4p2r3)/(GP2)
 Using the orbit of the Moon:
 rMoon = 3.84 x 105 km
 PMoon = 27.322 days = 2.36 x 106 seconds
 Mass of Earth is 6 x 1024 kg
How about Pluto?
 The same formula gives the mass of Pluto, too
MPluto = (4p2r3)/(GP2)
 Using the orbit of Pluto’s moon Charon:
 rCharon = 1.96 x 104 km
 PCharon = 6.38 days = 5.5 x 105 seconds
 Mass of Pluto is 1.29 x 1022 kilograms
Orbits tell us Mass
 We can measure the mass of any body that has
an object in orbit around it
 Planets, stars, asteroids
 We just need to know how fast and how far away
something is that goes around that object
 But we can’t determine the
mass of the Moon by watching
it go around the Earth
 To determine the mass of the
Moon, we need a satellite
orbiting the Moon
Revisions to Kepler's 1st Law
 Newton's law of gravity required
some slight modifications to
Kepler's laws
 Instead of a planet rotating around
the center of the Sun, it actually
rotates around the center of mass
of the two bodies
 Each body makes a small elliptical
orbit, but the Sun's orbit is much
much smaller than the Earth's
because it is so much more massive
Revisions to Kepler's 3rd Law
 Gravity also requires a slight
modification to Kepler's 3rd Law
 The sum of the masses of the
two bodies is now included in the
equation
 For this equation to work, the
masses must be in units of solar
mass (usually written as M)
 Why did this equation work
before?
3
a
P 
M1  M 2
2
Remember - for this
equation to work:
P must be in years!
a must be in A.U.
M1 and M2 must be
in solar masses
For Next Week
Surveying the Stars (Ch. 15)
Milky Way (Ch. 19)
Hand in EP2