Download A105 Stars and Galaxies

A100
Solar System
Start Reading NASA website (Oncourse)
2nd Homework due TODAY
IN-CLASS QUIZ NEXT FRIDAY!!
Today’s APOD
The Sun Today
 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
Newton's First Law
An object at rest will remain at rest, an
object in uniform motion will stay in
motion - UNLESS acted upon by an
outside force
Outside Force
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
Newton's
Third Law
 For every action, there is
an equal and opposite
reaction
 Simply put, if body A
exerts a force on body B,
body B will react with a
force that is equal in
magnitude but opposite
direction
 This will be important in
astronomy in terms of
gravity
 The Sun pulls on the Earth
and the Earth pulls on the
Sun
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 
2
r
• 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
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
Surface Gravity
 Surface gravity is the
acceleration a mass
undergoes at the surface
of a celestial object (e.g.,
an asteroid, planet, or
star)
Surface gravity:
Determines the weight of a mass at a
celestial object’s surface
Influences the shape of celestial objects
Influences whether or not a celestial object
has an atmosphere
Surface Gravity Calculations
 Surface gravity is determined from Newton’s Second Law and
the Law of Gravity:
ma = GMm/R2
 where M and R are the mass and radius of the celestial object, and
m is the mass of the object whose acceleration a we wish to know
 The surface gravity, denoted by g, is then:
g = GM/R2
 Notice dependence of g on M and R, but not m
gEarth = 9.8 m/s2
gMoon/gEarth= 0.18
gJupiter/gEarth = 3
Escape
Velocity
To overcome a
celestial object’s
gravitational force
and escape into
space, a mass must
obtain a critical
speed called the
escape velocity
Escape Velocity
Determines if a
spacecraft can move
from one planet to
another
Influences whether
or not a celestial
object has an
atmosphere
Relates to the nature
of black holes
Escape Velocity Calculation
The escape velocity, Vesc, is determined from
Newton’s laws of motion and the Law of
Gravity and is given by:
Vesc = (2GM/R)1/2
 where M and R are the mass and radius of the
celestial object from which the mass wishes to
escape
Notice dependence of Vesc on M and R, but not
m
Vesc,Earth = 11 km/s, Vesc,Moon = 2.4 km/s
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
TO DO LIST:
Start Reading NASA website (Oncourse)
Hand in 2nd Homework TODAY
IN-CLASS QUIZ NEXT FRIDAY!!