Astronomy 114
Lecture 7: Newton’s Law of Gravity, Tidal Force
Martin D. Weinberg
[email protected]
UMass/Astronomy Department
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—1/17
Announcements
Problem Set #1 solutions posted
Problem Set #2 posted last Friday, due this Friday
Today is Add/Drop day!
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—2/17
Announcements
Problem Set #1 solutions posted
Problem Set #2 posted last Friday, due this Friday
Today is Add/Drop day!
Today:
Newton’s Law of Gravity
Tidal Force
Wednesday: LIGHT, Chap. 5
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—2/17
Newton’s Law of Gravity (1/6)
Newton described the force of gravity mathematically
Explains Kepler’s laws
Every body in the Universe attracts every other body
with a force proportional to the product of their
masses and inversely proportional to the square of
the distance between them:
Fgravity
Gm1 m2
=
r2
G is the same here as it is in a distant galaxy. It is a physical
constant of the Universe.
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—3/17
Newton’s Law of Gravity (4/6)
Velocity
Planet
Acceleration
Combined with Laws of
Motion: explains orbits
Kepler’s Three Laws
Resulting
trajectory
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—4/17
Newton’s Law of Gravity (5/6)
Newton discovered that orbiting bodies may follow
any one of a family of curves called conic sections
The ellipse is only one possibility
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—5/17
Newton’s Law of Gravity (5/6)
Bound, finite orbits: circle, ellipse
Unbound, infinite orbits: parabola, hyperbola
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—5/17
Newton’s Law of Gravity (6/6)
Planets obey the same laws as objects on Earth
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—6/17
Newton’s Law of Gravity (6/6)
Planets obey the same laws as objects on Earth
Kepler’s laws: explained by force of gravity
Planets orbit around the center of mass of the
Solar System
Since most of the mass is the Sun, Sun is very
close to center of mass
Third law depends on the sum of the two masses:
P2 =
A114: Lecture 7—12 Feb 2007
"
4π 2
G(m1 + m2 )
Read: Ch. 4,5
#
a3
Astronomy 114—6/17
Newton’s Law of Gravity (6/6)
Planets obey the same laws as objects on Earth
Kepler’s laws: explained by force of gravity
Planets orbit around the center of mass of the
Solar System
Since most of the mass is the Sun, Sun is very
close to center of mass
Third law depends on the sum of the two masses:
P2 =
"
4π 2
G(m1 + m2 )
#
a3
New types of unbound orbits—hyperbolas and
parabolas—in addition to ellipses
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—6/17
Food for thought . . .
Inertial and gravitational mass . . . Are they the same?
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—7/17
Food for thought . . .
Inertial and gravitational mass . . . Are they the same?
Action at a distance?
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—7/17
Food for thought . . .
Inertial and gravitational mass . . . Are they the same?
Action at a distance?
Why does an astronaut in orbit feel weightlessness?
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—7/17
More consequences: tidal forces (1/5)
Does every point on the Earth feel the same
gravitation force from the Sun (or Moon)?
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—8/17
More consequences: tidal forces (1/5)
Does every point on the Earth feel the same
gravitation force from the Sun (or Moon)?
Differential force in near side and far side
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—8/17
More consequences: tidal forces (1/5)
Does every point on the Earth feel the same
gravitation force from the Sun (or Moon)?
Differential force in near side and far side
Stretches body along line joining body to Sun
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—8/17
More consequences: tidal forces (1/5)
Does every point on the Earth feel the same
gravitation force from the Sun (or Moon)?
Differential force in near side and far side
Stretches body along line joining body to Sun
Compresses body in 2 perpendicular directions
Results in “football” shape (prolate spheroid)
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—8/17
More consequences: tidal forces (2/5)
Both Sun and Moon influence tides on Earth
Which is bigger?
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—9/17
More consequences: tidal forces (2/5)
Both Sun and Moon influence tides on Earth
Which is bigger?
About the same (but not quite):
Sun is more massive but farther away
Moon is less massive but closer
Moon causes 70%, Sun causes 30%
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—9/17
More consequences: tidal forces (3/5)
Orientation of Moon and Sun relative to Sun–Earth
direction determines the strength of the tide
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—10/17
More consequences: tidal forces (3/5)
Orientation of Moon and Sun relative to Sun–Earth
direction determines the strength of the tide
Tidal force is reinforced when Sun-Moon-Earth are
along same line (Spring Tide)
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—10/17
More consequences: tidal forces (3/5)
Orientation of Moon and Sun relative to Sun–Earth
direction determines the strength of the tide
Tidal force is reinforced when Sun-Moon-Earth are
along same line (Spring Tide)
Tidal force is diminished if Sun-Earth force and
Moon-Earth force are perpendicular (Neap Tide)
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—10/17
More consequences: tidal forces (4/5)
At what time of day to neap tides occur?
a. Near sunrise
b. Near sunset
c. Near noon
d. Near midnight
e. More than one of the
above
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—11/17
More consequences: tidal forces (4/5)
At what time of day to neap tides occur?
a. Near sunrise
b. Near sunset
c. Near noon
d. Near midnight
e. More than one of the
above
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—11/17
More consequences: tidal forces (5/5)
Moon is receding from
Earth
Rotating bulge on
Earth accelerates
Moon in orbit
Earth
Moon
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—12/17
More consequences: tidal forces (5/5)
Moon is receding from
Earth
Rotating bulge on
Earth accelerates
Moon in orbit
Earth
Why does moon keep
same side toward
Earth?
Attraction of
Moon’s tidal bulge
by Earth locks with
Moon’s revolution
A114: Lecture 7—12 Feb 2007
Moon
Read: Ch. 4,5
Astronomy 114—12/17
Calculations with Kepler’s 3rd Law [1]
Newton’s generalization:
P2 =
A114: Lecture 7—12 Feb 2007
"
4π 2
G(m1 + m2 )
Read: Ch. 4,5
#
a3
Astronomy 114—13/17
Calculations with Kepler’s 3rd Law [1]
Newton’s generalization:
P2 =
"
4π 2
G(m1 + m2 )
#
a3
Set m1 = Msun , m2 = Mearth then P = 1year and a = 1AU.
Since Msun ≫ Mearth , m1 + m2 ≈ Msun .
P2
Msun
=
2
(year)
m1 + m2
a3
(year)3
or
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—13/17
Calculations with Kepler’s 3rd Law [1]
Newton’s generalization:
P2 =
"
4π 2
G(m1 + m2 )
#
a3
Set m1 = Msun , m2 = Mearth then P = 1year and a = 1AU.
Since Msun ≫ Mearth , m1 + m2 ≈ Msun .
P2
Msun
=
2
(year)
m1 + m2
a3
(year)3
or
Msun
2
P (year) =
a(AU)3
m1 + m2
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—13/17
Calculations with Kepler’s 3rd Law [2]
May solve for the period P of the planet, given a:
P (year) =
A114: Lecture 7—12 Feb 2007
s
Msun
a(AU)3/2
m1 + m2
Read: Ch. 4,5
Astronomy 114—14/17
Calculations with Kepler’s 3rd Law [2]
May solve for the period P of the planet, given a:
P (year) =
s
Msun
a(AU)3/2
m1 + m2
Example: radius of Mars’ orbit given the period
a(AU) = P 2/3 (year) = (1.88)2/3 = 1.52AU
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—14/17
Calculations with Kepler’s 3rd Law [2]
May solve for the period P of the planet, given a:
P (year) =
s
Msun
a(AU)3/2
m1 + m2
Example: radius of Mars’ orbit given the period
a(AU) = P 2/3 (year) = (1.88)2/3 = 1.52AU
Example: Quadruple mass of Sun, keep radius the
same. How does period of Earth orbit change?
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—14/17
Calculations with Kepler’s 3rd Law [2]
May solve for the period P of the planet, given a:
P (year) =
s
Msun
a(AU)3/2
m1 + m2
Example: radius of Mars’ orbit given the period
a(AU) = P 2/3 (year) = (1.88)2/3 = 1.52AU
Example: Quadruple mass of Sun, keep radius the
same. How does period of Earth orbit change?
P (year) =
A114: Lecture 7—12 Feb 2007
s
1
1
3/2
a(AU) = year
4
2
Read: Ch. 4,5
Astronomy 114—14/17
Newton’s 3rd law: how do rockets work?
1. People see: huge flame and hot gas pouring out the
back
2. Assume: rocket pushing against the ground or the air
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—15/17
Newton’s 3rd law: how do rockets work?
1. People see: huge flame and hot gas pouring out the
back
2. Assume: rocket pushing against the ground or the air
Wrong!
Controlled explosion
Material is ejected from nozzle
By 3rd law, rocket is accelerated in opposite direction
Rocket would work regardless of what is short out
the back!
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—15/17
Definitions (1/2)
scalar:
a simple numerical value
vector:
quantity described by both numerical value
and a direction
velocity:
the speed and direction of an object [vector]
acceleration:
a rate of change of velocity [vector]
inertia:
property of mass by which it resists change in
its motion
momentum:
a measure of an object’s inertia, equal to
product of object’s mass and velocity [vector]
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—16/17
Definitions (2/2)
force:
something which changes the momentum of an
object, equal to rate of change of momentum [vector]
mass:
a measure of the total amount of material (e.g.
atoms) in an object [scalar]
weight:
“downward” force on an object due to gravity
[scalar]
A114: Lecture 7—12 Feb 2007
Read: Ch. 4,5
Astronomy 114—17/17