Astronomy 114 - University of Massachusetts...
Transcript of Astronomy 114 - University of Massachusetts...
Astronomy 114
Lecture 7: Newton’s Law of Gravity, Tidal Force
Martin D. Weinberg
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 bodywith a force proportional to the product of theirmasses and inversely proportional to the square ofthe distance between them:
Fgravity =Gm1m2
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)
Acceleration
Velocity
Planet
Resulting
trajectory
Combined with Laws ofMotion: explains orbits
Kepler’s Three Laws
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 followany 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 theSolar System
Since most of the mass is the Sun, Sun is veryclose to center of mass
Third law depends on the sum of the two masses:
P2 =
[
4π2
G(m1 + m2)
]
a3
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 theSolar System
Since most of the mass is the Sun, Sun is veryclose to center of mass
Third law depends on the sum of the two masses:
P2 =
[
4π2
G(m1 + m2)
]
a3
New types of unbound orbits—hyperbolas andparabolas—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 samegravitation 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 samegravitation 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 samegravitation 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 samegravitation 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–Earthdirection 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–Earthdirection determines the strength of the tide
Tidal force is reinforced when Sun-Moon-Earth arealong 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–Earthdirection determines the strength of the tide
Tidal force is reinforced when Sun-Moon-Earth arealong same line (Spring Tide)
Tidal force is diminished if Sun-Earth force andMoon-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 theabove
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 theabove
A114: Lecture 7—12 Feb 2007 Read: Ch. 4,5 Astronomy 114—11/17
More consequences: tidal forces (5/5)
Moon is receding fromEarth
Rotating bulge onEarth acceleratesMoon 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 fromEarth
Rotating bulge onEarth acceleratesMoon in orbit
Why does moon keepsame side towardEarth?
Attraction ofMoon’s tidal bulgeby Earth locks withMoon’s revolution
Earth
Moon
A114: Lecture 7—12 Feb 2007 Read: Ch. 4,5 Astronomy 114—12/17
Calculations with Kepler’s 3rd Law [1]
Newton’s generalization:
P2 =
[
4π2
G(m1 + m2)
]
a3
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 =
[
4π2
G(m1 + m2)
]
a3
Set m1 = Msun, m2 = Mearth then P = 1year and a = 1AU.Since Msun ≫ Mearth, m1 + m2 ≈ Msun.
P2
(year)2=
(
Msun
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 =
[
4π2
G(m1 + m2)
]
a3
Set m1 = Msun, m2 = Mearth then P = 1year and a = 1AU.Since Msun ≫ Mearth, m1 + m2 ≈ Msun.
P2
(year)2=
(
Msun
m1 + m2
)
a3
(year)3
or
P (year)2 =(
Msun
m1 + m2
)
a(AU)3
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) =
√
Msun
m1 + m2
a(AU)3/2
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) =
√
Msun
m1 + m2
a(AU)3/2
Example: radius of Mars’ orbit given the period
a(AU) = P2/3(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) =
√
Msun
m1 + m2
a(AU)3/2
Example: radius of Mars’ orbit given the period
a(AU) = P2/3(year) = (1.88)2/3 = 1.52AU
Example: Quadruple mass of Sun, keep radius thesame. 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) =
√
Msun
m1 + m2
a(AU)3/2
Example: radius of Mars’ orbit given the period
a(AU) = P2/3(year) = (1.88)2/3 = 1.52AU
Example: Quadruple mass of Sun, keep radius thesame. How does period of Earth orbit change?
P (year) =
√
1
4a(AU)3/2 =
1
2year
A114: Lecture 7—12 Feb 2007 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 theback
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 theback
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 outthe 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 valueand 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 inits motion
momentum: a measure of an object’s inertia, equal toproduct 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 anobject, 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