Transcript Isaac Newton in 1689

Mass and Density
In the solar system
How do we know?
Isaac Newton
Isaac Newton discovered the relationship
between gravitational force, mass and
distance that we call the “law of gravity”.
Isaac Newton in 1689
July 5, 1687
Gravity and Orbits
The strength of the
gravitational force
that keeps one object
in orbit around
another depends on
two things.
If we could determine the strength of
the gravitational force and the
distance we could calculate mass.
The distance
between them . . .
. . . and their mass
Distance
q
d
R
d = R sinq
Distances can be found using astronomical
observations and trigonometry.
How can we find the gravitational force?
Thanks to Isaac Newton, there is a way
around this problem.
Isaac Newton
Newton also discovered three laws that
describe how the motion of an object is
changed by forces, including gravity. We
call these “Newton’s Laws of Motion”.
Isaac Newton in 1689
July 5, 1687
Isaac Newton
Combining Newton’s laws of motion with the law of gravity
for two objects orbiting each other . . . we get an equation
describing the motion of the objects relative to each other . . .
. . . and then with the aid of calculus (which Newton invented)
Mm
and some
algebra
F = - G . .2. (G is the Universal constant of gravitation.)
r
dv
(M + m)
=-G
dt
r2
r
M
FM = MaM
Fm = mam
FM = - Fm
m
Isaac Newton
Combining Newton’s laws of motion with the law of gravity
for two objects orbiting each other . . . we get an equation
describing the motion of the objects relative to each other . . .
. . . and then with the aid of calculus (which Newton invented)
and some algebra . . .
dv
2 (M
3 + m)
4
p
r
=-G
Pd2t = G
(M + m)r 2
r
m
M
P = orbital period
We obtain a relationship between orbital period, distance and mass.
Isaac Newton
For a planet with an orbiting moon, the mass of the moon is so
small compared to the planet that the sum of the moon’s mass and
the planet’s mass is about the same as the planet’s mass alone.
Ganymede, Jupiter’s largest moon and the largest moon in the
solar system has only 0.0078% the mass of Jupiter.
4 p2 r 3
P = G (M
G+
Mm)
r
2
m
M
P = orbital period
The Moon has a mass only 1.2% of Earth.
So, if Earth’s mass = 1.000, the mass of Earth + Moon = 1.012
Gravitational Force and Mass
So, if a planet has a
moon and we
measure both the
moon’s orbital period
and the distance
between the moon
and planet, we can
calculate the mass.
Here is an example:
Jupiter’s moon Io orbits Jupiter at about the
same distance as the Moon orbits Earth.
Earth
Moon
Io
Orbital Period
27.3 days
Orbital Period
1.77 days
Jupiter
However, Io takes MUCH less time for one orbit than the Moon.
Jupiter’s moon Io orbits Jupiter at about the
same distance as the Moon orbits Earth.
Earth
Moon
Orbital Period
27.3 days
MJ = ME (27.3 / 1.77)2 (1.10)3
Io
Orbital Period
1.77 days
Jupiter
Using the orbital periods we can compare the mass of
Jupiter and the mass of Earth.
Jupiter has a mass over 300 times larger than Earth’s mass!
Isaac Newton
In his book that announced his laws
of motion and gravity, Newton used
these laws to calculate the densities
of four objects in the solar system.
Isaac Newton in 1702
Only three planets were known to have
moons during Newton’s lifetime.
Earth
Credit: NASA/JPL/
Southwest Research
Institute
Saturn
Credit: NASA/JPL
Jupiter
Credit: NASA/JPL/Malin Space Science Systems
Newton calculated the density of the
these three planets and the Sun.
Earth
Credit: NASA/JPL/
Southwest Research
Institute
Saturn
Credit: NASA/JPL
Jupiter
Credit: NASA/JPL/Malin Space Science Systems
Newton used the orbit of Venus to
calculate the Sun’s density
This photograph shows the Sun and Venus during the Venus transit of 1882. The big white circle
is the Sun. Venus is the black dot on the Sun. Venus is near the top of the Sun, just left of center.
Image courtesy the U.S. Naval Observatory Library.
Newton’s Density Calculations
Newton wrote, “Thus from the periodic times [orbital
periods] of Venus around the Sun, . . . the outermost
satellite of Jupiter [Callisto] around Jupiter, . . . the
Huygenian satellite [Titan] around Saturn, . . . and of the
Moon around the Earth . . . compared with the mean
distance of Venus from the Sun and with [the measured
angles that would allow Newton to calculate the planetmoon distances] . . . , by entering into a computation . . .
The quantity of matter [mass] in the individual planets is
also found.”
Newton’s Cast of Characters
Venus
Titan
Moon
Saturn
Sun
Callisto
Earth
Jupiter
Newton’s Density Calculations
Newton could only calculate the masses of the planets
relative to each other because the gravitational constant
in his law of gravity had not yet been determined. With
the relative masses known, and by also calculating the
relative volumes, Newton wrote, “The densities of the
planets also become known.”
Delicate experiments performed by Henry Cavendish in
1797 and 1798 measured Earth’s average density,
allowing the determination of the gravitational constant.
Newton’s Density Calculations
As he could only calculate relative densities, he assigned
the Sun an arbitrary density of 100 and calculated the
densities of Jupiter, Saturn and Earth relative to the Sun.
Sun
Jupiter
Saturn
Earth
Newton’s
Calculation
100
94 ½
67
400
Modern
Value
100
94.4
50.4
390.1