4. In previous lecture, I introduced the transition from Geocentric(地⼼宇宙) model to Heliocentric
（⽇⼼宇宙） model.
Copernican revolution
However, in the era of Copernicus, Heliocentric model was not accepted.
How did human open modern astronomy??
Isaac Newton
(1642-1727)
Galileo Galilei
(1564-1642)
Johannes Kepler
(1571-1630)
Tycho Brahe
(1546-1601)
5. Galileo Galilei
(1564-1642)
•Galileo Galilei invented telescope in 1609.
Galileo’s telescope
•He discovered that the moon had mountains, valleys, and craters
(Galileo’s sketch)
•Galileo’s observation supports the idea of Copernicus.
Copernicus model Ptolemy model
6. Tycho Brahe
(1546-1601)
Johannes Kepler
(1571-1630)
•Brahe had measured the motion of the sun,
moon, planets and stars for 29 years.
•He collected much data of motion.
•He measured these data by eyes.
•Kepler inherited Brahe’s data.
•Based on heliocentric picture, he tried to explain the motion
of planetary motion described by Brahe’s data.
•In the end, he found it necessary to abandon Copericus’s
simple idea of circular planetary orbits.
•He finally developed the laws that can explain Brahe’s
data !
7. Kepler’s simple laws
(First law)
The orbital paths of the planets are elliptical(椭圆) , not circular. The Sun is located at
one focus point.
8. Kepler’s simple laws
(First law) The orbital paths of the planets are elliptical(椭圆) , not circular. The Sun is
located at one focus point.
Eccentricity(偏⼼率)
b
a
Homework
Please check the eccentricity of each planets in Solar system
0 < e < 1
small e > more circular
large e > more elliptical
9. Kepler’s simple laws
(Second law)
An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal
intervals of time.
For example, planet sweeps the area of A,B and C during 1 month. Then, the areas are equal
(A=B=C)
10. Kepler’s simple laws
(Third law)
The square of a planet’s orbital period is proportional to the cube of its semi major axis.
R
For example, the Earth takes 1 year to complete one orbit around the Sun. If another
planet, which has a major axis R, takes T to complete one orbit around the Sun, then
<latexit sha1_base64="IwZJQ8OYZJRAckwK9Rj60k5vMFE=">AAAB+HicbVDLTgIxFO3gC/HBqEs3jcTEFZlBjS6Jblyi4ZXAQDqlAw2dtmk7JjjhS9y40Bi3foo7/8YCs1DwJDc5Oefe3HtPKBnVxvO+ndza+sbmVn67sLO7t190Dw6bWiQKkwYWTKh2iDRhlJOGoYaRtlQExSEjrXB8O/Nbj0RpKnjdTCQJYjTkNKIYGSv13WK9V4FdqYQ0Aj70zvtuySt7c8BV4mekBDLU+u5XdyBwEhNuMENad3xPmiBFylDMyLTQTTSRCI/RkHQs5SgmOkjnh0/hqVUGMBLKFjdwrv6eSFGs9SQObWeMzEgvezPxP6+TmOg6SCmXiSEcLxZFCYP2yVkKcEAVwYZNLEFYUXsrxCOkEDY2q4INwV9+eZU0K2X/suzdX5SqN1kceXAMTsAZ8MEVqII7UAMNgEECnsEreHOenBfn3flYtOacbOYI/IHz+QNw+ZJM</latexit>
T2
/ R3 or
T2
/R3
= constant
T The time when planet goes around
11. •Kepler’s three laws, which simplified the solar system, was discovered empirically(经验
性的). This means that his results are not derived from any theory or mathematical model.
Issac Newton successfully explained the motion of planets.
Issac Newton
Isaac Newton
(1642-1727)
•Isaac Newton was born in England in 1642, the year Galileo died. He studied in Cambridge
university.
•He developed mechanics, optics, mathematics. He pioneered modern
physics
mechanics（力学） mathematics（导数、积分）
optics(光学)
12. Newton’s law of motion
•Every object continues in a state of rest or in a state of uniform motion in a straight
line, unless it is compelled to change that state by force acting on it.
Newton’s first law of motion
(a) An object at rest remains at rest.
(b) When a force does act, the object
moves uniformly.
(c)When second force is added, the
object changes its direction.
We would like to consider the motion of object more quantitatively.
13. Newton’s law of motion
Newton’s second law of motion
•When a force acts on a body of mass , it produces in it an acceleration equal to
the force divided by the mass.
F m a
a = F/m or
F = ma
⼒
加速度
质量
Newton’s third law of motion
At earth’s surface, the force of gravity produces a down forward acceleration of 9.8m/s2
•To every action, there is an equal and opposite reaction.
14. Gravity(万有引⼒)
Newton’s idea:
Any object having mass always have an attractive
gravitational force on all other massive objects.
Consider baseball. When we throw up ball on the
earth, the ball is pulled continuously downward(arrow)
by the gravity. The ball moves in a parabolic motion.
15. Gravity(万有引⼒)
The gravity between two objects can be expressed by
F =
GM1M2
r2
The gravity becomes stronger when…
(1) The mass of objects is heavier.
(2) The distance between two objects is closer.
(ex) If the distance between two objects becomes
1m from 2m, then the gravity becomes 4 times
larger
G = 6.67 × 10−20
km3
kg−1
s−2
16. gravity
centrifugal force
(离⼼⼒)
The motion of planet
The earth is pulled by sun’s gravity. Not only gravity, the earth feels centrifugal force(离⼼⼒)
due to its accelerating motion
Consider the motion of earth around the sun.
17. In order to keep that earth can orbit around the sun, gravity should be
equal to centrifugal force （ ）
Fg = Fc
The motion of planet
18. m
v2
r
= G
Mm
r2
Mass of the Sun
M =
rv2
G
r = 1.5 × 108
km
The distance between
Sun and earth
The speed of the earth
v = 30km/s G = 6.67 × 10−20
km3
kg−1
s−2
M =
rv2
G
=
1.5 × 108
× 30 × 30
6.67 × 10−20
= 2 × 1030
kg
Fg = Fc can be expressed by
Mass of Sun
Thus, we can evaluate the mass of the sun
19. Escaping velocity
When we throw a ball at point A, in order that the ball comes back to point A by going
through the earth, how much velocity do we need?
20. Escaping velocity
m
v2
r
= G
Mm
r2
Remember this equation Here, is the mass of the earth and is the mass of you.
M m
is the speed to escape from the earth and is the radius of
the earth.
v r
v =
GM
r
r = 6300km M = 6.0 × 1024
kg
G = 6.67 × 10−20
km3
kg−1
s−2
v = 7.9km/s
23. Derive Kepler’s 3rd law
r m
v2
r
= G
Mm
r2
Equation of motion
From equation of motion, we can get
T =
2πr
v
= 2πr
r
GM
In the case of circular motion, the orbital period is
T
Thus, we can derive
T2
r3
=
4π2
GM
= constant
Here we assume circular motion, we can also derive Kepler’s 3rd law in the case of elliptical
orbit.
v =
GM
r
24. Kepler’s law and Newton gravity
In previous page, I derived Kepler’s 3rd law from Newton’s gravity.
However
Historically, Newton discovered Newton’s gravity theory from Kepler’s law.
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T2
/ R3
F =
GM1M2
r2
25. Scientific direction
In the case of Newton, Newton discovered his gravity theory from Kepler’s data.
In science, there are 2 approaches.
(1) Data (phenomenon) > Theory
You discover a rule from observational data (phenomenon). (e.g Newton)
(2) Theory > Data (phenomenon)
With guidance of mathematics, you first establish theory and predict phenomenon (e.g. Einstein)
⿊洞
引⼒波
26. (Question 1) Please check the eccentricity of each planets in Solar system
(Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune) . What do you
learn from the eccentricity of each planet
Homework
(Question 2) Please check the orbital semi-major axis (R) and orbital period of
each planets (T) in Solar system (Mercury, Venus, Earth, Mars, Jupiter, Saturn,
Uranus, Neptune)
(Question 3) After Question2, calculate and check Kepler’s third law
holds.
T2
/a3
27. Summary
• After Copernican revolution, Brahe collected the data and
Kepler found the rules from the data
• Newton discovered the rule of gravity and established
mechanics. He successfully explained Kepler’s law.
• Their contributions opened modern astronomy.
