Introduction to modern astronomy 5
島袋隼⼠(Hayato Shimabukuro)（云南⼤学、
SWIFAR）
©GETTYIMAGES

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.

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 × 10−11
m3
kg−1
s−2

F =
GM1M2
r2
Let’s calculate gravity between us !
G = 6.67 × 10−11
m3
kg−1
s−2
Me: 78kg You: M kg
3m

F =
GM1M2
r2
(Answer)
G = 6.67 × 10−11
m3
kg−1
s−2
If your weight is M kg, then
F =
6.67 × 10−11
[m3
kg−1
s−2
] × 75[kg] × M[kg]
32[m2]
= 5.56 × 10−10
M[N]
Electrostatic force(静电⼒)
F ∼ 10−3
[N]
[N] = [kg ⋅ m/s2
]
F ∼ 102
N
Someone slaps you (average force)

F =
GM1M2
r2
(Answer)
G = 6.67 × 10−11
m3
kg−1
s−2
If your weight is M kg, then
F =
6.67 × 10−11
[m3
kg−1
s−2
] × 75[kg] × M[kg]
32[m2]
= 5.56 × 10−10
M[N]
Gravity is very weak at
human level !!
Electrostatic force(静电⼒)
F ∼ 10−3
[N]
[N] = [kg ⋅ m/s2
]
F ∼ 102
N
Someone slaps you (average force)

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.

In order to keep that earth can orbit around the sun, gravity should be
equal to centrifugal force （ ）
Fg = Fc
The motion of planet

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

Kepler’s law and Newton gravity
Newton’s theory can explain
Kepler’s law perfectly!
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T2
/ R3
F = ma
Kepler's law
F =
GM1M2
r2

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T2
/ R3
Derive Kepler’s 3rd law
Here, we derive Kepler’s 3rd law by Newton gravity theory.
For simplicity, we consider circular motion
r
m
v2
r
= G
Mm
r2
Equation of motion
or T2
/R3
= constant
Target

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

Is Newton theory perfect?
Newton explained the motion of planets. So, that’s all? Did we understand gravity?
No! This is the beginning of the modern physics.
Newton’s theory has continued for 250 years! But, in 1905, Einstein suggested a new theory of
gravity, called theory of general relativity (⼴义相对论)
250 years
Albert Einstein

Einstein gravity
•In Newton's theory, he did not explain what gravity is. He just explained the how we calculate
the gravity.
Rμν −
1
2
gμνR + Λgμν =
8πG
c4
Tμν
Einstein equation
•However, Einstein explained “Gravity is a distortion of space-time(引⼒是时空的扭曲).”
时空的扭曲 物质（能量）
If there exists material (energy), it distorts space-time and then gravity arises.

What General relativity predicts ①
•General relativity predicts very massive objects called “blackhole(⿊洞)”
•After relatively massive stars die, they become blackhole.
•In 2019, EHT first observed the shadow of
super massive blackhole !
•There are two kinds of black holes.
1. Star-size black hole (~太阳质量)
2. Supermassive blakchole ( 太阳质量)
105
− 1010

What General relativity predicts②
引⼒波
• General relativity predicts “gravitational wave (引⼒
波)”
•When massive objects (black holes, neutron stars)
move, they generate gravitational wave.
•We had not observed the gravitational wave for 100 years after Einstein predicted it.
•In 2016, LIGO & Virgo finally detected the gravitational wave!

What General relativity predicts ③
Distortion of image (Weak lensing) Multiple images (strong lensing)
• General relativity predicts the gravitational lensing
effect（引⼒透镜）and this effect is already observed.
•If there are massive objects (galaxy, galaxy clusters)
between background source and us, the path of light is
distorted.

What General relativity predicts④
•General relativity predicts “expanding universe (膨胀宇宙)”
In 1929, Hubble discovered that galaxies are moving
away from earth at speeds proportion to their
distance.
V = Hr
The velocity of galaxy How long distance galaxy is away from us
•This is due to expansion of the universe.
Our universe is expanding !!

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

(Question 1) : Let’s calculate the gravitational force between the sun and the earth.
Report task
F =
GM1M2
r2
G = 6.67 × 10−11
m3
kg−1
s−2
•The mass of sun :M1 = 2 × 1030
kg •The mass of the earth :M2 = 6.0 × 1024
kg
•The distance between the sun and the earth : r = 1.5 × 108
km
(Question 2) : Please explain how geocentric idea historically changed to heliocentric
idea.
(Question 3) : Please choose 1 topic below and report it.
1. Blackhole(⿊洞) 2. Gravitational lensing(引⼒透镜) 3. Gravitational wave(引⼒波)

•Please send your report to shimabukuro@ynu.edu.cn by PDF.
Report task
Please including following information
1. The tittle of your email is “现代天⽂学作业报告”
2. Your name (姓名)
3. Your student number（学号）
Deadline of the submission: 2023/3/27 (Monday)
I accept the report written by English or Chinese, but I recommend you to write the report by
English (and it is very helpful for me!). Even if your English is wrong, it does not affect your
score.

Summary
• Newton discovered the rule of gravity and established
mechanics. He successfully explained Kepler’s law.
• Their contributions opened modern astronomy.
• Einstein constructed a new gravity theory.