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### lecture5

1. Introduction to modern astronomy 5 島袋隼⼠(Hayato Shimabukuro)（云南⼤学、 SWIFAR） ©GETTYIMAGES
2. 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.
3. 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
4. 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
5. 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)
6. 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)
7. 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.
8. In order to keep that earth can orbit around the sun, gravity should be equal to centrifugal force （ ） Fg = Fc The motion of planet
9. 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
10. Kepler’s law and Newton gravity Newton’s theory can explain Kepler’s law perfectly! <latexit sha1_base64="IwZJQ8OYZJRAckwK9Rj60k5vMFE=">AAAB+HicbVDLTgIxFO3gC/HBqEs3jcTEFZlBjS6Jblyi4ZXAQDqlAw2dtmk7JjjhS9y40Bi3foo7/8YCs1DwJDc5Oefe3HtPKBnVxvO+ndza+sbmVn67sLO7t190Dw6bWiQKkwYWTKh2iDRhlJOGoYaRtlQExSEjrXB8O/Nbj0RpKnjdTCQJYjTkNKIYGSv13WK9V4FdqYQ0Aj70zvtuySt7c8BV4mekBDLU+u5XdyBwEhNuMENad3xPmiBFylDMyLTQTTSRCI/RkHQs5SgmOkjnh0/hqVUGMBLKFjdwrv6eSFGs9SQObWeMzEgvezPxP6+TmOg6SCmXiSEcLxZFCYP2yVkKcEAVwYZNLEFYUXsrxCOkEDY2q4INwV9+eZU0K2X/suzdX5SqN1kceXAMTsAZ8MEVqII7UAMNgEECnsEreHOenBfn3flYtOacbOYI/IHz+QNw+ZJM</latexit> T2 / R3 F = ma Kepler's law F = GM1M2 r2
11. <latexit sha1_base64="IwZJQ8OYZJRAckwK9Rj60k5vMFE=">AAAB+HicbVDLTgIxFO3gC/HBqEs3jcTEFZlBjS6Jblyi4ZXAQDqlAw2dtmk7JjjhS9y40Bi3foo7/8YCs1DwJDc5Oefe3HtPKBnVxvO+ndza+sbmVn67sLO7t190Dw6bWiQKkwYWTKh2iDRhlJOGoYaRtlQExSEjrXB8O/Nbj0RpKnjdTCQJYjTkNKIYGSv13WK9V4FdqYQ0Aj70zvtuySt7c8BV4mekBDLU+u5XdyBwEhNuMENad3xPmiBFylDMyLTQTTSRCI/RkHQs5SgmOkjnh0/hqVUGMBLKFjdwrv6eSFGs9SQObWeMzEgvezPxP6+TmOg6SCmXiSEcLxZFCYP2yVkKcEAVwYZNLEFYUXsrxCOkEDY2q4INwV9+eZU0K2X/suzdX5SqN1kceXAMTsAZ8MEVqII7UAMNgEECnsEreHOenBfn3flYtOacbOYI/IHz+QNw+ZJM</latexit> 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
12. 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
13. 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
14. 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.
15. 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
16. 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!
17. 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.
18. 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 !!
19. 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
20. (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)