Kepler s Laws and our Solar System

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1 Kepler s Laws and our Solar System The Astronomical Unit, AU Kepler s Empirical Laws of Planetary mo=on The mass of the Sun, M O. A very brief tour of the solar system Major planets Dwarf planets (defini=on) Minor bodies Asteroid Belt, Trojans, Centaurs TNOs Kuiper Belt Oort cloud The mass distribu=on and abundance of the solar system

2 The Astronomical Unit (AU) Approximately: the mean Sun- Earth distance: x m (i.e., ~1.5x10 11 m or 150 million kilometers) Precisely: Radius of a par=cle in a circular orbit around the Sun moving with an angular frequency of radians per Solar Day (i.e., 2π radians per year) Measured via: Transit of Venus (trad.), Radar reflec=ons from the planets, or the =me delay in sending radio signals to space missions in orbit around other planets and the applica=on of Kepler s Laws

3 The Earth- Sun distance Easy to measure rela=ve distance of inner planets, i.e., Venus Angle a can be observed (furthest angle Venus gets away from Sun) AB=AC cos(a) c Venus = 0.72AU [Can also use Kepler s 3 rd law to derive rela=ve distances to all planets, see later in lecture]

4 Transit of Venus (1769, Tahi=) 1. Observe Transit of Venus from two or more loca=ons. 2. Record displacement angle (E) of the Venus transit against the solar disc 3. Bring data together 4. Measure angle between transit points 5. Use basic geometry to derive Earth- Venus distance 1! tan# 1 " 2 v $ & = 2 d A'B % d Earth-Venus d Earth-Venus = (1' 0.72)d Earth-Sun

5 è 153million kilometers è 149million kilometers Microwave radarè 150million kilometers +/- 30m! Next transit: 6 th June 2012 Then: December 2117! Transit

6 Kepler s Empirical Laws of Planetary Mo=on REMINDER: Defini=on of an ellipse. The set points whose sum of the distances from 2 fixed points (the foci) is a constant. f b a a: semi- major axis b: semi- minor axis e: eccentricity e = f a b = a %1 e 2

7 Kepler s Empirical Laws of Planetary Mo=on REMINDER: Defini=on of an ellipse. The set points whose sum of the distances from 2 fixed points (the foci) is a constant. a f b 2a a a: semi- major axis b: semi- minor axis e: eccentricity e = f a!!!!!!!!

8 Kepler s First Law 1. Each planet moves about the Sun in an ellip=cal orbit, with the Sun at one focus of the ellipse. Perihelion (near Sun) distance: d perihelion = 1 2 (2a! 2 f ) d perihelion = a(1! e) Aphelion (away Sun) distance: d aphelion = d perihelion + 2 f d aphelion = a(1+ e) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Eccentricity A comet has a perihelion distance of 3AU and an aphelion distance of 7AU. What is the semi- major axis of the ellipse? What is the eccentricity?

9 Kepler s Second Law 2. The straight line (radius vector) joining a planet and the Sun sweeps out equal areas of space in equal intervals of =me. C B Planet Aphelion D Sun A Perihelion Area Sun- A- B = Area Sun- C- D if planet moves from C to D in same =me as from A to B.

10 Kepler s Second Law Area = 0.5 r.v o.t = constant From considera=on of areas being swept at Perihelion and Aphelion: v aph v per = ( )*+ (,) h

11 Kepler s Second Law Forced via conserva=on of energy and angular momentum When an object is closer to the Sun the radial gravita=onal force felt is greater (inverse square law) which induces faster circular mo=ons along the orbit path. v perihelion = ( 1+ e)gm (1! e)a, v = ( 1! e)gm aphelion (1+ e)a

12 Keplerʼs Third Law# 3. The squares of the sidereal periods (P) of the planets are proportional to the cubes of the semi-major axes (a) of their orbits: # # ## P 2 = ka 3 ()*h, = 4.2 /0 If we choose the year as the unit of time and the AU as the unit of distance, then k=1.# #

13 Kepler s Third Law - - verifica=on. Planet a in AU P in yr Mercury Venus Earth 1 1 Mars Jupiter Saturn Uranus Neptune

14 Kepler s Third Law - - verifica=on. Planet a in AU P in yr a 3 P 2 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

15 Kepler & Newton Newton s Laws of Mo=on and Gravita=on give Kepler s Law: Consider the simple case of a circular orbit, Centrifugal=gravita=onal forces (in equilibrium) => mv 2 r = GMm r 2 P = 2! " = 2!r v => 4! 2 mr => P 2 P 2! r 3 = GMm r 2 Can be shown to hold for ellipses as well via conserva=on of energy and angular Momentum where r becomes a the semi- major axis.

16 Mass of the Sun, M o. Kepler s third law allows us to derive the solar mass: 4! 2 P 2 = GM O. r 3 M O = 4! 2 r 3. GP 2 M O. = (4! 2 )(1.5!10 11 ) 3 (6.67!10 "11 )(365.24! 24! 60! 60) 2 M O. = 2.0!10 30 kg Using radar measurement of the AU The solar mass is the standard by which we measure all masses in astronomy, e.g., Milky Way central SMBH = 10 6 M o.

17 Inner planets Rocky

18 Outer planets Gassy

19 Planet sizes to scale (not distances)

20 Our backyard: Geophysics

21 Credit: Anima=on taken From the Minor Planet Centre hqp:// Permission to show granted: Gareth Williams 20/11/11

22 Inner Solar System

23 Major planet Def n: 1. Orbits the Sun 2. Enough gravity to be spherical 3. Master of orbit Dwarf planet Def n 1&2 above 3 Not cleared orbit 4 Not a satelliqe Dwarf planets ~9 known All else: Small solar System bodies. Ceres = planet ( ), then asteroid ( ), now dwarf planet (2006+)!

24

25 Kuiper Belt

26 Kuiper Belt s around other stars

27 Oort Cloud: source of comets?

28 Sedna an Oort cloud object?

29

30 Mass of solar system Sun: 99.85% Planets: 0.135% (Jupiter 0.09%) Comets: 0.01%? Satelliqes: %? Minor Planets: %? Meteoroids: %? Interplanetary Medium: %?

31 Chemical abundance of Sun Abundance of elements in the solar system, y- axis logarithmic Only H, He and Li are formed in Big Bang rest comes from stars

32

33

34

35 Our Galaxy

36 Herchsel s view of the Galaxy in 1860

37 Mapping the Galaxy from the stars

38

39

40 Our Galaxy

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