Physics 125 Solar System Astronomy

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1 Physics 125 Solar System Astronomy James Buckley Lecture 8 More Physics!

2 Reading Quiz??? -No Quiz! :) -Homework! :(

3 Physics (More)

4 Circular Acceleration ~a = ~v t ~v Recall: arclength is equal to radius times the angle, similarly ~v v ˆr ~v 0 ~v = ~a t t = t = r s/v /v ~a = v ˆr r /v ~a = ˆr v2 r

5 Kepler s law for circular orbits Force of Gravity: Circular Acceleration: Newton s Second Law: F = GMm r 2 a = v2 r ~F = m~a GMm r 2 = mv2 r Note: Mass of orbiting planet cancels! v 2 = GM r Period, T =circumference/speed T = 2 r v T 2 = 4 2 r 2 v 2 T 2 = 4 2 GM r3

6 Orbits This derivation was done for circular orbits. We need to modify things a bit for elliptical orbits - basically we must replace the radius of the circle r with the average distance between the two objects a. We also must take into account that it is not the Sun at one focus of the ellipse, but the center of mass of the Sun-Earth system. I won t do the math but this results in another modification where we need to replace the mass of the sun M with the sum of the mass of the Sun and the planet M+m T 2 = T 2 = 4 2 GM a3 4 2 G(M + m) a3 In fact the Sun makes very tiny orbits (within it s radius) about this center of mass.

7 Center of Mass Planets actually orbit around the center of mass of the Sun-planet system. Center of mass can be determined mathematically as above, or by placing two objects (or a scale model for planets!) on an imaginary balance (seesaw) structure (with some unspecified downward gravitational force) then finding the position of the fulcrum to achieve balance.

8 Demonstration A fun party trick to impress your friends (at parties) Analysis: Fulcrum + + Center of mass

9 Angular Momentum and Orbits ~L = ~r ~p ~L = ~r (m~v) L = mvr sin

10 Tides Note that the radius of the earth r = 12, 742km is much smaller than the distance between the earth and moon r moon = 384, 000km and the distance between the Earth and the sun r sun = 149, 000, 000 km r r = F 2 = GM m r 2 (1 + ) 2 2 =(0.03) 2 = GM m 0 = r 2 ( ) GM m 1 2 r 2 (1 + 2 ) 1 2 GM m r GM m(1 2 ) r 2 (define r/r) F 1 = GMm r 2 F 2 = GMm (r + r) 2 Sum of forces F 1 + F 2 + F other stu combines to keep Earth in circular orbit. Di erence, stretches oceans and makes tide Stretching, tidal force = F 1 F 2 Tidal force = 2GM m r GMm Tidal force = r 2 [1 (1 2 )] = 2GM r 3 r 2

11 Tides M= mass of moon or sun F 1 = GMm r 2 F 2 = GMm (r + r) 2 Difference in gravitational force on near and far side results in oceans being pulled more strongly than average gravitational force on the rigid body of the earth on near side, more weakly on far side

12 Spring Tides Note that on the near side, there is a slightly larger force than needed to keep the rigid planet in orbit around the sun, but on the far side slightly less - resulting in an effective stretching (centrifugal-like) force on both sides. So even when it is full moon, and it appears that the forces of sun and moon work against each other, both provide a positive effective stretching force, adding the tides. At full and new moon, we get stronger Spring tides

13 Neap Tides At first and third quarter moon, effects of Sun and moon work against each other producing weaker neap tides

Tidal Locking Near Side (we always see this face from Earth) Near Side (we always see this face from Earth) Moon is tidally locked to the Earth, we always see the same side (left) never the far side

14 Tidal Locking Near Side (we always see this face from Earth) Near Side (we always see this face from Earth) Moon is tidally locked to the Earth, we always see the same side (left) never the far side (right). How did this come to be? If the moon were not tidally locked, it would be compressed one way, then the next. All of this crunching dissipates energy and the moon is only happy when it is locked in place!

15 Tidal Locking We always see the same face of the moon because of tidal locking

16 Transfer of Angular Momentum You can t get rid of angular momentum unless you transfer it to something else. If someone slams the door on you, it imparts some angular momentum to you, and you to the floor (through friction) but you don t notice since you are much more massive than the door, and the earth is much more massive than you!

Earth s bulge back, slowing the rotation Time out! Foul!

17 Transfer in Eath/Moon System (1) Friction in rotating Earth pulls tidal bulge slightly ahead of moon (3) The Earth s bulge pulls the moon ahead, increasing its orbital distance (2) The moon gravity tries to pull the Earth s bulge back, slowing the rotation Time out! Foul! How do we know that the tug of the earth doesn t increase the velocity, and decrease the radius from Kepler s laws? If the radius increases does the velocity decrease?!

18 Angular Momentum ~L = ~r ~p ~v ~L = ~r (m~v) ~L = ~r (m~v) =mvr! ~r v 2 = GM r r GM v = r p pr L = GM m As r increases, the angular momentum increases (for a given mass m) So if angular momentum is conserved, and the Earth s angular momentum decreases, the angular momentum of the Moon s orbit must increase, so r increases and v decreases!

19 Summarizing The moon is slowing down the Earth s rotation and gradually moving away from the Earth. Early in the history of the Earth moon system, a day may have only been 5-6 hours long, and the Moon one tenth of its current distance. Eventually we may loose our moon, the Earth will become tidally locked to the moon, each month will only have one day and there will be fewer months per year. Is that weird or what?!

20 Discussion Question 7 lb 3 lb 5 lb 5 lb 5 lb 5 lb You have asked for a very specific 10 lb free-weight (bar bell) for your birthday, but worried your friend got the wrong one (there are actually 3 10 lb free weights at the sporting good store. What experiments can you do on the package to see which one you are getting (without opening the box, and ruining the surprise) - (Note: assumed the weights are well packed, with the point midway between the two extreme ends in the center of the box)

21 Phasesacceleration of the moon Circular (not to be confused with waning and waxing gibbons) v a = t Physics Lecture 1 p. 25/27

22 Eclipses

23 Eclipses

24 Summary Read Sections 4.4 and 4.5 (if you did not already)

25 Units of distance 1 a.u. = 93 million miles = km A light year is defined as the distance that light travels in one years time or 1ly sec m/sec = km = 63, 000 a.u. A parsec is defined as the distance at which 1 a.u. would subtend one second of arc. A star at distance d (measured in parsecs) will be observed to have a parallax angle of θ (in arcsecs) given by the formula: ( ) θ 1 arcsec = d 1 1 parsec 1 p.c. = 3.26 ly Physics Lecture 3 p. 12/12

26 Kepler s laws Kepler s First Law: A planet orbits the Sun in an ellipse with the sun at one focus of the ellipse. r + r =2a Kepler s Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals. Kepler s Third Law: P 2 = a 3 Physics Lecture 3 p. 7/12

27 Parallax distance S un Earth! Mercury

system Planet Period (years) Approx. Radius (a.u.) Earth 1.0 1.0 Mercury 0.

28 Distance to Planets Parallax distance S un Earth! Mercury Venus transit, APOD July 20, 2004 Relative scales of the solar system Planet Period (years) Approx. Radius (a.u.) Earth Mercury Venus Mars Jupiter Venus transit, Crow Observatory, June 5, 2012

29 List # 1 2 Map of nearby stars 3 Future and past 4 See also 5 References 6 External links Nearest Stars System Designation Star Star # Stellar class Apparent magnitude (m V ) Absolute magnitude (M V ) Epoch J Right ascension [2] Declination [2] Solar System Sun G2V [2]!26.74 [2] 4.85 [2] variable: the Sun travels along the ecliptic Alpha Centauri (Rigil Kentaurus; Toliman) Proxima Centauri (V645 Centauri) # Centauri A (HD ) # Centauri B (HD ) EZ Aquarii B 16 M? List of nearest stars and parallax [2] [2] EZ Aquarii C 16 M? (from Wikipedia) [2] [2] Parallax [2][3] Arcseconds(±err) Distance [4] Light-years (±err) Additional references 1 M5.5Ve [2] [2] 14 h 29 m 43.0 s!62 40! 46" (0 29)" [5][6] (16) [7] 2 G2V [2] 0.01 [2] 4.38 [2] 14 h 39 m 36.5 s!60 50! 02" 2 K1V [2] 1.34 [2] 5.71 [2] 14 h 39 m 35.1 s!60 50! 14" (1 17)" [5][8] (68) 2 Barnard's Star (BD a) 4 M4.0Ve 9.53 [2] [2] 17 h 57 m 48.5 s ! 36" (1 00)" [5][6] (109) 3 Wolf 359 (CN Leonis) 5 M6.0V [2] [2] [2] 10 h 56 m 29.2 s ! 53" (2 10)" [5] (390) 4 Lalande (BD ) 6 M2.0V [2] 7.47 [2] [2] 11 h 03 m 20.2 s ! 12" (0 70)" [5][6] (148) 5 Sirius (# Canis Majoris) 6 Luyten Sirius A 7 A1V [2]!1.46 [2] 1.42 [2] 06 h 45 m 08.9 s!16 42! 58" (1 28)" [5][6] (289) Sirius B 7 DA2 [2] 8.44 [2] [2] Luyten A (BL Ceti) Luyten B (UV Ceti) 9 M5.5Ve [2] [2] 01 h 39 m 01.3 s!17 57! 01" (2 70)" [5] (631) 10 M6.0Ve [2] [2] 7 Ross 154 (V1216 Sagittarii) 11 M3.5Ve [2] [2] 18 h 49 m 49.4 s!23 50! 10" (1 78)" [5][6] (512) 8 Ross 248 (HH Andromedae) 12 M5.5Ve [2] [2] 23 h 41 m 54.7 s ! 30" (1 10)" [5] (36) 9 Epsilon Eridani (BD!09 697) 13 K2V [2] 3.73 [2] 6.19 [2] 03 h 32 m 55.8 s!09 27! 30" (0 79)" [5][6] (27) 10 Lacaille 9352 (CD! ) 14 M1.5Ve 7.34 [2] 9.75 [2] 23 h 05 m 52.0 s!35 51! 11" (0 87)" [5][6] (31) 11 Ross 128 (FI Virginis) 15 M4.0Vn [2] [2] 11 h 47 m 44.4 s ! 16" (1 35)" [5][6] (49) EZ Aquarii (GJ 866, Luyten 789-6) Procyon (# Canis Minoris) EZ Aquarii A 16 M5.0Ve [2] [2] 22 h 38 m 33.4 s!15 18! 07" (4 40)" [5] (171) Procyon A 19 List of nearest stars - Wikipedia, the free encyclopedia has 8 planets has two proposed planets F5V- IV [2] 0.38 [2] 2.66 [2] 07 h 39 m 18.1 s ! 30" (0 81)" [5][6] (32) [2] [2] [2]

geographic latitude known as the tropic of cancer.

30 Tropics of Cancer and Capricorn Sun is directly overhead on the Summer Solstice along a line of geographic latitude known as the tropic of cancer. Further north, it may never reach this altitude angle.

31 Celestial Sphere Template

32 Earth Template

Physics 125 Solar System Astronomy

Physics 125 Solar System Astronomy James Buckley buckley@wuphys.wustl.edu Lecture 7 The Physics behind Astronomy Reading Quiz 4 -What conservation law explains why an ice skater spins faster when she pulls