1. H205 Cosmic Origins  Making Sense (Ch. 4)  EP2 Due Today APOD

2. Astronomy in the Renaissance  Could not reconcile Brahe’s measurements of the position of the planets with Ptolemy’s geocentric model  Reconsidered Aristarchus’s heliocentric model with the Sun at the center of the Solar system Nicolaus Copernicus (1473-1543)

3. Johannes Kepler (1571-1630)  Using Tycho’s precise observations of the position of Mars in the sky, Kepler showed the orbit to be an ellipse, not a perfect circle  Three laws of planetary motion

4. Kepler’s 1 st Law  Planets move in elliptical orbits with the Sun at one focus of the ellipse  Words to remember  Focus vs. Center  Semi-major axis  Semi-minor axis  Perihelion, aphelion  Eccentricity

5. Definitions  Planets orbit the Sun in ellipses, with the Sun at one focus  The eccentricity of the ellipse, e, tells you how elongated it is  e=0 is a circle, e<1 for all ellipses e=0.02 e=0.4e=0.7

6. Eccentricity of Planets (& Dwarf Planets) Mercury0.206Saturn0.054 Venus0.007Uranus0.048 Earth0.017Neptune0.007 Mars0.094Pluto0.253 Jupiter0.048Ceres0.079 Which orbit is closest to a circle?

7. Kepler’s 2nd Law  Planets don’t move at constant speeds  The closer a planet is to the Sun, the faster it moves  A planet’s orbital speed varies in such a way that a line joining the Sun and the planet will sweep out an equal area each month  Each month gets an equal slice of the orbital pie Kepler’s 2 nd Law

8. Kepler’s 2 nd Law:

11. If the planet sweeps out equal areas in equal times, does it travel faster or slower when far from the Sun? Same Areas

12. Kepler’s 3 rd Law The amount of time a planet takes to orbit the Sun is mathematically related to the size of its orbit The square of the period, P, is proportional to the cube of the semimajor axis, a P 2 = a 3

13. Kepler’s 3 rd Law  Third law can be used to determine the semimajor axis, a, if the period, P, is known, a measurement that is not difficult to make P 2 = a 3  Express the period in years  Express the semi-major axis in AU

14. Examples of Kepler’s 3 rd Law  Express the period in years  Express the semi-major axis in AU BodyPeriod (years) Mercury0.24 Venus0.61 Earth1.0 Mars1.88 Jupiter11.86 Saturn29.6 Pluto248 For Earth: P = 1 year, P 2 = 1.0 a = 1 AU, a 3 = 1.0 P 2 = a 3

15. Examples of Kepler’s 3 rd Law BodyPeriod (years) Mercury0.240 9 Venus0.61 Earth1.0 Mars1.88 Jupiter11.86 Saturn29.6 Pluto248 For Mercury: P = 0.2409 years P 2 = 5.8 x 10 -2 a = 0.387 AU a 3 = 5.8 x 10 -2 P 2 = a 3

16. Examples of Kepler’s 3 rd Law BodyPeriod (years) Mercury0.240 9 Venus0.615 2 Earth1.0 Mars1.88 Jupiter11.86 Saturn29.6 Pluto248 For Venus: P = 0.6152 years P 2 = 3.785 x 10 -1 What is the semi-major axis of Venus? P 2 = a 3 a = 0.723 AU

17. Examples of Kepler’s 3 rd Law BodyPeriod (years) Mercury0.240 9 Venus0.615 2 Earth1.0 Mars1.88 Jupiter11.86 Saturn29.6 Pluto248 For Pluto: P = 248 years P 2 = 6.15 x 10 4 What is the semi-major axis of Pluto? P 2 = a 3 a = 39.5 AU

18. Examples of Kepler’s 3 rd Law BodyPeriod (years) Mercury0.2409 Venus0.6152 Earth1.0 Mars1.88 Jupiter11.86 Saturn29.6 Pluto248 The Asteroid Pilachowski (1999 ES5): P = 4.11 years What is the semi-major axis of Pilachowski? P 2 = a 3 a = ??? AU

19. Comparing Heliocentric Models

20. Kepler’s 3 Laws of Planetary Motion  Planets move in elliptical orbits with the Sun at one focus of the ellipse  A planet’s orbital speed varies in such a way that a line joining the Sun and the planet will sweep out an equal area each month  P 2 = a 3 (the square of the period of a planet orbiting the sun is equal to the cube of the semi-major axis of the planet’s orbit) But WHY ????????

21. The Problem of Astronomical Motion  Galileo investigated this connection with experiments using projectiles and balls rolling down planks  He put science on a course to determine laws of motion and to develop the scientific method Astronomers of antiquity did not connect gravity and astronomical motion

22. inertia! Demonstrated the ideas of inertia and forces  Without friction…  a body at rest tends to remain at rest  a body in motion tends to remain in motion Galileo experimented with inclined planes

23. Isaac Newton, the Laws of Motion, and the Universal Law of Gravitation  Newton  Born same year Galileo died  Attempts to understand motion of the Moon  Leads him to deduce the law of gravity (as we still use it today!)  Requires him to invent new mathematics  Leads him to deduce the general laws of motion

24. A body continues in a state of rest or uniform motion in a straight line unless made to change that state by forces acting on it Galileo’s ideas of inertia became Newton’s First Law of Motion:

25. Newton’s First Law  Important ideas  What is a force? A push or a pull  The sum of all the forces on an object is the net force  If the forces all balance, the net force is zero, and the object’s motion will not change If the speed or direction of motion of an object changes, then a nonzero net force must be present

26. Newton’s 2nd Law: Acceleration  Acceleration  An object increasing or decreasing in speed along a straight line is accelerating  An object changing direction, even with constant speed, is accelerating  Acceleration is produced by a force  Acceleration and force are proportional (double the force, double the acceleration

27. Newton’s Second Law: Mass  Mass is the amount of matter an object contains  Technically, mass is a measure of an object’s inertia  Mass is generally measured in kilograms  Mass should not be confused with weight, which is a force related to gravity – weight may change from place to place, but mass does not

28. Newton’s Second Law of Motion  The amount of acceleration (a) that an object undergoes is proportional to the force applied (F) and inversely proportional to the mass (m) of the object  This equation applies for any force, gravitational or otherwise F = ma

29.  When a force acts on a body, the resulting acceleration is equal to the force divided by the object's mass  Acceleration – a change in speed or direction  Notice how this equation works:  The bigger the force, the larger the acceleration  The smaller the mass, the larger the acceleration Newton's Second Law or

30. F = ma

31. Newton’s Third Law of Motion  When two objects interact, they create equal and opposite forces on each other  This is true for any two objects, including the Sun and the Earth!

32. Newton and the Apple - Gravity  Newton realized that there must be some force governing the motion of the planets around the Sun  Amazingly, Newton was able to connect the motion of the planets to motions here on Earth through gravity  Gravity is the attractive force two objects place upon one another

33.  G is the gravitational constant G = 6.67 x 10 -11 N m 2 /kg 2  m 1 and m 2 are the masses of the two bodies in question  r is the distance between the two bodies The Gravitational Force

34. To explain the motion of the planets, Newton developed three ideas: 1.The laws of motion 2.The theory of universal gravitation 3.Calculus, a new branch of mathematics Newton solved the premier scientific problem of his time --- to explain the motion of the planets. “If I have been able to see farther than others it is because I stood on the shoulders of giants.” --- Newton’s letter to Robert Hooke, probably referring to Galileo and Kepler 2 21 r mGm F  F = ma

35.  Kepler's Laws were a revolution in regards to understanding planetary motion, but there was no explanation why they worked  The explanation was provided by Isaac Newton when formulated his laws of motion and gravity  Newton recognized that the same physical laws applied both on Earth and in space.  And don’t forget the calculus! Isaac Newton

36. Remember the Definitions  Force: the push or pull on an object that affects its motion  Weight: the force which pulls you toward the center of the Earth (or any other body)  Inertia: the tendency of an object to keep moving at the same speed and in the same direction  Mass: the amount of matter an object has

37. Astronomical Motion  Planets move along curved (elliptical) paths, or orbits  Speed and direction are changing Must there be a net force on the planets? Yes!

38. Gravity is that force  Gravity gives the Universe its structure  a universal force that causes all objects to pull on all other objects everywhere  holds the Earth in orbit around the Sun, the Sun in orbit around the Milky Way, and the Milky Way in its path within the Local Group

39. Everything attracts everything else!! Newton’s Law of Gravity

40. Orbital Motion and Gravity  Newton  Explained the Moon’s motion with force that pulls the Moon from a straight, inertial trajectory  Showed that the force must decrease with distance  defined the properties of gravity  wrote the equations of motion with gravity  The Moon moves “parallel” to the Earth’s surface at such a speed that its gravitational deflection toward the surface is offset by the surface’s curvature away from the projectile

41. Orbital Motion Using Newton’s First Law  At a sufficiently high speed, the cannonball travels so far that the ground curves out from under it.  The cannonball literally misses the ground!

42. Determining the Mass of the Sun  How do we determine the mass of the Sun?  Put the Sun on a scale and determine its weight???  Since gravity depends on the masses of both objects, we can look at how strongly the Sun attracts the Earth  The Sun’s gravitational attraction keeps the Earth going around the Sun, rather than the Earth going straight off into space  By looking at how fast the Earth orbits the Sun at its distance from the Sun, we can get the mass of the Sun

43. Measuring Mass with Newton’s Laws - Assumptions to Simplify the Calculation  Assume a small mass object orbits around a much more massive object  The Earth around the Sun  The Moon around the Earth  Charon around Pluto  Assume the orbit of the small mass is a circle

44. Measuring the Mass of the Sun  The Sun’s gravity is the force that acts on the Earth to keep it moving in a circle  M Sun is the mass of the Sun in kilograms  M Earth is the mass of the Earth in kilograms  r is the radius of the Earth’s orbit in kilometers  The acceleration of the Earth in orbit is given by: a = v 2 /r  where v is the Earth’s orbital speed

45. Measuring the Mass of the Sun  Set F = m Earth v 2 /r equal to F = GM Sun m Earth /r 2 and solve for M Sun M Sun = (v 2 r)/G  The Earth’s orbital speed (v) can be expressed as the circumference of the Earth’s orbit divided by its orbital period: v = 2  r/P

46. Measuring the Mass of the Sun  Combining these last two equations: M Sun = (4  2 r 3 )/(GP 2 )  The radius and period of the Earth’s orbit are both known, G and π are constants, so the Sun’s mass can be calculated  This last equation in known as Kepler’s modified third law and is often used to calculate the mass of a large celestial object from the orbital period and radius of a much smaller mass

47. So what is the Mass of the Sun? M Sun = (4  2 r 3 )/(GP 2 ) r Earth = 1.5 x 10 11 meters P Earth = 3.16 x 10 7 seconds G = 6.67 x 10 -11 m 3 kg -1 s -2 Plugging in the numbers gives the mass of the Sun M Sun = 2 x 10 30 kg

48.  The same formula gives the mass of the Earth  Using the orbit of the Moon:  r Moon = 3.84 x 10 5 km  P Moon = 27.322 days = 2.36 x 10 6 seconds  Mass of Earth is 6 x 10 24 kg What about the Earth? M Earth = (4  2 r 3 )/(GP 2 )

49.  The same formula gives the mass of Pluto, too  Using the orbit of Pluto’s moon Charon:  r Charon = 1.96 x 10 4 km  P Charon = 6.38 days = 5.5 x 10 5 seconds  Mass of Pluto is 1.29 x 10 22 kilograms How about Pluto? M Pluto = (4  2 r 3 )/(GP 2 )

50. Orbits tell us Mass  We can measure the mass of any body that has an object in orbit around it  Planets, stars, asteroids  We just need to know how fast and how far away something is that goes around that object  But we can’t determine the mass of the Moon by watching it go around the Earth  To determine the mass of the Moon, we need a satellite orbiting the Moon

51. Revisions to Kepler's 1st Law  Newton's law of gravity required some slight modifications to Kepler's laws  Instead of a planet rotating around the center of the Sun, it actually rotates around the center of mass of the two bodies  Each body makes a small elliptical orbit, but the Sun's orbit is much much smaller than the Earth's because it is so much more massive

52. Revisions to Kepler's 3rd Law  Gravity also requires a slight modification to Kepler's 3rd Law  The sum of the masses of the two bodies is now included in the equation  For this equation to work, the masses must be in units of solar mass (usually written as M  )  Why did this equation work before?  Remember - for this equation to work:  P must be in years!  a must be in A.U.  M 1 and M 2 must be in solar masses

53. For Next Week  Surveying the Stars (Ch. 15)  Milky Way (Ch. 19)  Hand in EP2