Orbits and Gravity v v v Tycho Brahe
Orbits and Gravity v v v Tycho Brahe: 1546 - 1601 Johannes Kepler: 1571 - 1630 Isaac Newton: 1643 - 1727 Orbits and Gravity 1
Tycho Brahe A Minor Noble Orbits and Gravity 2
Tycho Brahe The Last Great Non-Telescopic Observer v 1572: Found what is now called Tycho’s Supernovae v 1577: Comet Observations v Observations of these discoveries put their distances beyond the Moon ==> by around 1600 the unchanging crystalline (celestial) spheres were no more. Orbits and Gravity 3
Tycho Brahe v Observations of Planets and Stars with small errors: best were of order 2 minutes (Remember the Moon is 30 minutes!). v Was Not a Copernican. v Hired Johannes Kepler to analyze the data. Orbits and Gravity 4
Tycho’s Solar System Orbits and Gravity 5
Johannes Kepler v Court mathematician to various patrons in Germany. v Was a Lutheran and hence forced to move about at various times due to Catholic - Lutheran conflict. v Derived three laws of planetary motion: numbers 1 & 2 were published approximately 20 years before number 3 Orbits and Gravity 6
Conic Sections Orbits and Gravity 7
Ellipse Properties Eccentricity = (Distance Between Foci) / Major Axis Orbits and Gravity 8
Kepler’s First Law Planetary Orbits are Ellipses with the Sun at One of the Foci Planet Sun Foci Orbits and Gravity 9
Kepler’s Second Law Planets Sweep Equal Areas In Equal Times v This demands variable speeds v A result of the conservation of angular momentum Orbits and Gravity 10
Kepler’s Third Law P 2 = ka 3 P is the period of the planet. a is the semi-major axis. k is a constant. This must apply to the Earth where P = 1 yr and a = 1 AU so k = 1 yr 2/AU 3. v For Kepler this was an empirical result. Newton showed it to be demanded by the gravitational force law. v v Orbits and Gravity 11
An Empirical Law Planetary Data Planet Period Semimajor P 2 a 3 Mercury 0. 24 0. 39 0. 0576 0. 0593 Venus 0. 62 0. 72 0. 3844 0. 3732 Earth 1 1 Mars 1. 88 1. 52 3. 5344 3. 5112 Jupiter 11. 86 5. 20 140. 7 140. 6 Saturn 29. 42 9. 54 865. 5 868. 2 Uranus 83. 75 19. 19 7014 7066 Neptune 163. 7 30. 07 26800 27189 Pluto 248. 0 39. 48 61503 61536 Orbits and Gravity 12
Orbital Eccentricity The eccentricity of an ellipse is the ratio of the distance between the foci to the length of the major axis. The eccentricity of Mars is 0. 093 Major Axis = 11 Units Distance Between Foci = 1 Unit The eccentricity = 1/11 = 0. 09 Orbits and Gravity 13
Variables of Motion I Vectors and Scalars v Scalar: A quantity with only magnitude v Example: Speed (km/s) v Vector: A quantity with magnitude and direction. v Example: Velocity (km/s due east) Orbits and Gravity 14
Variables of Motion v Acceleration: m/s 2 - The time rate of change of velocity. v Uniform circular motion is constant speed but changing directions therefore the velocity changes constantly. It is a vector quantity. v Velocity: m/s - Distance per second in a particular direction Vector Quantity v v = at + v 0 (for a constant acceleration) v Distance: m - Vector Quantity v d = 1/2 at 2 + v 0 t + d 0 (for a constant acceleration) Orbits and Gravity 15
Momentum is a Vector Quantity v Linear Momentum: Product of Mass and Velocity: p = mv v A property of Objects in “Straight Line” Motion v When conserved during an interaction the interaction is called an elastic interaction: For example billiard balls have near-elastic interactions. Car crashes are highly inelastic. v Angular Momentum: Depends on Mass Configuration and Rotation / Revolution Speed. v For planets in orbit it is the product of Mass, Speed, and Orbital Radius: l = mvr v For a spinning sphere: l = (3 MR 2/5) ω where ω is the spin rate (cycles/second or Hertz). Orbits and Gravity 16
Isaac Newton v Set down the laws of motion (Newton’s 3 laws). v Set down the gravitational force law. v Developed calculus simultaneously with Cotes, Laplace, Euler, and Lagrange. Gauss was working at the same time. Orbits and Gravity 17
Newton’s First Law First Stated By Galileo Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. Note that v = 0 is a state of uniform motion! In uniform circular motion the velocity changes continuously; therefore, a force must be applied. Orbits and Gravity 18
Newton’s Second Law The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors; in this law the direction of the force vector is the same as the direction of the acceleration vector. Actually, Newton said that force is the time rate of change of momentum but in the case of constant mass the statements are equivalent. Orbits and Gravity 19
Newton’s Third Law For every action there is an equal and opposite reaction. Examples: Standing Boats Rockets Any Recoil Orbits and Gravity 20
The Law of Gravitation F = G M 1 M 2 / R 2 R is the vector distance between the masses G is the gravitational constant G = 6. 672(10 -11) Nm 2/kg 2 Orbits and Gravity 21
An Example of Gravity What is the Force of gravity between 2 100 kg masses separated by 1 meter? F = G M 1 M 2 / R 2 = 6. 672(10 -11) * 100 / 1*1 = 6. 672(10 -7) Newtons = ma a = 6. 672(10 -9) m/s 2 for each mass! but d = 1/2 a t 2 Assume they are point masses so 0. 5 = 0. 5 6. 672(10 -9) t 2 t = 12200 seconds (to come into contact) Orbits and Gravity 22
Weight and Mass v Mass is an Intrinsic Invariant Property of Matter: Unit = kg v Weight is a force and depends on location as the acceleration is that of gravity. v g = GMr/Rr 2 = 9. 8 m/s 2 v W = mg but g depends on where you are. Orbits and Gravity 23
Understanding Orbital Motions n Universal mutual gravitation allows us to understand orbital motion of planets and moons – For example, Earth and the moon attract each other through gravitation n n Earth is much more massive than the moon, thus, Earth’s gravitational force constantly accelerates the moon towards Earth This acceleration is constantly changing the moon’s direction of motion, holding it on its almost circular orbit Orbits and Gravity 24
Orbital Velocity n An object’s circular velocity is the lateral velocity it must have to remain in a circular orbit. – If mass of the spaceship << the mass of Earth, then the circular velocity is n n n M = mass of the central body in kg G = gravitational constant (6. 67ˣ 10 -11 m 3/s 2/kg) r = orbital radius Orbits and Gravity 25
Calculating Escape Velocity n Orbits and Gravity 26
Orbits and Freefall v An orbit is when the fall rate matches the curvature rate, ie, freefall. v Weightlessness occurs not because of where you are but because of what you are doing! v Think of a falling elevator - Your weight is less - If you freefall your weight is 0. The orbiting spacecraft is freefalling so the weight is 0! v The mass does not change - It still carries lots of momentum! Orbits and Gravity 27
Centripedal Acceleration Uniform Circular Motion V 1 R d = v∆t " V 2 Figure 1 V 1 " V 2 Figure 2 v The two triangles are similar v v ∆ t / r = ∆v / v v v 2 / r = ∆v/∆t = a ∆v Orbits and Gravity 28
Derivation of Kepler’s 3 rd Law Gravitational Force = Centripedal Force Consider Two Masses In Orbit About Each Other G m 1 m 2 / (r 1 + r 2)2 = m 1 v 12 / r 1 P v 1 = 2πr 1 So Object 1: G m 1 m 2 / (r 1 + r 2)2 = 4π2 m 1 r 1/P 2 Object 2: G m 1 m 2 / (r 1 + r 2)2 = 4π2 m 2 r 2/P 2 Orbits and Gravity 29
To Finish Up Cancel the common masses on each side: Object 1: G m 2 / (r 1 + r 2)2 = 4π2 r 1/P 2 Object 2: G m 1 / (r 1 + r 2)2 = 4π2 r 2/P 2 Now sum: G (m 1 +m 2) / (r 1 + r 2)2 = 4π2 (r 1 + r 2)/P 2 = (4π2 / G (m 1 +m 2)) (r 1 + r 2)3 This is Kepler’s Third Law! Orbits and Gravity 30
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