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Orbits & GravityOrbits & Gravity

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Tycho Brahe (1546 – 1601)Tee-ko Bra

A Danish nobleman and astronomer

His most significant contribution to science:Nearly 30 years of his life was spent accurately documenting the position of the planets against the backdrop of the Celestial Sphere.

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Tycho’s ‘luxurious’ pretelescope-era observatory on the island of Hven where most of his data was collected.

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After nearly 30 years of observations, Tycho could not find a satisfactory model that supported the Copernican universe.

He eventually settled for a more Aristotelian model.

66www.mhs.ox.ac.uk/tycho/

Eduard Ender (1855)

Rudolph II (sitting) contemplates Tycho’s Universe

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A Copernican

Introversive, referred to himself as ‘the mangy dog’

Brilliantly disciplined in the use of mathematics

Developed three laws of planetary motion based on Brahe’s extensive data collection

Johanes Kepler (1571 – 1630)

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Kepler’s Laws are based on the mathematics/geometry associated with conic sections

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Note that every ellipse has two focal points

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Eccentricity, e, is a quantitative measure of the degree to which an object deviates from being circular. Note that for a circle the eccentricity is zero and for an increasingly elliptical shape, the eccentricity is approaching 1. A parabolic path has an e equal to 1 and a hyperbolic path has an e greater than 1 but less than infinity.

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Kepler’s 1Kepler’s 1stst Law LawLaw of OrbitsLaw of Orbits

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Note that none of the planets within our solar system have orbits that are perfect circles. (the eccentricity is not zero)

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Planets are not the only objects that follow conic paths around the Sun

And the Sun is not the only object that has objects orbiting it. Kepler’s Laws apply equally well to the moon around the Earth and any other object orbiting another!

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Kepler’s 2Kepler’s 2ndnd Law LawLaw of AreasLaw of Areas

Note that equal areas are covered by a line extending from the Sun to the orbiting object in equal time intervals

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Kepler’s 2Kepler’s 2ndnd Law LawLaw of AreasLaw of Areas

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Kepler’s 3Kepler’s 3rdrd Law LawLaw of Periods Law of Periods

((Harmony of SpheresHarmony of Spheres))

Note that square of the orbital period (p) of each planet is equal to the cube of the semimajor axis (a).

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Kepler’s 3Kepler’s 3rdrd Law LawLaw of Periods Law of Periods

((Harmony of SpheresHarmony of Spheres))

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Isaac NewtonIsaac Newton

1642-17271642-1727Achievements:Achievements:CalculusCalculusLaws of MotionLaws of MotionLaw of GravitationLaw of GravitationNature of LightNature of LightAdv. TelescopesAdv. Telescopes

Just to name a few!!Just to name a few!!

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11stst Law of Motion Law of Motion

Every body continues in its state of rest, or of Every body continues in its state of rest, or of uniform motion in a uniform motion in a rightright [straight] line, [straight] line, unless it is compelled to change that state by unless it is compelled to change that state by a force impressed on it.a force impressed on it.

This law is often referred to as ‘The Law of Inertia’ and is credited This law is often referred to as ‘The Law of Inertia’ and is credited as an accomplishment of Galileo. as an accomplishment of Galileo.

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11stst Law of Motion Law of Motion

This is what Newton would refer to This is what Newton would refer to as ‘natural’ motion or inertial as ‘natural’ motion or inertial motion. motion.

Note that ‘Force’ is Note that ‘Force’ is notnot required for required for objects to be in or maintain motion.objects to be in or maintain motion.

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22ndnd Law of Motion Law of Motion

The change in motion is proportional to The change in motion is proportional to the motive force impressed; and is made the motive force impressed; and is made in the direction of the right line in which in the direction of the right line in which that force is impressed.that force is impressed.

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22ndnd Law of Motion Law of Motion

The change in motion (acceleration) is The change in motion (acceleration) is proportional to the motive force proportional to the motive force impressed; and is made in the direction of impressed; and is made in the direction of the right line in which that force is the right line in which that force is impressed.impressed.

                                                                       

       

F = ma

                                                                   

        

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22ndnd Law of Motion Law of Motion

The unit of force is given the name The unit of force is given the name Newton [N].Newton [N].

Based on F=ma, the unit of a newton Based on F=ma, the unit of a newton can be expressed as:can be expressed as:

1 Newton = (1 Newton = (1kilogram)(1meter)1kilogram)(1meter)(second)2

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33rdrd Law of Motion Law of Motion

To Every action there is always an To Every action there is always an equal reaction; or, the mutual actions equal reaction; or, the mutual actions of two bodies are always equal, and of two bodies are always equal, and directed to contrary parts [opposite directed to contrary parts [opposite directions].directions].

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33rdrd Law of Motion Law of Motion

If motion was If motion was bestowed upon one bestowed upon one object it must have object it must have been taken from been taken from another.another.

What if this astronauts jetpack fails? How can he get back to the ship? Knowledge of Newton’s 3rd will save his life!!

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33rdrd Law of Motion Law of Motion

FFBCBC = F = FCBCB

As the book leans and pushes on the As the book leans and pushes on the crate, the crate pushes with an equal crate, the crate pushes with an equal and oppositely directed force on the and oppositely directed force on the book.book.

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33rdrd Law of Motion Law of Motion

FFEarth on BallEarth on Ball= F= FBall on EarthBall on Earth

Each time the ball bounces off the Earth, the Earth Each time the ball bounces off the Earth, the Earth and ball exert forces on each other. According to and ball exert forces on each other. According to Newton’s 3Newton’s 3rdrd law, these forces are exactly the same law, these forces are exactly the same in magnitude and in opposite directions. Why is it in magnitude and in opposite directions. Why is it that the ball is the only object visible changing that the ball is the only object visible changing direction?direction?

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Newton’s Universal Law of Newton’s Universal Law of GravitationGravitation

Projectile motion on Earth had been well documented before Newton but the models of motion lacked a mechanism for the movement towards Earth.

Newton envisioned a force that gives every object with mass the ability to ‘reach across empty space’ and pull on neighboring bodies of mass.

GRAVITY

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The force that one massive body exerts The force that one massive body exerts on another such that the two bodies are on another such that the two bodies are drawn together is directly proportional drawn together is directly proportional to the masses of each body and to the masses of each body and inversely proportional to the separation inversely proportional to the separation distance of the two bodies.distance of the two bodies.

Written in the form of Newton’s 2Written in the form of Newton’s 2ndnd Law: Law:F = ma = FF = ma = Fgg= G(m= G(m11*m*m22)/r)/r2 2 = m = m11(Gm(Gm22/r/r22))

where G is a constant of proportionality given by the value 6.67x10where G is a constant of proportionality given by the value 6.67x10-11-11 Nm Nm22/kg/kg22

The value of G was never known to Newton! It would not be discovered until The value of G was never known to Newton! It would not be discovered until about 1800 by Henry Cavendish.about 1800 by Henry Cavendish.

Newton’s Universal Law of Newton’s Universal Law of GravitationGravitation

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The value of G was never known to Newton! It The value of G was never known to Newton! It would not be adequately measured would not be adequately measured

experimentally until 1798 by Henry Cavendish.experimentally until 1798 by Henry Cavendish.

Newton’s Universal Law of Newton’s Universal Law of GravitationGravitation

Henry Cavendish (1731-1810)Cavendish’s Torsion Balance

Used to estimate the universal gravitational constant, G

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How could Newton extend How could Newton extend his idea of Earthly his idea of Earthly gravitation to the motion of gravitation to the motion of the celestial bodies?the celestial bodies?

The moon was the key!The moon was the key!

Newton’s Universal Law of Newton’s Universal Law of GravitationGravitation

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If an apple falls towards If an apple falls towards the Earth due to gravity, the Earth due to gravity, does this same does this same gravitational force extend gravitational force extend into the heavens? Is it the into the heavens? Is it the same force that keeps the same force that keeps the Moon in orbit around the Moon in orbit around the Earth?Earth?

Newton’s Universal Law Newton’s Universal Law of Gravitationof Gravitation

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Consideration of Consideration of Uniform Circular MotionUniform Circular Motion

The acceleration of any object undergoing The acceleration of any object undergoing uniform circular motion is:uniform circular motion is:

This is based on simple motion experiments This is based on simple motion experiments performed performed herehere on Earth. The acceleration on Earth. The acceleration in this type of motion is known as in this type of motion is known as Centripetal AccelerationCentripetal Acceleration (or (or center-seekingcenter-seeking acceleration)acceleration)

r

va

2

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Acceleration of the Moon TowardsAcceleration of the Moon Towardsthe Earththe Earth

((Approximating it to be Uniform Circular motionApproximating it to be Uniform Circular motion))

The Moon moves around the Earth in a The Moon moves around the Earth in a nearlynearly uniform circular orbit with a speed of uniform circular orbit with a speed of ~1016m/s. ~1016m/s.

The distance between the Moon and Earth is The distance between the Moon and Earth is known to be ~3.8x10known to be ~3.8x1088m. m.

Approximating the motion to be uniform and Approximating the motion to be uniform and circular yields a centripetal acceleration of:circular yields a centripetal acceleration of:

aamoonmoon = v = v22/r = (1016m/s)/r = (1016m/s)22/(380 000 000m)/(380 000 000m)

aamoonmoon = 0.00272m/s = 0.00272m/s22

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Newton then imagined his ‘gravitational force’ extending out into the Newton then imagined his ‘gravitational force’ extending out into the heavens and pulling on the Moon.heavens and pulling on the Moon.

Since the distance to the Moon is about 60 times longer than the Since the distance to the Moon is about 60 times longer than the radius of the Earth, Earth’s gravity should be considerably weaker radius of the Earth, Earth’s gravity should be considerably weaker at the Moon’s location.at the Moon’s location.

Based on earthly experimentation, things accelerate downwards at a Based on earthly experimentation, things accelerate downwards at a rate of 9.81m/srate of 9.81m/s22 when near the surface of the Earth. when near the surface of the Earth.

Assuming an inverse square law relationship between gravity and Assuming an inverse square law relationship between gravity and distance, Newton supposed that the Earth’s gravitational distance, Newton supposed that the Earth’s gravitational acceleration should be about 60acceleration should be about 6022 times less (3600 times less) near times less (3600 times less) near the Moon than it is on Earth.the Moon than it is on Earth.

aamoonmoon = (9.81m/s = (9.81m/s22)/3600)/3600

aamoonmoon = 0.00271m/s = 0.00271m/s22

Acceleration of the Moon Acceleration of the Moon Towards the EarthTowards the Earth

((Using Newton’s Gravitational Inverse Square LawUsing Newton’s Gravitational Inverse Square Law))

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GravityGravity

Newton found the connectionNewton found the connection

The forces (gravity) that govern The forces (gravity) that govern projectile motion here on planet projectile motion here on planet Earth ALSO govern the motion of Earth ALSO govern the motion of the moon & the planets.the moon & the planets.

REVOLUTIONARY THINKING!!!REVOLUTIONARY THINKING!!!

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Newton’s Gravitation Newton’s Gravitation and Kepler’s Lawsand Kepler’s Laws

Kepler’s 3Kepler’s 3rdrd law (p law (p22 = a = a33) can be derived (with a little ) can be derived (with a little algebra & calculus) from Newton’s Law of algebra & calculus) from Newton’s Law of Gravitation.Gravitation.

G(MG(M11 + M + M22)P)P22 = 4π = 4π22aa33

Where a is the semimajor axis length, P is the Where a is the semimajor axis length, P is the period of orbit, and the M’s are the masses of the period of orbit, and the M’s are the masses of the

orbiting objectsorbiting objects

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Newton’s Gravitation Newton’s Gravitation and Kepler’s Lawsand Kepler’s Laws

G(MG(M11 + M + M22)P)P22 = 4π = 4π22aa33

Newton had given a mechanism for Newton had given a mechanism for Kepler’s planetary motion lawsKepler’s planetary motion laws

With this relationship, we can measure With this relationship, we can measure distances and periods by observation distances and periods by observation and then calculate the mass of any and then calculate the mass of any object orbiting another!!object orbiting another!!

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The masses of The masses of the stars and the stars and

planetsplanets

G(MG(M11 + M + M22)*p)*p22 = 4πa = 4πa33

It is from this It is from this relationship that relationship that we estimate the we estimate the mass of the Sun, mass of the Sun, the planets, and the planets, and any other celestial any other celestial body orbiting body orbiting another!!another!!

Keypoint: Masses of Keypoint: Masses of Celestial objects are Celestial objects are derived from observing derived from observing orbital motion of the orbital motion of the objects. objects.

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History recalls that nearly 20 years passed before Newton was urged by his contemporary, Sir Edmund Halley, to publish his results

Newton’s laws of motion and gravity were first published in 1687 as Philosophiae Naturalis Principia Mathematica.

Edmund Halley (1656-1742)

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Phases of the Moon

What causes them?Earth/Moon Gravitational Earth/Moon Gravitational

EffectsEffects

The Earth and Moon exert mutual gravitational forces on each other

Every particle on the Earth is influenced by the gravitational force of the Moon

This has a small effect on the hard surface of the Earth

However, the effect is quite noticeable on the Earth’s liquid surface (oceans)

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Phases of the Moon

What causes them? Ocean TidesOcean Tides

Two tidal bulges are caused by the pull of the moon

There is a tidal bulge on the moon side of the Earth due to strong lunar gravity and another directly on the other side due to weak lunar gravity and the fact that the Earth is accelerating towards the moon (‘essentially leaving the water behind’)

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Ocean TidesOcean Tides

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Phases of the Moon

What causes them?The Sun & Ocean TidesThe Sun & Ocean Tides

The Sun can also influence the tides (but to a lesser degree)When the Sun and Moon lie on the a line that passes through Earth

(either on different sides, Full Moon or on the same side, New Moon) the tides are higher and are called ‘Spring Tides’ (nothing to do with Spring season)

When the Sun and Moon lie on perpendicular lines relative to Earth (either first quarter or third quarter) the tides are lower and are called Neap tides.