Refined Refined Mathematics & Mathematics & Describing the Describing the

UniverseUniverse

ororHow Math Proved What All The Astronomers How Math Proved What All The Astronomers

and Physicists Thought Anywaysand Physicists Thought Anyways

Kepler’s ModelsKepler’s Models Kepler’s Laws were Kepler’s Laws were

proportionalproportional– They would tell us They would tell us

the the relativerelative, not , not absoluteabsolute sizes of sizes of planets’ orbitsplanets’ orbits

Measuring the Angular Diameter of Measuring the Angular Diameter of the Sun and Venusthe Sun and Venus

Parallax measures of Parallax measures of the the transitstransits of Mercury of Mercury and Venus allowed for and Venus allowed for more precise angular more precise angular measurementsmeasurements

RADARRADAR

Since the invention of Since the invention of radarradar, we can use radio , we can use radio signals to more accurately measure the distances signals to more accurately measure the distances to the Inner Solar Systemto the Inner Solar System

But…But…

Radar still doesn’t work towards the SunRadar still doesn’t work towards the Sun– The Sun gives off so much radiation at all The Sun gives off so much radiation at all

wavelengths that the signal gets scrambled!wavelengths that the signal gets scrambled!

Referred to as “The Referred to as “The PrincipiaPrincipia””

Explained Explained whywhy the the planets followed planets followed Kepler’s LawsKepler’s Laws

IncludedIncluded– Three Laws of Three Laws of

MotionMotion– Law of Universal Law of Universal

GravitationGravitation– Some basic Some basic

Calculus (invented Calculus (invented by Newton at the by Newton at the ripe old age of 20)ripe old age of 20)

Newton’s First Law of MotionNewton’s First Law of Motion

““An object at rest will remain at rest, and an An object at rest will remain at rest, and an object in motion will remain in motion, object in motion will remain in motion, unless acted upon by an outside force.”unless acted upon by an outside force.”

InertiaInertia Property of massProperty of mass Constant velocity requires no continuous Constant velocity requires no continuous

force – the planets require no “push”force – the planets require no “push”

Newton’s Second LawNewton’s Second Law

““Acceleration of a object is equal to the force Acceleration of a object is equal to the force applied divided by the mass.”applied divided by the mass.”

Force equals mass multiplied by Force equals mass multiplied by accelerationacceleration

F = m aF = m a Defines the Newton (N) as 1 kgDefines the Newton (N) as 1 kgm/sm/s22

Useful in determining many formulUseful in determining many formulææ concerning gravity and other forcesconcerning gravity and other forces

Newton’s Third LawNewton’s Third Law

““To every action, there is an equal and To every action, there is an equal and opposite reaction.”opposite reaction.”

Somewhat hard to recognize on the scales Somewhat hard to recognize on the scales involved in astronomyinvolved in astronomy– Planets’ gravities on each other, you, et al.Planets’ gravities on each other, you, et al.– Normal Force not easily recognizedNormal Force not easily recognized– Often shown as a negative force (for the Often shown as a negative force (for the

opposite direction)opposite direction)

Which has more inertia?Which has more inertia?

40

0%

0%

0%

0%

0%

1.1. An empty dump truckAn empty dump truck

2.2. A full dump truckA full dump truck

3.3. An empty dump truck at 50 kphAn empty dump truck at 50 kph

4.4. An F-150 truck at 50 kphAn F-150 truck at 50 kph

5.5. They are all the sameThey are all the same

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232

A man applies a 660 N force to a chair. A man applies a 660 N force to a chair. How hard does the chair push back?How hard does the chair push back?

40

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232

0%

0%

1.1. 660 N660 N

2.2. – – 660 N660 N

How much force is required to accelerate How much force is required to accelerate a 6 kg object to speed of 3 m/sa 6 kg object to speed of 3 m/s22??

40

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232

0%

0%

0%

0%

0%

1.1. 0.5 kg0.5 kgm/sm/s22

2.2. 2 kg2 kgm/sm/s22

3.3. 3 kg3 kgm/sm/s22

4.4. 9 kg9 kgm/sm/s22

5.5. 18 kg18 kgm/sm/s22

What is the acceleration of a 405 kg What is the acceleration of a 405 kg object having applied a 45 N force?object having applied a 45 N force?

40

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232

0%

0%

0%

0%

0%

1.1. 0.11 m/s0.11 m/s22

2.2. 9 m/s9 m/s22

3.3. 360 m/s360 m/s22

4.4. 450 m/s450 m/s22

5.5. 18225 m/s18225 m/s22

Which requires more force to accelerate Which requires more force to accelerate to the same speed?to the same speed?

40

0%

0%

0%

1.1. An empty dump truckAn empty dump truck

2.2. A full dump truckA full dump truck

3.3. They are the sameThey are the same

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232

Participant ScoresParticipant Scores

00 Participant 1Participant 1

00 Participant 2Participant 2

00 Participant 3Participant 3

00 Participant 4Participant 4

00 Participant 5Participant 5

The Inverse Square LawThe Inverse Square Law

All field forces (and energies, too) decrease All field forces (and energies, too) decrease at a rate equal to the inverse of the distance at a rate equal to the inverse of the distance between the objects squaredbetween the objects squared

Intensity = (Energy)Intensity = (Energy)1/d1/d22

How much sunlight does Saturn receive How much sunlight does Saturn receive at a distance of 9.54 AU?at a distance of 9.54 AU?

40

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232

0%

0%

0%

0%

0%

1.1. 0.01x the Earth receives0.01x the Earth receives

2.2. 0.1 x the Earth receives0.1 x the Earth receives

3.3. 1 x of what the Earth receives1 x of what the Earth receives

4.4. 10 x of what the Earth receives10 x of what the Earth receives

5.5. 50 x of what the Earth receives50 x of what the Earth receives

Participant ScoresParticipant Scores

00 Participant 1Participant 1

00 Participant 2Participant 2

00 Participant 3Participant 3

00 Participant 4Participant 4

00 Participant 5Participant 5

The Law of Universal GravitationThe Law of Universal Gravitation

Every particle of matter in the universe Every particle of matter in the universe attracts every other particleattracts every other particle

with a force that is directly proportional to with a force that is directly proportional to the products of the masses of the particles the products of the masses of the particles

and inversely proportional to the square of and inversely proportional to the square of the distances between them.the distances between them.

Big, Easy FormulaBig, Easy Formula

Gravitational Force = Gravitational Force = dd22

G G m m11 m m22

Where G is the universal gravitational Where G is the universal gravitational constant, 6.67 x 10 constant, 6.67 x 10 – 11 – 11 NNmm22/kg/kg22

Sometimes shown Sometimes shown as local gravity, gas local gravity, g

Kepler RevisitedKepler Revisited

Newton determined that the masses Newton determined that the masses rotated around each other at a rotated around each other at a common center of masscommon center of mass

This center of mass is at one focus of This center of mass is at one focus of the ellipse, not the center of the Sunthe ellipse, not the center of the Sun

Adjustments for massAdjustments for mass

Kepler’s Third LawKepler’s Third Law PP22 = a = a33

When adjusted for the mass, it becomesWhen adjusted for the mass, it becomes PP22 = a = a33/Mass total (in solar units, so it’s /Mass total (in solar units, so it’s extremelyextremely close to one) close to one)

Compare the masses!Compare the masses!

When compared with the mass of the Sun, When compared with the mass of the Sun, all other masses in the solar system pale in all other masses in the solar system pale in comparisoncomparison

When compared with the mass of the Earth, When compared with the mass of the Earth, all man-made objects are insignificantall man-made objects are insignificant

Escape VelocityEscape Velocity

Escaping a gravitational field is very difficultEscaping a gravitational field is very difficult Due to the sizes of the planets compared to Due to the sizes of the planets compared to

our vehicles’ thrustour vehicles’ thrust escapeescape = = √2GM/r√2GM/r

An object traveling at a speed greater than An object traveling at a speed greater than escapeescape has an “unbound” orbit has an “unbound” orbit

Lots of proofs!Lots of proofs!

Several formula describing properties of Several formula describing properties of motion and celestial bodies on p. 56motion and celestial bodies on p. 56

More Precisely 2 – 3More Precisely 2 – 3 but more on that later…but more on that later…