Refined Mathematics & Describing the Universe or How Math Proved What All The Astronomers and Physicists Thought Anyways

Kepler’s Models  Kepler’s Laws were proportional –They would tell us the relative, not absolute sizes of planets’ orbits

Measuring the Angular Diameter of the Sun and Venus  Parallax measures of the transits of Mercury and Venus allowed for more precise angular measurements

RADAR  Since the invention of radar, we can use radio signals to more accurately measure the distances to the Inner Solar System

But…  Radar still doesn’t work towards the Sun –The Sun gives off so much radiation at all wavelengths that the signal gets scrambled!

 Referred to as “The Principia”  Explained why the planets followed Kepler’s Laws  Included –Three Laws of Motion –Law of Universal Gravitation –Some basic Calculus (invented by Newton at the ripe old age of 20)

Newton’s First Law of Motion  “An object at rest will remain at rest, and an object in motion will remain in motion, unless acted upon by an outside force.”  Inertia  Property of mass  Constant velocity requires no continuous force – the planets require no “push”

Newton’s Second Law  “Acceleration of a object is equal to the force applied divided by the mass.”  Force equals mass multiplied by acceleration  F = m a  Defines the Newton (N) as 1 kg  m/s 2  Useful in determining many formulæ concerning gravity and other forces

Newton’s Third Law  “To every action, there is an equal and opposite reaction.”  Somewhat hard to recognize on the scales involved in astronomy –Planets’ gravities on each other, you, et al. –Normal Force not easily recognized –Often shown as a negative force (for the opposite direction)

Which has more inertia? 40 1.An empty dump truck 2.A full dump truck 3.An empty dump truck at 50 kph 4.An F-150 truck at 50 kph 5.They are all the same

A man applies a 660 N force to a chair. How hard does the chair push back? N 2.– 660 N

How much force is required to accelerate a 6 kg object to speed of 3 m/s 2 ? kg  m/s kg  m/s kg  m/s kg  m/s kg  m/s 2

What is the acceleration of a 405 kg object having applied a 45 N force? m/s m/s m/s m/s m/s 2

Which requires more force to accelerate to the same speed? 40 1.An empty dump truck 2.A full dump truck 3.They are the same

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The Inverse Square Law  All field forces (and energies, too) decrease at a rate equal to the inverse of the distance between the objects squared  Intensity = (Energy)  1/d 2

How much sunlight does Saturn receive at a distance of 9.54 AU? x the Earth receives x the Earth receives 3.1 x of what the Earth receives 4.10 x of what the Earth receives 5.50 x of what the Earth receives

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The Law of Universal Gravitation  Every particle of matter in the universe attracts every other particle  with a force that is directly proportional to the products of the masses of the particles  and inversely proportional to the square of the distances between them.

Big, Easy Formula Gravitational Force = d 2 d 2 G  m 1  m 2 Where G is the universal gravitational constant, 6.67 x 10 – 11 N  m 2 /kg 2 Sometimes shown as local gravity, g

Kepler Revisited  Newton determined that the masses rotated around each other at a common center of mass  This center of mass is at one focus of the ellipse, not the center of the Sun

Adjustments for mass  Kepler’s Third Law  P 2 = a 3  When adjusted for the mass, it becomes  P 2 = a 3 /Mass total (in solar units, so it’s extremely close to one)

Compare the masses!  When compared with the mass of the Sun, all other masses in the solar system pale in comparison  When compared with the mass of the Earth, all man-made objects are insignificant

Escape Velocity  Escaping a gravitational field is very difficult  Due to the sizes of the planets compared to our vehicles’ thrust   escape = √2GM/r  An object traveling at a speed greater than  escape has an “unbound” orbit

Lots of proofs!  Several formula describing properties of motion and celestial bodies on p. 56  More Precisely 2 – 3  but more on that later…