Birth of Modern Astronomy OR How Nerds Changed the World!!! The Motion of Planets Birth of Modern Astronomy OR How Nerds Changed the World!!!

Learning Outcomes (Students will be able to…):  explain qualitatively Kepler’s first and second laws and apply quantitatively Kepler’s third law explain and apply the law of universal gravitation to orbital notations by using appropriate numeric and graphic analysis  distinguish between scientific questions and technological problems as applied to orbital situations

Assumptions of Early Models of the Solar System (from the time of Aristotle…) Geocentric - Earth in the middle Everything orbits the Earth Stars are located on the Celestial Sphere Everything moves in uniform circular motions

Claudius Ptolemy (87-165) Epicycle Mars Equant Earth Deferent

Nicolaus Copernicus (1473-1543) Errors building up Must be a better way! Let’s try a Heliocentric (or Sun-centered) system! Not any better though

Tycho Brahe (1546-1601) Comet – beyond the Moon Supernova – far away Naked eye observations of planets Accuracy through repetition Best observations of planetary positions Hired “nerd” to help calculate model Died….

Three Laws of Planetary Motion Johannes Kepler (1571-1630) Worked for Brahe Took data after his death Spent years figuring out the motions of the planets Came up with… Three Laws of Planetary Motion

1st Law: Planets move in elliptical orbits with the Sun at one foci Perihelion Aphelion Foci (sing. Focus) Average distance from the Sun = 1 Astronomical Unit (1 A.U.) = approx. 150 000 000 km

2nd Law: Planets move faster at perihelion than at aphelion OR a planet sweeps out equal areas in equal time periods. 1 Month 1 Month

3rd Law: Period is related to average distance T = period of the orbit r = average distance kT2 = r3 Longer orbits - greater average distance Need the value of k to use the formula k depends upon the situation Can be used for anything orbiting anything else

T2 = r3 Special version of Kepler’s third Law – If the object is orbiting the Sun T – measured in years, r – measured in A. U., then…. T2 = r3

For planets A and B, Kepler’s 3rd Law can look like this…

Galileo Galilei (1564-1642) Knew of Copernicus’s & Kepler’s work Used a telescope to look at the sky What did he see?

The Moon was an imperfect object Venus has phases

Jupiter has objects around it Saturn is imperfect The Sun is imperfect

Isaac Newton (1642-1727) The Three Laws of Motion The ultimate “nerd” Able to explain Kepler’s laws Had to start with the basics - The Three Laws of Motion

1. Law of Inertia - Objects do whatever they are currently doing unless something messes around with them.

2. Force defined F = ma F=force m=mass a=acceleration (change in motion)

3. For every action there is an equal and opposite reaction. The three laws of motion form the basis for the most important law of all (astronomically speaking) Newton’s Universal Law of Gravitation

F=force of gravity G=constant (6.67 x 10-11 Nm2/kg2) M1, M2 = masses R=distance from “centers” Gravity is the most important force in the Universe

Newton’s Revisions to Kepler’s Laws of Planetary Motion: Kepler’s 1st and 2nd Laws apply to all objects (not just planets) 3rd Law rewritten: 4π2 and G are just constant #s (they don’t change) M1 and M2 are any two celestial bodies (could be a planet and Sun) Importance: if you know period and average distance of a planet, you can find mass of Sun (2 x 1030 kg) or any planet! Mass of Sun is 2 000 000 000 000 000 000 000 000 000 000 kg Mass of Earth is 6 000 000 000 000 000 000 000 000 kg Mass of Mr. J is 100 kg! WOW! 22

An Inverse Square Law…

But…before we proceed to examine the dynamics of orbital motion, take a look at uniform circular motion!

Uniform Circular Motion -planetary motion due to gravity requires understanding of circular motion (though most orbits are elliptical) V1 V2 ΔV Figure 1 Figure 2 -figure 1 shows the circular motion of an object with radius and speed constant -velocity is perpendicular to radius (tangential) -v1 and v2 are instantaneous velocities at A and B, respectively -figure 2 shows Δv = v2 – v1

-if A and B are very close together, then arc AB ≈ chord AB -thus, because of similar triangles in figures 1 & 2… -this describes centripetal (or “center-seeking”) acceleration (ac)

-since in a circle (where T is the period of revolution), substituting… -force that causes circular motion is called centripetal force (Fc)

Another way to look at “g”…

Another way to look at gravitational potential energy of an object… (h is height but since it is arbitrary, it can be chosen as the distance from the center of the Earth to the position of the object…or r)

Some important orbital applications… Geosynchronous means having an orbit around the Earth with a period of 24 hours

Einstein viewed gravity and the motion of celestial objects, like planets, VERY differently… Curved space-time effects both mass and light! (1875 – 1955)

Black Holes

In 1916, Einstein published his theory of general relativity (GR), which discussed gravity and explained how the presence of matter causes space and time to be warped.

Light travel in curved spacetime Photons of light passing near our Sun will move the same way through curved space. They will “bend” around the Sun.

Time runs slower in curved spacetime A fundamental tenet of GR is that time runs more slowly in curved spacetime.

The Principle of Equivalence, which states: All local, freely falling, non-rotating laboratories are fully equivalent for the performance of all physical experiments.

Gravitational Redshift Another important effect of Einstein’s theory of GR is gravitational redshift  photons lose energy as they try to escape from a strong gravitational field. energy of a photon is inversely proportional to its wavelength. characteristicof systems containing high-density objects such as neutron stars and white dwarfs, although the effect would be particularly strong if the system contained a black hole.

The Schwarzschild radius, Rs, is given by

Any star that collapses beyond its Schwarzschild radius is called a black hole. coined by the American mathematical physicist John Wheeler in 1968. A black hole is enclosed by an event horizon, the surface of which is described by a sphere with r = RS. At the centre of the event horizon is a singularity, a point of zero volume and infinite density where all of the black hole’s mass is located.