The heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which, through discordant tensions, through sincopes and cadenzas, as it were (as men employ them in imitation of those natural discords) progresses towards certain pre-designed quasi six-voiced clausuras, and thereby sets landmarks in the immeasurable flow of time. -- The Harmony of the Universe (Harmonice mundi) Johannes Kepler
Heavenly music was an idea that dated back to Pythagoras, and Kepler was the first to record the sounds of the planets in Harmony of the Universe
John Rogers and Willie Ruff of Yale University synthesized Kepler's music which became know as the "Harmony of the Spheres."
The Pythagorean theorem was used to find how fast the moon falls each second, just as the odd numbers add up to the perfect squares.
Studying the musical compositions of the heavens, Europeans developed distinct musical compositions in Europe.
Galileo found that each swing of a pendulum occurs in the same amount of time. This discovery led to keeping perfect time, an undeniably important aspect of music.
Galileo then discovered resonance through the application of a rythmically applied chorus.
Western scholars only made discoveries due to the foundations of science from the Pythagoreans and Greeks that ultimately led to the conclusion that nature is solely based off of mathematics.
Newton's Law F = m a describes all bodies on earth, in the heavens, and throughout the universe. This law united the physics of both earth and space.
All projectiles follow orbits that can be depicted by conic sections due to the fact that F = m a applies to every force in the mechanical universe, and every mass no matter what or where.
F= m a is an equation that relies on derivatives and can describe acceleration, velocity, and position of any projectile ever fired.
Copernicus challenged old theories and needed new mathematics of motion to describe the nature of new discoveries.
Whenever there is no outside force, momentum is constant. Force is the rate of change of momentum. Momentum is conserved and if r x f is 0, angular momentum is constant as long as there is no torque.
Energy is also constant. Work that lifts an object from one height to another becomes potential energy. Work that acclerates a block becomes kinetic energy. The kinetic energy of a falling object is transfered into heat, which leads to the kinetic energy among molecules and atoms.
E = U + K is always constant and determines the orbits of the planets:
e<1 - ellipse
e=1 - parabola
e=0 - circle
e>1 - hyperbola
The harmony of the mechanical universe, containing many mysteries yet to be solved.
Sunday, January 22, 2012
Lesson 25 Kepler to Einstein
The periods of the planets in their orbits, the ebb and flow of the tides, the acceleration of a falling body; all these phenomena are consequences of the force of gravity and they have inspired the labor of scientists from Kepler to Einstein.
Einstein was searching for the reason that gravity applies the same force for all bodies to fall at the same rate.
Galileo developed a theory of the tides that explained the tides of the earth. He concluded that it was due to the motion of the earth around the sun. However, this was a problem because for his theory to be true there would need to be one high tide at noon every day. In fact, tides rise and recede twice every day.
In the earth-moon system, the earth and the moon rotate around a common center of mass. This point is actually inside the earth, three-fourths of the distance from the center to the surface. The strange wobbling motion of the earth is a Keplerian orbit of the center of the earth around the center of mass of the earth-moon system. Only the center of the earth is in exactly the rigt orbit. On the side closest to the moon, the moon's gravity is too strong, and the water bulges as it is pulled towards the moon. On the opposite side the moon's gravity is too weak to hold it in place so it bulges as it tries to escape. As the earth wobbles around the earth-moon center of mass, it also rotates on its axis. As the earth rotates it passes beneath the bulges. At those locations where the rotating Earth passes under the rising water, high tides occur. At the points between, low tides occur; two high tides and two low tides. The sun plays a role in the tides at about half of the strength of the moon because of the distance between the earth and the sun. At "new moon" of full moon, the earth, moon, and sun fall in a staight line and the forces of gravity reinforce eachother to create the largest high tides and the smallest low tides.
1st - r = ed/ ecos(theta)+1
Each planet moves in an ellipse with the Sun at one focus.
2nd - dA/dt = constant
A line drawn from the Sun to a planet sweeps out equal areas in equal times.
The square of the period of a planet's orbit is proportional to the cube of the length of the semi - major axis.
Einstein was searching for the reason that gravity applies the same force for all bodies to fall at the same rate.
Galileo developed a theory of the tides that explained the tides of the earth. He concluded that it was due to the motion of the earth around the sun. However, this was a problem because for his theory to be true there would need to be one high tide at noon every day. In fact, tides rise and recede twice every day.
In the earth-moon system, the earth and the moon rotate around a common center of mass. This point is actually inside the earth, three-fourths of the distance from the center to the surface. The strange wobbling motion of the earth is a Keplerian orbit of the center of the earth around the center of mass of the earth-moon system. Only the center of the earth is in exactly the rigt orbit. On the side closest to the moon, the moon's gravity is too strong, and the water bulges as it is pulled towards the moon. On the opposite side the moon's gravity is too weak to hold it in place so it bulges as it tries to escape. As the earth wobbles around the earth-moon center of mass, it also rotates on its axis. As the earth rotates it passes beneath the bulges. At those locations where the rotating Earth passes under the rising water, high tides occur. At the points between, low tides occur; two high tides and two low tides. The sun plays a role in the tides at about half of the strength of the moon because of the distance between the earth and the sun. At "new moon" of full moon, the earth, moon, and sun fall in a staight line and the forces of gravity reinforce eachother to create the largest high tides and the smallest low tides.
Kepler's Three Laws:
2nd - dA/dt = constant
3rd - T^2 = (4pi^2/GM) a^3
dA/dt = L/2M Integrating Kepler's second law through one period, where the period of a planet's orbit is proportional to its area.
T = 2A/L/M holds true for many types of motion
F = -G M Mo/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
This motion depicts an elliptical orbit whose size depends on L/M ue to Newton's Law of Universal Gravitation.
L^2/DM/(1+ecos(theta))
T = 2A/sqrt(GMoa(1-e^2)) Area = pi a^2 sqrt(1-e^2)
T = 2pi/sqrt(GMo) (a^3) - 4pi^2 is the gravitational constant and the mass of the sun connects T^2 to a^3 for every planet.
T^2 = 4pi^2/GM0 (a^3)
Although this desribes the mechanical properties of the universe, Einstein used gravity, time, an space to solve the deepest mysteries of the cosmos.
Inertial mass and grvitational mass are the same for all bodies in the universe for the Law of Falling Bodies to occur. For this to happen, Einstein searched for a profound law of nature for the law to be veritable. This led to the Theory of Relativity.
The Principle of Equivalence states that there is constant gravity and acceleration that cause bodies to fall at same rate.
The Law of Falling Bodies and the mysterious equality of gravitational mass and inertial mass would be explained if the following fundamental principle were true:
Objects do not fall because of gravity, but appear to fall due to the acceleration of the Earth uppward into space.
Einstein used a beam of light to explain this theory. Einstein stated that in space light would bend downward slightly, and on earth light would bend downward slightly due to the force of the earth's gravity that causes light to not travel in straight lines.
Einstein observed light during an eclipse and was correct about the manner in which light travels.
In theory, there are no straight lines in space. Instead, the shortest distance is known as geodesic on any globe or surface.
Light is bent by the gravity of the sun, but travels inertially along the shortest distance between any two points in local curved space time.
Einstein eliminated gravity and attributed the order of universe to the curved space time. However, how does mass cause space time to curve? Einstein died before this question could be answered, however, under extreme conditions in the universe, only Einstein's Theory of Relativity holds true.
Monday, January 16, 2012
Lesson 24 Navigating in Space
Voyages to other planets require enormous amounts of energy. The amount of energy expended can be minimized using the same force that moves the planets through the solar system. The force of the sun's gravitational field is used to navigate through space.
1973 - Mariner 10 was sent to Venus and Mars
1975 - Viking was sent to Mars
1977 - Voyager was sent to Jupiter, Saturn, Uranus, and Neptune.
For an object to move in a straight line in space it requires a lot of propulsion energy. Instead, scientists recognized Kepler's second law - planets move around the sun in orbitals shaped as ellipses - as an operting principle to allow spacecraft to follow the same path and be projected from around the sun into space.
The spacecraft is put in an independent orbit around the sun that intersects the orbits of Earth and Mars In order for the spacecraft to coast after launch to its destination.
To travel between two points in space the craft coasts to its destination in orbit around the sun, just as if it were an orbiting planet. The path the craft travels from one planet to another is known as the transfer orbit.
The Viking mission was limited in the selection of a transfer orbit because it was the most massive interplanetary craft ever launched. The Viking mission followed a Hohmann transfer, which is the classic method for travel between two planets. The craft was launched when the Earth was at its point in its orbit closest to the Sun and arrived at Mars when Mars was at its point in its orbit farthest from the Sun.
T^2 = 4pi^2/GM (a^3) - Kepler's third law dictates the length of the Earth year and also how long a spacecraft coasting in its own orbit will take to get from the Earth to the position of the orbit of Mars, which is about eight and half months.
A spacecraft must be launched when Mars is 44 degrees ahead of the Earth; this is kown as an opportunity.
Opporunity:
Venus - 54 degrees - every 19 months
Mars - 44 degrees - every 2 years
Jupiter - 97 degrees - every 13 months
There are only certain times each day that a spacecraft can be launched due to the Earth's rotation. Any increase or boost in speed creates a larger more eccentric orbit. To go from the Earth's orbit to a Mars tranfer orbit the craft must be launched at 2.9 km/s. However, to do this the craft must escape the parking orbit around the Earth. A rocket thrust will boost the rocket into a hyperbola orbit at 2.9 km/s; this must occur at 7:56 p.m. The Sun will then bend the hyperbola orbit into a Mars transfer orbit. In order to travel to Venus, the rocket thrust must be activated at 7:40 p.m. in order for the craft to travel inward.
The four outer planets line up for a craft to visit all four every 175 years. The craft must overcome gravity in order to leave the planet and utilizes gravity assist to travel to other planets.
300 years after the discovery of classical mechanics, discoveries are still to be made in space, physics, and every field of science.
1973 - Mariner 10 was sent to Venus and Mars
1975 - Viking was sent to Mars
1977 - Voyager was sent to Jupiter, Saturn, Uranus, and Neptune.
For an object to move in a straight line in space it requires a lot of propulsion energy. Instead, scientists recognized Kepler's second law - planets move around the sun in orbitals shaped as ellipses - as an operting principle to allow spacecraft to follow the same path and be projected from around the sun into space.
The spacecraft is put in an independent orbit around the sun that intersects the orbits of Earth and Mars In order for the spacecraft to coast after launch to its destination.
To travel between two points in space the craft coasts to its destination in orbit around the sun, just as if it were an orbiting planet. The path the craft travels from one planet to another is known as the transfer orbit.
The Viking mission was limited in the selection of a transfer orbit because it was the most massive interplanetary craft ever launched. The Viking mission followed a Hohmann transfer, which is the classic method for travel between two planets. The craft was launched when the Earth was at its point in its orbit closest to the Sun and arrived at Mars when Mars was at its point in its orbit farthest from the Sun.
A spacecraft must be launched when Mars is 44 degrees ahead of the Earth; this is kown as an opportunity.
Opporunity:
Venus - 54 degrees - every 19 months
Mars - 44 degrees - every 2 years
Jupiter - 97 degrees - every 13 months
There are only certain times each day that a spacecraft can be launched due to the Earth's rotation. Any increase or boost in speed creates a larger more eccentric orbit. To go from the Earth's orbit to a Mars tranfer orbit the craft must be launched at 2.9 km/s. However, to do this the craft must escape the parking orbit around the Earth. A rocket thrust will boost the rocket into a hyperbola orbit at 2.9 km/s; this must occur at 7:56 p.m. The Sun will then bend the hyperbola orbit into a Mars transfer orbit. In order to travel to Venus, the rocket thrust must be activated at 7:40 p.m. in order for the craft to travel inward.
The four outer planets line up for a craft to visit all four every 175 years. The craft must overcome gravity in order to leave the planet and utilizes gravity assist to travel to other planets.
300 years after the discovery of classical mechanics, discoveries are still to be made in space, physics, and every field of science.
Lesson 23 Energy and Eccentricity
According to Newton's laws, all objects in a gravitational field trace out conic sections. The precise shape of an orbit depends on the interplay between energy and eccentricity.
Conic sections have mathematical and grammatical properties:
ellipse - ellipses
parabola - parable
hyperbola - hyperbole
Asteroids follow these conic sections and display how nature obeys mathematics, particularly in the universe.
Newton interpreted Kepler's three laws and developed the equation that models the Earth's orbit:
L^2/DM/(1+ecos(theta)) = r (closely related to the equation of an ellipse)
D = G M M(o)
One of the underlying mathematical properties of the universe is that planets move in ellipses determined by angular momentum, mass of a planet, the mass of the sun, and the eccentricity of the orbit. Not every orbit must be elliptic, circular etc.
The exact shape of an orbit is determined by energy. For example, when a planet falls close to the sun, the potential energy is low, but the planet speeds up so kinetic energy is high. Potential energy is high when a planet is far away from the sun, and kinetic energy is low. Since space is a vacuum, total energy does not change and this is the key to the shape of the orbit.
All the orbits of the planets in the universe are paths of constant energy. Pluto (discovered by Percival Lowell) is responsible for irregularities in Neptune and Uranus's orbit, and its orbit intersects that of Neptune. Halley's comet also has an orbit that is extremely eccentric and elliptical.
The total energy of the orbit is the sum of kinetic and potential energy.
Potential Energy:
U = -D/r
r = L^2/DM/(1 + ecos(theta))
U = D^2M/L^2 (-1-ecos(theta))
Kinetic Energy:
K = (1/2) m v ^2
K = MD^2/L^2((1/2) + ecos(theta) + (1/2)e^2)
E = U+K is constnt because the (cosine) terms cancel in the addition of potential energy and kinetic energy.
E = D^2M/L^2 (1/2) (e^2-1) - connection between energy and eccentricity.
The shapes of orbits depend on energy. For a given angular momentum, the energy determines the eccentricity, and the eccentricity determines the orbit.
Ellipse - if the total energy is negative, and the eccentricity is less than 1, then the orbit is an ellipse. Postive kinetic energy is too small to overcome negative potential energy so the body cannot escape a solar system.
Parabola - if the total energy is zero, and the eccentrcity is exactly 1, then the orbit is a parabola. If a body started with 0 kinetic energy and fell from infinity it would whiponce around the sun and return to infinity. This is very unlikely, however possible.
Hyperbola - if the total energy is positive and the eccentricity is greater than 1, then the orbit is a hyperbola. If an object was projected towards the sun from a great distance, the positive kinetic energy would overcome the negative potential energy. Some comets have this orbit.
Circle - if the total energy has a special value, the eccentricity 0 and the orvit is a circle. This is the lowest energy a planet can have for a given angular momentum. The rings of Saturn display this.
Conic sections have mathematical and grammatical properties:
ellipse - ellipses
parabola - parable
hyperbola - hyperbole
Asteroids follow these conic sections and display how nature obeys mathematics, particularly in the universe.
Newton interpreted Kepler's three laws and developed the equation that models the Earth's orbit:
L^2/DM/(1+ecos(theta)) = r (closely related to the equation of an ellipse)
D = G M M(o)
One of the underlying mathematical properties of the universe is that planets move in ellipses determined by angular momentum, mass of a planet, the mass of the sun, and the eccentricity of the orbit. Not every orbit must be elliptic, circular etc.
The exact shape of an orbit is determined by energy. For example, when a planet falls close to the sun, the potential energy is low, but the planet speeds up so kinetic energy is high. Potential energy is high when a planet is far away from the sun, and kinetic energy is low. Since space is a vacuum, total energy does not change and this is the key to the shape of the orbit.
All the orbits of the planets in the universe are paths of constant energy. Pluto (discovered by Percival Lowell) is responsible for irregularities in Neptune and Uranus's orbit, and its orbit intersects that of Neptune. Halley's comet also has an orbit that is extremely eccentric and elliptical.
The total energy of the orbit is the sum of kinetic and potential energy.
Potential Energy:
U = -D/r
r = L^2/DM/(1 + ecos(theta))
U = D^2M/L^2 (-1-ecos(theta))
Kinetic Energy:
K = (1/2) m v ^2
K = MD^2/L^2((1/2) + ecos(theta) + (1/2)e^2)
E = U+K is constnt because the (cosine) terms cancel in the addition of potential energy and kinetic energy.
E = D^2M/L^2 (1/2) (e^2-1) - connection between energy and eccentricity.
The shapes of orbits depend on energy. For a given angular momentum, the energy determines the eccentricity, and the eccentricity determines the orbit.
Ellipse - if the total energy is negative, and the eccentricity is less than 1, then the orbit is an ellipse. Postive kinetic energy is too small to overcome negative potential energy so the body cannot escape a solar system.
Parabola - if the total energy is zero, and the eccentrcity is exactly 1, then the orbit is a parabola. If a body started with 0 kinetic energy and fell from infinity it would whiponce around the sun and return to infinity. This is very unlikely, however possible.
Hyperbola - if the total energy is positive and the eccentricity is greater than 1, then the orbit is a hyperbola. If an object was projected towards the sun from a great distance, the positive kinetic energy would overcome the negative potential energy. Some comets have this orbit.
Circle - if the total energy has a special value, the eccentricity 0 and the orvit is a circle. This is the lowest energy a planet can have for a given angular momentum. The rings of Saturn display this.
Lesson 22 The Kepler Problem
Kepler's three laws describe the motion of the planets, but Isaac Newton's explanation of these laws was the culmination of the Scientific Revolution.
Isaac Newton hypothesized that the force of gravity gave rise to elliptical orbits:
F = -D/r^2(r^) - inverse square law of gravity
F = m a
m a = -D/r^2(r^)
a = -D/Mr^2 (r^)
d^2r/dt^2 = -D/Mr^2 (r^) - differential equation of any orbit because the solution is the algebraic equation of any conic section.
r = L^2/DM/ 1+ecos(theta)
Twist or torque cannot be applied by gravity rxF =0
F= m a
r x m a = 0
m r x a = 0
m r x dv/dt = 0
d(r x v) = dr/dt x v + r x dv/dt
dt
=v x v + r x dv/dt
d(r x v) = r x dv/dt
dt
m d(r x v) = 0
dt
Zero torque, together with Newton's second law leads to a differential equation to be integrated.
integral(d(mr x v)) = 0
dt
mr x v = L
When there is no torque a certain quantity is constant called angular momentum.
A planet orbiting with constant angular momentum stays in a single plane with orbital speed varying in a precisely determined way. The area swept out by its radius vector changes at a constant rate (Kepler's second law).
da/dt = (1/2) r x v
This is easily determined in polar coordinates where da/dt = 1/2r^2 d(theta)/dt k^ including planets moving in an ellipse under Newton's Universal Law of Gravity.
F = -G M M(o)/r^2 (r^)
= -D/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
>Cross these vectors
L = Mr^2 d(theta)/dt (k^)
a x L = -D/Mr^2 (r^) x Mr^2 d(theta)/dt (k^)
d(v x L) = D dr^
dt dt
v x L = D (r^ + e)
The orbit of heavens described by the perfect circle of underlying geometry. Product of these events:
r . v x L = Dr1 (r^ + e)
r x v . L = D r . (r^ + e)
Once order is exchanged, and mass allowed its play
L^2/M = Dr . (r^ + e)
L^2/DM = r . (r^ + e)
L^2/DM = (r . r^ + r . e)
= (r + recos(theta))
r = L^2/DM -conic section algebraic equation.
1 + ecos(theta)
The force of gravity moves all heavenly bodies along conic sections.
Isaac Newton hypothesized that the force of gravity gave rise to elliptical orbits:
F = -D/r^2(r^) - inverse square law of gravity
F = m a
m a = -D/r^2(r^)
a = -D/Mr^2 (r^)
d^2r/dt^2 = -D/Mr^2 (r^) - differential equation of any orbit because the solution is the algebraic equation of any conic section.
r = L^2/DM/ 1+ecos(theta)
F= m a
r x m a = 0
m r x a = 0
m r x dv/dt = 0
d(r x v) = dr/dt x v + r x dv/dt
dt
=v x v + r x dv/dt
d(r x v) = r x dv/dt
dt
m d(r x v) = 0
dt
Zero torque, together with Newton's second law leads to a differential equation to be integrated.
integral(d(mr x v)) = 0
dt
mr x v = L
When there is no torque a certain quantity is constant called angular momentum.
A planet orbiting with constant angular momentum stays in a single plane with orbital speed varying in a precisely determined way. The area swept out by its radius vector changes at a constant rate (Kepler's second law).
da/dt = (1/2) r x v
This is easily determined in polar coordinates where da/dt = 1/2r^2 d(theta)/dt k^ including planets moving in an ellipse under Newton's Universal Law of Gravity.
F = -G M M(o)/r^2 (r^)
= -D/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
>Cross these vectors
L = Mr^2 d(theta)/dt (k^)
a x L = -D/Mr^2 (r^) x Mr^2 d(theta)/dt (k^)
d(v x L) = D dr^
dt dt
v x L = D (r^ + e)
The orbit of heavens described by the perfect circle of underlying geometry. Product of these events:
r . v x L = Dr1 (r^ + e)
r x v . L = D r . (r^ + e)
Once order is exchanged, and mass allowed its play
L^2/M = Dr . (r^ + e)
L^2/DM = r . (r^ + e)
L^2/DM = (r . r^ + r . e)
= (r + recos(theta))
r = L^2/DM -conic section algebraic equation.
1 + ecos(theta)
The force of gravity moves all heavenly bodies along conic sections.
Lesson 21 Kepler's Three Laws
Studying the orbit of Mars around the Sun, Johannes Kepler discovered its path could be explained only if the planet traveled on an ellipse. This was the first of Kepler's three laws.
Eccentricity literally means somewhat off center while focus was used to describe a fireplace. Kepler utilized these words to describe characteristics of ellipses.
Kepler set out for the most accurate astronomical models and therefore sought Tycho Brahe. Brahe's family attempted to withold Tycho's studies after his death, but stole them in order to "wage his war on mars," and develop his three laws to interpret the universe. Kepler embraced the Copernicun system and had to study the orbit of Mars on a planet that was not central and stationary, but instead in motion with all the other planets; a planet that spins on its axis in a non-circular fashion with constantly varying velocity, a planet with an unknown center.
Kepler studied the orbit of the Earth by keeping track of Mars's orbit year after year an using triangulation.
Kepler discoered that Mars moved faster when closer to the Sun and more slowly when farther away. This later coincided with Kepler's second law.
Every two years the Sun, Earth, and Mars are in opposition. In other words, whether seen from the Earth or the Sun, the position of Mars is the same. With observations year after year, Kepler triangulated the orbit of Mars. A circular orbit could not fit all the points of the orbit of Mars even though it could be off center of the Sun. In fact, the orbit of Mars was an ellipse with a very small eccentricity of .09. Any circle viewed obliquely is an ellipse.
Circles, ellipses, hyperbolas, and parabolas are all conic sections. A moving point traces out a conic section if its distance from some fixed point (focus) and its directrix are contant. The ratio of distances is eccentricity, where an ellipse has an eccentricity that is less than one, a circle has an eccentricity equal to zero, a parabola has an eccentricity equal to one, and a hperbola has an eccentricty that is greater than 1.
All conic sections can be described using an algebraic equation that is also useful for studying planetary motion:
r = ed/ ecos(theta)+1
Kepler utilized this equation and his study of the conic sections in order to develop his three laws.
Kepler's Three Laws:
1st - r = ed/ ecos(theta)+1
Each planet moves in an ellipse with the Sun at one focus.
2nd - dA/dt = constant
A line drawn from the Sun to a planet sweeps out equal areas in equal times.
3rd - T^2 = (4pi^2/GM) a^3
The square of the period of a planet's orbit is proportional to the cube of the length of the semi - major axis.
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