ORBITS AND GRAVITY

GOALS

Towards the end of Tycho's life, Kepler became his assistant.

2. Ellipse

3. Kepler's 3 laws of planetary motion

Animation: http://www.wwnorton.com/college/astronomy/studyspace/_animations/keplers_laws.htm

3) Harmonic law (why is it called harmonic) = relates a planet's orbital period of revolution (a planet's year) with its distance from the Sun.

4. APPLICATION OF KEPLER'S SECOND LAW FOR THE EARTH-SUN SYSTEM

Earth is closest on Dec 22 and most distant on June 22.

2) Between March 21 and June 22 = 93 days

5. Newton's 3 laws of motion

Animation: http://www.wwnorton.com/college/astronomy/studyspace/_animations/projectile_motion.htm

1) a) On the concept of force: Almost always a force creates a direct pulling or pushing effect. We do not see forces, rather we see the effect they create on the motion of objects. A force has magnitude and direction.

b) First law (law of inertia)

2) Second law

b) Acceleration = describes a change in velocity (speed with direction)

c) F = ma

3) Third law (Action Reaction law)

1.  TYCHO AND KEPLER

At about the time Galileo was working on his experiments with falling bodies, Astronomers Tycho Brahe and Johannes Kepler helped with the understanding of the motion of the planets. Tycho was trying to figure out the mysteries of the skies by observing them, and Kepler was trying to do the same thing using mathematical models.

Tycho (1546-1601) was Danish and made important astronomical observations, such as exploding stars (called nova or supernova) and kept a 20-year-old record of the positions of the Sun, Moon and Planets. Kepler, trained in mathematics assisted Tycho in analyzing his extensive planetary data.

Kepler (1571-1630) born in Germany studied Tycho's data for more than 20 years and in 1609 he published the first results which included his first two (out of three) laws of planetary motion. These will be introduced below. Kepler’s most detail study was on Mars for which Tycho had an extensive set of data. Kepler discovered that planets’ orbits around the Sun were elliptical.

An ellipse is an elongated circle with two foci (singular focus) (figs.2.3 and 2.4). The more elongated the ellipse is, the closer the foci are to the ends of the ellipse.

(fig. 2.3)

(fig. 2.4)

A characteristic feature of all ellipses is that, the sum of the distances from any point on the ellipse to the two foci is always the same (fig. 2.4). The widest diameter of the ellipse is called the major axis and the smallest diameter is called the minor axis. Diameters pass through the center of the ellipse. The center of the ellipse is the point at the middle of the line that connects the two foci.

The roundness of the ellipse is measured by the eccentricity of the ellipse which is defined as the ratio of the distance between the foci to the length of the major axis. If eccentricity is zero then we have a circle. The most elongated ellipse is the one having the maximum eccentricity of one.

Kepler's 1^st law (law of the ellipses):

Kepler found that planets have an elliptical orbit with the Sun at one focus. The place where a planet is closest to the Sun is called perihelion and the place where the planet is farthest away is called aphelion. The more elliptical the orbit, the nearer the Sun is to one end of the ellipse. For a satellite (such as the Moon or spacecraft) orbiting the Earth, these terms become perigee and apogee.

Kepler’s 2^nd law (law of the equal areas): It relates a planet's orbital speed with its distance from the Sun. That is, a planet speeds up when closer to the Sun and slows down away from it. Specifically, at perihelion a planet moves the fastest and at aphelion the slowest. As a result of this, an alternative statement of the 2^nd law is the following: the imaginary line between a planet and the Sun sweeps over equal areas in space during equal time intervals (fig 2.5).

(fig. 2.5)

Kepler’s 3^rd law (harmonic law): This law relates a planet's orbital period of revolution (a planet's year) with its distance from the Sun. Specifically, a planet's year is longer the further it is from the Sun. Mathematically this law is a simple algebraic relationship between a planet’s semi-major axis and its period of revolution around the Sun.

(length of semi-major axis) ³= (period) ²

This equation makes sense dimensionally if the semi-major axis is expressed in astronomical units, AU (1 AU = the average distance between the Earth and the Sun = 1.5 x 10⁸ km) and the period is expressed in terms of Earth years. Why is the law called harmonic? Pythagoras (7 century BC) was convinced that there must be a mathematical harmony everywhere in nature and that simple numerical relationships must determine the motion of the planets which could be represented in a musical manner. He came to this conclusion when he saw the regularity in the motion of the planets. Pythagoras was the first one to use mathematics to describe a natural phenomenon, the sounds of music. Indeed today physics does exactly the same thing. But Pythagoras was not exactly sure how the motion of the planets could be represented mathematically. Kepler, 2000 years later showed how this can be done by forming the ration of the fastest to the slowest speed of the elliptical orbit of the planet.

Kepler’s laws are purely descriptive. In other words, Kepler's laws do not deal with the cause that makes the planets move that way. The force of nature that constrains planets to follow these rules was discovered by Newton.

APPLICATION OF KEPLER'S SECOND LAW IN THE EARTH-SUN DISTANCE

The Earth is closest to the Sun on December 22 and most distant on June 22. The spring equinox is on March 21 and the fall equinox on September 23. The number of days between December 22 and March 21 is 89 (or 90 when February has 29 days). The number of days between March 21 and June 22 is 93. The number of days between June 22 and September 23 is also 93. And finally, the number of days between September 23 and December 22 is 90. If we add these four time intervals we find the expected 365 days (or 366 when February has 29 days):

89 + 93 + 93 + 90 = 365.

Notice that between December 22 (the closest point of Earth Sun) and March 21 (spring equinox) the number of days, 89, is by 4 days less than 93, which is the number of days between March 21 until June 22 (the most distant point of Earth Sun). This is expected and is in accordance with Kepler's second law. That is, in 89 days the Earth does not reach its most distant point from the Sun, rather, it arrives there in (89 + 4) days on June 22. If the orbit of the Earth around the Sun were a perfect circle, then the time interval between the winter (of say the northern hemisphere, it does not really matter) and the spring equinox would have been exactly equal to the time interval between the spring equinox and the summer (of the northern hemisphere). (Note, in the case of a perfect circular orbit there is no closest and most distant points of the Earth to the Sun, and also it would not make sense to speak about particular dates (i.e. December 22 or March 21) as the year would have last a different number of days). On the other hand, if the orbit of the Earth around the Sun were very elliptical then, the number of days difference between the time interval of winter (of say the northern hemisphere, it does not really matter which hemisphere is considered) and the spring equinox, and the time interval between the spring equinox and the summer (of the northern hemisphere) would have been more than 4 days (which is roughly what we have in the actual orbit of the Earth). The 4 day difference is in fact an indication that the Earth's orbit around the Sun is almost circular.

2.  NEWTON'S 3 LAWS OF MOTION AND HIS LAW OF UNIVERSAL GRAVITATION

Newton (1642-1727) born the year Galileo died, did superb work on the motion of objects, on gravity and optics. About two years after reading a book in geometry written by Euclid around 3BC, Newton invented new mathematics (calculus) to help him deal with complex physical phenomena. His most famous book is called Principia was published in 1687. In his book he discusses his three laws of motion as well as the law of gravity which we will discuss later. But first let’s introduce Newton’s 3 laws of motion.

Newton’s 1^st law: If the net force acting on an object is zero, then an object at rest remains at rest and an object in motion continues moving with the same speed in a straight line. Examples are, either a book sitting at rest, or a car moving with constant speed in a straight line.

Let's try first to understand the notion of a force. Examples of a force are: gravity, friction, pushing a door with your hands, pulling a box, the electric force responsible for the interaction of charged objects, the magnetic force, etc.  From all these examples we notice that a force creates a direct pulling or pushing effect. (This is almost always the case; the only exception is with the weak nuclear force that is responsible for the transmutation of particles. We will talk about this more on chapter 15.) Also, the direction that a force acts, is important because it controls the direction the object will finally move in. We do not see forces. We see the effect they have on the objects they are applied. For example, we do not see gravity but we see the way a falling body moves due to gravity. A net force is the combined effect of more than one force acting on an object.

Newton’s 2^nd law: An object of mass m experiencing a non zero net force F moves in an "accelerated" manner with acceleration a. An example of the 2^nd law is a car initially at rest; pressing the gas petal, a force is applied on it, accelerating it to higher speeds.

F = m a

F = Force

m = mass = the amount of particles contained in an object or = mass is a measure of inertia; (inertia is the tendency of an object to resist a change in its state of motion). The unit of mass is kg (kilograms).

a = Acceleration = rate of change of velocity = acceleration describes a change in velocity

Velocity = speed with direction

Example of speed is 30 miles / hour

Example of velocity is 30 miles / hour, East

Example of acceleration is 30 miles / (hour)², East

Assuming negligible air resistance, then the Earth makes things fall with acceleration g = 9.8 m / sec². This is known as free fall. How do we interpret g = 9.8 m / sec²? It means that the speed of a falling object changes by an amount of 9.8 m / sec every second. That is, if an object is released from rest and we assume negligible air resistance, then after one second its speed will be 9.8 m /sec. After two seconds its speed will be 2 x 9.8 m /sec. After three seconds its speed will be 3 x 9.8 m / sec, etc.

The weight W of an object on the Earth is figured out by using W = m g. The mass of an object remains the same everywhere in the universe. On the other hand, the weight of an object depends on the g of each individual planet or moon. g depends on the mass and radius of a planet or moon. The g of the Moon is 1.7 m / sec².

Newton’s 3^rd law: For every action force there is an equal reaction force pointing in the opposite direction.

Example: The Earth attracts me, but I attract the Earth with the same force but opposite in direction. Why then if I jump up, I come down instead of the Earth coming up? Because even though the forces that I and the Earth are experiencing are the same, the downward acceleration I experience is g = 9.8 m / sec², but the upward acceleration the Earth experiences by my attraction is zero. Putting it differently, the inertia of the Earth is much larger than my inertia. Recall mass is a measure of inertia and the Earth's mass is much bigger than my mass.

A space flight occurs because of the validity of Newton’s 3 laws as well as Newton's law of universal gravitation.

The governing force in the universe is the force of gravity. The very structure of the universe is controlled by this force. What keeps the planets in orbit, Newton said, is the force of gravity. Like Earth which attracts objects and makes them fall, the Sun attracts the planets. Due to the Earth’s attraction falling objects accelerate as they fall. The Earth’s gravity extends as far as the Moon and beyond (up to infinity) and produces the acceleration necessary to curve the Moon’s path. Similar is the situation between the Sun and planets. In fact, Newton said that there is a universal gravitational attraction force between any two objects and went on to find a correct mathematical expression of that force indicating the factors it depends on. Specifically, the gravitational force F between two objects having masses m₁ and m₂ separated by a distance R is given by.

F = (G m₁m₂) / R²

G = constant of gravitation = a constant number

F decreases with increasing distance.

With his law of universal gravitation, Newton was able to derive Kepler’s laws. It is true that when the force of gravity acts between two objects, the objects tend to move towards one another. However, while planets are attracted towards the center of the Sun, they are also moving parallel to its surface (a motion resulting from the early conditions in the solar system, which we will discuss later) thus keeping them from crashing on the Sun.

As a result of Newton’s law of gravity all objects when dropped from relatively small height above the surface of the Earth, accelerate downwards with acceleration g = 9.8 m / sec². Objects dropped from high altitudes are falling with accelerations less than 9.8 m / sec². If you would like to know how strongly the Earth is attracting you (or how strongly you attract the Earth) simply jump on a scale and read your weight. Note, that even though there is an attraction between any two objects, when the combined mass of these objects is not sufficiently high, the force is negligibly small and not easily observed. For example, there is an attraction between two books but it is not easily observed.

Despite of the fact that the astronauts feel weightless, their gravitational attraction by the Earth is relatively as strong as when they are on Earth. However, because astronauts, space shuttles and everything else in the space shuttle all are falling freely (the same exact way) in relation to one another, objects appear as weightless. An orbiting space shuttle is an example of free fall as is the motion of a released ball from some height above the ground. The only difference is that unlike the released ball, which falls straight down, the space shuttle is falling towards the center of the Earth while simultaneously moving parallel to the surface of the Earth. As a result of that, the space shuttle is constantly free falling without ever crashing on the Earth. Note, it is the force of gravity that keeps the space shuttle in orbit around the Earth and not its engines. In fact, it does not need any engines to orbit, as the Moon does not need one to orbit the Earth, or as the planets do not need one to orbit the Sun.

More on gravity

Newton's force of gravity is assumed to be transmitted instantaneously in the universe at any distance. Specifically the force equation depends only on distance and not on time. But that means that the effect of gravity travels with infinite speed, a concept inconsistent with Special Relativity. Einstein solved the inconsistency with his General Theory of relativity. In that theory space-time is curved by mass and energy distributed in it, and gravity is not a force. With that conception, he found that gravity's effect travels with finite speed, which happened to be the same as the speed of light.

Newton did not really know how gravity really works. How, for example, two bodies separated by a distance influence each other's motion? Einstein answered that when in his General Theory of Relativity created a geometrical representation of space-time. Think of space and time to be like a piece of flat stretchable fabric. Just like the fabric's surface is stretched and warped by the presence of a mass placed on it, space-time warps by a mass in it as well. This warping or curving of space-time then creates what we feel as gravity. This warping travels at exactly the speed of light. The Earth moves in the warped space-time produced by the Sun. So even if the objects are separated by a distance, the curved space-time produced by their mass, makes them influence one another's motion. This is how Einstein explain why gravity influences objects.

3.  ORBITS IN THE SOLAR SYSTEM

Using the theory of universal gravitation, an object’s (star, planet, spacecraft) path through space as well as its position may be determined any time in the future.

Orbits of the planets

There are 8 planets in the solar system. These are with increasing distance from the Sun.

Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto (On August 24 2006 the international community of astronomers decided that Pluto is no longer considered a planet, rather a dwarf planet).

Even though Pluto has the largest semimajor axis of 40 AU = 5 x 10⁹ km, for the last 20 years of the 20^th century it has been near its perihelion, and because the orbits of planets are elliptical, Neptune happened to be the most distant planet during that period.

Planets’orbits are elliptical but with law eccentricities (which means that to a good approximation the orbits of planets may be considered almost circular). The closer a planet is to the Sun the fastest it revolves around it, and thus takes less amount of time to make one revolution about the Sun. For example one Mercury year lasts 88 Earth days, but one Pluto year lasts about 249 Earth years.

Planetary orbits are almost confined close to a common plane, which is near the plane of the Earth’s orbit (the ecliptic) and among the zodiac belt.

Orbits of Asteroids and Comets

In addition to the nine planets, there are many smaller objects in the solar system. Some of these are natural satellites, called moons, orbiting the planets. Note, Mercury and Venus do not have any moons. In addition two classes of smaller objects orbit the Sun. These objects are the asteroids and the comets which are believed to be small chunks of material left over form the formation of the solar system, and they will be discussed later in lecture 6.

Asteroids and comets differ from each other in composition and in the nature of their orbits. Asteroids have smaller semimajor axes than comets. The majority of asteroids is found in a region of the solar system called the asteroid belt which is between 2.2 and 3.3 AU from the Sun (roughly between Mars and Jupiter).

Comets on the other hand, have orbits of very large size, hence taking longer to orbit the Sun (sometimes thousands of years). One of the most popular comet is Haley's comet with orbital revolution time of 76 years.

Motions of Satellites and Spacecraft

Although there is no difficulty in maintaining an artificial satellite in orbit a great deal of energy and planning is required involving complex calculations to lift it off the Earth and accelerate it to its required orbital speed. Once the satellite is in orbit, if it is high enough to be free of retarding atmospheric frictional effects, it will remain in orbit forever.

Let’s illustrate how a satellite is launched. Having a mind fig. 2.11a, imagine an object is fired horizontally with some velocity V_a. As the object is flying horizontally is also falling vertically due to gravity. This simultaneous motion in the horizontal and vertical directions make the object move in a parabolic orbit until it strikes the ground at point a. In a second attempt, if the object is given a higher initial horizontal velocity, say V_b, then the object in general will strike the ground further away at point b. If now in a third attempt the object is given the proper horizontal velocity V_c the object will be continuously falling but it will never strike the Earth because its orbit will always be parallel to the surface of the Earth. In other words, the object's path will be turning exactly the same way as the Earth's surface is. That is, the amount of curving of the object's path due to the attraction for this particular velocity is neither too much nor too little and as a result, a circular orbit is created. Even though in this case the object is constantly falling, it never crashes on the Earth! The velocity needed to do this is called circular velocity (about 8 km / sec = 17,500 miles / hour). Once an object is given this initial velocity, it will achieve a circular orbit forever. The formula that determines this proper circular velocity is derived using Newton's laws.

V_c = (G m / R) ^1/2

As seen from the formula V_c depends on the mass m of the planet and the distance R the object is from the center of the planet.

Circular velocity is required to put an object in orbit around the Earth. In this case note that what keeps the object in orbit is the gravitational attraction force between the object and Earth. If we would like to send an object away from Earth so that it will forever escape its gravitational attraction, the object must be launched with the escape velocity, which is about 11 km / sec = 25,000 miles / hour. Going to our previous thought experiment, that means that the velocity with which the object must be launched to escape the Earth's attraction forever must, be greater than V_c . That means that the object's path will not be turning as much as the Earth's surface and as a result the object will be moving away from the Earth. Note, that not all velocities greater than V_c will result in escape. For the Earth, the minimum value of the escape velocity is 11 km / sec. When an object's launching velocity has a value between those of the circular and escape velocities, the object moves in an elliptical path. Each planet or moon has its own unique circular and escape velocities, which an object must be launched to either be put in a circular orbit, or forever escape the planet or moon. This is due to the fact that each planet or moon has its own unique mass and radius. 

Once away from the Earth, the orbit of the object can be modified by thruster rockets on board. Thruster rockets make use of the principle of conservation of momentum. Conservation of momentum is what happens a) when a figure skater picks up speed when her hands are brought closer to her body or b) when an ice skater standing still on the ice holding an object moves backwards if she throws the object forward. Also frequently, the path of a spacecraft moving around the solar system can be modified with the help of gravitational attraction between the spacecraft and a planet. This is known as gravitational boosting and is shown in fig 2.13.