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Gravity

The movement of apples and planets


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History

From very ancient times, the five naked eye planets were known to be different to the stars. Whereas the stars appeared to be fixed onto the sky (which was thought to resemble a crystal sphere), the planets moved around relative to the stars. Indeed, the word planet is from a Greek word meaning "wanderer".

The planets were originally thought to revolve around the Earth. Casual observation showed that a planet like Jupiter would travel slowly across the sky in a West to East direction (right to left in the Northern Hemisphere). The planet takes about 12 years to complete a circuit of the sky. Mars completes a similar circuit in just over 2 years.

By the time of the ancient Greeks, it was thought that all motion in the heavens was circular with a constant speed. The planetary orbits were called geocentric (meaning "centered on the Earth"). An example is shown below where a planet (P), orbits the Earth (E) in an anticlockwise direction (as seen from above the Earth's North Pole).

Geocentric Orbit

As observations became more accurate, it was noted that the planets would sometimes change direction in the sky and travel from East to West for a short while before resuming their general Easterly motion. The normal West to East motion is described as direct motion while the rarer East to West motion is called retrograde. During the 2 year movement of Mars around the sky it will spend a couple of months moving in a retrograde direction. An example of this motion is shown below.

Retrograde Motion

The ancient Greeks had great problems explaining this retrograde motion. However, they did not want to give up the idea of circular motion at a constant speed. To explain this complex motion they invented the notion of epicycles. These are circles on circles. In the diagram below, the point C (which is empty) orbits the Earth (E) at a constant speed. This point C is the centre of another smaller circle (an epicycle). The planet (P) orbits the moving point C at a constant speed. This combination of motions explains the complex paths of the planets.

Epicycles

The system became more complicated as more accurate observations were made. Epicycles had to added to the epicycles! The Greek thinker, Aristarchus, suggested around 250 BC that the Earth was moving around the Sun. This would explain the retrograde motions. If the Earth was on the same side of the Sun as a planet, it would appear to overtake it and leave it behind, causing it to appear to move "backwards" as seen from Earth. It took 1800 years before this idea became accepted. In 1543 the Polish astronomer, Nicholas Copernicus, published a book explaining the new system mathematically.

This heliocentric system ("sun centred") was better at explaining the motion of the planets and could be used for predicting their positions in the future. However, the planets were still considered to move in circular orbits with constant speed.

Heliocentric Orbit

At that time the ideas of force and motion were hazy. For example it was thought that heavier objects fell towards the Earth more quickly. It was also assumed that the planets had to be continually pushed around through space so that they could orbit the Sun.

The Italian mathematician, physicist and astronomer, Galileo Galilei, performed a series of experiments in the early 1600s. These experiments showed that all objects dropped from the same height fell to Earth at the same time regardless of their weight. They also indicated that objects accellerated (speeded up) as they fell to Earth. His work also showed that if objects were moving at right angles to the ground, the horizontal and vertical movements were independent of each other. In other words, there could be a constant speed in the horizontal direction but a varying speed in the vertical direction. This explained the curved paths that objects like cannon balls took when fired into the air.

Kepler's Laws

During the latter part of the 1500s, the Danish astronomer, Tycho Brahe had been observing and recording the planet Mars for 30 years. This planet has a relatively rapid motion so the data covered several circuits around the sky. The observations were the most accurate before the invention of the telescope. Brahe gave the information to a German astronomer, Johann Kepler.

Kepler studied the data for several years, using the newly discovered logarithms to attempt to calculate how the planet was moving. Between 1609 and 1619, he published his findings as three laws. These are called Kepler's Laws of Planetary Motion.

Kepler's First Law

Kepler's First Law, removed the need for circular orbits. The law reads:

A planet orbits the Sun in an elliptical orbit with the Sun at one focus.

The word elliptical describes a curve called an ellipse. An ellipse resembles a circle that has been stretched. In the diagram below, the centre of the ellipse is the point where the two reference lines meet. An ellipse has two foci (plural of focus), marked F1 and F2.

Ellipse: Foci

The position of the foci is determined by a mathematical property shown below. The foci are located in such a way that the sum of PF1 and PF2 is constant for any point on the ellipse.

This property can be used to draw the curve. Begin by placing a pair of drawing pins (thumb tacks) at F1 and F2 on a piece of paper resting on a hard surface. A piece of string or cotton, longer than the distance between F1 and F2 is tied to the pins. A pensil (at point P) is placed so that the string is fully stretched at all times. If a curve is drawn while keeping the string taut, it will be an ellipse.

Ellipse: Definition

The following diagram shows the position of the Sun at one of the foci. The other focus is empty. The planet moves along the elliptical orbit. Note that the Sun is not at the centre of the ellipse.

Ellipse: Position of the Sun

If the two foci are close together the ellipse will resemble a circle. If the foci are far apart, the ellipse will be more elongated. Ellipses can therefore be different shapes. The shape of an ellipse is determined by a quantity called eccentricity.

In the ellipse below the lines are the axes (plural of axis) of the ellipse. The longest axis (running horizontally) is called the major axis. The shortest axis (running vertically) is the minor axis.

Mathematicians work with half of these axes. The values a and b are called the semi major axis and semi minor axis respectively. The value of the semi major axis (a) represents the mean distance of a planet to the Sun.

Ellipse: Major and Minor Axes

The eccentricity of an ellipse (e) is given by:

Eccentricity

An eccentricty of 0 denotes a circle. As the eccentricty gets bigger, the ellipse becomes more elongated. An eccentricity of 1 denotes an open ended curve (called a parabola).

The part of an elliptical orbit closest to the Sun (point P in the diagram below) is called the perihelion. The point furthest from the Sun (point A) is the aphelion.

Ellipse: Perihelion and Aphelion

If the planet's mean distance (a) and the eccentricity of its orbit (e) are known, the perihelion distance (dp) and aphelion distance (da) can be calculated.

Perihelion Distance

Aphelion Distance

For the Earth, a = 149,600,000 km and e = 0.0167. The perihelion distance is just over 147,000,000 km while the aphelion distance is about 152,000,000 km.

Most planetary orbits have an eccentricity close to 0. The Earth's orbit has an eccentricity of 0.0167. The largest planetary eccentricity is Pluto's orbit (0.2444). The orbits of comets tend to be more eccentric (and hence more ellongated). Two ellipses with different eccentricities are show below.

Ellipse: Types

Kepler's Second Law

Kepler's Second Law, removed the constant speed of a planet. The law reads:

The line joining a planet to the Sun sweeps out equal areas in equal times.

This is best shown in the diagram below. The area of the fat triangualar shape on the left of the Sun is meant to be equal to the long thin triangle on the right of the Sun. If so, the planet will take the same time to travel from P1 to P2 as it does from P3 to P4. In the part of its orbit where it is close to the Sun the planet moves quickly. In the part of its orbit where it is far from the Sun the planet moves slowly.

Kepler's Second Law

A planet will be moving at its greatest speed at perihelion. At aphelion it will be moving at its slowest.

The Earth is at perihelion at the beginning of January and at aphelion in early July. The changes in the Earth's orbital speed at these times mean that the northern hemisphere winter (December to March) is three days shorter than the northern hemisphere summer (June to September). This is becasue the Earth is moving more quickly around the Sun in January and more slowly in July.

Comets have very eccentric orbits. The differences in speed at perihelion and aphelion are much greater. A comet typically spends years moving slowly close to its aphelion. At perihelion is whizzes around the Sun quickly spending only a short time in the inner solar system.

A planet's Perihelion Velocity (Vp) and Aphelion Velocity (Va) are given by:

Perihelion Velocity     Aphelion Velocity

Where a is the mean distance to the Sun, e, the eccentricity, and P the siderial period (in seconds). The siderial period of a planet is the time it requires to complete an orbit around the Sun.

For the Earth, a = 149,600,000 km, e = 0.0167 and P is 31,557,600 s. The value of the first part of the formula (2pa / P) is 29.79 km/s. This figure is actually the mean orbital velocity. Multipying by the factor in the root sign gives the perihelion velocity as 30.80 km/s, while the aphelion velocity is 28.81 km/s.

Rather than orbiting in perfect circles at constant speed, planets (and comets) orbit the Sun in elliptical orbits at varying speed.

Kepler's Third Law

Kepler's Third Law allows the scale of the solar system to be worked out. It gives a relationship between the period that a planet requires to orbit the sun (the siderial period) and the planet's mean distance to the Sun. In words, it states that for each planet:

The square of the siderial period is proportional to the cube of the mean distance.

This can be expressed as p2 a a3, where p is a planet's siderial period and a is its mean distance to the Sun. If the siderial period is given in Earth years and the mean distance in terms of the Astronmical Unit (the distance between the Earth and the Sun), the relationship can be written as:

Kepler's Thord Law (Earth Units)

Taking Jupiter as an example, its orbital period (p) is about 11.75 years. Squaring 11.75 gives 138.06. Taking the cube root of this gives a value of its mean distance (a) of 5.17.

From Kepler's Third Law, the distances of all the planets (in terms of the Earth's distance to the Sun) can be calculated simply by knowing their siderial periods.

The siderial period of a planet cannot be observed directly because the Earth itself is moving. However, there are two formulas for obtaining a planet's siderial period by observing its synodic period. One formula works for planets closer to the Sun than the Earth (the inferior planets) and the other for planets further from the Sun than the Earth (the superior planets).

A planet's synodic period is the time a planet takes to appear in the same aspect as seen from the Earth. If a planet is close to the Sun (in conjunction), after one synodic period it will once again be close to the Sun. If a planet is opposite the Sun (and visible all night long), it will be in the same situation after one synodic period.

Siderial Period (Inferior Planet)    Siderial Period (Superior Planet)

In these equations all quantities are expressed in years: E is the siderial perod of the Earth, p is the planet's siderial period, Si is the synodic period of an inferior planet, Ss is the synodic period of a superior planet.

As an example, Jupiter has a synodic period of 398.88 days. This is the time between successive oppositions of the planet. This can be written as 1.092 years. By substituting this into Ss in the formula on the right (Jupiter is a superior planet), and remembering that E is 1, the value for p comes out as 11.87 years.

Once the scale of the Solar System has been established, if any single distance can be measured, all the distances are then known. During the mid 1700s, an attempt was made to measure the distance between the Earth and Venus when the latter passed between the Earth and Sun (a transit). That measurement was then used to work out the distance from the Earth to the Sun (the Astronomical Unit) from which all other planetary distances could be calculated.

Newton's Laws of Force and Gravity

The English astronomer and mathematician Isaac Newton worked on the problem of force and gravity for several years before publishing his great work, Principia, in 1687. In this book he was able to show that the same force which pulled an apple to the ground on the Earth, also affected the Moon.

Newton's Laws of Motion

Newton described three laws of motion that completed the work of Galileo and removed the need for a constant push to keep the planets moving around the Sun.

Newton's First Law of motion states:

A body at rest remains at rest and a body in motion remains in motion as long as outside forces are not acting on the body.

This means that an object moving in space, with no forces acting on it, will carry on moving in a straight line. This removes the need for something to be pushing planets along. On the Earth, bodies in motion slow down because of forces like friction or air resistance. In space, where there is no air, no resistance exists to slow a body down, and it will carry on moving forever.

Newton's Second Law can be written:

A force acting on a body causes it to accellerate in the direction of the force.

The accelleration that a given force will impart on a body depends on the amount of matter that the body contains. This is called the body's mass. If a given force acts on two objects, the more massive object will be accellerated less; the less massive object will be accellerated more. This is common sense because it takes a stronger push to move a cannon ball that a football.

This is expressed by the famous equation:

Force

Where F is the force, m, the mass, and a the accelleration. This resistance to change in the state of motion is called intertia. All material bodies have inertia.

Newton was the first to distinguish between the mass of a body (something that does not vary) and its weight. Weight is a force acting on a mass. A person in space can be weightless but will always have the same mass. They are weightless because of the lack of a force acting on them. They continue to have mass because they are a material body with inertia.

On the earth the weight of a body (W) is its mass (m) multiplied by the accelleration of gravity, which has a value of 9.8 ms-2 and is given the symbol, g:

Weight

The value of 9.8 ms-2 for the accelleration of gravity means that, as a body falls, every second it will be moving 9.8 m/s faster than during the previous second. This value was measured by Galileo.

Newton's Third Law is so famous, it is almost a cliché:

To every action there is an equal and opposite reaction.

This means that if two bodies collide, they each impart a force on each other. It also governs the working of jet engines and rockets: gases escaping from a tube cause the object to move in the opposite direction.

Newton's Law of Universal Gravitation

Newton showed that the force of gravity pulling objects (like apples) onto the Earth reached up to the Moon. He showed that the Sun had a similar force that reached out and pulled the planets.

Consider two bodies with masses m1 and m2, separated by a distance of d.

Attraction of Bodies

The Law of Universal Gravitation can be can be stated as follows:

Two bodies attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

This can be written in the famous equation:

Newton's Law of Gravity

Where F is the force of gravity between the two masses (m1 and m2). The distance between them is d.

The quantity, G, is called the Gravitational Constant and can be measured in the laboratory. This was done by the English physicist, Henry Cavendish in 1798. G has the value 6.673 × 10-11 N m2 kg-2 and is a measure of the strength of the force of gravity.

Newton's equation shows that the force of gravity becomes stronger if the masses are larger. The force becomes weaker if the distance between the bodies increases. If the distance doubles, the force is reduced by 4 (because 4 is 2 squared).

This equation makes it possible to work out the orbits of the planets and to measure masses of celestial bodies.

Newton was able to show (by using the newly invented calculus) that, gravitationally, a planet behaves as if all its mass is concentrated in its centre. This is so for any spherical object. That property can be used to analyse the gravitational attraction between a planet like the Earth and an object on its surface.

Body and Earth

In the diagram above, a body of mass, m, is resting on the surface of the Earth, whose mass is M. The distance between the two bodies is R (the radius of the Earth) since the Earth behaves as if all its mass is concentrated in its centre.

The force between the body and the Earth (F) can be given by the Law of Universal Gravitation as F = GMm/R2. In addition, the force is equal to the body's weight (W) which is given by W = mg (g = 9.8 ms-2). Putting these two equations together gives:

Weight

Equating the two parts on the right, the body's mass (m) cancels from both sides leaving:

Accelleration of Gravity

This equation shows that the accelleration of gravity on the surface of the Earth (g) depends only on the mass of the Earth (M), its radius (R) and the Gravitational Constant (G). It does not depend on the mass of the body. This is verification of Galileo's experimental results that all bodies on the Earth fall with the same accelleration.

The above equation can be rearranged.

Mass of the Earth

The values of G and g can be measured. The radius of the Earth (R) is known (it was first measured by the ancient Greeks). The above equation can be used to calculate the mass of the Earth (M).

The value of the Earth's radius, is R = 6,371,000 m. Putting this value with the various constants gives the mass of the Earth as 5.9 × 1024 kg.

Newton and Kepler's Laws

Newton combined the Universal Law of Gravity with the laws of motion and derived Kepler's three laws.

He found that that path of a planet, moving under the gravitational attraction of the Sun, would, in general, be an ellipse with the Sun at one focus.

More exactly, the theory predicted that a body moving under gravitational attraction would take a path described by one of four possible curves: a circle, the ellipse, a parabola or a hyperbola.

The circle is really a special case of the ellipse with an eccentricity of 0 and the two foci coincident. The eccentricity of an ellipse lies between 0 and 1. Both are closed curves. The conditions required for an orbit to be exactly circular are never found. The ellipse is the path for planets, satellites, asteroids and comets within the Solar System.

Newton showed that the total orbital energy of a planet in a closed (elliptical) orbit around the Sun was given by:

Total Energy for Elliptical Orbit

As long as the total orbital energy is less than the quantity on the right, the planet remains in a closed elliptical orbit.

If the velocity of a body is too high it will follow a curved, but open path bending around the Sun. This path may be parabolic or hyperbolic. The conditions required for a parabolic orbit are too exact to be found in nature. The total energy for a planet in a hyperbolic path is given by:

Total Energy for Hyperbolic Orbit

Newton made Kepler's First Law more general. The four curves described are related to each other and are known as the Conic Sections. This is because they can all be produced by slicing a cone as shown in the diagrams below.

Circle   Ellipse   Parabola   Hyperbola

The parabola has an eccentricity of 1 while the hyperbola has an eccentricity greater than 1.

Newton showed that Kepler's Second Law is equivalent to saying that the angular momentum of a planet in orbit around the Sun remains constant. This is shown in the equation below:

Angular Momentum

J is the angular momentum. All the quantities on the right hand side of the equation are constants for a particular planet: its mass (m), its mean distance from the Sun (a), its orbital period (p) and the eccentricity of its orbit (e). This means that the angular momentum itself is constant.

Newton derived a formula for the velocity of a planet at any point in its orbit.

Velocity of a Planet

The quantity r is the distance of the planet to the Sun at the point at which its velocity (V) is required.

Newton also showed that Kepler's Third Law is equivalent to the conservation of the planet's total energy. From this, Newton derived the third law (p2 a a3) in a more exact form.

Kepler's Third Law

The term (MSun + MPlanet) is not an exact constant as thought by Kepler. It differs for each planet. However, the mass of the Sun is far greater that the mass of any planet. The difference in this term is less than 0.1% for the nine planets.

Once the mass of the Earth is known, this equation can be used to calculate the mass of the Sun. A similar equation is used to determine masses of planets and their satellites as well as star systems.

Newton's ideas about gravity survived until the early 1900s when the Swiss theoretical physicist, Albert Einstein refined them with his General Theory of Relativity.

But that is another story.

© 2004 by KryssTal

Conics figures from JB Conics

This section is dedicated to Johann Kepler and Isaac Newton.


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External Links

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MathWorld (The Ellipse)
Mathematical site about the ellipse.