Newton's law of Gravitation

I would guess that just about everyone has heard the story of Newton sitting in his garden when an apple fell on his head and he thus went on to formulate his theory of gravity. Well it probably didn't actually happen like that, although it is written that he saw an apple fall and as his mind wondered he thought why does the apple fall? From his first law of motion as the apple accelerates off the branch the change in velocity must be caused by a force, which he was to call gravity.

If we now consider celestial mechanics, it had long been accepted that the planets and the Moon did not move in straight lines but that they moved in circular/elliptical paths (if you haven't been there already have a look at our page on Kepler for more on the orbital paths of the planets) and any object following such a curved path is, by definition, constantly changing direction.

Now from Newton's first law a change in velocity (acceleration) is the result of a force acting on the body and as velocity is a vector, a change can be caused by either a change in magnitude or direction. Newton was thus able to conclude that as the planets and the Moon were constantly changing direction, they must be under constant acceleration, and therefore be constantly acted upon by a force.

Newton then asked himself if all the planets are constantly been acted on by a force, and the Moon was also been subjected to a constant force, as was everything on Earth, could it be that it was the same force? To answer this question he found himself asking another question,
Was it possible that the Earth's gravitational force could extend so far from the Earth that it was responsible for holding the Moon in its orbit around the Earth?

Of course the answer is clear to us now, but Newton was the first man to even consider this possibility. He came up with a rather novel idea for demonstrating how it might be possible. In thinking back to Galileo's projectiles (click here for our page on projectiles and would like to know more) he reasoned:
If a cannonball was fired from the top of a very high mountain it would go a certain distance and then fall back to Earth due to gravitational force. If another cannon ball was then fired with more velocity the projectile path would become flatter and thus it would travel further before falling to Earth. So there should be a critical velocity where the cannonball would travel around the Earth always falling (due to gravitational force) at the same rate that the Earth curves away (in other words put the cannonball into orbit!).
Therefore he realised it was possible that the force that held the moon in place could be the same force that makes objects fall to earth.

The Universal Law of Gravitation

It was at this point Newton unified terrestrial and celestial mechanics, as he realised Galileo's acceleration due to gravity and Kepler's laws of Planetary Motion are simply different aspects of the same thing. He thus began to formulate a Universal Law of Gravitation, rather than the separate theories of terrestrial and celestial gravitation.

Newton then started to analyse what he had found and apply it to his Laws of Motion, he made two crucial discoveries:

1. As already stated in the previous page, Newton and Motion gravitational force is proportional to mass, from Newton's second law, Newton generalised this to suggest that this was true for any gravitational force on a body.
   Now from Newton's third law, a force is generated from interaction of at least two bodies and to every action there will be an equal and opposite reaction, the symmetry of the forces allowed him to suggest that gravitational attraction is proportional to both masses involved and therefore must be proportional to the product of the masses.
   This can be written as:

   
   Where m1 and m2 are the two masses in question

   Newton went on to generalise this theory of symmetrical forces and suggest that all masses in the universe will exert a gravitational attraction upon one another. For example as the Earth attracts the apple to its centre, the apple in turn attracts the Earth towards itself with an equal force. Of course the relative mass of the Earth prevents a notable acceleration towards the apple (Newton's second law).
2. By Newton's time it was already established that the apple would fall 5 metres in 1 second at the Earth's surface (click here for an explanation if you are not sure about this). Newton then used geometry to calculate that the fall of the moon in 1 second was about 1.37mm and this led to his famous inverse square law. How? He began by theorising that the reason for this difference in distance of fall could be due to the greater distance that the Moon is from the Earth than the apple. The relationship between the two is that the Moon's fall is about 1/3600 of the apple's fall and the Moon is about 60 times further from the centre of the Earth than the apple, so he realised as 60² =3600 the gravitational attraction must be inversely proportional to the square of the distance between the centres of the bodies (the inverse square law). This can be written as:

   
   
   Where r is the distance between the centres of the two masses

   Hence if the distance, r, between the bodies is doubled then the strength of the gravitational attraction is reduced by a factor of 4

Drawing all this work together Newton developed his formula for the Universal Law of Gravitation (note this applied to the gravitational force between any two bodies):

The problem with this expression is that it deals only in proportionality, thus it can only give a relative result. This can, however, be converted into a more useful equation by the addition of a constant, and this constant was worked out by Henry Cavendish in 1798. He achieved this by measuring the gravitational attraction between two 1kg lead balls at a distance of 1 metre.
I would guess that you would assume that this experiment must have found the attraction to be very small, in fact it was so small it was a stunning achievement to measure the tiny acceleration caused and to therefore calculate the gravitational attractive force (using Newton's second law).
Cavendish found the force to be 6.67 x 10^-11 Nm²/kg² (or approximately 0.0000000000667 Nm²/kg²!).
Taking this together with the assumption that this value is constant for all bodies in the universe, allows us to add this as a gravitational constant to Newton's formula. This means that Newton's formula can now be written as an equation:

Where G is the gravitational constant

From his initial findings he was able to show that using his Laws of Motion and Universal Gravitation he was able to imply and therefore prove Kepler's Laws of Planetary Motion. This was an important step as before Newton the rules were only empirical no one knew how they worked, they just seemed to work.

Newton went further: as far as Kepler had been concerned his laws only applied to the planets but Newton had shown that it applied to all masses in the universe and that celestial and terrestrial mechanics were governed by the same laws (hence Universal Law of Gravitation).

Once Newton had begun to apply this law to the real universe, he was to make some discoveries that were to revolutionise man's understanding of the universe, as he began to expand Kepler's laws.

A Common Centre of Mass

The Commen Centre of Mass, also known as common centre of gravity, was the first major change Newton made to our understanding was to suggest that Kepler's idea of the Sun being at one focus point of the ellipse was incorrect.
Kepler had felt that the sun was a fixed point in space and the planets revolve around it, but his only reasons for placing such special emphasis on the Sun, was his belief that the Sun possessed almost god like powers, it was more of a religious argument than a scientific one. Newton, however, was about to introduce mathematical reasoning for the Sun's seemingly dominant position in our solar system.

From his third law that every action has an equal and opposite reaction Newton realised that a planet was not orbiting a stationary sun but the planet and the sun were orbiting a common centre of mass.
Ok but what is a common centre of mass?
It is easiest to visualise if we think of a seesaw, an object that we are all familiar with. If we wanted to balance the seesaw with two people of differing masses we could move the fulcrum of the seesaw off centre, towards the person with the greater mass. If we calculated the exact amount to move the fulcrum we could balance the seesaw exactly, and the fulcrum would be at the common centre of mass of the two people.

If we now look at the diagram, right, we can see a similar idea being applied to two general masses, which could represent anything from our two friends on the seesaw to a planet and the Sun (obviously this diagram is not drawn to scale!)
To find the centre of mass and therefore to achieve this balance the equation m1d1=m2d2 must be satisfied (again the heavier mass, the smaller distance from the centre).
Given d1+d2=r, from m1d1=m2d2 it can be derived that d1=(m2/m1+m2)r and d2=(m1/m1+m2)r.

(If you are interested click here for the proof, it is fairly simple and I have tried to to ensure that all the steps are laid out, so it can be followed).

Armed with these equations, if we know the two masses and the total distance we can find the common centre of mass:

If m1 = 5, m2 = 10 and r = d1+d2 = 270 then: d1 = (m2/m1+m2)r = (10/15)270 = 180 and d2 = (m1/m1+m2)r = (5/15)270 = 90

From this work that Newton had done we can now understand his thinking when he showed Kepler's third law was only an approximation, and modified it by providing quantitative corrections that were proved correct from observation.

(note this equation assumes the masses are measured in solar masses, the times in Earth years and the distances in AU's).

If you take Kepler's third law for two planets the masses cancel themselves out and you are left with the original equation, i.e.

(See Kepler's third law).

An important point to realise is if one of the masses (e.g. m2) is much greater than the other when they are added together the sum is still very close to m2, thus m2/(m1+m2) is approximately equal to 1 and m1/(m1+m2) is approximately equal to 0. The result of this will be that d1 will be approximately equal to d and d2 will be very small.
So in real terms this means the common centre of mass almost coincides with the centre of the massive object. This is the case for planetary orbits, the Sun is so massive that the common centre of mass occurs very close to its centre, therefore the Sun is seen as almost motionless, but it is not quite as still as Kepler believed!
Thus Kepler's law is very close to being correct due to the sun being so massive, but of course if m1 and m2 are very similar then Newton's correction would become very significant!

Newton used this theory of gravitation to explain various phenomena, such as:

• the eccentric paths of comets, he also suggested they may be on returning orbits
• tidal ebb and flow (Galileo had tried to understand this but never quite grasped it)
• the precession of Earth's axis
• the motion of the Moon as perturbed by gravity of the Sun

This was all contained within the third book of the Principia, which dealt exclusively with the Universal Law of Gravitation.

Orbital Paths in a Gravitational Field

Newton also discovered from his calculations and analysis that an elliptical orbit was not the only possible orbital path in a gravitational field. All the possible paths take the shape of a conic section and the determining factor as to what type of path an object takes is the relative speed of the object in orbit. Possible paths are:

1. Circle - Some of the planets orbits are virtually circular (e.g. Venus), artificial satellites may have circular orbits
2. Ellipse - Most planets fall into this category (all do in actual fact, but some have very low eccentricity (see 1)), as do returning comets
3. Parabolic - Some comets, they pass once and never return
4. Hyperbolic - Gravitational interaction of passing stars

1 and 2 are known as bound orbits, 3 and 4 are one time gravitational encounters. For a given central force increasing velocity results in flatter paths, for example if Earth's velocity was increased by a factor of 1.4 its orbit would change to a parabola and Earth would shoot off out of the Solar System.

Gravitational Perturbations

Take for example the Earth/Sun gravitational interaction, according the Universal Law of Gravitation every mass in the universe interacts with the Earth, so how is it possible for us to make any calculations concerning Earth's orbit of the Sun?
We know from Newton that the interaction is greatest when the masses are large and the distances between the centres of the masses are small. As luck would have it as all other masses in the solar system are virtually insignificant compared to the Sun and all other stars are too far away to have a significant effect, which means that what is known as the 'two body' approximation, such as the Earth/Sun relationship, is possible.

There are, however, small deviations from the predicted motion that can be explained by the fact that the much smaller bodies are having very minor gravitational effects. These are known as gravitational perturbations and may be calculated by considering the positions of the known masses in the solar system and applying Newton's Laws of Motion and Universal Gravitation.

Once all known perturbations are calculated if there is still any deviation in the observed motion of the body then there can be only two explanations:

1. there are previously undetected masses in the solar system, or
2. the Law of Universal Gravitation is inaccurate and needs modifying

Newton's laws proved to be very accurate as demonstrated by its precise identification of unknown masses in the solar system due to surprising deviations of the known planets orbits.
The planet Neptune was discovered in this manner in 1846, after calculation the prediction of where the mass must be was exact and it was spotted soon after.

In 1930 calculations suggested that there must be a planet outside of Neptune, and the hunt was on for Pluto.

There was however an error in the calculation because it turned out the Pluto was not where they had calculated it to be. It is now thought that the error was in the measurement of the deviations of Uranus and Neptune, because the now known properties of Pluto could not account for those measurements.

The power of Newton's theory was evident for everyone to see and increasing detailed calculations were made to allow an even greater precision in the accounting of the orbits of planets.
The continual success in the application of Newton's law of gravitation led to the theory becoming so indisputable that any observed deviations were seen as evidence for previously undetected masses.

Then came the bombshell, there was an observed irregularity found in Mercury's orbit and so it was immediately assumed there must be another planet inside the orbit of Mercury. This planet was christened Vulcan, and so the race to spot the planet began. Various people claimed that they had spotted it but finally it was accepted that this planet was simply not there.
The obvious consequence was that for the first time there were doubts about the theory and by the late 19th and early 20th century it was realised that the theory that had undisputed for nearly 200 years was in actual fact not the complete picture.

However Albert Einstein was on hand to save the day.

Go onto The Modern Understanding of The Universe
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