Gravity

Newton treated gravity in 1666, not by why it works, but by how it works. Essentially he gave a set of rules by which the Solar System's motions could be calculated.

He simply treated gravity as a "force" which worked in conjunction with his other Laws in a totally self consistent way.

His contemporaries of Europe disliked this approach immensely as it had a strong sense of "magic" to it - it works over a distance according to the rule

F is proportional to M₁M₂/r²

With his very powerful mathematical ability, he was able to show that Kepler's highly successful Laws (of orbits of the planets) followed directly. Confirmation of the gravitational law was through the measurement of the local acceleration of something falling to Earth ( 9.8 ms^-2 in modern notation) to the acceleration of the Moon using the known distance to the Moon and the size of the Earth. The values obtained with the times measurements were "close enough".

The inverse square law hypothesis was "around" at Newton's time - suggested by quite a few including archrival ( enemy ) Robert Hooke - but these other people had not the ability to derive Kepler's Laws from the hypothesis. Newton did so and much more. Newton undoubtedly thought of it first ( about 1666) but in his secretive fashion hid it. When others mooted it about 1680, Newton - under pressure from Edward Halley ( of Comet fame ) dug it back up and developed the old work as the Principia.

The modern statement

F = GM₁M₂/r² G = 6.67 x 10^-11 units

was not realised until 100 years later when the eccentric and rich Lord Cavendish used a torsion pendulum and lead weights to measure G.

With this measurement, we can now measure the masses of all objects in the Solar System starting with the Earth simply from the observation that

for a small mass M₁ near a large mass M₂,

M₁ x acceleration due to gravity, g IS GM₁M₂/r²

thus acceleration due to gravity, g = GM₂/r²

This explains why all small objects near Earth's surface accelerate IDENTICALLY at 9.8 ms^-2 - Galileo's observation of about 1600. The acceleration ONLY depends on the mass of the Earth.

(It is interesting to realise that Captain James Cook's original purpose of going to the Pacific in the First Voyage was not to "discover Australia" but to measure the "transit of Venus" - which was really part of an English international attempt to measure the size of the Solar System.)

Kepler's Third Law, discovered after years of studying the data Tycho Brahe got together ( and Kepler stole on Brahe's death ), unifies the Solar system.

(Period of a Planet )² / ( average radius of orbit )³ = same number for all planets of the Solar System

It applies to all "small" satellites going around a common central mass.

We can derive it from Newton's Gravitational Law for the case of circular motion in the following fashion

M_sat v² /r = centripetal force on satellite = gravitational force on satellite = GM_sat.M_centre/r²

Use v = 2πr/T , substitute and twiddle and you get

T² / r³ = 4π²/GM_centre

This is Kepler's Law and it is clear why it works - it depends on the central mass. ( When the satellite does not have a small mass compared with the other body, the "balance point" between them becomes important and the formulae must be modified.)

Newton derived this law even for elliptical orbits.

Gravitational Fields

After Faraday introduced the visual conception of "Force Fields" for firstly magnets then electrostatics, it became useful to introduce the idea to gravity.

We introduce the idea of "something" - a field - at a place "caused by" a mass - or "belonging to" a mass.

When another mass is placed at this point, it reacts to the field already there.

Why?

• It is an attempt to create a visual framework for the force.
• It turns out to REALLY exist in the sense it can carry gravity waves.

In the first conception, a gravity field can be thought of as a diagram of arrows of force everywhere with length and direction - a VECTOR FIELD.

We decide to make the arrow = gravitational force on 1 kg at that place.

This is called the GRAVITATIONAL FIELD STRENGTH, "g" at that place.

g = Grav Force on 1 kg = Grav Force on m
m

= local acceleration due to gravity Unit of "g" usually Nkg^-1 instead of ms^-2

From the work earlier,
for a single mass M, M creates a field strength g = GM/r².

This use of arrows makes for a clumsy picture so "Field Lines" are often used as an easier conception.

When many masses are present, each contributes its own field to each place in space.

To find a TOTAL FIELD STRENGTH, we must, unfortunately, use good old vector algebra to find the value and direction.

The total field lines then look like below (somewhat) - please note lousy artistry by page maker!

Earth Satellite Orbits

Geosynchronous or Geostationary orbit

Orbits locked to Earth's period ie 24 hour period

These are used largely by telecommunication satellites and are responsible for resending communications ( especially TV) from ground to large area surfaces.Aussat is one such family serving Australia andthe near Pacific.They MUST be placed over the Equator and they MUST have a period of 24 hours. This combination locks their position relative to any parabollic antenna which then does not have to be altered. (Many Australians use the Aussat family directly -if they have pay TV and have a dish antenna in their garden. Equally most pubs have these facilities.)

If we use the Kepler formula, the radial distance of the from the Earth's centre is easily shown to be 42 300 km.

Why over the Equator? Here is the only place where the centre of the orbit corresponds with the weight vector necessary to provide the centripetal force.

Suppose someone is stupid enough to try to place a satellite with 24 hour period over Tasmania at 42⁰ South. Then its orbit would be an impossibility as the centre of the necessary circle does NOT coincide with plane of the only force available to create the orbit, gravity.

Polar Orbit

Used a lot by weather satellites and land/sea survey satellites. These orbits are often very low and short period so that as they hurtle over the the poles, the Earth's surface beneath is spinning at right angles. The satellites subsequently see "strips" from pole to equator to pole which change each orbit, covering the whole Earth.

Mixed Orbits

GPS ref1 ref2 and the failed Iridium satellite phone system require at any one time many satellites above the a given position or any position. Multiple satellites are used. GPS uses a minimum of 24. Iridium has about 66.