h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hello Antimather.

How about this for a re-write of the first section? I've only touched it up slightly, in order to check that I follow it. Does it still make sense to you? If so, I will try the same exercise on the second section.

Ahem...

Childhood illusions of sunrise and sunset are soon explained by the rotation of the Earth, but not so the illusion of a ‘pull’ of gravity to keep it in orbit. It is not a thing you can easily, or safely, experiment with.

Faced with the question of what gravity is, Newton settled for a pull as an extension of Aristotle's `inherent tendency' of matter. This theory was still current after two thousand years, and bit out of date even in Newton's day. Advances in science may take a little while to filter through, but this is absurd. Newton eventually changed his mind, persuaded by Hooke of Galileo's argument for an outside force (instead of an internal 'pull'), but the theory stuck.

However, even while popular fascination with the unknown keeps the medieval notion of a pull alive, there is a move afoot to get us up to date by reviving the external force. Before the issue becomes clouded in mathematical wrangling over the small print, the physical argument for dumping an energyless pull needs to be aired and, for the sake of science, the illusion dispelled.

The force of gravity is characterised by an inverse square law. In fact the inverse square is not unique to the force of gravity. It crops up too in Kepler’s laws for elliptical orbits in planetary motion round the Sun, but they say nothing about where this inverse square comes from.

Kepler's laws were a Godsend to Newton as their existence enabled him to avoid ackowledging his debt to Hooke for his contribution. However, this evasion also obscured the crucial role of density in describing weight, which never appeared in Newton's ‘law of gravity’. (The ‘rigorous scientific method’ requiring verification came about later.)

By ignoring the geometry of Hooke's version, Newton could adopt an analogy with a false notion of suction for his force, and scientists still naively quote it. A century and a half later a value was put on his co-efficient G, which incorporated two elements: first, Hooke's geometrical component for calculating relevant volume in combination with the inverse square (more about this later), and second, the other variable component for specific force, which was found by trial and error.

Howzat?!

Oh! I agree your edited version is more likely to promote discussion, compared to my efforts to dot all the `i's and cross the `t's, leaving little room for discussion of the content. It seems rather to have provoked criticism of the style, unsuited to the casaual reader.

One or two specific points, though, I think need to be spelt out clearly, one of which, despite my twice mentioning it, FM managed to ignore:
1. The inverse square ratio is a fairly common feature in Nature so a mere correlation has no significance. Newton found a correlation between rates of acceleration which he attributed to a common force on the strength of his estimates of mass and distance of the Moon, which he extended to the planets by reason of the same ratio appearing in Kepler's laws, which are mathematical and give no insight into physical cause. This has the signs of a fudge since the distance was half the diameter in Hooke's equation, but twice the mass of the Moon as later measured. It justified his claim for a pull from the centre, but how do you explain all that in simple terms!?
2. Talk of `dumping the energyless pull' has to be qualified by adding `...by every particle on every other' as they just don't have the energy to spare. One correspondent objected in a circular argument that since that is where the force came from they must have the energy. On the other hand it is probably not worth trying to argue with that.
3. The final point is in finding the value of G `by trial and error', which might suggest it has a precise value when at best it is an average of local measurements.

I am left wondering whether it is worth the effort of trying to rewrite my initial entry in full, or to take two or three bites at the cherry to cover the field piecemeal. I do however applaud your ability to convey the essential message in more accessible terms than mine, which may well result in a better understanding of the issue.

Oh! I agree your edited version is more likely to promote discussion, compared to my efforts to dot all the `i's and cross the `t's, leaving little room for discussion of the content. It seems rather to have provoked criticism of the style, unsuited to the casaual reader.

One or two specific points, though, I think need to be spelt out clearly, one of which, despite my twice mentioning it, FM managed to ignore:
The inverse square ratio is a fairly common feature in Nature so a mere correlation has no significance. Newton found a correlation between rates of acceleration which he attributed to a common force on the strength of his estimates of mass and distance of the Moon, which he extended to the planets by reason of the same ratio appearing in Kepler's laws, which are mathematical and give no insight into physical cause. This has the signs of a fudge since the distance was half the diameter in Hooke's equation, but twice the mass of the Moon as later measured. It justified his claim for a pull from the centre, but how do you explain all that in simple terms!?
Talk of `dumping the energyless pull' has to be qualified by adding `...by every particle on every other' as they just don't have the energy to spare. One correspondent objected in a circular argument that since that is where the force came from they must have the energy. On the other hand it is probably not worth trying to argue with that.
The final point is in finding the value of G `by trial and error', which might suggest it has a precise value when at best it is an average of local measurements.

I am left wondering whether it is worth the effort of trying to rewrite my initial entry in full, or to take two or three bites at the cherry to cover the field piecemeal. I do however applaud your facility for conveying the message in terms more accessible than mine, which may result in a clearer understanding of the issue.

No problem! I'm not trying to re-write the piece really, except for my own benefit. Your specific points in the posting just above are very helpful, and I expect the best place for them (at the moment) is just there, where they clarify the issue. I will move onto the next section asap, which is probably tomorrow now.

In the meantime, I'm confused about something you wrote above:

<>

Do you mean, <>

I'm just checking that I understood it. Your style is very condensed.

Mea culpa.

The fudge consisted of adopting the radius of revolution taken from Kepler's law describing only the path of a planet but consistent with the assumption of a pull to the centre, and balancing Hooke's inverse-square law for the force based on the diameter of the orbit, by using a figure for double the mass of the Moon.

Newton had used the tidal effect of the Moon to calculate its mass, but perversely chose the tidal range at a site in the Severn estuary where it is about double that on an exposed coast, as at, say, Newhaven. If he had been working from Cambridge, the nearest coast would have been at Cromer or Great Yarmouth where the range is only half that at Newhaven, but an estimate of lunar mass based on that would have been disastrous for his theory. It is this that suggests he chose the range to suit his theory of a pull, and to avoid giving support to Hooke's theory.

You have to admit it spoils the flow of the argument to have to break off into long explanations of the skullduggery that went on in devising the mathematical support for the theory of a pull. All the gory details can be found at various sites on the internet.

Me a Mexican cowboy!

It must have been getting a bit late in the evening when I wrote to try to clarify the points you made, and succeeded in obfuscating instead.

The ending of my first para should have been turned round to read:
... and using a figure for double the mass of the Moon to give the force needed by Hooke's inverse-square law based on diameter, if it were to match Earth's gravity.

There, that's better.

Ok, I think I've understood you. He took the radius of the orbit instead of the diameter, and to compensate he deliberately used a faulty value for the mass of the moon. I agree, this is a difficult topic to simplify, and it is very interesting.

Continuing... Let me know if this makes sense to you.

Newton handed down his law in 1689, eight years after Hooke had used the reciprocal of the total area of a sphere to define the volume of two slender cones meeting at the centre as 2/3d(squared). This can be recognised as a major element of the law. The mass within this volume depends on its density, and a specific force of around 1 Nm per ten megatonnes applied across it yields a differential which is the force of gravity as given by the law. It gives no clue, however, to the strength of the gravitational field providing the force.

Hooke had solved Galileo's riddle from a century earlier by recognising that bodies in free fall were weightless. Galileo had asked (or 'riddled') whether two weights tied together would fall faster than either on their own. He knew the answer from experiment, but it took far longer than he expected for the 'penny to drop' among the academics of his time and in generations to come. (This is one of many examples that show that logic is no substitute for experimental verification.)

Where Newton's theory differed from Hooke's that Hooke’s gravitational `flux' passed clean through a body shedding a little of its energy on the way to account for a small differential in opposing forces. Newton's force was much smaller, by many orders of magnitude, being totally absorbed in a body and dissipated from it in some unspecified way, thus simulating an illusory ‘pull’ of suction. Is that how people see it now? Crucially, Newton needed a weak force for his calculations to work, as a weak force allowed approximations to be made where convenient. He was understandably reluctant to see his elaborately worked theory being so simply undermined.

Hooke’s theory scores by making gravity the equal and opposite reaction to ‘anti-gravity’, the cause of cosmic expansion, which is, like weight, a function of density and distance.

Incidentally, there can be no danger, as some would have us believe, of catastrophic results should either gravity or anti-gravity predominate. The greater likelihood is that both fail progressively in extremis as a critical density is reached, that can no longer support the release of energy needed for acceleration and cohesion. Now that would be a theme for science fiction!

Notes:
1) is weight a function of distance? I learned weight = mass x 'g'.
2) what does 'cohesion' mean here?
3) an example of how 'sci-fi' might exploit the theme would be useful, not for the entry, but for me to understand what you mean!

Starting with your notes first:

1. I believe it's the legacy of using 'pounds' to measure weight, mass and force that has preserved the pre-Galilean conflation of weight and force with mass. Weight and 'g' are alternative effects of the force: weight strongest at rest on the surface, declining below it with distance from the centre; 'g' inversely proportional to the square of distance (from the centre) above it. Kepler's laws being quadratic equations have two solutions, the one Newton rejected as irrelevant would describe linear variation in weight.

2. The loss of cohesion among particles would allow them to scatter into an inert cloud of the smallest division of particles, which might not even be able to survive on their own; result: total oblivion.

3. I am no fan of sci-fi, but its authors seem to be for ever seeking more extreme conditions for their fantasies, and I should like to see what they could make of this, perhaps in a desperate search for an alternative universe, but with no means of getting there!

In the absence of a diagram, it might help to give the picture in the first para with a small expansion: ...the volume of two slender cones meeting apex to apex at a focal point...

My line of thinking in regard to the second para sees Galileo conducting a thought experiment with his mischievous riddle, the solution to which was an exercise in logic. His contemporaries failed to take the bait though and responded with the infantile: <>. When he finally succumbed at Pisa, twenty or more years later, to their clamour for experimental verification, some still found the experiment inconclusive: an example of what might be termed 'invincible ignorance'.

When it comes to suggestions for the way multiple functions of the force might operate, your experience in wave mechanics might offer some insight into the role of the decay of pi mesons, with the release of neutinos, to add weight to the argument. However this could be taking the project too far, and maybe that section ought to be omitted.

Re your no.1, yes, I now remember the explanation we received during an 'A' level physics lesson, where the teacher drew a graph to show how 'g' varied above and below the surface of the earth. The penny didn't drop then that 'g' was a function of distance... some things go in and others don't.

I understand the other points too. I like the idea of a diagram showing where the cones meet. What I would like to ask though, is how the reciprocal of the area of a sphere comes into working out the volume of the 'focal point' of these cones. I'm quite happy with mathematical symbols and calculus, so don't feel you have to simplify this!


Alas, I don't think my wave mechanics will help with analysing the decay of pi mesons, but we'll cross that bridge when we come to it.

Let's hope invincible ignorance can be staved off...

<<His contemporaries failed to take the bait though and responded with the infantile: <>. When he finally succumbed at Pisa, twenty or more years later, to their clamour for experimental verification, some still found the experiment inconclusive: an example of what might be termed 'invincible ignorance'.>>

I think this part-paragraph above desribes what happened very well - it is better than the equivalent section of the entry!

It was a very neat calculation of Hooke's to describe the relevant volume of a sphere so succinctly using the single dimension of diameter.

He started with the equilibrium state of a medium coming from all directions penetrating a sphere along its diameters equally all round its surface measured as d and expressed a sample area of 1 as a fraction of the total as the base of a cone to be multiplied by 1/3 of its height, d, to find its volume, two cones of half the height, r, having the same value. The average density of these would fix the mass determining the force on the sample area, and the simplest form of the equation for this is F=Gd, but d can also take the form M/V x d, or M/d, providing M is correctly identified.

The geometry of it turns out to be breathtakingly simple!

It occurred to me overnight - in between snatches of sleep - that an even simpler equation for gravity could be constructed avoiding the unwelcome intrusion of (how much practical use it would be is another matter).

For hundreds, maybe even thousands, of years before the introduction of an approximate ratio between diameter and circumference, the precise area of the surface of a sphere was found by multiplying the two together (the area of a circle was 1/4 of c x d).

Hooke's unit area would be the reciprocal of the total area, and the combined volume of pyramids, rather than cones, subtended at the centre, d/3 x 1/cd. Multiplying that by their density to arrive at their mass, and by a factor for specific force gives: G'/3c.

The present value of G is a composite factor including a geometrical ratio of 2/3, which confuses the picture unnecessarily, and a plain figure of newtons per kilogram for specific force could aid an understanding of what exactly happens.

Give me a bit of time and I may be able to reconcile a figure for it with density, volume (as measured in terms of circumference) and acceleration. Are you with me still?

It's an interesting idea, getting rid of 'pi'. But given that using 'pi' is really a way of eliminating the diameter 'd' or radius 'r' from the equations, isn't it a step backwards? You would get rid of 'pi' at the cost of reintroducing 'r'.

To put it another way, in order to calculate the area of a circle (or surface area of a sphere, or whatever.) you would need to take two measurements instead of just one: you would need to measure the radius *and* the circumference instead of the radius alone.

Well, it might be worth it if it leads to a clearer understanding of what is going on physically...

I'm just working through posting no.13 above...

I entirely agree that for practical purposes calculations in the standard form are much more useful, but my suggestion is for a clearer way of describing the action of the force of gravity, which can then be shown to reconcile with the inverse-square form, whether relating to radius or diameter.

Which is really where I came in, as the standard form obscures the mode of operation, leading to the proliferation of theories, particularly in the field of cosmology, which get wilder by the hour which it was my intention to draw attention to.

Yes, it's an interesting approach from the angle of the physics as opposed to the maths. BTW, I made a mistake in posting no.15 above: I meant to say that 'pi' was a way of eliminating the circumference from the equations (not 'r' or 'd').

Hallo Andrew S

I fear I may have caused you to overdose on caffeine in trying to get to the bottom of my foray into mathematics as the result of my nighttime musing. I must have lost the thread somewhat in the morning leading me to over-egg the pudding.

The root of the problem seems to lie in being mesmerised by the term 'inverse square'. It may be valid in Kepler's law, though I cannot recall how it arose there. In Hooke's law it is the reciprocal of area as a proportion of total area, which is, of course, one (squared). The volume within the diameters extending between opposite unit areas may then be expressed in the simple d/3, but the units are cubic in whatever scale. Total volume doesn't come into the picture so neither pi nor the circumference need appear! Overall mass is only relevant to keeping station relative to other massive bodies nearby by virtue of its inertia.

For recognising the factors in play it is only necessary to see acceleration as the result of a specific force in N/kg, intercepted by the mass within a defined volume in m according to its density in kg/m. The figures for Earth's gravity would be:

9.8 m/s = 4.2x10<-10> N/kg x 4.24x10 m x 5.5x10 kg/m.

From that point the terms can be re-arranged to suit any practical means for measuring them or for making use of them. The value of 4.2 given for the factor G is 6.6 x pi/2 which reconciles it with the standard inverse-square law in relation to free fall. Weight needs slightly different treatment.

It all adds up quite neatly.

Thank you! Working through the above...

I think I'm getting there!