Tides in Free-fall

Why do we have two tides each day?

It was always a race against time.  Frantically digging on the beach to build up the sand wall that we were making ; convinced that this time we would create a barrier high enough and strong enough to withstand the incoming tide.
Of course we never did defeat the tide; the waves always washed away the sandy walls until the beach was made smooth again and all traces of our work was gone. But there was always tomorrow……
As a seven year old on holiday in West Scotland in the 1960s, we virtually lived on the seashore; a wonderful mix of rocky outcrops, small cliffs with sandy coves in between. Unknowingly, I was absorbing the first principles of civil engineering and fluid mechanics at first hand. My construction partner (my six year old cousin) and I gradually noticed that the tide became half an hour later each day; what started off on the first day of the holiday as high tide at 4pm had become 7:30pm by the start of the second week. Thwarted in our daily beach-works by the call into supper, we were delighted to find, when arriving at the beach early the next morning, that there was a second tide at 8am.
This was a big surprise- but with the adaptability of children we just accepted it and moved our battle with the Atlantic Ocean to the mornings.

Much later I started to wonder about this business of the tides. Why do the tides exist? What drives them relentlessly every day?  Why are they later each day? And most perplexing of all – why are there two tides every day?

In the late Sixteenth Century, Galileo started down the right path, musing that they were caused by the influence of heavenly bodies ; but incorrectly surmising that they were caused primarily by the sun. Later, Kepler developed a theory based on the moon’s influence; unfortunately this could only explain one tide per day.
It was Isaac Newton who, amongst all his amazing creative outputs, derived the correct model.  At the heart of it is the application of his theory of gravity.
Over the years, I’ve read many attempts to clearly explain Newton’s model – and I have to say that found them all confusing and I never felt I really ‘got-it’.  Then recently I was reading Roger Penrose’s ‘The Emperor’s New Mind’ and came across the clearest explanation yet – this time I really got it! Penrose’s explanation is part of his introduction to General Relativity – specifically concerning ‘the principle of equivalence’ (1)
So let’s see if I can relay this explanation simply and in a way that enables you to understand this beautiful model……

Step One

tides2Imagine a cloud of particles high up above the earth. Let’s make it tangible – let’s say it’s millions of sand grains and we’ve initially placed them 200 kilometres up in space so that they are equally distributed in a loose sphere shape, say 10m in diameter. Up at this height it is virtually a vacuum – so we can ignore any effects of air resistance.

Now these particles are not in orbit – we have managed to place them there with zero orbital velocity.
So what will happen? Of course they will start to fall towards the earth, pulled by the earth’s gravity. They will be in ‘free-fall’.
At this point we need a little maths. Newtons law of gravity says that the force acting on each sand grain particle is the inversely proportional to the square of the distance of the particle from the earth (actually from the centre of gravity of the earth – roughly the centre of the earth). We know this instinctively; although the sun is 330,000 times heavier than the earth, footballs don’t fly off to the sun when kicked because the Sun is much further away from the ball than is the centre of the earth.
equation

Newton also showed that the acceleration of a body is equal to the force acting upon it divided by its mass ( from  F = m  x  a).
So…. this means that the sand grains in our sphere closest to the earth ( at the bottom) will accelerate slightly faster than those 10m further away ( at the top of the sphere).  As the sphere of sand grains falls down toward the earth then it will gradually become extended into an ovaloid shape – like a rugby ball.
OK so far?

Step Two

Next we’re going to scale things up a bit. Let’s make the sand sphere bigger – say 1km in diameter. We’ve also inserted a solid sphere of rock , 0.95km in diameter inside it – so effectively we have a thin layer of loose sand approximately 0.025 km (25m) thick distributed around the rocky sphere.
tides3As this falls to earth, in free fall, it will take up the same shape as before, with the sand grains furthest from earth lagging behind those closest to the earth. Note that the rocky sphere moves as one object (it’s rigid) and its acceleration is the same as a sand grain would have if at the centre of the rocky sphere.
So hence we get a situation where we have a sphere covered in sand, with bulges in the thickness of the sand at the pints closest to and furthest away from the earth.
If you’re following this you may have got the gist of this and jumped ahead in your thoughts; replacing the sand by water and starting the rocky sphere rotating -creating two tides, each on opposite the sphere – but just hang on for one more step).

Step Three

tides4The next step is to give our rocky sphere some orbital velocity. If we give it enough orbital speed, it will become in orbit around the earth. Remember, we did nothing to stop it falling towards the earth, we just also gave it some ‘sideways’ velocity. That’s what objects in orbit, like the communications satellites, are doing; they are falling ‘around’ the earth. Their orbital velocity means that they continually miss the earth by the same amount. Too much orbital velocity and they’ll miss the earth by too much and drift off into space.

Step Four

Last step! Let’s now scale up that 0.95 km rock body to one, say, 12,750km in diameter ( i.e. the size of the earth) and replace the sand by water.  Let’s also replace our original earth by the moon. And let’s place the earth and moon 400,000km apart, orbiting each other once every 28 days.
Orbiting each other? Surely the moon orbits the earth! Actually no – they both orbit a point that is actually inside the earth about a quarter of the way from the surface to the centre.

So now we have the two objects orbiting around a common centre; both in continuous free-fall towards that point. The seas around the earth are distributed in the ovaloid shape as previously described with one bulge away on the opposite side to the moon and one on the side closest to the moon. And everything – seas, earth, moon you and me – in continuous free fall to the centre of orbit!

tides1

 References :

The Emperor’s New Mind : Roger Penrose Chapter 5 – the Classical world pp 261-266

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