Thursday, 3 March 2016

Life on a Spherical, Rotating, Tilted Earth - Curved 2D Space - Fascinating Insights into Geometry, Climate, Weather, Seasons, Gravity ...


(please click on a slide to view it bigger image)


The earth is spherical (almost) but we spend most of our life on a piece of land that is flat locally.  Our common sense develops in a way that is appropriate for a flat earth.  Even the geometry that we learn in school is Euclidean – that is – flat space geometry.  We hardly ever think about what it means to have a spherical earth.  Start delving into this question and a whole lot of fascinating insights spring up and that is the journey we shall make in this post – hope you will enjoy it and give some thought as to what life might be like on a flat earth.  Very different for sure – it may not even be  possible for life to survive on a flat earth.

General Background:  Earth is the 3rd planet in the Solar System – on average about 150 million km from the Sun.  It completes one revolution round the Sun in about 365 days 
The earth’s axis is tilted by about 23.4o relative to the direction perpendicular to the plane of the earth’s orbit. Average speed of the earth as it revolves round the Sun in its 970 million km long orbital path is 30 km/second or 67,000 mph.  


The next slide shows the spinning earth – it completes one rotation in 24 hours.  Because earth is spherical and it spins, we have nights and days - light from the Sun falls as a parallel beam on the earth and illuminates the part facing the Sun.  The radius of the earth is 6375 km; at the equator its circumference is 40,700 km and the speed of spin is 1670 km/h (1000 mph).  Because of its spin, the earth bulged a little at the equator; but it is spherical to a very good approximation and we shall consider it to be so in our discussion.  
Earth also has gravity and attracts all objects towards its centre.  The slide also shows how a person standing upright at the north pole finds that somebody stands upside down at the south pole; and sees people standing at all sorts of funny angles in between.
 Points on the surface of the earth are located by their latitude and longitudinal angles.  The equatorial plane (the circle that divides the earth in two equal halves) defines the zero degree latitude.  

Latitude of a point on earth’s surface is the angle that the line joining the point to the centre of the earth makes with the equatorial plane.  Longitudes run from 180oW to 180oE.  Half circles passing through the centre of the earth and the two poles are perpendicular to the equatorial plane and define the longitudes.  0o longitude (the prime meridian) passes through Greenwich, London.  Longitudes (meridian) run through from 0 to 180o in the East and West directions from the prime meridian and they meet at the International Date Line in the Pacific.
Spherical Geometry:  In order to really appreciate how spherical earth is different from a flat earth, let us look at the difference between flat (Euclidean) geometry and spherical (non-Euclidean) geometry.  The following slides demonstrate this difference.  (click on the slide to see its bigger image in a new window)


 On a spherical surface, our experience can be counter-intuitive and puzzling.  I have two puzzles here – see if you can figure them out. The answers are at the end of this post.
Puzzle 1.   After lunch, Euclid leaves his tent and walks 1 km to the South.  He then turns and walks 1 km to the East.  After a short tea break, Euclid walks 1 km to the North and finds himself at his tent.  How is it possible?
Puzzle 2.  Riemann leaves his tent at 10 am and start walking due North for 2 km.  After enjoying his lunch, he again walks due North for 2 kmSurprisingly, he finds that he has returned to his tent.   Can you explain this?
How far is the Horizon?: The ground around us always feels flat because our field of view is generally quite restricted – distances we see are only a few km – and with the earth’s radius being 6375 km – a few km patch of land will appear flat.  Flat (Euclidean) geometry will be a good approximation for our field of view and will work well over such short distances.  Even then, the curvature of the earth restricts the distance we can see – how far is the horizon? – In ancient times, this is how philosophers had concluded that the earth is not flat.
Navigation:  When we travel from point A to point B, we think of travelling along a straight line which is the shortest path between any two points on a flat surface.  This is not so on a curved surface; the shortest path on a 2D curved surface, like our earth, is a geodesic or the arc of the great circle passing through points A and B. This is shown in the slide 

 Airplanes and ships that travel over large distances choose to travel along geodesics to minimise travel time and fuel costs. See slide for an example.
A Parable:  Life on a spherical surface can be very confusing!
Climate:  The earth receives most of its energy from the Sun in the form of electromagnetic waves and lot of it in the visible region.  Rays from the Sun fall on the earth and deliver much more energy in the equatorial regions; amount of energy received per unit area reduces as we travel towards the poles.  This is shown in the following two slides: 

 The result is that the land and waters near the equator warm more than near the poles and a process of heat transfer ensues.  Our climate, atmospheric and ocean currents are largely determined by this process of heat transfer. I have discussed these subjects in detail in my outreach talks suitable for the general community and may be accessed in the links:      Climate          Atmosphere             Ocean

Consequences of Earth’s Spin:      The earth and its atmosphere spin (rotate) about the earth’s axis completing one rotation every 24 hours. The rotation of the earth gives rise to some very interesting effects; we shall look at a few of these here.  For ease of explanation, we shall talk in terms of centrifugal and Coriolis forces which are ‘fictitious forces’ observable in the frame of reference of the rotating earth.  They give the correct results and are widely used in general discussions.  
Earth bulges at the equator:  Because of the earth’s rotation, every object on the earth’s surface is moving with the earth at a speed of 465 m/sec or 1000 miles per hour and feels an outward force – the centrifugal force – directed away from the centre of the earth.  Its magnitude is proportional to the square of the linear speed divided by the radius of the earth at that point.   The linear speed of rotation v and hence the magnitude of the force is a maximum at the equator, decreases with latitude and is zero at the poles.
The centrifugal force causes the earth at the equator to bulge out and the earth has a larger radius at the equator than at the poles  - it is an oblate spheroid – the extent of the bulge is small; the earth’s radius measured across the equatorial plane is 6378 km compared to its value at the poles of 6357 km.
You are less heavy at the equator: All objects feel an attractive force directed towards the centre of the earth - gravitational attraction.  Gravitational attraction is largest at the poles and you weigh more there than at the equator. Your weight at any point on the earth is the product of your mass (m) and the acceleration due to gravity (g) at that point. 
Centrifugal force acts in exactly opposite direction and reduces the amount of earth’s gravitational attraction at all places – hence your weight is reduced.  This effect is greatest at the equator and your weight is reduced by about 0.5% relative to what you will weigh at the poles.  For a more detailed analysis, see the slide (your weight = mass x g)

Coriolis Effect (Force):  Is a force experienced by moving objects in a spinning earth.  The force depends on the speed of the object, the direction it is moving and on the earth's rotaional (angular) velocity.  It  is mathematically difficult to explain but for the particular case of an object moving from the Equator to North Pole, Coriolis Effect may be understood as follows:
All points on the earth complete one full rotation through 360 degrees in 24 hours (constant angular velocity) but a point at the equator has to go farther in a day than a point in Glasgow at a latitude of 56oN.   Object at the equator have a bigger eastward velocity than objects at more northerly latitudes.  As the object travels north, it will be going east at a faster speed (easterly speed it had at the equator) than the ground beneath and its path will deflect east  - it will seem as if some mysterious force is acting on the object.
Objects travelling towards the equator will have a smaller eastward speed than the ground beneath them and they will seem to be forced west. 
The angular velocity of Earth is 360 degrees per day (one rotation per day), or 73 microradians/s. At wind speeds of 20 m/s (or 50 m/h), the Coriolis Effect generates an acceleration of about 0.003 m/s2.   Over an hour, this is a total deflection of about 18 km.
Coriolis effect is responsible for the way atmospheric air circulation and ocean currents are. Because of the uneven heating of the different regions of the earth by the Sun, heat transfer happens and air and water masses move from hot equatorial regions towards the polar regions.  Coriolis effect causes the flow of air to be modified as explained above and creates a much more complex air circulation patterns than would exist if the earth were not spinning.  This is explained in the following slides:






With Coriolis force the pattern looks as in the above slide.

Earths Tilt:  The earth's axis is tilted at 23.4 degrees with respect to the ecliptic plane (the plane of earth's orbit around the Sun) and maintains this tilt during its orbital journey.

At any time of the year, the northern and southern hemispheres receive energy from the Sun in different amounts.  This is responsible for the earth to have seasons.  I have discussed this topic in detail in my post and I refer you to that post.  
Another interesting aspect of the earth's tilt is that the tilt changes slowly with a period of 41000 years.  

As the tilt changes, the amount of sunlight falling on the earth changes over the course of its rotation.  This can act as a trigger for the onset of ice ages.  The cause of ice ages is a bigger topic and is discussed under Milankovich Cycles in my post.  

SOLUTION TO PUZZLES:



POST SCRIPT:  We have gone over some of the fascinating aspects of life on a rotating spherical earth and looked at how this creates a large variety of events both over short and long time scales.  
The earth is spherical was established historically by observations of length of shadows, finite horizon distances, shape of earth's shadow in lunar eclipses and now by actually photographing the earth from space.  Against this background, I find it amazing that there are people who claim that the earth is flat.  This is a matter that I think falls in the realm of mental aberration - I am not a psychologist  but when otherwise sensible humans who lead a normal life can still believe that the earth is flat then I start to wonder about how our minds are organised.  To me, this represents a bipolar mind - switching between logical and illogical states . Is bipolar mind an illness or human mind is naturally organised in a multipolar mode?  I can think of many aspects of life - religious belief is another example - where a multipolar mind can quite happily live with contradictions.  Human society throws many contradictions; is multipolar mind just a strategy to cope or is it genetic or is this a real disorder (illness)?  I would like to understand.  
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