Flat Earth or round Earth?

 

We all know that the Earth is round. With the help of artificial satellites, it is quite easy now to take a picture of earth and ensue that it is round. But the situation was not like that over a century ago. Artificial satellites are recent discoveries, their history goes back to 66 years only. But we knew that earth is round much before that. How we came to know about it?

          In earlier days people didn’t know that the Earth is round. They believed that earth is a huge rectangular plane surrounded by oceans. Earlier navigators feared to go far in the ocean for the fear of falling off the edge of ocean. Where do they fall? That was known to them also, but they were afraid of falling into empty space.

          But some of the intellectuals began to think that earth must be spherical in shape, based on certain theoretical observations. Whenever someone observed the lunar eclipses, they observed that the shadow of earth falling on the moon was always round. It wouldn’t have been possible if the Earth was in some other shape. Even if the Earth was like a circular disc, the shadow would have been circular in only one direction when the plane of Earth was perpendicular between the sun and the moon. In any other position its shadow wouldn’t have been round.

          The next observation was made by the navigators who went into the sea. The distant ships coming towards the observer were showing the tops of their masts first. Slowly as they approached the observer, the lower part of the mast and finally the entire ship was becoming visible. This was possible only if the earth was curved. If the earth were flat, no matter however distant the ship is, the entire ship would have been visible at once. Now only the top of the mast is visible at a distance because the curvature of Earth hides the ship at a distance.

                If the Earth were flat, the entire ship would have been visible at once, no matter however distant it is. 

                As the Earth is round, the curvature of the planet hides the bottom of ship from our view when it is far away. 

          You can also easily experience the curvature of the earth by doing a simple experiment from wherever you are standing. Go near a tall building or a mountain. Stand at the base of the building during sunset. The sun slowly moves in the sky and disappears below the horizon. Once it disappears, climb to the top of the building and look towards it and you will be surprised that it is still visible. The reason for this is that the curvature of earth hides it from you when you were standing on the ground, but once you climb to the top of the building, you can still see it. But if the earth were flat, you wouldn’t have been able to see it any cost after it has set.

             If the Earth were flat, once the sun is set, it would not have been visible no matter what the elevation of your position is. 

                As the Earth were round, sun would still be visible if you climb to the top of the building or a mountain, even though it is not visible from the ground level. 

          The confirmation that earth was round was obtained by the famous round the globe expedition carried out by Ferdinand Magellan and his aides. A group of about 270 men along with Magellan left Spain in 1519 to circumnavigate the planet. Even though Magellan himself and majority of the others were killed on the way, about 18-19 survivors returned to their original place, proving that earth was a huge sphere indeed. This was truly a remarkable achievement at that moment. By navigating round the globe, they had thwarted the popular belief that existed for several thousand years that the Earth was flat.

          Despite all these, even still today many people believe that the earth is flat! In Facebook there is a page called flat earth society and there are several thousand followers for that page!

What is visual angle?

 

          What is visual angle? Visual angle is the angle that an object subtends in our eyes. In detail, it is the angle between the straight lines drawn from the two ends of an object to our eyes. If this angle is large, we say the object looks larger and if that angle is small, we say that the object is small. This depends on two things, i.e. the original size of the object and its distance from us. If two objects of same size are at different distances, the nearer object looks larger than the other one, because it subtends a larger angle in the eyes of the observer.

            

          There is another interesting possibility that we daily see in our sky. Two objects of different size may look similar in size because of the difference in their distance from us. The best example for this is our sun and the moon. Both look exactly the same size from Earth, not because they are of the same size, but because of the co-incidence of the ratio between their diameters and their distances from us. The sun’s diameter is about 400 times the diameter of the moon and it is exactly 400 times away from us compared to moon. So both of their visible disks subtend an angle of about half a degree in our eyes. Because of this coincidence, we can see total solar eclipse, which is one of the rarest celestial events that can never be seen from the surface of any other planet in our solar system. In their skies, either the image of moon is either too big and completely hides the image of sun behind them, or too small to cover the image of sun completely.

          There is a rough mathematical calculation that one can use to find the approximate visible angle of any planet. If a planet is at a distance of about 57 times their diameter, then that planet would subtend an angle of one degree in the eyes of observer. If its distance is 114 times its diameter, then it subtends half a degree and if its distance is 171 times the diameter, it subtends one third of a degree and so on. (This relationship is not as simple as arithmetic multiplication of this number, but we can easily ignore the error in case of any celestial body we see in our skies. If the objects are very close to the observer, this simple relationship no longer olds good and we need to get the help of complex trigonometric relationships to measure the visual angle, but none of the celestial bodies in our sky are so close to us. In case of planets whose distance is several thousand times their diameter, this error is only a miniscule part of decimal and that can easily be ignored as it makes no difference for our naked eyes.

          With the exception of sun, all other stars are quite far away from us, much further compared to even the farthest planets. Even the closest star to us, Proxima Centauri, is about 9000 times further away from our earth than the farthest planet, Neptune! You can easily understand why the planets appear to be having a visible disc outlined against the black background while the stars appear to be tiny points of light, no matter however powerful your telescope is.

Sidereal day and solar day. What is the difference?

 

          A day on our planet lasts 24 hours. That is what we have learnt from our school days. But is it the exact definition of a day? Have you ever wondered could it be exactly 24 hours or could there be few seconds, or even a few milliseconds of difference?

          To understand this, we have to know the definition of a day. A day is defined as the time our planet, or in fact any body in the solar system, takes to complete one complete rotation around its axis. But this rotation around the axis also has a vague definition. As we all know, motion is relative and while talking about the rotation of earth also, you can’t explain it without the reference of something outside Earth. While choosing any body outside the Earth, the first and foremost requirement is that the Earth has to be stationery with respect to that outside body. We usually define a day with respect to the sun, because our natural understanding of the day is the lapse of time between two successive sunrises or sunsets. But while taking sun as a reference for measuring the rotation of Earth, there is one fundamental problem. Let’s see what that problem is.

          Earth has two types of motion. One is the rotation around its own axis and the other is its annual journey around the sun. Now, let us consider the definition of a day as below: The time lapse between two successive sunrises, or we can tell it more precisely as the time that the sun takes to return to the exact position in the sky as it was before. It could either be the point at horizon, if you take sunrise or sunset as the moment, or it could be the zenith if you consider mid noon as the reference time, or it could be any other arbitrary point in the sky. But by that time, Earth also moves in its path around the sun. In a day, Earth travels about 1/365.25 of its path around the sun. Hence by that time, Earth would have moved slightly ahead in its path and hence sun wouldn’t be exactly in the same position. To compensate that, the Earth has to move a little bit ahead.

          Now let us see what the day of earth means with respect to some distant star, relative to which earth can be regarded as stationery. Stars are so distant from us that the daily motion of Earth around the sun, which involves a whopping 1.6 million miles which is by no way negligible, is still insignificant compared to their enormous distance. Hence taking stars as the yardstick for measuring the length of a day is a quite appropriate measure. It takes 23 hours, 56 minutes and 4.09 seconds for our planet to complete one rotation around its axis with respect to some distant star. In other words, if you mark a star which is directly overhead today and you measure the time it takes for that star to be seen exactly in the same position next time, that time is 23 hours, 56 minutes and 4.09 seconds. It is called a sidereal day. But as you can see it clearly, by that time Earth would have travelled a bit ahead in its path around the sun and hence the sun is still that much behind from its position of the previous day at that exact moment. The additional time it takes for the sun to reach that exact position is another 3 minutes and 55.91 seconds. That makes it exactly 24 hours. (In fact it is 24 hours and a few millionths of a second, but we can easily ignore that and consider it as exactly 24 hours for the sake of simplicity). This is called a mean solar day.

           In these three pictures, you can understand it very well. In the first picture the sun is directly overhead to the man. In the second picture, Earth has completed one complete rotation, but the sun is still not overhead. In the third picture, as the Earth rotates a bit more on its axis, the sun comes overhead. (The pictures here are not to scale and even Earth’s movement is also not to scale, as it is impossible to depict the celestial bodies and their movements exactly to the scale on paper. If you really try to do that, it would be totally incomprehensible).

          In case of any planet further than earth for which the planet completes several hundreds or even thousands of rotations for one revolution around the sun, the difference between sidereal day and a solar day is negligible. But in case of our first two planets, the case is quite different. Both these planets rotate very slowly on their axis and their sidereal day is an appreciable fraction of their year. Let’s see what the consequence of this is.

          Now consider the case of Mercury. This planet rotates from west to east and hence the sun travels from east to west in its sky. This means Mercury completes a 360 degree rotation around on its axis with respect to some distant star. The time that Mercury takes to complete this 360 degree is 58.7 days. Mercury also takes 88 days to complete one revolution around the sun. This means that the sun completes one rotation around Mercury in 88 days. This needs a bit of tricky mathematical calculations to understand it properly.

          The picture below shows how Mercury rotates on its axis and how it revolves around the sun. The consequence of these two motions is that the sun moves in two opposite directions in the sky of Mercury. But remember these two motions are not alternative, but simultaneous. So there will be a resultant motion of the sun in the sky of Mercury, and the direction of that motion is in the direction of which of the motion is faster. Let us explain it in detail.

          Firstly as the planet moves on its axis from west to east, the sun moves 360/58.7=6.13 degrees from east to west in the sky of the planet. Secondly, as the planet moves around the sun, the sun moves 360/88=4.09 degrees in the sky of the planet in the opposite direction. As a result of this, the sun moves 6.13-4.09=2.04 degrees from east to west in the sky of mercury, which means it takes 360/2.04=176 days for the sun to return exactly to the same position in the sky of Mercury! In other words, a day on Mercury lasts 176 earth days, or almost six months! Isn’t it wonderful?

 

Th       These two pictures elaborate Mercury’s situation very well. As we can see, in the first picture the sun is directly overhead the man who is standing on Mercury. In the second picture, you can see the position of Mercury has changed as it has moved in its orbit around the sun. By the time it completes one complete rotation on its axis, it would have completed two thirds of its revolution around the sun and the sun is to be seen nowhere in its sky.

The case of Venus is also a bit different. Its sidereal day is the longest for any planet in our solar system. It lasts 244.3 Earth days, or about eight months! But the year on Venus lasts 224.7 days. In other words, its sidereal day is longer than its year! But here the path of sun in its sky is quite different, because its rotation around the axis is quite opposite to that of Mercury or any other planet for that matter. So the corresponding movement of sun in the sky of Venus is the same for both its daily as well as annual motion. These two motions give sun 360/244.3=1.47 degree and 360/224.7=1.60 degrees of movement in the sky, both are in the same direction. So their motion add up to give a motion of 3.07 degrees for 24 hours (that is one earth day) from west to east in the sky of Venus. So to complete a 360 degree revolution in the sky of Venus, the sun takes 360/3.07=117 days.

          It is really wonderful, isn’t it? If you find these calculations, you can get some more funny results by using these statistics. What would be the length of a solar day on mercury if it were to rotate as slow as Venus, which is one rotation for every 224.7 Earth days? Try to solve this problem for yourself and you will find that the mathematics involving astrophysics is really interesting and amazing‼