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‼