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Posts Tagged ‘Kepler’s 2nd law’

How long is a day? It seems like a stupid question. As everyone knows, there are 24 hours in a day. The Earth rotates on its axis once every 24 hours. Or does it?

The difference between a ‘solar day’ and a ‘sidereal day’

In fact, there is a slight difference between how long the Earth actually takes to rotate 360^{\circ} (the ‘sidereal day’, the day as measured by the motion of stars in the sky) and how long it takes for the Sun to appear to go once around the Earth (the ‘solar day’). This is because, during the course of a day, the Earth has moved a little bit in its orbit about the Sun, and so the Earth has to rotate a little bit more than 360^{\circ} to bring the Sun back over the local meridian. We measure our day by the solar day, as otherwise the time of local noon would drift away from midday more and more, which we clearly do not want. (You may notice that this is related to the difference between a sidereal month and a synodic month, which I discussed here.)



The difference between a solar day and a sidereal day, which comes about because of the Earth's motion about the Sun.

The difference between a solar day and a sidereal day, which comes about because of the Earth’s motion about the Sun.



Kepler’s 2nd law

This difference is easy to measure, with a sidereal day being, on average, 4 minutes shorter than a solar day. This means that stars rise about 4 minutes earlier from day to day, or over the course of a month about 2 hours earlier. But, this 4 minute difference is not constant. It changes because the Earth is orbiting the Sun in an ellipse, not a circle. This means that the Earth’s speed in orbit changes, it travels faster when it is closer to the Sun (in January), and slower when it is further from the Sun (in July). This fact, which was first noticed by Kepler, is now known as Kepler’s 2nd law of planetary motion. It is illustrated below.



Kepler's 2nd law of planetary motion states that a planet will sweep out an equal area in equal time, so that in the same period of e.g. 1 month the three areas A will be equal. This means that a planet travels quicker when it is near the Sun, and slower when it is further away.

Kepler’s 2nd law of planetary motion states that a planet will sweep out an equal area in equal time, so that in the same period of e.g. 1 month, the three areas A will be equal. This means that a planet travels quicker when it is near the Sun, and slower when it is further away.



When the Earth is travelling quicker it has to rotate a little bit more to complete a solar day, and when it is travelling slower it has to rotate a little bit less. So, the length of the actual solar day changes in the course of a year, but in a cyclical fashion (this is known as the equation of time, something I will explain more in a future blog). The equation of time is the reason for the East-West motion of the Sun as shown in the analemma, which I discuss here.

Because of these changes in the difference between a sidereal day and a solar day at any given time of the year, we define something called the mean solar day, and it is the mean solar day which should be 24 hours, or 86,400 seconds long. But, it isn’t! In a blog next week, I will explain how the Earth’s period of rotation is not consistent, and this is why we had a leap second at midnight on the 30th of June this year.

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A few weeks ago I showed a photograph an an Analemma. As the Analemma in the photograph was vertical, I explained that it must have been taken at midday. Here is the photograph again, just to remind you.


A Solar analemma. Because it is vertical, this analemma was taken at midday.


In part 1 of my series on the Analemma, I also explained how the North-South motion of the Sun in the photograph was due to the changing elevation of the Sun at midday. This is, of course, due to the tilt of the Earth’s axis, as explained in this video below.



But, what about the East-West (left-right) motion? What is this due to?

It turns out that the East-West motion is due to two effects. One is the same inclination of the Earth’s axis in its orbit around the Sun which produces the North-South variation in the Sun’s elevation at different times of the year. I will explain how this affects the East-West position of the Sun at midday in part 3 of this blog.

But, the second effect is unrelated to this, it has to do with the details of the Earth’s path around the Sun.

The Heliocentric Universe

When Copernicus suggesed in 1547 that the Earth and the other planets went around the Sun, he argued that they would do so in perfect circles. He was, in fact, not the first to suggest that the Sun and not the Earth was at the centre of things. Aristarchus had suggested the same thing in the 3rd Century B.C., but his work had been largely ignored in preference to the teachings of Plato and Aristotle, who firmly held that the Earth was the natural centre of all things.

Building on Aristotle’s Geocentric Universe model, the Greek-Roman astronomer Ptolemy developed a sophisticated model of the Sun, Moon, planets and stars orbiting the Earth. This model was incredibly successful, and able to predict the positions of the celestial objects to a good degree of accuracy for some 1500 years.

In Copernicus’ 16th Century model, the planets orbited the Sun in circles. In the latter part of the 16th Century, the greatest observational astronomer was a Danish man, Tycho Brahe. Brahe had his own Observatory and research institute, Uraniborg, on the Danish (now Swedish) island of Hven, with a Royal patronage to fund his observing programme.


Tycho Brahe (1546-1601)


Brahe produced the most accurate observations of the planetary positions, and he found he got better agreement with Ptolemy’s geocentric model than he did with Copernicus’ heliocentric one. Towards the end of his career, a young mathematician by the name of Johannes Kepler came to work with him.

A very particular kind of curve

After Brahe’s death, Kepler set about seeing whether he could get a heliocentric model to agree with the observations. After over a decade of trial and error, he eventually found that, if he allowed the planets to move in ellipses rather than perfect circles, that very good agreement could be obtained. As Richard Feynman once said, “an ellipse is a very particular kind of curve”. To be more precise, it is the curve obtained when one passes a string about two drawing pins (“thumb tacks” as Americans call them) and draws the ensuing locus of points.


How to draw an ellipse


Kepler’s 2nd law states that a planet will “sweep out equal areas in equal times” in its orbit. What this means is that it will speed up when near the Sun (perihelion) and slow down when further from the Sun (aphelion).


Part of the East-West motion of the Sun in the Analemma is due to Kepler’s 2nd law, the fact that the Earth changes its speed of orbit as it goes around the Sun. The Earth moves quicker when it is closer to the Sun (perihelion), and slower when it is further from the Sun (aphelion). Kepler did not know why this happens, but it is a natural consequence of Newton’s law of gravity.

A mean Solar day

How do we measure the length of the day? It seems like a simple question. Surely, the answer is that it is the time it takes for the Earth to turn once on its axis. This is, in fact, the wrong answer. The time it takes for the Earth turn once on its axis is the sidereal day, and this is not how we measure our day. Why? The diagram below explains it.


The Earth has to turn a little bit extra for the Sun to cross the local meridian. We call this the Solar day. The sidereal day is the time for the Earth to rotate 360 degrees.


Because the Earth moves about the Sun in its orbit, the Earth has to rotate a little bit extra for the Sun to cross the local meridian on two successive occasions. This is how we define 24 hours, the solar day. But, there is an additional complication; because the Earth’s speed of orbit changes, the extra angle the Earth needs to turn to bring the Sun back over the meridian also changes. As the Earth approaches perihelion (closer to the Sun), it speeds up and so moves through a larger angle each 24 hours than when it is further from the Sun.

Of course, we cannot keep changing the length of the day, we fix it at 24 hours, which is what we call the mean solar day. This is the midday our watches will show. But, near perihelion, the Earth has to turn that little bit extra as I’ve explained, so when our watches say it is midday the Sun will still be to the East of the local meridian. So, if we were taking a photograph of where the Sun was at midday as shown by our watches, the Sun would have shifted eastwards of the mid-point.

The opposite effect happens near aphelion, when the Earth is moving more slowly in its orbit. This time, the Earth moves slightly less in 24 hours than it does in other parts of its orbit, and so the Earth does not have to turn through such a large angle to bring the Sun back over the local meridian. When our watch says midday, the Sun will have gone past the local meridian, and be to the West of it.

Hopefully, this now explains why there should be some East-West motion in the Solar Analemma. However, it turns out that there is a second effect which also causes an East-West motion, the tilt of the Earth’s axis in its orbit. I will explain this component and how it affects things in part 3 of this series.

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