# Interesting LaTeX Packages – Drawing the Solar System

Many times we need to quickly illustrate the solar system or the planets. Usually we use photos to illustrate the planets. But sometimes the photos can be an over kill. Also to draw the entire solar system is a typical and can be used for illustrative purposes. Though most of the illustrations of the solar system in the school and other text books are horribly out of scale, (with no indication that the figure is not to scale!). For example,  look at the illustration in the Class 6 Science Textbook from NCERT.

A good visualisation will always present a scale, and/or indicate whether the visualisation is to the scale or not. Though in the visualisation above the distances are given, they are not to scale.

Coming back to the topic of our post, a simple way to draw solar system diagram in latex is to use the PStricks package solarsystem. The package can create “Position of the visible planets, projected on the plane of the ecliptic” at a given time and date. This feature might be useful sometimes.

From the package manual:

As we can not represent all the planets in the real proportions, only Mercury, Venus, Earth and Mars are the proportions of the orbits and their relative sizes observed. Saturn and Jupiter are in the right direction, but obviously not at the right distance.
The orbits are shown in solid lines for the portion above the ecliptic and dashed for the portion located below.

The use of the command is very simple, just specify the date of observation with the following parameters, for example:
\SolarSystem[Day=31,Month=12,Year=2020,Hour=23,Minute=59,Second=59]
By default, if no parameter is specified, \SolarSystem gives the configuration day 0 hours to compile.

The resulting output for the above code:

The output also provides the longitude and latitude of the planets at the time given.

Another package that is useful to create free standing planets is the tikz-planets package which we will see next.

# Rotating Earth: the proofs or significance of Leon Foucault’s pendulum – Part 1

In an earlier post, we had discussed proofs of the round shape of the Earth. This included some ancient and some modern proofs. There was, in general, a consensus that the shape of the Earth was spherical and not flat and the proofs were given since the time of ancient Greeks. Only in the middle ages, there seems to have been some doubt regarding the shape of the Earth. But amongst the learned people, there was never a doubt about the shape of the Earth. Counter-intuitive it may seem when you look at the near horizon, it is not that counter-intuitive. We can find direct proofs about it by looking around and observing keenly.
But the rotation of Earth proved to be a more difficult beast to tame and is highly counter-intuitive. Your daily experience does not tell you the Earth is rotating, rather intuition tells you that it is fixed and stationary. Though the idea of a moving Earth is not new, the general acceptance of the idea took a very long time. And even almost 350 years after Copernicus’ heliocentric model was accepted, a direct proof of Earth’s rotation was lacking. And this absence of definitive proof was not due to a lack of trying. Some of the greatest minds in science, mathematics and astronomy worked on this problem since Copernicus but were unable to solve it. This included likes of Galileo, Newton, Descartes, and host of incredibly talented mathematicians since the scientific revolution. Until Leon Foucaultin the mid-1800s provided not one but two direct proofs of the rotation of the Earth. In this series of posts, we will see how this happened.
When we say the movement of the Earth, we also have to distinguish between two motions that it has: first its motion about its orbit around the Sun, and second its rotational motion about its own axis. So what possible observational proofs or direct evidence will allow us to detect the two motions? In this post, we will explore how our ideas regarding these two motions of the Earth evolved over time and what type of proofs were given for and against it.

Even more, there was a simple geometrical fact directly opposed to the Earth’s annual motion around the Sun and there was nothing that could directly prove its diurnal rotation. (Mikhailov, 1975)

Let us consider the two components of Earth’s motion. The first is the movement around the Sun along the orbit. The simplest proof for this component of Earth’s motion is from the parallax that we can observe for distant stars. Parallax is the relative change in position of objects when they are viewed from different locations. The simplest example of this can be seen with our own eyes.
Straighten your hand, and hold your thumb out. Observe the thumb with both the eyes open. You will see your thumb at a specific location with respect to the background objects. Now close your left eye, and look at how the position of the thumb has changed with respect to the background objects. Now open the right eye, and close the left one. What we will see is a shift in the background of the thumb. This shift is related by simple geometry to the distance between our eyes, called the baseline in astronomical parlance. Thus even a distance of the order of a few centimetres causes parallax, then if it is assumed that Earth is moving around the Sun, it should definitely cause an observable parallax in the fixed stars. And this was precisely one of the major roadblock

Earth moving around an orbit raised mechanical objections that seemed even more serious in later ages; and it raised a great astronomical difficulty immediately. If the Earth moves in a vast orbit, the pattern of fixed stars should show parallax changes during the year. (Rogers, 1960)

The history of cosmic theories … may without exaggeration be called a history of collective obsessions and controlled schizophrenias.
– Arthur Koestler, The Sleepwalkers

Though it is widely believed that Copernicus was the first to suggest a moving Earth, it is not the case. One of the earliest proponents of the rotating Earth was a Greek philosopher named Aristarchus. One of the books by Heath on Aristarchus is indeed titled Copernicus of Antiquity (Aristarchus of Samos). A longer version of the book is Aristarchus of Samos: The Ancient Copernicus. In his model of the cosmos, Aristarchus imagined the Sun at the centre and the Earth and other planets revolving around it. At the time it was proposed, it was not received well. There were philosophical and scientific reasons for rejecting the model.

First, let us look at the philosophical reasons. In ancient Greek cosmology, there was a clear and insurmountable distinction between the celestial and the terrestrial. The celestial order and bodies were believed to be perfect, as opposed to the imperfect terrestrial. After watching and recording the uninterrupted waltz of the sky over many millennia, it was believed that the heavens were unchangeable and perfect. The observations revealed that there are two types of “stars”. First the so-called “fixed stars” do not change their positions relative to each other. That is to say, their angular separation remains the same. They move together as a group across the sky. Imagination coupled with a group of stars led to the conceiving of constellations. Different civilizations imagined different heroes, animals, objects in the sky. They formed stories about the constellations. These became entwined with cultures and their myths.

The second type of stars did change their positions with respect to other “fixed stars”. That is to say, they changed their angular distances with “fixed stars”. These stars, the planets, came to be called as “wandering stars” as opposed to the “fixed stars”.

Ancient Greeks called these lights πλάνητες ἀστέρες (planētes asteres, “wandering stars”) or simply πλανῆται (planētai, “wanderers”),from which today’s word “planet” was derived.
Planet

So how does one make sense of these observations? For the fixed stars, the solution is simple and elegant. One observes the set of stars rising from the east and setting to the west. And this set of stars changes across the year (which can be evidenced by changing seasons around us). And this change was found to be cyclical. Year after year, with observations spanning centuries, we found that the stars seem to be embedded on inside of a sphere, and this sphere rotates at a constant speed. This “model” explains the observed phenomena of fixed stars very well.
The unchanging nature of this cyclical process observed, as opposed to the chaotic nature on Earth, perhaps led to the idea that celestial phenomena are perfect. Also, the religious notion of associating the heavens with gods, perhaps added to them being perfect. So, in the case of perfect unchanging heavens, the speeds of celestial bodies, as evidenced by observing the celestial sphere consisting of “fixed stars” was also to be constant. And since celestial objects were considered as perfect, the two geometrical objects that were regarded as perfect the sphere and the circle were included in the scheme of heavens. To explain the observation of motion of stars through the sky, their rising from the east and setting to the west, it was hypothesized that the stars are embedded on the inside of a sphere, and this sphere rotates at a constant speed. We being fixed on the Earth, observe this rotating sphere as the rising and setting of stars. This model of the world works perfectly and formed the template for explaining the “wandering stars” also.
These two ideas, namely celestial objects placed on a circle/sphere rotating with constant speed, formed the philosophical basis of Greek cosmology which would dominate the Western world for nearly two thousand years. And why would one consider the Earth to be stationary? This is perhaps because the idea is highly counter-intuitive. All our experience tells us that the Earth is stationary. The metaphors that we use like rock-solid refer to an idea of immovable and rigid Earth. Even speculating about movement of Earth, there is no need for something that is so obviously not there. But as the history of science shows us, most of the scientific ideas, with a few exceptions, are highly counter-intuitive. And that the Earth seems to move and rotate is one of the most counter-intuitive thing that we experience in nature.
The celestial observations were correlated with happenings on the Earth. One could, for example, predict seasons as per the rising of certain stars, as was done by ancient Egyptians. Tables containing continuous observations of stars and planets covering several centuries were created and maintained by the Babylonian astronomers. It was this wealth of astronomical data, continuously covering several centuries, that became available to the ancient Greek astronomers as a result of Alexander’s conquest of Persia. Having such a wealth of data led to the formation of better theories, but with the two constraints of circles/spheres and constant speeds mentioned above.
With this background, next, we will consider the progress in these ideas.
A stabilised image of the Milky Way as seen from a moving Earth.

# Can Stars Be Seen in Daylight?

The constellations that we saw at night half a year ago are now overhead in the daytime. Six months later they will again adorn the night sky. The sunlit atmosphere of the Earth screens them from the eye because the air particles-disperse the sun-rays more than the rays emitted by the stars. (The observer located on the top of a high mountain, with the densest and dustiest layers of- the atmosphere below, would see the brighter stars even in daytime. For instance, from the top of Mt. Ararat (5 km. high), first-magnitude stars are clearly distinguished at 2 o’clock in the afternoon; the sky is seen as having a dark blue colour.)
The following simple experiment will help explain why the stars disappear in daylight. Punch a few holes in one of the sides of a cardboard box, taking care, however, to make them resemble a familiar constellation. Having done so, glue a sheet of white paper on the outside. Place a light inside the box and take it into a dark room; lit from the inside; the holes, representing stars in the   night sky, are clearly seen. But, switch on a light in the room without extinguishing the light in the box and, lo, the artificial stars on our sheet of paper vanish without trace: “daylight” has extinguished them.
One often reads of stars being seen even in daylight from the bottom of deep mines and wells, of tall chimney-stacks and so on. Recently, however, this viewpoint, which had the backing of eminent names, was put to test and found wanting.  As a matter of   fact, none of the men who wrote on this subject, whether the Aristotle of antiquity or 19th-century Herschel, had ever bothered to observe the stars in these conditions. They quoted the testimony of a third person. But the unwisdom of relying on the testimony of
“eye-witnesses,” say in this particular field, is emphasized by the; following example. An article in an American magazine described daylight visibility of stars from the. bottom of a well as a fable. This was hotly contested by a farmer who claimed that he had seen Capella: and Algol in daytime from the floor of a  20-metre high silo. But when his claim was checked it was found that on the latitude of his farm neither of the stars was at zenith at the given date and, consequently could not have been seen from
the bottom of the silo.
Theoretically, there is no reason why a mine or a well should help in daylight observation of stars. We have already mentioned that the stars are not seen in daytime because sunlight extinguishes them. This holds also for the eye of the observer at the bottom of a mine. All that is subtracted in this case- is the light from the sides. All the particles in the layer of air above the surface of the mine continue to give off light and, consequently, bar the stars to vision.
What is of importance here is that the walls of the well protect the; eye from the bright sunlight; this, however, merely facilitates observation of the bright planets, but not the stars. The reason why stars are seen through the telescope in daylight is not because they are seen from “the bottom of a tube,” as many think, but because the refraction of light, by the lens or its reflection in the mirrors detracts from the brilliancy of the part of the sky under observation, and at the same time enhances the brilliancy of the stars (seen as points of light). We can see first-magnitude and even second-magnitude stars in daytime through a 7 cm. telescope. What has been said, however, does not hold true for either wells, mines, or chimneys.
The bright planets, say, Venus, Jupiter or Mars, in opposition, present a totally different picture. They shine far more brilliantly than the stars, and for this reason, given favourable conditions, can be seen in daylight.
From Astronomy for Entertainment – Yakov Perelman Pg: 135-137
Available here.