Measuring a Year

Here’s a puzzle:

You find yourself on an unknown uninhabited planet. You look up the sky and notice a giant star that looks like our sun. It rises in one particular direction and sets in the opposite, so you figure that this planet (that you’re on) orbits around this particular star. Now it’s easy to define a day on this planet: it’s the time between two subsequent sunrises. The question is: how would you define a year on this planet? (Assuming, of course, that your survival on this planet is not an issue.)

Someone posted this question on Quora, which got me thinking about the definition of a year. One earth-year is defined as 365 days, or 12 moon cycles, and these measures are not arbitrary. They (approximately) represent the time it takes for the earth to complete one orbit around the sun. With that in mind, the answer to the above puzzle is rather straight-forward for a planet that has an axial tilt.

If you landed on a planet with an axial tilt then the sun’s trajectory in the sky will change over time and you can track it to measure one year. The following approach can be used to measure one complete orbit: Mark the point at the horizon where the sun rises the next morning. Every subsequent morning the sun will rise at a slightly different point. (The amount of shift depends on the degree of axial tilt and how fast the planet is orbiting its sun.) For a while, the sunrise points will shift away from your initial mark in one particular direction (either left or right). And after a certain period, like a pendulum, they will start moving back towards the initial mark. (This U-turn marks a solstice on this planet.) It will keep moving and bypass the initial mark, move in the opposite direction, and then take another U-turn to come back to the initial mark. The morning when sun rises again from the original reference point would mark the completion of one year on this planet. (Note that this description changes slightly if you started observing the sunrises exactly on the day of one of the two solstices on this planet.)

Here’s how the sun’s trajectory looks like from Earth (looking southward from the Northern hemisphere):


In our solar system, most planets have an axial tilt. Earth’s current axial tilt of 23.4° is responsible for seasons, rain, and consequently, the existence of life on Earth. The most curious among these are Venus and Uranus; they are the only planets that rotate clockwise (while looking at the solar system from the top). Venus has flipped almost completely and is upside down as compared to other planets. While Uranus, with an obliquity of 98°, would look like a tilted rolling ball — as opposed to all other planets that look like tilted spinning tops.

Obliquity of the Nine Planets

Back to the question of measuring one year, if you landed on a planet without an axial tilt, I have no idea how you would measure a year.

The United States of Moon

Have you ever wondered how big the Moon would look if it were placed on the ground here on Earth? How much area do you think it would cover? Well, wonder no more. Here’s the US map projected on the surface of the Moon:


I agree with boreboarder8, who created this demonstration, that this does make the Moon appear much smaller than what I had imagined:

It was difficult for me to fathom the size of the moon, thus inspiring the creation of this map. For me, this map puts the scale of the moon much smaller than I previously imagined. But it’s really interesting hearing how others (already grasping the size of the moon) now see the US as larger.

Keep in mind that this is a rough estimation. It’s one thing to try to project spherical earth on a flat map, and another to project a spherical US map on another (smaller) sphere.

[Source: Reddit]


Previous “Sense of Proportion” posts: I, II, III, IV, V, VI.

Blue Moon

Today is our last chance to view the “blue moon”… before it occurs again in 2015. And I thought, may be once in a blue moon, I should recycle one of my blog posts. So here’s it goes — I am reproducing (a modified version of) a short blog post from 2007:

The time it takes for Moon to complete its orbit around Earth is 29.5 days. (The Moon would have needed only about 27 days, but due to Earth’s own orbit around the Sun, it needs a few more hours to catch up.) Since the length of one moon cycle (moonth?!) is pretty close to our calendar month, the Moon usually completes 12 orbits in a year. Hence, we normally see only one full moon in each calendar month. But because the moon cycle is few days shorter than an average calendar month, once in every two-three years two full moons would fall within a single month. When that happens, the second full moon is known as the blue moon. The phrase ‘once in a blue moon’ owes its origin to the rarity and inconsistency (in terms of which calendar month it will occur) of such event. Tonight (August 31st) is our once-in-a-blue-moon opportunity to enjoy an “extra” full moon!

And while you’re at it, take a moment to observe the Sea of Tranquility where Neil Armstrong took the legendary ‘small step for [a] man’ in 1969.

PS: I think the concept of extra month – called adhik māsa – in the Hindu calendar is created to fix this mismatch between lunar months and solar year. Because the months in Hindu calendar are based on lunar phase, they won’t completely align with the solar year and approximately every three years it will have to play catch up, i.e. adhik māsa.

The Ultimate High

It suddenly struck me that that tiny pea, pretty and blue, was the Earth. I put up my thumb and shut one eye, and my thumb blotted out the planet Earth. I didn’t feel like a giant. I felt very, very small.

Those are the words of Neil Armstrong [1930–2012] reflecting on the awareness-altering, mind-boggling experience of observing the Earth from outer space. The effect that many astronauts experience after viewing such breathtaking sight is called the overview effect: “[It] is a cognitive shift in awareness reported by some astronauts and cosmonauts during spaceflight, often while viewing the Earth from orbit or from the lunar surface.” It has been described as “a complete engulfment by a profound sense of universal connectedness”.

I imagine this would be the ultimate high.

Upside Down

When Apollo 17 (the last manned lunar mission) took the first complete color picture of illuminated Earth in 1972, the orientation of that picture challenged the prevalent notions about what’s considered as ‘up’. The north direction is so commonly perceived as ‘up’ that we often don’t realize that the attribution of ‘up’ to north is completely arbitrary.

Here’s the original Blue Marble picture (see below): due to the position of the spacecraft, the South Pole came out to be at the top, and Africa appears to be upside down. This picture had to be inverted to comply with the traditional view of the Earth.

In space, up and down are arbitrary — which makes one wonder why the world map always shows north at the top. Why should the North have monopoly over ‘up’? Well, it turns out that this convention started with the first century Egyptian astronomer Ptolemy. Putting north at the top of the map probably made sense, because at that time the most popular places belonged to the Northern hemisphere and keeping them towards the top would have made the study of these places physically more convenient. Also, the fact that most of the landmass of the world lies in the Northern hemisphere makes this orientation more desirable

Ptolemy wouldn’t have realized back then that there are psychological and behavioral consequences of this tradition that presumably started off as a matter of convenience. We associate north with good and south with bad. We say “things went south” to express failure. An experiment demonstrated that the participants preferred to live in the north side of the city even when the map was turned upside down. [link] In Hinduism, the South is ruled (or guarded) by the Yama, the lord of death — while the guardian of the North direction is Kubera, the lord of wealth.


World maps with non-standard orientation are available (check out the Wikipedia page) These maps can be used as educational tools and also to illustrate the Northern hemisphere bias. My favorite among these is the Dymaxion map. This method projects the spherical world map onto the surface of a polyhedron. Check out the illustration below. These maps do not have a “right way up”, it has less distortion of the actual areas and when unfolded, they show all continents as “one island earth”. These maps can be unfolded in many ways to see different aspects of our planet. Pretty cool, eh?


On a lighter note, check out this comic that proposes a north pole that has tacos, so when someone wants delicious tacos they know where to go: north!

The comic is taken from here.

Into the Cosmic Ocean

Earlier this week, Voyager I, the spacecraft that was launched in 1977 by NASA, was reported to have reached the edge of our solar system. Currently, it’s around 11 billion miles away from the Sun and hurling towards the interstellar space, the Great Unknown, at 130,000 mph.

In the last 33 years, Voyager I visited Jupiter and Saturn, made important discoveries about the Jovian system, and made close encounter with Titan (one of Saturn’s moons). After completing its primary mission, Voyager I turned its camera to take the first ever family portrait of our solar system (which included the the image known as ‘pale blue dot’ that I wrote about last month). Now it has reached a faraway point in our solar system where the solar wind speed has dropped to zero. Weakly grasped by the Sun’s gravitational pulls, and flung by the gravitational fields of the outer planets of our solar system, the probe will continue its ambitious  journey towards the other worlds, traversing the vast and mostly empty regions between stars. It will be the first man-made object to leave our solar system.

On this spacecraft lies a message in the form of a phonograph record: the Golden Record. This record contains sounds and images of our planet, intended to communicate our story to extraterrestrials, if it ever encounters one. In that sense, Voyager I is like a bottle with a message, floating in an Ocean. Hoping – against all odds – to find a shore one day!

Among other things, the record contains “115 images and a variety of natural sounds, such as those made by surf, wind and thunder, birds, whales, and other animals. To this they added musical selections from different cultures and eras, and spoken greetings from Earth-people in fifty-five languages.” A baby’s cry, a Peruvian wedding song, whale sounds, Bach, Beethoven and Mozart are on that record!

We will be in touch with the probe until around 2025, but after that, it will float into the Great Unknown and sail on for eons without any communication link with us. In that dark, cold and calm interstellar space, there’s almost nothing to erode the spacecraft, so theoretically it will wander around for billions of years.

In such long time-span, human beings would have become extinct, and the Earth would have been burned to ashes by the Sun. But far away from all that, unaffected and untouched, Voyager I will continue to march on, relentlessly carrying the memories of the world that doesn’t exist any more.

Pale Blue Dot

I am currently reading Carl Sagan’s inspiring book: Pale Blue Dot. In the first chapter (titled ‘You Are Here’) he discusses this phenomenal picture taken in 1990 by a NASA spacecraft Voyager that was passing through the orbit of Neptune, 3.7 billion miles away from the Earth. So far away was the spacecraft that it took more than 5 hours for each pixel to reach us! From this distance, the Earth, a luminous dot sitting in the illusionary beam of orange light, occupies a mere 1/8th of a pixel in this picture (see below).

Looking at this picture and trying to comprehend the enormity of the cosmos in which we live is a humbling, terrifying and mind-blowing experience — all at the same time! It also reminds me of this beautiful Hindi poem from the title track of Bharat Ek Khoj (an old television serial directed by Shyam Benegal):

सृष्टि से पहले सत नहीं था
असत भी नहीं
अंतरिक्ष भी नहीं
आकाश भी नहीं था
छिपा था क्या, कहाँ
किसने ढका था
उस पल तो
अगम अतल जल भी कहां था

सृष्टि का कौन है कर्ता?
कर्ता है वह अकर्ता
ऊँचे आकाश में रहता
सदा अध्यक्ष बना रहता
वही सचमुच में जानता
या नहीं भी जानता
है किसी को नही पता
नही पता
नही है पता
नही है पता

Here’s Sagan’s own reflections on this humbling illustration:

Look again at that dot. That’s here. That’s home. That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every “superstar,” every “supreme leader,” every saint and sinner in the history of our species lived there — on a mote of dust suspended in a sunbeam.

The Earth is a very small stage in a vast cosmic arena. Think of the rivers of blood spilled by all those generals and emperors so that, in glory and triumph, they could become the momentary masters of a fraction of a dot. Think of the endless cruelties visited by the inhabitants of one corner of this pixel on the scarcely distinguishable inhabitants of some other corner, how frequent their misunderstandings, how eager they are to kill one another, how fervent their hatreds.

Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light. Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves. […]

It has been said that astronomy is a humbling and character-building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.

[Source: Wikipedia]

P.S. All episodes of Bharat Ek Khoj are available online here.

A Sense of Proportion

Consider this:

  • The light travels around 300,000 kilometers per second.
  • An actual beam of light would take only 1.25 seconds to reach the the Earth from the Moon.
  • From the Sun, which is more than 1.3 million kilometers away from the Earth, it would take about 8 minutes and 19 seconds.
  • What about the next closest star Alpha Centauri? It takes 4.3 years!
  • And the nearest large galaxy Andromeda? Two and a half million light years!

Can you comprehend how far Andromeda is from us? I can’t!

This reminds me of the Total Perspective Vortex, the most horrible torture device (from Douglas Adam’s hilarious book The Hitchhiker’s Guide to the Galaxy):

When you are put into the Vortex you are given just one momentary glimpse of the entire unimaginable infinity of creation, and somewhere in it a tiny little mark, a microscopic dot on a microscopic dot, which says, “You are here.”

This devise was invented by a philosopher Trin Tagula just to annoy his wife who constantly nagged him about not having a “sense of proportion”. And the philosopher’s defense was:

In an infinite universe, the one thing sentient life cannot afford to have is a sense of proportion.

To show her what ‘having a sense of proportion’ really means, Trin subjects his wife to the Vortex. Unfortunately, the shock (of realization) destroys her brain!

Such is the vastness of the infinite universe we live in!


Here’s a brilliant portrayal of the observable universe (click on the image below to enlarge) by my most favorite cartoonist Randal Munroe – the creator of XKCD:


Also read my earlier “A Sense of Scale” posts: 1, 2

Lunar Eclipse and Lunar Phases

Few weeks ago a close friend of mine (Dev) asked me what the difference between a lunar phase and lunar eclipse is. I thought I knew the answer. But as I embarked upon an explanation, it dawned on me that I didn’t quite understand the lunar motions very well. Alas, all those years I have been looking at the moon… without realizing (and hence, marveling at) how it moved in space.

What I did know was: as the Moon orbits around the Earth, it’s always the same side of the Moon that faces the Earth. This is because the time it takes for Moon to rotate once is identical to the time it takes to complete an orbit around the Earth — 27.3 days. Hence, the Moon has a ‘near side’ and a ‘far side’ with respect to the Earth. We never observe the ‘far side’ from the Earth.

Now, at any given time, half of the Moon is illuminated by the Sun and the other half is dark due to Moon’s own shadow. When the Sun and the Moon are on the opposite sides of the Earth, the entire ‘near side’ is illuminated and thus we can see the entire ‘near side’: this is called full moon (poonam). When the Moon is between the Earth and the Sun, the ‘far side’ is illuminated and the ‘near side’ is dark, so we can’t see the Moon at all: this is called new moon (amavasya). And, the rest of the time (between full moon and dark moon) we’re able to see only a portion of the ‘near side’ — the portion that’s illuminated by the Sun.

Had the Moon revolved around the Earth in the same plane as the plane of Earth’s revolution around the Sun, every new moon would have resulted in a solar eclipse (i.e. the Moon covers the Sun and its shadow falls on the Earth) and every full moon would have caused a lunar eclipse (i.e. the Earth’s shadow falls on the Moon). But that doesn’t happen every time, because the plane of Moon’s orbit and the plane of Earth’s orbit are not the same (tilted by 5 degrees).

Every once in while, the Earth, Moon and Sun get aligned. If that happens during full moon, the Earth’s shadow on the Moon creates a lunar eclipse; and if it happens during new moon, a solar eclipse occurs.

So, that’s the difference between the lunar eclipse and lunar phases. The former is caused by Earth’s shadow on the Moon, while the latter by Moon’s illumination by the Sun (and how much of it we can see from the Earth).

Here’s a 3-minutes video explaining the phases of the Moon.

Related posts: One in a Blue Moon, Happy Leap Day!

Happy Leap Day!

Tomorrow is 29th February, a leap day in the leap year of 2008.

Refreshing the logic for determination for leap year:

  • If a year is divisible by 4 but not by 100 –> Leap year
  • If a year is divisible by 400 –> Leap year

The Gregorian calendar has 365 days in a year. But a solar year has 365 plus a little less than 1/4th day (i.e. 365 days + little less than 6 hours) in a year. In four years, the Gregorian calendar would fall behind by almost 1 day. Thus, it compensates for that “loss” by adding an extra day (29th February) every four years. However, at this pace Gregorian calendar would move ahead of solar year because it’s adding more (1 full day) than it should (little less than 1 full day). So to compensate for that, it does not add 1 additional day after every 100 years (except for every 400 years!).

Even after doing all these adjustments, it doesn’t completely match up to the solar cycle. But it’s close enough, we will be off by 1 day in about 8000 years!

Why in February? March 21st is the common date for vernal equinox. This happens when the Sun is positioned directly in front of the Earth’s equator (No shadow at noon on the Equator). The vernal equinox marks the beginning of Spring in the Northern hemisphere. To make sure that the vernal equinox happens exactly on (or as close to as possible) March 21st, the leap day is added in the prior month (i.e. February).

The Hindu calendar, which follows the Lunar year/cycle, has similar mechanism to compensate for the “lost” days. The lunar year is around 10 days shorter (approximately 355 days). The Hindu Calendar adds an extra month (called Adhik Maas) after every 3 years to match with the Lunar year.

The Islamic calendar, on the other hand, leap years or months are not used as they are forbidden (I don’t know for what reason) by the Qur’an.

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