## Make your own sky map

We live in the age of photo-realism, be it for maps of the world or of the sky – it is a realism that not long ago was beyond the reach of even our vision. That is why even though we have such a photo-realistic sky map just a click away we still have a deep nostalgia for the days before it. This is part of the reason we finally bit the bullet and wrote our own latest iteration of a good quality sky map. There is another reason which we shall come to later but for now we shall wander as part of recording our own memoirs into its biographical precedence. In our early childhood we were introduced to astronomy by our father from book which impressed upon us the great size of the sun relative to earth or even Jupiter and also introduced us to some constellations wherein shone stars that made the sun look small. To our luck the northern horizon was almost completely cleared by the demolition of a building leaving a space that was not built up for ten years due to the typical ways of the municipality in those days. Even before we had the book, we could see Ursa Major from our balcony and we had also internalized the unmistakable Pleiades and Orion. The said book had few more famous constellations like Leo which we learnt thereafter. Those days of halcyon youth were marked by relatively dark skies in our city with the occasional boon of complete power-cuts which brought the even greater darkness that we desired. It was then that it became apparent that a vast starry realm lay above above with more stars than our book ever showed. Beholding that, the idea took root in us to draw our own sky map marking all the stars based purely on visual observation. This endeavor went or for about an year or so after which for reasons unclear to us our interest in astronomy faded and the said book vanished.

But then as we have alluded to before, almost 4 years later our astronomical interests resurfaced with a vengeance and this was in the year before the apparition of the Halley’s comet. Then we obtained another book with slightly better maps and also embarked on making our own telescopes. This showed us that the maps in the book were woefully inadequate and the quest for new ones began. It was then that we found a kindred spirit a schoolmate from the 3rd varṇa from the Lāṭa-Ānarta country who was possessed with all the correct lakṣaṇa-s of his varṇa. He, with his mahādhana, procured a set of deep sky maps of Japanese provenance. To our eyes these charts had a great beauty in addition to their accuracy. Their symbols made a deep impression on our young minds distinguishing variable stars, planetary nebulae, globular clusters, open clusters, galaxies and other nebulosities with their own symbols. Armed with these we were among the first lay people in our city who caught sight of the Halley’s comet when it came out from behind the Sun. We wrote about this to our local news paper and our names and photos appeared in it. But the summer of the apparition of the comet had much more in store for us. Those maps led us to many observations of deep sky objects and variable stars as also in the subsequent year the great supernova of the Tarantula nebula. We caught sight of that on a single day close to the southern horizon when the weather cooperated. It was in this phase that the urge to sky maps surfaced again for our precious photocopies of the Japanese maps did not cover the whole sky. Our parents took us to an astronomer in a nearby city and we showed him our maps of drawn for the observations of the great supernova. He kindly commended the accuracy of our maps and showed those from a well-known American collection of maps for comparison. He wondered how we determined the relative positions of stars and we remarked that we had a fairly good memory of the sky and also used the time-honored instrument the yaṣṭi-yantra for getting various relative positions.

But a parallel thread ran in our lives. As we were studying the śruti, it was becoming increasingly clear that, much to our Yajurveda teacher’s chagrin, we lacked the conviction in the central pillar of mīmāṃsā – the doctrine of apauruṣeyatva. Instead we became very much intrigued by the question of the age of śruti or the various layers therein. There were numerous obvious astronomical references that were catching our ears, leave alone the less-obvious ones. It was then that we studied the great leader of the Hindus Lokamanya Tilak, the poorly known Kameshvara Ayyar and Shankar Dixit all of whom had used similar techniques to arrive a plausible date for the composition of various vaidika texts. This idea fascinated us and colluded with our star-chart making urge. One of the things that was sorely lacking was a nakṣatra-based map which divided the ecliptic into the nakṣatra-s and showed the sky using a nakṣatra-based ecliptic grid. For this purpose we wrote a primitive program which interfaced either with our paper star charts or in the last 21 years with an existing program to visualize the sky as usual.

Recently, an interlocutor on the internet expressed the wish of incorporating a nakṣatra grid in the open-source program sky map software Stellarium (something we have been using for several years for our regular observing needs) and asked some questions in that regard. It was in this context that our desire to make our own sky map resurfaced and we finally implemented something which recaptitulates the aesthetic of those old Japanese maps. Of course today we have loads of raw material available a click away so the task was nowhere as complex as it might have been before. We outline that below so that anyone with some computer skills can make their own in a straight-forward way:

1) We conceived our map as fitting in a 33×24 sq.inch (83.82 x 60.96 sq.cm) rectangle and covering the whole sky. The idea was to produce it in png, pdf and svg format for use on the web or for printing in large format.

2) We implemented it in the R because it was one of the easiest to use, open-source languages supporting vector operations. It is good for generating graphical outputs such as this and has a preexisting library for computing projections (see below). The code for plotting was written in base R and special fonts were imported and embedded using the R extrafont library. The conversion of coordinates from ecliptic to equatorial were done using a converter function that we wrote based on code written by Arnab Chakraborty in the AstrolibR package.

3) After some experimentation we chose the Eisenlohr projection for our map. This rather remarkable projection was introduced by German mathematician and physicist Friedrich Eisenlohr in 1870 (Incidentally his brother was the chemist and Egyptologist who studied the famous Rhind Papyrus). It has advantages over all the rest but was not used commonly perhaps because its formula was intensive on calculations in the pre-computer era.

Let $\lambda \in [-\pi,\pi]$ be the longitude; $\phi \in [-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ be the latitude and $R$ the radius of the globe which we wish to project. Then the Eisenlohr projection in terms of the x and y coordinates on the x-y plane is given by:
$S_1=\sin\left(\dfrac{\lambda}{2}\right)$

$C_1=\cos\left(\dfrac{\lambda}{2}\right)$

$T=\dfrac{\sin\left(\dfrac{\phi}{2}\right)}{\cos\left(\dfrac{\phi}{2}\right)+C_1 \sqrt{2 \cos(\phi)}}$

$V=\sqrt{\dfrac{cos\left(\dfrac{\phi}{2}\right)+(C_1+S_1)\sqrt{\dfrac{\cos(\phi)}{2}}}{\cos\left(\dfrac{\phi}{2}\right)+(C_1-S_1)\sqrt{\dfrac{\cos(\phi)}{2}}}}$

$C=\sqrt{\dfrac{2}{1+T^2}}$

$x=(3+\sqrt{8})\cdot R \cdot \left(-2\log(V)+C\left(V-\dfrac{1}{V}\right)\right)$

$y=(3+\sqrt{8})\cdot R \cdot\left(-2 \arctan(T)+C T \left(V+\dfrac{1}{V} \right)\right)$

It has the following notable features: i) it has no singularities so every point on the globe can be simultaneously represented; ii) it is completely conformal, i.e., it preserves angles across the globe completely; iii) it has the narrowest scale range for a conformal projection of $1 : 3+2\sqrt{2}=1 : 5.83$; iv) it has a constant scale along the boundary colures. Thus, even though we get some size distortion near the boundaries its preservation of conformality is a useful feature for astronomical depiction especially when we want to highlight the ecliptic which lies close to the equatorial aspect. Given the above formula provided by the US Geographical Survey, we wrote our own function for the Eisenlohr projection but we also implemented projection using the projection function from the R Mapproj library (which has several different projections) in the event we wished to try out alternative projections in the future.

4) For the plotted data we chose the following (the source is: Vizier unless specified otherwise): (i) The venerable Yale Bright Star Catalog with 9096 stars which is for practical purposes complete in its coverage till visual magnitude $m_V=7$. It also provides variability and double star data; (ii) For the background shading of the Milky Way we chose a catalog of 18693 of the brightest stars from the Milky Way; (iii) For the Milky Way and Magellanic cloud boundaries a set of 1073 positions giving the boundaries and constellation lines, i.e. stars to be connected, made available in electronic format by professor Dan Burton based on his article, which I think appeared in Sky & Telescope; (iv) Boundaries for all 88 constellations defined by the International Astronomical Union; (v) shading in and around the Magellanic clouds obtained from 526 bright electromagnetic sources in those regions; (vi) we extracted 196 deep sky objects from the NGC. It covers all genuine Messier objects and other brighter objects that are accessible to telescopes in the 6-10 inch range (e.g. the Centaurus A galaxy).

5) The sky map distinguishes variable and doubles stars recorded in the YBSC using a bull’s eye symbol and a dash passing through the star respectively, which we inherited from our old Japanese maps. The deep sky objects are shown thus: galaxies are marked with an ellipse; planetary nebulae with a fish eye symbol; globular clusters with a many pronged burst; open clusters with a circle and cross; other nebulosities with a dotted circle; miscellaneous objects with an empty circle. All these symbols are obtained using the unicode Symbola font made freely available by George Douros. Unfortunately, none of the Devanagari fonts render correctly for conjunct akṣara-s other than in the png format. The star-color was computed using the difference of the B and V band magnitudes wherever available.

6) Finally, one of the key motivations for making this map was to have the nakṣatra division on the ecliptic. But how does one get the nakṣatra divisions? While it is clear from the Vedāṅga Jyotiṣa that the early Hindus did conceive the ecliptic as having fixed nakṣatra divisions, these have not come down to us. Over the ages starting from the Vedic period the nakṣatra divisions were named according to yogatārā-s. The root yuj in this term might be taken to mean junction stars i.e., stars marking the divisions of the nakṣatra-s or it might be taken to mean stars united with a given nakṣatra division. Making note of the asterisms defined by the yogatārā-s over the ages, we feel that the latter definition is more likely and sensible. It seems that the yogatārā-s are a remnant of an early observational phase in Hindu astronomy where they were a rough guide to locate the moon within a nakṣatra division. This observation was likely carried out using a rod (yaṣṭi-yantra) which was projected against the cusps of the Moon to see which yogatārā they were aligned with. Based on that the Moon’s nakṣatra division was then assigned. However, as predictive algorithms improved starting with the VJ gradually the lay Hindu practitioner lost touch with the nakṣatra-s as actually asterisms, i.e. the yogatārā-s. The situation was so bad among the astrologers that in when in the 1800s the great historian of Hindu astronomy, Shankar Dixit, asked several astrologers if they could identify the nakShatra-s, not one of them could correctly point even a few out in the sky. This knowledge, however, survived among the rare śrauta ritualists. Dixit encountered one from near Kolaba who was able to correctly point out all of them in the sky. He further cited an incantation from tradition that went as below:

khau kha jā trī gu cu gai co cho bhū yuk
cha hi trī ku cū che ko dvi yuk
ṅau kha chā ṅī ku ghu tri yuk
kha jā ku cū ghe gho

 102 112 128 140 153 156 183 196 197 217 232 240 252 266 277 291 305 312 327 345 351 354 12  28  51  66  74  94 
This contains the above code in the kaṭapayādi system with the vowels (a, ā, i, ī, u, ū, e, ai, o, au) being 1:9 and 0. To determine the rising point of the ecliptic from the nakṣatra asterism at the meridian at midnight one does the following: the first is Aśvayuji. If it is at meridian then the rising point of ecliptic is 102 degrees on the ecliptic. Thus, the separation of the rising point is given for each nakṣatra (note they are taking Betelguese to be Ārdra and it comes really close to Mṛgaśīrṣa. This illustrates that the Hindus were clearly aware of the positions of the yogatārā-s in ecliptic coordinates and that they were unevenly positioned. However, the nakṣatra divisions themselves are a regular $13 \tfrac{1}{3}^o$ (800 minutes of an arc) since the removal of abhijit; so the question remains where to start them. One could use the above incantation to calculate one such starting point. That was for the 1800s. Somewhat later, following up on the monumental work of Shankar Dixit, his acquaintance, the great patriot, BG Tilak, proposed that there should be a pan-Indian astro-calendrical reform the use of modern astronomical calculations to determine the pañcāṅga and fixing of nakṣatra divisions in a standard way across Hindudom. This project was taken up after the English tyrants left by the calender committee constituted under the great physicist Meghnad Saha and the jyautiṣa Lahiri. They produced a nakṣatra division recommendation, which while not exactly equivalent to the ancient division, is an entirely usable as a modern substitute. They began with with the premise of having the star Spica (Citra; $\alpha \; Virginis$) in middle of the citra division of the nakṣtra in ecliptic coordinates for 1956 CE. This resulted in the equinoctial colure passing through uttara-proShTapada. Examining this proposal we find that it has issues with respect to the yogatārā-s at Ārdra, which we have argued was originally Sirius, Uttarāṣāḍa, and Śroṇa but it is clear that some such issues will be there irrespective of the juggling of boundaries one might do. Hence, we took these divisions and converted them to the J2000 coordinates currently used in astronomy. This gives us the start of the Revati division at $9.112253 ^o$ in decimal ecliptic coordinates. Starting from there we took 27 divisions of 800 minutes of an arc each.

With the nakṣatra divisions in place we created 2 types of maps – one with the nakṣatra points as defining ecliptic longitudes and another with the same defining equatorial longitudes. Finally, one could also use the AstrolibR or any equivalent in your favorite system to get the positions of the Moon and plot it on the map as a visual pañcāṇga if you like.

Finally one may ask why write so much and make such ado about just plotting a bunch of points from coordinate files. Yes, that is all there is to this exercise but do it yourself and if you are a mere mortal like us you might find the process interesting.

The maps in PDF format(view with high magnification 150% or more):
1) Ecliptic longitudes
2) Equatorial longitudes
3) With Full moon(filled) of Sep 6 2017 and New Moon (empty) of Sept 21 2017 plotted on the map.

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