Some meanderings among golden stuff

•August 23, 2016 • Leave a Comment

There are some angles that we often encounter in the construction of the golden ratio and its use in religious art. The first is the most obvious is the angle \dfrac{2\pi}{5}=72^o which is the angle made by the diagonals connecting a side of the regular pentagon to a vertex. Thus, for a unit pentagon \phi is the length of its diagonals. This captures the quintessence, literally the five-ness of \phi. This angle was used in Hindu tradition as can be seen in the figure below. Here the angle and \phi are deployed in one of the most remarkable religious works from the Vaṅga country (region of modern Balurghat) — a liṅga on the four sides of which are the four kula-yoginī-s of the directional āmnāya-s. This is keeping with the deployment of the golden ratio in other figures of the kaula tradition like the Kubjikā and śrī yantra-s.


In the previous note on finding the translation vectors for the golden construction we encountered the famous angle \theta=\sin^{-1}\left(\dfrac{2}{2\phi-1}\right)=\cos^{-1}\left(\dfrac{1}{2\phi-1}\right)=\tan^{-1}(2) \approx63.43^o. This angle is the supplementary angle of the dihedral angle of a dodecahedron that quintessentially quintessent object. This angle comes up often in the construction of golden ratio with the bhujā-koṭi-karṇa-nyāya, which we place here along with other miscellany for the record. Indeed, Posamentier and Lehmann point out in their wonderful monograph on the golden ratio that Johannes Kepler had said: “Geometry harbors two great treasures: One is the Pythagorean theorem, and the other is the golden ratio. The first we can compare with a heap of gold, and the second we simply call a priceless jewel.” While these are rather elementary constructions, we are just putting down a few of them for own record and the instruction of those in need.

Construction 1:

1) Draw two unit squares sharing a common side \overline{DB}.
2) Bisect \overline{DB} to get midpoint H. With H as center draw a circle of radius 0.5
3) Draw \overline{AF}. \overline{AF} cuts the circle at points I and J. Now \overline{AJ}=\phi. Also \angle BHA=\tan^{-1}(2)
From the construction we have:

Construction 2:

1) Draw three unit squares with each adjacent one sharing a common side.
2) Draw \overline{ED}. Note \angle BED=\tan^{-1}(2). Bisect \angle BED to get point K. \overline{AK}=\phi
From construction we have:
Using the angle bisector theorem we have:
\dfrac{\overline{BK}}{\overline{DK}}=\dfrac{\overline{BE}}{\overline{DE}}. Let \overline{BK}=x\; \therefore \dfrac{x}{2-x}=\dfrac{1}{\sqrt{5}}

\therefore x=\dfrac{2}{1+\sqrt{5}}=\dfrac{1}{\phi}\therefore \overline{AK}=1+\dfrac{1}{\phi}=\phi

Construction 3

1) Draw a square ABCD with sides of length 2 units.
2) Obtain midpoint E of \overline{CD} and join it point A and B to obtain \triangle ABE . Note \angle ABE=\tan^{-1}(2)
3) Draw incircle of \triangle ABE having determined incenter I. It touches \overline{AB} at F.
4) Using \overline{FI} as radius and F as center draw another circle to obtain point J. \overline{AJ}=\phi.
From construction we have:
\overline{AE}=\overline{BE}=\sqrt{5}\; \therefore perimeter(\triangle ABE)=2+2\sqrt{5}

Radius of incircle r_i=\sqrt{\dfrac{(1+\sqrt{5}-\sqrt{5})(1+\sqrt{5}-\sqrt{5})(\sqrt{5}-1)}{1+\sqrt{5}}}=\dfrac{1}{\phi}
\therefore \; \overline{AJ}=\phi

A construction such as these can be also used to easily construct a model of the great pyramid of king Khufu of Egypt. We illustrate this below using construction 2 with 3 unit squares:

1) The first of these is used to construct a square of side 2 units BEDC, which forms the base of the pyramid.
2) We then use the above construction to obtain \overline{OF}=\phi.
3) We then deploy the geometric mean theorem in the vertical plane on \overline{OG} and \overline{OF} to obtain \overline{OA}=\sqrt{\phi}, which will be the height of the pyramid.
4) We then join the vertices of the base square BEDC to form the model pyramid of Khufu.

shrIchakraOne will notice that the cross-section of the great pyramid is the \triangle GAL with base 2 and isoceles sides \phi. This triangle is the same as the great śrīkaṇṭha triangle of the śrī-cakra. Possibly the Hemiunu used such a construction to model his pyramid. There has been some debate since the modern rediscovery of the golden ratio in the great pyramid as to whether the Egyptians really knew of it or not. When we account for the the remarkable casing stones and the damage over the centuries the dimensions of the great pyramids does come strikingly close to the above constructed model. The clinching factor is the statement of the yavana Herodotus who evidently recording an Egyptian tradition noted that the square of the height of the great pyramid is equal to the area of triangular faces. From the above construction one can see that this implies that the pyramid had to have the height \sqrt{\phi} and the isoceles sides of the cross-section of through the middle of the square base being \phi. Thus, some form of expressing the golden ratio, perhaps in the form of a construction like the above was known to the Egyptians.

This makes the great pyramid at ~2560 BCE perhaps the first monument in world history to encode this famed ratio. It is simultaneously a stark reminder of a great ancient civilization and how it can vanish despite it achievements. But due to the encoding of this information on a monumental scale a glimpse of ancient Egypt’s knowledge has come down to us. Indeed the scale and the brilliant engineering of the pyramids has helped them survive the attempts to erase them: Al-Malik al-Aziz Uthman the son of the counter-crusader Salah ad-Din who wanted clean Giza of its jāhilīyah devoted a large amount of labor to demolish the pyramids but only having succeeded in damaging the smallest of the big three eventually gave up. Then the Mamluqs of Nasir-ad-Din al-Hasan carved out the casing stones of the great pyramid to build masjids but could not get down the colossus. However, there is no guarantee that in the coming years a Mohammedan upheaval in Egypt does what their predecessors wished but failed to do.

This brings us to whether the Hindus and yavana-s obtained the idea of using \phi from the Egyptians. This is plausible since they are seen using a construction that embodies this ratio much before the other two. The yavana-s show a clear record of the ratio and specific constructions for the first time in the work of Euclid. Given Euclid was recording earlier geometric knowledge it might have been known sometime before him as suggested by \phi used by Phidias in several architectural features of the Parthenon. This was around the same time Herodotus records relationship concerning the great pyramid. Hence, we at least have evidence for Greek-Egyptian contact where a construction involving this ratio is recorded supporting the possibility that the yavana-s obtained the concept from the Egyptians before it was mathematically formalized in Euclid.

The Indian situation is less clear but offers certain interesting clues. The as yet mysterious Harappan civilization was undoubtedly in contact with Egypt at time reasonably close to when the great pyramid was constructed. Yet, I have so far not found any evidence for the use of golden ratio-related constructions in the Harappan artefacts that I have seen. The Indo-Aryan tradition abounds in extensive geometric constructions related the śrauta ritual. There is no evidence for the use of golden ratio-related constructions in the early Indo-Aryan śrauta tradition among the bricks, altars or ritual halls. Since this tradition was something derived from the common ancestral tradition that also seeded Greek geometry, it is likely that in old Indo-European tradition the golden ratio did not play any major role. However, the late Atharvavedic tradition we find the construction of a regular pentagonal altar for Vāyu where the ratio could have been involved (“vartulaṃ pañcakoṇaṃ…vāyavyāṃ pañcakoṇaṃ tu vāyavyeṣv api karman ||“).

On the other hand we see a completely distinct mathematical approach which yields a series whose limiting ratio is \phi: the Meru of Piṅgala. This at its latest would have been close to Phidias in antiquity but there is no evidence that the numbers of the Meru were used in a geometric construction. With the rise of recognizable Indian iconography, i.e. the iconographic tradition that has remained rather conservative since then, suddenly the golden ratio appears in certain examples. Below is one of its early appearances in a tāthāgata context from Sarnath.


While iconographic examples such as these might be disputed, the śrīcakra is a rather clear illustration. Any direct connection to yavana tradition for these Hindu examples can hardly be established. Moreover, the ratio does not play any role to our knowledge in the classic Hindu successor mathematical tradition of the śrauta constructions. Thus, it appears to have been a para-mathematical tradition that primarily survived in a religious context illustrating that classical Hindu mathematical tradition does not encompass all of Hindu mathematical knowledge. Other examples of such include the Platonic solids in bead manufacture in ancient India or the ellipse in temple architecture.

Mysterious fractal polyhedra

•August 20, 2016 • Leave a Comment


A golden construction

•August 17, 2016 • Leave a Comment

Anyone with even a small fancy for geometrical matters would have at some point in their lives played with the golden ratio (\phi=\dfrac{1+\sqrt{5}}{2}\approx 1.61803398875). Indeed, we too have had our share of fun and games with the golden ratio. In course of this we stumbled upon what seemed to us an interesting geometrical problem although it is likely to be seen as a trivial issue by mathematicians. \phi was apparently rather important to the yavana-s. Among the Hindus as we have noted before, it was used in the construction of the śrīcakra, the primary yantra of the śrīkula tradition.

Problem and the rules of the game
Starting material: A unit square. For convenience we place one vertex at origin and one side on the x-axis.
Objective: To construct a recursive golden rectangle and a golden spiral of any given resolution using this unit square. At the risk of sounding slow-witted I must emphasize this is not the same as drawing a golden rectangle and sectioning it.
It is well-known that the recursive golden rectangle is constructed by starting with a unit square and repeatedly drawing squares which are scaled by a factor of \phi-1=\dfrac{1}{\phi} \approx 0.61803398875 and arranging them in an inwardly spiralling fashion. The golden spiral is obtained by drawing a quadrant arc using one vertex of each of these squares as the center and the side of the square as the radius such that we get a smooth curve. Figure 1 shows a few iterations of this being done manually.

golden_manualFigure 1

Now doing the above is a tedious manual procedure with its limitations. So the question was can we do it relatively automatically taking advantage of a modern construction software like GeoGebra. One could technically compute the coordinates of each new square and draw them out with such a program but this is tedious too and needs some rather unappetizing programming sleights. Instead, we wish to achieve this construction using only the following three allowed operations, which can be easily automatically repeated in construction software (e.g. GeoGebra) to obtain a recursive golden rectangle of spiral of any desired resolution:
1) A single vector which can translate (rectilinear displacement) the starting unit square by a given distance in a given direction. Figure 2 shows the displacement of our unit square (square_1 by the vector u to get square_2.
u=\begin{bmatrix} -1.66\\ 0.73 \end{bmatrix}

golden_translationFigure 2

2) Dilation (scaling) with the origin as the center of dilation. This operation allows one to scale an object (in our case the unit square) by any factor such that ratio of the distance of any point on scaled square from origin to distance of its equivalent point on the the unit square from origin is equal to the scaling factor. Figure 3 shows such a dilation operation performed on square_2 by scaling\;factor=\dfrac{1}{\phi} to get square_3.

golden_dilationFigure 3

3) Rotation of an object about origin by a constant angle in one direction. Figure 4 shows the rotation of square_3 by 90^o to get square_4.

golden_rotationFigure 4

To get a flavor of how this works when done recursively we start with our unit square square_1, which is what we need to construct the golden rectangle, and inscribe a circular quadrant inside it. We then apply following series of operations recursively on these two object: Translate by vector v, then dilate by a certain scaling factor and then rotate by a given angle. For obtaining the golden rectangle, one can find the rotation angle easily. As one can see from the manual construction in Figure 1, for a recursive rectangle each successive square must be rotated by 90^o. The scale-factor is also obvious because we are aiming for a golden rectangle; the square must be successively scaled by the factor of \left ( \dfrac{1}{\phi}\right )^n, where n=0,1,2,3\;... for each iteration. But these two are not enough as can be seen in Figure 5. There we apply the above rotation and dilation transformations on square_1 with an arbitrary translation vector v thus:
Rotate[Dilate[Translate[square_1, v],\;b^n], \dfrac{n\pi}{2}], n=0\;to\;10
where the vector is:
v=\begin{bmatrix} -1.23\\ 0.42 \end{bmatrix}

golden_transformationFigure 5

We get a spiral arrangement of the squares and the quadrant arcs but clearly this is not the golden rectangle or spiral. Hence, the big question is how do we find the right translation vector and how many such vectors exist which can produce a recursive golden rectangle with the unit square.

Determination of translation vector and construction of the desired golden entities
The determination of the translation vectors to produce the golden rectangles and spirals involves an interesting construction. We are not going to repeat the well-known constructions of the yavana-s by which they obtained \phi and \phi-1=\dfrac{1}{\phi}. We start with segments of these values already pre-constructed:
1) Let the unit square be ABCD with point A at origin.
2) From point A in the direction opposite to side \overline{AB} along the same straight line mark the point G at distance 1-\dfrac{1}{\phi} and point H at distance \dfrac{\phi}{2}.
3) From point A in the direction opposite to side \overline{AD} along the same straight line mark the point F at distance \dfrac{2-\phi}{2} and point E at distance \dfrac{1}{\phi}.
4) Draw \overleftrightarrow{GF}. Drop perpendiculars to \overleftrightarrow{GF} from points H and E to meet it at points I and J.
5) One will notice that \overline{IJ}\cong \overline{AB} i.e. it is congruent to the sides of the unit square.
6) Now use \overline{IJ} to complete the construction of square IJKL which is congruent to our unit square ABCD.
7) Draw vectors u_1, u_2, u_3, u_4 that connect point A to points I, J, K and L. Any of these 4 vectors can serve as translation vectors with the above dilation and rotation factors to give us 4 golden rectangles from the unit square ABCD and 4 golden spirals from the 4 quadrant arcs inscribed in square ABCD (Figure 6 and 7).

golden_construction_1Figure 6: single example of golden rectangle/spiral

golden_construction_2Figure 7: All four golden rectangles and spirals

This construction reveals some interesting features:
1) The angle between the vectors u_1 and u_2 is \dfrac{3\pi}{4}=135^o and each vector is separated from the adjacent one by \dfrac{\pi}{4}=45^o. Thus the 4 vectors together form a trisection of the angle \dfrac{3\pi}{4}.
2)As noted above the square IJKL which determines the end points of the four translation vectors is congruent to the starting unit square but from the construction we can see that it is rotated with respect to it by an angle of \theta, where \theta=\sin^{-1}\left(\dfrac{2}{2\phi-1}\right)=\cos^{-1}\left(\dfrac{1}{2\phi-1}\right)=\tan^{-1}(2) \approx63.43^o. This angle is the supplementary angle of the dihedral angle of a dodecahedron.
3) From the construction we can show the shortest of the vectors is:
u_2=\begin{bmatrix} \dfrac{2-\phi}{2\phi-1}\\[10 pt]-\dfrac{2-\phi}{2\phi-1}\phi \end{bmatrix}

3) The six ratios of the magnitudes of the 4 translation vectors can be expressed as relationships featuring \phi thus:






In the gray zone

•August 15, 2016 • Leave a Comment

Among the thinkers of the modern Occident one may take note of the Englishman John Gray. He is firmly an insider of the Anglosphere not just by origin but also by worldview: he is quick to spot the barbarisms in the transitions to modernity of the Germans and Rus while being *apparently* silent about those of higher magnitude of the Anglosphere. He also apparently acknowledges that the first two Abrahamisms were positive contributors to human thought thus squarely falling inside the Abrahamosphere. On these matters, especially as heathens who have directly faced the genocides committed by his nation and more generally Abrahamism, we cannot disagree more. Yet, there are other matters in which he seems to be rather distinctive in that occidental space in being among the few who explicitly recognize secularism as being a resurfacing of a blend of the first two Abrahamisms.

He also makes several other notable points that are not commonly understood in the western academia:
-He notices and criticizes the “physics envy” among the economists. He notices how a large body of investigation that tries to provide an economic explanation for every human endeavor, including religious conversion; he points out that this is fraught with erroneous assumption.
-He points out that science cannot and is unlikely to replace religion.
-A related point he seems to make is that not all science or more broadly human endeavor in the realm of knowledge would come within the ambit of the usages of hard science (i.e. the most mathematized science).
-He notices and points out that the theory of the telos [the end] of history is fundamentally flawed and history will continue with all the basic causes of conflict spurring it on in the future.
-A related point he makes is that the transition to modernity, while widely assumed in the west to be a universal in form, is unlikely to be so.

Regarding the last point, for a heathen it is easy to notice that the universality is merely a secularized version of the second Abrahamism in turn deriving its roots from the Mosaic distinction of the first one. For people like us who have had a more expansive experience of the 3rd Abrahamism can again see how closely the universal modernism hews to the universal peace that is sought by the 3rd Abrahamism, the religion of peace, be it of the type pedaled by the leaders of the Anglosphere or the Marxians. Thus, unlike several in the west, Gray is willing to admit that Russia has its own modernism. Thus, one can say that the anti-Russian block of the Occident’s idea that Russia is somehow caught in a barbaric Tsarist past (currently represented by Vladmir Putin) is a rather flawed characterization. We would go further to insist that an analysis of the Russian genius and transition modernism and needs an eye that is entirely unaffected by Anglospheric bias.

In contrast to Gray there is a whole cluster of thinkers which represents what we characterize as the opposite side of the occidental thought spectrum: Jared Diamond, Steven Pinker, Steven Levitt, Dick Dawkins, Stephen Gould, Daniel Dennett and the like. While I am acutely aware of the differences between members of this cluster, I still group them together because they all display a certain proclivity for secular universalism in overt or covert ways. Indeed, Gray in his review of Dawkins’ biographical narrative brings out this issue. In addition to Dawkins, Gray had earlier criticized the claim of Pinker’s claim of the trend of decreasing violence. In this aspect of Pinker’s thought one may see a secularized form of Abrahamistic eschatology – comparable to the telos of the second coming of the preta or the unmatta.

Another facet of this claim of Pinker relates to the “criminal tribes” view of Diamond. Diamond presents the case that tribal societies are quintessentially criminal, in essence violent and homicidal, a trait that has been since attenuated in humans. However, as we have said before, and suggested in Armand Leroi’s review of Diamond’s sweeping universalism, Diamond seems to deny a role for genetics. To paraphrase Diamond’s words that would simply be “racist”. So it is not genetic change but something more superficial that has effected a transition from a past as “criminal tribes” to a peaceful present. This clearly smacks of a secularized form of the Mosaic distinction being received by a tribal group in the form of a prophetic revelation from the ekarākṣasa there is a transition of the tribe to a state of religion distinct from the evil heathen past – what the marūnmatta expresses as the jāhilīyah.

This is rather easily noticed in secularizing Indian Mohammedans, Pakistanis and Bangladeshis, who typically eagerly lap up the same secular condition of the Occident, rather than returning to their past in the dharma. They would continue to condemn the latter as something primitive, disgusting or in the least as unpalatable as the overt Abrahamism they are shedding. In fact even the more discerning and informed among them feel more threatened by the unabashed display of Hindu religiosity than homicidal Abrahamism. On a personal note, we have interacted with a powerful and dangerous man of intelligence and originality (not a marūnmatta but originating in another Abrahamism), who was extremely enthralled by Diamond’s, Levitt’s, and Pinker’s theses. We gingerly questioned him as to how he so easily bought into all of it given all that he knew and the paradox it raised with respect to his own knowledge. His answer was interesting in that it drew not from logic but from a purposeful ignorance stiffened by an underlying rock-firm foundation of credo – a secular manifestation of his ancestors’ credo in Abe and his successor unmatta-s.

The above cluster of players in modern occidental thinking have a notable parallel that is little known outside Hindu circles – a subsection of white indologists and their fellow travelers. This subsection, epitomized by Sheldon Pollock, Wendy Doniger, Robert Goldman, Johannes Bronkhorst, Giovanni Verardi and the like differs from their earlier European counterparts. Their analysis of Hindu tradition (of course they would doubt something like that exists) is essentially the application of secularized Abrahamistic constructs in guise of objective theorizing. This cluster, while professing scientific objectivity with almost the zeal characteristic of Abrahamism, deep down do not seem to care much for that. Rather they play a game similar to what Aristotle is said to have taught Alexander – eristics – something the Macedonian warrior is said to have enjoyed. In eristics you can successively take either side of an argument and argue for it being the correct one. Similarly, members of this cluster, who are particularly common in elite American academia, can take up a viewpoint and argue for it without really being interested in investigating the truth – seemingly it is an intellectual game for their high IQs. At the long end of history much of this cluster can be seen as making little lasting intellectual contribution beyond creating ferment in their times.

In conclusion we find Gray’s thoughts in regard to the dominance of secularized Abrahamism in occidental intellectual endeavor rather consonant with ours.

This brings us to a gray zone, something which John Gray expresses obliquely in his critique of his compatriot Dawkins as well as in his talks, i.e., man is not a machine. Many old Hindus would have said that man is a machine but inside that machine or permeating that machine is something non-mechanical, which they termed the ātman. The bauddha-s would say that that ātman is a mere illusion, which may not be required for explaining things but would they accept that the organism is machine? It is not entirely clear. While Gray might be entirely willing acknowledge these possibilities, his bigger concerns seem to be at the level that early Hindus did not doubt much: yes, man is a machine in so far as what they defined as his śarīra goes, i.e, his biochemical body. Gray seems see that there is a certain complexity with things like the body or at least aspects of its function that it is not simply analyzed like a machine (i.e. even before reaching the issue of a philosophical zombie).

This brings us to another domain which we have thought about due to our own performance of science and ritual. How “algorithmizable” is science/mathematics or to put it more simply can we make a machine that replaces the “expert”. One can easily see that the performance of mathematics often features the virtuoso who obtains remarkable insights that are quite unattainable by the average human. Even with advanced computational mathematical tools of today, which make the mathematical experimental procedure more accessible to lesser mortals like me, it is easy to see that the mathematical virtuoso cannot be recapitulated by just a machine or a lesser mortal running these tools. Conversely, there are certain computer-heavy proofs which some have questioned as even constituting real mathematical activity as they merely involve a gigantic scale of exhaustive mechanical computation that is best done by machine.

Beyond mathematics one could argue that the sciences heavy on more mechanical activities like chemistry and biology, which need that prolonged training in the mechanical aspect (people usually say about 10 years of practice), are less amenable to virtuosity. Yet common knowledge and psychological investigation suggest that not just these sciences but other activities clearly has a class of experts who can categorized as virtuosos even if not recognized as such. Such have been noted in many domains which are clearly distinct from each other: 1) expert physicians with extraordinary special diagnostic capacity; 2) chess players; 3) Bird identification by bird watchers; 4) Aircraft identification. 5) chicken sexing; 6) Auditing; 7) Financial investment (e.g. see Richard Horsey: The art of chicken sexing; Ericsson, K. A. & A. C. Lehmann (1996): Expert and exceptional performance: Evidence of maximal adaptation to task constraints. Annual Review of Psychology, 47: 273–305).

Indeed, this has been explored in a domain removed from “Science” in the case of sexing chickens, which relates the barbarism that arises from the industry catering to dinosaur-eating barbarians – male chicks need to be killed ASAP for the industry to cut costs; hence, there is a great premium on identifying the sex of chicks when they are day old or so. Here interestingly machines have been replaced by the human often receiving training in Japan. Those coming from that school make out the sex often with very subtle cues which others can hardly notice. When asked to explain how they do it, their answer is not different from the expert who “sees” the answer. This is also seen in the medical science where the “great” physician can diagnose difficult cases, which would take the average physician with the same formal training much more information and trouble to diagnose if at all. Indeed, the negligence of this aspect in modern medical culture (e.g. in USA) has resulted in physicians of a lower rung being accorded powers much more than they are really deserving to the detriment of patients. In bird-watching the virtuoso can often recognize birds correctly with the smallest of cues gleaned from the briefest of glances. The average watcher might merely register the presence of something up there with the same stimulus.

The experts, to different degrees in such domains, can often perform at a remarkable level of virtuosity and display particular special features: 1) They often “catch” or “see” the right answer or solution with cues far fewer than what others need to achieve the same result, if at all they can achieve it. 2) Often they are unable to easily explain how exactly they reach the result – thus it might almost look like magic to the rest. The expert often says that “he just sees the right path” or it just “flashed in his mind”. For instance, in my own domain of activity I have seen people working with me, who are doing things in an adjacent rather the same activity, think I am somehow magically producing the result. Even if I explain how I arrived at the result to them they are often unable “to get it” and are unable to “see” what I am seeing so clearly in my head like a real object.

So now the question is whether a machine can be created that does the same as such experts. There can be potentially several different lines of answers to this:
1) Yes, a machine can be created. Such tasks are merely very complex, hence the machine would need to be similarly very complex and our technology might eventually get there to achieve such artificial intelligence.
2) A conventional machine cannot be created but one which exploits some deeper physical principle such as one stemming from the quantum theory can.
3) No machine can be directly engineered for this. One can only recapitulate the evolutionary process in some way to create a biological system where such capacity might emerge as a higher order system property. The lower order system components by themselves will not have the property but it will emerge from their interactions – i.e. the conglomerate is greater than the parts, like the assemblage of neurons or cells having properties beyond those of individual cells. Hence, the emergent property cannot be determined (at least easily) by the underlying parts.
4) It needs a special system so far not seen among engineered machines that has this capacity to somehow “connect” deep into a system’s information and read it directly. This recapitulates the subconscious element of virtuosity described above.
5) No machine can achieve it. It is actually related to the hard problem of consciousness, which lies outside the domain of objective inference.

Now, in the modern scientific world conditioned by secularized Abrahamism some of these alternatives could be seen as a blasphemy by some. However, we feel these should be evaluated without offhand dismissal. First, we hold the last possibility of the above to be unlikely. Even though there is a first person experience associated with the act of the expert having “seen” the answer, that experience is produced only after sensory activity much like qualia produced by the act of seeing more mundane things like red color, and it is this sensory activity not the qualia which matters. Second, not all people with similar or higher IQ than the expert can be trained to become one. There is a vector with a non-zero angle with respect to that of IQ though the two might show some correlation in certain domains of expertise. In particular, many of these tasks and IQ have a relationship to memory. Moreover, the propensity for such expertise in several domains shows genuine difference between the sexes. These things suggest that there is ultimately a component rooted deep in the biology of the individual.

Next, like with IQ, we think this capacity is actually distinct from being able to perceive some kind of mathematical regularity or beauty. In fact such patterns often fool the seeker. We can cite the example of Kepler seeing the wrong pattern of the relationship between planetary orbits and the Platonic solids. Similarly, great minds like Gamow and Maynard-Smith had wrong ideas for the genetic code which were moved by such considerations seeking a certain mathematical regularity of patterns. So we can say that it cannot be achieved by a machine that is merely very good at capturing such patterns. The genetic code in particular offers a great case of how the biological solution is rather distinct from a seemingly mathematically regular one.

However, at this moment we do not know if a really complex engineered machine can do the same thing as a non-engineered evolutionary product i.e. the biological machine. Our hunch is that it might not be able to do the same kinds of computations as our immune systems or or neurons. We see science, the human endeavor, as a product of the latter type of system. If this were the case then indeed the line of thinking of Gray might have a deeper validity.

Additionally, we also hold that the same features that result in an expert in the domain of science also leads to an expert in the domain of religion – the capacity to “see” certain things with very subtle cues. Expert practitioners of rituals and what today primarily survives in the indosphere, the mantraśāstra know this well whereas the layperson sees nothing at all. That is why the lay person who has been detached from his heathen system by Abrahamism can easily see all this as mumbo-jumbo. In the past, even in the Abrahamistic world, certain scientists/mathematicians like Newton or Euler had a deep religiosity. Had they been in the Hindu world this would have been seen as as unsurprising as the domains of knowledge are not disjoint for the heathen. This dismemberment is yet another secularized aspect of the Mosaic distinction.

Some words on “para-Rāmāyāṇa-s”-I

•August 12, 2016 • Leave a Comment

I use the hybrid Greek-Sanskrit word para-rāmāyāṇa for all rāma-kathā-s other than that of Vālmīki. These span a great diversity of literature going back to relatively early times in Indo-Aryan tradition. Lots of people have said a lot of things on this matter and we are hardly going to review any of that here. The Rāmāyaṇa itself is enigmatic in some ways. There is no allusion to the main characters of the epic in the Vedic corpus just like the central pāṇḍu heroes of the Mahābhārata. Yet the late Vedic tradition belonging to the Paippalāda branch of the Atharvaveda remembers king Hiraṇyanābha Kauśalya a descendant of Rāmacandra Aikṣvākava. On the other side a remote dynastic predecessor of Rāma emperor Trasadasyu is one of the greatest heroes of the ṛgveda. Another mighty dynastic predecessor who is remembered in the ṛgveda is emperor Bhajeratha Aikṣvākava who is known by his more widely used name as Bhagīratha from the time of the Samaveda brāhmaṇa-s. Another great hero and dynastic predecessor of Rāma who is remembered in the RV is emperor Mandhātṛ. The founder of this dynastic throne Ikṣvāku appears as a rājarṣi in the RV. However, of the later occupants of the famed Ikṣvāku throne we know little from Vedic literature, which often talks of the occupants of the thrones of the Kuru and the Pañcāla. This, we suspect, reflects the dynamics of the invasions and the subsequent conquest of the sub-continent by the Indo-Aryans warriors coming in from their ultimate homeland in the steppes – Ikṣavāku group branched away from the pañcajana group and moved eastwards without major participation in the later Vedic consolidation of the śrauta ritual. However, their hero Rāmacandra was to have a major impact of Hindu tradition from his para-Vedic sphere.

The first of the para-rāmāyaṇa-s is the Rāmopākhyāna of Mārkaṇḍeya from the Mahābharata. Many have suggested that this mini-Rāmāyaṇa was the precursor of Vālmīki’s text. We however feel this is rather incorrect. The Rāmopākhyāna has several text fragments in common with Vālmīki while being just over 700 verses in length in its most basic form reconstructed by Sukthankar. Thus, we posit that it simply represents a para-Rāmāyaṇa tradition that might have branched off by contraction from an earlier “Ur-Vālmīki-rāmāyaṇa from which the extant Vālmīki text also evolved subsequently as part of the epic system (by epic system I mean the phase when the two national epics were similarly handled by a common set of transmitters, probably sūta-s, as evidenced by a certain similarity in phrase usages between them). It also appears that the rāmāyaṇa itself was orally transmitted longer than the rāmopākhyāna itself. The Rāmopākhyāna has feel of of being a “refresher” for people who knew the general rāmāyaṇa story and we suspect this prior knowledge was due to the existence of an earlier Vālmīki text that was widely known at that time. It is important to note that the mahābhārata repeatedly uses rāmāyaṇa as a source of metaphors and allusions but not vice versa. Hence, there is no reason to give into a white indological assertion that the Rāmāyaṇa is a post-Mahābhārata tradition. However, we do suspect that the old Vālmīki that was around at the time of the Mahābhārata’s old composition was distinct from the current version.

The Rāmopākhyāna has several distinct features from Vālmīki-s version; hence, we suspect that it had already branched off and preserved as a separate tradition again providing indirect evidence for the antiquity of the original Vālmīki. Its notable features includes:

-The god Tvaṣṭṛ is said to have made Sītā specially as a wife for Rāma. This is a reflection of the Vaidika tvaṣṭā rūpāṇi piṃśatu; tvaṣṭā rūpeva takṣyā; tvaṣṭā rūpāṇi hi prabhuḥ
-No mention of any putrakāmeṣṭi for the birth of the four Ikṣvāku princes.
-Viśravas has 3 wives: Puṣpotkaṭā who bears Rāvaṇa and Kumbhakarṇa; Mālinī who bears Vibhiṣaṇa; Rākā bears Khara and śūrpankhā as twins.
-While well-versed in the śruti, with the exception of Vibhiṣaṇa, they were paradoxically anti-brāhmaṇa.
-Vibhiṣaṇa joined the yakṣarāṭ Kubera who made him the lord of the rakṣas-es who were under him (they are alluded to in the Vaidika ritual to Kubera specified in the Taittirīya-śruti )
-The remaining piśāca-s and man-eating rakṣas-es elected Rāvaṇa as their overlord. He is said to have 1.4*10^8 piśāca-s and 2.8*10^8 rakṣas under his command.
-Brahmā sent the gandharva woman Dundubhi to be born as the hunchback to set the rāmāyaṇa in motion.

-Rāvaṇa meets Mārīca at Gokarṇa, a Rudra-tīrtha to recruit him for his cause.
-Rāmā pursuing the golden deer is described with the simile of him looking like Rudra pursuing the stellar deer with his bow.
-Rāma kills Vālin with a huge bow with a mechanical device (yantra; a cross-bow?).
-Sītā informs the ape Hanūmat of an old respected rakṣas known as Avindhya who had told her of the emperor of the apes and his councillers.
-No indications that Vibhīṣaṇa was in Lankā with Rāvaṇa. He simply arrives with his four ministers and joins hands with Rāma against his brothers.
-Many apes suggest crossing the sea by means of boats but Rāma goes for the causeway strategy arguing that they did not possess sufficient boats for landing a huge army in Lankā.
-Rāma places his ape and bear force in the midst of a forest in Lankā so that they would have ample supplies.
-A simile of Añgada smashing four rakṣas-es like a tiger in conflict with four hawks.
-The bears under their king Jāmbavant are explicitly described as being sloth bears.
-Several imp-like piṣāca-s launched the first wave of attack named: Parvana, Pūtana, Jambha, Khara, Krodhavaśa, Hari, Praruja, Aruja and Praghasa who were invisible to the apes and bears. But Vibhīṣaṇa broke their invisibility and they were killed by the vānara-s.

-Rāvaṇa is said to have arrayed his troops in the battle formation invented by the great bhārgava of yore Uśanas Kāvya. Rāma arrayed his troops using the method of the god Bṛhaspati.
-Vibhīṣaṇa kills Prahasta (in VR the great ape Nīla kills him). Battle between Dhūmrākṣa and Hanūmat is described as being like that between Indra and Prahlāda.
-Kumbhakarṇa leads the assault accompanied by the brothers of Dūṣaṇa named Pramathin and Vajravega. He captures Sugrīva. Lakṣmaṇa pursues him and after a fierce fight kill him with brahmāstra. Then Hanūmat and Nala kill the other two rakṣas-es.
-After the fall of Kumbhakarṇa, Megannāda enters the field and after a fierce fight became invisible in the sky and struck down Rāma and Lakṣmaṇa with his missiles. Then he tied them with his magical arrow net. Vibhīṣaṇa having successfully accomplished victory against the rakṣas on another front came to the place and seeing the fallen heroes revived them by using the prajñāstra. Then Sugrīva treated them using the viśalya herb and mantra-s. The coming of Garuḍa is not found in the original Rāmopākhyāna. It was only inserted into some recensions – it is clearly an interpolation without basis because the preceding text of the Rāmopākhyāna does not talk of the nāgapāśa.
-Then Vibhiṣaṇa informed Rāma that Kubera has sent a guhyaka with magic water that will allow them to see invisible objects. Upon applying the water to their eyes the ikṣvāku brothers and the apes and bears were able to see the invisible Meghanāda. Under the guidance of Vibhīṣaṇa, Lakṣmaṇa then attacked Meghanāda and after a fierce fight fired three āgneya weapons that respectively cut his bow-wielding hand, the other hand which was holding a naraca missile and finally his head.

-Rāvaṇa then wanted to kill Sītā but Avindhya persuaded him from doing so and urged him to go and fight Rāma directly. After a some fierce fighting with the army of Sugrīva, Rāvaṇa started emitting numerous warriors from his body who resembled Rāma and Lakṣmaṇa. Rāma saw through this illusion and killed the impostor Rāma-s while directing Lakṣmaṇa to kill those who looked like him.
-Then Indra sent down his own chariot with his charioteer Mātali. Rāma this time thought it to be an illusion of Rāvaṇa and did not want to board the car. However, Vibhśaṇa informed him that it was no illusion and really Indra’s chariot.
-In course of the great battle that followed the vānara-s scattered in terror from the fury of Rāvaṇa but Rāma finally deploying the mighty brahmāstra “turned Rāvaṇa to plasma” for the lack of a better usage. The Sanskrit goes thus:

sa tena rākṣasa-śreṣṭhaḥ sarathaḥ sāśva-sārathiḥ |
prajajvāla mahājvālenāgninābhi-pariṣkṛtaḥ ||

He the lord the rākṣasa-s with his car, horse and charioteer were set ablaze by [the brahma missile] and surrounded with a great fiery conflagration.

tataḥ prahṛṣṭās tridaśāḥ sagandharvāḥ sacāraṇāḥ |
nihataṃ rāvaṇaṃ dṛṣṭvā rāmeṇākliṣṭa-karmaṇā ||

Thus, seeing Rāvaṇa slain by Rāma of unperturbed deeds, the gods with the gandharva-s and cāraṇa-s were gladdened.

tatyajus taṃ mahābhāgaṃ pañca bhūtāni rāvaṇam |
bhraṃśitaḥ sarvalokeṣu sa hi brahmāstratejasā ||

By the energy of the brahma missile the five elements abandoned him [Rāvaṇa] of great opulence and he was deprived of all the worlds.

śarīra-dhātavo hy asya māṃsaṃ rudhiram eva ca |
neśur brahmāstra-nirdagdhā na ca bhasmāpy adṛśyata ||

All his bodily substances, indeed his flesh and blood, were burnt by the brahma missile such that not even ash was seen.

-Rāma then conferred Lankā on Vibhīṣaṇa. He along with Avindhya came out bringing Sītā.
-Rāma then tells Sītā that she is free and may go wherever she would like. He says that he has done his duty of freeing her but irrespective of whether she was chaste or not he was not going to consort with her again.
-The gods then arrived in person and Sītā swore by the elements constituting her body that she was pure. This was confirmed by the gods Vāyu, Agni and Varuṇa. Then the god Brahmā explained to Rāma that Sītā had been protected from being raped upon abduction by Rāvaṇa due the curse of Nalakūbara the son of Kubera.
-Then the ghost of Daśaratha appeared and congratulating Rāma asks him to return to Ayodhyā with Sītā. There is no Agniparīkṣa of Sītā in the original Rāmopākhyāna. It has been inserted into a regional variant and is clearly an insertion for it simply does not fit with the rest of the coherent narrative of this part.
-Rāma then bows to all the deva-s and Kubera and confers boons on the demon Avindhya and the demoness Trijaṭā for being good to Sītā.
-The gods offered Rāma boons: He asks for firm adherence to dharma, invincibility in battle and restoration of the lives of the apes killed in the battle. They also offered the yellow-eyed ape Hanūmat the boon of life as long as the Rāmāyaṇa is known, and access to divine food and drink.

-Rāma and his retinue then crossed the ocean by the same bridge by which the came to Lankā and on reaching the other shore he rewarded the apes richly for their services.
-The with the Puṣpaka air-ship he returned to Kishkindha with just Sugrīva and Vibhīṣaṇa to show it to Sītā and install Aṅgada as the crown-prince.
-Upon being united with Bharata, Rāma was crowned the king by Vāmadeva and Vasiṣṭha. After which he respectfully returned the Puṣpaka airship to Kubera.

In conclusion, the upākhyāna clearly represents a distinct tradition but generally recapitulates all the key elements of the extant VR. Notably, in the this version Vibhīṣaṇa is a major positive figure – he does not appear a like traitor, having joined Kubera early in his life he plays a major role in the war. Lakṣmaṇa-s profile is also higher in this version. No special role is allotted to Hanūmat such as bringing the medicinal mountain from the Himālaya or in the war to the exclusion of the rest. Yet, he is recognized as a key figure for his reconnaissance leap to Lankā.

Some astronomical experiments

•August 8, 2016 • Leave a Comment

Our interest in stellar evolution began sometime in the 8th year of our life after reading an old but very readable account by the famed Russian scientist George Gamow. It captivated us beyond words but we had to wait until the 12th year of our life when our mathematics came up to speed for a major astronomical renaissance. We spent every clear night observing stars in particular certain variables and recorded their magnitudes hoping to re-live the excitement of the astronomers of the past. The excitement hit a peak as we caught a fading R Coronae Borealis and the out-bursts of SS Cygni when it reached the range of our homemade telescope. Those were the heady days when the Hipparcos-Tycho project was in preparation and it was to be a great bonanza of astronomical data once complete. We read of the launch of the satellite and hoped that some day we could directly play with the Hipparcos data doing some trivial things for our own edification. Now years down the line the successor of Hipparcos, Gaia is in the sky and has already completed the second year of observations with the data release scheduled for next month. Gaia is of a much vaster scale than Hipparcos. It is set to survey a billion stars and that number is still just one hundredth of the number of stars in Milky Way. So at least before that happens we decided to put out the little examination of some trivial things in the Hipparcos data. When the Hipparcos data first came out the computers we possessed or had easy access in Bhārata were not powerful enough to handle the data of about 118322 stars. Today with the R language on our laptops we can quite easily deal with that. That said, we must reiterate that we are just presenting here some trivial stuff, which people have done over and over again, but we are doing it for ourselves just because there is no substitute to pratyakṣa when it comes to science.

One of the main objective of the Hipparcos project was the measurement of parallax of stars. The concept of parallax (Sanskrit: lambana) was first discovered by early ārya (among whom it was believe that it was taught by Sūrya himself to the demon Maya) and yavana astronomers like Hipparkhos. This trigonometric principle allows you to compute the distance of an astronomical object by measuring its angular shift from two positions separated by known distance against an even more distant background. It was first used to find the distance of a star (61 Cygni) other than the sun by the astronomer and mathematician Friedrich Bessel, one of the greatest intellectuals of all times.

So having downloaded the 118322 stars of the Hipparcos project we cleaned up the data to remove those stars with 0 or negative parallax. We also removed those stars that lacked the visual magnitude or B-magnitude, i.e. magnitude measured using a blue filter (with wavelength midpoint of filter being 445 nm). Thus we ended up with a total of 112823 stars to work with. Of course this still includes some stars with bad parallax measurement which result in unusual results but we are not bothered too much by that for we are mainly looking only at the bulk data and we have a great deal of reasonable measurement therein.

All parallax in the Hipparcos dataset is measured as milliarc seconds. Hence, we first we calculate distance of the stars in parsecs using formula:

d=\dfrac{1000}{p};1 parsec=3.26156 light years.


If we plot a histogram of the distance of the stars in the Hipparcos data we get a strongly right-skewed distribution with a distinct peak. The median distance is 208.7 parsecs with the peak of the distance distribution being 100-150 parsecs. The median value is ~.007 times the diameter of the Milky Way showing that large as the number of stars are Hipparcos has merely explored a very local neighborhood of the Milky Way. But the arms of our barred spiral galaxy being quite similar in general terms, and we being located midway from the center to the outer arms, we can consider our sample fairly representative for the arms. Of course some of extreme values of distance in the data are likely the result of defective parallax determination; hence we truncate the curve at a 1000 parsecs.

Once we have computed the distance of the star, given that the project has measured the visual magnitude (m) of the star in addition, we can next calculate its absolute magnitude (M). The absolute magnitude of a star is the magnitude with which it would shine if it were placed at 10 parsecs (~32.6 light years away).

M=m-5(log_{10}(d)-1); where d is parsecs


While the Sun blazes away in our sky its absolute magnitude is a dim 4.83, which would require us to retire to dark skies in order to see it if it were 10 parsecs away. In contrast Deneb (\alpha Cygni) would blaze away at the magnitude of about -8.4 from the same distance and would be seen in broad daylight. The plot of M is revealing in more than one way. The Sun lies at the beginning of the dim tail of the distribution considerably dimmer than the median M of 1.54 – that would be about as bright as \epsilon Canis Major appears in our sky and just a bit dimmer than the absolute magnitude of Sirius A. The distribution has a clear peak around 1 and a bit of a shoulder around 3. The bottom line is we are on dim side, in the 4th quartile of the distribution, keeping with the the yellow dwarf standing of our star.

The absolute magnitude is a measure of the luminosity*(L) of the star. Hence we can represent the same data directly as luminosity in terms of Sun units i.e. L(Sun)=1 using the formula:


This is obviously a similar distribution with a median luminosity around 11 Suns and peak distribution of stars with the luminosity of 13-15 Suns. The shoulder occurs at the luminosity of around 5 Suns. A big caveat for the absolute magnitude and luminosity data from Hipparcos is that it mostly measured only bright stars. Thus, the very numerous dim red dwarfs and even dimmer smoldering brown dwarfs are vastly under-represented in this data, thereby artificially moving the peak to the bright side. Just to give a feel for this the nearest star to the Sun is Proxima Centauri, which shines at a dim 11.05 magnitude while being a mere 4.25 light years away. Thus, it is unsurprising that Hipparcos does not have many of them.

The Hipparcos project also obtained the B magnitudes for the stars thus we have the B-V values which is the difference in magnitude with a blue filter and the visual magnitude. The stars with negative B-V are the blue stars and those with high B-V values are the red stars. The Sun, which is an archetypal yellow star, has a B-V value of 0.656.


This distribution is interesting in showing two prominent well-separated peaks in the middle flanked by shoulders. The higher peak is at B-V=.5, which contains the stars with a whitish yellow color, but to our eyes simply appear white. The lower peak is at B-V=1 which contains the orange stars. Between them is the valley at around B-V=.8. Thus the deep yellow stars are generally under-represented. The prominent first shoulder (almost a mini-peak) is at B-V=0 and contains the white stars. The second shoulder is at B-V=1.5-1.6 and represents the red stars. The dominance of the shoulder at B-V=0 and peak at B-V=.5 is why most stars in the sky look white to us. Here again the paucity of red dwarfs in the data has shifted the color plot in the direction of whiteness.

We can then calculate the approximate surface temperature of a star from its B-V value. While there are multiple formulae to do this we shall use the Ballesteros formula. While hardly perfect, it gives a better approximation among the simple formulae that just use B-V:


The distribution of temperatures shows 3 distinct peaks. The first and the lowest is around 3600-3700K, the second and the highest around 4600K and the 3rd is around 6400K. The Sun by this method gets T=5766K which is quite a good approximation of the typically cited T=5777K. The distribution is strongly right-skewed with a fat tail of hot stars. We can check a few other stars to see how this formula performs using the Hipparcos data with respect to commonly reported temperatures:

Star T/Formula T/Reported
Betelgeuse 3793K 3650K
Aldebaran 3737K 3910K
Sun 5766K 5777K
Sirius 10014K 9940K
Rigel 10515K 12100K
Bellatrix 14192K 22000K

The inaccuracy seems to rise with increasingly lower B-V values. Thus, the actual distribution onf the right tail is likely to be more stretched out rightwards.

Once we have the temperature and luminosity of a star we can next calculate its radius in Sun radius units using the formula:


The radius mostly corrects itself despite the issues with temperature we encounter with the above formula. The distribution is bimodal with median radius of ~3 Sun radii, again illustrating the bias towards larger stars as opposed to the tiny dwarfs in the Hipparcos survey. The first and higher peak is close to 2 Sun radii and second and lower peak is around 10 Sun radii thus separating the dwarfs from the giants.

With temperature and luminosity at hand we can also generate different Hertzsprung-Russell diagrams for the Hipparcos data. The first of these uses absolute magnitude and scales the points as a positively correlated function of the diameter. Second of these uses luminosity and does not scale the stars so as to give an an unobstructed view of the basic H-R diagram.


In both of these the main sequence, where majority of stars lie, comes out very clearly. The other prominent feature that comes out is a branch that emerges from the main sequence and climbs upwards between 6-5000K. This the orange-red giant branch with a prominent “red clump” which comprises of stars close to 5000K and absolute magnitude ~0-.5 which are fusing helium in their cores. This red clump population seems to be clearly common in our general galactic vicinity. The diameter-scaled plot gives allows to visualize the several groups of relatively rare giants and supergiants that lie above the main sequence and the few white dwarfs below it. Overall the H-R diagrams we have plotted here are similar to those with much larger number of stars which have just come in from Gaia. This suggests that for the Milky Way this can be considered a fairly representative H-R (barring the dim, small stars). One of the densest areas of this H-R diagram is the general area around which the Sun lies in it. Newer data from Kepler and Gaia are beginning to tell us the distribution of planets around such stars but I do not fully understand it. However, it does seem like there should be a great abundance of stars with earth-like planets and I am of the opinion contrary to some others that life should be rather common on those.

The Hipparcos data also has a large number of variable stars identified. Hence we shall next extract the variable stars from the data and plot at H-R diagram from just the variables. On extraction we get a dataset of 10312 variable stars. The H-R diagram for this dataset with stars scaled as a positively correlated function of diameter is shown below.


One thing that becomes rather apparent from this plot is the proclivity for giants as opposed to lower main-sequence stars to be variable. In particular, the red giants and supergiants are particularly enriched in intrinsic variables, as is well-known to anyone with an elementary knowledge of astronomy. Among the red giants/supergiants we have several different types of variables: 1) Mira-type long period variables, which are the so-called asymptotic branch giants, the stars that have left the main sequence and bloated up to become red giants in the later period of their lives; 2) the semi-regular variables like Betelgeuse with low amplitude pulsations; 3) irregular variables like Aldebaran.

On the upper side of the yellow-orange main-sequence we have the T Tauri variables, which are young stars often showing protoplanetary disks. Among the white and light yellow supergiants we have the famed Cepheid or \delta Cephei variables. Below them all the way from the white to lighter yellow just above the main-sequence to giants we have several similarly pulsating stars like the RR Lyrae, W Virginis and \delta Scuti variables. Further upwards on the plot among the yellow supergiants we have the RV Tauri variables. These “Cepheid-like” variables have various positively-correlated relationships between their periods and luminosity, which allow them to be used to measure distances. Interestingly, while classic Cepheids tend to occur only along the equatorial plane of the galaxy the RR Lyrae variables are common at all latitudes and in globular clusters. On the blue end near the main-sequence and giants we have the \beta Cephei and slowly pulsating blue variables.

Now we shall use the data to do a simplistic exercise of trying to get the data on Hyades and Pleiades clusters to compare them on the H-R diagram. For isolating the Hyades approximately we choose the star HIP 20484 from the cluster and take all stars in a circular 3 degree field around it from the Hipparcos data. Then we clean up the data compute all the above described values. We then plot our stars as rough star map to check if we have got it right. As one can see below the Hyades comes out suggesting that we have got the approximate field right.


Next we plot the distances of the stars in our field and notice a peak around 40-52 parsecs which is where the actual Hyades members are located.


We can see that Aldebaran lies much closer to us and is not physically part of the Hyades cluster. Once we have the distance of the actual Hyades stars we can isolate them as a separate data set.

Now for the Pleiades we take our central star as Alcyone (\eta Tauri) and do the same procedure as above. Upon plotting the rough star map we get the below picture showing that we are generally on the right track.


Then we make a similar distance plot to note that according to the Hipparcos data the Pleiades ~115-120 parsecs.


This distance resulted in one of most famous controversies in modern astronomy. It was much lower than what was thought to be the real distance of the Pleiades. After much discussion the astronomers do think that the distance of 136.2 parsecs is likely the correct distance, embarrassingly pointing to some systematic error in Hipparcos data. Hopefully, the final verdict will be given by Gaia in the coming year or two. However, for our experiment let us continue with the Hipparcos data keeping in mind that it would imply a dimmer Pleiades cluster. Using the above inferred distance we extract the Pleiades members as we did for the Hyades.

With all this in place we can now can now make a combined H-R diagram of the Hyades and Pleiades which we have extracted from the Hipparcos data. The stars of the Hyades are shown in empty circles and those of the Pleiades are shown in filled circles. The circles are scaled according to the diameter of the stars.


We can see right away that together they generally recapitulate the overall H-R diagram with most stars lying on the main sequence and a few moving into the red giant branch with 4 prominent red clump stars from the Hyades. As is visually apparently the Pleiades are in general bluer and hotter than the Hyades (this would be even more accentuated if the Hipparcos distance is an underestimate) with its prominent stars occupying the blue giant end of the main sequence. Of course with the Hipparcos data we do not have the dim red stars of the Pleiades. Notably the Hyades have more stars in the Sun-like region of the H-R diagram including some quite Sun-like stars. Thus, in a very general sense in the long-past days the sun could have been part of a Hyades-like cluster when it was about ~13% its current age. Of course with a more comprehensive collection of members of these cluster we could get a more complete picture.


•August 7, 2016 • Leave a Comment

Varoli had just completed the synthesis of the xanthine coupled with hydroxymethyluracil and diaminotriazine to test an interesting hypothesis of Somakhya regarding a particular class of DNA-binding proteins. She was putting in considerable effort to finish it off by the time of the autumnal break and hand it over to her student to mop up the rest of the operation. After having been delayed by the arson caused by a post-doc in the neighboring lab she had managed salvage some of the lost time. But all this had left her so tired that after fetching her kids she fell fast asleep. She felt she had not slept so well in a while and a dream unfolded of what seemed like the ancient past.

Varoli was in her last but one year of school. They never taught astronomy in school but it was perhaps as a part of geography that they had some basics of astronomy. Her teacher who taught that subject had a soft-corner for her and her sister. She came up Varoli and said: “Your eldest sister Lootika knew a lot of things about the sky and had showed her class many of those things which helped them make more sense of what was in the textbook. It was a lot of fun. I wonder if you may be able to do the same.” Varoli: “Perhaps. I don’t think I know as much as her but I could try.”

That evening session of elementary star-gazing conducted by Varoli was quite a success. The next week the teacher approached her and said: “Varoli, a major competition in astronomy has been announced at the big planetarium in the city of Visphotaka where scholarly students from all over the state or even country would be participating. When it was held four years ago your sister Lootika had brought the school great honor by winning it. I have entered your name name for the same and hope you might be able to achieve something similar.”
Varoli tried to deflect this: “I am not sure I can do it. Moreover, my parents are unlikely to let me travel to the big, bad city of Visphotaka, which is only next to Indraprastha, nowadays Shakrashapta.”
The teacher: “No No. You seem good for this. Moreover, I will be taking you all; So, I can talk to your mother and tell her that we can take the best care of you when journeying to Visphotaka.”
Varoli: “But this competition needs a partner. I don’t have any in our class.”
The teacher: “Don’t tell me that. In your sister’s day we had another student Somakhya who had made a telescope and was rumored by many to be even better than Lootika. We had asked him to partner with her but he adamantly refused saying that he would not do so. So we just found another girl who was just a dummy. Likewise, we can get Shukavati to go with you – she’s quite good at learning things by rote and you can delegate some stuff to her.”

Varoli easily breezed through the written qualifying round which had to be submitted online to the planetarium. Eventually, the teacher convinced Varoli to go and her parents to let her. For the first time her mother handed her a phone and asked her to call them twice a day. She also handed Varoli a several packets of dried berries and nuts and told her to eat them in lieu of any snacks and to be very careful in eating only well-cooked food. Her sister Vrishchika showed her the pictures of the scans of a brain of a man from Visphotaka which she had obtained from her father. She excitedly pointed to lesions telling her sister that they were the result of cysticercosis arising from him eating street food in Visphotaka. This greatly alarmed Varoli who asked Vrishchika to close those images right away. Lootika took her aside and gave her a garala-śaṅkula just before she left.

On the said day, after a journey of a few hours, Varoli and Shukavati reached the venue with their teacher. After the registration they were offered snacks and some time for socializing with other students. Thereafter they were all to receive a free planetarium show. While Shukavati was all excited and busy socializing Varoli sat silently in one corner. Before leaving Lootika had taken her to Somakhya, who had brought her to up to speed with some difficult stuff. So she felt quite good about herself and was brimming with quiet excitement of the impending clash. Some schools from big city of Visphotaka were very competitive and holding mock rounds in preparation for the competition that was to occur the next day. Her teacher came up to Varoli and said that she had arranged with the teachers of those schools for her to compete in the mock rounds too. The dormant competitive spirit that ran in all the four sisters was suddenly kindled like the god Agni praised in the mantras “kṛṇuṣva pājaḥ…” by her ancestors of yore. So she decided to participate in the mock rounds.

It soon became apparent that indeed she was ahead of most in the mock rounds but there was one guy who clearly seemed quite a challenger to her. He was clearly being acknowledged by the teachers as a hotshot and had an entourage of other guys who were vying either to get answers from him or his attention. At one point he solved a particularly difficult problem concerning mass overflow from a \beta Lyrae-type eclipsing variable, which seemed quite beyond even any of the teachers out there. Varoli remarked to herself: “He’s somebody to watch out. Hope Bṛhaspati lays him low before me.” Later that evening as Varoli was wandering among the planetarium exhibits she ran into the hotshot guy at an exhibit on emission spectra of elements from the periodic table. One spectrum showed up on the screen and the beholders had to identify it. Varoli and the hotshot guy called almost simultaneously. He said: “Aluminium”. Varoli: “Manganese”. No one else even answered. On pressing the button the correct answer showed as Manganese. Varoli could hardly conceal her triumphant smile and she remarked: “Do you not see the smearing of the indigo and green lines”. It was the first time the hotshot guy took notice of her. He was smarting from being embarrassed in front of his retinue by a girl. He looked at her again. A strange feeling passed through him. It was only for the second time in his life he had such a feeling – he had felt this strange feeling that his astronomical knowledge came from somewhere within him as though what the old Hindus would call a recollection from from a former birth. But now he felt as though he had known this girl the same way though he had never seen her before – she was not even from his city. He wondered what he should say to her but by then Varoli had drifted away to some other exhibit.

The next day the competition initially went well for Varoli; she and her partner Shukavati, who did nothing all, were one of the teams who made it to the finals. Then came a series of problems concerning relativistic jets, black hole entropy and the like which rather blew the wind out of their sails and the hotshot guy and his partner solved all of them to win the competition. Nevertheless, Varoli took her school to the second place. Her teacher was happy with that and congratulated her but she was hardly thrilled and in utter disappointment retired to her room rather than attend the prize-distribution session and the star-gazing session that was to occur thereafter. Her partner Shukavati proudly collected the prize on behalf of her school despite having been a mere spectator in the team. Just then a clump of rowdy students from a couple of schools smashed and vandalized many of the exhibits in the planetarium and stole a couple of binoculars. Hence, the star-gazing session that was summarily canceled and the students were sent back to the residences as a search was organized by the cops for the stolen property.

The hotshot guy and his friends were hanging out in the compound of the residences. While they all thought he should be celebrating his great win and the offer for a scholarship to pursue a special program in college, he remained ill at ease with his mind repeatedly going back to Varoli. Just then he suddenly saw Shukavati, who was basking in borrowed glory, pass by. He recognized her from the day’s events and went up to her: “Where is your partner?” Shukavati: “She should be in our room I believe.” The hotshot guy: “Would it be possible to meet her?” Shukavati: “I can try calling her and you can ask her.” Shukavati called Varoli and gave the phone to the guy. He said he wanted to talk to her briefly. Varoli: “I am not in a mood for any conversation.” The guy gave her his email: “I fully understand but just send me your email whenever you get the chance. Hope to meet you again sometime.” Varoli: “You have won fair and square and know what needs to be known on things that matter. Hence, there’s nothing much you can gain from talking to me. So let’s just go our ways like two people who cross each other on a railway platform.” The guy: “Then get on to your train and chug away.”

Varoli then called Vrishchika and gave a run down of the days events which were depressing from her view point. Vrishchika: “Never mind dear Varoli. Only when one attains the wholeness of knowledge does true insight dawn on one. This is so for the śāstra-s as well as mantra-s. Hence, it is good to strive for the highest standards; however it is not possible to be equally good in all the sciences and mathematics at once. Neither Lootika nor I are like that. Belonging to a high brāhmaṇa clan we should have a core competency in some śāstra-s and a general knowledge beyond that which takes you close to the wholeness of knowledge. On that path you are doing well so far. It never appeared that jyotiṣa was your primary forte in any case. Finally, there is no need to be unduly rude to winners; being better than us they could be of use – in the least merely to learn from their non-genetic aspects of success. So there is no harm briefly talking to that hotshot guy who was the vijayin and checking out his real mettle.”

Varoli said to herself: “upāgrajā has a point.; now that guy was redoubtable.” So she sent him a mail asking if he was still around and that they could meet in the general area of the residence compound. Even as they met he displayed the chomma of the crocodile. Varoli responded with that of the two birds. He responded by showing the chomma of the umbrella. She responded with that of the four faces. He said: “vāṃ deṃ vāṃ. Varoli: “yaṃ namaḥ svāhā hroṃ. He: “aham eva mitrāyur asureṇa mitreṇābhirakṣitaḥ ।” Varoli: “aham varolī devena tumburuṇābhirakṣitā ।. We had to meet, right…” Thereafter, they kept chatting about various interesting scientific issues for a couple of hours. Finally in course of that conversation Varoli said: “The evolutionary course of star after star, in galaxy after galaxy, once placed in the H-R diagram can be cleanly predicted. Now, could a comparable diagram exist which would tell us, planet after planet, where life takes root, what its course would be.” Mitrayu: “That’s a tough one to even begin comprehending.” Just then their supervisors announced that the respective sections of residences for males and females would be shut for the day. Hence, they had to get in leaving the question unanswered and depart to their hometowns the next day without seeing each other again.

Varoli just then heard her door open and woke up in a scare thinking her elder kid might be trying to get out of the house for some mischief. But to her relief it was just Mitrayu who had come home. Mitrayu: “You were so fast asleep that you did not perceive the first time I came. Seeing you so deep in sleep I decided to go to the devālaya with the kids and got food from there.” Varoli: “That was kind of you. For some reason I had a long dream of every detail of the unbelievable day we met the first time.” Mitrayu: “That’s really strange Varoli – just today Somakhya and I came up with this plan of conducting a course on the origin or origins of life the coming summer, where I would cover the astrochemistry and he the biology. I was thinking we should recruit you too for this to cover the chemistry of replicating and templating molecules. Varoli: “Sounds like fun. We should talk more of it. But talking of all this it has been a while since we have done our childhood activity of seeing the sky. We should take out our telescope and take it along for the autumnal break.”


Many years ago, when they were still young, Vidrum had shown Somakhya a strange tooth he had found near the village of his ancestral family. Somakhya had identified it as being that of a herbivorous fossil crocodile but never got the chance to investigate it further as Vidrum sadly lost the tooth. Now Vidrum floated the idea that they may make a joint trip for their autumnal break to the village so that they could investigate further. He told them that he had set up a medical center there for the less-privileged and that they could stay in the guesthouse that served it, which he used to board noted visiting physicians to provide consultation and treatment for difficult cases in course of ad hoc visits. Having been advised by his friends he had come to see the importance of fighting for the Hindu cause which he had ignored earlier. He now saw this as one means of keeping out the śavārādhaka-s from the place. Somakhya, Lootika, Varoli, Mitrayu, and their kids accordingly joined Vidrum and Kalakausha to go to that place for some days. In addition to the paleontological survey they also hoped to use the clear skies for astronomical observations and education.

After a day in the field where they had to their great excitement found the tooth of a large abelisaur they returned to the guesthouse to cook food for the evening. As Somakhya’s son Tigmanika was milling around in the kitchen in great anticipation and excitement of the night’s astronomical observations he asked his mother and aunt several questions as they prepared dinner.
Tigmanika: “I recall you showing me the two close stars in the Sapta-ṛkṣa and mentioning there were actually more stars in that system which went around each other. Can we see any of those tonight?
Lootika: “Yes, you will see that the brighter of them Vasiṣṭha will actually resolve into two close stars when we look at it through the telescope. You will see one more faint more star between Vasiṣṭha and Arundhatī but that is not part of the system.”
T: “How do we know that Vasiṣṭha and Arundhatī are connected but the other one is not?”
L: “That was a difficult question for a long time and people did not know for sure. But eventually they figured out that Vasiṣṭha and Arundhatī are likely at the same distance from the solar system and that they share the same motion with respect to other more distant background stars which is known as proper motion. From that could infer that they must be real companions.”
T: “How did they measure their distance?
L: “They measured their distance using parallax which the angular shift of a star against the background of more distance stars as the earth goes round the sun. Then you can use trigonometry to calculate distance from the angle as we know the major axis of earth’s orbit. As my hands are busy I cannot draw it out for you but you may ask your father or uncle to help you.”
T: “OK. But did you not say there were more stars in that system?”
L: “Yes, using observation of spectra, like I showed you for the sun with a prism, you can infer that there are actually six stars there. Vasiṣṭha you will see is a double with the telescope. Further, each of that pair of Vasiṣṭha and Arundhatī can be shown spectroscopically to be doubles.”
T: “Can we make out any differences between these stars visually other than brightness?”
Varoli: “Visually they are not very distinct as they are all A type stars mostly white in color. You may recall that your mother and I had sometime ago explained to you the types of stars based on stellar spectra. From the spectrum we can see that the brighter doublet of the Vasiṣṭha is enriched in the element Silicon. However, the fainter doublet in addition to being enriched in Silicon is also enriched in the rare earth metals Cerium and Samarium. In contrast, the Arundhatī dyad has one which is an A type star with no peculiar elemental enrichment and the other one is faint red dwarf. Of course we would not be able to see that with our telescope.”
T: “That sounds really exciting. So all the rare earth metals for our magnets and silicon in the sand must have formed in such stars that existed before the sun!”
V: “That’s partly correct. In smaller stars in their earlier life you only form some light elements coming from the Helium, Neon and Argon rows of the periodic table. Then when the stars age and enter the red giant state they can use iron nuclei as a seed to undergo slow neutron capture to yield some heavier elements including some rare earth metals like Lanthanum, Cerium and Samarium. When such giant stars eject their mass these elements can condense in solid grains known as star dust which is usually enriched in silicon carbide. Such star dust can seed the nebulae where future generations of stars form with heavier elements. However, not all elements can be made this way. Some rare earth metals like Europium and other metals like Lead or Uranium can only be made by the explosive process of neutron capture in supernovae.”
T: “Are there other stars with unusual element compositions?”
V: “You can ask Mitrayu about that; he can keep you busy for the whole day telling you about it. But just to give you a flavor there are stars like R Geminorum which are rich in technetium. What would that tell you Tigmanika?”
T: “Technetium is unstable so it must be made recently by these stars.”
V: “Indeed, with a half-life of just 4.2 million years it was the first incontrovertible evidence that heavier elements are actually continually made in stars because it is present in red giant stars much older than the order of millions years. Similarly, there is another interesting star in Centaurus which is rich in multiple lanthanides and actinides and has radioactive elements like Promethium and Curium illustrating the same process.”

Thus, primed Tigmanika was all excited and ran out see if his father and Mitrayu had already assembled the telescope on the limestone eminence in estate of the guesthouse.


After dinner they had all assembled on the limestone eminence. The elders gave a quick orientation of the sky above them to those who were yet young or unfamiliar. Vidrum remarked to Kalakausha: “As a youth I was always puzzled as to what Somakhya found interesting in the white spots of lights in the sky. He would always point to one or the other and say something”. Today Vidrum was more inclined to hangout a bit with them and learn a little about what it was all about. Kalakausha agreed that it was indeed a peculiar form entertainment but might be worth trying out as a change from their usual stuff. Kalakausha: “I too never understood much of this. Lootika one day showed some of us many objects when we were in the same school but I don’t remember any of them.” Seeing his friends and their families Vidrum sensed that were experiencing something profound under the great canopy of the heavens adorned with stellar frescoes. While he and Kalakausha were unable understand that feeling, they at least felt that there was perhaps something of awe up there. Mitrayu informed those who needed to know that the vast heavenly realm, while extending as though endlessly above them, was still largely governed by forces at the other end of the scale of existence – the sub-atomic world with its ionizations and nuclear fusions which was powering all that they were able to see. The only exception he remarked was the mysterious gravitation which acted at altogether different scale and was the only signal of that even more mysterious thing called dark matter.

It was a glorious night with the sprawling Prajāpati (Orion) having mounted the autumnal sky like the allusion found in the mantra used by the dvija-s to change their yajñopavīta-s. But in this yuga, having been slain by Rudra, Prajāpati was united with his dear Rohiṇī (\alpha Tauri), while his old constellation was now handed to over King Soma, the king of the vipra-s. Rohiṇī shone with the unmistakable orange luster, like the eye of the angry Vṛṣabha (Taurus), the sign of the great god Indra who was of the form of the universe. The approximately straight line formed by the three Invaka stars (\zeta, \; \epsilon, \; \delta Orionis), the triple-jointed arrow of the god Rudra, that slew Prajāpati, worshiped with the Invaka-saman, pointed on one side to the orange Rohiṇī and further ahead to the brilliant Kṛttikā-s (Pleiades) – that heap of stars verily shone forth with a bluish light as the herald of the god Agni in the sky. On the other side the Invaka-s pointed to the brilliant ārdrā (Sirius) presided over by the shooter śarva; hence the vipra-s of yore said: “ārdrayā rudraḥ prathamānameti |”. This star embellished the constellation also known as the dog (Canis Major) endowed with many bright stars, which was praised by the Bhṛgu-s of yore in the sūkta known as śuno divyasya. Between the head of the sprawled Prajāpati, who tried to flee in the form of a deer, and twin stars of the asterism of the goddess Aditi stretched the galactic band of the heavenly Sarasvatī or the foam with which Indra had slain the dānava Namuci. On this heavenly foam sailed the great ship (Argo Navis) of the gods Savitṛ, Bhaga and Puṣaṇ, which journeyed to the southern realm of the black Yama who slays the martya in a thousand ways.

Hence it was no surprise that to get started Mitrayu and Somakhya brought the ten inch telescope to the Orion nebula (M42) and showed everyone the great trapezium star cluster of (\theta Orionis) in the middle of the nebula. While the two of them and their wives had seen it innumerable times in their lives the magnificence of it was still no different from the first time they had seen it. However, Vidrum and Kalakausha having seen it could not understand why their friends used terms like “most wondrous”, “the greatest sight” and the like. Then Mitrayu asked them all if they could discern any color. While Lootika agreed with her saṃbandha-s about the glory of the sight, she remarked that she saw only gray. In contrast, while most of Lootika saṃbandha-s saw some green color, only her daughter Prithika, though small, claimed to see green and a bit of red making Somakhya remark that at least her eyesight was better than her parents’. Mitrayu: “The green comes from doubly ionized oxygen while the red is a type of excited hydrogen emission in most part.”
Tigmanika: “Normally oxygen takes electrons. If it has lost two electrons then it must be rather energized. How did they figure out that it must be from ionized oxygen?”
Mitrayu: “Indeed, the stars of the Trapezium supply the energy. The main star you see there has the temperature of about 45,000 degrees and it emits a good part of its energy in the ultraviolet part of the spectrum, which is what makes the Orion nebula glow. In the past they thought that the spectral lines of doubly ionized oxygen belonged to some unknown element. Only much later where people able to perform calculations based on the improved understanding of electronic transitions between different energy levels of an atom for elements across the periodic table that showed that these lines should come from doubly ionized oxygen.”


Now Somakhya wanted to train his putra in using the telescope the hard way: “Tigma, when I was young my first telescope was something I made myself. That is something you will have to do someday if science interests you. That telescope did not have any computer to help point at the object of interest. We had to do it by forming a mental map of star patterns from a star map and guide the scope by hand. Now look at the star map and take the scope to \iota Orionis.”

Being relatively close to the Trapezium, Tigmanika was able to quickly take it there. To his surprise he found that it was a multiple star with a closer and farther companion, and he called out to the rest to see the triple star at the eyepiece.
Mitrayu: “The brighter component of \iota Orionis which you see there blazes away at the brightness of 11,000 suns or more! Tigma, I had earlier explained magnitudes to you. So can you now tell us what you might think is the brightness of the fainter closer companion?”
Kalakausha: “You guys are really stressing the kid…”
Lootika: “That’s OK. He’ll have to learn sooner than later that science is not about just looking at fascinating things but being able to perform with ones own head and hands.”
By then Tigma had gotten an estimate of the difference in brightness of the companions visually and had done the calculation taking help from the calculator on his mother’s phone: “I would say the closer companion would be about 250 times as bright as the sun.”
Mitrayu: “That sounds about right. The brighter component will be brighter than Venus even from a distance of 32.6 light years and the closer of the fainter companions would be nearly about as bright as Sirius! The magnitude at that distance is called absolute magnitude. Somakhya, Lootika have you taught him that?”
Lootika: “I guess he should read up on all that by himself, as we had done as kids. Alright, now look closely at the star map and locate \mu Columbae.”

This took Tigmanika more time but he eventually took the telescope down south and located it: “Ah this sort of looks of a similar to \iota Orionis! May be it to has a hint of blue.”
Somakhya: “Good. Now look at AE Aurigae on the map and take the scope there.”
Tigmanika: “That’s way back north again.” After having located an unmistakable asterism in Auriga he remarked: “That asterism looks interesting, like a parallelogram with an orange star (16 Aurigae) at the tail !”
Somakhya: “Indeed, that was one of the asterisms on which I tested the first telescope I made.”
Lootika: “O dear, that was perhaps one of the first things you showed me through your scope too on the way to M37…” After a little struggle Tigmanika got to AE Auriga from that asterism and having spied it for some time was dazzled by the sight: “It seems to have some nebulosity!”
Mitrayu: “That’s the famed Flaming Star nebula. Take a good look and let us all have a look too.”
After everyone had had their look Somakhya asked his son: “What do you think of the positions of AE Aurigae and \mu Columbae with respect to the Orion nebula?”
Tigmanika: “Well, they are on opposite sides of it. Why, they seem very roughly at a comparable distance from the Nebula to the north and south.”
Somakhya: “Amazing as it might sound determination of their proper motion suggests that they were forcibly ejected from the region of the Trapezium from the Orion Nebula cluster in opposite directions. The main component of \iota Orionis, where you saw a single bright star, is further a spectroscopic double. This, it seems was also part of this system. Thus, four stars seem to have come close as part of the original celestial event in the Orion nebula cluster. Two stars appear to have captured each other to form the main component of the \iota Orionis system, while two others got flung in opposite directions as AE Aurigae and \mu Columbae.”

Tigmanika: “That’s most remarkable! I find it rather wonderful that we have been able determine so dramatic an event. If they know the motions they should be able to figure out when it happened?”
Mitrayu: “That’s correct. It seems to have happened very recently around 2-3 million years back.”
Varoli: “That would about the time of the earliest Homo. If the earliest members of our line had any interest gazing upwards they could have in principle caught sight of this.”
Mitrayu: “In fact there was an even earlier event that might have kicked a star out of this system about 4-5 million years ago. That is 53 Arietis. Tigma see if you can get hold of that one next.”
Lootika: “These stars have all fled from corpse Prajāpati like his blood splattering upon being struck by The god’s arrow. (tasya devasya śareṇābhihataḥ।)”.
Tigmanika: “That’s quite striking. Why is it that we have multiple ejections from the same site? Did some of these young stars explode as supernovae throwing out the rest?”
Mitrayu: “Indeed, that is one possibility, especially, if one stars goes off as a supernova in close multiple star systems. The other possibility is that such stars are formed in dense star-forming regions close to each other and are thrown out due to a gravitational slingshot.”
Tigmanika: “How would such a slingshot work?”
Somakhya: “For that you have study more of gravitation and complexities of multi-body problems. Something for which your mathematics has to go a long away but we might be able to show you some simulations on a computer to get a feel for it.”
Tigmanika: “This gives me a thought – perhaps we don’t see such dazzlingly brilliant stars in our sky because they are not conducive for life to develop in their vicinity? Being blasted by a supernova or being part of unstable multiple star systems does not seem like something too conducive for life?”
Mitrayu and Varoli caught each other’s eye and smiled. Somakhya: “Tigma, that something you may want to think more about to decide if it might be exactly true or not.”
Tigmanika: “Now that you say so, it does seem more complicated because aunt Varoli had explained to me how radioactive elements are formed in supernovae. So we would also need a supernova in the vicinity to get the Uranium and Thorium.”

By then Vidrum’s eyes had started making out some patterns in the sky. He pointed up and asked the rest: “What is that tight group of stars up there. May be Mitrayu mentioned them but only now it is striking me that I have seen those as youth more than once.”
Kalakausha: “I too seem to have seen them. I recall even Lootika saying something about them.”
Tigmanika: “They are the Kṛttikā-s, which tradition holds to be the constellation of the mothers of the god Kumāra – the Pleiades or M45. Let me take the telescope over there.”
Having changed the eyepiece to the lowest power widefield one he swung the scope over to bring the Kṛttikā-s in view.” Then they all took a look. Vidrum: “Never knew there were so many stars in there.” Prithika and her cousin Vikranta , Mitrayu’s son, noticed some nebulosity that some of those present there were able to see. Somakhya then asked his putra to take the scope to AB Doradus.


With some difficulty he located it close to the southern horizon. Somakhya: “Its not often we get to see this star that keep so south.” T: “Why are looking at it. It looks rather unimpressive, though I guess it is some kind of variable from its name.” S: “Unimpressive it might look but, teaches you an important lesson. It was actually born along with the Pleiades in the same star-forming cloud.” T: “That is rather surprising to learn! How is it the Pleiades are all up there close together while this one has drifted so far away?” Mitrayu: “Actually there are bunch of such which were born along with the Pleiades but have drifted away. During star-formation some stars remain together as part of the birth cluster but some drift away either due to ejection from dynamical gravitational interactions or due the initial supernovae. Indeed a major fraction of the bright O and B stars show signs for being ejected thus. However, AB Doradus itself is a dim orange dwarf that occasionally flares up, which simply seems to be part of a group that disassociated early from the Pleiades cluster went their own ways.”

T: “That’s rather interesting indeed. So what about the Sun? Do we know the stars of the cluster with which it was born? Is \alpha Centauri one such?”
M: “Indeed the sun was born in some ancient open cluster that has now disintegrated. By studying the motion and chemical compositions we can say that \alpha Centauri and the sun have merely drifted close to each other and were not connected at birth. The stars with which the sun was born are known as solar siblings. Finding them has proved generally challenging but there is one star in Hercules that might be a sibling (HD 162826). You can find it some day when Hercules is up. It is somewhat whiter than the sun. Now you can spend some time thinking about what kind of cluster the sun must have been born in.”
T: “May be something like Hyades rather than the Pleiades with fewer superhot stars?”
M: “Good line of reasoning. Investigate it further.”

T: “Conversely there must be stars which were not born with the Sun but are exactly like it. Do we know of any such?”


Somakhya: “There are bunch of stars that are generally similar though not identical to the Sun. They might be the same spectral type but might have different chemical compositions or ages than the Sun.” Looking up his catalog he continued: “If you might like try to locate this star HIP 11915 in Cetus, which is quite Sun-like and even has a Jupiter-like planet around it. It may house a solar system around it like ours.”
After Tigmanika had located they all took a look. Vidrum remarked: “that really reinforces our insignificance. A star so dim is now said to be just another Sun among the so many we see up there. That’s really depressing and deflating.”
Tigmanika however took a much brighter view of the same: “Now can we say that around a significant number of these G-dwarfs which have their own solar systems there are planets housing life? And would that life on each of those planets play out like ours?”
Varoli and Mitrayu looked at each other a bit amused and almost simultaneously remarked: “Wish we could answer that more precisely.”
Tigmanika: “When you describe the fate of stars based on just their temperature and luminosity is it not strange we don’t have something simple about where and how life’s fate would be?”
Lootika: “tasyopari cintaya cintaya toka!”


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