Wednesday, May 25, 2016

I wrote this bunch of reflections when in the Valley, returning after 29 years...

My land
Consider the constant migrations of people all over the world , from the smallest groups to the largest, out of Africa, eddying and swirling across continents, even crossing oceans, and it becomes clear that no particular land ‘belongs’ to any specific group. The flows of genes (Y- and mitochondrial), languages and cultures - all of them continuously changing – have been used to map the migrations, often with contradictory results.

What does it mean to be the ‘original’ inhabitants of Africa, if these autochthons just barely qualify as being (recognizably) ‘human’? Even hunter-gatherer societies are territorial, but with the advent of agriculture and ‘settled’ populations, land ownership became even more of an issue. So, to define ‘original’ inhabitants, do we stop at ‘recorded history’ – or do we carry on digging into the often mythical past?

Plants and animals often seem to be more rooted than humans to particular geographical areas – but even they have travelled extensively, sometimes on their own, most often using birds and insects, and sometimes with human aid.

Still we are tied to the land we live in by quirks of geography, weather and food which affect our customs, clothes and language in odd ways. (Bengalis must have hilsa, mutton for Kashmiris, maple syrup for the Canadian…) There are ‘geographical indicators’ in our genes e.g. the lung capacity of Sherpas and Tibetans and the sickle cell anaemia (malaria-resistant) genes in parts of India and Africa. Other traits may not be so obvious. Some mutations may be neutral.

Although we are ‘tied’, we are not truly bound to a given place: one sibling may stay a whole lifetime in one place, another may land up in some other continent, and yet another may be travel all over the world and never settle anywhere – while eyeing the other planets in insatiable wanderlust. In principle, anyone could live anywhere – but in practice there are many barriers to overcome.

But in what ways does a land alter your perceptions, your language? The story that the Eskimos have 40 words for snow turned out to be an exaggeration. Yet it is true that the seasons and the plants (some of them carried along with us), leave their imprints upon our languages. Landmarks acquire historical overtones and memories that last centuries if not millennia – like Mt. Kailash for Hindus or the Wailing Wall for Jews - even if they have been 'lost'. These can arouse feelings of connection with the land, and feelings of ownership. Most of these seem to be for religious reasons, which would naturally get entrenched in the collective memory of a community or group.

But in many places this sacred feeling can be fixed on a river (the Ganga, the Nile, etc) or a lake, that would mostly be adjacent to a temple. It could be a place of pilgrimage (e.g. the Amarnath Yatra, Kailash Manasrover Yatra - which is a mountain and a lake – or Lake Baikal,…).  Often the river has sufficed to demarcate ‘us’ and ‘them’… but the paths of neither rivers nor men are invariable and stable.

Imagine belonging to a nomadic community that has always gone South in the winter, and returned North in summer, along well-worn paths marked and traced by numerous generations of ancestors, along with ‘their’ herds of reindeer, or cattle, or whatever…

The feeling of being attached to some place could also be related to some purely individual memory: like some place you went to with your parents as a child… or it could be an accretion of memories, just the sights, sounds and smells of the village or town that you grew up in…

For many a Kashmiri Pandit, his or her identity is bound up with temples of Shankaracharya, Martand, Tulla Mulla,.. And with the lakes and gardens of the Valley, the apples, the mulberries, the rainbow trout … yet, it may just be the very quotidian, mundane place you grow up in…it doesn’t have to be a putative paradise! 

But it is very hard to pin it down: why do we have these feelings for our ‘homeland’? For me, the song from the film Kabuliwala, “Ae Mere Pyare Watan…” comes closest to expressing the ineffable feeling of nostalgia and homesickness. I do not relate to the ridiculous anthropomorphism of “Bharat Mata”…

Farmers and fishermen are closest to the land and the sea. They are tuned to the seasons and the soil, and the tides and currents of the sea. Local knowledge is important: take the case of the Turkish farmers who refused to remove stones from their fields, because they – counter-intuitively – boosted their yields. That knowledge, often accumulated over generations, ties farmers to their land.

But this is a paradox: we don’t ‘own’ the land – it doesn’t even need us! – and yet we have this feeling of ownership. And particularly in the Anthropocene, humans have made irreversible changes at the global level, leaving behind detectable residues of plastic, concrete and radioactivity.

Food is sourced from so many countries today that we are in denial of the seasons, of the annual and decadal rhythms of the land. The elites imagine themselves as global citizens – and think that there is no price to be paid for these extravagant, profligate ways of global citizenry.


Does the land ‘belong’ to us? No matter how many fences we weave, how many walls we erect, how many canals we dig, dams we build, whether we map it with rulers, theodolites or GPS – the land does not need us – as in Nevil Shute’s “Earth Abides”, it will be there long after we are dead and gone and long forgotten, both as individuals and as a species, that tends to think too much of itself. We may irrigate the land with water, tears or blood, and we may indeed belong to the land – as do countless other organisms, ranging from bacteria to whales, but it does not belong to ‘us’.

Tuesday, May 24, 2016

Maximum height of (mostly) mountains and a flagpole:

Victor Weisskopf gave a derivation of the limit to the height of a mountain. This can be found in different sources (with minor numerical differences) [1].
The simplification is that he assumes a silicon dioxide (rock) mountain of area A and height h on a SiO2 base. Its height is at its limit when its weight Mg bearing down on the base has a total potential energy Mgh that is greater than the liquefaction energy of SiO2 El. That is, it would cause the base to turn into liquid, and it would sink into the liquid. The value of EL is assumed to be 0.05X the binding energy of SiO2 EBE, and EBE is O.2X the Rydberg energy (13.6 eV/atom). So, EL = (0.05)(0.2)(13.6) = 0.136.
The mass of the mountain can be obtained from the volume of the mountain (V = Ah) multiplied by the number density of SiO2 nSiO2 (atoms/m3) and the mass of the protons & neutrons (assumed equal) mp (1.67x10-27 kg) and the mass number AMS (the number of protons and neutrons in SiO2):
M = AhnSiO2mpAMS
The mountain sinks by a distance z and displaces a volume Az of SiO2 in the base, and the potential energy (on the l.h.s. equals the energy of liquefaction of the volume Az on the r.h.s.):
Mgz = nSiO2 Az EL
Or, cancelling z:
Mg = nSiO2 A EL
But M itself involves h, so:
AhnSiO2mpAMSg = nSiO2 A EL
The number density nSiO2 and the area A both cancel, leaving:
h = EL/(mpAMSg)
Numerically, this becomes:
h = [(0.136)(1.6x10-19)]/[(1.67x10-27)(60)(9.8)] = 21.6 kms
a)      For a conical mountain, the mass M will depend upon the volume, which is now Ah/3. So for this case, the height will go up by a factor of 3X.
b)      Weisskopf’s argument takes into account energy of liquefaction, whereas much before the base turn liquid, it would undergo plastic flow. This would lead to a lower and more realistic value of the maximum height.
A similar, though much simpler argument is given by Gnadig et al [2]:
The base of a mountain that does not melt under its load if:
h < L/g
where L = latent heat of melting of metals (200-300 kJ/kg), and the answer for Earth is 20-30 kms.
Gnadig  [2] also points out that on Mars g ~ 4 m/s2, and in fact, the highest mountain on Mars is Mt.Olympus which is 26 kms high, according to Gnadig, but 21.9 kms high according to Caplan [3].
P.A.G.Scheuer [4] was more interested in the question of how high a mountain can be…on a neutron star (he gets a range of 0.04 to 0.4 mm for the maximum height of his mountain range!). But on the way, he deigns to look at Earth too. Scheuer gives two answers h1 and h2, where h2 is much higher and refers to a broad-based pyramid:
h1 =4 (1.5x107)/[(2650)(9.8)] = 2.31 kms
Scheuer gives this number as 2.25 kms assuming a value of yield stress of Y = 1.5x106 kg/m2 which equals 15 MPa ( the latter is the kg force multiplied by g).
h2 = (h1b/4)1/2 where b is the base of the mountain, such that b >> h.
Using an unconfined compressive strength value of 100-250 MPa for granite gives a much larger number for h1 than the one Scheuer got:
h1 = 4Y/(rg) where Y is the yield stress, Y = 250 MPa for granite and granite density r = 2650 kg/m3.
h1 = 4(2.5x108)/[(2650)(9.8)] = 38.5 kms
Miskinis [5] gives a somewhat different argument, using the Hooke’s law relation between the shear stress sr on compressed rock and the shear angle q:
sr = Gq
where G is the rock shear modulus.
Then he uses the weight of the mountain argument on the surface area S, to find the maximum height, giving:
s = mg/S = grhmaz
Solving:
hmaz = Gq/(rg)
Substituting values of G = 12.5x109 Pa for basalt, r = 104 kg/m3 for the increased density of compressed rock, and q = 0.1 rad (or 6°), he gets a maximum height of 12.5 kms.



The mountain argument can now be applied to an iron flagpole, assuming it to be an iron mountain on a silica base, adapting Weisskopf’s argument ( so why bring up Gnadig, Scheuer and Miskinis? Never mind!).
A similar argument can be used for an iron ‘mountain’ of area A and height h, on a SiO2 base, giving:
AhnFempAMSg = nSiO2 A EL
In this case, the number densities are different on both sides of the equation: nSiO2 = 2.66x1022 and nFe = 8.49x1022 cm-3, and AM, Fe = 55. The number densities are obtained from the density, the mass number AM and the Avogadro number: n = rNA/AM. The areas still cancel, so:
h = [(2.66x1022)(0.136)(1.6x10-19)]/[( 8.49x1022)(1.67x10-27)(55)(9.8)] = 7.6 kms
So, if one goes by Weisskopf’s argument there is plenty of scope for building higher flagpoles, since even the Jeddah flagpole is just at 170 m!
Of course, a structural engineer could probably come up with a much lower – and more realistic -- limit by invoking other problems such as yielding or buckling… but that stuff is way too complicated… I found one reference on “Metal Flagpole Design” which is the American National Standard but it is much too iterative and demands too much knowledge of material properties.
And, why build a high flagpole when you can just as well stick your flag on top of some convenient hill or mountain? Tell that to the BSF!

References:

[2] Peter Gnadig et al ,”200 Puzzling physics problems” (Cambridge Univ.Press, 2001) p.197
[3] M.E.Caplan arXiv 1511.04297v1 “Calculating the potato radius of asteroids using the height of Mt.Everest”
[4] P.A.G.Scheuer “How high can a mountain be?” J.Astrophy.Astr.2 (1981) 165-69
[5] Paulius Miskinis “Mathematical modeling of mountain height distribution on Earth’s surface“ Geologija 53 (2011) 21-26

Wednesday, May 4, 2016

Flags, striped shirts and zebras
The Olympic motto is “Citius, altius, fortius” (faster, higher, stronger).
Our nationalist flag-wavers have decided to emulate this motto: our flags must go higher, and since the wind there is faster, the flags must be stronger.
Firstly, the Centre decreed (to inculcate ‘nationalism’) that every Central University must have a 207 foot high flag.
The proposed flags will be modeled on the one in New Delhi’s Central Park, which is 207’ high, made of knitted polyester, is 90’x 60’ and weighs 35 kgs.
More recently, the Border Security Force has decided to put up a flag at the Attari-Wagah border by 2017 which will be 350’ high.
“The current record for the tallest national flag is held by the one atop a 293-ft-high mast in Jharkhand’s Ranchi. This 66-ft x 99-ft flag was hoisted in January by Defence Minister Manohar Parrikar — it was, incidentally, brought down today for repairs after being stuck at half-mast 10 days ago.”
The new flag, to be hoisted in Jan.2017, a top Border Security Force official said, that at 350 feet, “it can be seen both from Lahore as well as Amritsar (Amritsar and Lahore are approximately 18 km away from the international border)”.
“At that height, the flag would have to be proportionately sized, therefore it would also be the largest Tricolour. Officials say the material to be used for the flag is also being discussed because at such heights it may get damaged by rain and high winds.”

The first question is: will the flag be visible in Amritsar and Lahore. A little simple trigonometry shows that the distance to the horizon is given by:
S = Ö2hR
Where the height h of the flag is much less than the radius R of the Earth (assumed as 6,400 kms). If h is not much smaller, an h2 term is to be added within the square root. With h as 106 metres, this distance is 36.8 kms. So the flag should indeed be visible, as claimed.

The second question is: will the flag be visible to the unaided eye (no binoculars or telescopes)?
The minimum angle resolved by the human eye is 0.5 milliradian, as opposed to the diffraction limited resolution of 0.2 milliradians under optimum conditions, according to the following website:


At a distance of 18 kms, the smallest visible size would clearly be: (18x103)(0.5x10-3) = 9 metres.
Even the Ranchi flag is 99’x 66’ wide (30x20 metres) which would translate to about 3x2 pixels – and the flag planned will be proportionately bigger.

However, to be a bit more precise, a flag like the American flag has a pattern of Stars and Stripes. The Indian flag has just 3 stripes (never mind the Chakra!). So can these stripes be seen?

According to the following website, the appropriate parameter is the visual acuity of the human eye:


“The ability of the eye to resolve detail is known as "visual acuity." The normal human eye can distinguish patterns of alternating black and white lines with a feature size as small as one minute of an arc (1/60 degree or π/(60*180) = 0.000291 radians). That, incidentally, is the definition of 20-20 vision. A few exceptional eyes may be able to distinguish features half this size. But for most of us, a pattern of higher spatial frequency will appear nearly pure gray”.

This angular resolution is almost the same as that mentioned by the hyperphysics.phy site, so nothing much changes.

Of course, one might note that the colours in our flag are not black and white, and that would make the analysis more complicated. And to be absolutely rigorous one should take into account the modulation transfer function (MTF) of the eye, because, according to Norman Koren: “The statement that the eye cannot distinguish features smaller than one minute of an arc is, of course, oversimplified. The eye has an MTF response, just like any other optical component.

The only way I’m gonna do the MTF analysis is:
 a) somebody pays me, or
b) I’m feeling really, really bored and I’ve got nothing to do…

Anyway, let me just start ending this post by noting that the world’s highest flag is in Jeddah, Saudi Arabia:


and it is 560 feet (170 m) high. The flag weighs 570 kilograms (1,260 lb) and is 49.5 metres (162 ft) long and 33 metres (108 ft) wide. The area of the flag is equal to ‘half a football field’.


At 400 ft, the flagpole will be the tallest the Shebogyan, Wisconsin, the company Acuity has constructed — and the tallest in North America. The flagpole will support a 60 ft wide by 120 ft long American flag, which will be weighted at the bottom to prevent it from wrapping around the pole. 

3rd question: can the unaided eye see the Jeddah flag from outer space, say the International Space Station? The ISS height is about 400 kms above the Earth, so the minimum size needed would be about:
 Wmin = (0.5 x10-3)(400 x103) = 200 metres.
I guess it would not be visible – even assuming that the flag were laid flat on the ground – which would not be countenanced by any self-respecting nationalist. Maybe if we allow for a satellite just at the edge of outer space (100 kms altitude), the Jeddah flag may just be visible as (almost) one pixel.

Can we expect an arms race at the Attari-Wagah border, with each country trying to push up its flagpole even higher? Anyway, such a competition is better than war…

Which leads to: ….Q4: what is the limit? With existing materials, how high can the flagpole go? Victor Weisskopf gave a derivation for the maximum size of a mountain, which can be tweaked a bit…. Let’s go into that after we touch on striped shirts and zebras – in the next post (whenever that happens!).

Striped shirt:

 If we assume a shirt with 5 mm wide alternating black and white stripes, and taking the visual acuity (VA) of a person with 20/20 vision as 0.5 mrad, then the maximum distance at which the stripes can still be resolved is (0.005)/(0.0005) = 10 metres, under optimum lighting. At a greater distance, the shirt will just look gray – a very common experience that almost everyone would have had.
The dependence of VA on background illumination, for 100% contrast between stripes, according to Norman Kopeika in his book, ”A system engineering approach to imaging” (SPIE Press, 1998) p.389
 is:

Illumination (foot-lamberts, fL)
VA (mrad)
1
0.36
10
0.28
100
0.20
This applies to the photopic system i.e. to the cones in the human visual system.
1 fL = 10.76391 lux equivalent. Because 10.76391 = (3.2808)2, the conversion factor from metres to feet.

Zebra stripes:

There is a paper on the ease with which predators such as lions, hyenas and humans can perceive zebra stripes – which are alternating white & black of different widths. The paper is entitled: “Zebra stripes through the eyes of their predators, zebras and humans” by Amanda Melin et al (March 2016). 

A human with 20/20 vision should in photopic (daylight) conditions be able to resolve the widest stripes (~3 cms wide) at about 180 m, while the lion should resolve the stripes at 80 m, and a hyena at 48 m – while a zebra would resolve the other zebra’s stripes at 140 m.

The authors argue that the ideas that the B&W stripes serve to ‘break up’ the outline of the zebra and make it difficult for predators to see them are unlikely to be correct because the vision of predators is not that good anyway. What I’m interested in is that the authors use three numbers for the visual acuity of humans:

Illumination level
Visual acuity (mrad)
Photopic (daylight)
0.145
Mesopic (dusk)
0.377
Scotopic (night)
6.77

The numbers for mesopic and photopic are close to those for the higher illumination levels (10 and 100 fL) above. As expected the zebra would have to be a lot closer for one to see the stripes by moonlight…


The above links to a photo taken of a zebra at night and the stripes are clearly visible … but the distance is not stated, and besides the camera has optics…


To conclude: flags, striped shirts and zebras have something in common: the eye of the beholder.