Monthly Archives: May 2000

stable and unstable resonance orbits

Please broaden ur e-mail viewers vause we have diagrams today :

 

      COntinuing with VERITAS on orbital resonance that we had yesterday.

Now there are two configurations for resonance : stable and unstable .

 

      To understand these we will first have to understand some terms :

aphelion , perihelion . All planetary( or moon ) orbits are elliptic.

SO at times the planet is near the sun . This position is called perihelion .

When the planet is farthest from the sun the position is called aphelion.

The same applies to moon-planet orbits . When the moon is closest to  the planet it is said to be in perihelion.

 

 

      Now imagine a SUN and two planets p and P . The planet P is

a large planet and p is a small planet . the planets p and P have

a 1:2 resonance and the time period of rotation of p is 4 years .

The time period of P is 8 years ( obviously ! )

 

 

There can be two configurations :

 

 

Configuration 1 :

 

                                                                                                 P

 

                                             p

 

        SUN                              SUN               P                           SUN

         p                                                                                        p

 

         P

 

 

      time = 0                               time= 2 yrs                                  time = 4 yrs

 

 

 

 

 

 

Configuration 2 :

 

                                                                                                   P

 

                                               p

          SUN                             SUN                P                          SUN

 

           p                                                                                      p

 

           P

 

 

          time =0                      time = 2yrs                          time = 4yrs

 

 

 

 

      The difference between these 2 configurations is that in

Configuration 1 the planet p is close to P when p is close to SUN ( time = 0 )

ie is at its periheilion . In configuration 2 p is close to P when ( time=0)

p is far from the sun ie is at is aphelion .

 

 

         The configuration 2 is dangerous for p because it comes under

the influence of P when it is far from the sun . Thus P can disturb

p’s orbit and give it more influence than SUN . Bad position.

 

 

         Configuration 1 is better for p because when it is close to P ( time0)

it is also close to SUN and its orbit is mainly under the influence of

the large SUN . This configuration is more stable  resonance position .

 

 

      This is similar to a studious student meeting the class bully when

the teacher is close ( configuration 1 ) or when the teacher is far away(

configuration 2 ! very scary )  🙂 

Resonance in Orbits

First a little Astronomy poetry and then the serious subject :

 

WHEN I HEARD THE LEARN’D ASTRONOMER

 

When I heard the learn’d astronomer,

When the proofs, the figures, were ranged in columns before me,

When I was shown the charts and diagrams, to add, divide, and measure them,

 

When I sitting heard the astronomer

where he lectured with much applause in the lecture-room,

How soon unaccountable I became tired, and sick,

Till rising and gliding out I wander’d off by myself,

In the mystical moist night-air, and from time to time,

Look’d up in perfect silence at the stars.

 

 

            Resonance in Orbits

            ——————–

 

      The moon goes around the earth in 28 days . It takes the same amount of time to rotate around its own axis . We thus always see the same face of the moon . The moon is said to be caught in a 1:1 orbit-spin resonance by the earth .

      It was believed that Mercury is also caught in a 1:1 resonance by the sun . Astronomers had observed it and found that they always see the same face of Mercury. This was later proven wrong . There is a time of the year when observing Mercury is particularly convenient and astronomers looked at Mercury only during that period . Mercury actually goes around the sun in 88 days and it revolves around its axis in 59 days . So there is a 2:3 resonance ratio ( 59/88 ~ 2/3 ) . But then how did astronomers always see the same face of Mercury year after year ? The answer was that if u look at mercury every year at the same time then it would have completed 365/88 ~ 4 revolutions around the Sun and 6 rotations around its axis and u will again see the same face ( u would have seen the other face if there were 6.5 or 5.5 rotations ) .

 

      This was one kind of resonance – the ratio between a body’s rotation time and revolution time . There is another kind of resonance – the ratio of 2 body’s revolution time .

An example of this is found in the moons of Jupiter. Ganymede has a revoution period around Jupiter twice that of Europa which in turn has a revolution period twice that of Io. So it is a 4:2:1 resonance .

 

      Another example is that of Neptune and Pluto . The orbit of Pluto is very different from the rest of the planets . While the rest of the planets revolve in one plane , the orbit of Pluto is 17 degree tilted to the plane of the rest of the planets . Also the orbit is very eccentric( ie more elliptical than the rest of the planets ) .

And the orbit crisscrosses the orbit of Neptune . So for about 24 years of its orbit Pluto is the 8th planet and Neptune is the Nineth .

So will Pluto and Neptune ever collide ? Nope ! The reason is that Pluto and Neptune have their orbits in a 3:2 resonance . So when Pluto’s orbit crosses Neptune’s orbit then Neptune in far away ( on the other side of the sun ) . And it will stay like this forever . So Pluto and Neptune are forever tied in a cosmic dance with each other.

 

      Some Astronomers feel that these resonances make the orbit of the bodies more stable . There is a new theory which says that planets were not actually formed where we see them now . They migrated to their present positions  because the present orbits are the most stable because of the resonances.

Holy Roman Empire

 

Today we will talk about the Holy Roman Empire . It was more of a political institution than an empire . It lasted from Dec 25 800 AD to 1806 . The “empire” shifted  its boundries many times during this period.

It was not Roman . For a long time it didnt even have Rome within its terrotiry. The only “holiness” in the empire was that the pope and religious heads tried to assert political authority .

 

      By the 8th century the popes tried to get more political power . They thought of the kings as the political arms of the church . on Dec 25 800 the pope Leo III crowned Charlemange the emperor . This was the birth of the Holy Roman Empire . After charlemange his empire split . A succession of emperors( who were nominated by the pope ) followed .

 

      The “core” of the empire then shifted to Germany . Otto II gave himself the title of the Roman Emperor . Then there was a controversy . The popes gave themselves the right to “appoint” emperors . But the rulers did not line this idea . And a long “battle” followed .

 

      By the 13th century italy ceased to be a part of the empire .

For several centuries the empire maintained itself in germany . It was little more than a coalition of some german stetes . With each state having its own ruler .

 

      The thirty years war ( in the 16 century ) devestrated Germany

and the “empire” became really weak .

 

      The end of the empire came when Napoleon marched east . The then

emperor Francis II gave up the title of Emperor and the “empire” died .

It persisted for some more time as an ideal which Bismark and later Hitler

tried to revive . But in vain .  

 

      The empire was always an idea used by the emperors to associate

themselves with history . Any emperor could generally give himself

the title of the Roman Emperor ( if he could convince the pope ) . And

for the church : it gave it political power .

 

Leonhard Euler: Analysis Incarnate

Leonhard Euler was one of the greatest mathematicians of all time .

He made a huge number of contributions in almost all fields of mathematics.

He was the most prolific mathematical writer of all times. It is said that “Euler calculated without effort , as men breathe … ” .

 

      Euler was born in Basel,Switzerland on April 15 1707 . His father wanted him to become a clergyman and send him to University of Basel . Euler graduated from the univ in 1724 where he had studied theology and Hebrew. But he came under the influence of a famous mathematician Johann Bernoulli and Bernaulli was impressed with the mathematical abilities of Euler and convinced his dad to let him become a methematician .

 

      In 1727 he joined the math dept of Academy of St.Petersnerg , Russia.

He married and had kids . But the amount of math he did was just too much !

While in Russia he became blind in one eye , because he worked on a problem for for 3 days and nights without rest !

 

      In 1741 he moved to Berlin and took up a position there .

In 1766 he went back to Petersburg to become the director of the academy there . There is an interesting story here : The queen of Russia Catherine was disgusted by a philosopher Diderot who would find new arguments to prove that GOd does not exist . Catherine invited Euler to shut up Diderot . Someone told Diderot that Euler has a mathematical proof to prove the existance of God . Diderot asked to hear it .

Euler said “Sir  (a + b^n)/n = x , hence God exists . Reply .”

Diderot had no idea of what hit him and soon returned to france .

 

      something about his contributions : He combined Leibniz’s Diffrenential calculus and Newton’s Method of fluxions to form what we know as calculus.

He invented the theory of functions . He invented Calculus of Variations .

He made NUmber theory into a science . He made significant contributions to the mechanics of rigid bodies . He was responsible for many commonly used mathematical notation like e , pi , the sigma symbol . He also did a lot of work in goemetry .

 

      He produced more mathematical work that any other dude . His combined works will fill 70 large volumes . It is said that he could produce a Mathematics paper in 30 minutes . And he had a huge pile lying on his desk . Whenever some Mathematics journal ran short of papers they would just pick up a few from the pile and publish them without review !! And Euler could work anywhere . He didnt need peace or quiet ! He would have a child in his lap and be thinking of cutting edge mathematics .

 

      In 1766 he became completely blind . But that didnt slow his mathematicsl pace . He had a wonderful memory and could produce mathematics even without writing . He would think and talk and his son would write !

 

      He was just too good . Prople called him “Analysis Incarnate ” .

He died in 1783 .

 

The History of Pizza

 

Today I thought we will have a fun and paushtic VERITAS . so I discovered the history of Pizza . Sharing it with you .

The idea of using the bread as a plate came to the Greek . They

thought that it was a cool idea to eat the food along with the

baked plate(the base ) .

 

The Pizza has been a basic food of the italian peasants since the stone age .

The italian dictionary has an entry for piza which means : hot pie

was plucked from the oven .

 

But it was considered a food of the poor and was not really very popular.

 

Queen Margherita of Italy got curious about the “flat bread ” that she saw

people on the streets of Napoli eat.SHe just had to try that!( my kinda gal !)

There was a Pizza maker by the name of Esposito who received royal summons

to prepare a special pizza for Queen Margherita . And he wanted to really

impress the queen( who wouldnt ? ) . So he decided to salute the colors

of the italian flag by adding white mozzarella cheese to red tomato and green

basil . THis kinda pizza is still popular today . So he prepared and delived

the pizza to King Umberto and Queen Margherita . That was the first

pizza delivery . This was in 1889 . The pizza was called Pizza Margherita.

The queen loved it so much that she wanted this “flat bread” whenever

she left her palace . The queen was very popular and decided the fashions

of the day . SO pizza started getting popular among the aristrocats of the

day . Pizza shops opened all over italy and people started experimenting

with various toppings .

 

True mozzarella cheese is made from the milk of a  water buffalo .

 

The first pizzeria opened in US in 1905 in New York . But it was not

until second world war that it became really popular . The GIs returning

from the war demanded their wives and bakeries to bake pizzas . They had

eaten the pizza while posted in Italy and loved it . This started a

nationwide demand for the pizza . In 1950’s a number of pizza

only shops like pizza hut started and the pizza became really popular.

 

      Baking a great Pizza is an art . The dough has to be prepared fresh .

The oven has to be heated hours before you start baking . The cheese has

to be right . Paneer wont do !!!!!

 

      So next time you bite into a pizza give it some respect . It has

royal history on its side !

 

                Ps : My favourite toppings :pineapple, black olives,lamb ,sweet corn

Human Genome Project

Before I start today’s VERITAS I am reminded of a few lines by Goethe which run opposite to the subject of today’s VERITAS :

 

      He who would study organic existence

      first drives out the soul with rigid persistence

      Then the parts in his hands he may hold and class

      but the spiritual link is lost . Alas !!

                              Goethe ( Faust )

 

                  The Human Genome Project

                  =========================

 

Each of our cells contain 23 pairs of chromosomes . One set that we got

from Dad and another we got from Mom . Each of these chromosomes contain

genes . These genes control the way we look , behave etc . These genes

are a sequence of chemical building blocks represented by letters :

A( adenine ) , T(thymine ) , G(guanine) , C(cytocine ) . So you get a pair

of genes of each characteristic ( one from Mom and another from Dad ) and

one of them dominates and decides the way we are .

 

 

In 1990 a project was started to study each of the chromosomes and find

how many genes each contains and where they are on the chromosomes and

what their functions are . This is called The Human Genome Project .

The task was huge . There are approximately 10,000 human genes and

each has a long chemical structure . The project should be finished in 2003.

If the entire DNA sequence of an individual is written is a book it

would have 200 books of 1000 pages each . It is that big .

 

 

At the end of all this we would know what diseases are caused by what

gene . What each gene is responsible for and how we can treat ourselves

genetically . But most importantly we will be able to understand ourselves

from a basic chemical level .

 

 

How far has the project progressed ? The 5th , 16th , 19th , 21 st and the

22 nd chrosomes have been completely mapped( studied ) . The rest of

the chrosomes are beiong studied parallely at different labs in the world .

 

 

      The first chromosome to be completely studied was the 22nd . THis

was because of its small size . It reveled 545 genes and 134 pseudogenes

( these are genes that once functioned but no longer do ) .

 

      One very important development occurred when the 21st chrosome

was decoded . It contains 225 genes . Mutations in 14 known genes of

this chromosome cause diseases like Alzeimer’s syndrome , epilepsy .

And this may be a boon to control these diseases before they occur

in individuals ( right at the embroyonic stage ) . Mutation is some kinda

garbar in sequence of a gene . Another disease we may control by the

map of the 21st chrosome is Down’s syndrome which is responsible for

mental retardation in one out of every 700 kids .

 

      The Human Genome project is funded by Department of energy US

and has scientists all over the world working on it . IT also maintains

a very inteseting website : http://www.ornl.gov/hgmis . You can even see

some chrosomes and see the sequences . Amust see !

What orbit can I get you?

Mount where Science Guides

Go measure earth , weigh air and state the tides

Instruct the planets in which orbs to run

Correct Old time and regulate the Sun .

            Lord Kelvin

 

      If you go a new universe and someone asks you : What is the law of gravity here ? Is it the inverse square law or not ( F = GmM/r^2 ? )

 

      To answer this question all you have to ask is : Are the orbits of planets closed ? ie Do the planets retrace their paths revolution after revolution ?

 

      If they do then the law for gravity is inverse square ie depends inversely on the square of the distance between the planet and the star .

 

      Only the inverse square law has the ability to generate orbits

that retrace themselves . No other law can do that ! An inverse cube ,

or 1/r^4 etc laws can make planets go around stars but then the planets

keep changing their paths revolution after revolution .

 

      And every object in the universe retraces its path because the

law in our universee in inverse square . Of course at times it does not but

that is because of some object nearby disturbing it !

 

      That is the power of Physics : By doing some calculations on a

piece of paper u can talk about a planet revolving around a star

billions and billions of miles away .

 

      Do you know which is the smallest object in our solar system to

have a moon of its own . It is an asteroid named Ida which lies between

Mars and Jupiter . It is only 36 miles long and 14 miles wide . And

it has a small moon revolving around it named Dactyl . The moon is is

only a mile across . The asteroid Ida is not even big enough that its

gravity can make it spherical but he wants a moon of his own . cho chweet !

Godel’s Theorem

First of all : thanks to Sandeep Seth for lending me a wonderful book that introduced me to Godel’s theorem ( and Escher’s Paintings ) The name of the book is “Godel,Escher,Bach The Eternal Golden Braid”.

A wow book !

      After the theory of sets was invented Mathematicians started facing

a new kind of problem . Problems related to statements that were neither

true or false . Statements whose truth was indeterminate . FOr example take

the following statement :

   A set G that contains all sets that dont contain themselves .

         Now does G contain itself ?

Lets suppose it does . But then there is a problem because then the

definition of G would be wrong .

Now lets suppose that it does not . But then the set G would not contain a

set G ( which does not contain itself ) . And that is incomplete .

 

Paradox ! Paradoxes of this kind are called Russel’s Paradox .

 

Mathematicians didnt like such inconsistencies in their Math . So

Bertrand Russel and Whitehead tried to make a new system that has

no inconsistiencies. They called it Principia Mathematica .

 

Another Great Mathematician Hilbert had bigger plans . He challenged

Mathematicians to make a formal system that is complete and consistent

and we could base all mathematics on the top of that .

 

What is a formal system :

 

Any system in mathematics contains some axioms . These axioms are

things you assume to find theorems . For example lets invent a new

system . We have a string : KO . The rules of the system are :

 

a) if we have O at the end then we can add a T at the end of the

   string .

 

b) if we have OT ate the end of the string we can add another OT at the

   end .

c) If we have 2 sets of OT at the end then we can remove them .

 

so our three rules and our strings are axioms . Stuff we start off with .

 

And a new string that you may invent is a theorem .

 

Is the string K a theorem in our system ? lets check :

 

start off with KO and apply rule a to it . Our string becomes KOT .

Now apply rule b we get KOTOT . NOw apply rule c we get K .

 

So K is a theorem in our system . Any mathematical system proceeds

in a similar way . Theorems and axioms . SO either a statement can be

a theorem( it is true ) or it is false ( its opposite is a theorem ) .

 

How Hilbert’s program was to reduce mathematics to string analysis .

So that all you need to know is the axioms and you can prove anything .

Just keep applying axiom after axiom and u get a new string which you

call a theorem . You can express any theorem in math as a string :

 

for example :

there exists no x,y,z : x^3 + y^3 = z ^3

 

is a statement of fermat’s last theorem

 

So Hilbert’s program tries to reduce Mathematics to string analysis .

And he propopsed to start off with Principia Mathematica .

 

But Kurt Godel published a theorem in 1931 that destroyed such hopes .

He said 2 important things :

 

1) YOU CAN REDUCE ANY FORMAL SYSTEM TO A SYSTEM INVOLVING NATURAL NUMBERS.

   for example in our system we can give K a no 10 . O is 2 and T is 3 .

   so KO is 1023 and rule c) is that if you have anything followd by 2323 then

   you can remove that . SO K is a theorem means that 10 is a valid

   no in our system . Similarly all formal systems can be expressed in

   terms of natural numbers and then in terms of each other .

 

 

b) YOU CAN ALWAYS CREATE A NEW THEOREM THAT CANNOT BE PROVEN ( AND ITS

   NEGATION CANNOT BE PROVEN TOO . Ie ANY COMPLETE FORMAL SYSTEM CONTAINS

   UNDECIDABLE THEOREMS ) . THis means that there will always be theorems

   that will escape the net of our formal methods . These are kind of

   theorems that reference themselves . Godel showed this for Principia

   Mathematica and then since all formal systems are equiivalent thus

   it applies to all branches of Mathematics . Thus provability

   is weaker than a formal system .

 

So what this showed was that Mathematics cannot be made into a formal

system and that the role of intution is still very important .

THis was a great blow to Hilbert’s program . Now no one tries to make

systems that dont have inconsistencies . Because Godel showed that

you cannot do that . If a system is strong enough then there will

always have inconsistencies . 

Three Body Problem

What is the motion of 3 bodies moving under the gravitational effect of each other ?

 

 

This is the three body problem . The problem looks simple but nobody

has( yet ) found a solution of this problem ! The problem was first

proposed by Newton .

 

Newton solved the “two-body” problem geometrically . Then Daniel Bernaulli

solved the “two-body” problem in 1734 using calculus .

 

Many mathematicians have tried to tacke this problem but in vain .

Poincare in early 20 century said that the solution of this problem

is chaotic ( involves chaos )

 

This problem is so important because solving this problem would help us

understand if our solar system is stable ! The solar system is a n-body

problem . The sun , planets , moons moving with respect to each other .

But this n-body problem is much much more harder than the 3 body problem .

 

 

The only way ( at present ) to know the motion of the 3 bodies under the

influence of each other is by Computer Simulations .