To celebrate the birthday of the famous mathematician Shrinivas Ramanujam or Mathematics Day (22nd December), I would like to share some interesting stuff,
First of all this year, Ramanujam's birth date is more special because 22-12-2017 is a prime number! And it would have been his 130th birthday.
What is shown here is a Magic square. It is an n×n array whose row, column and the diagonal numbers add up to the same number.
The magic square shown above was made by Ramanujam. Its very first row contains his Birth Date. So its also known as Ramanujam's magic square.
If you add the numbers along the row, column or along the diagonal they will add up to 139 (a prime number).
Inspired by this standupmaths video I made some shapes and figured out whether they can be made without taking pen off the paper.
So if a shape has got all odd number of lines going from its every point you can never make it without taking your pen off.
If it has only two such odd points it's called Semi-Eulerian Graph and then you can make the shape by beginning on one odd point and ending on another, as in the very first shape of the following picture.
When we see a picture of scientists in lab or at a research
centre, we notice that there are computers everywhere. And we think that they
use these computers to analyze the data from the experiments in particle
physics or cosmology. But there are lots of other things that are done with
the aid of these computers, like landing of a rover, communication with a
satellite orbiting mars, etc.
I was curious how physicists, cosmologists, and
mathematicians use computers to solve problems. Like how the simulations of
colliding black holes or galaxies is produced. How can it reveal something
which has not happened yet? Like how our sun will explode at the end of its
life and become a white dwarf.
The stars we see at night are just a small part of our
galaxy. And our closest star, the sun can be seen throughout the day.
But how a black hole looks like?
We know that the planets are illuminated by star. Similarly
Black holes are illuminated by the accretion disc surrounding them. So how does
a black hole look like with its accretion disc?
So now we should
have a real picture of black holes, right?
But still we do not have one.
Although the simulations of black hole have become more
beautiful and precise, and Luminet’s original work has now been done with a
computer and shows 3D model of similar picture he created with his own hands.
And the black hole Gargantua in the movie INTERSTELLAR with
its distorted accretion disc due to gravitational lensing is also now a famous visualization of such black holes.
But we do not have to be disappointed, because the Event Horizen Telescope, which is a large collection of telescopes, has started
working on it and within few years we’ll see the first picture of a black hole!
“Astronomy is older than physics. In fact, it got physics started, by showing the beautiful simplicity of the motion of stars and planets, the understanding of which was the beginning of physics. But the most remarkable discovery in all of astronomy is that the stars are made of atoms of the same kind as those on earth.*”
Richard Feynman
How can a star turn into a Black Hole? From an object that
shines due to nuclear fusion to something which does not allow light itself to
escape from it. It becomes something so mysterious that cosmologists need to combine
one of the greatest theories – General theory of relativity and Quantum
mechanics to understand what is inside it.
As Michael describes in his video "Travel INSIDE a black hole", theoretically anything, you, me or the Earth you are sitting right now can become a black hole if you shrink it under the Schwarzschild Radius. fortunately there is no means of doing it. But a star can do it at the end of its life.
But
how? How can a star turn into a black hole.
To understand it we have to see what happens inside a star.
Stars emit electromagnetic radiation during nuclear fusion.
In our Sun during nuclear fusion Hydrogen nuclei fuse into Helium nuclei
and produce electromagnetic radiation which we feel as heat. Photosynthesis starts as the radiation falls on leaves.
Thus everything we eat here on earth is just Bottled Sunshine!
Here
are some of my favourite lines from Carl Sagan’s book “cosmos”,
It was more awesome to listen it in his beautiful voice inthe video seriesCosmos Episode 2 - One Voice in the Cosmic Fugue
Stars
are very stable due to the thermal equilibrium that exists between the outward pressure and gravitational force. Outward pressure is maintained by heat
generated in the nuclear reactions. But as more heavier elements form in the
center of star, different layers start forming, star begins to swell and becomes a red giant. In the end it collapses into a dense core.
This is how we are told that the stars die, but its still described as one of the unsolved problems of physics, "What is the exact mechanism by which an implosion of a dying star becomes an explosion?"
So, every star becomes a black hole in the end?
No, two
things are important for a star to become a black hole,
Chandrashekhar limit
Schwarzschild radius
If
a star’s mass is less than 1.4 times the mass of sun (Chandrashekhar limit), it
will settle down to a white dwarf, which will remain stable due to the balance
between gravitational force and Pauli Exclusion Principle which is the
repulsion between electrons. That is the the dense core left at the end will be so dense, it will have only electrons!
But if its mass is above Chandrashekhar limit, it
will become a neutron star which is stable by the Pauli Exclusion Principle
followed by neutrons. The dense core will be made of neutrons! But if the mass of a star is more than 10 or 20 times
that of the sun it will collapse in on itself to a point! This is what is
called a BLACK HOLE!
So
what is inside a black hole?
We don't know!
One
of the solutions of Einstein’s field equations is “Schwarzschild metric”. It
describes spacetime surrounding a non-rotating massive object. Schwarzschild
black holes which are simplest kind of black holes are described using this metric.
EINSTEIN'S FIELD EQUATIONS
Schwarzschild metric described by Matthew O'Dowd of PBS space time,
Metric
is another word to describe the distance between two points in space. Metric
can be different for different space.
In
spherical polar coordinate system the Schwarzschild metric is,
In the above equation, 2m=Rs (Schwarzschild Radius).
According to this metric, there is a Schwarzschild Radius surrounding a black hole where the escape velocity is greater than speed of light. This Schwarzschild Radius is actually whats written in many books as event horizon of a NON-ROTATING black hole.
As nothing can travel faster than light, this region surrounding a black hole looks black.
Usually
when we see a picture of a black hole, we see the surroundings distorted and the
black hole itself shown literally black. But as we know, all these images are
not real. But they tell us what really is true. What we actually see
when we look at a picture like this is the Schwarzschild radius or the sphere,
from which nothing can escape, not even light.
And because of black hole's immense gravity, the galaxy behind it also seems to be distorted due to gravitational lensing.
In gravitational lensing, when a heavy object comes in
front of a star, galaxy or quasar it bends the light coming from them and thus
distorts is totally. Sometimes it forms a ring and sometimes multiple images. Like
in yesterday’s APOD - https://apod.nasa.gov/apod/ap170227.html
And the following picture shows other gravitational lenses taken by Hubble Space Telescope.
The video shows how the bottom of a glass can be used to see a ring of light when the source of light and the bottom of glass are aligned.
As this experiment was inside my mind, I saw another way of looking at the effect using only water in a bowl.
You can see in the image below, when the light from LED falls in the center of bowl a ring is formed even though its not a full circle.
Schwarzschild
black hole is the simplest black hole. Its assumed to be non-rotating and with no charge. So what
would it be like to travel inside it? There are some mindbending simulations on
this page made by Andrew J. S. Hamilton.
But there
are other black holes which are not formed after death of stars, they are called
supermassive black holes.
And astronomers have found these supermassive black holes at the centre of every
galaxy. So how these black holes are formed? And what came first galaxy or
black hole?
As theoretical physicist Michio Kaku explain in this video, the latest theory tells us that supermassive black holes formed first and then the galaxy itself formed around it. But its still difficult to tell which formed first.
Theoretical
physicist Carlo Rovelli in a course at World Science U describes that a quantum
theory of gravity is needed to explain the interior of black holes.
So we have still many mysteries to unravel as in Newton's time. HAPPY SCIENCE DAY!
