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!

“The black holes
collide in complete darkness. None of the energy exploding from the collision
comes out as light. No telescope will ever see the event. That profusion of
energy emanates from the coalescing holes in a purely gravitational form, as
waves in the shape of spacetime, as gravitational waves.” -Janna Levin in “Black
Hole Blues and Other Songs From Outer Space”

In 1969 Joseph Weber announced that he had detected
Gravitational Waves. He used a large Aluminum bar which he thought would ring
when a GW passed through it. But later he was proved wrong.

The basic principle behind LIGO was actually invented in a
course of general relativity taught by professor Rainer Weiss. It was a
Gedanken problem he gave to students.

Later he built a small 1.5m prototype. And he realized that
instrument needed to be big.

Construction of LIGO began in 1994 and completed in 1999.
The name LIGO was actually suggested by Rainer but Kip Thorne wanted it to
be “beam detector”.

There are two LIGO observatories: LHO (LIGO Hanford
Observatory) and LLO (LIGO Livingston).

And on September 14, 2015 the detectors caught the final
four orbits of a black hole 29 times the mass of the sun in a pair with a black
hole 36 times the mass of the sun.

Even though gravitational waves are not sound waves, they
can be converted to sound. In the video you can hear that chirp of merger
of two black holes.

Same exciting moment came again on December 26, 2015
when GWs were detected but this time it was due to merger of two smaller
black holes.

The best thing about the GW astronomy would be that it will allow
us to see the earliest moments of the Big Bang, because early universe was opaque
to light but it was not opaque to GWs. The first light which we now detect as
CMB radiation was free to travel only 300000 years after the Big Bang. But by
detecting GWs we will be able to see what actually happened in the earliest moments
of the Big Bang!

These words were first aired on a BBC radio show when Sir Fred Hoyle was describing the two theories of creation of the universe
at that time: Dynamic evolving model and Steady state model. He used these two
words to describe the Dynamic evolving model of the universe which he opposed throughout
his life even when there were much observational evidence and almost every
scientist accepted it.

At first Einstein was also in favor of the steady state
model. He used cosmological constant so that the equations do not predict a collapsing
universe.

But Alexander Friedmann used equations of general relativity
to show that different values of cosmological constant give rise to different
fate of universe,

1.It expands and then contracts

2.Continuous expansion

3.Neither collapses nor expands

Even though Einstein found Friedmann’s calculations correct,
he refused to believe in such a dynamical universe. So Friedmann’s work didn’t become popular.

Later Georges Lemaitre rediscovered all
these facts without knowing Friedmann had already gone through same thought
process. Using the concept of radioactive decay Lemaitre speculated that on a
greater scale a similar process might have given birth to the universe. By extrapolating
backwards in time the universe began in a small compact region from which it
exploded outward he found all the stars squeezed into a super compact universe,
which he called primeval atom.

In his words,

“The evolution of the universe can be likened to a display
of fireworks that has just ended: some few wisps, ashes and smoke. Standing on
a well-cooled cinder, we see the fading of the suns, and try to recall the
vanished brilliance of the origins of the worlds.”

But at that time there were no observational evidence to these
theories. Later when Edwin Hubble using his observational data (of red shifted
galaxies) drew a graph which showed a linear relationship between the distance
and velocity of galaxies, scientists concluded that if galaxies are moving away
from us, in past they must have been closer to each other.

v = Hd

Using this equation, if we know the
speed of a galaxy, its distance can be calculated. And we can also evaluate the
time when the galaxies were closer to each other. As the measurements became
more accurate, age of the universe came out bigger and it was concluded between
10 – 20 billion years.

Ralph Alpher and Robert Herman proposed
that the oldest light in the universe which spread everywhere 300000 years
after the creation can be taken as the test for big bang model.

Later Robert Wilson and Arno
Penzias detected signal from space which was proved to be the echo from the big
bang: the cosmic microwave background radiation (oldest light in the universe).
Since steady state model does not predict this radiation, it was now clear that
the universe started billions of years ago with Big Bang.

But still scientists including Fred
Hoyle along with Jayant Vishnu Narlikar were working on steady state model, and
developed it to a new Quasi-steady state model.

Since the CMB radiation detected is
just a few thousand years older than the universe’s creations. So any density
variation at that time would have given rise to the density variation later
like galaxies. After many efforts and other experiments COBE (Cosmic Background
Explorer Satellite) was launched on 18 Nov. 1989.

A comparison of the sensitivity of WMAP with COBE and Penzias and Wilson's telescope. Simulated data : NASA

COBE had four detectors and
its main purpose was to observe the density variations in CMB radiation. Later
WMAP (Wilkinson Microwave Anisotropy Probe) launched in 2001, provided more
data and thus the age of the universe was calculated to be 13.8 billion years and
it also became known that universe 23% Dark Matter, 73% Dark Energy, 4%
Ordinary Matter.

Another space observatory was launched by ESA
in 2009, Planck. Its data provided most accurate measurements of these cosmological
parameters.

As I have written in my previous post, Edwin Powell Hubble expanded our understanding of the
universe by calculating the distance of Andromeda using Cepheid Variable stars
using 100 inch Hooker telescope at Mount Wilson. It was believed that it’s a nebula
like many others inside our own galaxy but he calculated that its distance is
90,000 light years from earth.

Later Walter Baade
studied the RR Lyrae stars using the same telescope. RR Lyrae are also variable
star similar to Cepheids but less luminous. It was shown before that like
Cepheids, the variability of RR Lyrae stars can be used to measure distances.

The movie shows RR Lyrae stars in a globular cluster. You
can see their brightness changing; they look blue as they become brighter.

Walter
Baade wanted to use these stars to measure the distance of Andromeda as it was
done before using Cepheids. But the 100 inch telescope was not good enough to
detect those stars. So he had to wait until the 200 inch (~5 meter) telescope was
ready which was being built by George Hale but sadly he died two years after
the project started. Later the telescope was named after him.

When the new Hale telescope became operational Baade used
the telescope to search the faint variable stars in Andromeda but even after
searching for a long time he was not able to find any sign of these stars.

He concluded that the only possible reason for this can be
the distance of Andromeda previously measured is not correct!

At that time it was becoming evident that stars can be categorized
into two broad types called populations. Older stars fall in Population2 and
younger and brighter stars in Population1.

So Baade assumed that Cepheid variable stars will also have
two different types. Thus he reasoned his argument that the previously measured
distance of Andromeda was wrong using two points,

1.Population1 Cepheids are brighter than
Population2 Cepheids.

2.Astronomers only saw the brighter Population1
stars in Andromeda and compared it to the dimmer population2 stars in Milky Way.

That’s what lacked in the calculations made by Hubble. And
that’s why he measured the distance of Andromeda little less.

Baade calculated that Population1 stars are on average 4
times more luminous than Popluations2 stars of same period of variations. So if
a star is moved twice as far away it will appear 4 times fainter. Thus Andromeda
Galaxy should be twice as far away - approximately 2 million light years away!

More accurately,

How far is Andromeda? Its 2.537
million light years away.

I was wondering how you can know that your formula is correct
when you are discovering it. In this post I am sharing some of my findings related to it.

The triangle can be thought of as half of rectangle, so its
area should also be half,

½ ar(rectangle) = ½ (ab)

But in this sense, to discover this formula you need to know
the area of a rectangle.

Heron’s formula gives the area of triangle without using it,

√s(s-a)(s-b)(s-c)

Where s = ½ (a+b+c)

Now let’s take another formula, the area of a cuboid which
is,

2(lb+bh+hl)

This formula also involves the area of rectangle. There are three
pairs of identical rectangles, so their area lb, bh, hl adds up twice to give
the formula 2(lb+bh+hl).

Similarly in a cube all the sides have equal length so the
area of cube is,

2(3a²) = 6a²

I tried to discover formula of triangle without using any
other formula but I failed to do so. So how our ancient mathematicians discovered them and how they knew that
their formulas are correct? I will try to find answers of these questions in future.

But the formulas
discovered in ancient times were not always correct. Like an Egyptian formula
for finding the area of a circle was to take the square of 8/9 of the circle’s
diameter. It’s not correct because if we compare it with the formula we now know, then we get very less accurate value of pi.

I am going to share some of my last month’s discoveries, which
I found very interesting…

1All of us have made stars or hexagon using two
triangles. But while doing a physics question, I realized that we
can make hexagon using three rectangles. Inspired by this discovery, I started
thinking what properties the rectangles should have. Can they be all squares or
different rectangles? After experimenting I came to believe that
all three rectangles can be different or same but we cannot make a hexagon
using 3 squares, only one of them can be a square.

A hexagon made with two equal and one different rectangle

An octagon made with all equal rectangles

2 Even though I loved reading the chapter “Violins
to Videos” in Ian Stewart’s “Nature’s Numbers” book but I found the following lines
from a previous chapter “Constants of Change” really amazing… It changed my perspective towards differential equations.

“Because a rate of change is about the
difference between some quantity now and its value an instant into the future, equations
of this kind are called differential equations. The term
"differentiation" has the same origin. Ever since Newton, the
strategy of mathematical physics has been to describe the universe in terms of
differential equations, and then solve them.

However, as we have pursued this strategy into
more sophisticated realms, the meaning of the word "solve" has undergone
a series of major changes. Originally it implied finding a precise mathematical
formula that would describe what a system does at any instant of time. Newton's
discovery of another important natural pattern, the law of gravitation, rested
upon a solution of this kind. He began with Kepler's discovery that planets
move in ellipses, together with two other mathematical regularities that were
also noted by Kepler. Newton asked what kind of force, acting on a planet, would
be needed to produce the pattern that Kepler had found. In effect, Newton was
trying to work backward from behavior to laws, using a process of induction
rather than deduction. And he discovered a very beautiful result. The necessary
force should always point in the direction of the Sun; and it should decrease
with the distance from the planet to the Sun. Moreover, this decrease should
obey a simple mathematical law, the inverse-square law.”

I saw a beautiful video “Donald Duck - Mathmagic land” and inspired by it, performed some experiments with Rubber strings. I will observe their vibrations more closely this month.

3

4 Squares inside squares

I accidentally made something in my copy when I was in classroom. I found it really beautiful.

Draw a square and then using its diagonal
as the length of new side, make another square and then repeat… If you start with side 1cm
then the diagonal is √2 so all the sides of next square will have length √2.