LIGO

“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”

www.ligo.caltech.edu 

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).

www.ligo.caltech.edu

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.

Youtube: Journey of a Gravitational Wave

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!

There are some great lectures of WorldScienceU by Rainer Weiss, Gabriela Gonzalez and Nergis Mavalvala.

Big Bang

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.”

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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. 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.

How far is Andromeda

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.

astro.princeton.edu

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.

Formulas of area

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.

Pi r² = (8/9)²(2r)²

Pi = 256/81 = 3.1604…

The Man Who Knew Infinity...

I read the last pages of “The man who knew infinity” (Biography of Srinivasa Ramanujam) yesterday night. 

Even though he didn't have lots of books as a student, he discovered so many theorems in mathematics which could only be understood by Mathematicians.

I would conclude by sharing a small piece of work.

A document in which I have written how I understood the limit of a sequence.

Document: Proof of limit of a sequence

Galaxies

Its really fascinating that once humans didn’t know what is beyond our galaxy. But a discovery by Edwin Powell Hubble changed everything.

The 100-inch telescope at Mount Wilson was the largest in the world in the early 1900s. At that time small patches of light in the sky were termed as Nebulae and were thought to be existing inside the Milky Way.

We know that when a star shows fluctuation in its brightness the reason may be a planet orbiting it. But there are some variable stars named Cepheid variables which show fluctuations in their brightness for some different reason. 

When the star is relatively cool it contracts thus its core contracts and it generates more heat and then star begins to expand. Energy is released before and after the expansion thus the star cools and contracts. The process repeats and that’s how we see fluctuation in its brightness.

Our pole star, Polaris is also a Cepheid Variable.

Edwin Hubble also discovered a Cepheid Variable in  a Nebula named Andromeda and thus using its fluctuations and luminosity he calculated its distance and thus the distance of Andromeda Nebula from Earth. It was 900,000 light years away. And American astronomer Harlow Shapley had determined that the Milky Way was only 100,000 light-year across. So it must be outside our galaxy. 

And later many stars were found in the nebula and thus it was proved to be a galaxy.

(Later it was proved that the distance measured by Edwin Hubble was not correct, please read this post.)

Picture of Andromeda by Edwin Hubble
Image credit: http://obs.carnegiescience.edu/PAST/m31var

Now we know that our galaxy is only a part of 100 billion other galaxies!

Image credit: http://hubblesite.org/newscenter/archive/releases/2012/37/image/a/

 More beautiful pictures can be found on Hubble website. 

Tessellations

Tessellations will appear everywhere if you have built a habit of seeing them. In shirts, dresses, on the bed-covers, curtains, buildings, flowers, etc.

But mostly in nature we do not see tessellations but symmetry. So how tessellations and symmetry are related?

I remember butterfly wings which are symmetrical, and tiles in my home which repeat (or tessellate). 

The building block or tile used in tessellations is repeated over and over again to fill the whole space.

For example here are few tessellations with similar tiles:

Sometimes the shape used is similar but the way it is arranged or its colour is different. And that’s how different tessellations can be made using similar shapes. The detailed description can be found here: Tessellation symmetry

Triangles, squares and hexagons are the only shape which can tessellate regularly. And others can be made to tessellate for example by using two different shapes.

The work of Maurits Cornelis Escher is very famous in the world of art and mathematics.

M.C.Escher
Image source: http://www.mcescher.com/gallery/italian-period/hand-with-reflecting-sphere/

As written on the website of Tessellations:

During his life, he became obsessed with filling surfaces with pictures that did not overlap or leave spaces. Aged 68, he stated, “Filling two-dimensional planes has become a real mania to which I have become addicted and from which I sometimes find it hard to tear myself away.”

First Tessellation of M.C.Escher
Image source: http://www.tessellations.org/tess-escher6.shtml

He also made tessellation which grew smaller on the outside, and similarly in reverse.

Image source:  http://tessellations.org/eschergallery24.shtml

So let’s keep on seeing patterns in nature! Happy World Tessellation Day.