String Theory

Lagrange Points The Beauty of Deep Space

Lagrange Points

A Lagrangian point is a point between two large bodies where if you place a small body with negligible influence on the two larger bodies (such as, say, a satellite between the Sun and the Earth), the gravitational force exerted by the two large bodies is equal in magnitude and opposite in direction such that they cancel each other out, thus enabling the small object to stay in equilibrium in that position in space relative to the bigger object.

                                 

Typically, a spacecraft sent from the Earth into the vacuum of space will continue moving in the direction we send it, unless it's affected by the gravity of a celestial object, like a star or a planet. Even if we stopped a spacecraft in the middle of space, the gravity of objects around it would eventually pull it in some direction. 

Enter Joseph-Louis Lagrange. He theorized that at certain points the gravity of two bodies, combined with the third body's centripetal force, would keep the third body in a constant location relative to the other bodies. 

There are five Lagrange points located at sun earth system.

                                          

L1

Kepler’s laws require that the closer a planet is to the Sun, the faster it will move. Any spacecraft going around the Sun in an orbit smaller than Earth's will also soon overtake and move away, and will not keep a fixed station relative to Earth.

However, there is a loophole. If the spacecraft is placed between Sun and Earth, Earth's gravity pulls it in the opposite direction and cancels some of the pull of the Sun. With a weaker pull towards the Sun, the spacecraft then needs less speed to maintain its orbit.

If the distance is just right - about a hundredth of the distance to the Sun - the spacecraft, too, will keep its position between the Sun and the Earth and will need just one year to go around the Sun. This is L1.

                                                   L2

An effect similar to that which causes the L1 point occurs on the ‘night’ side of Earth (further away from the Sun but about the same distance from Earth).
A spacecraft placed there is more distant from the Sun and therefore should orbit it more slowly than the Earth; but the extra pull of the Earth adds up to the Sun's pull, and this allows the spacecraft to move faster and keep up with Earth.
At a certain point, the spacecraft’s orbital period equals that of Earth’s. This is L2. It is located 1.5 million kilometres directly 'behind' the Earth as viewed from the Sun. It is about four times further away from the Earth than the Moon.
 L2 is a great place from which to observe the larger Universe. A spacecraft would not have to make constant orbits of Earth, which result in it passing in and out of Earth's shadow and causing it to heat up and cool down, distorting its view. Free from this restriction and far away from the heat radiated by Earth, L2 provides a much more stable viewpoint.

                                                     L3

L3 lies on a line defined by the Sun and Earth, but beyond the position of the Sun.
On the opposite side of the Sun, just outside the orbit of Earth, the combination of the Sun’s and Earth’s gravity would cause a spaceraft’s orbital period to equal that of Earth.
Since the position of this Lagrange point lies behind the Sun, any objects which may be orbiting there cannot be seen from Earth.

L4 and L5

 

The L4 and L5 points lie at 60 degrees ahead of and behind Earth in its orbit as seen from the Sun. Unlike the other Lagrange points, L4 and L5 are resistant to gravitational perturbations. Because of this stability, objects tend to accumulate in these points, such as dust and some asteroid-type objects.
A spacecraft at L1, L2, or L3 is ‘meta-stable’, like a ball sitting on top of a hill. A little push or bump and it starts moving away. A spacecraft at one of these points has to use frequent rocket firings or other means to remain in the same place. Orbits around these points are called 'halo orbits'.
But at L4 or L5, a spacecraft is truly stable, like a ball in a bowl: when gently pushed away, it orbits the Lagrange point without drifting farther and farther, and without the need of frequent rocket firings. The Sun's pull causes any object in the L4 and L5 locations to ‘orbit’ the Lagrange point in an 89-day cycle. These positions have been studied as possible sites for artificial space stations in the distant future.

Group of asteroids orbiting at L4 & L5

Misconception about zero gravity

Misconception about zero gravity

if i ask a question ,why the astronaut floats inside the space station or feels weightlessness ? most of the people says because at the space station gravity is zero,but it's absolutely wrong !!! although the gravity at the altitude of the Station is about as strong as it is on Earth's surface (it is about 90% of full gravity).
in fact,zero gravity doesn't exist at all.

Suppose you climbed to the top of a ladder that's about 300 miles tall (altitude of ISS). You would be up in the vacuum of space, but you would not be weightless at all. You'd only weigh about ten percent less than you do on the ground. While 300 miles out in space, a 100kg person would weigh about 90kg. Yet a spacecraft can orbit 'weightlessly' at the height of your ladder! While you're up there, you might see the Space Shuttle zip right by you. The people inside it would seem as weightless as always. Yet on your tall ladder, you'd feel nearly normal weight. What's going on?

The reason that the shuttle astronauts act weightless is that they're inside a container which is FALLING! If the shuttle were to sit unmoving on top of your ladder (it's a strong ladder,) the shuttle would no longer be falling, and its occupants would feel nearly normal weight. And if you were to leap from your ladder, you would feel just as weightless as an astronaut (at least you'd feel weightless until you hit the ground!)

So, if the orbiting shuttle is really falling, why doesn't it hit the earth? It's because the shuttle is not only falling down, it is moving very fast sideways (at 7.71 km/s at that altitude) as it falls, so it falls in a curve. It moves so fast that the curved path of its fall is the same as the curve of the earth, so the Shuttle falls and falls and never comes down. Gravity strongly affects the astronauts in a spacecraft: the Earth is strongly pulling on them so they fall towards it. But they are moving sideways so fast that they continually miss the Earth. This process is called "orbiting," and the proper word for the seeming lack of gravity is called "Free Fall." You shouldn't say that astronauts are "weightless," because if you do, then anyone and anything that is falling would also be "weightless."

Where do comets come from & how ?

Where do comets come from & how ?

Comets are small celestial bodies that appear in the night sky as a ball inside a long, thin envelope of visible light. They are composed of water, frozen gasses and dust.often referred to as "dirty snowballs."
Comets are actually collections of materials left over from the formation of the solar system. This makes them especially interesting to astronomers because the solar system formed around 4,600 million years ago! Studying the composition of comets can tell us something about the history of the solar system. 

comets are found in two regions of solar system.

1. Kuiper belt
2. Oort cloud

also there are two types of comet Short period and long period comets.

Short Period Comet :-

These types of comets frequently appears in the solar system and comes from kuiper belt, beyond the orbit of Neptune.

Long Period Comet :-

These types of comets takes thousand of years to complete the orbit and comes from oort cloud,a vast group of frozen bodies that surrounds the solar system.

okay.now we know the location of comets but one of the interesting question is how they enters in the solar system.
answer is the gravity of the outer planets of solar system cause the comet to enter in the solar system.
below animation from ESA will help you to understand more clearly.

as shown in the animation comets gains its tail because of the radiation from sun which cause the comet to burn and releaseas gas and dust.

As comets pass through the inner solar system, the radiation from the Sun causes them heat up, evaporating the dusty-icy materials of the comet. These particles are left in the wake of the comets passage creating a stream of small debris that is strewn along the comets orbital path. If the orbit of the Earth intersects the orbital path of a comet, then at regular predictable times throughout the year the Earth will pass through the
stream of debris creating a meteor shower.

Below are some example of the comets that passed through solar system in past time.

Gravity Assist

Gravity Assist

How the spacecrafts travels so long distance in solar system and beyond the solar system with less propellant mass (fuel) ? when travelling among the planets, it's a good idea to minimize the propellant mass needed by your spacecraft and its launch vehicle. That way, such a flight is possible with current launch capabilities, and costs will not be prohibitive. The amount of propellant needed depends largely on what route you choose. Trajectories(path) that by their nature need a minimum of propellant are therefore of great interest.

 Overcoming gravity is all about velocity. Escaping Earth's gravity requires approximately 25,000 mph. Escaping the Solar System needs more than 45,000mph. We don't have a large enough rocket to achieve that speed so spacecraft can use a planets gravity to increase its velocity and then the planet can "slingshot" it onto a new trajectory toward the next target. The gravity of a large object can "pull" something to a higher velocity and then, rather than crash into the object, the craft can just miss the planet or moon and, for a moment, go into orbit. The centripetal force of the orbit will increase the craft's velocity and "shoot" it off on a new trajectory.It's called Gravity Assist.

Normally, it would take almost twenty-four years to get from the Earth to Pluto.That’s travelling at about 17,000 mph.Gravity assist can cause the spacecraft to speed up/speed down or change the direction without firing the rockets in the space. With the use of the gravitational slingshot around as many planets as possible, that time could be cut in half and that speed could be increased by upwards of 5000%. For instance, the New Horizon’s craft is using only the gravitational assist from Jupiter and is managing speeds of over 32,000 mph. If we used the full potential of something as large as the sun, that speed could be increased to over 1.1 million mph. There are comets that do it, so there is nothing that could stop a shuttle from doing the same thing.

How Gravity assist works ?
 

to be continued....

Eigenvalue & Eigenvector

Eigenvalue & Eigenvector

Have you ever heard the crazy names eigenvalue & eigenvector ? probably you have found that in maths (matrices).
i have solved many problems of matrices without even knowing what exactly this means & what is the real life application of it.

some of applications are :-

• Face recognition system (biometric system) which is based on Principle Components Analysis (PCA)
• Vibration analysis
• Google page-rank algorithm and many more...

What is Eigenvalue & Eigenvector ?

a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with thateigenvector

Definition

If A is an n × n matrix, then a nonzero vector x in Rn is called an eigenvector of A if Ax is a scalar multiple of   x;that is ,

Ax= λx

 for some scalar λ. The scalar λ is called an eigenvalue of A , and x is called the eigenvector
of A corresponding to the eigenvalue λ.

(1) An Eigenvector is a vector that maintains its direction after undergoing a linear transformation.

(2) An Eigenvalue is the scalar value that the eigenvector wasmultiplied by during the linear transformation.

lets take an example :-

Here A is our Matrix and x is vector

  

 lets take another example :-

Methods to place Satellite into Geostationary orbit

Methods to place Satellite into Geostationary orbit

There is a considerable amount of expertise and technology used to ensure that satellites enter their orbits in the most energy efficient ways possible. This ensures that the amount of fuel required is kept to a minimum; an important factor on its own because the fuel itself has to be transported until it is used. If too much fuel has to be used then this increases the size of the launch rocket and in turn this greatly increases the costs.

Many satellites are placed into geostationary orbit, and one common method (1st method) of achieving this is based on the Hohmann transfer principle. This is the method use when the Shuttle launches satellites into orbit. Using this system the satellite is placed into a low earth orbit with an altitude of around 180 miles. Once in the correct position in this orbit rockets are fired to put the satellite into an elliptical orbit with the perigee at the low earth orbit and the apogee at the geostationary orbit as shown. When the satellite reaches the final altitude the rocket or booster is again fired to retain it in the geostationary orbit with the correct velocity.

as shown in the fig the perigee of the transfer orbit intersects the LEO orbit.This point is 1st pro-grade burn i.e rocket thrust (called delta V) in the same direction as of orbit and transferred to elliptical orbit i.e Hohmann Transfer orbit or Geostationary Transfer orbit.

at half of the transfer orbit is apogee point which intersects the geostationary orbit after that point satellite starts to fall i.e we have to fire our thrust in the same direction of the orbit which is 2nd pro-grade burn to remain in the geostationary orbit.

( 2nd method )Although now-a-days almost all the county uses Expandable Launch vehicle which directly place satellite into geostationary transfer Orbit without using LEO. ( PSLV , GSLV is expendable launch vehicle use by India)

But hohmann Transfer orbit is very useful for interplanetary travel (travels to diffrent planets by including one more method called gravity Assist )

America NASA's Space shuttle and Russia's Soyuz are Partial reusable launch vehicle to carry payloads to LEO orbit (ISS is in LEO orbit) while PSLV GSLV (india), Ariane 5 (ESA),Proton M (Russia) are expandable launch vehicle i.e non reusable vehicle.

no true reusable launch system is currently in use.

Spacecraft Orbit Transfer or Orbital Maneuver

Orbital Maneuver

have you ever wonder how spacecraft travels to other planet ?

Launching the spacecraft required  launch vehicle. To escape from earth gravity or to achieve escape velocity (11.2km/s) required much fuel. we separate out the fuel tanks in stages to decrease the payload or in other words to reach at escape  velocity( high mass = much fuel).once we are out of earth's gravity i.e we are not going to fall back to earth and so we are in the orbit. satellite/spacecraft stays in the orbit until no external force is applied.

now suppose we want to change orbit to other planets orbit.but to change the orbit we required to propel the engine i.e required the fuel but at this stage we have not enough fuel to travel so ling distance. we can design our space craft to carry more fuel but our payload increases and require more fuel at launch vehicle i.e more space.

so how can we achieve require orbital change with minimum fuel the answer is Orbital Maneuver.
( maneuver is military word means change the place )
by using Orbital maneuver we can tackle the space constrains.


have you ever wonder how satellite/spacecraft placed into orbit



Where there is righteousness in the heart

There is beauty in the character.

When there is beauty in the character,

there is harmony in the home.

When there is harmony in the home.

There is an order in the nation.

When there is order in the nation,

There is peace in the world

-APJ Abdul Kalam

Fourier Transform

Fourier Transform

Fourier Transform  is a very powerful tool that can able to convert any Time Domain Signal (periodic / non periodic) as a function of  Frequency called Frequency Domain.
we can also convert from frequency domain to time domain by inverse fourier transform.

Here, w = 2.pi.f

F(w) is called as Fourier Transform & f(t) is called as Inverse Fourier Transform.

The time domain description tells you what sound you hear every instant, while the frequency domain description tells you, roughly,what instruments are involved in the ways & how they are played.

one of the most important properties of the Fourier transformation is that it converts calculus i.e differentiation and integration into algebra i.e multiplication and division. This underlies its application to linear ordinary differential equations and further to partial differential equation

Fourier Transform has many Application in many fields of science such as 

Signal Analysis,Image Processing,Sound Filtering/Digital Filter-high pass low pass band pass etc,Data Compression,Solving Linear Partial Differential Equations,Communication,Optics,Geology,Astronomy,Antenna Designing and many moreand many more.

in fact any field of physical science that uses sinusoidal signals, such as engineering, physics, applied mathematics, and chemistry.

let's take an example :-

Fig 1

lets observe fig 2 by looking at time domain we can't determine the frequency of resultant sound wave signal but by taking the transformation we can say that it contains three harmonic frequency of 50Hz,100Hz and 150Hz. similarly by taking the transformation of modulated signal at the receiver one can able to determine the carrier frequency and information signal frequency.

Fig 2

in modulation process i.e while signal is passing through the channel,noise is superimposed on the signal.by  taking the Fourier transformation we can suppress the noise signal by designing the filter.that is the use of filter in signal processing.we can able to eliminate the unwanted frequency (particular or band of frequency ) by designing high pass, low pas , band pass filter. click here

Square

Sawtooth

Pulse

above fig shows how many harmonics of sine wave are used to approximate/generate the square wave,saw tooth & pulse.

Fourier Series

Fourier Series

Any Periodic function can be represented as sum of sine and cosine wave this representation is known as Fourier series named in the honor of Joseph Fourier.

Fourier series is given by :-

where,

n = 1 , 2 , 3 , ... and T is the period of function f(t). 

a_n and b_n are called Fourier coefficients and are given by

now lets we want to represent Square wave in the form of sine & cosine wave.
  periodic square wave function f(t) defined by

here suppose T = 6 (approx) now lets find coefficient a₀ which is  DC value    a₀ = 0 
i.e our waveform will oscillation either side of 0 = horizontal axis X line. now  lets find coefficient a_n which is Even Part

a_n = 0
 i.e our waveform will not contain any of cosine Part (even) now, lets find coefficient b_n which is Odd Part Note that cos (n pi) may be written as

cos (n pi) = (-1)ⁿ

and that b_n = 0 whenever n is even.

The given function f(t) has the following Fourier series   so, our waveform contains only of sum of sine function i.e odd part. now put N=1 and observer the waveform, we got 1st harmonic of sine wave also called fundamental harmonic.

now put n=2  at 2nd harmonic we observe no change in the waveform because our function is zero when ( i.e bn = 0) whenever n is even

now put n=3

its a addition of 1st harmonic + 3rd harmonic(shown in black at centre) = resultant red waveform

n=4 which is even so there is no change in waveform

n=5 note we are closer to approximation of square wave

n=29 we have almost approximated the square wave

Interfaces In GSM

Interfaces in GSM

BSS subsystem consist of :-

1. A Interface
2. Ater Interface
3. Abis Interface
4. Um Interface

A Interfaces :-

At MSC uses PCM30(8000 samples/second) or E1 links to carry the data to and from transcoder. Suppose 4-E1 links between MSC – Transcoder.

1 E1 = 32ch

4 E1=128ch

Each channel is of 64kbps.

128*64 = 8.192Mbps

but out of 128 channel only 120 is used (8 for signalling i.e 2 from each E1 link)
so there are 120 channel available for Traffic.

Ater interface :-

as shown above Ater interface consist of 32 channel.
but remember CH0 & CH16 is not converted into 16 kbps i.e it is transparent.

so total of 120 sub channel (30*4) each of 16Kbps.

Abis interface & Um interface:-

2 channel from Abis interface are mapped to 8 timeslot of TDMA Frame.
here in fig it is mapped at full rate of 16kbps but its depends on vendor equipment configuration.one can also use 32kbps or 64kbps. i.e Abis interface is proprietory.



So,maximum 15 TRX = 15 ARFCN = 15*8 timeslots =120 user calls can be handled by 1 BTS with 1 E1.
but if all the channels are used for voice then there wouldn't be any channel for signaling.so, signaling information is carried over in specific Abis timeslots of 64kbps each,or in 16kbps sub timeslots to atleast 1 TRX per cell.

Signaling CAS & CCS

        Signaling 

In today's world Two types of signaling is used:-

1.        CAS - Channel Associated Signaling
2.        CCS - Common Channel Signaling

In CAS,signaling bits are associated inside the voice channel i.e. no dedicated channel for signaling also called in band signaling.

T – Carrier CAS

T1 in CAS à  24 voice channel (1.544Mbps)

Basic single channel rate is 64 kbps also called DS0. (as per sampling theorem sampling rate should be twice the maximum frequency of the signal. We know that human voice range is 4Hz - 4kHz of frequency so to reproduce the signal we must sample it twice i.e. 2*4KHz=8KHz=8000  sample/sec  & each channel is of 8 bits so channel rate is 8000*8=64Kbps)

So, T1 contains 24 DS0 voice channel in CAS. but all channel is used for voice so  how can we obtain the singling. Here is the concept CAS in T1 use every last bit (Least Significant Digit)of each channel of every 6^th frame for signaling (not total channel but only a bit).this is known as robbed bit signaling or RBS.

fig1
T1 CAS

Note one thing here ,we know that T1 is of 1.544 Mbps.

lets calculate 24ch*8bits=192bits

192bits*8000samples = 1536 Kbps=1.536 Mbps so how come its 1.544 Mbps.

here, is the conclusion we use 1 bit for framing at the end of each frame so, its 192+1=193bits and 193bits*8000samples=1544 Kbps=1.544 Mbps which is our T1 Line.

8000 samples means each frame is repleted 8000 times to obtain 1.544Mbps T1 line.

T1 CAS uses two types of framing :-

Super Frame sends 12 frame at a time.in SF we get 2 bits A,B for signaling

Extended Super Frame 24 frame at a time.in ESF we get 4 nits A,B,C,D for signaling.

The meaning of these bits depends on what type of signaling is used on the channel.The most common types of signaling are loop start,ground start and E&M.

Loop Start

Ground Start

E&M

E – Carrier CAS

E1 in CAS à  30 voice channel + 2  dedicated signalling channel (2.048 Mbps)

E1 CAS has :-

• CH1 dedicated channel for framing 
• CH17 for signaling
• CH2-16 & CH18-32 are dedicated for voice

why is it called CAS even if it uses dedicated channel for signaling ? answer is E1 CAS uses same signaling type as that of T1 CAS as shown in below fig.

so, E1 CAS is compatible with T1 CAS.one can able to convert E1 to T1 or vice-versa.

T – Carrier CCS & E – Carrier CCS

T1 in CCS à  23 voice channel + 1 signaling channel (1.544Mbps)

T1 in CCS uses 1 dedicated channel for signaling which operates at 64Kbps.

E1 in CCS à  30 voice channel + 2  dedicated signalling channel (2.048 Mbps)

E1 in CCS uses 1 dedicated channel for signaling(ch 17 in our diagram above) & 1 dedicated for sync purpose ch 1 (which is neither bearer(voice) ch nor data ch(signaling))

so, what is new in CCS ? answer is instead of using bits (A,B,C,D) CCS uses packets for exchanging short messages for signaling i.e it uses protocol for communication on signaling channel (64kbps)

some of the available protocols are ISDN & SS7(signaling system 7).
note here applications/protocols such as SS7 have the flexibility to define any of channel as signaling channel.