Particle Physics

aParticle PhysicsThe branch of physics which deals with property, interaction and structure of elementary particles is called particle physics. Particle physics deals with smallest things in the universe whereas cosmology deals with the biggest thing in the universe.Elementary ParticlesThe particles which are structure less, indivisible and not regarded as made up of some other particles is known as elementary particles. In 1897, after the discovery t electron by JJ Thomson, it was assumed that atoms were considered as fundamental particles of all the matter. Thomson’s discovery of electron and Rutherford’s discovery of atomic nucleus and proton in 1811 made it apparent that atom where not fundamental in the sense that they have an internal structure.A brief description of some important particles is as follows:Electron The electron is the first fundamental particle to be discovered and it revolves around the nucleus of an atom in different orbits. Its charge is -1.6×10-19 C${\displaystyle -1.6\times {10}^{-19}C}$ and mass is ${\displaystyle }$9.1×10-31 C${\displaystyle 9.1\times {10}^{-31}C}$.ProtonProton was discovered by Rutherford in 1911. Its charge is +1.6×10-19 C${\displaystyle +1.6\times {10}^{-19}C}$ and mass is 1.6726×10-27${\displaystyle 1.6726\times {10}^{-27}}$ kg which is 1836 times the electronic mass.NeutronNeutron was discovered by Chadwick in 1932. It carries no charge but its mass is 1.6749×10-27${\displaystyle 1.6749\times {10}^{-27}}$ kg which is 1839 times the electronic mass. In free state the neutron is unstable, but it constituents a stable nucleus along with proton.PositronPositron was discovered by Anderson. Its charge and mass are same of the electron, the only difference is it is positively charged. Whereas electron is positively charged.AntiprotonIt was discovered in 1955. Its charge and mass are same as those of proton, the only difference is it is negatively charged whereas proton is positively charged.Antineutron It was discovered in 1956. It has no charge and its mass is equal to the mass of the neutron. The only magnetic moments will be in opposite direction.Neutrino and antineutrinoThe existence of these particles was predicted by Pauli while explaining the emission of β-particles from radioactive nuclei, but they were observed in 1956. Their rest mass and charge both are zero but they have energy and momentum. Both neutrino and antineutrino are stable particles. The only difference between them is their spins are in opposite directions.Pi-mesonsThe existence of these particles was predicted by Yukawa in 1935 as originator exchange forces between the nucleons, but they were actually discovered in 1947 in cosmic rays. Pi-mesons are of three types: 1. positive pi-mesons, 2. negative pi-mesons and 3. neutral pi-meson.PhotonsThese are the bundles of electromagnetic energy and travel with the speed of light. If the frequency of waves is 𝜈${\displaystyle 𝜈}$, then the energy of a photon is h𝜈${\displaystyle 𝜈}$ and momentum is h𝜈${\displaystyle 𝜈}$/c. its symbol is γ.Classification of Elementary Particles Elementary particles can be classified on the basis of different properties of particles. They can be classified on the basis of mass (massless, light, intermediate and heavy), charge (positive, negative, neutral), spin or statistics (Bosons and Fermions), interaction (Gravitational, strong, weak and electromagnetic), lifetimes (stable and resonance).The classification of massive elementary particles.

Characteristic Properties of Elementary ParticlesMass: The elementary has always the same rest mass. The magnitude of the rest mass serves as the principle label which identifies the particles uniquely.Charge: All known elementary particles have charge positive negative or zero. Further, the charge is always conserved in any collision process.Average lifetime: All known elementary particles except photon, electron, proton and neutrinos are unstable and undergo decay into elementary particles of similar mass. The decay probability of a particular particle is, however, independent of the length of the time has lived. Spin: Many elementary particles spin in a manner analogous to that of the earth on its axis, but with certain differences. The spin property forms a basis for the classification of elementary particles.Interactions: Four kinds of interactions between elementary particles are known: gravitational, weak, electromagnetic and strong.Particles and AntiparticlesA subatomic particle that has the same mass as another particle but opposite value of some other properties is called antiparticle. For example, the antiparticle of the electron is a positron, which has the same mass as that of the electron but has a positive charge equal to the proton's positive charge. The existence of antiparticles is predicted by relativistic quantum mechanics. When a particle and its antiparticle collide annihilation takes place. When an electron meets a positron the energy produced due to annihilation is ${\displaystyle }$2mc2${\displaystyle 2m{c}^2}$ where m is the mass of electron or positron and c is the velocity of light.particles Symbols antiparticles mass average lifes(s) (times me${\displaystyle m_{e_{}}}$) Proton p+ Antiprotons p¯ 1836 StableElectron e- positron e+ 1 stableNeutron n Antineutron n¯ 1839 1010Neutrino v Antineutrino v¯ 0 stablePion π+ π- 274 2.6 ×108${\displaystyle 2.6\times {10}^8}$ Pair-production and Pair-AnnihilationWhen an energetic γ-ray photon falls on heavy substance, it is absorbed by some nucleus of the substance, and its energy gives rise to the production of an electron and a positron. This phenomenon, in which energy is converted into mass, is called ‘pair-production’ and represented by following equation:

According to Einstein’s energy-mass relation, a body in the state of rest also has some energy, called its rest-mass energy. If the rest mass of the body be ${\displaystyle }$m0${\displaystyle m_0}$, then its rest-mass energy is E0=m0c2${\displaystyle E_0=m_0{c}^2}$The rest-mass of each of the electron and the positron is 9.1 × 10-31 kg.${\displaystyle 9.1\times {10}^{-31}kg.}$ so, the rest-mass energy of each of them is

For pair production, it is essential that the energy of γ-photon must be at least 2× 0.51 = 1.02${\displaystyle 2\times 0.51=1.02}$ MeV. If the energy of γ photon is less than this, it would cause photoelectric effect or Compton effect on striking the matter. If the energy of γ-photon is more than 1.02 MeV, then electron and positron are produced and the energy in excess of 1.02 MeV is obtain as a kinetic energy of these particles. The converse phenomenon of pair-annihilation is also possible. Whenever an electron and a positron come very close to each other, they annihilate each other by combing together and two γ-photons are produced. This phenomenon, in which mass is converted into energy, is called ‘pair-annihilation’, and s represented by following equation:

Fundamental Forces/InteractionsThere are four types of forces/interactions in the nature. They are:Strong Force:This force acts between hadrons/quarks and is mediated by mesons/gluons. This force is charge and mass independent and saturative. Its range is small and it is responsible for stability of nucleus.Electromagnetic Force:The force which acts in between all charged particles is called an electromagnetic force. It is stronger than gravitational force but weaker than strong force. This force is attractive for unlike charge and repulsive for like charge. It is responsible for the stability of atoms, binding atoms in a matter and chemical reaction.Gravitational Force:This force acts between the particles having mass and is always attractive. It is the weakest force of nature (1039${\displaystyle {10}^{39}}$ times weaker than strong force). It is mediated by graviton and is responsible for the stability of the universe.Weak Force:This force acts between leptons and hadrons. It is stronger than gravitational force but weaker than electromagnetic and strong force. This force is responsible for decay.LeptonsThe leptons are the lightweight elementary particles which do not have strong interactions. There are six types of leptons. They are electron (e-), the muon(µ-), the tau particle (τ-)${\displaystyle electron\left(e^{-}\right),themuon\left({\mu }^{-}\right),thetauparticle\left({\tau }^{-}\right)}$, electron neutrino (ve), muons neutrino (vµ) and tau neutrino (vτ${\displaystyle electronneutrino(v_e),muonsneutrino(v_{\mu })andtauneutrino(v_{\tau }}$). Each of the six particles has a distinct antiparticle. All leptons have spin and thus are fermions.The types of leptons are shown in the tableParticle Name Symbol Anti-particle Mass (MeV/c2${\displaystyle MeV/c^2}$) Lifetime (s) aElectron e- e+ 0.511 StableElectron neutrino ve ⏨𝜈e < 3× 10-6 Stable Muon µ- µ+ 105.7 2.20 ×10-6Muon neutrino vµ ⏨⏨vµ <0.19 Stable Tau τ- τ+ 1777 2.9 ×10-13Tau neutrino vτ ⏨vτ <18.2 Stable ${\displaystyle \begin{array}{l}Electrone-e^{+}0.511Stable\\ Electronneutrino{v}_e\overline{{𝜈}_e}<3\times {10}^{-6}Stable\\ \\ \\ Muon\mu -\mu +105.72.20\times {10}^{-6}\\ Muonneutrino{v}_{\mu }\overline{v_{\mu }}<0.19Stable\\ \\ \\ Tau{\tau }^{-}{\tau }^{+}17772.9\times {10}^{-13}\\ Tauneutrino{v}_{\tau }\overline{v_{\tau }}<18.2Stable\\ \\ \end{array}}$ HadronsHadrons are the strongly interacting particles. Each hadron has an antiparticle. There are two subclasses of hadrons: mesons and baryons. Properties of some hadrons are shown in the table below.Hadrons and their proportiesParticle Mass (MeV/c2) Charge RatioQ/e Spin Mean Lifetime(s) Quark ContentMesons π0 135.0 0 0 8.4 × 10-17 u ⏨u, d ⏨d${\displaystyle 8.4\times {10}^{-17}u\overline{u},d\overline{d}}$π+ 139.6 +1 0 2.60 × 10-8${\displaystyle 2.60\times {10}^{-8}}$ u⏨d${\displaystyle u\overline{d}}$π- 139.6 -1 0 2.60 × 10-8 d ⏨u${\displaystyle 2.60\times {10}^{-8}d\overline{u}}$K+ 493.7 +1 0 1.24 × 10-8 u ⏨s${\displaystyle 1.24\times {10}^{-8}u\overline{s}}$ K- 493.7 -1 0 1.24 × 10-8 s⏨u${\displaystyle 1.24\times {10}^{-8}s\overline{u}}$η0${\displaystyle {\eta }^0}$ 547.3 0 0 a≈10-18 u⏨u, d⏨d, u⏨s ${\displaystyle \begin{array}{l}\approx {10}^{-18}u\overline{u},d\overline{d},u\overline{s}\\ \end{array}}$Baryons P 938.6 +1 ½ stable uudn 939.6 0 ½ 886 uddΛ0${\displaystyle {\Lambda }^0}$ 1116 0 ½ 2.633 × 10-10 ${\displaystyle 2.633\times {10}^{-10}}$ uds∑+ 1189 +1 ½ 8.02 × 10-11${\displaystyle 8.02\times {10}^{-11}}$ uus∑0 1193 0 ½ 7.4 × 10-20${\displaystyle 7.4\times {10}^{-20}}$ uds∑- 1197 -1 ½ 1.48 × 10-10${\displaystyle 1.48\times {10}^{-10}}$ ddsΞ0 1315 0 ½ 2.90 × 10-10${\displaystyle 2.90\times {10}^{-10}}$ ussΞ- 1321 -1 ½ 1.64 × 10-10${\displaystyle 1.64\times {10}^{-10}}$ dssΔ++ 1232 +2 ½ ≈10-23${\displaystyle \approx {10}^{-23}}$ uuuΩ- 1672 -1 ½ 8.2 × 10-11${\displaystyle 8.2\times {10}^{-11}}$ sss∧+ 2285 +1 ½ 2.0 × 10-13${\displaystyle 2.0\times {10}^{-13}}$ udcMesonsMesons are intermediate mass particles. They are heavier than leptons but lighter than baryons. Mesons include pions, kaons, eta particles etc. They are all bosons with spin 0 and 1. There are no stable mesons.BaryonsBaryons include nucleons and hyperons. They are heavy particles and they have half-integer spin and, therefore, all are fermions. The only stable baryon is a proton.QuarksQuarks are fundamental constituents of all the hadrons. They are strongly interacting particles. No isolated existence of quark is discovered so far.Original Quark ModelAccording to this model all hadrons are a composite system of two or here fundamental constituents called quark. There are three types of quarks up, down and strange. Each quark has anti-quark of opposite charge, baryon number and strangeness.Main properties of quarks and anti quarks are given in the table:Name Symbol Spin Charge Baryon Number StrangenessQuarks Up u ½ +2e/3 1/3 0 Down d ½ -e/3 1/3 0 Strange s ½ -e/3 1/3 -1Anti Quarks Anti-up ⏨u${\displaystyle \overline{u}}$ ½ -2e/3 -1/3 0 Anti-down ⏨d${\displaystyle \overline{d}}$ ½ +e/3 -1/3 0 Anti-strange ⏨s${\displaystyle \overline{s}}$ ½ +e/3 -1/3 -1The composition of all hadrons could be completely specified by three simple rules.Mesons consist of one quark and one anti-quark.Baryons consist of three quarks.Antibaryons consist of three antiquarks.Composition of several baryons and mesons are given belowBaryons Quarks compositionP uudn uddλ0${\displaystyle {\lambda }^0}$ uds∑+ uus∑- dds∑0${\displaystyle {}^0}$ uds Ξ0${\displaystyle {}^0}$ ussΞ-${\displaystyle {}^{-}}$ dssΩ-${\displaystyle {}^{-}}$ sssΔ-${\displaystyle {}^{-}}$ dddΔ++${\displaystyle {}^{++}}$ uuuMesons Quarks compositionaπ0 uπ+ uπ- dK+ uK - sK0 u ${\displaystyle \begin{array}{l}{\pi }^{0}u\\ {\pi }^{+}u\\ {\pi }^{-}d\\ K^{+}u\\ K^{-}s\\ K^{0}u\\ \end{array}}$ The third quark is needed only to construct charged particle with strangeness.Charm and Other Recent Development Although quark model is successful in classifying particles into families, there were some discrepancies. so fourth quark was proposed by several physicists in 1967. The fourth quark was given the new property or quantum number called charm.In 1975 researcher reported strong evidence of the τ${\displaystyle \tau }$ lepton. This discovery led to the more elaborate quark model and purposed of new quarks called top (t) and bottom (b)The properties of overall quark with additional quark.

Particle Name		Symbol	Spin	Charge	Baryon Number	Charm ness	Bottom ness	Top ness
Quarks	Charm	c	1/2	+2e/3	1/3	1	0	0
	Bottom	b	1/2	-e/3	1/3	0	1	0
	Top	t	1/2	+2e/3	1/3	0	0	1
Quarks	Anti-Charm	⏨c${\displaystyle \overline{c}}$	1/2	+2e/3	-1/3	-1	0	0
	Anti-Bottom	⏨b${\displaystyle \overline{b}}$	1/2	-e/3	-1/3	0	-1	0
	Anti-Top	⏨t${\displaystyle \overline{t}}$	1/2	+2e/3	-1/3	0	0	-1

After the discovery of top quark according to the standard model all the matter is composed of six strongly interacting particles the quarks (u, d, s, c, b, t) and six weakly interacting particles the leptons (e, µ, τ, ne, nµ, nτ) together with their antiparticles. Universe All matter, energy, and space that exists, is called the universe. The branch of physics which deals with the study of the universe is called astronomy. The branch of astronomy, which deals with physical processes connected with the celestial bodies and the intervening region of space, is called astrophysics. The study of the origin, evolution, and nature of the universe is called cosmology.The Solar SystemThe solar system, in which we live, is part of the universe which includes the planets, asteroids, and comets revolving round the sun, in an elliptical orbit.The SunSun is a star which mainly contains extremely hot hydrogen gas and radiates energy in all directions. The temperature of the outer region is photosphere which is about 6000 K and has a diameter of 1.4 x 109${\displaystyle {}^9}$ m. the diameter of the invisible part called chromospheres, is much greater than this. The mean distance between the sun and earth is 1.496 x 1011${\displaystyle 1.496x{10}^{11}}$ m. and is called one astronomical unit (A.U.). The mass of the sun is 2⨯ 1030${\displaystyle 2⨯{10}^{30}}$ kg which is called one solar mass.The planetsThere are eight planets revolving round the sun in their elliptical orbits. They are Mercury, Venus, Earth, Mars, Jupiter, Saturn and Neptune. Mercury is the nearest to the sun while Neptune is the farthest one.CometsComets are astronomical bodies, moving round highly elongated elliptical orbits. It consists of frozen gases, like ammonia, methane, water and nuclei of solid particles. When moving near the sun, a comet has a head and a tail. Substances like water in the comet get vaporised and the radiation pressure forces the vapour away in the shape of the tail.AsteroidsMinor planets revolving in elliptical orbit around the sun, mostly between the orbits of mars and Jupiter, in the same plane as that of the earth are called asteroids. The diameter of the asteroids varies from 1.6 km to 1000 km. Among the ten thousand asteroids Ceres is the largest with the size of 1000 km, Pallas about 600 km and Vesta is about 540 km. Meteor, Meteoroids, and MeteoritesMeteor is an object orbiting the sun which when enters the earth’s atmosphere is vigorously accelerated, due to gravity and becomes incandescent. Meteors are collectively called as meteoroids. When these objects come close to the earth’s surface they are heated to high temperatures due to friction and look like bright lines of fire and are called shooting stars. Some of the meteoroids survive while passing through the earth surface and they hit the surface. They are called meteorites. The starsA star is a self-luminous celestial body which converts nuclear energy into heat and light through nuclear fusion. There are billions of stars in the universe which are not uniformly distributed but collected together in the groups, called galaxies.Stellar EvolutionBirth of star: The inter-steller space contains an enormous amount of dust particles and gas, which come closer and closer due to the gravitational force of attraction and a cloud is formed. Within the cloud large clumps are formed, which attracts more mass and they get heat up due to contraction. The temperature of the central core heats up with the occurrence of nuclear fusion of hydrogen. During fusion of hydrogen atoms helium atom is produced with the release of energy. The gravitational force of attraction towards the centre of the star, due to its own mass is balanced outward pressure due to the heat generated due to the nuclear fusion.Death of a star:When the central core is completely converted into helium, there is no more production of heat. There is no outward pressure to balance the inward gravitational pull. So, star contracts and temperature increases causing the outer layer of the star to expand. Due to expansion outer layer is cooled and after certain time star looks red. The red star is called red giant. Due to further contraction, the temperature may rise to such high value, that fusion of helium will take place, forming heavier elements. Once again, the fuel will be exhausted and then there will be a violent explosion, called a supernova. Due to this explosion, a large portion of star’s envelope is thrown into interstellar space and this is the death of a star.White dwarf:For stars like the sun, the gravitational compression will leave the core, composed of protons and electrons, flying around a gas-like phase, called the electron gas. The electron gas is able to withstand the inward gravitational pull. Under this situation, the star is called white dwarf. White dwarf starts cooling and changes its color from white to yellow and yellow to red. Finally, it becomes a black dwarf, without emission of any radiation.Black hole:If the original mass of the star is greater than 5 solar mass, the gravitational pull inward will be so high that the core contracts to a radius R given by R = 2GM

c2${\displaystyle R=\frac{2GM}{c^2}}$, where M is the mass of the star and c is the velocity of light. After meeting above criteria, a black hole is formed. The gravitational pull, within this, is so strong that even a light photon cannot escape from it.Neutron StarIf the mass of the star is between 1.4M and 5M the core may end up as a neutron star. When the mass is greater than 1.4 solar mass, due to gravitational compression, the electrons are forced into the nuclei by the process called the inverse 𝛽${\displaystyle 𝛽}$-decay. In this process, the electrons and protons combine to form neutrons. Since the core contains only neutrons, it is called a neutron star.Supernova:When the neutron star is formed, the compression of the core produces a tremendous amount of gravitational energy. Due to the release of energy, the outer layers of the star explode. The phenomenon of the brightness of the star sharply increasing for some time and then decreasing is called a supernova.Galaxy:Galaxy is a large collection of a group of stars and is the building block of the universe. The galaxies are of different shapes, like a spiral, elliptical or irregular. Normal galaxies emit an only little amount of radio waves, but radio-galaxies give out million times more radio waves than normal galaxies. Our galaxy is a milky way and Andromeda is the nearest galaxy to us. Redshift and Expanding UniverseOn the basis of observations on the galaxies the distant galaxies are receding from us and also away from each other at a very high speed. According to Doppler effect in light, if source of light is receding from an observer, its wavelength appears to increase 𝜆0=𝜆sc+v

c-v${\displaystyle {𝜆}_0={𝜆}_s\sqrt{\frac{c+v}{c-v}}}$, that is the light emitted by it appears to have a longer wavelength than when it was at rest. Thus, a receding source of light should show a shift towards longer wavelength region in the lines of its spectrum. This phenomenon is called the red shift.If a radiation of particular wavelength l emitted by a star/galaxy is observed through a electroscope, then due to velocity v of the star/galaxy with respect to the earth, the wavelength recorded or observed will be l0 which is quite different from the real wavelength l. Then the red shift is denoted by z and is defined by z=𝜆0-𝜆

𝜆 = ±v

c$$z=\frac{{𝜆}_0-𝜆}{𝜆}=\pm \frac{v}{c}$$ 0r, z= Δ𝜆

𝜆 = ±v

c$$0r,z=\frac{\Delta 𝜆}{𝜆}=\pm \frac{v}{c}$$ Where positive sign means star/galaxy is receding away from the earth and negative sign means moving towards the earth. Hubble’s lawThis law states that the speed of recession of the galaxy is proportional to its distance from us. If v is the speed with which a galaxy recedes and r is its distance from earth,v= Hr${\displaystyle v=Hr}$ where H is called Hubble constant. Its value is 2.3 X 10-18 s-1.${\displaystyle 2.3X{10}^{-18}{s}^{-1}.}$ The Big BangHubble’s law suggests that at some time in the past, all the matter in the universe was far more concentrated today. It was then blown apart on an immense explosive called big bang, giving all observable matter more or less the velocities that we observe today.According to Hubble law, the matter at a distance away from us is travelling with speed v=Hr${\displaystyle v=Hr}$The time needed to travel a distance r is given by, t= r

Hr =1

H${\displaystyle t=\frac{r}{Hr}=\frac{1}{H}}$ By this hypothesis, the big bang occurred about 14 billion years ago. It assumes that all speeds are constant after the big bang; that is it neglects any changes in the expansion rate due to gravitational attraction or other effects.The events that went on during succeeding time after the big bang are summarised as follows:1. 14 billion years ago initial expansion began. This event was referred to be singularity because at this time the volume of the universe was zero and density of mass-energy was infinite.2. At t = 10-43${\displaystyle {}^{-43}}$ s the temperature of the universe was about 1032${\displaystyle {}^{32}}$ K, the average energy per particle was about 1019${\displaystyle {}^{19}}$ GeV. The entire universe was much smaller than a proton.3. At t = 10-35${\displaystyle {}^{-35}}$ s the temperature had decreased to about 1027${\displaystyle {}^{27}}$ K and average energy to about 1014${\displaystyle {}^{14}}$, The universe has undergone a tremendously rapid inflation increasing the size by a factor of about 10 30${\displaystyle {}^{30}}$.4. At t = 10-32${\displaystyle {}^{-32}}$ s the universe was a mixture of quarks, leptons and mediating bosons.5. At t = 10-6${\displaystyle {}^{-6}}$ s the temperature was about 1013${\displaystyle {}^{13}}$ K and the typical energies were 1GeV. At this time, quarks began to bind together to form nucleons and antinucleons.6. At t = 1 min, the universe has now cooled enough so that protons and neutrons in colliding, can stick together to form the low mass nuclei H2, He3, He4, and Li7${\displaystyle H^2,H{e}^3,H{e}^4,andL{i}^7}$ the predicted relative abundance of these nuclides are just what we observe in the universe today.7. At t = 300,000 years, the temperature was about 104 ${\displaystyle {}^4}$ K and electrons can stick together to bear nucleus when they collide forming atoms.Atoms of hydrogen and helium under the influence of gravity begin to clump together, starting the formation of galaxies and stars. Early supernovas spewed out the various elements heavier than helium that later became incorporated in stars and in their satellite planets.Critical DensityThe average density of matter in the universe determines whether the universe continues to expand indefinitely or not. The particular density needed just to stop the expansion of the universe is called critical density.Expression for critical densityConsider a large spherical volume of the universe with the radius R and mass M containing many galaxies as shown in the figure. Let the mass of our universe m and is located at the surface of the sphere. According to the cosmological principle, the average distribution of the matter within the sphere is uniform. Let r be the density of matter inside this sphere.The total energy E of the galaxy is the sum of its kinetic and potential energies, that is E = 1

2mv2 + (-GMm)

R(1)= 1

2mv2- GMm

R$$\begin{array}{c}E=\frac{1}{2}m{v}^2+\frac{(-GMm)}{R}\\ =\frac{1}{2}m{v}^2-\frac{GMm}{R}\end{array}$$ If E is positive, our galaxy has enough energy to escape from the gravitational attraction of the mass inside the sphere, in this case the universe should keep expanding forever. If E is negative, our galaxy cannot escape and the universe should eventually pull back together. The cross over between these two cases occurs when E = 0, and 𝜌 = 𝜌c.${\displaystyle 𝜌={𝜌}_c.}$ 1

2mv2 = GMm

RBut, M = 4

3𝜋R2𝜌c $$\begin{array}{c}\frac{1}{2}m{v}^2=\frac{GMm}{R}\\ But,M=\frac{4}{3}𝜋R^2{𝜌}_c\end{array}$$ If v be the speed of our galaxy relative to the centre of the sphere, then by Hubble’s law,v= HR${\displaystyle v=HR}$so, equation (1) beccomes 1

2m(HR)2 = Gm

R(4

3𝜋R3𝜌c)pc = 3H2

8𝜋G= 6.3× 10-27 kgm-3$$\begin{array}{c}\frac{1}{2}m(HR{)}^2=\frac{Gm}{R}(\frac{4}{3}𝜋R^3{𝜌}_c)\\ p_c=\frac{3H^2}{8𝜋G}\\ =6.3\times {10}^{-27}kg{m}^{-3}\end{array}$$ The mass of a hydrogen atom is 1.67 x 10-27${\displaystyle {}^{-27}}$ kg so this density is equivalent to about four hydrogen atom per cubic meter. Dark MatterDark matter is the non-luminous material distributed throughout the universe that cannot be directly detected by observing any form of electromagnetic radiation but whose existence is suggested by gravitational effects on the visible matter. According to present observations of a structure larger than the galaxy, big bang cosmology, dark matter and dark energy account for the vast majority of the mass in the observable universe.The galaxies near the Milky Way appear to be rotating faster than the rotation rate expected from the amount of visible matter that appears to be in these galaxies. Many astronomers believe that 96% of the matter in a typical galaxy is invisible. Some astronomers argue that the cluster galaxies are bound together from billions of years by the gravity due to the presence of enough mass which include up to 96 % dark matter and energy (72% dark energy, 24% dark matter).Black HoleA black hole is a region of space in which the gravitational pull is so strong that nothing can escape. General relativity predicts that if a star of mass more than solar masses have completely burned its nuclear fuel; it should collapse into configuration known as black hole. The resulting object is independent of the properties of matter that produced it and is completely described by its mass and spin. The most striking feature of this object is the existence of a surface called horizon, which completely encloses the collapsed matter. The horizon is an ideal one way membrane i.e. particles and light can go inward through the surface but not outward. As a result, the object is dark i.e. black and hides from view a finite region of space. The escape velocity, v=√(2GM/R) ${\displaystyle v=\surd (2GM/R)}$shows that a body of mass M will act as a black hole if its radius R is less than or equal to a certain critical radius.Karl Schwarzschild in 1916, derived an expression for the critical velocity from Einstein’s general theory of relativity, known as Schwarzschild radius Rs. This is given asc=2GM

Rs ${\displaystyle c=\sqrt{\frac{2GM}{R_s}}}$or, Rs=2GM

c2 ${\displaystyle or,R_s=\frac{2GM}{c^2}}$ If a spherical, non-rotating body of mass M has a radius smaller than Rs, then nothing –not even light can escape from the surface of the body. The body is then a black hole. Any other body within a distance Rs from the centre of the black hole is trapped by the gravitational attraction of the black hole and cannot escape from it. The surface of sphere with radius ‘Rs’, surrounding a black hole is called event horizon. We cannot see events occurring inside it. All that can be known about a black hole is its mass (from its gravitational force on other bodies), its electric charge (from electric forces on other charged bodies). Since light cannot escape from a black hole, then how can we know about black holes? The answer is that any gas or dust near to the black tends to be pulled into an accretion disc that swirls around and into the black hole, rather like a whirl pool. The friction within the accretion disc’s material causes it to lose mechanical energy and spiral into the black hole. As it moves inward, it is compressed together and this causes heating of the material, just as air compressed in a bicycle pump gets hotter. Temperature in excess of 10 6${\displaystyle {}^6}$ K can occur in the accretion disc so that the disc emits x-rays. Astronomers look for these x-rays emitted before the material crosses the event horizon to signal the presence of a black hole. Gravitational WaveGravitational waves are 'ripples' in space-time caused by some of the most violent and energetic processes in the Universe. Albert Einstein predicted the existence of gravitational waves in 1916 in his general theory of relativity. Einstein's mathematics showed that massive accelerating objects (such as neutron stars or black holes orbiting each other) would disrupt space-time in such a way that 'waves' of undulating space-time would propagate in all directions away from the source. These cosmic ripples would travel at the speed of light, carrying with them information about their origins, as well as clues to the nature of gravity itself.

The strongest gravitational waves are produced by events such as colliding black holes, supernovae (massive stars exploding at the end of their lifetimes), and colliding neutron stars. Other waves are predicted to be caused by the rotation of neutron stars that are not perfect spheres, and possibly even the remnants of gravitational radiation created by the Big Bang. A gravitational wave is an invisible (yet incredibly fast) ripple in space. Gravitational waves travel at the speed of light (186,000 miles per second). These waves squeeze and stretch anything in their path as they pass by.Einstein predicted that something special happens when two bodies—such as planets or stars—orbit each other. He believed that this kind of movement could cause ripples in space. These ripples would spread out like the ripples in a pond when a stone is tossed in. Scientists call these ripples of space gravitational waves.Gravitational waves are invisible. However, they are incredibly fast. They travel at the speed of light (186,000 miles per second). Gravitational waves squeeze and stretch anything in their path as they pass by.How are gravitational waves detected?When a gravitational wave passes by Earth, it squeezes and stretches space. LIGO can detect this squeezing and stretching. Each LIGO observatory has two “arms” that are each more than 2 miles (4 kilometers) long. A passing gravitational wave causes the length of the arms to change slightly. The observatory uses lasers, mirrors, and extremely sensitive instruments to detect these tiny changes.