Stars and Black Holes

Stars and Black Holes Manjunath.R #16/1, 8th Main Road, Shivanagar, Rajajinagar, Bangalore560010, Karnataka, India *Corresponding Author Email: manjunath5496@gmail.com *Website: http://www.myw3schools.com/ Abstract A star is a massive self-luminous celestial sphere of plasma that produces light and heat from the churning nuclear forges inside its core and represents the most fundamental building block of galaxies. And black holes are the corpse of a dead massive star. This paper discusses the concepts of stars and black holes. 1 Subrahmanyan Chandrasekhar was an Indian-American astrophysicist who spent his professional life in the United States. He was awarded the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and evolution of the stars". 2 “The element carbon can be found in more kinds of molecules than the sum of all other kinds of molecules combined. Given the abundance of carbon in the cosmos — forged in the cores of stars, churned up to their surfaces, and released copiously into the galaxy — a better element does not exist on which to base the chemistry and diversity of life. Just edging out carbon in abundance rank, oxygen is common, too, forged and released in the remains of exploded stars. Both oxygen and carbon are major ingredients of life as we know it.” ― Neil deGrasse Tyson For a spherical star of uniform density, the gravitational binding energy U B is given by the formula: UB = − 3GM2 5R where G is the gravitational constant, M is the mass of the star, and R is its radius. UB Mc2 − where rs = 2GM =− UB 0.3Mc2 3𝑟𝑠 10R = 𝑟𝑠 R is the Schwarzschild radius of the star. Any star with Radius smaller than its c2 Schwarzschild radius will form a black hole. If R < rs: − UB > 0.3Mc2 The star will form a black hole. 142 3 The core pressure of a star of mass M and radius R is given by: Pc = Pc = Pc = − 5GM2 4πR4 5 4πR 25 UB 9 V GM2 × 3 =− R 25 9 ρB where ρB is the gravitational binding energy density of the star. ______________________________________________________________________________ Since − 9Pc V 25 where ρE = If R < rs: 10UB = UB: =− 3Mc2 90Pc V = 75Mc2 Mc2 V 90Pc 75ρE = 𝑟𝑠 R 𝑟𝑠 R 𝑟𝑠 R is the mass energy density of the star. Pc 0.833ρE = 𝑟𝑠 R Pc > 0.833 ρE 2 143 4 The star will form a black hole. The core density of the star is given by: 3M ρc = πR3 The core temperature of the star is given by: Tc = 5μmH GM 3kB R where kB is the Boltzmann constant, μ denotes mean molecular weight of the matter inside the star and mH is the mass of hydrogen nucleus. 4μmH Pc ρc × T c = Pc = Since − 9Pc 25 = UB V kB ρc Tc kB 4μmH = ρB : ρB = − 9ρc Tc kB 100μmH The ideal gas equation PV = NkBT does not hold good for the matter present inside a star. Because, most stars are made up of more than one kind of particle and the gas inside the star is ionized. There is no indication of these facts in the above equation. We need to change the ideal gas equation, so that it holds good for the material present inside the star. It can be shown that 3 144 5 M kBT where μ denotes mean molecular weight μmH of the matter inside the star, M is the mass of the star and mH is the mass of hydrogen nucleus. kB 4P Since = c: μmH ρc Tc the required equation can be written as PV = P Pc ρ = 4× ρc × T Tc The Einstein time scale, tE, is given by: Mc2 tE = L The Kelvin-Helmholtz time-scale, tKH, is given by: GM2 tKH ≈ tE 2RL tKH = 4R rs where G is the gravitational constant, M is the mass of the star, R is the radius of the star, L is the 2GM star's luminosity and rs = 2 . c R3 Dynamical timescale: tdyn = √ GM tdyn = 145 6 R c tE √2t KH Binary Stars – a pair of stars in orbit around their common center of gravity.  Apparent Magnitude – a star's brightness as it appears to Earth  Absolute Magnitude – how bright a star actually is. (Apparent Magnitude − Absolute Magnitude) = 5log10 ( Distance between the star and the earth 10 Distance modulus Measured in parsecs  Red Giants – a large, cool star of high luminosity  Super giants – a very large, very bright red giant star  Nebulae – clouds of dust and gases in space  Hydrogen in core is depleted. A star's color reveals its surface  Core contracts and heats up. temperature.  Heat causes the outer layers to expand.  Expanding causes the layers to cool. Sun Stars are classified by their spectra as:  O, B, A, F, G, K, and M spectral types OBAFGKM hottest to coolest bluish to reddish 7 Yellow 5,500 K ) O hotter than 25,000 K B 11,000 - 25,000 K A 7500 - 11,000 K F 6000 - 7500 K G 5000 - 6000 K K 3500 - 5000 K M cooler than 3500 K Hertzsprung - Russell diagram → Graph of luminosity (or absolute magnitude) versus temperature (or spectral class) How Far Away? Distance How Bright? Luminosity How Hot? Spectral Type How Massive? Mass Many of the stars in our universe come in pairs. Ordinary stars orbiting around a black hole will appear to "wobble" in the sky. 8 Parallax Formula: P= 1 d  P = the parallax angle of star (in arcseconds)  d = the distance to star (in parsecs) The closer the star, the more its apparent position shifts as observed from earth. 1 parsec = 3.26 light years Luminosity and Inverse square Law: Apparent brightness = Luminosity 4×(distance)2 Apparent brightness ∝ 1 (distance)2 Red color → cooler → lower energy radiation → lower luminosity Blue color → hotter → higher energy radiation → higher luminosity 9 Stellar nebula → Average star → Red giant → Planetary nebula → white dwarf Neutron star Massive star → Red Super giant → Supernova Black hole Life Cycle of a star  A quasar is an enormously bright, distant galaxy with a giant black hole at its center.  A black dwarf is a theoretical stellar remnant, specifically a white dwarf that has cooled sufficiently that it no longer emits significant heat or light.  A supernova is a powerful and luminous stellar explosion. Released by contraction The virial theorem: U + 2K = 0 2K = −U Produced both by contraction and by fusion and other internal processes The negative gravitational energy of a star is equal to twice its thermal energy.  A decrease in total energy E of the star leads to a decrease in U but an increase in K and hence temperature T increases, i.e. when a star loses its total energy, it heats up. 10 The rate of change of the total energy of star (rate of nuclear energy generation in the deep interior − rate of energy loss in the form of radiation from the surface) In a state of thermal equilibrium: rate of nuclear energy generation in the deep interior = rate of energy loss in the form of radiation from the surface The total energy of star remains constant 4 possible sources of energy generation in stars: • cooling • contraction • chemical reactions • nuclear reactions If the temperature in the contracting core reaches values close to 1010 Kelvin, the energy of the photons becomes large enough to break up the heavy nuclei into lighter ones − in particular 56Fe is disintegrated into α particles and neutrons: 56 Fe + γ ↔ 13 α particles + 4 neutrons 11 (Photo-disintegration) For ultrarelativistic particles (v ∼ c) the rest mass of the particle may be neglected and the equation: E = √p2 c 2 + m20 c 4 take the form: E ≃ kBT = pc λ≃ hc kB T Gas pressure: Pgas ∝ ρT Radiation pressure: Prad ∝ T4 Degeneracy pressure → resistance of electrons (or neutrons) against compression into a smaller volume. Pdeg ∝ ρ5/3 (Non-relativistic case) Pdeg ∝ ρ4/3 M < 0.3Msun (Relativistic case) Star completely convective 0.3Msun < M < 1.5Msun M > 1.5Msun 12  Core radiative  Envelope convective  Core convective  Envelope radiative  Brown dwarfs → heavier than a planet (13 × mass of Jupiter) and lighter than a star. Hydrogen burning: 4p + 2e− → 4He + 2νe  proceeds by pp chains and CNO cycle  no heavier elements formed because no stable isotopes with mass number A = 8  neutrinos from proton → neutron conversion  typical temperature 107 K (∼ 1 keV) Helium burning: 4He + 4He + 4He ↔ 8Be + 4He → 12C  triple alpha reaction builds up Be with concentration ∼ 109 C + 4He →16O 12 16  O + 4He → 20Ne typical temperature 108 K (∼ 10 keV) Carbon burning:  Many reactions like 12C + 12C→ 20Ne + 4He etc.  typical temperature 109 K (∼ 100 keV) A star generally cannot reach hydrostatic equilibrium if its surface is too cool. 13 William Alfred Fowler was an American nuclear physicist, later astrophysicist, who, with Subrahmanyan Chandrasekhar won the 1983 Nobel Prize in Physics. He is known for his theoretical and experimental research into nuclear reactions within stars and the energy elements produced in the process. 14  If the distance between nuclei < 10−15m, the strong nuclear force overpowers the electromagnetic repulsion. Hydrostatic equilibrium keeps the fusion process at a constant rate:  If the nuclear fusion process speeds up: More energy would be produced and pressure would increase. This increased pressure would cause the core to expand and cool, and the fusion rate would slow down to normal.  If the core temperature drops: The nuclear fusion process slows down The pressure would decrease and the core would contract As the core shrinks Temperature would increase Fusion rate would return to normal The force of electrostatic repulsion between 2 protons is: F= and the potential energy is: e2 4πε0 r2 EP = e2 4πε0 r 15 So if e2 3 kBT = , the two protons can exceed the electrostatic force of repulsion and fuse 4πε0 r 2 together. rmin = r = e2 6πε0 kB T rmin → the distance of closest approach at which the nuclear attractive force becomes dominant to bind the two protons together.  At low velocities, the electrostatic force of repulsion prevents the collision of protons.  At high velocities, protons come close enough for the strong nuclear attractive force to bind them together. There is a small (but non-zero) probability that two protons can overcome their repulsion and fuse even if their velocities are too low. This is called quantum mechanical tunneling. mv2 2 λ= = 3 2 Probability ∝ exp (− rmin ) λ Probability ∝ exp (− rmin ) λ kBT h √3mkB T 16 Probability ∝ exp (− Fine structure constant π mc2 ×√ ) 3kB T Probability is highest at fast speeds v (or high temperatures T) kB Tcore  υ=  For a photon traveling from the center of Sun to the surface, with constant mean free path h is the frequency corresponding to the core temperature of the star. "ℓ" and assuming no destruction and recreation, the time taken for it to traverse this path is: t = R2sun cℓ , where c is the speed of light and Rsun is the radius of the sun. For ℓ = 3.01×10−5 m: t = 5.36 × 1013 s = 1.70 × 106 yr. If the photon didn't interact at all with matter, then t = Hawking Radiation:  Proposed by Professor Stephen Hawking in 1974.  Reduces mass and energy of a black hole. 17 Rsun c . Stephen William Hawking was an English theoretical physicist whose theory of exploding black holes drew upon both relativity theory and quantum mechanics. He also worked with space-time singularities. Black hole entropy: SBH = Entropic density = SBH V = kB A 4L2Planck kB 4L2Planck Entropic density = 18 × A V == kB 4L2Planck 3kB 2 8GtPlanck M × 3 𝑟𝑠 Entropic density ∝ Black hole energy density: uBH = uR uBH = 4 ×(σT4 × A) 3Mc3 M Mc2 Hawking radiation energy density: uR = 1 V 4σT4 c × rs Given that power emitted in Hawking radiation is the rate of energy loss of the black hole: P=− uR uBH If P = = dMc2 dt = σT 4 A 8P 3 × Planck power 3 × Planck power 8 : uBH = uR 19 P∝ uR uBH The time that the black hole takes to dissipate is: tev = tev SBH = 5120πG2M3 1280GM kB c3 ℏc4 = = 480c2 V ℏG 1280 kB × Planck power × Mc2 "Nature and Nature’s laws lay hid in night: God said, Let Newton be! and all was light." ― Stephen Hawking References:  The Physics of Stars By A. C. Phillips.  Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects By  Stuart L. Shapiro, Saul A. Teukolsky.  Introduction to Astrophysics: The Stars By Jean Dufay, Owen Gingerich. 20