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Black Holes Lecture 20 Lec 20: Black Holes 1 Lecture Topics ■ Schwarzchild radius ■ black hole radius The density of black holes ■ Properties of black holes ■ Falling into a black hole ■ Lec 20: Black Holes 2 Making a “Dark Star” ■ Suppose the escape velocity of an object was equal to the speed of light. Rs = Schwarzchild radius Putting in numbers: Rs = 3M Lec 20: Black Holes Rs in km M in solar masses 3 Dropping rocks ■ ■ Suppose we drop a rock from very far out in space. How fast is it going when it hits? The potential energy of the ball is: M, R = mass & distance from center of Earth m = mass of rock Lec 20: Black Holes 4 Gravitational Potential Energy ■ The potential energy difference between two points surrounding a mass is: GMm GMm ∆ PE = − R1 R2 ■ R2 > R1 When R2 -> infinity (very large distance) Lec 20: Black Holes 5 Potential ⇒ Kinetic Energy ■ The kinetic energy of the rock when it hits is: where v = velocity of the rock at impact m = mass of the rock. ■ This KE comes from the conversion of PE into KE (by gravity). Lec 20: Black Holes 6 Converting PE to KE ■ All the PE the rock had when it started is converted to KE at impact. ■ which means Lec 20: Black Holes 7 The Escape Velocity ■ ■ ■ Reverse the problem: What is the minimum speed upward the rock must have to escape the earth. It’s the same as if you let if fall (only going the other way)! Lec 20: Black Holes 8 A Black Hole is Born Black holes bend space Clip from “Space 1999” – bad 60’s SciFi series How big are black holes? Object Star Star Sun Earth Lec 20: Black Holes Mass (Msun) 10 3 1 3 x 10-6 11 Rs 30 9 3 9 km km km mm How dense are black holes? ■ The average density of a black hole is: but More massive black holes are less dense! Lec 20: Black Holes 12 Densities (cont’d) ■ For a black hole with M = Msun. ρ = 2 x 1016 g/cm3 ■ For M = 10 Msun. ρ = 2 x 1014 g/cm3 Lec 20: Black Holes 13 Very Massive Black Holes ■ Suppose we could make a black hole a big as the solar system, e.g Rs = 40 AU. Then and M = 2 x109 Msun ρ = 0.005 g/cm3 (!) - A 2 x109 Msun black hole can not be formed by a single star. Lec 20: Black Holes 14 Warped Space Time The Event Horizon ■ ■ The event horizon is located at Rs. Rs Anything inside the event horizon is gone from sight forever (nothing can escape). Lec 20: Black Holes 16 Event Horizon Time Dilation Recall that clocks run slower on the surface of the earth than on a mountain top. ■ Viewed from space clocks slow down as they approach the event horizon. ■ At the event horizon, the clock stops! ■ Lec 20: Black Holes 18 Gravitational Redshift The gravitational redshift gets larger and larger as objects approach the event horizon. ■ At the event horizon the redshift becomes infinite! ■ Lec 20: Black Holes 19 Falling in Falling into a black hole What happens if you fall in? Black Hole Clock A “A” falls in while “B” stays outside. Lec 20: Black Holes B 21 Person A falling into BH Person outside BH sees 1. Photons from A redshifted. 2. Clock A slow down. 3. Person A stretched and ripped apart by tidal forces. Black Hole Lec 20: Black Holes 22 Person falling in sees ■ If person a “paused” while falling in then he would see: ■ ■ Clock B is running very fast. Photons coming from person B and the rest of the universe are blueshifted. ■ ■ Visible photons become X-rays and γ-rays! The tidal forces will be very bad for the person falling into the black hole. Lec 20: Black Holes 23 Tides Tidal forces are due to the difference in the gravitational force across an object. ■ Near a black hole gravity changes very rapidly with distance. ■ ■ ■ neutron stars too! Tides pull on the object and stretch it in the direction of the star. Lec 20: Black Holes 24 Tides People lying in 12 ft. spaceship near the walls. Tides pull the two people towards the outer walls. Black Hole Lec 20: Black Holes Distance from BH 5000 km 1000 km 100 km 20 km 25 Tidal Force ( g’s ) 1.2 144 1.4x105 1.8x107 Circling a Black Hole Orbiting Lec 20: Black Holes Flying along event horizon 26 Are black holes dangerous? BHs don’t go around scooping up people and stars. ■ Only if you get very close to one is there a problem. ■ Replacing the sun with a 1 Msun black hole would not change the orbits of the planets! ■ ■ But we’d have a problem keeping warm. Lec 20: Black Holes 27 What’s on the other side? The singularity Relevance of Black Holes ■ Black holes can be formed when a massive star collapses ■ ■ (star: M > 15 Msun) Center of the Milky Way (our Galaxy) ■ ■ Mcore > ~ 4 Msun A 2x106 Msun black hole Centers of Quasars ■ Black holes up to 2x109 Msun Lec 20: Black Holes 30 In-Class Question Rs = 3 × M Rs in km, M in Msun 1) What is the radius of a 1,000,000 Msun black hole (in km)? a) 106 b) 30 c) 3x106 d) 3x107 ⇑ Lec 20: Black Holes 31 In-Class Question Rs = 3 × M Rs in km, M in Msun 1) What is the radius of a 1,000,000 Msun black hole (in km)? a) 106 b) 30 c) 3x106 d) 3x107 ⇑ 2) What is the event horizon of a Black Hole? a) Place where tides rip things apart ⇒ b) Place from which nothing can escape c) Place to go for a drink d) Place where photons are emitted Lec 20: Black Holes 32 Nearing a Black Hole Approaching Lec 20: Black Holes “Entering” 33 The impact velocity ■ Putting in some numbers Mearth = 6 x 1024 kg Rearth G Lec 20: Black Holes = 6400 km = 6.4x106 m = 6.67x10-11 N-m2/kg2 (m3/kg/s2) 34 Escape velocity ■ ■ ■ Escape velocity is the speed an object would need to escape from a celestial body. The escape velocity depends on mass. Examples: ■ ■ ■ ■ ■ ■ Earth: Moon: 1 km asteroid: Sun: White Dwarf: 11.2 km/sec (25,000 mph) 2.4 km/sec 1.3 m/sec (you could jump off it!) 618 km/sec 6000 km/sec !! How high can the escape velocity get? Lec 20: Black Holes 35 Dark Stars Rev. John Mitchell - 1783 ■ An object more massive than the Sun could have an escape velocity greater than the speed of light! ■ ■ Today we call this object a black hole. ■ An object from which no light can escape. Lec 20: Black Holes 36 Making a “Dark Star” ■ Suppose the escape velocity of an object was equal to the speed of light. Rs = Schwarzchild radius Lec 20: Black Holes 37