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**White Dwarf: The Quantum Mechanical Star (ch 39,40)**

The Universe provides a laboratory for physics under conditions more extreme than any we could replicate on Earth. New theory, Quantum Mechanics, under development while Universe already revealing its predicted, strange states of matter. Take the white dwarf, a star the size of the Earth, revealing quantum mechanics in action. This is a repeated theme--we look at the Universe to gets Clues about the way physics really is. piece of lead, piece of aluminum

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**A wobble, a mystery. 1844-F. Bessel saw that Sirius, brightest star in**

7.6” A wobble, a mystery. 1844-F. Bessel saw that Sirius, brightest star in Sky (mag=-1.5) had a wave-like motion indicating possible, unseen companion 1862-telescope maker A. Clark resolves Sirius B, (18” refractor). Kepler’s law & distance from Center of mass, Sirius B ~ 1 solar mass, but 10 mags or 1010/2.5=10,000 times fainter than Sirius A Recall from H-R diagram (next slide) that stars that faint should be red and cool. Good Sirius homework exercise at Group ONE:･(a) Use this graph of the orbit of Sirius B about Sirius A for parts (a), (b), and (c). Note that in reality both stars orbit about their common center of mass - this graph has shifted their positions so that Sirius A stays stationary, so that you can see the orbit more clearly. Also, the Sirius system is close enough to face on to Earth that we will ignore tilt. First, from the dates of the observations, determine the period (in years) of the orbit.
･(b) Use the scale key on the graph to carefully measure the semi-major axis of the orbit, in arc-seconds (you should probably print the page out). This is half the distance across the long diameter of the ellipse. Also measure periastron, the shorter distance between Sirius B and A along the long diameter of the ellipse, and apastron (the longer distance).
･(c) The parallax for Sirius is arc-seconds. How far away is Sirius in parsecs? (How far is this in light years, using 1 pc = 3.26 ly ?)
･(d) Use the distance to Sirius (in parsecs) that you found in part (c) to convert the orbital elements (semi-major axis, periastron, apastron) that you found in (b) from arc-seconds to AU. To do this, you can use a variation of the parallax formula: distance (in parsecs) = size (AU) / size (arc-seconds).
･(e) Finally, use Kepler's third law to compute the total mass of the Sirius system, as in the lab we did, using your measurement from (a) for the period in years and from (d) for the semi-major axis in AU to get the total mass in solar units.Group TWO:･(a) Given that the wavelength at which the emission of Sirius A peaks is 293 nm, use this applet to determine its effective temperature. First set the temperature range to K, then set the wavelength range to nm, then play around with the temperature slider on the right until the blackbody curve peaks at about 293 nm. You will need to determine roughly how far apart the tick marks are to be accurate.
･(b) Check your answer by using Wein's law to calculate the effective temperature. You should get a similar answer to what you found in part (a).
･(c) Now use this applet to determine the luminosity from the temperature of Sirius A. Click on the "100" button several times, until there are stars all along the main sequence, then click around until you find the star closest to the temperature you measured from (b). Record the luminosity and the mass for this star.
･(d) Calculate the radius of Sirius A in solar units. First divide the temperature you found in part (b) by 5800, to get the temperature of Sirius A in solar units, then use L = R^2 * T^4 (with everything in solar units) with the luminosity you found in part (c) (already in solar units) to solve for R.
･(e) Finally, given that the radius of the sun = 6.96e10 cm and the mass of the sun = 1.99e33 g, convert your radius and mass for Sirius A from solar units to cm and g. Then calculate the density of Sirius A in g/cm^3, where density is mass divided by volume, and volume is (4/3)*pi*R^3. Compare your answer to the density of water (1.0 g/cm^3).Group THREE:･(a) Given that the temperature of Sirius A is about 10,000 K, determine the mass of Sirius A from this applet. Click on the "100" button several times, until there are stars all along the main sequence, then click around until you find the star closest to K. Record the mass.
･(b) Group ONE will calculate that the total mass of Sirius A and B is roughly 3.2 solar masses. Given your answer from (a), what is the mass of Sirius B?
･(c) The apparent bolometric (all wavelengths) magnitude of Sirius B is 5.69 and the distance is 2.64 pc. The apparent bolometric magnitude of the sun is and one AU is 4.85e-6 pc. This is the annoying math part... calculate the luminosity, in solar units, of Sirius B. Start with the equation m1 - m2 = -2.5*log(f1/f2), where m1 = apparent magnitude of star 1 and f1 = flux from star 1, then use f = L/(4*pi*d^2). This may be helpful but ask me if you get stuck. Show all your work.
･(d) Using the HR diagram found here, estimate the radius of Sirius B. To double check your answer, note that the temperature of Sirius B is 24,790 K; divide this by 5800 K to convert to solar units, then use L = R^2 * T^4 (with everything in solar units) to find the luminosity of Sirius B based on the radius you measure. Your result should be reasonably close to the answer you got in (c).
･(e) Finally, given that the radius of the sun = 6.96e10 cm and the mass of the sun = 1.99e33 g, convert your radius and mass for Sirius B from solar units to cm and g. Then calculate the density of Sirius B in g/cm^3, where density is mass divided by volume, and volume is (4/3)*pi*R^3. Compare your answer to the density of water (1.0 g/cm^3).Group FOUR:･(a) Given that the mass of Sirius A is 2.14 solar masses, and the luminosity is times that of the sun, use lifetime = M/L to calculate the main sequence lifetime of Sirius A as a fraction of that of the sun. If the sun's MS lifetime is 10 billion years, what is the MS lifetime for Sirius A in years?
･(b) What does the HR diagram found here suggest the MS lifetime for Sirius A should be? Compare to your answer from part (a).
･(c) Habitable planets should receive roughly the same amount of energy from their star as Earth gets from the sun, so that water is liquid. Using the equation f = L/(4*pi*d^2), with the luminosity from part (a), calculate the distance (in AU) a planet would need to orbit around Sirius A so that the flux at the surface of this planet would be the same as that on Earth. Your answer should be between 2 and 10 AU.
･(d) The closest distance between Sirius A and B is only 8.07 AU. The mass of Sirius B is 1.03 solar masses. The force of gravity is proportional to mass divided by distance squared. When Sirius A and B are closest together, how close could a planet orbiting Sirius A at the distance you found in part (c) come to Sirius B? Which would have a stronger gravitational pull on the planet at this location, A or B? (In other words, is M/d^2 higher for A or B?) What does this tell you about the possibility of forming stable orbits at this distance?
･(e) Comment on the likelihood of Earth like planets with intelligent life existing in the Sirius system.Group FIVE:

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**The Hertzsprung-Russell (H-R) Diagram**

Original: Russell 1914 Sirius A “Absolute Magnitude” scale of luminosity: M= log L/L, so L2=100L1 M2=M1-5 Modern, Local H-R diagram, 23,000 stars

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**A wobble, a mystery. 1844-F. Bessel saw that Sirius, brightest star in**

7.6” A wobble, a mystery. 1844-F. Bessel saw that Sirius, brightest star in Sky (mag=-1.5) had a wave-like motion indicating possible, unseen companion 1862-telescope maker A. Clark resolves Sirius B, (18” refractor). From Kepler’s law and distance from Center of mass, Sirius B ~ 1 solar mass, but 10 mags or 1010/2.5=10,000 times fainter than Sirius A ! Recall from H-R diagram (next slide) that stars that faint were supposed to be red and cool. 1915-Walter Adams at Mt. Wilson gets spectrum: (Its not in his paper so lets look at an HST Spectrum of Sirius B: Good Sirius homework exercise at Group ONE:･(a) Use this graph of the orbit of Sirius B about Sirius A for parts (a), (b), and (c). Note that in reality both stars orbit about their common center of mass - this graph has shifted their positions so that Sirius A stays stationary, so that you can see the orbit more clearly. Also, the Sirius system is close enough to face on to Earth that we will ignore tilt. First, from the dates of the observations, determine the period (in years) of the orbit.
･(b) Use the scale key on the graph to carefully measure the semi-major axis of the orbit, in arc-seconds (you should probably print the page out). This is half the distance across the long diameter of the ellipse. Also measure periastron, the shorter distance between Sirius B and A along the long diameter of the ellipse, and apastron (the longer distance).
･(c) The parallax for Sirius is arc-seconds. How far away is Sirius in parsecs? (How far is this in light years, using 1 pc = 3.26 ly ?)
･(d) Use the distance to Sirius (in parsecs) that you found in part (c) to convert the orbital elements (semi-major axis, periastron, apastron) that you found in (b) from arc-seconds to AU. To do this, you can use a variation of the parallax formula: distance (in parsecs) = size (AU) / size (arc-seconds).
･(e) Finally, use Kepler's third law to compute the total mass of the Sirius system, as in the lab we did, using your measurement from (a) for the period in years and from (d) for the semi-major axis in AU to get the total mass in solar units.Group TWO:･(a) Given that the wavelength at which the emission of Sirius A peaks is 293 nm, use this applet to determine its effective temperature. First set the temperature range to K, then set the wavelength range to nm, then play around with the temperature slider on the right until the blackbody curve peaks at about 293 nm. You will need to determine roughly how far apart the tick marks are to be accurate.
･(b) Check your answer by using Wein's law to calculate the effective temperature. You should get a similar answer to what you found in part (a).
･(c) Now use this applet to determine the luminosity from the temperature of Sirius A. Click on the "100" button several times, until there are stars all along the main sequence, then click around until you find the star closest to the temperature you measured from (b). Record the luminosity and the mass for this star.
･(d) Calculate the radius of Sirius A in solar units. First divide the temperature you found in part (b) by 5800, to get the temperature of Sirius A in solar units, then use L = R^2 * T^4 (with everything in solar units) with the luminosity you found in part (c) (already in solar units) to solve for R.
･(e) Finally, given that the radius of the sun = 6.96e10 cm and the mass of the sun = 1.99e33 g, convert your radius and mass for Sirius A from solar units to cm and g. Then calculate the density of Sirius A in g/cm^3, where density is mass divided by volume, and volume is (4/3)*pi*R^3. Compare your answer to the density of water (1.0 g/cm^3).Group THREE:･(a) Given that the temperature of Sirius A is about 10,000 K, determine the mass of Sirius A from this applet. Click on the "100" button several times, until there are stars all along the main sequence, then click around until you find the star closest to K. Record the mass.
･(b) Group ONE will calculate that the total mass of Sirius A and B is roughly 3.2 solar masses. Given your answer from (a), what is the mass of Sirius B?
･(c) The apparent bolometric (all wavelengths) magnitude of Sirius B is 5.69 and the distance is 2.64 pc. The apparent bolometric magnitude of the sun is and one AU is 4.85e-6 pc. This is the annoying math part... calculate the luminosity, in solar units, of Sirius B. Start with the equation m1 - m2 = -2.5*log(f1/f2), where m1 = apparent magnitude of star 1 and f1 = flux from star 1, then use f = L/(4*pi*d^2). This may be helpful but ask me if you get stuck. Show all your work.
･(d) Using the HR diagram found here, estimate the radius of Sirius B. To double check your answer, note that the temperature of Sirius B is 24,790 K; divide this by 5800 K to convert to solar units, then use L = R^2 * T^4 (with everything in solar units) to find the luminosity of Sirius B based on the radius you measure. Your result should be reasonably close to the answer you got in (c).
･(e) Finally, given that the radius of the sun = 6.96e10 cm and the mass of the sun = 1.99e33 g, convert your radius and mass for Sirius B from solar units to cm and g. Then calculate the density of Sirius B in g/cm^3, where density is mass divided by volume, and volume is (4/3)*pi*R^3. Compare your answer to the density of water (1.0 g/cm^3).Group FOUR:･(a) Given that the mass of Sirius A is 2.14 solar masses, and the luminosity is times that of the sun, use lifetime = M/L to calculate the main sequence lifetime of Sirius A as a fraction of that of the sun. If the sun's MS lifetime is 10 billion years, what is the MS lifetime for Sirius A in years?
･(b) What does the HR diagram found here suggest the MS lifetime for Sirius A should be? Compare to your answer from part (a).
･(c) Habitable planets should receive roughly the same amount of energy from their star as Earth gets from the sun, so that water is liquid. Using the equation f = L/(4*pi*d^2), with the luminosity from part (a), calculate the distance (in AU) a planet would need to orbit around Sirius A so that the flux at the surface of this planet would be the same as that on Earth. Your answer should be between 2 and 10 AU.
･(d) The closest distance between Sirius A and B is only 8.07 AU. The mass of Sirius B is 1.03 solar masses. The force of gravity is proportional to mass divided by distance squared. When Sirius A and B are closest together, how close could a planet orbiting Sirius A at the distance you found in part (c) come to Sirius B? Which would have a stronger gravitational pull on the planet at this location, A or B? (In other words, is M/d^2 higher for A or B?) What does this tell you about the possibility of forming stable orbits at this distance?
･(e) Comment on the likelihood of Earth like planets with intelligent life existing in the Sirius system.Group FIVE:

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B-Sirius!! Peak color=blue HST spectrum Its A0, can’t be as late as F because no CH visible at 4330, nor Ca II so earlier than A5, Sirius-B should have been a cool, red M-type, what is it?

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**Sirius Problem! Sirius A Sirius B**

Eridani B Sirius B is hot (25,000 k) and faint, and so was Eridani B! Russell:”I was flabbergasted, baffled trying to make out what it meant” And, if Sirius A and B had same temperature (surface flux), why was B 10,000 fainter? How could Sirius B be so faint?

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**White Dwarf, A Star The Size of the Earth!**

Sirius A B Remember the Stefan-Boltzmann law: L = 4pR2 sT4 Luminosity = (Surface Area) x ( ) energy emitted cm2 sec Let’s compare the size of Sirius B to A: RB/ RA=(LB/ LA)1/2(TA/TB)2 and (TA/TB)2~1, LB/ LA=10-4, so RB/ RA~ Now RA=1.7 R, so RB=0.01R=104 km So Sirius B has mass of the Sun but size of the Earth! Density=2x1030 kg/(4/3 (104km)3)=500kg/cm3, 1000 lb in a sugar cube! styrofoam=0.03 gr/cm3, Water=1 gr/cm3, lead=11 gr/cm3, gold/uranium= 19 gr/cm3 What could a white dwarf be made of to be a 100,00 times denser than the densest known elements?! In 1914 unknown..

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**Question What would happen if held a pinch of white dwarf**

in your hand? it would explode b) you would explode c) it would cut your hand c), assuming a pinch is ~1/10th of a sugar cube, that’s ~100 lb sitting on ~a millimeter of skin

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**How Did the White Dwarf Get So Small? The Incredible Shrinking Star**

What happens when a star runs out of fuel? 1 M Crudely speaking central pressure in a star goes as GM<rho>/R as derived from hydrostatic equilibrium (remember a WD is 105 times denser any known material, so stiffness not sufficient) Recall: star is held up by pressure generated by heat which balances gravity. When the furnace goes out, gravity wins, star contracts. Might heat up again and radiate, but eventually star runs out of fuel and keeps shrinking… A white dwarf becomes so compact (I.e., size of Earth, mass of Sun), so gravity at surface (and the support pressure needed) is ~10,000x the Sun’s! So, what is holding the star up after it runs out of fuel for its fire?

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**Quantum Mechanics to the Rescue!**

A white dwarf so dense, needs tremendous support pressure to keep from being crushed by its own gravity (gas not hot enough to supply it, and out of fuel anyway). How does it hold itself up? By 1926 the new field of quantum mechanics gives a clue. The Pauli Exclusion Principle (W. Pauli: 1924 empirical, 1925 Explained by QM). No two electrons can occupy the same “quantum state” (location, speed, etc). So electrons must “stack”, they cannot be jammed too close together. Protons and neutrons are also fermions like electrons, photons are bosons which can be jammed together. The reason for the principle is that in QM fermions have antisymmetric wavefunctions and the expression Of a two particle state for identicle particles cancels and thus has probability of zero, therefore fermions can never be identicle, see Wikipedia This explains the filling of electron shells in chemistry and properties of the periodic table. QM works!

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**Uncertainty Heisenberg Uncertainty principle (1927):**

x=positional uncertainty, v=velocity uncertainty x v>h, A particle’s location and speed cannot both be completely confined, there is a limit. So this sets the smallest possible spacing between electrons. i.e., x =h/v Velocity e- Let’s get together, where will you be at 8:00? I can tell you where I am going or where I am now, but not both! Protons and neutrons are also fermions like electrons, photons are bosons which can be jammed together. The reason for the principle is that in QM fermions have antisymmetric wavefunctions and the expression Of a two particle state for identicle particles cancels and thus has probability of zero, see Wikipedia Position

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Degenerate Matter or How stuff moves and creates pressure even when its cold R. Fowler, 1926, Electron Degeneracy Pressure Gravity is crushing but free electrons cannot get closer than x ~h/v so they begin to stack into a super-dense state, the gaps in matter (usually 1 in 107 states filled) are gone! Even cold, electrons forced to have large velocities because these are unoccupied states (if x small, v is big). The tighter its squeezed the more it pushes back! (The smaller it becomes, the harder to compress) Stacking of positions and velocities provides pressure (like jumping jellybeans in a jar) without heat (not thermal pressure)! Called degeneracy pressure (pressure to be different, like anti-peer pressure!) How stuff keeps moving even when temperature=0

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**In R. Fowler’s Own Words, “On Dense Matter” 1926:**

We recognize now that matter can exist in such a dense state that the electrons are not bound in their ordinary atomic orbits but are free…there may come a time when a very curious state of affairs is set up…As the dense matter radiates its energy away…the absolutely final state is one in which there is only one possible configuration left …the star is analogous to one gigantic molecule. Not only has Fowler explained what holds up WD, he has pointed to a way stars can gracefully die (not fire but ice)!

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**Retired Stars or What to do With the Stellar Corpse**

The degenerate electrons in a white dwarf are moving very fast but the laws of quantum mechanics prevent them from losing or conducting away energy. They provide the stiffness to hold up the white dwarf. Atomic nuclei (positive ions) are not degenerate at this density, they remain in classic, gas state. So they can and do lose energy. So white dwarfs can shine as the nuclei lose energy, slowly cool down, and turn black: Black dwarfs. Size remains the same. Like Earth-sized lumps of coal with the mass of the Sun. Electron degeneracy pressure is the like the scaffolding of a stellar corpse

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**Subrahmanyan Chandrasekhar**

In 1930 Chandrasekhar was 19, traveling by boat to England and discovered an even stranger consequence of Fowler’s theory of dense (degenerate) matter. The more massive such a star was (imagine adding matter teaspoon at a time or finding a heavier one) the smaller it became, until finally it disappeared! Eddington ridiculed Chandra’s idea in 1935 at an RAS meeting, “there should be a law of nature to prevent a star from Behaving in this absurd way!”. This unfortunately set back the acceptance of Chandra’s work for ~20 years! (he won Nobel 1983)

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Strange Matter Add more beans Jarful gets smaller, beans move faster! If more matter placed on a white dwarf, gravity--pressure get out of balance, so star contracts, x gets smaller, v gets larger which increases pressure. Star gets back to equilibrium but at a smaller size now then before! Can’t keep getting smaller as v approaches speed of light! At the speed of light (and a change to a relativistic formula). R0, density∞,MMCH. Thus, there is a maximum mass, the Chandrasekhar mass (limit).

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**Chandrasekhar Limit Chandrasekhar’s Prediction WD Mass and Size limit**

The white dwarf no longer has a way to hold back gravity (can’t get its electrons closer together and jiggle them faster than speed of light). Degeneracy pressure allows small stars (like the Sun) that end their lives with M<1.4M to die gracefully, leave a corpse. (Bigger stars can’t get their electrons closer together and jiggle them faster than speed of light). So what happens to stars much bigger than the Chandrasekhar limit when they run our of fuel? How do they hold their massive, bloated corpse up?

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Supernova! Remember Tyco, Kepler, and Galileo’s “guest star”? By 1930’s indications were that L~109xSun! SN in 1054 mag=-6, visible by day In 1934 Baade and Zwicky suggested that these could be the death of stars too massive to use the “graceful exit” possible for stars below Chandrasekhar’s limit. When a very massive star runs out of fuel it would be unable to hold back gravity by degeneracy pressure and implode smashing electrons and protons to make neutrons. The result: a neutron star (neutron degeneracy allows greater density), a solar mass just 10 miles across! Then the star would rebound and an explosion would result. Demo! Alternatively, if you (or its companion) adds one more teaspoon of matter to a Chandrasekhar-mass white dwarf it might explode. These sounded like crazy ideas, but then how else could you explain a supernova in 1934? A good place to do the demo of a tennis ball and a basketball

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**Bang! A star explodes… 1987: The galaxy next door**

Only 170,000 light years away: a cosmic stone’s throw. Big-time story (“see: big TIME…”)! More important that the Israel Spy Case Furor: things like that happen all the time, but how often does a star explode in your own back yard?!

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**Supernovae From what you’ve seen so far you might have guessed we look**

For supernovae at the ends of giant arrows in space! (No!) Actually they are very rare.

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**Poem lyrics of When I Heard The Learn'd Astronomer by Walt Whitman, 1900**

When I heard the learn'd astronomer;
When the proofs, the figures, were ranged in columns before me;
When I was shown the charts and the diagrams, to add, divide, and
measure them;
When I, sitting, heard the astronomer, where he lectured with much
applause in the lecture-room,
How soon, unaccountable, I became tired and sick;
Till rising and gliding out, I wander'd off by myself,
In the mystical moist night-air, and from time to time,
Look'd up in perfect silence at the stars.

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**The stars are made of the same atoms as the earth**

The stars are made of the same atoms as the earth. I usually pick one small topic like this to give a lecture on. Poets say science takes away from the beauty of the stars - mere gobs of gas atoms. Nothing is "mere." I too can see the stars on a desert night, and feel them. But do I see less or more? The vastness of the heavens stretches my imagination - stuck on this carousel my little eye can catch one-million-year-old light. A vast pattern - of which I am a part - perhaps my stuff was belched from some forgotten star, as one is belching there. Or see them with the greater eye of Palomar, rushing all apart from some common starting point when they were perhaps all together. What is the pattern, or the meaning, or the "why?" It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined! Why do the poets of the present not speak of it? What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?
-Richard Feynman

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Question: Do you feel more like Whitman (A) or Feyman (B)?

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