Review: magnitudes distance modulus & colors

1. Review: magnitudesdistance modulus & colors • Magnitudes: either apparent magnitude (flux) m = -2.5 log(F) + const or absolute (luminosity). M = -2.5 log (L) + const in bands (e.g. B, V, R) • distance modulus, or difference between apparent and absolute magnitudes, which is a direct measure of distance from the inverse square law: m – M = 5log Dpc -5, where m & M are in given band (e.g. V) & D in pc • Get practice in manipulating m and M values! This is what you need to get M (and thus L) for Vega in EL2… Oct. 25, 2007

2. Colors as difference in magnitudes • Last class we saw (again) how colors and therefore temperature of stars can be measured by ratio of fluxes (e.g. ratio FB/FV • Given our definition of m = -2.5log F, we can define a color in magnitude units as the difference in mag, e.g. B-V: B-V = -2.5log (FB/FV) +const’ = -2.5logFB – [-2.5log FV] • Since absolute mags are just apparent mag corrected for distance = 10pc, the same B-V value applies to abs mag • So colors of stars most easily expressed as B-V (or V-R, etc. – i.e. difference of any two magnitudes. LARGER value of B-V means REDDER color (think about it…) Oct. 25, 2007

3. Review: Nuclear Fusion powers the Sun (and stars) • Origin of solar (and stellar) luminosity is THE major question • Both gravitational “settling” (Helmholtz’s theory ), or tapping into potential energy of gravity, E = GM/R, and “chemical burning”, or tapping energy of electron orbits in atoms, would give lifetime of Sun as only ~104 years and ~107 years which are each much shorter than known age of Earth (and Sun) • Concept of Stellar Lifetime, τ: τ = Mass/Mass-to-Luminosity rate = [mass-req]/[mass/sec] = [sec] where mass-req is required mass to supply obs luminosity if mass is converted to luminosity at some rate So energy per atom available from “chemistry” is only ~1 eV ~10-19Joules & Sun Luminosity of L ~ 4 x 1026 Joules/sec would then “burn” 4 x 1045 atoms/sec but Sun has only ~1057 atoms to burn, so lasts only for lifetime ~1057/4 x 1045 seconds or ~3 x 1011 = 104 years! • So nuclear fusion (continuous H bombs in central Sun) powers the Stars Oct. 25, 2007

4. Proton-proton cycle in Sun and stars • Hans Bethe (Cornell…) discovered the secret of the Sun and stars’ luminosity: nuclear fusion • Proton-proton cycle: convert 4H 2p + 2n = 1 He nucleus as described in Text, Box 16-1 • Energy released per He nucleus creation is Ep-p = [mass(4H) – mass(1He)] c2 where mass difference is due to difference in proton vs. neutron mass and the binding energy of the He nucleus • Ep-p = 4 x 10-12 Joules = 0.7%(mass(4H)), vs ~ 10-19 Joules from chemistry (“burning”). This is why nuclear fusion power plants are the dream… (provided their by-products can be disposed) Oct. 25, 2007

5. What is required for nuclear fusion? • Getting protons (H nuclei) to fuse means getting them very close together. They have to overcome their electric charge repulsion (both pos. charges) and the strong nuclear force • To overcome the repulsion between them, protons have to be able to smash into each other with enough energy • Its (high) time to define Kinetic Energy: the energy of motion of an object with mass m moving at velocity v with temp T is EK = ½ m v2 = 3/2 k T where k = Boltzmann’s constant. So incr. v is incr. T. So increased energy means higher temperature, and p-p cycle requires T ~ 107 K in center of Sun!! Oct. 25, 2007

6. Requirements for p-p cycle, cont. • But its not only high T required; we also need high density! That is, there must be protons very close to each other, and at high velocity or temperature, to sustain the fusion reaction • Density in center of Sun must be ρ ~ 1.6 x 105 kg/m3 to sustain nuclear burning! • So center of Sun is very hot and very dense. But surface of Sun, the Photosphere (i.e. “limb” of Sun we see in DL), is only at T ~6000K. And density of gas in solar atmosphere also very low…So temp. & density of Sun must decrease from center out! Oct. 25, 2007

7. So what holds the Sun “up”? • IF the Sun must be so hot and dense in its center vs. “cool” and “fluffy” at its surface, it must have enormously higher Pressure in its center than at its surface, since Pressure = P = 3/2 n k T where n is density of matter (no./cubic cm), k is a constant (Boltzmann’s constant; used above) and T is temp • But this enormous pressure “pushing out” is balanced by the enormous “weight” of all the mass of the Sun (which you can derive! From Kepler’s Law…) that is “pushing in”. IF they didn’t match, the Sun would either expand or collapse. But it doesn’t; its in hydrostatic equilibrium, which allows us to build models of the variation of density & temp. from center to surface of Sun and which also explains why the Sun is round! Oct. 25, 2007